cosmology with cmb polarization - research explorer

COSMOLOGY WITH CMB

POLARIZATION: IMPACT OF

FOREGROUND RESIDUALS

A THESIS SUBMITTED TO THE UNIVERSITY OF MANCHESTER

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

IN THE FACULTY OF SCIENCE AND ENGINEERING

2018

Carlos Eduardo Hervías Caimapo

School of Physics and Astronomy

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Contents

Abstract 11

Declaration 13

Copyright Statement 15

Acknowledgements 17

1 Cosmology and the CMB 19

1.1 Modern cosmology: the ΛCDM model . . . . . . . . . . . . . . . . . . 21

1.1.1 The cosmological constant Λ . . . . . . . . . . . . . . . . . . . 23

1.1.2 Dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.2 Cosmic inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.2.1 Primordial perturbations . . . . . . . . . . . . . . . . . . . . . 32

1.2.2 Models of inflation . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.2.3 Current observational constraints . . . . . . . . . . . . . . . . 37

1.3 The Cosmic Microwave Background Radiation . . . . . . . . . . . . . 38

1.3.1 The CMB polarization . . . . . . . . . . . . . . . . . . . . . . . 39

1.3.2 Sources of E and B anisotropies . . . . . . . . . . . . . . . . . . 43

1.3.3 Cosmological parameters . . . . . . . . . . . . . . . . . . . . . 45

1.3.4 Current observations of CMB B-modes . . . . . . . . . . . . . 48

2 CMB foregrounds 51

2.1 Astrophysical foregrounds . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.1.1 Synchrotron radiation . . . . . . . . . . . . . . . . . . . . . . . 53

2.1.2 Thermal dust radiation . . . . . . . . . . . . . . . . . . . . . . . 54

2.1.3 Anomalous Microwave Emission (AME) . . . . . . . . . . . . 55

2.1.4 Point sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.1.5 Other unpolarized foregrounds . . . . . . . . . . . . . . . . . . 56

2.2 Component separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3

2.2.1 Parametric methods . . . . . . . . . . . . . . . . . . . . . . . . 58

2.2.2 Blind methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3 Simulated observations of the microwave sky 63

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Sky model components . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.1 CMB component . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.2 Foreground templates . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.3 Baseline foreground model . . . . . . . . . . . . . . . . . . . . 67

3.2.4 Spatially-varying spectral indices . . . . . . . . . . . . . . . . . 67

3.2.5 Curved synchrotron spectral index and multiple thermal dust

components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.6 Additional polarized component: AME . . . . . . . . . . . . . 69

3.3 Simulated observations of CMB polarization experiments . . . . . . . 71

3.3.1 Simulating the instrumental response . . . . . . . . . . . . . . 71

3.3.2 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . 72

3.4 Comparison with data from Planck and WMAP . . . . . . . . . . . . 72

3.4.1 Foreground model . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.4.2 Data maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.4.3 Comparison with foregrounds only . . . . . . . . . . . . . . . 76

3.4.4 Including the contribution from noise and CMB . . . . . . . . 76

3.4.5 Optimal spectral index test . . . . . . . . . . . . . . . . . . . . 81

3.5 Forecast and Monte-Carlo capabilities . . . . . . . . . . . . . . . . . . 85

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4 Foreground uncertainties 89

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 Simulated observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.3.1 Component separation . . . . . . . . . . . . . . . . . . . . . . . 91

4.3.2 Power spectra estimation . . . . . . . . . . . . . . . . . . . . . 92

4.3.3 Cosmological parameters likelihood . . . . . . . . . . . . . . . 95

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.4.1 Sky model with constant spectral indices (Simple model) . . . 98

4.4.2 Sky model with spatially variable spectral indices (Complex

model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4

5 Forecast for Simons Observatory 113

5.1 The Simons Observatory . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2.1 Modelling the foreground spectral parameters errors . . . . . 114

5.2.2 Estimate of power spectrum residuals . . . . . . . . . . . . . . 114

5.2.3 Fisher matrix information . . . . . . . . . . . . . . . . . . . . . 116

5.3 σ(r) forecast results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.3.1 Simons Observatory instrument specifications . . . . . . . . . 117

5.3.2 Simulated observations . . . . . . . . . . . . . . . . . . . . . . 118

5.3.3 Forecasted results for the baseline SO specification . . . . . . 119

5.3.4 SO baseline configuration + high-frequency bands . . . . . . . 124

5.4 Optimizing the fraction of the sky for measuring r . . . . . . . . . . . 126

5.5 σ(r) forecasts for SO-EBT . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.5.1 Spatially-varying foreground spectral indices . . . . . . . . . 130

5.5.2 Estimation of r bias . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 CORE study on foregrounds 137

6.1 The CORE proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.2 Results from the CORE foreground study . . . . . . . . . . . . . . . . 138

6.2.1 Simulated observations . . . . . . . . . . . . . . . . . . . . . . 138

6.2.2 Results: Component separation and power spectra estimation 139

6.2.3 Results: Cosmological parameters posterior probabilities . . . 144

6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7 Conclusions 153

7.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 153

7.1.1 A new model of the polarized microwave sky . . . . . . . . . 153

7.1.2 Characterizing the impact of foreground residual contamina-

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.1.3 Forecasting the performance of the Simons Observatory . . . 155

7.1.4 CORE proposal forecasts . . . . . . . . . . . . . . . . . . . . . 157

7.1.5 Overall conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Word count 57105

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List of Tables

4.1 COrE satellite specifications . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2 Summary of runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3 Bias on r for simple model . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.4 Bias on r for complex model . . . . . . . . . . . . . . . . . . . . . . . . 107

5.1 Baseline configuration for Simons Observatory Small Aperture Camera117

5.2 Sensitivities per band for the SO configurations considered . . . . . . 132

5.3 Forecasts for the baseline and improved SO-EBT configuration. First

case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.4 Forecasts for the baseline and improved SO-EBT configuration. Sec-

ond case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.5 Forecasts for the baseline and improved SO-EBT configuration.

Third case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.1 Instrumental specifications used in the CORE proposal to simulate

full-sky observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2 Summary of simulations considered for the reconstruction of B-

modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.3 Measured tensor-to-scalar ratio values from the posterior probabili-

ties for all 5 simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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List of Figures

1.1 Discovery of accelerated expansion of the Universe with supernovae

type Ia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.2 Current constraints on inflation cosmological parameters from Planck 38

1.3 Planck 2015 CMB angular power spectra . . . . . . . . . . . . . . . . . 40

1.4 Generation of polarization by quadrupole anisotropies . . . . . . . . 41

1.5 Polarization pattern produces by pure E and B modes . . . . . . . . 42

1.6 Illustration of scalar and tensor perturbations pattern . . . . . . . . . 44

1.7 Current measurements of the CBB` angular power spectrum . . . . . 48

2.1 SEDs at microwave frequencies in temperature and polarization . . . 52

3.1 Q polarization maps of the foregrounds templates . . . . . . . . . . . 64

3.2 EE power spectra for the foregrounds templates . . . . . . . . . . . . 65

3.3 Maps of spectral indices βdust and βsyn . . . . . . . . . . . . . . . . . . 67

3.4 AME template map constructed in this work . . . . . . . . . . . . . . 70

3.5 Intensity scatter plot of model and observations . . . . . . . . . . . . 73

3.6 Side by side comparison of Q maps between model and observed sky 74

3.7 Side by side comparison of U maps between model and observed sky 75

3.8 Power spectrum comparison for Planck bands . . . . . . . . . . . . . 77

3.9 Power spectrum comparison for WMAP bands . . . . . . . . . . . . . 78

3.10 Correlation between our model and Planck observations at high

Galactic latitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.11 Probability for βdust and βsyn . . . . . . . . . . . . . . . . . . . . . . . . 82

3.12 Forecast for BB modes detectability . . . . . . . . . . . . . . . . . . . 83

3.13 Zoomed-in region of intermediate foreground contamination . . . . . 84

3.14 Example capabilities of the creation of random small-scale features . 86

4.1 Example maps of the simulated polarized sky at 105 and 555 GHz . . 90

4.2 Default Galactic mask . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.3 Comparison of foreground residuals and foreground templates . . . 95

9

4.4 Example reconstructed BB power spectrum . . . . . . . . . . . . . . 98

4.5 Example likelihoods for runs on the simple model . . . . . . . . . . . 99

4.6 Bias measured for runs in the simple model . . . . . . . . . . . . . . . 101

4.7 Histograms of βdust and βsyn for the complex model with respect to

the mean values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.8 Optimized Galactic masks . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.9 Posterior for r for the complex model, and assuming constant spec-

tral indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.10 Posterior for r for the complex model . . . . . . . . . . . . . . . . . . 108

4.11 Example of the correlation between the ∆βdust residual and the inten-

sity of polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.1 SO mask with fsky = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2 Example Modified Black-Body spectral law for thermal dust . . . . . 120

5.3 Fit of a Modifidied Black-Body spectral law to the simulated thermal

dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.4 Histogram of the synchrotron and thermal dust spectral indices

residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.5 CBB` residual power spectrum from foregrounds residuals, noise and

cosmic variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.6 Empirical correlation between the thermal dust parameters . . . . . . 124

5.7 CBB` residual power spectrum from foregrounds residuals, noise and

cosmic variance for the baseline+high-frequency configuration . . . . 126

5.8 Statistics of the foreground parameter residuals as estimated by CCA 127

5.9 Estimates of σ(r) in the observable sky from the SO site . . . . . . . . 129

5.10 SO mask with fsky = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.1 EE power spectrum reconstruction by component separation methods140

6.2 BB power spectrum estimations for simulation 1 and 3 . . . . . . . . 142

6.3 BB power spectrum estimations for simulation 4 . . . . . . . . . . . . 143

6.4 Posterior probability for τ . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.5 Posterior probability for r, for simulations 1 and 3 . . . . . . . . . . . 146

6.6 Example low-` BB power spectrum of the reconstructed thermal

dust and foreground residuals . . . . . . . . . . . . . . . . . . . . . . . 147

6.7 Posterior probability for r, for simulations 4 and 5 . . . . . . . . . . . 148

10

The University of ManchesterCarlos Eduardo Hervías CaimapoDoctor of PhilosophyCosmology with CMB polarization: Impact of foreground residualsAugust 20, 2018

In this thesis, I present my work related to the characterization of diffuse Galac-tic foregrounds for observing the polarization of the Cosmic Microwave Back-ground (CMB) radiation, and the impact of these foregrounds on the measurementof cosmological parameters.

One of the most important future challenges for cosmology is the potential de-tection of polarization B-modes of the CMB. Inflation is a theory that explains theextremely early Universe, and solves several problems that were present in clas-sical cosmology. It describes the anisotropies observed in the current Universe asprimordial quantum fluctuations stretched by rapid exponential expansion. A keyprediction of inflation is the production of a background of primordial gravitationalwaves, which could be detected through the associated large-scale B-mode signalin the CMB polarization. The amplitude of the B-mode signal, which depends onthe energy scale of inflation, is parametrized by the tensor-to-scalar ratio r. Diffuseemission from within our Galaxy, and other extra-Galactic sources, collectively re-ferred to as CMB foregrounds, obscure a fraction of the cosmological signal fromthe CMB radiation. This is a huge problem, because they have to be cleaned usingdata analysis methods, called component separation. A significant challenge forthe potential detection of the primordial B-mode signal is that it can be extremelysmall, to the extent that it can be dominated even by the residual foreground con-tamination after component separation.

In this work, we characterize these foreground residuals and assess their impacton the cosmological parameters. We create a method to simulate observations ofthe microwave sky, including diffuse Galactic foregrounds, CMB realizations andinstrumental noise. These simulations are used to propagate errors on the charac-terization of foregrounds through the analysis procedures employed in the obser-vations of the CMB, including component separation, angular power spectra calcu-lation and cosmological parameter estimation. We estimate the bias and the σ errorfor the tensor-to-scalar ratio, to quantify the impact of the foreground residuals inthe cosmological signal. We also propose a novel method to model these residu-als when determining cosmological parameters, in order to avoid a bias on the rparameter.

We performed forecasts and optimization analyses for two proposed CMB po-larization experiments: the Simon Observatory, a funded ground-based telescopethat will observe the polarization of the CMB from the Atacama desert in Chile, andCORE, a proposed next-generation CMB satellite experiment.

All of our work shows that the issue of foreground residuals must be consideredvery carefully in future studies. Foreground spectral parameters must be modelledvery accurately, with errors . 0.5%, if we wish to measure a value r ∼ 10−3. Theseforeground residuals can easily be mistaken as primordial cosmological signals, soour work motivates further research into developing new data analysis techniques.

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Declaration

No portion of the work referred to in the thesis has

been submitted in support of an application for an-

other degree or qualification of this or any other uni-

versity or other institute of learning.

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Copyright Statement

i. The author of this thesis (including any appendices and/or schedules to this the-

sis) owns certain copyright or related rights in it (the “Copyright”) and s/he has

given The University of Manchester certain rights to use such Copyright, includ-

ing for administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic

copy, may be made only in accordance with the Copyright, Designs and Patents

Act 1988 (as amended) and regulations issued under it or, where appropriate,

in accordance with licensing agreements which the University has from time to

time. This page must form part of any such copies made.

iii. The ownership of certain Copyright, patents, designs, trade marks and other in-

tellectual property (the “Intellectual Property”) and any reproductions of copy-

right works in the thesis, for example graphs and tables (“Reproductions”),

which may be described in this thesis, may not be owned by the author and may

be owned by third parties. Such Intellectual Property and Reproductions cannot

and must not be made available for use without the prior written permission of

the owner(s) of the relevant Intellectual Property and/or Reproductions.

iv. Further information on the conditions under which disclosure,

publication and commercialisation of this thesis, the Copyright

and any Intellectual Property and/or Reproductions described in

it may take place is available in the University IP Policy (see

http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=24420),

in any relevant Thesis restriction declarations deposited in the

University Library, The University Library’s regulations (see

http://www.library.manchester.ac.uk/about/regulations/) and in The Uni-

versity’s Policy on Presentation of Theses.

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Acknowledgements

I would like to thank all the people that helped me at some point or another during

my PhD, especially my supervisor, Michael Brown, and my co-supervisor, Anna

Bonaldi. Thank you for all your support and guidance. Without it, I could not have

finished my thesis. I also wish to thank Mathieu Remazeilles for his advice and

help during my research.

I thank the Comisión Nacional de Investigación Científica y Tecnológica (CONICYT)

and the Chilean tax-payers for their financial support through the Becas Chile schol-

arship program.

Finally, I would like to thank my family. Para mi familia, especialmente mi mamá

Cecilia, mi hermana Javiera y mi tía Marcela. Muchas gracias por todo el apoyo.

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Chapter 1

Cosmology and the CMB

Our understanding of our Universe as a whole has dramatically increased over

the last 30 years. The study of cosmology has become a precise science thanks to

many astrophysical observations and surveys, such as Supernovae Ia (SNIa), Large

Scale Structure (LSS) redshift surveys, Baryon Acoustic Oscillations (BAOs) mea-

surements, and more recently Gravitational Waves (GWs), among others. One of

the main observables that contributes to this cosmological revolution is the Cos-

mic Microwave Background (CMB) radiation, first discovered by Arno Penzias and

Robert Wilson in 1964 (Penzias and Wilson, 1965). The CMB photons are the left-

over radiation that were able to free-stream, at a time when the Universe was very

young (∼ 380 thousand years old). The primordial particle soup, made of nucleons,

electrons and photons, cooled down enough so that neutral hydrogen atoms could

be formed by the union of protons and free electrons. This time is referred to as the

epoch of recombination. The photons, which previously were (Thomson) scattered

by the ionized free electrons, could then travel freely. Therefore, the Universe went

from being opaque to being transparent, and radiation travelled uninterrupted as

the Universe evolved to our present day and to us (except for small secondary ef-

fects). This background of radiation that was the last to scatter when atoms were

formed, at the so-called last scattering surface, is the CMB we observe today. It is very

close to an uniform and isotropic radiation background across the sky, whose fre-

quency spectrum is consistent with a black-body at a temperature of 2.725 K (Fixsen,

2009), implying that it peaks at ∼ 160 GHz by Wien’s displacement law. However,

it is not the same black-body across the sky, since it exhibits small anisotropies with

a primordial origin at a level of 10−5. These anisotropies are the main observable

of the CMB, since they were set up, for the most part, by primordial mechanisms

when the Universe was born. The statistical properties of the anisotropies allow us

to measure parameters in the cosmological concordance model, known as ΛCDM.

19

20 CHAPTER 1. COSMOLOGY AND THE CMB

Besides the temperature of the CMB, we can also observe its polarization. The

ΛCDM model also predicts the statistical properties of the CMB polarization, which

complement the measurement of its temperature. Both the anisotropies in temper-

ature and polarization can be explained by the theory of inflation, a mechanism of a

very rapid expansion of the Universe shortly after it was born, which solves several

problems that were present in classical cosmology, such as the horizon and flatness

problems. In this framework, these temperature and polarization anisotropies have

its origin in quantum perturbations that get amplified as the Universe expands.

These perturbations can be of two types: scalar, which correspond to density fluc-

tuations; and tensor, which are fluctuations of the space-time, also known as pri-

mordial gravitational waves. The scalar perturbations spectrum is measured by

recent experiments, e.g. Planck (Planck Collaboration et al., 2016i). The tensor-to-

scalar ratio, r, is the ratio between the primordial tensor and scalar power spectra.

A detection of a non-zero value for r would confirm the predicted background of

primordial GWs, and in the process, would prove the nature of inflation, at energy

scales well above what is accessible from Earth particle accelerators. Finally, this

measurement would probe particle physics, reaching energy scales of the Grand

Unified Theory (GUT), ∼ 1016 GeV.

The cosmological community is putting significant effort towards the measure-

ment of r, by the means of planning future experiments, such as CMB-S4 (Abaza-

jian et al., 2016), LiteBIRD (Ishino et al., 2016), CORE (Delabrouille et al., 2018), and

PIXIE (Kogut et al., 2011), among others. This potential detection will depend on

our ability to measure the CMB polarization B-modes, which the inflation theory

predicts are only produced by primordial tensor perturbations. Different models

of inflation predict different values for r, which are tied to the different values of

the energy scale. In principle, r can take any value from ∼ 1 to infinitesimally

small, which obviously would be too small for our instruments to measure. The in-

strumental challenge is that the primordial signal scales linearly with r, so for low

values (∼ 10−2 or smaller), the B-mode angular power spectrum is a few orders of

magnitude smaller than the polarization E-modes, and several orders of magnitude

smaller than the temperature spectrum. This will test the current and future instru-

mentation and sensitivity required to measure such a small signal. Parallel to this,

diffuse Galactic and extra-Galactic foregrounds radiate at the same frequencies as

the CMB. This requires special analysis and techniques to remove this contaminat-

ing signal, known as component separation (Delabrouille and Cardoso, 2009). The

level of foreground residuals left after component separation is likely to be com-

parable to the primordial B-mode signal we are trying to measure, so we need to

1.1. MODERN COSMOLOGY: THE ΛCDM MODEL 21

improve foreground characterization and component separation analysis.

This chapter will describe the basic cosmological framework: the current ΛCDM

model is described in Section 1.1, the theory of inflation and its predictions in Sec-

tion 1.2, and the CMB analysis in Section 1.3.

1.1 Modern cosmology: the ΛCDM model

Our current understanding of how the Universe works, as a single entity, is the

ΛCDM model, which is a geometrically flat (zero curvature) universe, where the

mass/energy density is accounted for by baryonic matter, made of protons and

neutrons (∼ 4 %), by dark matter (∼ 28 %), by dark energy in the form of cosmologi-

cal constant (∼ 68 %), and a very small percentage of photons and neutrinos. This

model takes its name from the cosmological constant Λ and the model of Cold Dark

Matter (CDM), where cold describes it as slow and non-relativistic. This model

can describe successfully the evolution of our Universe, which is probed by several

observables, among them, the CMB radiation. We will delay the discussion of cos-

mological parameters, that characterize the ΛCDM model, until Section 1.3, once

we introduce several concepts regarding the CMB.

We know empirically that the Universe is expanding. Distant galaxies are mov-

ing away from our Milky Way. We know this by measuring their redshift z, which

increases as we measure more and more distant galaxies. This is know as the Hub-

ble flow. The theory implies that, at some point in the past, the Universe must have

been extremely dense, hot and concentrated in a small point, after being born in

the Big-Bang. From there, it expanded through time. The Universe is a dynamical

system, which can be described in a relatively simple way. First, we introduce the

equations of motion that govern the homogeneous isotropic Universe. The starting

point for cosmology are the Friedmann equations, which are derived from the Gen-

eral Relativity (GR) field equations Gµν = 8πTµν . Gµν is the Einstein tensor, which

can be calculated explicitly. The chosen metric contains the geometry information

of the space-time of the universe. Tµν is the energy-stress tensor, which describes the

distribution of matter/energy and momentum of the system. The energy content

of the Universe determines how the space-time bends, and in turn the curvature of

the space-time dictates the mass/energy content’s trajectories. We consider a Fried-

mann–Lemaître–Robertson–Walker (FLRW) metric in 4-dimensional space-time, in

spatial spherical coordinates,

ds2 = dt2 − a(t)

[1

1− kr2dr2 + r2(dθ2 + sin2 θdφ2)

], (1.1)


where we have set c = 1, and k is the spatial curvature, with units of length−2. It

can be positive for an elliptical geometry (like the surface of a sphere), where the

interior angles of a triangle sum more than 180, negative for a hyperbolic geometry,

where the sum of the interior angles in a triangle is less than 180, or zero for an

Euclidean geometry, also known as flat geometry. a(t) corresponds to the unitless

scale factor which can evolve with time and reflects the expansion or contraction of

the Universe. The relation between redshift and scale factor is given by a = 11+z

,

where we normalize a = 1 when z = 0, i.e. at present time. Then, we consider

the Universe to be filled with different components with specific equations of state.

The Einstein field equations originates the Friedmann equation

a2

a2= H2 =

8πGρ

3− k

a2=

ρ

3M2pl

− k

a2, (1.2)

where the Planck mass Mpl is defined as M2pl = ~c

8πGwith ~ = c = 1; and the fluid

equation, which describes the evolution of the density of the component of the Uni-

verse with time

ρ+ 3a

a(ρ+ p) = 0, (1.3)

where G is the gravitational constant, ρ is the density, p is the pressure, and the

derivative is with respect to time. We define the Hubble parameter as H(t) = a(t)a(t)

.

The equation of state of a given component is parametrized as p = wρ. These two

equations can be combined to derive a third equation, the acceleration equation

a

a=−4πG

3(ρ+ 3p) . (1.4)

From now on, we assume a flat Universe for simplicity (k = 0). In this simple

model Universe, the components that constitute it are matter and radiation. Matter

is pressureless, because its energy content is mostly stored in the form of rest matter,

so its kinetic energy is very small. This means w = 0 and p = 0. By solving the fluid

equation for a model universe which only contains matter, we find ρm(a) = ρm,0a−3,

where the sub-index 0 indicates the value at the present time (a0 = 1). Solving

the Friedmann equation, we find a ∝ t2/3. For radiation, the equation of state is

p = ρ/3. If we consider a model universe filled with only radiation, the solution

of the fluid equation is ρr = ρr,0a−4. Solving the Friedmann equation, we find a ∝

t−1/2. In general, the Universe would contain a mixture of components, each with

a different equation of state. Next, we assume a Universe with a mixture of matter

and radiation. The critical density ρc is defined as the density such that the Universe

is flat, so from the Friedmann equation, ρc = 3H2

8πG. The value for it at the present

time is ρc,0 ∼ 10−26 kgm−3. Then, with the assumption of k = 0, we can divide the


Friedman equation by H20 , so we have

H2

H20

=ρ

ρc,0

=ρm,0a

−3

ρc,0

+ρr,0a

−4

ρc,0

= Ωm,0a−3 + Ωr,0a

−4, (1.5)

where we define ΩX,0 =ρX,0

ρc,0for component X. In this case, with the flat Universe,

the sum of the Ωs must be 1. It is not straightforward to derive a(t) with an equation

such as this. It can be integrated to yield∫ a

0

da′√Ωr,0a′−2 + Ωm,0a′−1

= H0t. (1.6)

Radiation is composed of the CMB photons, which have an energy density of

ρCMB,0 = 4.6 × 10−31 kgm−3 (with a temperature of 2.725 K), so ΩCMB = 4.6 × 10−5.

The ultra-relativistic neutrinos at the early age of the Universe would create a Cos-

mic Neutrino Background Radiation, which theory predicts should have an energy

density similar, but not equal to CMB photons. The total radiation density param-

eter is then Ωr ∼ 10−4, so radiation is not important today, but it was the dominant

component very early in the Universe when a 1. At some point, the radiation

and matter energy density were equal (this epoch is called equality). This happened

when

ρr = ρm = Ωr,0a−4rm = Ωm,0a

−3rm , (1.7)

where arm is the scale factor at the equality. This is arm = Ωr

Ωm. Later, we will show

that Ωm,0 ∼ 0.3, and therefore arm ∼ 1.5 × 10−4. This corresponds to a redshift of

z ∼ 6500, which is long before the time of last scattering (which we have measured

to be z ∼ 1100).

1.1.1 The cosmological constant Λ

The Einstein field equation can be written as Gµν + Λgµν = 8πTµν , where a constant

(called the cosmological constant) is proportional to the metric. The energy density

is constant, so the principle of conservation is maintained. This constant was in-

troduced by Einstein to allow a steady state Universe, which was supported at the

time, and the equation is unaffected by its presence. Then, equation 1.2 becomes

a2

a2= H2 =

8πGρ

3− k

a2+

Λ

3, (1.8)

and the fluid equation 1.3 is unaffected. The extra term in the Friedmann equation

is equivalent to a new component of the Universe, with an energy density ρΛ = Λ8πG

constant-across-time. The fluid equation then states that for ρ = 0, the equation of

state of this new component must be p = −ρ, or negative pressure. Solving for a


Figure 1.1: Distance modulus (m−M ) as a function of redshift for many supernovaetype Ia. The theory distance modulus as a function of redshift is fitted as a functionof Ωm and ΩΛ. The best fit is for Ωm ∼ 0.3 and Ω ∼ 0.7, which allows for a Universewith an accelerated expansion with a dominant cosmological constant component.Taken from Riess (2000).


pure Λ model universe means ρΛ = constant and a(t) = exp(H0(t− t0)), meaning an

exponential (accelerated) expansion of the universe. We can add this new cosmo-

logical constant component to the mixture of the Universe, so equation 1.6 becomes∫ a

0

da′√Ωr,0a′−2 + Ωm,0a′−1 + ΩΛ,0a′2

= H0t. (1.9)

This cosmological constant became unnecessary in the early epoch of cosmology,

when cosmologist realized that the Universe was apparently expanding rather than

stationary. It was reintroduced to cosmology with the discovery of the accelerated

expansion of the Universe in the late 1990s. The luminosity distance in a FLRW

metric universe filled with matter, radiation and cosmological constant, at a given

redshift z is

dL(z) =1

H0

(1 + z)

∫ z

0

dz′√Ωr,0(1 + z′)4 + Ωm,0(1 + z′)3 + ΩΛ,0

, (1.10)

expressed in terms of redshift rather than scale factor.

On the other hand, the luminosity distance to a standard candle (object with

a known absolute flux) is dL =√

L4πF

, where L is the intrinsic luminosity of the

source and F is the flux measured at a distance dL. In term of optical magnitudes,

the relation is

m−M = 5 log10

(dL

Mpc

)+ 25, (1.11)

where m and M are the apparent and absolute magnitude, respectively. Riess et al.

(1998) and Perlmutter et al. (1999) used supernovae type Ia as standard candles to

measure the acceleration of the Universe. Figure 1.1 shows the distance modulus

m−M as a function of redshift z for several supernovae type Ia. The distance mod-

ulus can be calculated by integrating numerically equations 1.10 and 1.11, with the

fitting of Ωm and ΩΛ (the Ωr value is negligible). The best fit values are Ωm ∼ 0.3 and

ΩΛ ∼ 0.7. This means that today, the Universe is composed by roughly 70% of some

component that has negative pressure and has a constant energy density. Since the

Universe is expanding, to maintain a constant density, new energy is created from

vacuum. This is labelled dark energy.

Dark energy is dominant today, but at some previous time matter was the dom-

inant component. At the scale factor of equality amΛ, we have

ρm = ρΛ = Ωa−3mΛ = ΩΛ, (1.12)

so amΛ =(

0.30.7

)1/3= 0.75. This corresponds to a redshift of z ∼ 0.33. We can iden-

tify distinct periods in the evolution of the Universe: following the Big Bang, in

the very early Universe, as we will see in Section 1.2, a very sudden and rapid


expansion of the Universe, the cosmic inflation, that finished ∼ 10−32 s after the be-

ginning. Inflation is followed by a period of reheating, where it is populated by

elementary particles. After the first 10−12 s of the Universe, we have the so called

early Universe, explained by classical cosmology, where we can identify three peri-

ods: radiation-dominated, followed by matter-dominated, followed by cosmologi-

cal constant-dominated, which corresponds to the present time.

1.1.2 Dark matter

There are reasons to believe that the matter content of the Universe is not all bary-

onic (the familiar matter to us, made of protons and neutrons), but also contains

dark matter, meaning that we only know about its existence from its gravitational

effect and not from its interaction with radiation. We have several pieces of ob-

servational evidences that point to the existence of this dark matter. Among them,

the galaxy rotation curves. The tangential velocity of gas and stars in the disk of a

spiral galaxy can be measured, where it is found that it is roughly constant as the

radius increases. On the other hand, from Newtonian dynamics we know that for

tangential velocity from gravitational centrifugal force

GM(< r)m

r2= m

v2

r, (1.13)

whereM(< r) is the mass of the gravitational field within a radius r, v is the tangen-

tial velocity and m is the mass of the particle. From this, v(r) =√

GM(<r)r

, therefore

for v to be constant, the mass must be M ∝ r. However, the mass inferred from

the luminosity of the stars is not enough to explain this mass profile. The gas in

the outskirts of the galaxy still has a constant velocity, even though there are not

enough stars or gas mass to justify this behaviour.

Further evidence comes from galaxy clusters. We can measure their mass from

X-ray observations, from the velocity dispersion of its components, and from grav-

itational lensing of background high redshift sources. All these estimates of mass

cannot be explained by the sole presence of luminous matter. The virial theorem

states that in a system in equilibrium, where the components are bound by a poten-

tial force (gravity), in average over a long period of time, the kinetic energy is −1/2

the amount of potential energy. This allows to derive an estimate of the mass of a

galaxy cluster as a function of its velocity dispersion and its radius. For example, for

the Coma cluster, the mass estimated from this method is ∼ 1015M (Zwicky, 1937;

The and White, 1986). The mass estimate coming from stars is ∼ 1013M, and from

hot intracluster gas (from X-ray observations) is ∼ 1014M. So, about 10 − 15% of


the matter content can be explained, with the rest constituting the dark matter. An

independent measurement of the distribution of mass of the galaxy structure is the

use of gravitational lensing effects. The strong gravitational lensing is the deflection

of light from a distant source by the gravitational potential of an intervening mas-

sive galaxy cluster. This creates one (or multiple) arcs of the distant source, whose

analysis allows the estimation of the cluster mass. Weak gravitational lensing is the

same effect, but to a smaller degree. An intervening gravitational field creates small

distortions in the shape of the background galaxies. The latter can be measured sta-

tistically to estimate the mass from the intervening dark matter distribution. In both

lensing cases, the total mass measurements agree with the virial theorem, most of

the matter cannot be explained by known sources.

The current favoured model for dark matter is CDM, meaning that matter is not

relativistic at the early Universe, so its velocity dispersion is very small. This idea

was proposed in 1982 by Peebles (1982); Bond et al. (1982); Blumenthal et al. (1982).

The Large Scale Structure (LSS) of our (local) Universe is made of galaxies and

galaxy clusters, arranged in filaments and sheets with voids within them. This

distribution of visible matter is observed empirically by galaxy redshift surveys,

such as the 2 degree Field Galaxy Redshift Survey (2dFGRS) (Colless et al., 2001) or

the Sloan Digital Sky Survey (SDSS) (York et al., 2000). The picture that the CMB

paints is of very small perturbations on an otherwise smooth background. From

these anisotropies, the LSS formed through the gravitational evolution of dark (and

a small fraction of baryonic) matter. This transition between the z ∼ 1100 and the

z ∼ 1 Universe can be studied through numerical simulations (Springel et al., 2006).

Theory predicts a hierarchical formation of structure, where big dark matter haloes

grow by the aggregation of smaller ones. When gas cools at the centre of overden-

sities, they form stars, which together make up galaxies. The difference between

CDM and stars can be seen in realistic numerical simulations like the Millenium

simulation (Springel et al., 2005), where galaxies outline the clustering nature of

the LSS early on, while the dark matter distribution grows into the LSS from a

smooth distribution. Warm Dark Matter (WDM), for example, with more kinetic

energy than CDM, does not agree with observations of the LSS, as it cannot repro-

duce the necessary degree of clustering of structures. CDM at early times, on the

other hand, brings predictions by simulations and the observations of the LSS into

agreement (e.g. White et al., 1983; Davis et al., 1985; White et al., 1987). However,

simulations and observations do not agree on all aspects. There are the cusp-core

and the missing satellites problems (Weinberg et al., 2015). The former is the fact

that simulations predict the dark matter density profile at the centre of galaxies to


be cuspy, meaning a steep increase at smaller radii, while observations of rotation

curves show that it is more similar to a core, or a flat dependence with radius. The

latter problem is the prediction by simulations that in dark matter haloes, clumpy

substructures are formed. This would mean that the number of satellite galaxies in

a galaxy group, formed from those sub-haloes, should be relatively high. However,

in our Local Group, we only observe ∼ 40 satellites of the Milky Way, when theory

predicts about an order of magnitude more.

Today, the cosmological model measures Ωm ∼ 0.32 (Turner, 2002), with Ωb ∼0.04 and Ωcdm ∼ 0.28. The baryon fraction is determined from Big Bang Nucle-

osynthesis (BBN) and the primordial fraction of deuterium in high-redshift clusters

(Burles et al., 2001a,b). BBN is the very early epoch (∼ 300 s) of the Universe when

nucleons, protons and neutrons, were fused to form light elements (deuterium, He

and Li isotopes), at energies below the binding energy of deuterium (2.2 MeV). Fol-

lowing the rates of the reactions between nucleons, under the conditions of temper-

ature in the early universe, nuclear physics predicts that the primordial fraction of

helium YP is close to 0.25, and the dependence of the primordial abundances of light

elements. One that is significant is the deuterium fraction, which depends critically

with the baryon-to-photon ratio η. The primordial deuterium fraction is measured

by Lyman-α absorption lines of distant quasars by intervening high-redshift clus-

ters. It is measured to be [D/H]P ∼ 3 × 10−5 (Burles and Tytler, 1998a,b), which

means that the baryon density is measured to be Ωbh2 ∼ 0.02.

The total matter density parameter Ωm = Ωcdm + Ωb is measured from multiple

sources, from which we can deduct Ωcdm. The location and ratio of odd and even

acoustic peaks in the CMB angular power spectrum can constrain both Ωm and Ωb

(e.g. Hancock et al., 1998; Efstathiou et al., 1999). The mass-to-light ratio from

galaxy clusters, described above, can constrain the ratio Ωb/Ωm. Also, the matter

3D power spectrum estimated from the LSS galaxy redshift surveys can constrain

Ωm and Ωb/Ωm (Cole et al., 2005; Percival et al., 2007).

1.2 Cosmic inflation

The development of the theory of inflation has been one of the most important

breakthroughs in physics in the last 30 years. Inflationary cosmology was formu-

lated to try to solve several problems that the classical formulation of cosmology

had. Some of these were

1.2. COSMIC INFLATION 29

• The flatness problem The measurement of cosmological parameters has con-

strained our Universe to be geometrically flat, so Ωtotal = 1. The Friedman

equation with curvature can be written as

1 =8πGρ

3H2− k

a2H2, (1.14)

and using the definition of the critical density ρc = 3H2

8πG, we can rewrite the

Friedmann equation as

(1− Ω)(a2H2) = −k, (1.15)

where we use the definition Ω = Ωtotal = ρρc

. This represents the total density

parameter by the sum of components. In practice, when saying that the Uni-

verse is flat, we mean |1− Ω| ∼ 0 to within some small uncertainty. For exam-

ple, in the time of matter radiation, a ∝ t2/3, therefore H ∝ t−2 and a2 ∝ t4/3,

the factor a2H2 ∝ t−2/3. To keep the left hand side constant, 1 − Ω ∝ t2/3, so

during matter epoch the uncertainty increases with time. Even earlier, when

the Universe was dominated by radiation, a ∝ t1/2, so by the same analysis,

the factor a2H2 ∝ t−1, and 1− Ω ∝ t. If we measure 1− Ω0 ∼ 0 today, then in

the beginning of the Universe Ω should have been extremely close to 1, since

as the Universe evolves, it grows to a present day uncertainty of ∼ 0.01. More

complex calculations say that |1 − Ω| ≤ 10−60 should be true close to the be-

ginning of the Universe. It seems unrealistic that Ω so close to 1 was set just

by coincidence.

• The horizon problem The CMB radiation is a nearly perfect black-body. Since

the anisotropies in temperature are very small, of the order ∼ 10−5, this is

consistent with all the regions of the sky having been causally connected at

some point before the last scattering surface, and therefore, share a common

temperature. However, unless we assume a period of very rapid expansion

in the early Universe, this is not the case. This concept is best explained in

terms of the Hubble radius, which is the radius of the volume around an

observer at which objects recede at a speed greater than the speed of light

because of the expansion of the Universe in between them. It is given by

dHubble = 1H(t)

(remembering that c = 1). The comoving Hubble radius is given

by (aH)−1, once we factor in the scale factor. From equation 1.5, we know that

H(a) = H0

√Ωm,0a−3 in a matter-dominated Universe, where the last scatter-

ing happened. We know that als ∼ 11100

, so we can calculate dHubble ∼ 0.2 Mpc

at last scattering. The Hubble volume was 0.4 Mpc in diameter, which corre-

sponds to an angle in the sky of ∼ 1 at the distance of last scattering. There-

fore, points in the sky, as observed by us, were not causally connected with


each other if they are separated by more than 1 in the sky today, yet they all

have the (almost) same temperature.

• The monopole problem This is related to the standard model of particle

physics, and the Grand Unified Theory (GUT), which unifies the electro-

magnetic, weak and strong forces into a single force at very high energies,

∼ 1016 GeV, at the epoch of inflation. The phase transition at the GUT scale

creates magnetic monopoles, with a considerable energy density. This is so

large that they should be the dominating component of the Universe today,

however we do not observe magnetic monopoles in nature. In principle, these

are hypothetical particles, and there are alternative Grand Unified Theories in

which they are not produced.

The idea of cosmic inflation is that the Universe went through an epoch, just

after being born, of very rapid exponential expansion a small fraction of a second

after the Big Bang ∼ 10−34 s, driven by a scalar field denominated the inflaton. It

was proposed by Guth (1981) to solve the problems outlined above.

The comoving horizon for an observer is the maximum distance light would

travel and that would be able to reach another observer at that distance in the fu-

ture. Light travels at speed c, but as it travels through the Universe, space-time is

also expanding in between. For an observer, it defines the size of the observable

Universe from its point of view. It is given by dH(t) =∫ t

0dt′

a(t′). Using the definition

of the Hubble parameter daa

= Hdt, we have dH(a) =∫ a

0d ln(a′)a′H(a′)

. Today, the comov-

ing Hubble radius is smaller than the comoving horizon. If the comoving Hubble

radius was much larger (than today) in the early Universe, then all points would

be within the comoving Hubble radius and in causal contact. Then, the dramatic

exponential decrease in the comoving Hubble radius pushed out the largest scales

of perturbations (which today are separated by more than ∼ 1). These were left

outside the comoving Hubble radius when the inflation period ended. Eventually,

as the Universe evolved, these large-scale modes re-entered the comoving Hubble

radius. This solves the horizon problem, and moreover, it simultaneously solves

the flatness one. Since we showed that the Hubble radius increases with time as

the Universe evolves, the solution to the this problem is that the Hubble radius

decreases initially, so that a2H2 increases with time and 1 − Ω can decrease. The

Universe could have had any arbitrary curvature value at the beginning, not even

close to flatness. The fact that 1aH

decreases makes 1−Ω converge to∼ 0 eventually.

The monopole problem is solved by considering the fact that inflation expands the

Universe dramatically, so a high density of particles created before inflation would


dilute many orders of magnitude. In the period of reheating, after inflation, the

potential energy of the inflaton field decays into Standard model particles, repop-

ulating the Universe with particles. The monopoles are not created after inflation,

since they originate at much higher pre-inflation energies.

The condition for the decrease of the comoving Hubble radius is ddt

1aH

< 0. Per-

forming the derivative, we find ddt

1aH

= −a/(aH)2, which must be negative, mean-

ing that a > 0, in other words, accelerated expansion. The acceleration equation 1.4

implies

H2 + H =a

a=−4πG

3(ρ+ 3p), (1.16)

and dividing by H2 and using the Friedmann equation 1.2 (for a flat Universe), we

findH

H2= −3

2

(1 +

p

ρ

). (1.17)

The accelerated expansion a > 0 in the acceleration equation 1.4 implies that what-

ever is driving inflation, its equation of state must be p/ρ = w < −1/3. This has

negative pressure, like the cosmological constant Λ, which drives an accelerated

expansion.

We do not know what drives inflation. In the simplest models, the idea of a

single scalar field is postulated, the inflaton introduced above, which in principle

can depend on time and space φ = φ(x, t). The energy density and pressure of such

a field is given by

ρφ =1

2

(dφ

dt

)2

+ V (φ) (1.18)

pφ =1

2

(dφ

dt

)2

− V (φ), (1.19)

where V (φ) corresponds to the potential energy associated with the field. The equa-

tion of state p = wρ then is

wφ =12

(dφdt

)2 − V (φ)

12

(dφdt

)2+ V (φ)

, (1.20)

so in order for the pressure to be negative, V > 12φ2 and the potential energy must

dominate over the kinetic energy. Taking the fluid equation for the inflaton, equa-

tion 1.3, replacing aa

= H and using equation 1.18, we find

φφ+ V + 3H(φ2) = 0. (1.21)

Making the change of variables dV/dt = φdV/dφ, we find the equation of motion

that governs the dynamics of the inflaton

φ+ 3Hφ+dV

dφ= 0. (1.22)


The condition for the slow evolution of the kinetic energy of the inflaton is la-

beled slow-roll inflation, so that the potential always dominates. From equation 1.22,

we need |φ| dVdφ

. To enforce the slow-roll conditions, the slow-roll parameters are

defined. These should be 1. Taking equation 1.16, we can write aa

= H2(1 − ε)with

ε = −H/H2 < 1, (1.23)

which defines the first slow-roll parameter. From equation 1.17

ε =3

2(1 + wφ) =

3

2

φ2

12φ2 + V

=1

2

φ2

H2. (1.24)

The condition for equation 1.22, that the acceleration of the inflaton is small, is en-

forced by the definition of the second slow-roll parameter

η = −2H

H2− ε

2Hε= 2ε− ε

2Hε. (1.25)

In the slow-roll approximation, the kinetic energy of the inflation is much

smaller than its potential energy, which is approximately constant. Using the Fried-

mann equation H2 = 13M2

plρφ = 1

3M2pl

(12φ2 + V ) ≈ 1

3V = constant. With a constant

Hubble parameter, maintained during inflation, the scale factor grows exponen-

tially a(t) = a(tinitial)eHt.

In terms of the potential of the inflaton, the slow-roll parameters are

εV (φ) =M2

pl

2

(dV/dφ

V

)2

ηV (φ) = M2pl

d2V/dφ2

V. (1.26)

Inflation generally ends when the conditions for these parameters is violated ε, η ∼1. Another quantity we need to define is the number of exponential folds Ne, or

how many e-folds the scale factor is expanded. It is defined as

Ne = ln(aend/ainitial) =

∫ tend

tinitial

Hdt, (1.27)

between the beginning and end of inflation. To solve the horizon and flatness prob-

lems, the minimum number of e-folds is Ne ∼ 60, but it can be larger than that.

1.2.1 Primordial perturbations

When inflation is happening, we can write in the linear regime small perturbations

in the inflaton (which fills the energy-stress tensor Tµν) and in the metric.

φ(t,x) = φ(t) + δφ(t,x) (1.28)

gµν(t,x) = gµν(t) + δgµν(t,x). (1.29)


Because the perturbations are very small, and using the fact that the early Universe

is linear, the Einstein equations are δGµν = 8πδTµν . We can decompose the pertur-

bations into three modes: scalar, vector and tensor, or SVT decomposition. Scalar

and tensor modes corresponds to density perturbations and gravitational waves,

respectively. Vector modes are not created in models of inflation, and will decay

after the rapid expansion process. Mathematically, they can be explained in Fourier

space. For example, for the inflaton perturbation

δφk =

∫δφ(t,x)eix·kdx, (1.30)

where k is the corresponding vector of the wavenumber of the perturbation. If a

rotation around k takes place, by an angle α, the perturbation transforms as δφk →eimαδφk, where m can be either 0 (scalar), ±1 (vector) or ±2 (tensor).

For the metric perturbations δgµν , there are ten possible degrees of freedom.

Four of these can be eliminated by gauge choices, so they are not physical. Of the

remaining six, two are tensor and two are vector. The two last ones are scalar, but

they can be reduced to one. A gauge invariant quantity, is the curvature perturbation

on uniform-density hypersurfaces ζ , which is a combination of inflaton and metric

perturbations. During slow-roll inflation,

−ζ = Ψ +H˙φδφ (1.31)

where Ψ is the perturbation to the spatial section of the metric gij . The comoving

curvature perturbation R is another gauge invariant variable, which in the slow-roll

approximation is equal to ζ (equation 1.31). An important property is that ζ and Rare constant at horizon crossing and at super-horizon scales, when k aH .

Scalar perturbations

A small perturbation in the inflaton is related to a perturbation in the energy-stress

tensor, which will induce a perturbation in the metric by the Einstein field equa-

tions. We can choose a gauge for the inflaton and the metric

δφ = 0, gij = a2[1− 2R]δij . (1.32)

The power spectrum of the comoving curvature perturbation is defined through

the variance

〈Rk(t)Rk′(t)〉 = (2π)3δ(k + k′)H2

hc

2k3

H2hc

φ2hc

, (1.33)

where the subscript “hc” means it is evaluated at horizon crossing (when the size

of the perturbation k−1 is equal to the Hubble radius, k ≈ aH). The dimensionless


power spectrum ∆R is defined as 〈Rk(t)Rk′(t)〉 = (2π)3δ(k+k′)2π2

k3∆2R(k). Then, we

can write

∆2s (k) =

1

8π2H2 1

ε|k=aH , (1.34)

using equation 1.24 and evaluating at horizon crossing k = aH . Also, we changed

fromR to s to reflect the scalar nature of the perturbations.

Finally, for cosmological parameter estimation, the power spectrum is

parametrized by a power law with an spectral index ns, defined by

ns − 1 =d ln ∆2

s

d ln k, (1.35)

and by an amplitude As, defined in

∆2s (k) = As(k?)

(k

k?

)ns(k?)−1+ 12αs(k) ln(k/k?)

, (1.36)

where k? is a predefined pivot scale, and an additional running of the spectral index

αs can be included if needed.

Tensor perturbations

In the case of gravitational waves, the perturbed metric is parametrized by hij ,

which is chosen to be Gauge-invariant,

gij = a(t)2[δij + hij], (1.37)

where hij = h+e+ij +h×e×ij has two polarization modes. For both of them, analogous

to the scalar perturbations, we define a power spectrum through the variance of

h = h+,h×

〈hk(t)hk′(t)〉 = (2π)3δ(k + k′)2π2

k∆2h(k). (1.38)

In this case, the power spectrum is given by ∆2h = 4

(H2π

)2|k=aH . The tensor pertur-

bations power spectrum, ∆2t is equal to twice ∆2

h, because there are 2 polarization

states.

The tensor power spectrum is also parametrized as a power law, with a spectral

index nt defined by

nt =d ln ∆2

t

d ln k, (1.39)

and an amplitude At

∆2t (k) = At(k?)

(k

k?

)nt(k?)

. (1.40)

The tensor-to-scalar ratio r is defined as

r =∆2

s (k?)

∆2t (k?)

, (1.41)


with the preferred scale usually being k? = 0.002 Mpc−1 or k? = 0.05 Mpc−1.

Under the slow-roll approximation, we have H2 ≈ V (φ)/3M2pl, as shown above,

so the scalar and tensor spectra can be written as

∆2s (k) ≈ 1

24π2

V

M4pl

1

ε|k=aH ∆2

t (k) ≈ 2

3π2

V

M4pl

|k=aH , (1.42)

evaluated at horizon crossing. From this, the tensor-to-scalar ratio is

r = 16ε. (1.43)

Using the definition of ε and η, the slow-roll parameters, in equations 1.23 and 1.25,

and considering that ∆2s (k) ∝ H2/ε, plus the fact that during inflation H is very

close to constant, we can then make a change of variables between k and a, which

are related by the horizon crossing condition k = Ha. This means d ln(k) ≈ da/a =

Hdt. From equation 1.35, we can derive that

ns − 1 = 2η − 6ε. (1.44)

Under similar arguments, we can find that

nt = −2ε. (1.45)

The last equation and equation 1.43 imply that

r = −8nt, (1.46)

which is a consistency relation between two observables. This relation should be

fulfilled for single field slow-roll models of inflation. The fact that the scalar and

tensor perturbations power spectra are not perfectly scale-invariant (ns = 1 and

nt = 0) supports the idea of slow-roll inflation. If measured, as we will see below,

a significant departure from ns = 1 is strong evidence for the inflationary origin of

our Universe.

1.2.2 Models of inflation

The inflaton determines a given shape for the potential, based on standard or non-

standard physics. Once the potencial V (φ) is given, we can determine the energy

scale of inflation, and further predict the values of the observables ns and r.

The Lyth bound (Lyth, 1997) sets a useful limit to classify models of inflation.

Differentianing the Friedmann equation (and using equation 1.18), we find 2HH =1

2M2plφ[φ+dV/dφ], and using the equation of motion of the inflaton, equation 1.22, we


find H = φ2

2M2pl

. We can rewrite this as dHdφ

= 12M2

plφ, and using this to make a change

of variables (from t to φ) in the definition of the number of e-folds, equation 1.27,

we have

Ne(φ) = − 1

2M2pl

∫ φend

φ

H

dH/dφdφ =

1

Mpl

∫ φ

φend

1√2ε

, (1.47)

where the definition of ε is used in the last equality. Finally, using equation 1.43, we

find the Lyth bound,

∆φ

Mpl

&

√r

10−3

(Ne

10

), (1.48)

where ∆φ is the excursion of the inflaton field, which is the inflaton difference be-

tween the start and end of inflation. Inflation models are usually divided into sub-

and super-Planckian, when ∆φ is smaller or greater than the Planck mass, respec-

tively. For Ne ∼ 60, the boundary between sub- and super-Planckian models is

r ∼ 10−2. A difficulty for large-field super-Planckian models is that it would be dif-

ficult for them to maintain smallness of the slow-roll parameters ε,η 1 at all times

over a large field excursion ∆φ > Mpl. However, this is possible under scenarios

of symmetry of quantum gravity, with models of inflation based on string theory.

Determining whether the excursion of the inflaton is sub- or super-Planckian will

give information on state-of-the-art quantum-gravity physics.

Large-field models

Some of the large-field super-Planckian models of inflation include power-law or

chaotic potentials, with the form

V (φ) ∝(

φ

Mpl

)n. (1.49)

These models predicts ns − 1 ≈ −n(n + 2)M2pl/φ

2? and r ≈ 8n2M2

pl/φ?, with φ2? ≈

nM2pl(4Ne + n)/2. Models based on particle physics have been proposed for n = 2

(Linde, 1983, one of the first and simplest model of inflation), n = 1 (McAllister

et al., 2010), among others.

Natural inflation (Freese et al., 1990) is a model where the potential has the form

V (φ) ∝[1 + cos

(φ

f

)], (1.50)

where f is a periodicity that can have any value in principle. For f & 1.5Mpl, it is a

large-field model. These model are based on axion fields.


Small-field models

The Hilltop models have the potential of the form

V (φ) ∝(

1− φp

µp

), (1.51)

where µ Mpl. The value p = 2 is a large-field model, and p > 2 for small-field

excursion of the inflaton. This models predict ns − 1 ≈ − 2Ne

p−1p−2

and an upper limit

r . 8 pNe(p−2)

(8π

Nep(p−2)

)p/(p−2)

.

Another important model is the Starobinsky R2 inflation (Starobinsky, 1980). In

this model, the action has an extra quadratic term in the Ricci scalarR. The potential

has the form

V (φ) ∝(

1− e−√

2/3φ/M2pl

)2

. (1.52)

The predictions of this model are ns−1 ≈ −8(4Ne+9)/(4Ne+3)2 and r ≈ 192/(4Ne+

3)2.

1.2.3 Current observational constraints

The current constraint on the scalar spectral index is ns = 0.968 ± 0.006, from the

Planck 2015 data (Planck Collaboration et al., 2016i). A scale-invariant spectrum

(ns = 1) is ruled-out by more than 5σ, which is a strong evidence for inflation.

In the case of tensor modes, we only have upper limits. The tensor-to-scalar ra-

tio is constrained to r0.002 < 0.11 (2σ confidence level) by Planck Collaboration et al.

(2016h). The joint analysis from Planck and BICEP2/KECK (BICEP2/Keck and

Planck Collaborations et al., 2015) finds an upper limit of r0.05 < 0.12. A more re-

cent analysis from the BICEP2/Keck data from two bands (95 and 150 GHz), yields

r0.05 < 0.09 (BICEP2 Collaboration et al., 2016).

Figure 1.2 shows the current joint constraint on ns and r by Planck Collaboration

et al. (2014d) and by Planck Collaboration et al. (2016h). The two circles represent

the predictions for ns and r for Ne = 50 and Ne = 60 for different inflation models.

The chaotic model ∝ φ2, one of the first models proposed, is ruled out, lying out-

side the 95% confidence level. Natural and Hilltop models are shown as a range,

since they contain a free parameter. Starobinsky R2 inflation is fully within the 68%

confidence interval.

In the future, proposed experiments like CORE could further improve the de-

tection limits of r, down to limits of σ(r) ∼ 4 × 10−4 (Finelli et al., 2018). This is

enough to improve the current measurement of r up to 2 order of magnitude. The

next milestone for inflation would be to reach the Lyth bound limit of r ∼ 0.01.


Figure 1.2: Current constraints on ns and r by the Planck 2013 and 2015 data release(Planck Collaboration et al., 2014d, 2016h). The predictions (for Ne = 50 and Ne =60) from different models of inflation are shown.

This would provide evidence to confirm or discard large-field inflationary models.

In the not-so-distant future, ∼ 5 years, experiments will target this value, like the

Simons Observatory.

1.3 The Cosmic Microwave Background Radiation

To use the CMB radiation temperature as a cosmological probe, in principle we

need a full-sky map of it. The very small spatial anisotropies can be described com-

pletely by harmonic space statistics (power spectra), if the anisotropies are Gaus-

sian in nature. Current limits are consistent with Gaussianity (Planck Collaboration

et al., 2016g), but the errors still would allow for a small non-Gaussianity. To de-

scribe the statistics on the surface of the sphere, spherical harmonics are used. The

temperature anisotropies field is described as

∆T

T= Θ(θ,φ) =

∞∑`=1

∑m=−`

a`mY`m(θ,φ), (1.53)

where Y`m(θ,φ) are the spherical harmonics in the direction θ,φ, and the coefficients

a`m are the analogous to the coefficients in a Fourier series (harmonic functions in

flat space), which are multiplied by sin and cos. The variance of the a`m coefficients

1.3. THE COSMIC MICROWAVE BACKGROUND RADIATION 39

is the angular power spectra C`, defined as

〈a`m, a∗`′m′〉 = δ``′δmm′C`, (1.54)

where the brackets indicate a statistical average over many realizations in m for a

fixed `. If the field is Gaussian, then C` contains all the information.

The angular power spectrum estimator from the a`m coefficient is calculated as

C` =1

2`+ 1

∑m=−`

|a`m|2. (1.55)

Since for a given `, there are only (2` + 1) a`m coefficients, there is an intrinsic un-

certainty in the calculated value of C` for low-` values because of the low number

of samples. This uncertainty is called Cosmic Variance, and it is a consequence of the

fact that the CMB is a single realization of a random Gaussian field. We only have

a single Universe to observe, therefore a single realization. It is calculated as

∆CCV` =

√2

2`+ 1C`. (1.56)

In Figure 1.3, we show the temperature angular power spectrum CTT` , in black,

as measured by Planck in its 2015 data release (Planck Collaboration et al., 2016m).

1.3.1 The CMB polarization

In the epoch of recombination, the quadrupole temperature anisotropies generate

linear polarization. The fraction of polarization is small (∼ 10%), since only the

photons with low optical depth can scatter in these conditions. The optical depth

decreased as the Universe went from opaque to transparent, so the photons with

low optical depth were the last ones to scatter, right before the moment the optical

depth became small. The polarization anisotropies pattern is imprinted by scalar

and tensor perturbations in different ways, which we could use to our advantage

to constrain cosmological theories.

The polarization of an electromagnetic wave is usually parametrized in term of

Stokes parameters: I , Q, U , and V . I measures the total intensity of the radiation,

which is the temperature field of the CMB, previously described. The total polar-

ization intensity is P =√Q2 + U2 + V 2. V measures the amount of circular polar-

ization, which corresponds to the rotation of the electrical (and magnetic) field with

respect to the propagation axis of the wave. The circular polarization is expected to

be 0 for the CMB, since the mechanism that polarizes it would only produce linear

polarization, entirely described by the Q and U parameters. They both measure the


101 102 103

multipole

10 2

10 1

100

101

102

103

104

(+

1)/2

C

K2

E-modes auto-spectrumTemperature E-modes cross-spectrumTemperature auto-spectrum

Figure 1.3: TT , EE and TE angular power spectra as measured by the 2015 datarelease of Planck. The best fit theory spectra are shown as solid lines. The absolutevalue of the TE spectrum is shown for clarity.

intensity difference between two perpendicular modes of linear polarization, but Q

is rotated 45 with respect to U .

The origin of polarization in the CMB is Thomson scattering (Hu and White,

1997), which is the low-energy limit of Compton scattering. Photons strike a free

electron, which are accelerated. The photon is re-emitted with the same frequency

but in a different direction. The Thomson scattering cross-section is σT/dΩ ∝ |ε ·ε′|2, where ε and ε′ are the incoming and scattered directions. This means that the

polarization of the scattered wave is maximal perpendicular to the direction of the

incoming wave. The anisotropies can have a dipole pattern, which roughly means

that variations from hot to cold spots are over 180, as shown in the cartoon in

Figure 1.4, left. In the X direction, there are two incoming photons, one hot and one

cold over 180. The resulting scattered photon in the Z direction will average both

incoming directions and the scattered photon will be unpolarized. However, when

there is a quadrupole pattern, as shown in the right of the figure, the variation from

hot to cold spot is within 90. Therefore, the hot photon in the X direction will keep


Figure 1.4: Left: Dipole anisotropy pattern. When incoming radiation (in the XYplane) is scattered by the electron, the resulting wave in the Z direction is unpo-larised. Right: Quadrupole anisotropy pattern. In this case, the incoming radiationpattern allows that the scattered wave in the Z direction is polarised in the Y direc-tion. From Dodelson (2003).

the X polarization of the scattered photon, and the cold photon from the Y direction

will keep the less intense Y polarization in the scattered photon. The net result is

that the scattered wave is polarized.

E and B modes

For the CMB polarization, we measure the Stokes parameters Q and U . However,

these in general depend on a reference coordinate axis. If we consider another ref-

erence axis, rotated by an angle α, then Q and U transform as components in a

symmetric trace free tensor. In complex notation, the field Q ± iU will be trans-

formed as (Q + iU)e∓2iα for an axis rotation by an angle α. This means that Q and

U is a tensor (spin-2 field), which can be thought of as a directionless vector. For

cosmology, it is more convenient to define a linear combination of the two, the so

called E and B fields. The anisotropy field Q + iU is expanded in harmonic space

as

(Q± iU)(θ,φ) =∞∑`=1

∑m=−`

(aE`m ∓ iaB`m)∓2Y`m(θ,φ), (1.57)

where ∓2Y`m are the spin-2 spherical harmonics. Equivalently, we can define two

spin-0 or pseudo-scalar fields, analogous to the temperature field

E(θ,φ) =∞∑`=1

∑m=−`

aE`mY`m(θ,φ), B(θ,φ) =∞∑`=1

∑m=−`

aB`mY`m(θ,φ). (1.58)

Because the pseudo-scalar field B is multiplied by i = e−iπ/2, it will be oriented 45

with respect toE. In Figure 1.5, the polarization pattern produced by either the E or


Figure 1.5: Top: Polarization pattern produced by pure E modes. The colour mapis the temperature field. Around hot (cold) spots, the polarization is oriented tan-gentially (perpendicularly). Bottom: Polarization pattern by pure B modes. Aroundhot or cold spots, the polarization is oriented 45 with respect to pure E. Taken fromKamionkowski and Kovetz (2016).

B modes is shown. A pure E mode polarization will produce a pattern shown in the

top. The temperature field is on colour scale. The orientation around a hot (cold)

spot is tangential (perpendicular). This is said to be curl-free. The pure B mode

pattern shown in the bottom is oriented 45 with respect to the pure E pattern. In

this case, there is a characteristic right or left-handed curl around the spots, which is

divergence-free. The gravitational waves will have two polarizations, either right or

left-handed, analogous to cold and hot spots in the scalar, or density perturbations.

Another difference between these modes is how they transform under parity. A

transformation of the coordinate axis with respect to the normal unit vector on the

sphere, for a given `, does not change the sign of the pure E mode polarization

((−1)` parity), but it changes the sign of the pure B mode ((−1)`+1 parity). Another

way to understand this is to mirror the polarization pattern shown in Figure 1.5.

The pure E mode will look the same, but the B mode will invert the direction of the

vorticity.


Just like with the temperature field, we can define the variance of the a`m coeffi-

cients as the auto and cross angular power spectra

〈aE`m, a∗E`′m′〉 = δ``′δmm′CEE` (1.59)

〈aB`m, a∗B`′m′〉 = δ``′δmm′CBB` (1.60)

〈aT`m, a∗E`′m′〉 = δ``′δmm′CTE` (1.61)

〈aT`m, a∗B`′m′〉 = δ``′δmm′CTB` (1.62)

〈aE`m, a∗B`′m′〉 = δ``′δmm′CEB` . (1.63)

The three fields T , E, and B define 6 auto and cross angular power spectra, but

since B has opposite parity with respect to T and E, we expect CTB` and CEB

` to

be zero. The CEE` and CTE

` angular power spectra as measured by the Planck 2015

data release (Planck Collaboration et al., 2016m) are shown in Figure 1.3 in blue and

green, respectively.

1.3.2 Sources of E and B anisotropies

Before the last scattering, the photons are tightly coupled to the baryons in the

baryon-photon fluid. After the photons free-stream, the perturbation to the pho-

ton distribution evolves from the primordial perturbations set up by inflation. A

large-scale super-horizon mode, whose wavevector k is larger than the horizon at

recombination, will hardly evolve at all. The perturbation of another smaller mode,

whose wavevector is of the same size as the horizon at recombination, will peak ex-

actly at this time. There will be a maximum in the fluctuations at this scale, which

will be translated to the first peak in the angular power spectrum of the CMB. The

perturbation of an even smaller mode, which enters the horizon earlier, will have

been through its maximum before recombination, and the oscillation will be going

back to zero just at the time of recombination. Therefore, this mode will be at a

minimum of fluctuations, which will be translated as the first trough in the angular

power spectrum. An even smaller and earlier mode will enter the horizon before re-

combination and will go through a full oscillation at the time of last scattering. The

fluctuations will be at a second maximum at recombination, which will be trans-

lated as a second peak in the angular power spectrum. This will continue forever,

with smaller modes that enter the horizon earlier and earlier, showing as the peaks

and troughs in the CMB angular power spectrum, as shown in Figure 1.3. These

are the acoustic oscillations of the photon perturbations, showing up as anisotropies

in the CMB.


Figure 1.6: Left, scalar perturbation spherical pattern. The direction of the bulk mo-tion is shown. Right, tensor perturbation spherical pattern. A gravitational wavepasses through upwards. A circle being streched is shown as reference. From Huand White (1997)

If the quadrupole temperature anisotropies are decomposed in spherical har-

monics Y m`=2, then m = 0, ±1 and ±2 correspond to scalar, vector, and tensor pertur-

bations, described in Section 1.2.1. First, consider the scalar perturbations. The tem-

perature pattern and the gravitational potential will create a bulk motion. Matter

will move from hot to cold spots (this is cold in the sense of effective temperature,

also considering the effect of the gravitational potential). The pattern created by

this is proportional to Y m=0`=2 ∝ 3 cos2 θ−1. This pattern is shown in Figure 1.6, left. θ

is the colatitude angle of observation, so the temperature pattern peaks at θ = π/2,

when the observer is perpendicular to the bulk flow. It is zero for θ = 0, looking

from the poles. The polarization is all-Q, with U(θ,φ) = 0 and Q(θ,φ) = sin2 θ. With

this, aE`m 6= 0 and aB`m = 0, therefore, scalar modes only produce E-mode polariza-

tion.

In the case of tensor perturbations, the temperature pattern is proportional to

Y m=±2`=2 ∝ sin2 θe±2iφ. This is produced by a single plane gravitational wave, prop-

agating in the z direction, with wavelength k, as shown in Figure 1.6, right. At

the equator, the quadrupole polarization patter is full-Q, with alternating cold and

hot lobes along the azimuthal angle φ. At the poles, the polarization pattern is at

maximum. The full polarization pattern is (Hu and White, 1997)

Q(θ,φ) = (1 + cos2 θ)e2iφ (1.64)

U(θ,φ) = −2i cos θe2iφ. (1.65)

In this case, aB`m 6= 0, as shown by Zaldarriaga and Seljak (1997); Kamionkowski


et al. (1997), so both E and B-modes are non-zero for a gravitational wave. By sum-

ming all Fourier modes, we can obtain the B-mode power spectrum from a back-

ground of gravitational waves (see details in Kamionkowski and Kovetz, 2016)

CBB` =

1

2π

∫dkk2

[`+ 2

2`+ 1∆Q,`−1(k) +

`− 1

2`+ 1∆Q,`−1(k)

]2

, (1.66)

where ∆Q,` = (1/2)∫ 1

−1dµ∆Q(µ)P`(µ), µ = cos θ and ∆Q is the amplitude of the

perturbed distribution function of the photon, which in turn depends on the per-

turbation to the metric (h+ and h×) by the gravitational wave. We note that weak

gravitational lensing, a late time effect, transforms E-mode into B-mode signal.

1.3.3 Cosmological parameters

The ΛCDM cosmological model depends on several parameters. The accepted min-

imum number of independent parameters that fits well all cosmological observa-

tions is six. This is usually labelled as a baseline ΛCDM model. One possible choice

for these six parameters is: Ωbh2 (the density parameter for baryonic matter), Ωch

2

(the density parameter for cold dark matter), As and ns (the amplitude and spectral

index of the power spectrum of inflationary primordial scalar perturbations), τ (the

optical depth to reionization), and θMC (the sound horizon at last scattering). In

this case, some reasonable assumptions are made, such as a flat universe (therefore

ΩΛ = 1−Ωb−Ωc) or an equation of state for dark energy wΛ = −1. The parameters

Ωm, Ωb, ns and As were explained previously.

The cosmological parameters can be measured with the acoustic oscillations in

the CMB angular power spectra (Dodelson, 2003). The morphology of the peaks

and troughs depends on the parameters. For example, the content of baryons in

the Universe (Ωbh2) affects the locations and the relative heights of even and odd

peaks. If the baryon-photon fluid has more baryons, it is heavier, which makes the

sound speed smaller. This means that the acoustic oscillations will have smaller

frequencies (or larger wavenumber), which shifts the locations of the peaks. An-

other example is changing the Λ or matter content of the Universe. If there is no

cosmological constant, the large-scale modes, which are entering the horizon only

recently, are shifted towards smaller scales, which moves the peaks to higher mul-

tipoles. If the total matter of the Universe is lowered, then the acoustic oscillations

fluctuate even more, since there is more radiation content, which facilitates this

process, as opposed to matter which by gravity tends to cluster and slow down

the oscillations. Then, the heights of the peaks increase in the angular power spec-

trum. In the following, some of the cosmological parameters that have not been


introduced will be described.

The epoch of reionization (Zaroubi, 2013) corresponds to the period after the

dark ages (which goes from the CMB last scattering to the formation of the first

stars), when the firsts galaxies formed and their UV radiation eventually ionized

the neutral hydrogen in the Universe. This did not happen instantaneously, but as

a process, at redshifts 15 ≥ z ≥ 6. The Gunn-Peterson trough is the first observa-

tional evidence for a reionization of the neutral hydrogen. Predicted by Gunn and

Peterson (1965), the absorption spectra of distant quasars should show an absorp-

tion trough at wavelengths shorter than the corresponding Lyman-α line, by the

intervening neutral hydrogen. This was not observed for quasars at redshift z . 6,

instead observing a series of small absorption features denominated the Lyman-α

forest, by the intervening ionized hydrogen. However for objects at higher redshift,

the flux beyond the Lyman-α line is effectively zero (Becker et al., 2001), which is

evidence of the Universe going through a process where the neutral hydrogen ion-

ized at around z ∼ 6. Another way to probe the reionization epoch is by using the

CMB, because free electrons cause Thomson scattering of the CMB photons at later

times, while they travel from the last scattering surface to us. This effect is the same

that occurred at recombination, but now it occurs at a much lower rate, since the

density of free electrons has decreased at late times because of the expansion of the

Universe. The parameter τ is the optical depth of the Thomson scattering for the

CMB photons, defined by

τ(η) =

∫ η0

η

σTnea(η′)dη′, (1.67)

where η is the conformal time, σT is the Thomson scattering cross-section, and ne is

the free electron density. Therefore, the reionization optical depth provides an inte-

grated constraint over the entire reionization process, not enabling us to distinguish

the details in reionization history. The effect on the temperature anisotropies of the

CMB is suppression of the power spectra at small scales, while in polarization it

creates features at small scales.

The matter in the primordial fluid, before recombination, created gravitational

potentials along the density perturbations. The photon-baryon fluid is compressed

along by the gravity. At the same time, photons push back with radiation pressure,

and so forth. The acoustic oscillations in the photon-baryon fluid were imprinted

once the photons decoupled at last scattering. The sound horizon to last scattering

θMC is the “angular distance” of the Universe at last scattering. It is defined by θ∗ =rsds

, where rs is the comoving sound horizon of the Universe at recombination (the

maximum distance a wave could have travelled in the primordial photon-baryon


fluid), defined by

rs =

∫ t=t∗

t=0

cs/a∗dt, (1.68)

where cs is the sound speed of the photon-baryon fluid. ds is the comoving angular

distance to the last scattering surface, given by

ds =1

a∗H0

1

1 + z∗

∫ z

0

dz′√Ωr,0(1 + z′)4 + Ωm,0(1 + z′)3 + ΩΛ,0

. (1.69)

By doing some approximations, θ∗ can be measured to a value∼ 1/200. In harmonic

space, the multipole associated with its real space angle is ` = π/θ, so the sound

horizon at last scattering multipole is `∗ ∼ 200, which is approximately where the

first peak of the temperature CMB angular power spectrum is located. In cosmo-

logical models, an approximation of θ∗ is used, labelled θMC.

Based on a six-parameter cosmology, others parameters can be derived, such

as the Hubble constant h, the age of the Universe, the redshift of recombination

z∗, the normalization of the matter power spectrum σ8 (defined as the standard

deviation in the matter density distribution inside a comoving sphere of diameter

8 Mpc), among others. Some other parameters are fixed at fiducial values, and can

be varied when fitting for extensions to the baseline ΛCDM model. For example,

the canonical value of the dark energy equation of state is w = −1, but extended

models can fit a free w parameter, either constant or depending on a (e.g. Planck

Collaboration et al., 2016k).

Using other physics, extensions can be made to the baseline ΛCDM model. Neu-

trino physics can be constrained with cosmology. The baseline model usually as-

sumes a normal mass hierarchy (one neutrino species much more massive than the

other two) with a mass of mν = 0.06 eV, and Neff = 3.046, the effective number

of neutrino-like species (which parametrizes the contribution of neutrinos to the

energy density). We know that the presence of neutrinos will slow the agglomera-

tion of CDM when the mass structure is forming (Lesgourgues and Pastor, 2006), so

there will be an imprint on the CMB weak lensing potential, BAOs, etc. Current con-

straints are∑mν . 0.17 eV and Neff ∼ 3.04± 0.2 (Planck Collaboration et al., 2016i;

Couchot et al., 2017). Another example is primordial nucleosynthesis. The primor-

dial Helium fraction YP can be set to a fiducial value (e.g. YP = 0.2534± 0.0083 Aver

et al., 2012) in the baseline model, however, for example, the CMB spectra damping

tail shape is sensitive to YP. Planck constrains the value to YP = 0.253+0.025−0.026 (Planck

Collaboration et al., 2016i). As a final example, the ΛCDM model is extended regu-

larly to include early Universe physics, that is inflation, as explained in Section 1.2.


Figure 1.7: Current measurements of the CBB` angular power spectrum, from PO-

LARBEAR (The POLARBEAR Collaboration et al., 2017), ACTpol (Louis et al.,2017), SPTPol (Keisler et al., 2015a) and BICEP2/Keck array (BICEP2 and Keck Ar-ray Collaborations et al., 2015).

In this case, the parametrization of the primordial scalar perturbation power spec-

trum, As and ns, and the tensor-to-scalar ratio r, are parameters that are be fitted.

1.3.4 Current observations of CMB B-modes

The recent history of CMB observations has been defined by series of instruments,

both ground-based as well as satellites. The Planck satellite (Planck Collaboration

et al., 2014a, 2016a) observed the CMB from 2009 for 4 years. Ground-based ex-

periments that started observing after 2010 have also measured the CMB at small

and intermediate scales. The South Pole Telescope (SPTpol) polarization instru-

ment (Austermann et al., 2012) and the BICEP2/Keck Array (BICEP2 Collaboration

et al., 2014a) are observing from Antarctica. The Atacama Cosmology Telescope

(ACTpol) polarization instrument (Niemack et al., 2010; Thornton et al., 2016) and

the POLARBEAR (Arnold et al., 2010; The Polarbear Collaboration: P. A. R. Ade

et al., 2014) instrument are observing from the Atacama desert in Chile.

An indirect measurement for B-modes is to cross-correlate a predicted map of


lensing B-modes with observed B-mode maps. The lensing potential φ can be re-

constructed from external data (e.g, the Cosmic Infrared Background is the most

correlated observable, Planck Collaboration et al., 2014c). This potential allows us

to estimate the gravitational lensing contribution from EE to BB spectra. This can

be correlated with a measured B map. Hanson et al. (2013) detected this correla-

tion for the first time with SPTPol, with a 7.7σ non-zero significance. POLARBEAR

(Ade et al., 2014), ACTpol (van Engelen et al., 2015) and Planck (Planck Collabora-

tion et al., 2016j) have also detected this correlation.

The ground based experiments have detected the small/intermediate-scale CBB`

power spectrum from gravitational lensing. POLARBEAR have measured it from 2

seasons of data, in the multipole range 500 ≤ ` ≤ 2100 (4 bins) (The POLARBEAR

Collaboration et al., 2017). This is consistent with gravitational lensing signal (zero

B-mode signal is rejected with 3.1σ), and they measure an amplitude for the lensing

power spectrum CBB,lens` of AL ∼ 0.6± 0.45. ACTPol measured directly the lensing

B-mode spectrum (Louis et al., 2017). This is only in two bins in the multipole

range 500 ≤ ` ≤ 2500, fitting a lensing power spectrum amplitude of AL = 2.03 ±1.01. The SPTpol instrument has measured the lensingBB power spectrum (Keisler

et al., 2015a) at 4.1σ (for B-modes from lensed E-mode polarization). The CBB` are

measured in the multipole range 300 ≤ ` ≤ 2300 for five bins, based on data taken

in a 100 deg2 patch over two seasons.

The BICEP2 experiment measured the large/intermediate-scaleBB power spec-

trum in nine bins at multipoles 20 < ` < 340 (BICEP2 Collaboration et al., 2014b).

They found an excess over the pure lensed BB power spectrum with a significance

of ∼ 5.2σ. If this were due to primordial tensor perturbations, the best fit tensor-

to-scalar ratio would be r = 0.20+0.07−0.05. However, they underestimated the contami-

nation by thermal dust in the observed sky patch. A joint analysis with the Planck

observations at higher frequencies of dust showed that the excess is entirely con-

sistent with foreground contamination (BICEP2/Keck and Planck Collaborations

et al., 2015). Since then, the Keck Array also measured the B-mode spectrum (BI-

CEP2 and Keck Array Collaborations et al., 2015), and also combined with BICEP2

(BICEP2 Collaboration et al., 2016). Their estimate of the lensing power spectrum

is AL = 1.20± 0.17.

The current measurements of the (lensing) CBB` angular power spectrum, de-

scribed above, are shown in Figure 1.7.

BLANK PAGE

50

Chapter 2

CMB foregrounds and component

separation

When trying to extract any cosmological signal from the CMB, the instrumental

sensitivity is not the only issue to be considered. Other sources of electromagnetic

radiation emit at the same frequencies as the CMB, in the local and low-redshift

Universe. Of special consideration is the emission from our own Milky Way, which

is all around us and blocks our clean view of the CMB. Any source of contami-

nation are collectively known as CMB foregrounds. The algorithms and methods

designed to separate these foregrounds from the clean CMB (the primary signal

plus secondary effects), with the objective of reducing (or eliminating all together)

the impact of systematics in the derived cosmology, are known as CMB component

separation (Delabrouille and Cardoso, 2009).

2.1 Astrophysical foregrounds

Several emission mechanisms radiate at the same frequency range where the CMB

peaks. Figure 2.1, top, shows the spectral energy distributions (SEDs) of the CMB,

as well as several of the diffuse Galactic foregrounds for temperature observations1. The CMB is dominating around 50 − 150 GHz outside the Galactic plane, but

the thermal dust is the strongest foreground at higher frequencies. At lower fre-

quencies, synchrotron, free-free and spinning dust emission dominate. Figure 2.1,1The units correspond to brightness temperature TB, using the Rayleigh-Jeans approximation

hν kT , given by TB = Iνc2

2kν2 , where Iν is the intensity (energy per surface area per solid angle perfrequency), c is the speed of light, k is the Boltzmann constant and ν is the frequency. This unit oftemperature is also referred to as Rayleigh-Jeans units. These units are different from Thermody-namic units, where the Rayleigh-Jeans units are normalized with a Planck law with a temperatureequal to the CMB temperature. Therefore, in this units the SED of the CMB is constant across fre-quency and all other components are distorted.

51

52 CHAPTER 2. CMB FOREGROUNDS

Figure 2.1: SEDs of the CMB and diffuse Galactic foregrounds. On top, tempera-ture, and on the bottom, polarization. The Planck observation frequencies are alsoindicated for comparison. The upper and lower limits on each component are con-sidering maks with different sky fractions (81 and 93% outside the Galactic plane).Taken from Planck Collaboration et al. (2016b), figure 51.

2.1. ASTROPHYSICAL FOREGROUNDS 53

bottom, shows the SEDs of the components in polarization, where the CMB is a few

orders of magnitude fainter than its temperature. We know that thermal dust and

synchrotron are strongly polarized and they are more intense that the CMB over

most of the regions of the sky. They constitute the main foreground components

that must be accounted for. Spinning dust is slightly polarized, but no more than

a few percent in active star formation clouds. There are also extra-Galactic sources,

mainly Active Galactic Nuclei (AGNs) or young dusty star forming galaxies, which

are slightly polarized. In the following, we will describe these foregrounds in some

detail.

2.1.1 Synchrotron radiation

This emission is produced by the acceleration of charged particles in interstellar

magnetic fields. The electrons are deflected by the magnetic field, that exerts an

acceleration perpendicular to its velocity. This provokes a spiral trajectory around

the magnetic field lines, through which the particles emit radiation with power

proportional to the magnetic field squared. The SED of synchrotron radiation is

very close to a power law, given by

TA,syn(ν) ∝ νβsyn+C log(ν), (2.1)

with a negative slope βsyn ∼ −3 and an optional curvature parameter C (Davies

et al., 2006; Kogut et al., 2007), to account for the steeping as the frequency increases.

βsyn will vary across the sky, because it is dependent on the particular acceleration

of electrons and their velocities distribution. The synchrotron polarized intensity

consists of filamentary structures, the so-called spurs, which follow the lines of the

Galactic magnetic field away from the Galactic plane. The polarization fraction

can be as high as ∼ 40 percent in some of the filaments (Vidal et al., 2015). How-

ever, most of the Galactic plane is expected to be polarized to smaller levels, from a

few percent to . 20 percent (Page et al., 2007). The morphology of polarized syn-

chrotron and low-frequency foregrounds is constrained with Planck in its 2015 data

release (Planck Collaboration et al., 2016e).

A further complication is the fact that at low frequency, ν ∼ 20 GHz, the spec-

tral laws of synchrotron, free-free and spinning dust are very hard to distinguish,

having a similar slope, as shown in Figure 2.1 for temperature. If spinning dust is

slightly polarized, it can be easily confused with synchrotron, which complicates

the modelling of both components. To overcome this, present and future low-

frequency experiments are being developed to map the sky in this frequency range,


such as C-BASS (Irfan et al., 2015), at 5 GHz; and QUIJOTE (Rubiño-Martín et al.,

2010), at 10-30 GHz.

2.1.2 Thermal dust radiation

This is produced by the presence of dust grains of various sizes in the Interstellar

Medium (ISM). In regions of star formation, the UV radiation from newly formed

stars (compact and ultra-compact HII regions) heats up the surrounding medium,

including dust grains, which in turn radiate in the IR and microwave frequencies.

In principle, each grain at a given temperature should radiate very close to a ther-

malized black-body. In practice, for a given line of sight, there are many different

dust grains populations at different temperatures, all emitting at the same time. In

the simplest model, the SED of thermal dust is well fitted by a Modified Black-Body

(MBB) of the form

TA,dust(ν) ∝ νβdust+1[exp(hν/kTdust)− 1]−1, (2.2)

where Tdust and βdust are a single temperature that collectively describes the dust

grains, and a positive slope representative of the ∼hundreds of GHz frequency

range, respectively. Along the Galactic plane, the parameters are consistent with

Tdust ∼ 19 K, and a spectral index of βdust ∼ 1.51 (for temperature) and βdust ∼ 1.59

(for polarization) (Planck Collaboration et al., 2015b). However, the different dust

populations mean that every pixel has a different MBB spectral law, even away

from the Galactic plane (Planck Collaboration et al., 2017). The 2015 data release

of Planck finds maps of the thermal dust spectral parameters, with average values

across the sky 〈Tdust〉 ∼ 21 K and 〈βdust〉 ∼ 1.53 (Planck Collaboration et al., 2016b).

More complex models have also been proposed, in particular the use of multi-

component MBB spectral laws (e.g. Meisner and Finkbeiner, 2015). In this case,

usually two (or more) populations of thermal dust are considered for each line of

sight. The dust grains are not perfectly spherical, so they tend to align with the

Galactic magnetic field. This means that the microwave emission they radiate is

polarized. In some areas of the Galactic plane, the polarization fraction can reach

up to 20 percent (Planck Collaboration et al., 2015a). The modelling of polarized

thermal dust is more complicated that a MBB spectral law. 3D models of the Galac-

tic emission try to overcome this, by considering that along a line of sight there

is a superposition of MBB laws, whose polarization properties will depend on the

physical conditions of the dust and on the local magnetic field (e.g, O’Dea et al.,

2012; Martínez-Solaeche et al., 2017; Chluba et al., 2017).

2.1. ASTROPHYSICAL FOREGROUNDS 55

2.1.3 Anomalous Microwave Emission (AME)

The Anomalous Microwave Emission (AME) is an emission mechanism that has

been discovered relatively recently. It was first detected directly by Leitch et al.

(1997), where a low-frequency emission strongly correlated with dust was detected.

Now we know that the most likely origin of it are grains of dust rapidly spinning,

with a dipole moment, that emit at microwave frequencies (Draine and Lazarian,

1998). Hence, this emission is sometimes also referred to as spinning dust. Since

then, a lot of knowledge has been gathered on AME from WMAP and Planck,

among others (Génova-Santos et al., 2011; Planck Collaboration et al., 2011, 2014b).

The SED of spinning dust is uncertain. There are numerical templates based

on the dust properties (e.g. Ali-Haïmoud et al., 2009), but also analytical spectral

laws that depends on a few parameters (see e.g. Davies et al., 2006; Bonaldi et al.,

2007; Stevenson, 2014). The polarization of AME is not constrained, and we only

have upper limits on its polarization (Dickinson et al., 2018). It is likely polarized

. 3 percent below 30 GHz, but at higher frequencies it falls to < 1 percent (Rubiño-

Martín et al., 2012). Upper limits have been measured in Galactic star formation

clouds, of only a few percent (Dickinson et al., 2011; López-Caraballo et al., 2011;

Génova-Santos et al., 2017).

2.1.4 Point sources

The extra-galactic sources that contribute in the CMB frequency range are dusty

star-forming sub-mm galaxies and radio galaxies (De Zotti et al., 2015, 2018).

Radio galaxies are distant and bright quasars and blazars that are active. They

emit mostly through the synchrotron mechanism, with the interaction of energetic

output (jets) and the intra-galactic medium. Their spectral properties are consistent

with a power-law Sν ∝ ν−α (Massardi et al., 2016). The slope of the population

emitting in the microwave range is close to flat in flux units, but above 30 GHz, it

steepens to values α ∼ 0.6 2.

Star-forming galaxies are dominated by dust emission in the microwave fre-

quency range. Therefore, they are consistent with MBB spectral laws, with spectral

indices ∼ 1.8 (Clemens et al., 2013).

In polarization, the properties of points sources in the microwave range have

only been recently constrained. Massardi et al. (2013) obtain a complete sample of

polarized sources at few tens GHz with Australia Telescope Compact Array (ATCA)

2As explained above, flux and brightness temperature are related by Iν ∝ Tbν2, therefore the α

index in flux is related to the β index in synchrotron by β = α+ 2


and WMAP data. Most of the sources are weakly polarized, with a distribution of

polarization fractions of a few percent. More recently, Galluzzi et al. (2017) studied a

complete sample of 53 radio sources and characterized their properties in intensity

and polarization. The spectral laws are well described by power-laws, but spectra

in polarization can differ significantly from temperature intensity, since some of the

sources show variability with time scales of a few years.

Bonavera et al. (2017a) and Bonavera et al. (2017b) study the polarization frac-

tion properties of the radio and dusty point sources, respectively, detected by

Planck in their second catalog of compact sources (Planck Collaboration et al.,

2016f). They conclude that the distribution of polarization fractions can be approx-

imated by a log-normal function with a mean value of few percent.

2.1.5 Other unpolarized foregrounds

The other major diffuse Galactic foreground is free-free emission, also known as

bremsstrahlung. This is originated by the scatter and deceleration of electrons by

charged particles in the ISM. Free-free emission is very close to a power-law for

frequencies above 1 GHz, with a spectral index ∼ −2. At even lower frequencies

there is a break in the SED. As mentioned above, at low-frequency free-free is de-

generate with synchrotron. However, free-free is more intense at higher-frequency

since its slope is flatter. This helps to constrain its emission. Free-free is expected to

be essentially unpolarized, since the directions of the scatter processes are random.

Over all the sky, it is measured to be polarized by < 1% (Macellari et al., 2011).

The Sunyaev-Zel’dovich (SZ) clusters are extra-Galactic sources that are de-

tected through their effect in the CMB. The thermal SZ effect is the inverse-Compton

scattering of CMB photons by electrons in the ionized hot intra-cluster medium of

galaxy clusters (Sunyaev and Zeldovich, 1972). The photons gain energy, which dis-

torts the perfect black-body spectrum by a small amount depending on the electron

density of the cluster, shifting the intensity towards higher frequencies. If a black-

body is subtracted from the observed CMB signal, then a lack of CMB photons is

observed at lower frequencies and an excess at higher frequencies. The transition

frequency is located upwards of 200 GHz. Planck mapped the thermal SZ distortion

(Planck Collaboration et al., 2016n). The kinetic SZ effect is produced by the scatter

of CMB photons with electrons that have high-energy because of the bulk motion

of the galaxy cluster with respect to the CMB.

2.2. COMPONENT SEPARATION 57

2.2 Component separation

As explained above, the CMB is obscured by several sources of emission at the same

frequencies, from our Galaxy and from external sources. Several component sep-

aration methods have been proposed to extract the clean CMB (Delabrouille and

Cardoso, 2009). In general, these methods can be categorized as blind or non-blind.

In non-blind methods, usually there are predetermined spectral functions that char-

acterize the SEDs of the components. Therefore, this assumes that we know what

the spectral dependence of a given component is, even though in the real sky it

might not be exact. Blind methods use minimal prior information about the compo-

nents, and mainly use their statistical properties (e.g. their statistical independence,

since we know that the CMB is a Gaussian field, while the diffuse Galactic compo-

nents clearly are not) to disentangle them. We will refer as parametric methods to

a category that is somewhere in between, since they model explicitly the SED of

the foreground components, but allow the estimation of these parameters from the

data directly.

In general, we will assume that in an observed pixel p, we will measure a super-

position of components emitting with their own spectral law (this superposition

is convolved with the beam of the instrument observing) plus some instrumental

noise. This is usually labelled a Linear Mixture. That is, the observed signal y at

band ν is given by

yν(p) =

∫B(p, ν)t(ν)

Nc∑j=0

xj(p, ν)dpdν + n(p, ν), (2.3)

where the upper ∼ means an observed/estimated variable, y is the total intensity

observed by the instrument, xj is the true signal from component j, B(p, ν) is the

beam pattern function, t(ν) is the window function from the instrument at band ν,

n(p, ν) is the instrumental noise, and ν =∫B(p, ν)t(ν)νdνdp is the frequency of each

band (the mean frequency weighted by the beam and window function). Each indi-

vidual band is indexed by an integer ν. In practice, we make some simplifications:

the beam function B is constant within the passband, meaning it only depends on

space and is usually parametrized with the Full Width at Half Maximum (FWHM)

of the main beam, and the true signal of component j has a spatial component s

(an spatial amplitude), and a frequency component a which scales it in frequency

(a SED),

xj(p, ν) = aj(ν)sj(p). (2.4)


Then, we can write a single beam that convolves a superposition of components

yν(p) = Bν(p) ∗Nc∑j=0

ajνsj(p) + nν(p), (2.5)

where ∗ denotes convolution (integration in p), ajν =∫t(ν)aj(ν)dν is the weighted

spectral law of component j, where the weight is the window function of band ν.

In matrix form, it can be written (for every pixel p) as

ˆy = B ∗ As+ n, (2.6)

where ˆy is a Nbands vector that contains the yν elements, A is the mixing matrix,

with size Nbands ×Nc, where Nc is the number of components, and contains the ajνelements, s is a Nc vector that contains the sj elements and n is a Nbands vector that

contains the noise elements nν .

The objective is to get an estimator for s, given by ˆs = Wˆy, where W is the recon-

struction matrix, chosen appropriately. Sometimes, this is calculated as a function

of the mixing matrix. In other methods, especially blind ones, W is estimated it-

eratively to maximize the independence between components. From now on, we

will ignore the beam window function B when working in real space. Usually the

resolution of all the maps at different bands is equalized by smoothing the maps,

to absorb the beam into the mixing matrix. In harmonic space, the convolution is

transformed into a multiplication, which is even simpler. We will also drop the

upper hats to denote vector quantities.

2.2.1 Parametric methods

Usually, the simpler approach is to parametrize analytic spectral laws for each com-

ponent. Then, the mixing matrix A is calculated in a straightforward way, by in-

putting the normalized spectral laws at Nbands frequencies by Nc components. This

requires previous knowledge on how the different mechanisms of emission behave

spectrally. Moreover, in principle we do not know if the parameters assumed or

estimated are a reflection of the true sky. Once a mixing matrix is estimated, it can

be used to compute a suitable W matrix. The accuracy of it will depend on the

accuracy of the mixing matrix.

Choosing the reconstruction matrixW

The simplest way of choosing the reconstruction matrix is to invert the mixing ma-

trix W = A−1 and s = s + A−1n. This is noisy when the signal-to-noise ratio is


low, and A has to be square (same number of components and bands), which is in-

efficient, since usually we have more bands than components, and therefore some

information is not going to be used.

The Generalized Least Squares (GLS) solution is

W =[A†R−1

n A]−1

A†R−1n , (2.7)

where Rn is the noise covariance matrix. In this case, s = s +[A†R−1

n A]A†R−1

n n.

The solution is unbiased, keeping a noise contribution, and it is also the solution

that minimizes the variance of the error when the signal s is deterministic.

The Wiener filter solution is given by W =[A†R−1

n A + R−1s

]−1A†R−1

n . As op-

posed to the GLS solution, it is assumed that the noise and the data are stochastic,

with known statistics in the form of the noise and signal covariance matrix, Rn and

Rs, respectively. The estimated signal is given by s =[A†R−1

n A + R−1s

]−1A†R−1

n As+[A†R−1

n A + R−1s

]−1A†R−1

n n, which means that it is not unbiased, introducing fore-

ground residuals in s. It is the solution that minimizes the variance of the error for

stochastic signals.

Bayesian parametric methods

In this class of algorithms, the observed sky spectral law is modelled as a sum of

components, controlled by amplitudes and spectral parameters, and noise. The

ability to accurately subtract the foregrounds from the CMB depends on accurately

describing the observations through analytical spectral laws. The COMMANDER

code (Eriksen et al., 2008) uses the Bayesian framework to fit foreground com-

ponents in pixel space through Gibbs sampling and Monte Carlo Markov Chain

(MCMC). The posterior distribution of the parameters Ai (the amplitudes of the

components in both Q and U ), βj (the spectral parameters of the components,

e.g. spectral indices), and C` (the EE and BB angular power spectra of the fitted

model), is given by the Bayes theorem

P (s,β,C`|d) ∝ L(d|s,β,C`)P (s,β,C`), (2.8)

where d is the observed sky data. P (s,β,C`) corresponds to the assumed prior on

parameters and the likelihood minimizes the residuals between the data and model

in pixel space, using the noise covariance matrix (calculated assuming Gaussian

uniform noise uncorrelated across bands).

To sample the multi-dimensional parameter space in order to compute the pos-

terior distribution is prohibitively expensive, and therefore an alternative scheme


is to use Gibbs sampling, where the full joint posterior is sampled by alternatively

sampling the simpler conditional posterior probabilities of each parameter. This

is done iteratively, as steps of a Markov Chain. If the chain is long enough, the

sampled posterior will converge to the true posterior (Wandelt et al., 2004).

A nice feature of this method is that since the angular power spectra is a param-

eter in the sampling of the joint posterior distribution, the cosmological parameters

that control the shape of the angular power spectra of the CMB can be included

as parameters to fit. Therefore, in the same Markov chain, the foreground and

cosmological parameters can be fitted at the same time. This method is compu-

tationally expensive, since the estimation is performed pixel-by-pixel. If all possi-

ble parameters are being fitted (CMB and foregrounds, both spectra and templates,

plus fiducial CMB spectra), then the code is limited to low resolution maps, usu-

ally Nside = 16. However, when fitting for only foregrounds, it can be run at much

higher resolutions.

Correlated Component Analysis (CCA)

This is a method to estimate the spectral parameters of the components assumed to

be present in a set of multi-frequency maps of the sky (Bonaldi et al., 2006; Ricciardi

et al., 2010).

It exploits the second order statistics of the data, that is, the covariance matrices.

For the observations y, we have its covariance matrix given by

Ry(τ ,ψ) = ARs(τ ,ψ)A† + Rn, (2.9)

where A is the mixing matrix, Ry, Rs, and Rn are the covariance matrices of the

observations, true signal, and noise, respectively. (τ ,ψ) represents a shift in the

coordinates of the variable of the covariance matrix. For a given variable X , that

depends on generic coordinates (ε, η), the “shifted” covariance matrix is given by

RX(τ ,ψ) = 〈[X(ε, η)− µ(X)] [X(ε+ τ , η + ψ)− µ(X)]†〉, (2.10)

where µ(X) is the mean of the variable X , and 〈...〉 represents the average over

realizations. Covariance matrices are defined by shifts in the coordinates.

We want to estimate Rs and A. The mixing matrix is parametrized with spec-

tral parameters of the components considered. To this purpose, the functional Φ is

defined

Φ[Rs,A] =∑τ ,ψ

|ARs(τ ,ψ)A† − Ry(τ ,ψ) + Rn(τ ,ψ)|2. (2.11)

This must be minimized for sufficient coordinates shifts.


The Correlated Component Analysis has also been implemented in the har-

monic domain, where there are many advantages over pixel space. The observed

signal is the angular power spectra C`, and (ε, τ) = (`,m). We do not need to equal-

ize the beams for frequency bands, as we have to do in pixel space. The end result

for the method is the estimation of the spectral parameters of the components mod-

elled and of the mixing matrix A. Then, the reconstruction of the components is

done with the GLS solution of the reconstruction matrix, using the estimated mix-

ing matrix.

2.2.2 Blind methods

These methods assume minimal prior information on the foreground component.

Independent Linear Combination (ILC)

This method does not exploit the spectral information of the foregrounds, but it

does exploit the known black-body spectral shape of the CMB. We represent the

data in thermodynamic units, which is a normalized unit of temperature, normal-

ized to the SED of the CMB. In this way, the CMB SED is a constant across frequency.

ILC assumes that the other components have different frequency spectra. The sky

signal y is a mixture of CMB s, noise n and foregrounds f (everything else), that

is y = s + f + n, written in matrix notation, where the elements of the vector are

the different frequency bands. The reconstruction matrix W (in this case it is square

with size Nbands), such that s = Wy, is given by

s =a†C−1

a†C−1a(s+ f + n), (2.12)

where a is actually a vector of ones, representing the CMB SED in thermodynamic

units, which is constant, and C is the signal covariance matrix. The method tries

to adjust the weights of the reconstruction matrix, such that the contribution of

the foregrounds is minimized. In practice, however, this methods requires perfect

decorrelation between the CMB and everything else, which is not true when dealing

with finite resolutions of maps, and by chance correlation. Examples of this method

are found in Bennett et al. (2003); Tegmark et al. (2003); Eriksen et al. (2004).

Another issue is the fact that the weights are dominated by the low Galactic

latitude emission, where the foregrounds are important, while at high Galactic lat-

itude the noise is dominant. This will bias some of the scales of the CMB, which is

not optimal, since all scales are needed. One implementation of ILC uses spherical


wavelets called needlets, method known as Needlet ILC (NILC Remazeilles et al.,

2011; Basak and Delabrouille, 2012, 2013).

Independent Component Analysis (ICA)

The only assumption this method makes is that each of the components are statisti-

cally independent of each other and of the noise. We do not assume any information

on the SED or mixing matrix A. The signal estimate is s = Wy, where the measured

sky is a sum of Nc = Nbands components and noise, yν =∑

iCi,ν + nν . The Nc × Nc

reconstruction matrix (weights) has to be calculated in such a way that the elements

of the s vector are independent. The condition for independence of the elements of

the s vector, in pixel space, is

1

Np

Np∑p=1

si(p)sj(p) = 0 i 6= j ≤ Nc, (2.13)

where Np is the total number of pixel in the map, p indexes pixels, and i,j in-

dexes components. These conditions are symmetrical, therefore there are only

Nc(Nc − 1)/2 constraining equations, and we need N2c to calculate W. To break

the symmetry, for example, non-linear functions of si are used in the condition 2.13.

Another way is to use local decorrelation, where the sum in condition 2.13 is di-

vided by a number σ2ip, which depends on the pixel p.

An implementation of ICA in harmonic space is Spectral Matching ICA (SMICA

Delabrouille et al., 2003; Cardoso et al., 2008). The Nbands×Nbands covariance matrix

of the observations is calculated at each `

C` =1

2`+ 1

∑m=−`

y`my†`m (2.14)

The weight to mix the bands, the reconstruction matrix, is the same as the one used

for ILC, but in harmonic space, given by

w` =C−1

` a

a†C−1

` a, (2.15)

where a is a vector with the spectral law of the CMB. SMICA models the observed

covariance matrix C` with a model C`(θ), that depends on parameters θ. The model

is assumed to include a CMB component, a general foreground to model for all

Galactic emission, and a noise model given by a noise spectrum N`. The general

foreground model has an arbitrary number of independent templates, with arbi-

trary spectral laws and power spectra. The number of components for this general

foreground can be adjusted to better represent the observations.

Chapter 3

Simulated observations of the

microwave sky

3.1 Introduction

To simulate observations by a CMB experiment in microwave frequencies, we first

need a model of the sky emission, i.e., the intensity and polarization of some of

the sources of emission (components) at the frequencies of interest. Until recently,

full-sky polarization maps of the Galactic emission were based on total intensity

measurements and models of the polarization physical properties, polarization an-

gles and fractions (e.g. Miville-Deschenes, 2011; Delabrouille et al., 2013; O’Dea

et al., 2012). However, the uncertainties in such modelling makes it difficult to cre-

ate polarization templates accurately reproducing the observed morphology in the

sky. The 2015 Planck data release has improved this situation, by providing for the

first time foreground maps extracted directly from the polarization data (Planck

Collaboration et al., 2016b). Before this information can be used to forecast future

polarization experiments, however, it is necessary to overcome the limitations due

to the Planck resolution and noise levels. Moreover, a suite of foreground models

needs to be explored, to reflect the current uncertainties on polarized foregrounds.

In this chapter, we deliver a new sky model of diffuse polarized emission in the mi-

crowave frequency range, using the templates computed by the latest Planck data

release.

In contrast to previous work (e.g. Delabrouille et al., 2013), the model we present

is not a comprehensive model that includes all point-like and diffuse emission in

the microwave sky. Instead, we focus on diffuse polarized emission only and aim

to provide a simpler and more flexible tool, to allow model selection for forecast

purposes, as well as to test and debug data analysis methods on simulated data of

63

64 CHAPTER 3. SIMULATED OBSERVATIONS OF THE MICROWAVE SKY

-20 20µK -40 40µK

Figure 3.1: Left: Template map of thermal dust Q polarization intensity as derivedin Planck Collaboration et al. (2016b) at 353 GHz. Right: Template map of syn-chrotron Q polarization intensity as derived in Planck Collaboration et al. (2016b)at 30 GHz.

varying complexity. We also introduce the capability to vary the foreground mor-

phology for Monte-Carlo purposes. The content of this chapter has been published

in Hervías-Caimapo et al. (2016).

3.2 Sky model components

3.2.1 CMB component

The CMB is generated starting from a set of input C` angular power spectra from

theory: TT , EE and BB, and TE. These can be produced starting from a set

of cosmological parameters1, for example with the codes CAMB (Howlett et al.,

2012) or CLASS (Lesgourgues, 2011). The map is generated using the synfast task

of HEALPIX (Górski et al., 2005). It is then converted from thermodynamic to an-

tenna temperature units at various frequencies with the usual black-body law with

TCMB = 2.72548 K.

3.2.2 Foreground templates

The simplest model of diffuse polarized foregrounds that is compatible with the ob-

servations has two Galactic polarized foregrounds: synchrotron and thermal dust,

as described in Section 2.1.

We construct templates of these emission components based on the synchrotron

and thermal dust polarization maps extracted from Planck observations with the

1We use Ωb = 0.0461, Ωcdm = 0.2286, h = 0.7, ns = 0.96, and r with various values.

3.2. SKY MODEL COMPONENTS 65

101 102 103

multipole `

10-6

10-5

10-4

10-3

10-2

10-1

100

C` µK

2

EE, all-`

EE, template

EE, high-`

102 103

multipole `

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

C` µK

2

EE, all-`

EE, template

EE, high-`

Figure 3.2: Synchrotron (left) and thermal dust (right) templates EE power spec-tra. The BB spectra is very similar, so it is omitted for clarity. The green curvecorresponds to the original template. The blue curve is the high-` extension, wherethe slope is extended to higher multipoles by a power-law fit. The red curve cor-responds to the power spectra of the template map including the high-` features.Note that the red curve includes a 5′ beam smoothing, whereas the blue curve is anextrapolation without smoothing.

Bayesian component separation method COMMANDER (Planck Collaboration et al.,

2016b) and publicly available through the Planck Legacy Archive 2.

The synchrotron template has a reference frequency of 30 GHz and a resolution

of 40′ Full Width at Half Maximum (FWHM). However, the pixel size of ∼ 14′ (cor-

responding to Nside = 256) means that pixelization artefacts are visible on the maps.

We eliminated these artefacts by resampling the map, upgrading it to Nside = 512

and smoothing it to a final 1 resolution.

The thermal dust template obtained by Planck Collaboration et al. (2016b) has

a reference frequency of 353 GHz and a resolution of 10′ FWHM. Figure 3.1 shows

both Q intensity templates.

Adding high-` features to the foreground templates

Since the synchrotron and thermal dust templates have finite resolution, they do

not have power at small scales. In our models, we would like to simulate high-`

power since the real sky thermal dust and synchrotron emissions are expected to

have these features.

The approach that we follow is to generate a random map using a suitable power

spectrum, based on the extrapolation of the angular power spectra of the original

templates to higher multipoles. Since this procedure involves a random realization,

2http://pla.esac.esa.int/pla/

http://pla.esac.esa.int/pla/


it can be also used to create variations over different foreground maps, for example

for Monte Carlo purposes (this aspect is discussed in Section 3.5). However, this

additional features that we add to the foreground templates are only Gaussian, and

do not consider non-Gaussianity.

A common assumption is that the power spectrum of the foreground maps has

a power-law behaviour in `. Figure 3.2 (left) shows the EE power spectra of the

synchrotron template in green. The power-law behaviour is not a good approxima-

tion at the highest multipoles, where the slope flattens with respect to lower `s. We

nonetheless adopted the power-law approximation and computed the best-fitting

slope at the lowest multipoles. We used a least squares polynomial fit, that mini-

mizes the difference between model and data, added in quadrature within a given

multipole range. Our model is a straight line in the logC`–log ` space. This proce-

dure also outputs the covariance matrix for the fit parameters, and we adopted the

square root of the diagonal terms as errors on each of them.

For the EE power spectrum, we fitted for the slope in the multipole interval

10 ≤ ` ≤ 120 and obtained a value of −1.7 ± 0.04; for the BB power spectrum, we

fitted for it in the interval 4 ≤ ` ≤ 40 and obtained a flatter slope of −1.4± 0.05. We

obtained power spectra for the high-` features as the difference between the original

spectrum and its extrapolation computed using the best-fitting slope. This proce-

dure creates a smooth high-` power spectrum, shown in blue in Figure 3.2 (left).

After this, we create a realization map with the artificial high-` power spectrum,

using synfast and using a Gaussian beam appropriate for the resolution of the sim-

ulation. We finally multiply the resulting random map by a normalized version of

the original template map. This reproduces the anisotropy of the foreground map

(a Galactic plane mask in a sense), where regions in the Galactic plane are typically

much brighter than at high Galactic latitudes. The high-` random map is finally

added to the original template, but multiplied by an amplitude chosen to give a

continuous power spectrum at multipoles corresponding to the original beam. The

power spectrum of the resulting map (for a 5′ final resolution) is shown in Fig. 3.2

(left) in red.

We follow the same procedure for the dust template. Fitting for the EE and BB

slope in the range 60 ≤ ` ≤ 600 yields −2.36± 0.005 and −2.16± 0.007, respectively.

Figure 3.2 (right) shows the EE power spectra for the dust template, for a final

resolution of 5′. The colour code is the same as in Figure 3.2 (left). The slopes of

the high-` power spectrum, the beam and the amplitude of the high-` maps are free

parameters of the model and can be chosen to give different small-scale features, as

needed for the simulation.


1.3 1.7 2.6 3.3

Figure 3.3: Left: map of thermal dust spectral indices based on Planck Collaborationet al. (2016b) and smoothed to 3 FWHM to reduce noise. Right: map of synchrotronspectral indices based on Giardino et al. (2002) with βsyn increased to better fit thePlanck frequency range (see text).

3.2.3 Baseline foreground model

We model the frequency scaling of the dust and synchrotron components in antenna

temperature as

TA,dust(ν) ∝ νβdust+1[exp(hν/kTdust)− 1]−1 (3.1)

TA,syn(ν) ∝ ν−βsyn , (3.2)

where ν is the frequency. The parameters Tdust, βdust and βsyn are the dust tempera-

ture, dust spectral index and synchrotron spectral index, respectively.

The best-fitting values of Planck Collaboration et al. (2016b) are spatially-

varying with average values across the sky 〈Tdust〉 = 21 K and dust spectral index

〈βdust〉 = 1.53. For the synchrotron component, Planck Collaboration et al. (2016b)

uses a template spectrum obtained with the GALPROP code (Orlando and Strong,

2013) instead of a power-law model; the slope of the spectrum between ∼ 19 and

∼ 97 GHz corresponds to a βsyn ∼ 3.10. For our baseline model we use spatially-

constant parameters derived by the Planck Collaboration et al. (2016b) analysis:

Tdust = 21 K, βdust = 1.53 and βsyn = 3.10.

3.2.4 Spatially-varying spectral indices

In order to add complexity to the models, we also considered using spatially vary-

ing spectral index maps for both dust and synchrotron emission. For thermal dust,

we started from the map of best-fitting spectral indices calculated using the Planck

temperature maps from COMMANDER in Planck Collaboration et al. (2016b). This

map has a resolution of 7.5′ FWHM but it is very noisy. We therefore smooth it to 3.


The final map of βdust in our model is shown in the left panel of Figure 3.3. For our

test model, we do not consider spatially varying Tdust, since there is a degeneracy

between βdust and Tdust. With no ∼THz data, it is very difficult to constrain both at

the same time, so we only consider spatially varying βdust, which has a larger effect

on the spectral law in the frequency range we consider.

For synchrotron, we use the map of spectral indices by Giardino et al. (2002).

This map was derived using the full-sky map of synchrotron emission at 408 MHz

from Haslam et al. (1982), the northern-hemisphere map at 1420 MHz from Reich

and Reich (1986) and the southern-hemisphere map at 2326 MHz from Jonas et al.

(1998). The Giardino et al. (2002) map has a resolution of 10.

One possible problem with the Giardino et al. (2002) map is that is was derived

at radio frequencies, where the synchrotron spectral index is typically flatter. We

corrected for this effect by computing the expected steepening between ∼ 490–

2120 MHz and 20–30 GHz using the same GALPROP template used in the Planck

analysis and applying it to the Giardino et al. (2002) map. The result is shown in

the right panel of Figure 3.3; the steepening applied is ∆βsyn = 0.13. The mean and

standard deviation of this map over the full sky are 2.9 and 0.1, respectively.

3.2.5 Curved synchrotron spectral index and multiple thermal

dust components

As mentioned in Section 2.1.1, there is evidence that the synchrotron spectral law

is not a simple constant power-law, instead having a curvature as the frequency

increases (Kogut, 2012). In order to model this, we replace equation 3.2 by

TA,syn(ν) ∝ (ν/ν0)−βsyn+C log(ν/νpiv), (3.3)

whereC is the curvature amplitude, ν0 is the reference frequency of the synchrotron

template and νpiv is a pivot frequency. Positive values of C flatten, and negative

ones steepen the spectral law for increasing frequency. For example, Kogut et al.

(2007) finds a slight flattening of the spectrum with C ∼ 0.3 for νpiv = 23 GHz for

WMAP data.

The thermal dust spectral law might be better modelled using more than one

modified black-body, (e.g., Finkbeiner et al., 1999; Meisner and Finkbeiner, 2015).

The physical motivation is that different types of dust grains would be character-

ized by a different emission law. For this reason, we allow an arbitrary number of

components, provided the user specifies βdust (or a map of coordinate-dependent

βdust), Tdust, and an amplitude Edust for each component. We replace equation 3.1


with

TA,dust(ν) ∝Nmbb∑i=1

Edust,i νβdust,i+1[exp(hν/kTdust,i)− 1]−1, (3.4)

where Nmbb is the number of modified black-body components. We note that our

parameterization is equivalent to that in Meisner and Finkbeiner (2015) once our

Edust,i is their fiqi. In that work, qi is a physical parameter describing the dust com-

ponent, specifically the ratio of far-infrared emission cross-section to optical absorp-

tion cross-section. The parameter fi is the relative contribution (or fraction) of each

component to the total (normalized such that∑Nmbb

i fi = 1). Our amplitude param-

eter Edust,i accounts for both, and it is therefore a phenomenological, rather than a

physical, parameter. For example, the best-fitting model (model 8) of Finkbeiner

et al. (1999) has two modified black-body components that, in our parametrization,

are described by Tdust,1 = 9.4 K, βdust,1 = 1.67, Tdust,2 = 16.2 K, βdust,2 = 2.70, and

intensity ratios Edust,1/Edust,2 = 0.49.

3.2.6 Additional polarized component: AME

There is evidence that AME due to spinning dust is polarized, with a polarization

fraction of few percent (see Section 2.1.3). We consider the polarization intensity of

the AME as an additional feature to simulate observations by future experiments

with a better accuracy.

To construct our AME template we used the Planck 2015 total intensity AME

template (with a resolution of 1) and the thermal dust polarization maps. By as-

suming that the polarization angles for AME are the same as for the thermal dust,

we can obtain polarization Q and U maps for AME as

QAME = fp,AME TAME cos(2χTD) (3.5)

UAME = fp,AME TAME sin(2χTD), (3.6)

where fp,AME is a spatially-constant polarization fraction (we used a default value

of 1%), TAME is the total intensity AME template, and χTD is the thermal dust polar-

ization angle. The Pearson correlation coefficient between the thermal dust and the

AME template (at 1 resolution andNside = 64) is 0.71±0.01 (Qmap) and 0.73±0.01

(U map). The errors were calculated with jackknife resampling. Figure 3.4 shows

the Q intensity of the constructed AME template, with a reference frequency of

23 GHz.

As a spectral law, we adopt a parabola in the logarithmic flux-frequency space,


-4 4µK

Figure 3.4: Template map of Q AME component, as derived from the total inten-sity AME and the thermal dust polarization maps from Planck Collaboration et al.(2016b). This map has 1 resolution, a reference frequency of 23 GHz and assumeda polarization fraction of 0.01.

proposed by Bonaldi et al. (2007), given by

log(TA,ν) = const.−[

m60 log(νmax)

log(νmax/60GHz)+ 2

]log(ν)+

m60

2 log(νmax/60GHz)(log(ν))2, (3.7)

where the free parameters are m60 (the slope at 60 GHz) in the log(ν)-log(S) space,

and νmax is the peak frequency (for the spectrum in flux units). We adopt as default

values νmax = 19 GHz, from Planck Collaboration et al. (2016b), and 4.0 form60 from

Bonaldi et al. (2007).

Adding dust-correlated high-` features to the AME maps

Similarly to what is done for the synchrotron and thermal dust components in Sec-

tion 3.2.2, the AME polarization maps can be upgraded in resolution by adding

high-` features. However, in this case, we want the high-` thermal dust and AME

maps to exhibit the same level of correlation measured at low resolution. We there-

fore developed a special procedure for this case, that generates both dust and AME

high-` correlated random maps at the same time.

3.3. SIMULATED OBSERVATIONS OF CMB POLARIZATION EXPERIMENTS 71

We followed the procedure described in Brown and Battye (2011), which uses

as input the power spectra and cross-spectra of the set of correlated maps (in our

case, dust E and B, and AME E and B). This information is used to generate

4 correlated random a`m fields, which are finally transformed to Q and U with the

HEALPIX alm2map function. Extending what is described in Section 3.2.2, the spectra

and cross-spectra for the high-` maps are constructed by extrapolating those of the

dust and AME templates to higher multipoles. The last step of our procedure, the

modulation of the random high-` maps with a mask enhancing the Galactic plane,

is unchanged.

3.3 Simulated observations of CMB polarization ex-

periments

3.3.1 Simulating the instrumental response

To simulate the observation of the microwave sky in polarization by a given exper-

iment, we need to know the frequency bands of observation and, for each of the

bands, the point-spread function and the noise level. Each of these properties can

be simulated with different level of complexity, specified by the user, as detailed in

the following.

The frequency response can be either simulated as a single frequency or a more

general transmission window. In the more general case, the Q or U of the sky com-

ponent i at the frequency band νj is given by

Xi(νj) =

∑kWj(νk)XrefSi(νk)∑

kWj(νk), (3.8)

where Si(ν) is the spectral law of the component, Xref is either the Q or U ampli-

tude of the corresponding template, Xi is the Q or U weighted intensity of each

component, and Wj(νk) is the transmission of band j for a set of frequencies νk.

In practice, when simulating a band response, the signal needs to be simulated for

a set of frequencies νk and averaged over the entire band, with weights given by

the transmission Wj(νk). The effect of the instrumental resolution is simulated by

convolving the maps with a Gaussian beam of specified FWHM.

The noise can be modelled either as a uniform white noise, described by an rms

value constant over the sky, or as an anisotropic white noise specifying a map of rms

varying in the sky. For Planck, we model this using the 3 × 3 noise covariance per

pixel containing the Stokes parameter covariance elements TT , QQ, UU , TQ, TU ,


and QU . In this case, for each pixel, a Cholesky decomposition is performed over

the covariance matrix; the diagonal elements of the decomposition finally yield the

standard deviations per pixel for T , Q, and U , respectively.

3.3.2 Simulation procedure

Once the experiment is specified, by means of a set of frequencies, resolution and

noise, the simulation procedure is the following:

• A CMB map is generated using synfast, up to a resolution equal to θ∗, which

should be at least equal to the smallest instrumental beam of the experiment.

• High-` features are optionally added to the synchrotron, thermal dust and/or

AME templates up to a resolution θ∗ (if θ∗ is smaller than the intrinsic resolu-

tion of the template).

• The CMB map and foreground templates are scaled in intensity according

to the frequency behaviour to each frequency band, added together and

smoothed to match the resolution appropriate for that channel.

• A noise map is generated and added to the frequency band for each channel

to obtain the simulated frequency map.

The outputs are the frequency maps, but also the component maps at all re-

quired frequencies. Some of the components can be easily deactivated to obtain

noise-only, signal-only, foreground-only or CMB-only simulations, for example for

Monte Carlo purposes.

3.4 Comparison with data from Planck and WMAP

3.4.1 Foreground model

For the comparisons shown in this section, we used the baseline foreground model

described in Section 3.2.3. This includes synchrotron and thermal dust with fixed

spectral indices in the sky. We do not include the polarized AME component, that

was not detected by Planck due to its weakness compared to the noise levels (Planck

Collaboration et al., 2016b).

3.4. COMPARISON WITH DATA FROM PLANCK AND WMAP 73

Figure 3.5: Scatter plot inside the Galactic plane |b| ≤ 20. The Nside is 64. The toprow corresponds to Q intensity, and the bottom row to U intensity. The black linerepresents the perfect one-to-one match. Both the templates and the observed skywere smoothed to a common 1 resolution. In parenthesis, we show the factor bywhich we multiply each data point for display purposes.

3.4.2 Data maps

We compared the output of our model with Planck sky observations in polariza-

tion at 30, 44, 70 and 353 GHz, complemented by the WMAP W band at 94 GHz

and K band at 23 GHz. In the following, we carried out the comparison between

model and data smoothing to a 1 common resolution, which is the resolution of

our synchrotron template. Such resolution is also good for display purposes be-

cause it reduces the noise and allows an easier visual inspection of the foregrounds

morphology.

The Planck frequency maps have been corrected for the polarization leakage

due to bandpass mismatch with the correction maps available on the Planck Legacy

Archive. The WMAP maps have been downloaded from the LAMBDA-WMAP

archive 3.

3http://lambda.gsfc.nasa.gov/product/map/dr5/

http://lambda.gsfc.nasa.gov/product/map/dr5/


Sky 2360453015

015304560

Model 2360453015

015304560

Sky 305040302010

01020304050

Model 305040302010

01020304050

Sky 4420161284

048121620

Model 4420161284

048121620

Sky 7020161284

048121620

Model 7020161284

048121620

Sky 9420161284

048121620

Model 9420161284

048121620

Sky 35360453015

015304560

Model 35360453015

015304560

Figure 3.6: Maps comparison between the observed sky and the foreground modelfor Q. The rows are six bands, the left column corresponds to the observed skyand the right one to our foregrounds model. The units are µKA. The maps aresmoothed to a common 1 resolution and degraded to Nside = 32 to suppress thenoise for display purposes.


Sky 23302418126

0612182430

Model 23302418126

0612182430

Sky 30252015105

0510152025

Model 30252015105

0510152025

Sky 44108642

0246810

Model 44108642

0246810

Sky 70108642

0246810

Model 70108642

0246810

Sky 94108642

0246810

Model 94108642

0246810

Sky 353302418126

0612182430

Model 353302418126

0612182430

Figure 3.7: The same as Fig. 3.6, but for U maps.


3.4.3 Comparison with foregrounds only

We first compare the data with a model of the sky including only the foregrounds

(with no high-` features) and without CMB and noise. In this way, we only com-

pare the deterministic components of the model, without any random realization.

Also, we subtract the observed CMB from the Planck and WMAP frequency obser-

vations. To this end, we use the CMB map as estimated by COMMANDER (Planck

Collaboration et al., 2016c), with a resolution of 10 arcmin atNside = 1024, smoothed

to 60 arcmin. The true Planck and WMAP frequency responses have been used to

create the model sky as described in Section 3.3.

Figure 3.5 shows a pixel-by-pixel comparison of true versus model Q (top row)

and U (bottom row) maps. We show only the pixels inside the Galactic plane

(|b| ≤ 20) for Nside = 64 to reduce the effect of noise and CMB. Figures 3.6 and

3.7 show the maps in pseudo-colour scale, for 6 frequencies and in Q and U inten-

sity. The modelled foregrounds and the observed sky have the same colour scale.

As expected, the agreement is very good for the foreground-dominated frequen-

cies. At 70 and 94 GHz the agreement is less good, because CMB and noise become

important at these frequencies. The direct comparison of the maps shows that the

foreground model is quite good once the noise is reduced (by means of degrad-

ing to Nside = 32). The U intensity of the 94 GHz band is noisy, which makes the

comparison difficult.

3.4.4 Including the contribution from noise and CMB

For the comparisons presented in this section, we included noise and a CMB re-

alization, which are present in the sky observations, in order to assess the match

of the model when all components are included. In this case we only consider the

power spectra, since the different CMB and noise realizations do not allow a mor-

phological comparison.

The CMB map has been generated starting from the best-fitting model of Planck

(including polarization information, Planck Collaboration et al. (2016i)) and with a

tensor-to-scalar ratio r = 0.1.

In this case, we consider the 30, 44, 70, and 353 GHz Planck bands and all five

of the WMAP bands. The WMAP noise is simulated using the hit counts maps

and RMS information available from the Lambda website. Planck noise has been

simulated using the pixel covariance information.


101

102

EE 30 GHz BB 30 GHz

100

101

EE 44 GHz BB 44 GHz

10-1

100

101

EE 70 GHz BB 70 GHz

101 102

101

102

EE 353 GHz

101 102

BB 353 GHz

multipole `

`(`+

1)C`/

2π µ

K2 A

Figure 3.8: EE (left) and BB (right) full-sky power spectra comparison betweenthe complete model (foregrounds+noise+CMB, in green) and the observed sky (inorange) in four Planck bands, for Nside = 256. The error for the observations isplotted as the orange shaded region. The full-sky maps are smoothed to a common1 resolution.


102

103

EE 23 GHz BB 23 GHz

101

102

EE 33 GHz BB 33 GHz

100

101

EE 41 GHz BB 41 GHz

10-1

100

101

EE 61 GHz BB 61 GHz

101 102

10-1

100

101

EE 94 GHz

101 102

BB 94 GHz

multipole `

`(`+

1)C`/

2π µ

K2 A

Figure 3.9: EE (left) and BB (right) full-sky power spectra comparison for the fiveWMAP bands. The error for the observations is plotted as the orange shaded re-gion. The full-sky maps are smoothed to a common 1 resolution. The conventionis the same as in Fig. 3.8.


Figure 3.8 shows the comparison of the EE and BB full-sky power spectra be-

tween the complete model (foregrounds model + noise realization + CMB realiza-

tion) and the Planck bands. Figure 3.9 shows the same for the five WMAP bands.

Since these spectra are computed over the full sky, there is no leakage between EE

andBB modes and no correction is needed. In this case, the error on the data power

spectrum (orange shaded region) is just due to sample variance in the noise and the

CMB (the latter corresponds to the Cosmic Variance, explained in Section 1.3), and

it is calculated as

∆C` =

√2

2`+ 1(CCMB

` +N`), (3.9)

where CCMB` is the input CMB power spectrum and N` is the noise bias power spec-

trum, calculated from 100 noise Monte Carlo realizations based on the noise covari-

ance matrix information of each band. This error is generally very small compared

to the foreground signal (< 1%), and in most cases not visible in the figures. The

maximum observed error, considering multipoles up to ` = 100, is∼ 10% at 70 GHz

for Planck and ∼ 23% at 61 GHz for WMAP.

The agreement between the intermediate frequencies (44 and 70 GHz) benefits

from the inclusion of CMB and noise in the comparison. At 70 GHz the CMB po-

larised intensity is strong, so the match improves. The inclusion of noise is particu-

larly important to reconcile model and data towards ` = 100.

Comparison on small patches at high Galactic latitude

As a final assessment of our model, we compare the local angular power spectra

to the latest Planck observations in several small sky patches at intermediate and

high Galactic latitude. This is particularly useful for ground-based CMB polariza-

tion experiments, which target these areas. For example, the BICEP2/KECK array

(BICEP2 Collaboration et al., 2014a) observes with two bands at 100 and 150 GHz.

They target a high Galactic latitude patch visible from the South Pole, with a size of

∼ 800 deg2. The South Pole Telescope has measured the sub-degree scales lens-

ing BB power spectrum in a southern 100 deg2 patch using two bands (95 and

150 GHz) (Keisler et al., 2015b). Another example is the POLARBEAR experiment,

which measured the lensing BB spectrum in three small patches with a total area

of 25 deg2 at 150 GHz (The Polarbear Collaboration: P. A. R. Ade et al., 2014). Given

the frequency coverage of these experiments, observing around 150 GHz, where

the CMB peaks, it is crucial to model correctly the contamination from polarized

thermal dust emission.

For our comparison, we followed a similar procedure to the one described in


0.0 0.5 1.0 1.5 2.0

AXX for sky model µK2353GHz

0.0

0.5

1.0

1.5

2.0

AX

X f

or

sky o

bse

rvati

ons µK

2 353

GH

zAEE

ABB

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0

αXX for sky model

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0.5

1.0

αX

X f

or

sky o

bse

rvati

ons

αEE

αBB

Figure 3.10: Correlation of the fitted power-law between our model and Planck353 GHz observations for several intermediate and high-latitude small patches.Top, correlation for the value of the amplitude AXX at ` = 80. The stars are theBICEP2 field. Bottom, correlation between the values of the power-law slope αXX .


Planck Collaboration et al. (2016d) for assessing the contamination by thermal dust.

We produced disk-shaped masks with a radius of 11.3 (equivalent to 400 deg2).

Each patch is located on the centre of a pixel of a Nside = 8 map and we considered

patches whose centre has a latitude of |b| > 45. This leaves 48 circular patches. We

also added another mask that selects the region targeted by BICEP2/KECK. The

masks are apodized by smoothing with a 2 FWHM beam.

We used the same maps assessed in Figure 3.8 (Nside = 256, 1 FWHM resolution,

modelled as foregrounds + noise + CMB). On each small patch, we calculated the

pseudo-C` for both our model and the Planck observations at 353 GHz, in order to

compare the thermal dust polarization intensity. Then, we correct for the effect of

masking with a pseudo-C` approach (e.g., Brown et al., 2005) (this will be detailed

in Section 4.3.2). Following Planck Collaboration et al. (2016d), we fitted for the

power law DXX` = AXX(`/80)αXX+2, where XX is either EE or BB, using the same

fitting method described in Section 3.2.2 over the range of multipoles ` = 40–100.

To assess the match, we plot the fitted AXX and αXX from the model and from

Planck 353 GHz observations in Figure 3.10. The crosses represent each one of the

disk-shaped patches, while the star represents the BICEP2 field for either EE or

BB. The agreement is better on the foreground amplitude AXX than on the slope

αXX , our model being generally a bit steeper than the Planck 353 GHz band. The

mismatch is most likely due to errors in modelling the noise and matching the CMB

signal. The match on individual 400 deg2 sky areas may vary, but there is a good

agreement when considering a sample of sky patches.

3.4.5 Optimal spectral index test

In the previous tests, we used the best fit values for the spectral indices βsyn = 3.10

and βdust = 1.53, according to Planck Collaboration et al. (2016b). We might wonder

if this is the optimal choice. Here, we explore the possibility that changing the

spectral index of synchrotron and/or dust may improve the match. This analysis

will also provide indications of what is a reasonable range within which to vary

the synchrotron and dust spectral indices, for example for Monte Carlo purposes,

while preserving a good agreement with the data.

We do this by defining a χ2 statistic in the pixel domain. The sky observations

and the model, with a common resolution of 1, are degraded to Nside = 64. Then,

we compare the maps pixel by pixel, using as the σ error the Planck noise maps


0.5 1.0 1.5 2.0 2.5

βdust

1.5

2.0

2.5

3.0

3.5

4.0

βsy

n

0.5 1.0 1.5 2.0 2.5

βdust

1.5

2.0

2.5

3.0

3.5

4.0

βsy

n

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Norm

aliz

ed p

robabili

ty

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Norm

aliz

ed p

robabili

ty

Figure 3.11: Normalized probability for βdust and βsyn obtained by adding the χ2

values for two Planck bands: 44+70 GHz. In this case, the pixels inside a Galacticlatitude b±20 are used. The black curve represents the 1 σ confidence interval. Theleft panel shows the constraints given by Q maps, and the right one those given byU maps.

discussed in Section 3.3.1, properly smoothed and degraded. Therefore, for an ob-

served band ν, we minimize

χ2ν(βdust, βsyn) =

Npix∑i=0

(Qi,ν −Qi,model(βdust, βsyn))2

(σQi,ν)2

, (3.10)

and analogous for U , where i cycles through the Npix pixels of the map being con-

sidered.

We mapped the χ2 values for a grid of (βdust, βsyn) parameters and for both the

70 and 44 GHz bands. We could not use the Planck 30 and 353 GHz because those

are the frequencies at which the synchrotron and dust templates are normalized,

therefore are unaffected by changing the spectral indices. Also, we did not consider

the WMAP channels because of their higher noise levels.

We finally obtained a joint likelihood for the 44 and 70 GHz channels as

exp(−χ2/2), where χ2 = χ244GHz + χ2

70GHz. In Figure 3.11 we show the results for the

pixels inside a b = ±20 Galactic strip. The left panel corresponds to theQmaps, the

right one to the U maps. The black curve corresponds to the 1σ confidence interval


101 102 103

multipole `

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

`(`+

1)C

BB

`/2π

mK

2

Prim. r=0.1

Prim. r=0.01

Prim. r=0.001

G. Foregrounds 45 GHz



Figure 3.12: Forecast for primordialBB modes detectability. The black curves showprimordial CBB

` for three values of the tensor-to-scalar ratio. We also show theBB power spectra for polarized foregrounds for three frequencies: 45, 105, and315 GHz. The foreground maps were masked with the WMAP polarization dataanalysis mask and deconvolved from pseudo-C` following Brown et al. (2005).

in the βdust-βsyn space considered. Note that we assume a prior on the range of val-

ues these indices can take. The probability distribution is flatter in U (and therefore

the 1σ contour is wider) because the U maps have weaker foreground emission at

the Galactic plane, as can be seen in Figure 3.7.

The best fit values for the 1D marginalized probability of each parameter and

its 1σ confidence interval, adding both bands and both polarizations (44 + 70 GHz

and Q + U ), are βsyn = 3.4+0.7−0.5 and βdust = 1.6+0.5

−0.2 for the pixels inside the Galactic

plane strip, and βdust = 1.6+0.6−0.3 and βsyn = 3.4+0.9

−0.6 when we consider the full sky.

The reason why we obtain relatively weak constraints is because we could only

use data at central frequencies, where foregrounds are not so strong and the two

spectral indices are more degenerate. A test of the model with much lower (higher)

frequency would give a much stronger constraint of the synchrotron (dust) spectral

index.


15

20

25

30

35

40

45CMB COrE simulation, 45 GHz

165 170 175 180 185 190 19515

20

25

30

35

40

45COrE simulation, 105 GHz

170 175 180 185 190 195

COrE simulation, 315 GHz

10

8

6

4

2

0

2

4

6

8

10

Q Inte

nsi

ty µK

G. Longitude

G. La

titu

de

Figure 3.13: Zoomed-in region of intermediate foreground contaminations in a sim-ulation of COrE observations. The maps are patches ofQ intensity, 30×30, centredat l = 180, b = +30. The top left panel shows a CMB realization (in thermody-namic units), and the remaining panels show the simulated observations for threefrequencies: 45 GHz (dominated by synchrotron emission), 105 GHz (dominated byCMB), and 315 GHz (dominated by thermal dust emission). These panels have An-tenna units. The resolution in each panel is different and corresponds to the COrEresolution quoted in Table 4.1.

3.5. FORECAST AND MONTE-CARLO CAPABILITIES 85

3.5 Forecast and Monte-Carlo capabilities

To illustrate the simulation capabilities of our sky model, we consider the speci-

fications of the first proposal of the Cosmic Origin Explorer (COrE) experiment,

described in The COrE Collaboration et al. (2011). The instrumental specifications

used are listed in Table 4.1. This experiment has 15 frequency bands with frequency

ranging from 45 to 795 GHz and resolution of 23–1.3 arcminutes. The noise is simu-

lated as Gaussian and uniform, with standard deviation of a few µK per arcminute,

as quoted in the Table.

Figure 3.12 shows the forecasted B-mode for foregrounds for COrE as obtained

with our sky model. The CMB BB power spectra for different tensor-to-scalar ra-

tios are compared with the foreground power spectrum at three COrE frequencies:

45, 105 and 315 GHz. The foreground power spectra have been computed with the

WMAP polarization mask (excluding ∼ 37% of the sky); it has been corrected for

again using the pseudo-C` approach of Brown et al. (2005). Although this figure is

qualitatively similar to other forecasts in the literature (e.g. The COrE Collaboration

et al., 2011), the fact that our model is based on actual polarization observations en-

sures a better match with the real sky and therefore improved forecast capabilities.

In Figure 3.13 we show the Q intensity on a 30×30 sky patch located at

l = 180, b = 30 and exhibiting intermediate foreground contamination. The

top left panel shows a CMB realization with 4′ resolution. The remaining pan-

els show the simulated COrE observations in this region at 45 GHz, dominated by

synchrotron emission, 105 GHz, dominated by CMB emission, and 315 GHz, domi-

nated by thermal dust emission. The resolution in each panel is different and corre-

sponds to the COrE resolution quoted in Table 4.1. The features of the foreground

components are easily viewed. On the smallest scales, the foregrounds structures

are those generated with the procedure described in Section 3.2.2, which allows

overcoming the limits imposed by the intrinsic resolution of the templates (1 for

synchrotron and 10′ for thermal dust).

As mentioned, a valuable feature of our sky model is the capability to randomize

to a certain degree the polarized foreground components, for example for Monte-

Carlo purposes. This can be particularly useful to produce simulations that re-

flect our uncertainties on the polarization of the foregrounds. The randomization

method exploits the same procedure described in Section 3.2.2 to upgrade the res-

olution of the templates. By changing the parameters describing the power spec-

trum of the high-` map (such as the slope and normalization), it is possible to ob-

tain fainter/stronger foreground contamination at intermediate and small scales.


27

28

29

30

31

32

33Original template Added high-`

177 178 179 180 181 182 18327

28

29

30

31

32

33Added high-`, different seed

178 179 180 181 182 183

Added high-`, different C` power law

20

16

12

8

4

0

4

8

12

16

20

Q Inte

nsi

ty µK

G. Longitude

G. La

titu

de

Figure 3.14: Illustration of the foregrounds randomization capabilities of the codeon a 5×5 sky patch centred at l = 180, b = 30. The top left panel shows the origi-nalQ synchrotron template, which has a FWHM of 1; the top right panel shows thesame template upgraded in resolution to 5′ by adding high-` features. The bottomleft panel shows the same as the top right, but with a different random realization.The bottom right panel shows the same as the top right one, but with a steeperpower spectrum at high multipoles (and therefore fainter foreground features).

3.6. CONCLUSIONS 87

Moreover, by changing the seed of the random realization, it is possible to obtain

independent patterns for the same configuration. This is illustrated in Figure 3.14,

where we show the Q intensity for the synchrotron emission in a 5×5 patch of the

sky centered in the same coordinates as Figure 3.13. The top panels show the origi-

nal synchrotron emission template at 1 resolution on the left and a high-resolution

(5′) one on the right. The bottom panels show two alternative versions of the top-

right panel: a different realization with the same power spectrum on the left and

the same random realization with steeper power spectrum (and therefore fainter

foreground features) on the right.

3.6 Conclusions

We have constructed and validated a new model of the microwave sky in polar-

ization, based on the most recent results from the Planck experiment (Planck Col-

laboration et al., 2016l). Our model features multiple choices for the frequency-

dependence of the polarized synchrotron and dust foregrounds of increasing com-

plexity. The templates, based on the Planck observations, are upgraded in reso-

lution by means of a random map modulated by the large-scale foreground emis-

sion. This allows simulating experiments with higher resolution than Planck. At

the same time, it allows randomizing over the small-scale foreground features, for

example in a Monte-Carlo approach. We also include curvature in the synchrotron

spectral law, and the capability to model an arbitrary number of modified black

bodies for the thermal dust law. Finally, we include the choice of a polarized AME

emission template, constructed from the thermal dust polarization angles and from

the AME total intensity template provided by Planck.

We demonstrated that our baseline model (power-law synchrotron with fixed

spectral index βsyn = 3.10, modified blackbody thermal dust with fixed temperature

Tdust = 21 K and spectral index βdust = 1.53) gives a very good match with both

WMAP and Planck data. We also found good agreement between the dust model

and the data on small high-latitude regions, typically targeted by ground-based

experiments. When changing the parameters of the model, βsyn = 2.9 − 4.2 and

βdust = 1.4− 2.1 also provide a good fit to the data.

We finally showed the capabilities of our model for forecast and Monte-Carlo

purposes by simulating data for the COrE experiment. Our easy to use python

package, which is available at http://www.jb.man.ac.uk/~chervias, will be

a useful tool for the CMB polarization community.

http://www.jb.man.ac.uk/~chervias

BLANK PAGE

88

Chapter 4

Impact of foreground uncertainties on

future CMB experiments

4.1 Introduction

The astronomical community has put a significant effort on the search for primor-

dial gravitational waves. Planned future experiments will aim at detecting primor-

dial large-scale B-modes. However, if the true signal turns out to be small, it would

definitively test the limits of our instrumentation and analysis capabilities. Achiev-

ing the required sensitivity is not enough for detecting r, because of the presence

of diffuse polarized Galactic foregrounds. Therefore, component separation tech-

niques must be employed, in order to obtain the cleanest possible CMB maps. The

question is then, how accurately can we model and clean the foregrounds to the

level required for measuring r = 10−2–10−3?

Several forecasts of tensor-to-scalar ratio measurements, including foreground

residuals, have been performed for different experiments in the recent literature

(Betoule et al., 2009; Armitage-Caplan et al., 2012; Errard and Stompor, 2012;

Bonaldi et al., 2014; Remazeilles et al., 2016; Alonso et al., 2017; Remazeilles et al.,

2017b). In this chapter, we study how the error in the diffuse foregrounds compo-

nent separation modelling propagates into the measured cosmology (the tensor-to-

scalar ratio). We model a full pipeline, going from multi-frequency sky observations

to measuring the cosmological parameter r. Our approach is quite agnostic from

the point of view of physical modelling of the Galactic emission, and it focuses

on quantifying the bias on r corresponding to some arbitrary modelling error lev-

els. We also consider component separation and error mitigation techniques with

different levels of complexity. The content of this chapter was published in Hervías-

Caimapo et al. (2017).

89

90 CHAPTER 4. FOREGROUND UNCERTAINTIES

0 20µKCMB 100 3400µKCMB

Figure 4.1: Polarization intensity P =√Q2 + U2 maps of the simulated sky at

105 GHz (left) and 555 GHz (right), for the sky model with variable spectral indices.Both maps are dominated by thermal dust emission. The maps of the model withspatially constant spectral indices look very similar.

4.2 Simulated observations

For our analysis, we use the specifications of the Cosmic Origins Explorer (COrE),

as outlined in an earlier version of the experiment presented to ESA in 2010 (The

COrE Collaboration et al., 2011), as a representative of the capabilities of a future

CMB polarization experiment. The frequencies, beam sizes, and sensitivities used

in this work are listed in Table 4.1. Recently, a second proposal for this mission,

renamed CORE, was submitted to ESA in 2016 (Delabrouille et al., 2018). In partic-

ular, this design has different specifications (more frequency channels, more tightly

packed in the 60-600 GHz frequency range), and it is described in Chapter 6.

We perform our simulations using HEALPIX maps with a resolution parameter

of Nside = 512, corresponding to a pixel size of ∼ 7 arcmin. Some of the actual

COrE bands have better resolution than the one allowed by such a pixel size, so

we limit the band resolution to 7 arcmin in these frequencies, marked with ∗ in

Table 4.1. This modification does not change our results appreciably because we

focus on diffuse foreground components and primordial B-modes, both dominant

at low multipoles.

To simulate the full-sky observations of the microwave sky, we use the model

presented in Chapter 3. We consider three polarized sky components: CMB, ther-

mal dust, and synchrotron. The details of each component are described in Sec-

tion 3.2. We generate two distinct models: one with spatially-constant spectral in-

dices (simple model), with βsyn = 3.1, βdust = 1.53, and Tdust = 21 K; and a sec-

ond model with spatially-variable spectral indices (complex model), using maps

4.3. METHODOLOGY 91

Band [GHz] 45 75 105 135 165 195 225 255 285Beam FWHM [arcmin] 23.3 14.0 10.0 7.8 7.0∗ 7.0∗ 7.0∗ 7.0∗ 7.0∗

Noise [µKA·arcmin] 8.61 4.09 3.5 2.9 2.38 1.84 1.42 2.43 2.94

Band [GHz] 315 375 435 555 675 795

Beam FWHM [arcmin] 7.0∗ 7.0∗ 7.0∗ 7.0∗ 7.0∗ 7.0∗

Noise [µKA·arcmin] 5.62 7.01 7.12 3.39 3.52 3.60

Table 4.1: COrE satellite specifications used in this work to simulate observations,taken from The COrE Collaboration et al. (2011). As explained in the main text, thebands marked with a ∗ have better resolution than 7 arcmin, but have been simu-lated with a 7 arcmin pixel size (Nside = 512 Healpix maps) to limit the computa-tional complexity of our analysis.

described in Section 3.2.4. Some example maps for the sky model with spatially-

variable spectral indices are shown in Figure 4.1.

For each model, we produce 100 sets of FITS maps of the observed sky at each

band. Each set has the frequency bands, resolution and white noise levels as spec-

ified in Table 4.1. Each set has a different CMB and white noise realization, but the

same foreground components. We produce them with a HEALPIX resolution pa-

rameter of Nside = 512. We also produce 100 low-resolution sets with Nside = 16. In

this case, the modelled sky is produced with a resolution of 3.5 across all bands,

corresponding to the larger pixel size. Although the beam is not very well sampled

by this pixel size, we have verified that, once both beam and pixel window function

are deconvolved, the pipeline described in Section 4.3.2 yields an unbiased recov-

ery of the CMB polarization power spectra. We used the same random number

generator seed for both the high-resolution and low-resolution CMB map in each

iteration, therefore having a consistent CMB realization for each set.

4.3 Methodology

In this section, we describe the various steps of our pipeline: component separation,

power spectra and likelihood estimation.

4.3.1 Component separation

To perform the component separation, we rely on the linear mixture model, as de-

scribed in Section 2.2. In our particular case, we use the GLS solution, given by

equation 2.7. We need then an estimate of the mixing matrix A. This is typically


calculated by parametrizing the spectral laws of the CMB and foreground compo-

nents; and by estimating the relevant parameters from the data. In this case, we skip

such estimation. Instead, we assume some errors on the estimation of the spectral

parameters describing the true mixing matrix and propagate them through the full

pipeline.

In the pixel domain, the effect of the beam must be equalized across multiple

bands. In practice, this is achieved by smoothing all the frequency maps to the

beam resolution of the band with the lowest resolution. This assumption, that the

instrumental beam does not depend on frequency, is not true in general, nor it is

for COrE, as shown by Table 4.1. Therefore, we pre-process all maps by smoothing

them with a Gaussian beam, thus equalizing their resolution to 23.3 arcmin (the

resolution of the lowest frequency channel, for the high-resolution sets Nside = 512)

or 3.5 (the resolution sampled by the Nside = 16 maps for the low-resolution sets).

4.3.2 Power spectra estimation

We estimated the polarization angular power spectra from CMB maps with a hy-

brid approach: using a Quadratic Maximum Likelihood (QML) estimator at low

multipoles (` < 30), and a pseudo-C` estimator at the remaining intermediate and

high multipoles. The QML estimator is optimal at low multipoles, and it can be

used to recover the reionization bump at ` < 10. The pseudo-C` estimator is appro-

priate for high multipoles, which are unobtainable for the QML estimator, where

it can recover the first acoustic peak at ` ∼ 100 in the BB spectrum. This hybrid

approach has been shown to be nearly optimal in the whole ` range and at the same

time computationally feasible (e.g. Efstathiou, 2004a, 2006). The simulated obser-

vations at Nside = 512 are used for estimating the pseudo-C` angular power spectra,

while the low-resolution maps with Nside = 16 are used for the QML estimator.

QML

This method is based on Tegmark (1997); Tegmark and de Oliveira-Costa (2001), see

also Efstathiou (2004b); Gruppuso et al. (2009). It works in pixel space, constructing

an estimator based on the covariance matrices of the data. Specifically, if x is the

CMB map in pixel space, we aim to recover a set of parameters qi with a quadratic

estimator, Qi

qi = xtQix = tr[Qixxt]. (4.1)

4.3. METHODOLOGY 93

A good choice for the quadratic estimator is

Qi ∝∑j

(B)ijC−1PjC

−1, (4.2)

where C is the covariance matrix of data, B is an arbitrary invertible matrix (we

used the identity matrix, B = I) and

Pj =∂C

∂pi. (4.3)

The normalization of equation (4.2) is such that all window functions sum to unity

Np∑j=1

tr[QiPj] = 1, (4.4)

where Np is the number of parameters (binned power spectra values) considered.

This method gives minimal error bars but it is very computationally demanding,

since it requires matrix inversions and multiplications, of order O(N3d), where Nd

is the number of pixels outside the mask. In practical terms, our limit is to run the

QML estimator for Nside = 16 maps.

Pseudo-C` deconvolution estimator

We use the method described in Brown et al. (2005) and Brown et al. (2009), which

extended to polarization the technique proposed by Hivon et al. (2002). First, we

calculate the pseudo-C` of the masked sky using the HEALPIX anafast function. The

limited sky coverage creates leakage of polarization between E and B modes. The

pseudo-C` (C`) is related to the full sky estimator (C`) through

C` =∑`′

M``′B2`′W

2`′C`′ , (4.5)

where M``′ is the coupling matrix and B` and W` are the beam and pixel window

functions, respectively. Details on the calculation of M``′ are given in Appendix A of

Brown et al. (2005). Once a binning scheme is adopted, the binned coupling matrix

is given by

Kbb′ =∑`

Pb`∑`′

M``′B2`′W

2`′Q`′b′ , (4.6)

where Pb` and Q`b are the binning and inverse binning operators, respectively. Fi-

nally, the deconvolved binned power spectra is given by

Pb =∑b′

K−1bb′

∑`

Pb′`

[C` − 〈N`〉MC

], (4.7)


0 1 0 1

Figure 4.2: Default Galactic mask used for the power spectrum estimation, retaininga fraction of the sky fsky = 0.513. Left: Nside = 16 mask used for the QML powerspectrum estimation; right: Nside = 512 apodized mask used for the pseudo-C`power spectrum estimation.

where 〈N`〉MC is the noise bias. This is calculated by averaging the pseudo-C` of

100 noise-only realizations of the simulated observations.

The bandpass window functions are also an output of this procedure, and they

are needed for binning a model power spectra in the same way as the reconstructed

spectra for comparison. These are given by

Wb`

`=

2π

`(`+ 1)W 2` B

2`

∑b′

K−1bb′

∑`′

Pb′`′M`′` (4.8)

for bin b. A given model power spectra C` should be binned with Pb =∑``(`+1)

2πWb`

`C`.

Galactic mask

To exclude the foreground residual contamination due to the Galactic emission, we

estimate the power spectrum outside a Galactic mask. The default mask is con-

structed using the dust and synchrotron polarization templates from Planck Col-

laboration et al. (2016b), smoothing them to a FWHM of 3, and masking every

pixel with an intensity of 14µK or higher. We repeat this procedure for both Q

and U maps, and dust and synchrotron. We combine all of them to produce a fi-

nal mask. For the pseudo-C` power spectrum estimation, it is beneficial to use an

apodized mask, because sharp edges make the deconvolution kernel more compli-

cated. Therefore, we apodize the Nside = 512 mask by using the function

f(d) =

1− cos3(dπ2s

) d ≤ s

1 otherwise, (4.9)

4.3. METHODOLOGY 95

101 102

multipole `

10-4

10-3

10-2

10-1

100

101

`(`+

1)/

2πCBB

` [µK

2 45G

Hz]

C` Residuals (fullsky)

C` Thermal Dust (fullsky)

C` Synchrotron (fullsky)

Figure 4.3: Left: polarization intensity maps of the foreground residuals(reconstructed-true CMB, top) compared to the thermal dust (middle) and syn-chrotron (bottom) maps reconstructed by the component separation. Right: full-sky BB power spectrum of foregrounds residuals compared to the full-sky powerspectrum of the reconstructed thermal dust and synchrotron foregrounds. Noticethe similar shape between the residuals and the thermal dust.

where d is the distance between the pixel of interest and the closest masked pixel

(with value 0), and s is the distance scale of apodization (the scale in which the

function goes from 1 to 0, s = 20 in our case). The resulting apodized mask is

shown in the right panel of Figure 4.2. The sky fraction retained is fsky = 0.513. The

Nside = 16 version of this mask, needed for the QML estimator, is not apodized, and

it has been constructed by rounding theNside = 512 mask and degrade toNside = 16.

This mask is shown in the left panel of Figure 4.2.

4.3.3 Cosmological parameters likelihood

We calculate the likelihood for the tensor-to-scalar ratio based on the power spec-

tra averaged over the 100 realizations of simulated observations, where we varied

both the CMB and noise realizations. This effectively averages out the effect of the


cosmic variance, and only leaves the foreground residuals bias, which is of interest

in this case. We define a standard Gaussian χ2 likelihood to calculate the posterior

distribution of the tensor-to-scalar ratio. We define the χ2 as

χ2(r) =∑bb′

[PBBb − CBB,theory

b (r)]C−1bb′ [P

BBb′ − C

BB,theoryb′ (r)], (4.10)

where PBBb is the measured B-mode bandpower at bin b, CBB,theory

b (r) is the binned

B-mode theory spectrum and C−1bb′ is the inverse of the binned signal+noise covari-

ance matrix.

We construct PBBb from the low-multipole and high-multipole analysis, by join-

ing at ` = 30 (with no overlap) the results from the QML and the pseudo-C` estima-

tors.

The theory power spectrum is binned using the bandpass window functions at

the high multiple range and using a top hat function centred at each bin in the low

multipole range. The theory power spectrum is calculated as

CBB,theory` (r) =

r

r?CBB,prim` (r?) + CBB,lensing

` , (4.11)

where CBB,prim` is the primordial (scalar+tensor perturbations) power spectrum at

a given r, and CBB,lens` is the weak gravitational lensed power spectrum, which we

assume as known. r? is a fiducial value for the tensor-to-scalar ratio.

The covariance matrix is calculated using the 100 realizations signal+noise com-

plete runs of the pipeline (including the component separation). Therefore, it ac-

counts for cosmic and noise variance but also foreground residuals effects.

Modelling foreground residuals with nuisance parameters

The likelihood presented in equation (4.10) assumes that the measured power spec-

tra contains only CMB and noise. In reality, there are also some foreground resid-

uals, due to non-perfect component separation. We will extend this likelihood to

explicitly model a foreground residual contribution, using

CBB,newb (r) = CBB,theory

b (r) + AdustCBB,dustb + AsynC

BB,synb , (4.12)

where CBB,dustb and CBB,syn

b are models for the BB power spectrum of thermal dust

and synchrotron residuals, respectively, and the amplitudes Adust and Asyn are two

free nuisance parameters that can be varied, together with r, and marginalized over.

The need for adding one or both such extra parameters can be checked by seeing

whether they improve the fit, by means of the reduced χ2 value.

4.4. RESULTS 97

Simulation run Sky model Componentseparation model

Reference

Simplemodel

r = 0.01 Spatially constantβdust, βsyn

Constant βs, witha ±1,2,3% error 4.4.1; Table 4.3

r = 0.001

Complexmodel

r = 0.01 Spatially variableβdust, βsyn

Constant βs (the average ofthe true variable β maps)

4.4.2;Table 4.4, topr = 0.001

r = 0.001 Variable βs, with aglobal error of 1% and 0.5%

4.4.2;Table 4.4, bottom

Table 4.2: Summary of the different runs performed in this work.

In practice, a good choice for the foreground residual template models CBB,dustb

and CBB,synb is to assume that they are proportional to the dust and synchrotron

power spectrum. In our case, these can be computed from the foreground maps

which are the output of the component separation, as exemplified by Figure 4.3. To

the left, on top, we show the total foreground residual map for one run. The middle

and lower maps corresponds to the dust and synchrotron templates, respectively.

In harmonic space, to the right, the correlation between foregrounds residuals and

foregrounds maps is clearer. The full-sky foreground residual BB angular power

spectrum (black) is compared with the same spectrum for the thermal dust (blue)

and synchrotron (red) templates. The residual and dust spectra are very similar,

which indicates that the foreground residuals are essentially a scaled down dust

template.

In the analysis that follows, CBB,dustb and CBB,syn

b are the binned power spectra

of the thermal dust and synchrotron, respectively, reconstructed by the component

separation. We process these maps through the same procedure we use for the

reconstructed CMB, that is, the power spectra estimation with pseudo-C` for the

high-resolution map and the QML estimator for the low-resolution map, under the

same conditions.

4.4 Results

We run the component separation pipeline, described in Section 4.3, for the two sky

models, with constant and spatially-variable spectral indices. The summary of all

the runs performed in this chapter, together with descriptions and the referenced

section where the results appear, is shown in Table 4.2.


101 102

multipole `

10-6

10-5

10-4

10-3

10-2

10-1

`(`+

1l)/

2πCBB

` [µK

45G

Hz]

Ctheory` (r=0.01)

AdustCdust` (foreground residual)

AsynCsyn` (foreground residual)

Reconstructed C`Reconstructed C` minus foreground residuals

Figure 4.4: Reconstructed BB power spectrum for the simulation with r = 0.01,constant foreground spectral indices and a +2% estimation error on both βdust andβsyn. The reconstructed CMB (on 10 frequency bands, with ν ≤ 315 GHz) is biased(black circles). The modelled foreground residuals are shown as the diamonds andstars. The reconstructed CMB minus the modelled foregrounds residuals is shownas the green triangles, which can be compared with the theory power spectrum,shown as the grey curve. The multi-parameter likelihood yields r = 0.0088±0.0020.The vertical dotted line is the limit between the QML and pseudo-C` power spec-trum.

4.4.1 Sky model with constant spectral indices (Simple model)

We use the model described in Section 4.2, where the foregrounds have spatially

constant spectral indices (βdust = 1.53 and βsyn = 3.10) and we run the component

separation assuming fixed errors on these spectral indices. These are ±1, 2 and

3 percent errors on both spectral indices βdust and βsyn. As a reference, we also

examined the case of perfect knowledge on the foregrounds, i.e., 0 percent error on

the spectral indices.

We consider two cases for the GLS reconstruction: a linear mixture of all the 15

COrE frequency bands, and one of only the lowest 10 bands, having ν ≤ 315 GHz.

This is motivated by the fact that high-frequency bands are strongly contaminated

by thermal dust, so including them in the CMB reconstruction increases the dust

residuals for a given error on the dust spectral index. The drawback of limiting the

number of frequency channels is an increase in the mixed noise level, that needs

to be weighted against the reduction in the foreground residuals. In any case, it is

4.4. RESULTS 99

0.00 0.01 0.02 0.03 0.04 0.05 0.06tensor-to-scalar ratio

0

100

200

300

400

500

600

700

Norm

aliz

ed lik

elih

ood

15 bands, 1 parameter likelihood, 0% error

15 bands, 1 parameter likelihood, +3% error

15 bands, multi-parameter likelihood, +3% error



0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014tensor-to-scalar ratio

0

200

400

600

800

1000

1200

1400

Norm

aliz

ed lik

elih

ood

15 bands, 1 parameter likelihood, 0% error




Figure 4.5: Left: tensor-to-scalar ratio likelihoods for the model with constant spec-tral indices and r = 0.01. The dashed grey curve shows the likelihood for the per-fect knowledge of foreground spectral indices, centered in the correct r. The dashedcurves show the likelihood when an error of +3% is made on both spectral indices,using the entire frequency range (blue) and limited (ν ≤ 315 GHz) one (red). Thesolid curves show the results in the same cases when the multi-parameter likeli-hood is used. Right: same as the left panel, for the model with constant spectralindices and r = 0.001. We do not show the 15 bands, 1 parameter likelihood casewith +3% error, since it is extremely biased, measuring r = 0.0470± 0.0013.

worth pointing out that the whole frequency range should be used in order to esti-

mate the spectral indices before the GLS reconstruction, as this strategy in general

achieves the smallest errors on βdust and βsyn.

For all the assumed error cases, we calculate both the likelihood of equa-

tion (4.11), where the only parameter is r, and the multi-parameter likelihood of

equation (4.12), where we include either one or both of the foreground parameters

Adust and Asyn, depending on what achieves the lowest reduced χ2 value.

As an example of the multi-parameter likelihood method, we show in Figure 4.4

theBB power spectrum for the case with r = 0.01, +2 percent error on both spectral

indices and using only the frequency bands ≤ 315 GHz. The reconstructed CMB

(black circles) contains extra power because of the foreground residuals. However,

we are able to model the foreground residuals, shown as diamonds for the thermal

dust and as stars for the synchrotron. The 3-parameter model yields and unbiased

value of r = 0.0088 ± 0.0020 despite the large foreground residuals present. The

1-parameter model gave the highly biased result of r = 0.0153± 0.001 for the same

case.


In Figure 4.5, we show some example tensor-to-scalar ratio likelihoods for the

simple simulations, with r = 0.01 (left) and r = 0.001 (right), respectively. The

grey curves show the perfect knowledge component separation using all the 15 fre-

quency bands, which always yields an unbiased result (r = 0.0099 ± 0.0009 and

r = 0.00095 ± 0.00037). All the other curves assume a +3 percent error on both

spectral indices. The blue curves correspond to a CMB reconstruction using 15 fre-

quency bands, and a 1-parameter (dashed) and multi-parameter (solid) likelihood.

The red curves are the same for a CMB reconstruction using only the first 10 fre-

quency bands.

For both the r = 0.01 and r = 0.001 cases, the multi-parameter likelihood on

the 10 frequency bands case allows removing the large bias corresponding to the

+3 percent spectral index error. However, there is a degradation in the measured

error σ(r). For the r = 0.001 case, this does not allow a detection at the 2σ level any

more, but only corresponds to an upper limit.

We show the summary of all the results for the simulated observations with

r = 0.01 in the top half of Table 4.3 and in Figure 4.6. In the top panel, we show the

measured bias (estimated minus true r); in the bottom panel, we show the width

of the likelihood, σ(r). As expected, when we assume perfect knowledge of the

foregrounds, the result is unbiased. However, when we introduce some error in

the component separation, the likelihood is biased towards higher values of r. If

we adopt the multi-parameter likelihood instead of the 1-parameter one, the bias is

either reduced or removed.

Limiting the frequency bands used in the CMB solution to ν ≤ 315 GHz is also

effective in reducing the bias. In fact, a large fraction of the foreground residuals

is introduced by the high frequency bands that are strongly dominated by thermal

dust. By comparing the red stars (multi-parameter likelihood, full frequency range)

to the green circles (1-parameter likelihood, limited frequency range), we see that

they give similar biases for the same error on the spectral parameters. However,

the green circles have smaller σ(r) values than the red stars, showing that, in this

case, it is preferable to limit the bands used in the component separation than to

introduce a multi-parameter likelihood. Even so, in some cases, when the bias is

large (e.g. ±3 percent spectral index error), both approaches must be used at the

same time (shown by the yellow triangles).

The summary of all the cases for the simulations with r = 0.001 is reported in the

bottom half of Table 4.3 and shown in Figure 4.6 (right). It follows the same scheme

from Figure 4.6, for the same assumed component separation error cases. Getting

an unbiased result is much more difficult in this case, due to the small value of r. In

4.4. RESULTS 101

3 2 1 0 1 2 30.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Tenso

r-to

-sca

lar

rati

o b

ias

15 bands, 1 parameter likelihood

15 bands, multi-parameter likelihood



3 2 1 0 1 2 3% error on βdust and βsyn

0.0005

0.0010

0.0015

0.0020

0.0025

σr

3 2 1 0 1 2 3

0

10-2

10-1

Tenso

r-to

-sca

lar

rati

o b

ias





3 2 1 0 1 2 3% error on βdust and βsyn

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040σr [

or

95

% C

L upper

limit

]

Figure 4.6: Tensor-to-scalar ratio bias (estimated – true r, top) and error (σ(r), bot-tom) for different cases of fixed constant errors on both spectral indices. Left, forthe simulation with constant spectral indices and r = 0.01, and right, for the simu-lation with constant spectral indices and r = 0.001. In the case of multi-parameterlikelihoods, the empty symbols are the ones using onlyAdust, and the filled symbolsare the ones using both Adust and Asyn. If the measured r value is less than 2 σ(r)away from r = 0, instead we plot the 95% upper limit, with an arrow down symbol.


particular, for errors in the spectral indices larger than ±1 percent, we always need

both the multi-parameter likelihood and the limited frequency range. We note that,

for the perfect knowledge case, the value of σ(r) = 3.7× 10−4 is only just below the

value allowing a 2σ detection. Therefore, the multi-parameter likelihood increases

σ(r) and only allows for a 95% upper limit.

4.4.2 Sky model with spatially variable spectral indices (Complex

model)

Now, we consider a more realistic model of the sky, where the spectral indices of

the foreground components are spatially variable, as explained in Section 4.2. We

simulate the measurement of the tensor-to-scalar ratio for two levels of component

separation modelling complexity, as detailed in the two following sub-sections.

Modelling the component separation with spatially constant βdust and βsyn

The first approach we adopt is to model the spatially variable spectral indices as a

constant value across the sky. We set this value to be the true average of the βdust

and βsyn maps outside the Galactic mask of Figure 4.2. As such, these values (βdust =

1.53 and βsyn = 2.89) better represent most of the pixels used for the analysis. The

histograms of the spectral index residuals, defined as the difference between the

true indices (at a given pixel) and the true average value, for all the pixels in the sky

outside the mask are shown in Figure 4.7. The standard deviation of these residuals

are 0.0253 for βdust and 0.1074 for βsyn, which corresponds to a 1.7 % and 3.5 % error,

respectively. These errors are qualitatively similar to the ones we considered in

Section 4.4.1.

We start by considering the 1-parameter and multi-parameter likelihood on

the CMB reconstructed using the first 10 frequency bands. This was the best-

performing case in the previous (constant spectral index) exercise. The results are

reported in the top half of Table 4.4 as the base case. As we can see, this case is

not good enough any more, because the bias on r is still significant. We therefore

proceeded to optimize the analysis to reduce the bias on r, and obtained the results

quoted in Table 4.4 as the best case. The modifications we introduced are:

1. Secondary dust component: we use the idea presented in Stolyarov et al.

(2005), which is that a component with spatially variable spectral dependence

can be modelled as a series of components with constant spectral dependence,

4.4. RESULTS 103

rva

lue

∆βdust

,∆βsy

nU

sing

all1

5ba

nds

Usi

ng10

band

s(ν≤

315

GH

z)1-

para

met

erm

ulti

-par

amet

er1-

para

met

erm

ulti

-par

amet

erbi

asσ

(r)

[10−

4]

bias

σ(r

)[1

0−4]

bias

σ(r

)[1

0−4]

bias

σ(r

)[1

0−4]

r=

0.01

0%1.

2×

10−4(0

.1)

8.8

––

––

––

+1pe

rcen

t6.

5×

10−3(7

.0)

9.2

1.4×

10−3(1

.1)

12

1.3×

10−

3(1

.3)

9.8

−1.6×

10−3(−

0.9

)17

-1pe

rcen

t7.

2×

10−3(6

.6)

115.0×

10−5(0

.0)

18

1.1×

10−

3(1

.0)

11

−1.5×

10−3(−

0.9

)16

+2pe

rcen

t2.

2×

10−2(2

0)11

5.2×

10−3(3

.6)

15

5.3×

10−

3(5

.0)

10

−1.1×

10−3(−

0.5

)21

-2pe

rcen

t2.

4×

10−2(1

8)13

4.7×

10−3(2

.5)

19

4.4×

10−

3(3

.7)

12

−8.8×

10−4(−

0.5

)19

+3pe

rcen

t4.

2×

10−2(3

2)13

1.1×

10−2(6

.2)

17

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10−

2(9

.3)

12

1.0×

10−3(0

.5)

19

-3pe

rcen

t4.

7×

10−2(3

1)15

9.9×

10−3(4

.6)

22

9.4×

10−

3(7

.0)

13

−1.6×

10−4(−

0.1

)21

r=

0.00

1

0%−

5.0×

10−5(−

0.1)

3.7

––

––

––

+1pe

rcen

t6.

7×

10−3(8

.3)

8.0

3.7×

10−4

<31.5†

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10−4

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10−4

<19.4†

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3)10

3.0×

10−3(2

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<24.6†

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1)11

3.8×

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10−4

<26.7†

+3pe

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7)13

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4<

36.4†

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5×

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4)13

9.1×

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171.0×

10−2(1

1)9.

4−

3.6×

10−4

<35.0†

Tabl

e4.

3:M

easu

red

tens

or-t

o-sc

alar

rati

obi

ases

andσ

(r)

valu

esfo

rru

nson

the

sim

ulat

ion

wit

hsp

atia

llyco

nsta

ntsp

ectr

alin

dice

s.Th

eco

mpo

nent

sepa

rati

onis

mod

elle

dus

ing

the

true

spec

tral

indi

ces

wit

ha

smal

lerr

or.I

nth

er

bias

colu

mns

,the

bias

expr

esse

das

num

ber

ofσ

(r)

issh

own

inpa

rent

hesi

s.T

heva

lues

wit

h†

are

95%

uppe

rlim

its.


0.15 0.10 0.05 0.00 0.05 0.10

βdust(p)−βdust residual

0

2

4

6

8

10

12

14

16

18

Norm

aliz

ed c

ount

0.3 0.2 0.10.0 0.1 0.2 0.3 0.4 0.5

βsyn(p)−βsyn residual

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure 4.7: Histograms of the dust (left) and synchrotron (right) spectral indexresiduals (true – average spectral index) outside the Galactic mask. The true spec-tral indices are the ones in the sky model from Hervías-Caimapo et al. (2016). Thetrue average βdust is calculated on a Nside = 2048 map, while the true average βsyn

is calculated on a Nside = 512 map. The histograms are normalized so that they in-tegrate to 1. The standard deviations of the distribution of spectral index residualsare equivalent to a 1.7% error for thermal dust and to a 3.5% error for synchrotron.

each one corresponding to a term in a Taylor expansion. In this way, the first-

order thermal dust is the usual grey-body spectral law, and we add a second-

order thermal dust component, whose spectrum is the derivative of that spec-

tral law with respect to βdust. We also explored the possibility to add a second

synchrotron component to account for the variability of the synchrotron spec-

tral index, but this achieved no significant improvement in our case.

2. Optimization of the Galactic mask: we produced a new mask specifically

optimized for B-modes and tailored to the component separation approach

we used. Specifically, we produced an estimate of foreground errors on the

CMB B map with a Monte Carlo (MC) approach, and then excluded all pixels

for which this map was over some threshold. To derive the error B map,

we repeated the GLS CMB reconstruction 100 times by varying randomly the

4.4. RESULTS 105

0 1 0 1

Figure 4.8: Same as Figure 4.2, but now showing the optimized Galactic mask forthe runs described in Section 4.4.2 and labelled as best in the top half of Table 4.4.The mask has fsky = 0.48.

assumed spectral indices with Gaussian distributions (having σβdust = σβsyn =

0.1 and σTdust = 1 K). We transformed each MC output CMB from Q/U to E/B

and finally computed the standard deviation of the 100 MC B maps for each

pixel. The optimized Galactic mask used in the best case analysis is shown in

Figure 4.8. This mask has fsky = 0.48, which is very similar to that of the mask

used before.

The results for this run are reported in the top of Table 4.4 and shown in Fig-

ure 4.9. The left panel shows the measured likelihoods for the simulated observa-

tions with r = 0.01. The base case measurements (the blue curves) are biased, even

when foreground residuals are modelled in the likelihood. However, the improve-

ments we introduced to the analysis allow for an unbiased detection (shown by the

solid red curve). This detection is only 2σ(r) away from r = 0.

Figure 4.9, right, shows the same result likelihoods for r = 0.001. In this case, the

results are all biased, even with the improvements in the best case (the red curves).

This means that, for such a small value of r, the systematic error introduced by

neglecting the spatial variability of the spectral indices is too big to be compensated.

In this case, the spatial variability needs to be modelled directly in the component

separation, as we do in the next subsection.

Modelling the component separation with spatially variable spectral indices

For the model with r = 0.001, which did not give an unbiased result in the previ-

ous subsection, we model the spatial variability of the spectral indices during the

component separation step. Indeed, most component separation methods are able


0.000 0.005 0.010 0.015 0.020 0.025 0.030tensor-to-scalar ratio

0

50

100

150

200

250

Norm

aliz

ed lik

elih

ood

1 parameter likelihood, base case

multi-parameter likelihood, base case

1 parameter likelihood, best case

multi-parameter likelihood, best case

0.000 0.002 0.004 0.006 0.008 0.010tensor-to-scalar ratio

0

100

200

300

400

500

Norm

aliz

ed lik

elih

ood

1 parameter likelihood, base case

multi-parameter likelihood, base case

1 parameter likelihood, best case

multi-parameter likelihood, best case

Figure 4.9: Same as Figure 4.5 for the simulation with spatially varying spectralindices and component separation assuming spatially constant spectral indices, forr = 0.01 (left) and r = 0.001 (right).

to perform a local estimation of the foreground spectral properties, either pixel-by-

pixel (e.g. COMMANDER Eriksen et al. 2008, MIRAMARE Stompor et al. 2009), on sky

patches (e.g. CCA) or by means of other kind of spatial localization (e.g. NILC). As

a drawback, estimation errors might be larger on a local estimate than on a global

one, especially where foregrounds are weaker. That is, in lines of sights where the

(polarized) intensity is stronger, the error in the determination of spectral properties

would in general be smaller, since there is a higher signal-to-noise ratio.

In order to model such error properties in our analysis, we investigate the spa-

tial correlation of errors in βdust and βsyn made with the COMMANDER algorithm.

These simulated observations were produced with the Planck Sky Model (PSM, De-

labrouille et al., 2013) for the “Exploring Cosmic Origins with CORE” foregrounds

paper (Remazeilles et al., 2018, described in Chapter 6), and using a Galactic mask

with fsky = 0.54. Although the instrumental specifications and the sky model are

slightly different from what we use here, this allows us to derive the basic error

properties that are needed for our modelling.

We find that the error on the synchrotron spectral index, ∆βsyn, is consistent with

a random distribution and it is not significantly correlated with the synchrotron

polarized intensity. This is because of the frequency coverage of CORE, which

does not include many synchrotron dominated channels. This means that the syn-

chrotron estimation error is essentially noise-dominated. We then modelled the

synchrotron spectral index as having a Gaussian distribution with standard devia-

tion given by

σ∆βsyn = εβsyn/100, (4.13)

4.4. RESULTS 107

r value Case Measured values1-parameter multi-parameter

bias σ(r) bias σ(r)

r = 0.01Base 1.1× 10−2(6.6) 1.7× 10−3 5.7× 10−3(2.2) 0.00257Best 4.7× 10−3(1.9) 2.5× 10−3 −2.4× 10−3(−0.6) 0.00390

r = 0.001Base 7.1× 10−3(8.2) 8.6× 10−4 5.9× 10−3(5.1) 0.00115Best 4.1× 10−3(3.3) 1.2× 10−3 1.7× 10−3 < 0.00566†

r value ∆βdust,∆βsyn global error

r = 0.001

0,0 percent 1.0× 10−5(0.0) 4.5× 10−4 – –1,1 percent 4.6× 10−4(0.9) 5.2× 10−4 −1.2× 10−4 < 1.9× 10−3 †

0.5,0.5 percent 2.1× 10−4(0.4) 4.9× 10−4 – –1,0.5 percent 3.1× 10−4(0.6) 4.9× 10−4 – –0.5,1 percent 2.3× 10−4(0.5) 4.9× 10−4 – –

Table 4.4: Measured tensor-to-scalar ratio biases and σ(r) values for runs on thesimulation with variable spectral indices. On the top half, we show the results formodelling with constant spectral indices in the component separation. In the bot-tom half, we show the results for modelling with the true spatially variable spec-tral indices with a small level of error in the component separation. In the r biascolumns, the bias expressed as number of σ(r) is shown in parenthesis. The valueswith † are 95% upper limits.

where ε is the error percentage we assume in our analysis and βsyn = 2.89 is the

average synchrotron spectral index outside the Galactic mask.

We find that the thermal dust error, ∆βdust, is clearly anti-correlated with the

polarized dust intensity. We plot this for the COMMANDER results in Figure 4.11,

top. To model this property, we binned the pixels outside the Galactic mask into

ranges of polarized intensity P having roughly the same number of data points and

fitted Gaussian density functions to the error distribution in each bin. The standard

deviation as a function of P is well modelled by a power-law, shown in Figure 4.11,

bottom, and given by

σ∆βdust(P ) = εA1%(P/µK353GHz)−b, (4.14)

where A1% = 0.019±0.003 is the normalization corresponding to σ∆βdust = βdust/100

outside the Galactic mask, b = 0.019 ± 0.003 is the slope of the anti-correlation of

σ∆βdust with P , and ε is the assumed error percentage.

To simulate the estimation of spatially variable spectral indices on our study,

we start from the true input spectral indices maps, and add random error maps

consistent with the error characterization of equations 4.13 and 4.14. We assume

error levels of 1 percent (ε = 1) and 0.5 percent (ε = 0.5).

We generated 100 realizations of both ∆βsyn and ∆βdust at Nside = 16. We run our

component separation pipeline on each of the 100 simulation sets, having a different

CMB, noise and random spectral index error realization, on the ν ≤ 315 GHz limited

frequency range. We used the Nside = 16 spectral index maps directly for the low-

resolution pipeline, and we upgraded them to Nside = 512 for the high-resolution


0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035tensor-to-scalar ratio

0

200

400

600

800

1000

Norm

aliz

ed lik

elih

ood

1 parameter likelihood, perfect knowledge

1 parameter likelihood, 1% global error

multi-parameter likelihood, 1% global error

1 parameter likelihood, 0.5% global error

Figure 4.10: Tensor-to-scalar ratio likelihoods for a complex model (with spatiallyvariable spectral indices) and modelling the component separation as spatially vari-able spectral indices. All the runs are made with the optimized mask shown in Fig-ure 4.8 and limiting the frequency coverage to ν ≤ 315 GHz. The case with 1 percentglobal error has a 1 percent error (standard deviation of error in pixels outside theGalactic mask) modelled with a spatially uniform random Gaussian error for ∆βsyn

and with a spatially correlated (following equation 4.14) random Gaussian error for∆βsyn. The case with 0.5 percent global error is analogous.

pipeline.

The likelihoods for the reconstructed power spectrum averaged over the 100

realizations for both the 1 and 0.5 percent global error cases, along with the case

assuming perfect knowledge on the spectral indices, are shown in Figure 4.10 and

reported in the bottom part of Table 4.4. As usual, assuming a perfect knowledge

(spectral indices errors equal to 0) yields an unbiased result, r = 0.00101± 0.00045.

With a 1 percent global error on both spectral indices, we measure a bias on r of

4.6 × 10−4, and a σ(r) = 5.2 × 10−4 for the 1-parameter likelihood. The multi-

parameter likelihood yields a very small bias of −1.2× 10−4, but with a 95% upper

limit of 0.00189.

With 0.5 percent global error, we still measure a small bias, of 2.1× 10−4 and an

4.4. RESULTS 109

0.10

0.05

0.00

0.05

0.10

Err

or

∆β

dust

=β

model

dust−β

estim

ated

dust

0 2 4 6 8 10 12 14 16

P=√Q2 +U2 from thermal dust template [µK353GHz]

0.00

0.01

0.02

0.03

0.04

0.05

0.06

σ∆β

dust f

rom

fit

to h

isto

gra

m

Figure 4.11: Example of the correlation between the ∆βdust residual and the inten-sity of polarization. These residuals are obtained with the CORE analysis using theCOMMANDER code. On top, we show for each pixel inside the Galactic mask ofFigure 4.8, the spectral index residual as a function of the polarization intensity. Wedivide the polarization intensity into 6 bins, whose limits are shown as red dashedlines. In each bin, we calculate the standard deviation of the βdust error as a functionof the polarization intensity. This is shown in the bottom, along with a power lawfit, in green.


σ(r) error of 4.9× 10−4 for the 1-parameter likelihood. Fitting with the foregrounds

nuisance parameters is not well motivated, since the marginalized likelihoods for

Adust and Asyn are consistent with 0.

We have verified that the residual systematic error on r is due mostly to thermal

dust residual contamination (as opposed to synchrotron residuals). In fact, the 1-

parameter likelihood yields r = (13.1 ± 4.9) × 10−4, and r = (12.3 ± 4.9) × 10−4,

respectively, if we consider a 0.5 percent global error only on βsyn and βdust and we

leave the other foreground at 1 percent global error.

4.5 Conclusions

We have performed an analysis on the tensor-to-scalar ratio bias produced by the

mis-modelling of foreground spectral parameters, taking into account a realistic

model of the sky and a full data analysis pipeline.

We have considered two sky models: a very simple one, where the foregrounds

(synchrotron and dust) have spatially-constant spectral parameters across the sky,

and a more complex one, where the spectral dependence is spatially-varying, while

the Tdust parameter is constant. We modelled component separation strategies and

likelihood estimations of increasing complexity. The main results of our analysis

can be summarized as follows: For r = 0.01, we obtain an unbiased estimation of

r for all simulations considered. The requirements on the accuracy of foregrounds

modelling for component separation purposes are not too stringent (for example

modelling spatially-varying foreground spectral indices as spatially-constant still

gives a successful measurement).

Depending on the error level on the synchrotron and thermal dust spectral in-

dices (from 1 to 3 percent), the best results may require exploiting a limited set of

“cleaner” frequency maps to reconstruct the CMB. The use of all channels is always

recommended to obtain accurate estimates of the foreground spectral indices. We

achieve significant improvements by explicitly modelling the synchrotron and dust

foreground residuals in the likelihood, and marginalizing over foreground ampli-

tude nuisance parameters. Furthermore, an important role is played by the Galactic

mask, which needs to be optimized for the component separation method used and

for B-mode detection.

For r = 0.001 and a simple sky model, using a suitable mask, modelling fore-

ground residuals in the likelihood and limiting the frequency range for CMB recon-

struction always yields an unbiased r value. The error on r often does not allow

a detection but just an upper limit; this result is however conservative because the

4.5. CONCLUSIONS 111

Gaussian likelihood we adopted is not optimal in the low-multipole regime.

When increasing the complexity of the sky, large modelling errors, such as ap-

proximating a spatially-varying spectral index with a constant, should not be used,

as they give rise to biases on r that are too large to be corrected for (at the likelihood

level). In such cases, modelling the spatial variability of the spectral indices at the

component separation level is required. For a global error of 0.5 (1) percent in βdust

and βsyn, we obtain an unbiased detection (upper limit) on r. We show that the

foreground residuals biasing the measurement of r = 0.001 are due in greater pro-

portion to thermal dust emission than synchrotron emission, due to the particular

frequency coverage of CORE.

Such level of accuracy in the determination of foreground spectral parameters

is very challenging, and motivates further research on component separation and

foreground characterization. However, our analysis does not take into account po-

larization ancillary data that will become available, such as C-BASS (Irfan et al.,

2015). Our method could also be used to further optimize the instrumental specifi-

cations of future CMB B-modes experiments.

BLANK PAGE

112

Chapter 5

Forecasts on measuring r for the

Simons Observatory

In this chapter, I discuss my work related to the forecasting of the detection levels of

the tensor-to-scalar ratio for the Simons Observatory. In this case, we use a simple

strategy, known as Fisher matrix forecasting, because it uses the Fisher information

matrix to estimate the 1σ error of a parameter given some observable data. This

work corresponds to my contribution to the Simons Observatory collaboration.

5.1 The Simons Observatory

The Simons Observatory is a planned stage-3 CMB experiment that will observe

from the Chajnantor plateau in the Atacama desert in Chile. The collaboration is the

joint effort of two existing experiments that have been observing from the same site:

the Atacama Cosmology Telescope (Swetz et al., 2011) and POLARBEAR/Simons

array (Kermish et al., 2012), and includes more than 45 institutions from 8 countries.

The experiment is planned to begin observing around 2021, and will consist of an

array of microwave polarization-sensitive telescopes: a large aperture telescope of

∼ 6 m, and one or more small aperture telescopes of ∼ 0.5 m.

The science cases the experiment aims to address are the detection of primordial

gravitational waves with polarization B-modes, and to be able to better constrain

the neutrino sector. Also, the dark energy sector, CMB weak lensing and cluster

counts with the SZ effect, among others. It will be an essential stepping stone into

the ultimate ground-based CMB experiment, CMB-S4 (Abitbol et al., 2017).

At the moment, the collaboration is focused on evaluating the performance of

the planned instrumentation and detectors, together with forecasting the impact on

the cosmology that such an instrument could have. I have done some work for the

113

114 CHAPTER 5. FORECAST FOR SIMONS OBSERVATORY

measuring r working group, which is presented in the following sections.

5.2 Methodology

The pipeline to estimate the detection limits of the tensor-to-scalar ratio, that is, σ(r),

consist of several steps: the modelling of foreground spectral parameter errors, the

estimation of power spectrum residuals, and the estimation of the error on r. These

are detailed in Section 5.2.1, 5.2.2, and 5.2.3, respectively.

5.2.1 Modelling the foreground spectral parameters errors

We obtain an estimate of the residual error on the foreground spectral indices by

running the CCA algorithm (detailed in Section 2.2.1) on a set of SO simulated ob-

servations. We tessellate the full sky in small square patches, and therefore we

calculate the statistics using the results from the patches within the SO mask con-

sidered. We estimate an average residual error for each spectral parameter with the

standard deviation of the statistics. More details of the practical application of this

is described in Section 5.3.3.

5.2.2 Estimate of power spectrum residuals

We need a way to estimate the residuals left in the angular power spectra by the

foregrounds in the cleaned CMB once a component separation method has been

used over the multi-frequency maps. In a parametric component separation frame

work, this means estimating a reasonable error on the fitting of the spectral pa-

rameters of the components. This depends on several factors, such as frequency

coverage, ancillary data we could have on the components at lower and higher fre-

quencies, how good our models reflect the true components, etc. The error on the

foreground spectral parameters propagates to a residual in the CMB power spectra;

this error is added to the noise and cosmic variance to give a final error bar on the

CMB angular power spectra (Bonaldi and Ricciardi, 2011; Bonaldi et al., 2014).

We use the parametric linear mixture scheme as our component separation

method and we adopt the GLS solution. The estimated mixing matrix A is cal-

culated with the estimated parameters of the components considered, that is, with

the parameters including errors. Then, the estimated reconstruction matrix by the

GLS solution is W, calculated as shown in Section 2.2.1. The true and estimated

5.2. METHODOLOGY 115

signal are s and s, respectively. The residuals are given by

s− s = (WA− I)c, (5.1)

where A is the true mixing matrix (which uses the true spectral parameters of the

components being considered in the simulated sky), I is the identity matrix (with

dimensions Ncomponents × Ncomponents), and c are the template maps of the compo-

nents in the true simulated model (with dimensions Ncomponents × 1). There will be

a mismatch between the true mixing matrix A and the estimated mixing matrix A

(which enters in the computation of the estimated reconstruction matrix W). This

mismatch of estimated foregrounds spectral parameters will result in residuals in

the CMB cleaned map. In practice, for every pixel in the map, the row correspond-

ing to the CMB in the (WA − I) matrix is multiplied with the vector c, containing

the intensity of each component model template, to obtain T , Q and U maps of

foreground residuals on the CMB.

As an additional step, we need to compute the power spectra of the residual map

to get the contamination on theBB angular power spectrum. We use the pseudo-C`procedure over the residual map, using a mask. This is because the forecasts are be-

ing considered for ground-based experiments, with a partially observed sky away

from the Galactic plane. The pseudo-C` procedure is explained in Section 4.3.2.

Assuming that the noise is Gaussian and uniform over the sky, the angular

power spectrum of the instrumental noise for a given frequency channel ν is given

by

Nν =4πfsky

Npix

σ2νB

2ν , (5.2)

where fsky is the sky fraction observed, Npix is the total number of pixels in the map

(note that the standard deviation of the noise changes depending on the pixel size),

σν is the standard deviation of the white noise at the band ν, corresponding to the

pixel resolution chosen, and Bν is the beam window function, corresponding to the

beam size at band ν.

In the GLS linear mixture solution, the noise bias N on the CMB component is

given by

N =

Nbands∑ν

w2νNν , (5.3)

wherew2ν are the elements of the squared reconstruction matrix W2. Accounting for

sample variance, the uncertainty in the power spectrum measurement due to noise

is given by

∆Cnoise` =

√2/(2`b + 1)

fskynbN, (5.4)


where `b is the multipole in the middle of bin b, and nb is the number of multipoles

included within bin b.

The binned cosmic variance residual is calculated as

∆CCV` =

√2/(2`b + 1)

fskynbC`, (5.5)

where C` is the fiducial angular power spectra of the modelled CMB. This fiducial

spectrum contains both the primordial signal (e.g. the low-` primordial spectrum

that scales with r for B-modes) and signal due to lensing. This last issue is important

to note, since further ahead we will scale this lensing signal to simulate the impact

of delensing to some degree.

To estimate the foreground residual power spectrum, we use a Monte Carlo ap-

proach. We run NMC realizations of foreground errors. Based on a value of the

standard deviation error for the foreground spectral parameters, we generate NMC

realizations of errors for each parameter. Then, we have NMC realizations of the

mixing matrix A and of the reconstruction matrix W. This generates NMC resid-

ual map realizations, and NMC foreground residual angular power spectra. In the

last step, we average over the NMC realizations to calculate the mean foreground

residual power spectra ∆Cforegrounds` .

5.2.3 Fisher matrix information

This is a way of quantifying the amount of information a given observable random

variable has on a given parameter(s), which models the distribution from which the

observable random variable is drawn. In practice, we can determine the 1σ error on

cosmological parameters, propagating the residuals of the cosmological observable

(which is the CMB angular power spectra).

Given the errors on the observable value, i.e. the residuals on the angular power

spectra, ∆C`, the Fisher information matrix is defined as

Fij =m∑`=1

1

σ2`

∂2d`∂pi∂pj

, (5.6)

where ` cycles through the observables data points, pi are the parameters that model

the distribution of the observable d`. The inverse of the Fisher information matrix

corresponds to the covariance matrix of the set of parameters pi. Therefore, the

diagonal of the inverse of the Fisher matrix are the variances of each parameter pi.

In our particular case, σ` are the binned residual angular power spectra ∆CXX`

(with XX is either the EE or BB), m being equal to twice the number of bins (be-

cause we consider as observables both the EE and BB power spectra), d` being

5.3. σ(R) FORECAST RESULTS 117

Frequency [GHz] Number of detectors Sensitivity per detector Sensitivity20 883 284.5 9.5730 883 206.3 6.9440 883 207.1 6.9795 11867 226.9 2.08150 11867 253.9 2.33220 5933 697.2 9.05270 5933 1157.7 15.03

Table 5.1: Baseline configuration for the Simons Observatory Small Aperture Cam-era. The sensitivity units are µKCMB·arcmin.

the fiducial angular power spectra that depends on one or more cosmological pa-

rameters pi. The partial derivative of CXX` with respect to a given cosmological

parameter pi has been calculated numerically. We use the CAMB code to generate

angular power spectra at the fiducial value of pi, and also at a small step above

and below the fiducial value. We calculate with CAMB the fiducial power spectra at

r = 0 and at r = 0 + ∆r = 10−5, fixing all other cosmological parameters values at

fiducial values. The numerical derivative is given by

∆CXX`

∆r(r = 0) =

CXX` (r = 10−5)− CXX

` (r = 0)

10−5. (5.7)

Since we are working with multipole bins, the above equation is used with the

binned fiducial power spectra. Note that we are underestimating the r error since

we are fixing all other parameters at their fiducial values, and we are ignoring the

degeneracies between the parameters. However, this approximation is enough for

our purpose since the uncertainty on r dominates the error budget.

5.3 σ(r) forecast results

In this section, we detail the first σ(r) forecast that was performed for an early Si-

mons Observatory specification. This is outdated now, since the planned configura-

tion of the experiment has changed. This would correspond to observations by the

Small Aperture Camera in a∼ 1 m telescope, whose target is to do a survey as deep

as possible in a small fraction of the sky, targeting small/intermediate multipoles

to constrain the primordial B-mode signal.

5.3.1 Simons Observatory instrument specifications

We use the instrumental specifications for the experiment listed in Table 5.1. In

this configuration, the projected total number of detectors for the Small Aperture


Equatorial

0 1

Figure 5.1: Simons Observatory mask with fsky = 0.05. This mask is plotted inEquatorial coordinates, highlighting the southern Galactic hemisphere patch, andthe northern Galactic hemisphere small patch that is observable from the site inChile.

Camera instrument is ∼ 40, 000. The sensitivity for a given band scales down with

the number of detectors Ndet as σν = σ1ν/√Ndet, where σ1

ν is the sensitivity for a

single detector for band ν.

The simulated noise is Gaussian, and the frequency range includes a 20 GHz

channel which would help to constrain the synchrotron emission. This channel was

discarded in later configurations of the experiment, considering the poor spatial

resolution at this frequency of a 1 m telescope; the idea being to rely on ancillary

low-frequency data, such as C-BASS, for the same goal.

5.3.2 Simulated observations

The Simons Observatory simulated observations are created with the PYSM code

(Thorne et al., 2017), which is a sky simulations code alternative to the one pre-

sented in Chapter 3.

The baseline model includes polarized thermal dust and synchrotron emission,

with spatially variable spectral index maps. The template for the synchrotron po-

larization is the 23 GHz 9-year observations of WMAP. The spectral index βsyn map


corresponds to ‘model 4’ of Miville-Deschênes et al. (2008). The thermal dust polar-

ization template is the same as our model in Chapter 3, the Planck 353 GHz band.

The input CMB map is a realization with a fiducial spectra with r = 0 and the map

is created with the code TAYLENS (Næss and Louis, 2013) to include realistic lensing

signal.

The mask we use, representative of the area of the sky available from the SO site

in Chile, has a sky fraction fsky = 0.05. It is shown in Figure 5.1. The potential areas

of the sky to observe is a patch close to the south Galactic pole, and a small patch

close to the north Galactic pole that lies at the limit of the observable sky from the

site.

5.3.3 Forecasted results for the baseline SO specification

To estimate the foregrounds spectral parameters residual errors, we run the CCA

algorithm on the simulated observations and compare the results with the true in-

put parameters. In order to get a good statistical characterization of these errors,

we split the full sky in 3072 patches, with a size of 20 × 20 square degrees, with

centres coinciding with the pixels of a Nside = 16 pixellization. We run the CCA

algorithm to estimate the foreground spectral parameters in these patches. We cal-

culate statistics on the residual errors within the mask representing the observed

sky.

The limited range of frequencies of the Simons Observatory baseline specifi-

cations brings complications for the fit of the thermal dust spectral law. This is

because the peak of the Modified Black-Body (MBB) is controlled by the dust tem-

perature Tdust, which is only constrained by frequencies higher than a few hundred

GHz. As Figure 5.2 shows, the difference between a fiducial Modified Black-Body

and one with a shift of±2 K in the Tdust parameter is very small. For the Simons Ob-

servatory frequencies considered here (shown as vertical dashed lines), the differ-

ence in MBBs is less than ∼ 4%. This means that the Simons Observatory channels

do not cover high enough frequencies to constrain the Tdust parameter accurately.

In virtually all of the CCA runs on the simulated observations, the algorithm has

problems to converge to a result.

To deal with this limitation, we run CCA to fit parameters, with βsyn and βdust

as spatially variable parameters, but with Tdust to a fixed value across the sky. In

particular, we use the average true Tdust within the portion of the sky observable

from the Simons Observatory in Chile (referred to as the observable strip). Since we

have shown that the dust temperature is a parameter which is difficult to constrain


101 102 1030.0

0.5

1.0

1.5

2.0

2.5

Therm

al dust

MB

B

βdust =1.55, Tdust =23 K

±1 K

±2 K

101 102 103

Frequency (GHz)

0.90

0.95

1.00

1.05

Norm

aliz

ed T

herm

al dust

MM

B

Figure 5.2: Example Modified Black-Body spectral law for thermal dust. On top, theblack line shows a fiducial MBB with βdust = 1.53 and Tdust = 23 K. The shaded areasshow the MBB with different values of Tdust. The dashed vertical lines show the SObaseline bands. Bottom, the same MBB laws normalized by the fiducial MBB.

when there are high-frequency bands missing, the error in the uncertainty of this

parameter is transferred to the error in the correlated βdust parameter. In this way,

our estimate is pessimistic in the sense that we consider the error on the Tdust twice,

once due to the fact that we consider it constant even though it is not, and once due

to its correlation with βdust. There is a residual error in the temperature value, thus it

will shift the estimation of βdust by a corresponding amount, since the thermal dust

parameters are correlated with each other. We illustrate this in Figure 5.3. We show,

to the left, the true thermal dust SED in the modelled observations at the baseline

Simons Observatory channels. The grey points are the individual pixels within the

observable strip, and the blue stars are the mean dust SED within the observable

strip. To the right, we show the likelihood of the MBB fit to the mean dust SED

for fixed values of Tdust and variable values of βdust. This fit is performed with a

standard χ2 minimization, where the likelihood is given by exp(−χ2(Tdust)/2). Both

parameters of the thermal dust, βdust and Tdust, are anti-correlated when we try to fit

a MBB. For a given range of Tdust values, the MBB spectral law is a good fit because


50 100 150 200 250Frequency (GHz)

100

101

102

Therm

al dust

Spect

ral Energ

y D

istr

ibuti

on

10 20 30 40 50 60 70Tdust parameter (K)

0.0

0.2

0.4

0.6

0.8

1.0

Like

lihood o

f fit

to D

ust

SED

0.00

0.05

0.10

0.15

0.20

His

togra

m o

f tr

ue T

dust

Figure 5.3: Fit of a Modifidied Black-Body spectral law to the simulated thermaldust. Left: the spectral law of the simulated dust component. The grey points arethe spectral law of the individual pixels within the observable strip at the SO base-line frequencies. The blue stars are the average spectral law within the observablestrip. Right: posterior distribution of the fit to the average dust spectral law of aMBB, for given values of Tdust (βdust is fitted as the variable). The red line is thehistogram of the true Tdust values in the simulations within the observable band.

we compensate by fitting the value of βdust. In red, we show the histogram of the

true Tdust values (used in the simulated observations) within the observable strip.

For most of the pixels, any reasonable value of Tdust can be used in the fit because

the error will be injected into the βdust parameter. This justifies the choice of using

a constant Tdust in the CCA run, since the residual error in the parameter will be

included in the next step.

The results of the CCA run of the baseline configuration of the Simon Observa-

tory is shown in Figure 5.4. We calculate the residual error for the βsyn and βdust

spectral indices as the subtraction between the estimated (by CCA) index and the

true index in the model, in each small patch within the mask. To the left, the his-

togram of the βsyn residual. To the right, the histogram of the βdust residual. In this

case, there is a shift in the histogram, because of the Tdust-βdust degeneracy described


0.06 0.04 0.02 0.00 0.02 0.04 0.06Residual βsyn

0

5

10

15

20

25

30

Norm

aliz

ed f

requency

0.15 0.10 0.05 0.00 0.05 0.10 0.15Residual βdust

0

2

4

6

8

10

12

14

16

Figure 5.4: Histogram of the synchrotron (left) and thermal dust (right) spectralindices residuals for the SO baseline configuration. The residuals are calculatedwithin the mask, between the value estimated by CCA and the true value in thesimulated observations.

above. We estimate the standard deviation of the spectral index residuals from the

sample of CCA patches within the corresponding mask. These are σ(∆βsyn) = 0.012

and σ(∆βdust) = 0.126. These standard deviations have been computed from the

samples of estimates, discarding outliers > 3σ.

To estimate the foregrounds C` residuals, we generate NMC = 100 realizations

for residual errors in the spectral indices. This is with a Gaussian function with

the standard deviations computed previously. Since we do not have an empirical

estimate for the residual of the Tdust parameter, we use the correlation from the Tdust-

βdust degeneracy. We calculate it from the fit shown in Figure 5.3. We fit a 4th order

polynomial function to the correlation.

Tdust(βdust) = Aβ4dust +Bβ3

dust + Cβ2dust +Dβdust + E, (5.8)

which yields A = 226935.94, B = −1437199.23, C = 3412300.35, D = −3599935.52,

and E = 1423942.62. We then used the correlation to compute an error on Tdust for

each realization of the error on βdust. The estimated residual BB angular power


102 103

multipole `

10-7

10-6

10-5

10-4

10-3

10-2

10-1

`(`+

1)/

2πCBB

` (µK

2)

Fiducial power spectrum

Cosmic variance residuals

Noise residuals

Foregrounds residuals

Figure 5.5: CBB` residual power spectrum from foregrounds residuals, noise and

cosmic variance. The dashed blue line is the fiducial angular power spectrum,which is a pure lensed spectrum since the fiducial value of r is 0. The grey pointsare the individual binned power spectrum estimates from foregrounds residuals inthe Monte Carlo run. The blue circles show the average binned power spectrumfrom the residuals. The green stars are the binned noise bias power spectrum. Thered crosses are the binned cosmic variance power spectrum, calculated with thefiducial angular power spectrum.

spectrum is shown in Figure 5.5. The individual binned power spectrum from each

of the 100 Monte Carlo runs are shown as the grey points. The average foreground

residual power spectrum is shown as blue points. The error bars are the standard

deviation from the 100 Monte Carlo runs. The noise and cosmic variance angular

power spectrum are shown as the green stars and red crosses, respectively. We add

all three sources of residuals to make a full residual power spectrum. We use this

as the observable to estimate the error in the detection of cosmological parameters

in the forecasting. The Fisher information matrix estimate of the 1σ error limit is

σ(r = 0) = 2.1 × 10−3, which is consistent with independent estimates performed

in parallel within the Simons Observatory collaboration. The cosmic variance C`is roughly the same as the foregrounds residuals. In this forecast, we consider the


1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

estimated βdust

10

15

20

25

30

35

est

imate

d T

dust

Figure 5.6: Empirical correlation between the thermal dust parameters. The param-eters are fitted by CCA on several patches within the mask. The straight line is a fitto the Tdust(βdust) correlation.

fiducial theory power spectrum (dashed blue line in Figure 5.5) to be 100 percent

lensed. In an actual data analysis however, we would perform some delensing

procedure in the weak-gravitationally-lensed CMB map (e.g. Yu et al., 2017; Re-

mazeilles et al., 2017a; Sehgal et al., 2017). In an optimistic case for satellite ob-

servations, using external tracers of the LSS that lenses the CMB, ∼ 60 percent of

the lensing power spectrum signal could be removed (Challinor et al., 2018). This

would improve the σ(r) value, since the cosmic variance power spectrum is scaled

down.

5.3.4 SO baseline configuration + high-frequency bands

In this subsection, we consider the effect of adding extra high-frequency bands

to improve the constraint on the thermal dust temperature Tdust. We consider

two extra bands, at 320 and 500 GHz, on top of the baseline configuration. These

bands correspond to roughly band 7 and 8 of the Atacama Large Millimiter Array

(ALMA). We project the sensitivity for these two extra bands as the same one we use


for the 270 GHz channel from the baseline configuration. We generate simulated

observations with PYSM in these two extra bands following the same guidelines in

Section 5.3.2.

Running CCA on these simulated observations allows to fit for the Tdust param-

eter. We run on the baseline configuration plus two high-frequency bands sim-

ulated observations, using the same mask. The statistics of the residuals of the

foreground component parameters are σ(∆βsyn) = 0.015, σ(∆βdust) = 0.056, and

σ(∆Tdust) = 2.7 K. In this case, to calculate the residual foreground power spectrum,

we also need to correlate the generated error for the two thermal dust parameters.

We estimate the correlation between Tdust and βdust by plotting the values in each

patch, as estimated by CCA. This is shown in Figure 5.6. This correlation is fitted

by the linear relation

Tdust(βdust) = (−44.28)βdust + (92.81). (5.9)

The generation of random errors for the thermal dust parameters is done in the

same way as in Section 5.3.3, so as to capture the correlation.

The residual foreground, cosmic variance and noise BB power spectrum are

shown in Figure 5.7. In this case, the foreground residuals are much lower. This

is mainly because that the thermal dust component is better constrained and sepa-

rated. This is the conclusion of Hervías-Caimapo et al. (2017), shown in Chapter 4:

at the frequencies probed by current experiments, ∼ 100 − 200 GHz, reducing the

thermal dust contamination is crucial for the detectability of the tensor-to-scalar

ratio.

Running the Fisher matrix forecasting, we estimate σ(r = 0) = 1.34× 10−3. This

is certainly better than the performance of the baseline configuration alone. By com-

paring Figures 5.5 and 5.7, we note that the foreground residuals are much smaller

in the second case, while the cosmic variance is the same in both cases, consider-

ing the fully lensed signal. In the baseline configuration, the level of foregrounds

is higher, but similar, to the cosmic variance. The baseline + high-frequency con-

figuration on the other hand is cosmic variance dominated. We conclude that with

this improved configuration, we are reaching the limit of what can be done with

foregrounds. A delensing procedure would further improve the results for the de-

tection of the tensor-to-scalar ratio.


102 103

multipole `

10-7

10-6

10-5

10-4

10-3

10-2

10-1

`(`+

1)/

2πCBB

` (µK

2)

Fiducial power spectrum

Cosmic variance residuals

Noise residuals

Foregrounds residuals

Figure 5.7: CBB` residual power spectrum from foregrounds residuals, noise and

cosmic variance. This is analog to Figure 5.5, but for the baseline+high-frequencysimulated observations.

5.4 Optimizing the fraction of the sky for measuring r

In this section, we apply our forecast pipeline to test if the Simons Observatory

default mask is centred on the best portion of the sky to measure the tensor-to-scalar

ratio. An issue that will surface in future polarization CMB experiments, with very

high sensitivity, is the fact that there is a balance between how well foregrounds

can be measured versus how clean the sky patch to observe must be. If we wish

to remove the polarized foregrounds from the CMB with component separation,

we need to be able to measure the properties of the foregrounds with good enough

signal-to-noise. In order to do that, the foregrounds polarization intensity must be

as high as possible. At the same time, if the foregrounds are more intense, they

contaminate more and contribute to the residuals that leak into the cleaned CMB

map.

The approach we follow is to run a forecast for σ(r) across several masks in the

sky, in a declination strip coinciding with the observable sky strip from the SO site

5.4. OPTIMIZING THE FRACTION OF THE SKY FOR MEASURING R 127

Equatorial

0.000

0.015

0.030

0.045

0.060

0.075

0.090

0.105

σ(∆β

syn)

Equatorial

0.000

0.015

0.030

0.045

0.060

0.075

0.090

0.105

σ(∆β

dust)

Figure 5.8: Statistics of the foreground parameter residuals as estimated by CCA.At the top, σ(∆βsyn) and at the bottom, σ(∆βdust). Each point corresponds to adisc mask with fsky = 0.05. The background map is the corresponding template ofsynchrotron and thermal dust polarization intensity.


in Chile. This strip is defined within declinations −22 < δ < +68. We approx-

imate the actual SO mask with a disk-shaped one with the same 5% sky fraction

(which has a 25 radius). The centres of the masks are defined as the coordinates of

the pixels in aNside = 8 HEALPIX map. The number of disc masks that are contained

within the declination strip is 256.

First, we run the estimation of foreground spectral parameters assuming the

same baseline instrument configuration as in Section 5.3.1. We use CCA in patches

of sizes 20 × 20 degrees, whose centres are the pixels of a Nside = 16 map. We run

the estimation of parameters with CCA for the full sky. For each disc mask, we

estimate the statistics of the βsyn and βdust residuals by using the CCA patches that

intersect in area 90% or more with the corresponding disc mask. For each disc mask,

we calculate the statistics using ∼ 65 CCA patches. We use the same technique

described in Section 5.3.3: we calculate the statistics of the errors on the estimated

parameters and discard the patches that lie outside 3σ.

In Figure 5.8, we show the standard deviation of the residuals calculated for ei-

ther βsyn (top) or βdust (bottom). For reference, we show the polarized intensity P of

the corresponding template in the background. The values for the standard devi-

ation of the spectral indices residuals will depend on how intense the foregrounds

are in the corresponding disc mask.

The next step is to calculate the foreground residuals in the EE and BB power

spectra. To do this, we run 200 Monte Carlo samples of a GLS inversion using as

statistics the standard deviation values for β residuals, as described in Section 5.3.3.

We repeat this procedure for each disc mask in the declination strip. For each disc

mask, we also calculate the power spectra for the mixed noise bias and for the cos-

mic variance term with full lensing.

Figure 5.9 shows the main result from this exercise. In the background, we show

the polarized intensity of the thermal dust template at 353 GHz, since dust is the

dominant foreground component in the cleaning process. Each dot corresponds to

the value of σ(r), measured in the corresponding fsky = 5% disc mask, in logarith-

mic scale.

We can see in Figure 5.9 that trying to measure r in the Galactic plane results

in a very high σ(r) error. The centre of the Galaxy, towards the right of the map,

evidences σ(r) ∼ 0.1. The anti-centre of the Galaxy, towards the left of the map,

shows slightly smaller values. The best results are towards the Galactic South Pole,

towards the centre of the map. In this area of the sky, the foregrounds are weaker.

The estimation of the foregrounds spectral indices is not the best, especially for

βdust, but it is accurate enough to allow low foreground residuals power spectra.

5.5. σ(R) FORECASTS FOR SO-EBT 129

Equatorial

P = Q2 + U2 353 GHz thermal dust template

2.75

2.50

2.25

2.00

1.75

1.50

1.25

1.00

0.75

log 1

0((r)

)

Figure 5.9: Map of σ(r) estimated for several disc-shaped masks with fsky = 5%in the observable sky from the SO site in Chile. On the background, the map cor-responds to the polarized intensity of the thermal dust template. Each dot is thecenter of each mask.

Therefore, we conclude that the best portion of the sky to measure r is close to the

Galactic South Pole, where presumably the foregrounds intensity is the weakest.

Probably, the North Galactic Pole is just as good to measure r, but only a small

fraction of it is observable from the SO site in Chile. The best value is σ(r) = 0.0015,

with the disc mask located at α = 22.5, δ = −35.7 in equatorial coordinates, or

l = 261, b = −78 in Galactic coordinates. This patch of sky overlaps with the

original SO mask, shown in Figure 5.1. The difference is not significant, so we

support the choice for the SO mask.

5.5 σ(r) forecasts for SO-EBT

As part of Simons Observatory, I am involved in a proposal from the European in-

stitutions of the SO collaboration to build a new telescope, the Simons Observatory-

Europe B-mode Telescope (SO-EBT). This telescope aims to be a fundamental part of

the overall SO collaboration effort, by providing three Small Aperture Cameras

with similar sensitivities as the ones already considered in the SO project, which

essentially will double the detectors for the deep sky survey that aims to detect the


polarization B-modes. This project includes the design and construction of detec-

tors using state-of-the-art materials and technology. The proposal will add∼ 40, 000

detectors in three cameras, housed in one Small Aperture Telescope. Therefore, the

SO-EBT would contribute a new and independent telescope capable of doing a deep

survey to detect polarization B-modes. Since it will be part of the SO collaboration,

it will use the infrastructure and resources that will be developed for the already

funded SO effort.

As part of this proposal, I contributed to the tensor-to-scalar ratio forecasting

efforts. The addition of ∼ 40k detectors will eventually double the number of de-

tectors, and therefore the σ(r) error could improve significantly. By running the

forecast pipeline, as the one described in the previous sections, we can quantify the

improvement in the value of σ(r).

We use as a sky model the same model used in the previous runs, described in

Section 5.3.2. This is a simplified model, that contains thermal dust, synchrotron,

and CMB as polarized components, with spatially variable spectral parameters

maps. The basic model also includes Gaussian noise, as specified by the different

configurations and spread of detectors. Later, we will also consider more realistic

atmospheric noise.

We forecast the performance of different configurations by measuring σ(r) using

the forecasting pipeline described in Section 5.2, with two improvements described

in the following subsections: one to account for the pixel-to-pixel spatial variability

of the foreground spectral indices, and one to account for the systematic error for

the estimation of r.

5.5.1 Spatially-varying foreground spectral indices

A more realistic modelling of the estimated foreground spectral parameters would

capture their spatial variability, rather than assuming a constant spectral law across

the sky, as we did in Section 5.2.2. To do this, we perform the following procedure.

For a given foreground realization, we measure the statistics by running CCA

in the same way described above in Section 5.2.1. Then, an error for βsyn and βdust is

generated based on the measured standard deviations, and added to a Nside = 512

map of spatially-variable true spectral indices, described in the simulated observa-

tions, Section 5.3.2. For each pixel, an estimated reconstruction matrix W is calcu-

lated. Then, for the estimation of the residual map, we use Equation 5.1, but using

a different reconstruction matrix for each pixel. To save time, we estimate the resid-

ual map only on the pixels within the corresponding mask, rather than the full sky.


In the next step, the power spectra of the masked map of the residual is calculated.

Then, the pipeline is applied in the same way as described above.

5.5.2 Estimation of r bias

Previously, we calculated the σ(r) error due to random uncertainties, but we did not

consider the systematic uncertainties, which will cause a bias in the calculation of a

cosmological parameter. We are improving our analysis to account for both. If the

data analysis was naive, and assuming we would have complete ignorance on the

foreground residuals, we would mistake these as primordial cosmological signal

that would be measured as an effective r. To compute this value, we ran a likelihood

for r on the residual foreground power spectrum (as performed in Section 4.3.3).

We use the foreground residual power spectrum as the observed data points,

and a fiducial primordial power spectrum (without the lensing signal) as the theory

data points. To estimate the errors, we use the standard deviation from the Monte

Carlo realizations of foreground residuals. The BB power spectrum is used, at the

same multipole bins as used before.

5.5.3 Results

First, we consider a baseline configuration for Simons Observatory: a single Small

Aperture Telescope with three optical tubes, which will constrain the values for

sensitivities at each band. We consider a 2-year survey and an area of observation

of fsky = 0.05, with 6 frequency bands (30, 40, 95, 150, 220, and 270 GHz), and one

optic tube in each frequency range. The sensitivities are listed in the first row of

Table 5.2.

The SO-EBT proposal considers doubling the number of detectors, by adding

three optic tubes in another Small Aperture Telescope. There are several ways to

distribute these extra detectors. We consider 3 improved configurations: SO++ v1

adds all the detectors to the middle frequency range, which effectively is to put

all the extra sensitivity in the CMB-dominant channels. SO++ v2 adds one tube

in each frequency range, spreading the sensitivity over the synchrotron, CMB, and

thermal dust-dominated channels. SO++ v3 adds two tubes to the CMB-dominated

channels and one to the thermal dust-dominated channels. The sensitivities for

these 3 upgraded configurations are listed in the second, third , and fourth rows of

Table 5.2.

For the first run of forecasts, we consider the nominal SO mask with fsky = 0.05,

shown in Figure 5.1. We use the foreground model detailed above, and adding


Experiment Freq ranges No of tubes 30 GHz 40 GHz 95 GHz 150 GHz 220 GHz 270 GHz

SO baselineLF 1

14.98 8.82 3.25 3.83 5.66 13.98MF 1UHF 1

SO++ v1LF 1

14.98 8.82 1.62 1.91 5.66 13.98MF 4UHF 1

SO++ v2LF 2

10.59 6.24 2.29 2.71 4.00 9.89MF 2UHF 2

SO++ v3LF 1

14.98 8.82 1.87 2.21 4.00 9.89MF 3UHF 2

Table 5.2: Sensitivities per band for the SO configurations considered. The unit ofthe sensitivities is µKCMB·arcmin.

Experiment Foregrounds r bias σ(r) Alens = 1.0 σ(r) Alens = 0.5 σ(r) Alens = 0.25

SO baseline ∆βsyn = 0.058 3.57 2.91 2.25 1.91∆βdust = 0.084

SO++ v1 ∆βsyn = 0.059 2.40 2.54 1.87 1.54∆βdust = 0.072

SO++ v2 ∆βsyn = 0.043 2.64 2.55 1.89 1.55∆βdust = 0.061

SO++ v3 ∆βsyn = 0.052 2.90 2.50 1.84 1.50∆βdust = 0.062

Table 5.3: Forecasted bias and σ(r) for the baseline and improved SO configurations.The 1σ errors for the foreground spectral parameters are listed. The bias and σ(r)values are in units of 10−3. These results are for a Galactic mask with fsky = 0.05and Gaussian white noise.

Gaussian noise using the sensitivity values listed in Table 5.2. In Table 5.3, we tab-

ulate the results for this exercise.

We estimate the errors in the spectral indices of the foregrounds made by the

CCA algorithm with the baseline and improved sensitivities. Going from baseline

to SO++ v1, the improvement on the error is not significant, since the sensitivity on

the synchrotron and dust channels is the same. However, from baseline to SO++

v2 the error is improved significantly (∆βsyn goes from 0.058 to 0.043, and ∆βdust

goes from 0.084 to 0.061). In the case of the SO++ v3 improved configuration, the

synchrotron-dominated channels are not improved, but the dust-dominated chan-

nels are. Therefore, the ∆βdust error is significantly improved. The next columns of

Table 5.3 list the r bias and the σ(r) error for the detection.

With these numbers, the power spectrum levels of the foreground residuals are

comparable to the level of B-modes due to lensing. Therefore, there is some im-

provement, ∼ 20%, going from baseline to an improved configuration. This is with

the full lensing signal. We also include the values when we reduce the lensing

signal artificially by calculating the cosmic variance power spectra with a reduced


Equatorial

0 1

Figure 5.10: Simons Observatory mask with fsky = 0.1. This is the same morphologyas the mask shown in Figure 5.1. The southern and northern Galactic patches arehighlighted.

lensing signal, multiplying with a Alens factor, equal to 0.5 or 0.25. When we simu-

late delensing, the σ(r) error is significantly better. The bias values are of the same

order of magnitude as the σ(r) error. This corresponds to the most pessimistic case,

however, since we use the full foreground residuals being mistaken as primordial

B-mode signal, when in reality there are several ideas we can apply to mitigate the

impact of foreground residuals.

We consider a second run with a second SO mask, with more sky coverage,

fsky = 0.1. This will make the survey shallower, but would increase the signal-

to-noise ratio at the large-scale multipoles. We again consider a two-year survey,

which means that the sensitivities are the ones listed in Table 5.2, but multiplied by√

2, since the integration time at each line of sight is half the original time. The new

sky mask is shown in Figure 5.10.

The results are tabulated in Table 5.4. In this case, since the sensitivity is worse,

the residual error on the foreground spectral parameters is larger than in the pre-

vious case. The performance of the baseline configuration is considerable worse

for the synchrotron error, ∆βsyn = 0.14. For the dust, the performance is about

the same. When upgrading to a SO++ configuration, the dust residual error is im-

proved in a similar way as in the previous case with fsky = 0.05, and in the case




SO++ v1 ∆βsyn = 0.137 3.90 2.71 2.20 1.94∆βdust = 0.073

SO++ v2 ∆βsyn = 0.098 4.36 2.77 2.27 2.00∆βdust = 0.060

SO++ v3 ∆βsyn = 0.128 4.35 2.77 2.26 1.99∆βdust = 0.062

Table 5.4: Forecasted bias and σ(r) for the baseline and improved SO configurations.The details are the same as in Table 5.3, for a Galactic mask with fsky = 0.1 andGaussian white noise.

of synchrotron the performance is improved to some degree in SO++ v2 and v3,

where the sensitivity is improved in the low-frequency channels. In the SO++ v1

configuration, the synchrotron residual error is not really improved, by the lack of

extra sensitivity at the appropriate bands. It must be stressed that the thermal dust

emission is dominant above∼ 70 GHz in polarization, but synchrotron is only dom-

inant at low frequencies, dropping in polarized intensity very rapidly according to

a power law. Therefore, in essence, we only have two channels to constrain the syn-

chrotron properties (30 and 40 GHz), and these channels are close together, which

makes it even more difficult. This explains why the values of the ∆βsyn residual

error seem to be fluctuating.

The σ(r) values are overall worse than the previous run considered with fsky =

0.05, due to having worse sensitivity. With full lensing signal, the improvement

is only ∼ 20%. With a delensing of Alens = 0.25, the improvement from baseline

to SO++ increases slightly to ∼ 25%. The r bias is about twice that the one mea-

sured in the first run. This is because the residual errors on the foreground spectral

parameters are large. The bias is directly proportional to the foreground residuals

power spectrum.

We consider a third run, where we add realism to the forecast of r by considering

atmospheric pink noise, rather than Gaussian noise. Even though a ground-based

experiment would observe in the transmission windows at millimetre wavelengths,

there are several factors that should be accounted for, for example the oxygen and

water vapour in the atmosphere, which varies as a function of time and line-of-

sight. Other molecules also contribute to an overall emission of the atmosphere.

The incident photons on the detectors are also increased. The overall effect is fre-

quency and spatial power spectra correlation (Errard et al., 2015).

We model the atmospheric noise by modifying the spectrum from equation 5.2




SO++ v1 ∆βsyn = 0.104 3.14 2.54 1.88 1.54∆βdust = 0.082

SO++ v2 ∆βsyn = 0.053 2.83 2.50 1.84 1.50∆βdust = 0.062

SO++ v3 ∆βsyn = 0.087 3.43 2.79 2.12 1.78∆βdust = 0.077

Table 5.5: Forecasted bias and σ(r) for the baseline and SO++ configurations. Thedetails are the same as in Table 5.3, for a Galactic mask with fsky = 0.05, but theinput model contains more realistic atmospheric noise.

to

Nν =4πfsky

Npix

σ2ν

(1 +

`

`knee

)nknee

, (5.10)

where `knee = 60 and nknee = −2.6, and where σν corresponds to the sensitivities

per band listed in Table 5.2. After this, we create a correlated noise map at each

band by running the synfast function from HEALPIX with the 1/f noise power spec-

trum, which will be spatially correlated. This noise map is added to the simulated

foregrounds and CMB maps.

However, we run the rest of the pipeline in the same way as before, including

the use of the mask with fsky = 0.05. We are aiming to measure the improvement

on the measurement of r by going from the baseline to the SO++ configurations. In

the CCA code, the noise is assumed to be Gaussian in the input, even though we

are feeding it a more complicated noise.

The results are tabulated in Table 5.5. The baseline configuration performs

poorly in terms of the thermal dust residual error, which is ∆βdust = 0.13, the worst

performance by any of the runs. When going to an improved SO++ run, the dust

residual error is noticeable better, ∆βdust ∼ 0.07. This translates into the σ(r) error.

With full lensing, the improvement is ∼ 25% from baseline to SO++ v2. With a

Alens = 0.5 delensing, the improvement is ∼ 30%, and with a Alens = 0.25 delensing,

the improvement is ∼ 35%. The bias is worse than the first case considered with

Gaussian noise. In this case, we can attribute this to the fact that we are assuming

a simplified noise in the component separation. The level of noise is comparable

to the first case (even though the noise is correlated), which means that the SO++

improved σ(r) errors are similar. Therefore, in terms of absolute σ(r) value, the best

values are obtained by improving the depth of the survey.

The SO-EBT proposal will have a significant impact on the overall SO effort.

It will double the number of detectors, and it can improve the σ(r) uncertainty

upwards of ∼ 20%. This can potentially have a major impact in a deep survey


searching for primordial B-modes.

Chapter 6

CORE study on foregrounds

This chapter discusses my contribution to the proposal for the future CMB polar-

ization satellite CORE, submitted in response to the “M5” call for experiments from

the European Space Agency (ESA). Several articles were prepared to support this

proposal, on different aspects (the “Exploring Cosmic Origins” series, Delabrouille

et al., 2018). Among these, the potential improvements on cosmological parameters

(Di Valentino et al., 2018), on inflation (Finelli et al., 2018), or the mitigation of sys-

tematics (Natoli et al., 2018). My contribution was to the analysis of the impact of

foregrounds on the potential detection of a primordial B-mode signal, published in

Remazeilles et al. (2018). This work is described in this chapter.

6.1 The CORE proposal

The proposed experiment is a space-based telescope, with an array of 2100 cooled,

polarization-sensitive detectors, at the focus of a dish with an aperture of 1.2 m.

The 19 frequency bands proposed are between 60 and 600 GHz, with resolutions

between 18 and 2 arcmin. The aggregate sensitivity of the array in polarization is

1.7µK × arcmin. The experiment is designed to be mounted on a dedicated space

satellite, to observe for a period of 4 years from the L2 Lagrange orbital point, spin-

ning one revolution every 2 minutes with a precession of ∼ 4 days. This strategy

allows observing half of the sky every few days. The broad frequency coverage

ensures that the foregrounds are accurately mapped below and above the CMB

frequencies. The observation of ancillary data at lower frequencies (e.g. C-BASS)

could however be helpful to improve the mapping of synchrotron radiation.

137

138 CHAPTER 6. CORE STUDY ON FOREGROUNDS

Frequency [GHz] 60 70 80 90 100 115 130 145 160 175Beam FWHM [arcmin] 17.9 15.4 13.5 12.1 10.9 9.6 8.5 7.7 7.0 6.5Sensitivity [µKCMB × arcmin] 10.6 10.0 9.6 7.3 7.1 7.0 5.5 5.1 5.2 5.1Frequency 195 220 255 295 340 390 450 520 600Beam FWHM 5.8 5.2 4.6 4.0 4.0 4.0 4.0 4.0 4.0Sensitivity 4.9 5.4 7.9 10.5 15.7 31.1 64.9 164.8 506.7

Table 6.1: Instrumental specifications used in the CORE proposal to simulate full-sky observations. The beam of the higher frequency bands are limited to 4 arcminbecause of the pixel size of the HEALPIX pixelization.

6.2 Results from the CORE foreground study

The objective of the analysis presented in Remazeilles et al. (2018) is to demon-

strate that the proposed instrument can achieve foreground cleaning to the level

necessary to detect a tensor-to-scalar ratio of r = 10−2 − 10−3. This has been done

by performing a full end-to-end data reduction analysis on simulated observations,

including component separation, angular power spectra estimation, and cosmolog-

ical parameter estimation.

6.2.1 Simulated observations

The simulated observations maps are produced with the Planck Sky Model (PSM,

Delabrouille et al., 2013). The instrumental configuration used is shown in Table 6.1.

The maps are created with a resolution Nside = 2048, so we limited the resolution of

the higher frequency bands to 4 arcmin. The emission components included in this

model are CMB, synchrotron, thermal dust, AME and extragalactic point sources,

along with instrumental Gaussian noise.

The CMB is generated from a set of fiducial spectra generated with the CAMB

code, with several values of r in the range r = 10−2 − 10−3 and the optical depth to

reionization, τ = 0.055, based on the recent estimate by Planck Collaboration et al.

(2016o). The input model also included the effects of lensing which are modelled

using a Alens amplitude parameter. Alens = 0 means that only the primordial signal

is included, while Alens = 1 adds the full lensing signal. In practice, an intermediate

Alens value can mimic the effect of subtracting some of the lensing signal using a

delensing procedure.

The synchrotron radiation template is derived from 23 GHz WMAP polariza-

tion observations (Miville-Deschênes et al., 2008). The spectral law we assume is

the usual power law with spectral indices βsyn ∼ −3. The spatial map of the syn-

chrotron spectral index is derived by Miville-Deschênes et al. (2008) extrapolating

6.2. RESULTS FROM THE CORE FOREGROUND STUDY 139

Simulation number r value Foregrounds Lensing1 10−2 Only dust and synchrotron Alens = 0.02 5× 10−2 Only dust and synchrotron Alens = 0.03 10−3 Only dust and synchrotron Alens = 0.04 10−3 Dust, synchrotron, AME, PS Alens = 1.05 10−3 Dust, synchrotron, AME, PS Alens = 0.4

Table 6.2: Summary of simulations considered for the reconstruction of B-modes.

between the Haslam et al. (1982) 408 MHz map and the 23 GHz WMAP observa-

tions.

The thermal dust radiation template is from the 2015 data release of Planck at

353 GHz, using the GNILC algorithm to separate the Cosmic Infrared Background

(CIB) from the thermal dust (Planck Collaboration et al., 2016p). The spectral law

assumed is the usual modified black-body, with spatial maps of spectral parameters

derived also by Planck Collaboration et al. (2016p).

The AME radiation template is built using the thermal dust template from

Planck Collaboration et al. (2016p), where the intensity is scaled down to 23 GHz,

and the polarization angles are assumed to be the same. The spectral law is a theo-

retical spinning dust model, from Ali-Haïmoud et al. (2009).

Both radio and IR point sources are included. The radio sources included in the

PSM are based on surveys at 4.85, 1.4 and 0.843 GHz. The spectral law depends on

the classification of the sources, with either a flat or steep power law. The dusty

point sources are taken from the IRAS catalogue (Beichman et al., 1988). The spec-

tral law assumed is a modified black-body.

The noise is white and uniform at each band, using the sensitivities from Ta-

ble 6.1.

6.2.2 Results: Component separation and power spectra estima-

tion

We consider 3 different component separation methods to clean the CMB from fore-

ground contamination and we combine them using a hybrid approach to estimate

the angular power spectra. At low multipoles, we apply the COMMANDER code,

described in Section 2.2.1. This algorithm works on pixel space, on low resolution

maps (Nside = 16). We used a procedure to generate the low-resolution maps data

set, in order to avoid introducing spurious contamination. The issue we are trying

to avoid is to introduce a bias in the spectral law of the components (see e.g. Chluba

et al., 2017). In fact, in a Nside = 2048 pixel of the modelled map, the synchrotron


Figure 6.1: EE power spectrum reconstruction by component separation methods.The COMMANDER, NILC and SMICA estimates of the power spectrum are shownat the top left, top right and bottom, respectively. These results are particular forsimulation 1, but there is no difference for other simulations in the reconstruction oftheEE spectrum. On the bottom of each panel, the residuals between the estimatedand fiducial spectrum at each bin is shown, in number of σ. The bins that are biasedby more than ±3σ are shown as arrows.

will be a perfect power law in any given line of sight, by definition. If we degrade

the resolution of several adjacent pixels into a single Nside = 16 pixel, the resulting

spectral law of that pixel will no longer be a perfect power law. This is because

we are averaging temperatures, not power law indices 1. An effective curvature for

synchrotron is introduced if we degrade the temperature of the components. This

is true for all the other components spectral laws. To avoid this, we degrade in res-

olution the temperature of the templates of each component, and we also degrade

1If we have two pixels whose synchrotron temperature is modelled with power-laws given byA1ν

−β1 and A2ν−β2 , then the average temperature is 1

2 (A1ν−β1 + A2ν

−β2), which is different thanA1+A2

2 ν−12 (β1+β2)


the maps of the spectral parameters, for example, the map of βsyn. Only after this,

we extrapolate in frequency using the spectral laws of the components.

The analysis on the full resolution maps has been performed with two compo-

nent separation methods, that work on harmonic space: NILC and SMICA, both

described in Section 2.2.2. Both of these methods are blind, exploiting the indepen-

dence between components. The EE and BB power spectra are estimated between

multipoles ` = 48−349 for NILC and between ` = 48−600 for SMICA. Each method

estimates the power spectra using its own procedure.

Figure 6.1 shows the reconstruction of theEE power spectrum for COMMANDER

at low-` (top left), NILC (top right) and SMICA (bottom) for high-`. The fiducial

angular power spectrum is shown in black. The bottom panel of each plot shows

the residuals (in number of σ) at each multipole bin, between the reconstructed

and fiducial spectrum. The residuals that are biased by more than ±3σ are plotted

as arrows. The COMMANDER result is unbiased. The high-` pipelines show large

scatter among bins, but in general within ±2σ of the fiducial spectrum. However,

the SMICA result shows a systematic bias for some of the bins ` > 500.

For the BB angular power spectrum, to measure the tensor-to-scalar ratio, we

consider several cases, summarized in Table 6.2. First, we consider the simplest

model, where the only foreground components are thermal dust and synchrotron.

The CMB realization has no lensing, and three values for the tensor-to-scalar ratio:

realistic values we can approach in the next∼ 5 years, r = 10−2 and 5×10−2, and r =

10−3, which is the challenging value that CORE aims to measure in 10 to 15 years.

These simulations are referred to as 1, 2, and 3, respectively. Then, we consider a

more realistic simulation, including AME and extragalactic point sources, together

with Galactic dust and synchrotron. Simulation 4 includes the full lensing signal,

with Alens = 1.0, and simulation 5 has Alens = 0.4, which simulates the effect of

applying delensing to the reconstructed CMB, removing 60 percent of the lensing

signal from the reconstructed CMB map with a delensing technique. This level of

delensing is approximately what CORE could potentially achieve (Challinor et al.,

2018).

The measured BB angular power spectra for simulations 1 and 3 are shown in

Figure 6.2. The top, middle, and bottom row are the results from COMMANDER,

NILC, and SMICA, respectively. The left column corresponds to simulation 1, and

the right column corresponds to simulation 3. For simulation 1, r = 0.01 allows a

good reconstruction of the power spectrum. SMICA is more sensitive to the effect

of foreground residuals, so there is a small bias. For simulation 3, it is more diffi-

cult. The value r = 0.001 means that the primordial B-mode signal is smaller, so


Figure 6.2: BB power spectrum estimations for simulations 1 and 3, at the left andright column, respectively. The COMMANDER, NILC and SMICA estimates of thepower spectrum are shown at the top, middle and bottom rows, respectively. In theNILC spectrum, the foreground residuals are also shown, thermal dust in magentaand synchrotron in green.


Figure 6.3: BB power spectrum estimations for simulation 4. The COMMANDER,NILC and SMICA estimates of the power spectrum are shown at the top left, topright and bottom, respectively. In the NILC spectrum, the foreground residuals arealso shown, thermal dust in magenta, synchrotron in green and point sources onyellow.

the foreground contamination is much more significant. This is illustrated in the

NILC power spectrum reconstruction. The dust (magenta) and synchrotron (green)

residual power spectrum are also shown. These are calculated by running NILC on

simulated maps with only the respective foreground present and noise. This shows

that the NILC and SMICA reconstructed signal are contaminated by foreground

residuals.

Figure 6.3 shows the BB power spectrum reconstruction for simulation 4 for

COMMANDER (upper left), NILC (upper right) and SMICA (bottom). The total

fiducial spectrum, including lensing, is shown as the black solid line. The pri-

mordial signal, without the lensing, is shown as the dashed line. In the case of

NILC and SMICA, the foreground residuals are clearly evident, as the bias in the

spectrum at low-`. In the NILC panel, we also show the power spectrum of the


foreground residuals, for thermal dust (magenta), synchrotron (green) and point

sources (yellow). At this level, the AME radiation is accounted for together with

the synchrotron radiation, because it has similar spectral properties. The impact of

AME is not significant in the foreground residuals.

6.2.3 Results: Cosmological parameters posterior probabilities

Using the power spectrum estimation from the previous section, we run a cosmo-

logical parameter likelihood to estimate the posterior probability for the tensor-to-

scalar ratio (using theBB power spectrum) and for the optical depth to reionization

τ (using the EE power spectrum). The cosmological parameter estimation was my

main contribution to the collaborative work presented in Remazeilles et al. (2018).

We adopt a hybrid approach (e.g. Dunkley et al., 2009; Planck Collaboration et al.,

2016m), using the COMMANDER internal Blackwell-Rao likelihood (for multipoles

` ≤ 47) together with a standard χ2 Gaussian likelihood for either NILC or SMICA

(for multipoles ` > 48). The χ2 likelihood formalism is the same to what is de-

scribed in Section 4.3.3. Each algorithm provides its own method to estimate the

covariance matrix to use in the likelihood (for specific details on the methods and

how they calculate covariance, see Appendix A.2 of Remazeilles et al., 2018). The

EE fiducial spectrum we use in the Gaussian likelihood is

CEE` (τ) =

exp(−2τ)

exp(−2× 0.05)CEE` (τ = 0.05). (6.1)

The BB fiducial spectrum used in the Gaussian likelihood is

CBB` (r) =

r

0.01CBB,prim` (r = 0.01) + AlensC

BB,lens` , (6.2)

where CBB,prim` is the primordial (scalar and tensor perturbations) fiducial spec-

trum, CBB,lens` is the fiducial lensing spectrum, and Alens can be varied depending

on the specific simulation. Both components of the total fiducial spectrum are pro-

vided by CAMB. The joint likelihood for either COMMANDER+NILC or COMMAN-

DER+SMICA is the multiplication of the low-` Blackwell-Rao posterior with the

high-` Gaussian posterior. We must note that the likelihood in the BB spectrum

for r is dominated by the COMMANDER part. This is because the lensing signal

dominates the B-mode spectrum at the higher multipoles, so there is limited con-

straining power on r. The key is to constrain the reionization bump, at scales ` . 10,

where the primordial B-mode signal dominates.

The posterior for τ , calculated with the χ2 likelihood, is shown in Figure 6.4.

The measured values are very close to the fiducial value of τ = 0.055. For NILC,


Figure 6.4: Posterior probability for τ , calculated from the spectra shown in Fig-ure 6.1. The NILC result is shown to the left, and the SMICA result is shown to theright. The fiducial value τ = 0.055 is shown as the vertical dashed line.

shown to the left of the figure, we measure τ = 0.054 ± 0.002 in the multipole

range 20 ≤ ` ≤ 359. For SMICA, shown to the right of the figure, we measure

τ = 0.056± 0.008 in the multipole range 20 ≤ ` ≤ 199.

The results for the joint likelihood for the tensor-to-scalar ratio using the esti-

mated B-mode power spectrum for simulations 1 and 3 are shown in Figure 6.5.

The estimated values for r, along with the 1σ(r) error, are tabulated in Table 6.3.

The r posterior probability for simulation 1 for the joint COMMANDER-NILC and

COMMANDER-SMICA are shown to the top left and top right, respectively, of Fig-

ure 6.5. The fiducial value of r = 0.01 is measured in both cases, to within less than

1σ(r). The results for simulation 2, with a fiducial value r = 0.005, are not shown,

but they are qualitatively similar. For simulation 3, the bottom left and right of the

figure corresponds to the joint COMMANDER-NILC and COMMANDER-SMICA like-

lihoods, respectively. The r posterior is shown as a dashed line. Since the fiducial

value of r = 0.001 is challenging, the foreground residuals are comparable to the

primordial cosmological B-mode signal, and therefore the posterior is biased.

We modelled the foregrounds residuals in the likelihood in terms of nuisance

parameters, as described in Section 4.3.3. The thermal dust residuals have the most

impact on the overall foreground residuals, and they are more important at low

multipoles. In Figure 6.6, the BB power spectrum of the thermal dust, as recon-

structed by COMMANDER, is shown in orange, for simulation 3. In blue, the fore-

ground residual power spectrum is shown, multiplied by 107 to compare them at

the same scale. This is calculated by subtracting the input true CMB map with the


Figure 6.5: Posterior probability for r, calculated using the joint likelihood betweenCOMMANDER and either NILC or SMICA. The top row corresponds to simulation1, the bottom row is simulation 3. The left column are the results for the joint COM-MANDER-NILC likelihood, and the right column for the one with SMICA. In thecase of simulation 3, we also plot a second posterior where the COMMANDER like-lihood also fits for nuisance parameter AFG for the dust residuals. The fiducial rvalue is marked with the vertical dashed line.


101

multipole

102

103

(+

1)/2

CBB

[K2 CM

B]Power-law fit to thermal dustForegrounds residuals ×107

Reconstructed thermal dust

Figure 6.6: Example low-` BB power spectrum, for simulation 3, of the recon-structed thermal dust by COMMANDER, shown in orange. To compare, we alsocalculated the power spectrum of the foreground residuals, in blue, calculated bysubtracting the input true from the reconstructed CMB map. The residuals are mul-tiplied by 107 to compare them at the same scale. A power-law, fitted to the recon-structed dust, is shown as the dashed line.

reconstructed CMB map, and estimating the power spectrum using a QML esti-

mator (see Section 4.3.2). We fit a power law to the thermal dust power spectrum.

Then, in the COMMANDER r likelihood, this power law is scaled by the nuisance

parameter AFG, which is also fitted to model-out the residuals. The joint likelihood

using the COMMANDER posterior fitting for foregrounds is shown in the bottom

row panels of Figure 6.5 as the solid lines. The values are tabulated in Table 6.3 in

parenthesis. In the case of simulation 3, we are able to remove the bias on r.

The posterior probabilities for r for simulations 4 and 5 are shown in Figure 6.7.

For simulation 4, we are able to measure r with a bias, in the top row of the figure.

In this model, there is full lensing signal. In the likelihood, we assume we know the

fiducial lensing power spectrum, so it is included in the fiducial spectrum. Fitting

for the parameter AFG, the bias on the posterior is slightly reduced. The peak of

the posterior is within the 1σ(r) error. The results from the COMMANDER-NILC


Figure 6.7: Posterior probability for r, calculated using the joint likelihood betweenCOMMANDER and either NILC or SMICA. The top row corresponds to simulation4, the bottom row is simulation 5. The left column are the results for the jointCOMMANDER-NILC likelihood, and the right column for the one with SMICA. Thedashed and solid lines correspond to the fit of r, or to the fit of r and AFG, respec-tively, in the same way plotted in Figure 6.5.


Simulation 1`min `max r [10−3] σ(r) [10−3]

COMMANDER+NILC 2 349 10.0 0.9

COMMANDER+SMICA 2 600 10.6 0.7

Simulation 2`min `max r [10−3] σ(r) 10−3

COMMANDER+NILC 2 349 4.7 0.4

COMMANDER+SMICA 2 600 4.9 0.4


COMMANDER+NILC 2 349 1.3(1.0) 0.1(0.1)

COMMANDER+SMICA 2 600 1.3(1.0) 0.1(0.1)


COMMANDER+NILC 2 349 1.4(1.2) 0.5(0.6)

COMMANDER+SMICA 2 600 1.4(1.2) 0.5(0.6)


COMMANDER+NILC 2 349 1.6(1.1) 0.5(0.5)

COMMANDER+SMICA 2 600 1.7(1.2) 0.4(0.5)

Table 6.3: Measured tensor-to-scalar ratio values from the posterior probabilities forall 5 simulations. We list the multipole ranges used (COMMANDER corresponds to` = 2 − 47 and NILC/SMICA corresponds to ` > 47), the measured value and theerror.

and COMMANDER-SMICA joint likelihoods are virtually the same. For simulation

5, we simulate a process of delensing by removing 60% of the lensing BB power

spectrum in the input simulation. The likelihood is then run with Alens = 0.4 in

the fiducial spectrum. In theory, this should help with the foreground residuals

and bias, and also improve the σ(r) error, since we are effectively removing cosmic

variance. The resulting posterior probabilities for r are shown in the bottom row

of Figure 6.7. The two joint likelihoods give very similar results. Fitting the AFG

parameter is effective in removing the bias on r. However, the σ(r) is not improved,

therefore making the posterior only a 2σ detection of r.

All the measured tensor-to-scalar ratio values, along with their respective σ(r)

errors, are tabulated in Table 6.3 for simulations 1 through 5.


6.3 Conclusions

We have measured cosmological parameters, r and τ , in simulated polarization ob-

servations of the CMB by the proposed CORE experiment, as described in Table 6.1.

We use a realistic model for the full-sky observations, including foregrounds such as

thermal dust, synchrotron, AME and extra Galactic point sources. We also include

Gaussian uniform noise, and a CMB realization with only primordial cosmological

signal, or also including the dominant lensed signal, that can be scaled with a Alens

parameter.

The E-mode spectrum is reconstructed with a high signal-to-noise ratio. The

NILC and SMICA methods can reconstruct the spectrum and measure τ with a

significance of 24 and 6σ, respectively. The SMICA power spectrum is affected by

bias at high multipoles, which are excluded in the calculation of the τ posterior.

Further work is needed to improve these results.

For the B-mode results, we find that r can be measured without bias for values

r & 0.005. This is addressed by our simulations 1 and 2, with values r = 0.01 and

r = 0.005, respectively. In the absence of lensing, r is measured to a significance of

∼ 10σ, as listed in Table 6.3. Separate tests, made with COMMANDER, indicate that

r can still be measured without bias in the presence of lensing. However, since the

cosmic variance is significantly larger when the dominant lensing signal is present,

the detection drops to a significance of ∼ 4σ.

For r = 0.001, the target that CORE is trying to measure, the analysis is more

difficult. All the considered component separation methods struggle to achieve an

unbiased detection of r. In simulation 3, with a fiducial value r = 0.001, r is biased

by 3σ. In this case, there is only primordial signal in theBB power spectrum, which

makes σ(r) ∼ 10−4. This result can be improved if we model for the foreground

residuals in the likelihood, by means of a nuisance parameter AFG that scales the

reconstructed thermal dust (the dominant source of residuals), in a similar manner

to what is described in Section 4.3.3. Once we apply this procedure, we obtain

an unbiased estimate of r with a ∼ 10σ significance. However, the caveat is that

simulation 3 is lacking the lensing signal.

For realistic scenarios, that include the lensing signal at a 100% and at a 40%

level (simulation 4 and 5, respectively), we measure r with a ∼ 1σ bias. In both

cases, the σ(r) error is∼ 5×10−4. We note that reducing the lensing by 60% does not

improve the results. This probably occurs because the lensing is not the dominant

component of the residuals. By removing 60% of the lensing BB power spectrum,

we effectively reduce a large fraction of the cosmic variance uncertainty, but the

6.3. CONCLUSIONS 151

foreground residuals are still larger and they are the dominant component in the

uncertainty. The joint likelihood for r is dominated by the COMMANDER likelihood

at low multipoles. For the NILC/SMICA-only likelihood, for example, the effect

of delensing is more noticeable. Since at multipoles ` & 100, the signal is virtually

all coming from lensing, removing a fraction of it will improve σ(r), regarding of

the bias. A more extended discussion of this issue is presented in Section 6.1 of

Remazeilles et al. (2018).

In summary, with the proposed CORE instrument, we predict an unbiased de-

tection for r & 0.005. When we try the challenging value r = 0.001, the dominant

source of error is foreground mis-modelling, which makes the naive analysis bi-

ased. We can model-out the foreground residuals at the likelihood level, and we

can remove the bias to a certain degree, but further research is needed in the com-

ponent separation techniques and in the modelling of foregrounds to improve the

detection error σ(r).

BLANK PAGE

152

Chapter 7

Conclusions

In this thesis, the main issue we have investigated is the impact of Galactic diffuse

foregrounds on the observations of the polarization of the Cosmic Microwave Back-

ground, which entails the measurement of cosmological parameters. The problem

we have tried to address is the contamination by residuals after cleaning said fore-

ground emission. We know that these residuals could be mistaken for polarized

primordial B-mode signal if not properly characterized when we try to measure

cosmological parameters. The work presented in this thesis is a step forward in the

direction of characterizing these residuals and search for ways to limit their impact.

7.1 Summary and Conclusions

7.1.1 A new model of the polarized microwave sky

In order to study the issue of the residual diffuse foreground contamination in the

primordial B-mode signal in a flexible way, we have constructed our own polarized

sky model, which we use to simulate the observation of the full sky at millimetre

wavelengths by a CMB experiment. This model includes the common Galactic dif-

fuse polarized foregrounds: synchrotron, thermal dust and anomalous microwave

emission, as well as CMB realizations. Our model can also simulate the effect of

an observing instrument in a simple way, as Gaussian noise (which can be spatially

constant or pixel-by-pixel), Gaussian beams, and frequency bandpasses.

This model is presented in a very simple code written in PYTHON. The novel as-

pects include the creation of small-scale features for each of the foregrounds compo-

nents. The templates that are the basis of the model are from the 2015 data release of

Planck, so their small-scale signal is dominated by noise. This procedure is achieved

153

154 CHAPTER 7. CONCLUSIONS

by extrapolating the power spectrum of the corresponding template to higher mul-

tipoles. The fact that the foreground emission is concentrated in the Galactic plane

is accounted for by weighting each pixel according to the polarized intensity in the

template. We create the small-scale features of the AME and thermal dust polarized

emission with the same random seed, since these two sources of emission should

be correlated.

We have tested our model against real observational data, both from Planck and

WMAP. We have show that it is a good fit at low/intermediate frequencies. We have

also shown that the model is a good fit to intermediate and high Galactic latitudes

away from the Galactic plane. We looked at the thermal dust emission in these

areas, whose characterization is critical for the analysis of polarized foreground

contamination. Also, we have provided this model publicly, which can be used

to generate multiple random realizations of foregrounds (by generating different

realizations of small-scale features) for Monte Carlo purposes.

7.1.2 Characterizing the impact of foreground residual contamina-

tion

The analysis presented in Chapter 4 is critical for forecasting the cosmological con-

strains obtainable from future polarized CMB experiments. As an example of future

observations, we use the 2010 proposal of the COrE satellite. We have run an end-

to-end simulation that goes from simulated observations maps, to component sep-

aration, power spectra estimation and cosmological parameters likelihood. In the

last step, when calculating the measured tensor-to-scalar ratio, we have developed

a method to model the polarized foreground residuals directly into the likelihood,

to prevent them from biasing the primordial B-modes.

We measure the random and systematic error bias on the tensor-to-scalar ratio

in three cases:

1. Using as input a simplified model with foregrounds with spatially constant

spectral indices.

2. Using as input a more complex model with spatially variable spectral indices,

but assuming spatially constant indices in the component separation.

3. Using the same input as 2., but including spatially variable error in the spec-

tral indices when performing the component separation.

We have found that the amount of r bias is very sensitive to the input value

of r. For a value of r = 0.01, the bias can be managed. We inject a systematic

7.1. SUMMARY AND CONCLUSIONS 155

error in the component separation and we propagate through the simulation. For

errors of . 3% for the simplified case, the estimates are unbiased. Even when we

consider a complex case we are successful; if we approximate the spectral properties

as spatially constant in the component separation, and if using our method to model

the foreground residuals into the likelihood. The value r = 0.01 is small, but with

a future satellite experiment with such a good sensitivity, it can be measured to at

least a 2σ level detection in the most pessimistic case.

For a value of r = 0.001, the analysis is much more challenging. Even for a

simple sky, the bias on r is very sensitive to errors on the foreground spectral pa-

rameters. When trying to correct it, we can only recover an upper limit detection.

For a complex sky, we note that assuming spatially constant spectral parameters

for the component separation results in a large bias that cannot be corrected for.

Therefore, the spectral parameters must be modelled as spatially variable. Simulat-

ing a global residual error of half a percent in the spectral parameters allows a 2σ

detection of r = 0.001.

In this work, we have quantified how bad a problem that we already know ex-

isted really is. If we aim to detect r = 10−3 from a future high-sensitivity satellite,

which is the most advanced experiment we can conceive in the near future to ob-

serve the CMB, we need to characterize and model the Galactic diffuse foregrounds

very carefully. It is easy to mistake foreground contamination for primordial B-

mode signal in the data analysis, so our main message is the critical need to develop

studies of polarized foregrounds in the future.

7.1.3 Forecasting the performance of the Simons Observatory

As part of the Simons Observatory collaboration, I performed forecasts for the de-

tection of the tensor-to-scalar ratio, using the Fisher information matrix and realistic

estimations of foreground spectral parameter errors. We considered instrumental

configurations for a Small Aperture Camera, that would survey a small patch of the

sky as deep as possible to try to detect primordial B-modes.

For an early configuration of the instrument, we forecast σ(r = 0) ∼ 2.1× 10−3,

which is consistent with other forecasts within the collaboration. The thermal dust

properties, in particular Tdust, are hard to constrain with a narrow frequency cov-

erage, up to ∼ 270 GHz. If we add high-frequency bands, to ∼ 500 GHz, the dust

temperature is better constrained and the performance improves.

We also conducted an exercise to determine the patch of the sky where it is

best to measure r. Here, we have tried to address the balance between how clean


of foregrounds the sky must be versus how well we can measure the foregrounds

properties in the same region in order to clean them. We forecasted the value of

σ(r) for all the observable sky from the site in Chile, tilling the sky in fsky = 0.05

disc masks. The best results we found, with σ(r) = 1.5×10−3, is the patch of the sky

centred in l = 261 b = −78 in Galactic coordinates, very close to the South Galactic

Pole. In general, we conclude that the value of σ(r) is better when the sky is cleaner

of polarized thermal dust emission, which makes the most important impact at the

SO frequencies.

Further work I have done within the SO is related to a proposal for a new

European-funded Small Aperture Camera in a separate telescope to complement

the current planned instrument. The Simons Observatory European B-mode Tele-

scope (SO-EBT) would effectively double the detectors of the deep survey that aims

to measure primordial B-modes. We consider the baseline configuration for the pro-

posal already funded, with 3 optical tubes, and three improved configurations, that

add and spread extra three optical tubes in different frequencies to evaluate how

much the detection of r improves.

We estimated realistic foreground errors with the CCA component separation

method, and then we propagated the foreground residuals into the power spectra.

Using the Fisher information we forecast the uncertainty in the tensor-to-scalar ra-

tio. We considered a survey with a fsky = 0.05 coverage and Gaussian noise. The

improvement in σ(r) is ∼ 20% going from a baseline configuration to an improved

one. The foreground residual level is similar to the cosmic variance lensed signal,

so with these specifications we are leaving the foreground-dominated regime and

almost entering the cosmic variance-dominated regime. Considering this, the effect

of applying delensing to the CMB is important. We show that by simulating de-

lensing by 50 and 75%, the σ(r) error is reduced. Going from baseline to improved

configurations and applying delensing, the enhancement is ∼ 21% at best. We also

consider two more runs, the first one considering a fsky = 0.1 coverage. In this case,

the improvement made by the proposed SO-EBT experiment is ∼ 20% with the full

lensing signal, and ∼ 25% with a delensing of Alens = 0.25. The second run con-

siders realistic atmospheric noise on the simulated observations. The improvement

in the σ(r) error is ∼ 25% with the full lensing signal and ∼ 35% with a delensing

of Alens = 0.25. This is the best improvement we have obtained for forecasting the

effect of doubling the number of detectors.

We note that the value of σ(r) depend on the sensitivity of the channels. The

worst errors we can forecast are in the case with fsky = 0.1 coverage, since the sen-

sitivities are√

2 higher. Also, we note that the spread of the extra optics tubes does

7.1. SUMMARY AND CONCLUSIONS 157

not appear to be an important factor, since the results of the 3 improved configu-

rations are similar. If we compare the forecast runs with fsky = 0.05, Gaussian and

atmospheric noise, the results of σ(r) are close, but the run with Gaussian noise has

better results. We conclude that adding another Small Aperture Camera will have a

significant impact on the detection of r, potentially improving the value of the σ(r)

error by ∼ 20% or more.

In conclusion, we note that we have developed a tool to forecast the performance

of an experiment quickly, which we can use to complement the analysis pipeline

we have also developed, which has an end-to-end approach. Using such a tool,

we conclude that for measuring primordial B-modes, a patch of the sky with no

polarized foregrounds will always be best. Based on our sky model, we find that

the best areas of the sky are located near the South Galactic Pole. In addition, we

have to consider that the sensitivity of the instruments in the future will improve,

so the signal-to-noise ratio will always improve for polarized foregrounds.

7.1.4 CORE proposal forecasts

Part of my work was within the CORE collaboration, and aimed at forecasting the

potential of the proposed satellite experiment to measure the primordial B-mode

signal. We simulated observations of the microwave sky and we performed the data

analysis, considering different component separation algorithms, both parametric

and blind. My particular contribution was the cosmological parameter estimation,

evaluating the posterior probabilities for r and τ . We reach similar conclusions to

the ones we reach in Chapter 4: We can achieve an unbiased measurement of r for

values & 0.005. For a lower value, of r = 10−3, the analysis is more difficult. For a re-

alistic scenario, including synchrotron, thermal dust, anomalous microwave emis-

sion, extragalactic point sources, and CMB lensing signal, we can measure r with

an uncertainty σ(r) = 5 × 10−4, and a ∼ 1σ bias. Using the technique that models

the foreground residuals in the parameter estimation, described in Section 4.3.3, we

can reduce the bias and make a ∼ 2σ detection of r = 10−3.

We also note that there are no significant differences between the blind methods

(SMICA and NILC) tested at intermediate and high multipoles. However, the r pos-

terior probability is dominated by the low-multipole signal, which is estimated by

the COMMANDER code. The dominant source of errors in the cosmological parame-

ter estimation is polarized foregrounds, above the weak lensing signal, so we stress

the need to improve our techniques in the future to deal with this problem.


7.1.5 Overall conclusions

The main conclusions of this thesis are:

1. The measurement of r is very sensitive to the presence of foreground contam-

ination, especially for future satellite missions, as shown in Chapters 4 and 6.

It is critical that the foreground spectral parameters are estimated with a very

high accuracy. To detect a value of r ∼ 10−3, the spectral parameter errors

must be ≤ 0.5%. We can detect r = 10−3 with a 2σ significance with the 2011

configuration of COrE. We need to account for the foreground residuals suc-

cessfully. In order to do this, an essential step is improving our methods to

characterize and model polarized foregrounds.

2. The issue of modelling foreground residuals at the likelihood stage is impor-

tant. In this work, we explore one method to do this. It is a relatively simple

procedure, but we show how powerful it is to effectively reduce a large frac-

tion of the bias in r. This stresses the need to improve the methods used in

CMB observation and data analysis. As previously remarked, the foreground

residuals are very problematic because they can easily be mistaken as primor-

dial B-mode signal. Therefore, a procedure to mitigate the contamination by

residuals, directly in the component separation step, or further ahead in the

power spectra calculation/cosmological parameter estimation, is essential for

future observations of the polarized CMB.

3. The potential results we could obtain from a ground-based experiment are

really good. The Simons Observatory is a Stage-3 experiment, which will be

observing in the next 4-5 years. It most likely will be capable of detecting

B-modes at least at the level of r = 0.01 at 3σ significance, which is a huge

improvement, of essentially one order of magnitude, over current upper limits

of r . 0.08 (95% C.L.). Considering that in the next few years we will improve

our analysis techniques, we could be able to detect smaller values of r. Given

some reasonable premises, we show that by doubling the number of detectors,

the uncertainty on r improves & 20%. CMB-S4 is the ultimate goal in the

long run, but Simons Observatory will be a very important move in the right

direction.

4. We have developed a quick tool to forecast the performance of cosmological

parameters estimation from CMB observations. This method can be used for

several purposes, for example searching for the best patch of the sky to mea-

sure r, or for the use in the optimization of future experiments.

7.2. FUTURE WORK 159

7.2 Future work

There are several issues we could further develop in the work presented in this

thesis. Also, we can add certain features that were not considered here, such as the

effect of weak gravitational lensing.

A key aspect of the work we presented in this thesis is the modelling of fore-

ground residuals during the cosmological parameter likelihood. As stated before,

the foreground residuals will dominate the sources of error for B-modes signal de-

tection, so developing this technique is important. We can improve the model for

the residuals, by re-processing the foreground component templates or improving

the analytic model in the likelihood by introducing extra parameters, for example.

Even further, we could plan the study of new methods to reduce the r bias, maybe

directly at the CMB map stage, or at the estimation of the angular power spectra.

Overall, the CMB analysis pipeline, described in Chapter 4, can be developed

further to include more features. The simulated observations can be improved to

include more complex components, such as polarized point sources, among others.

We can also account for ground-based atmospheric noise, including correlations.

For satellite experiments, a more realistic noise can be achieved, including scanning

strategies and hits maps, which change the noise considering how much time is

spent in each line of sight. We can also include instrumental systematics effects,

such as bandpass and beam mismatch, among others.

The component separation analysis can be expanded, by testing several meth-

ods, both parametric and blind, as previously mentioned in Section 2.2. We can

expand the cosmological parameter estimation by considering multiple parameters

and using Markov Chain Monte Carlo codes, e.g. CosmoMC (Lewis and Bridle,

2002; Lewis, 2013). We can expand our analysis to include intermediate and high-

multipoles, and we can consider forecasts for other parameters, such as the neutrino

physics ones, or the dark energy equation of state, among others.

One issue that was not considered in detail is the effect of the B-mode lens-

ing signal. We only consider it by scaling up and down the dominant lensing BB

spectrum in the cosmic variance term, therefore simulating a given percentage of

delensing. However, as stated before, there are sophisticated methods to remove

the lensing effect from a CMB map, using either tracers of the Large Scale Struc-

ture or the CMB signal itself. This will add realism to our end-to-end CMB analysis

pipeline or for the Fisher forecasting pipeline.

The Fisher matrix forecasting procedure can be further refined to apply it to

Simons Observatory or any other planned experiment. We can add features to


the Correlated Component Analysis algorithm to manage more foreground com-

ponents and correlated noise. For the delensing procedure, we can upgrade the

analysis to use more sophisticated methods as described above.

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