Light-Cone Averaging and Dispersion of the

dL – z RelationIdo Ben-Dayan

DESY1202.1247, 1207.1286, 1209.4326,

1302.0740 + Work in Progress

IBD, M. Gasperini, G. Marozzi, F. Nugier, G. Veneziano

UChicago, 2013

OutlineBackgroundLight-Cone Averaging PrescriptionApplication: Luminosity-distance (dL) Redshift (z) Relation in the Concordance Model.

Hubble ParameterLensing Dispersion of dL

2011 Nobel Prize in PhysicsThe 2011 Nobel Prize in Physics is

awarded "for the discovery of the accelerating expansion of the Universe through observations of distant supernovae" with one half to Saul Perlmutter and the other half jointly to Brian P. Schmidt and Adam G. Riess

Attempted ExplanationsChanging the energy content of the Universe: Cosmological Constant, Dark Energy…

*Challenge: Fine-tuned, coincidence?, fundamental theory?

Changing Gravity: f(R), scalar-tensor theory…

*Challenge: Solar system tests, fine-tuned, fundamental theory?

Changing the metric: void models

*Challenge: Matching with CMB and other probes, fine-tuned in space =>giving up the Copernican principle

o Changing the equations of motion: averaging, smoothing out small scale inhomogeneities…

*Challenge: Proper Averaging, matching with data, magnitude of the effect

Main stream CosmologyGR is a non-linear theory, Averaging and solving do not commute.Assume FLRW = homogeneous and isotropic metric.

Þ Implicit averagingModeling the energy momentum tensor as a perfect fluid.Pert. give rise to structure, highly non-linear at some scale. Background unchanged.

Þ Implicitly neglected the possibility of backreaction.Holz & Wald 1998, Green & Wald 2010

Consistency?

SDSS

2.73 K

PLANCK

Perturbations at Low RedshiftMeasurements of SNIa Mostly neglected, naively argued as irrelevant ~10-10 (Amplitude of the primordial power spectrum)The concordance model of cosmology: (before PLANCK)

~73% of the critical energy density is not accounted for by known matter, dark matter or curvature.

PLANCK: DE is only 68% of the critical energy density!

Average dL on the past LC as a function of redshift

A= redshift, V= light-cone coordinate

LC AveragingUseful for interpreting light-like signals in cosmological observations. Hyper-surfaces using meaningful physical quantities: Redshift, temperature etc.Observations are made on the light-cone. Volume averaging give artefacts and the matching with data is not clear.Past attempts: Coley 0905.2442; Rasanen 1107.1176, 0912.3370

Light Cone Averaging 1104.1167

A-priori - the averaging is a geometric procedure, does not assume a specific energy momentum tensor.

The prescription is gauge inv., {field reparam. A->A’(A), V->V’(V)} and invariant under general coordinate transformation. A(x) is a time-like scalar, V(x) is null.

This gives a procedure for general space-times.

Prescription PropertiesDynamical properties: Generalization of Buchert-Ehlers commutation rule:

For actual physical calculations, use EFE/ energy momentum tensor for evaluation. Example: which gravitational potential to use in evaluating the dL-z relation.Averages of different functions give different outcome

GLC Metric and Averages

Ideal Observational Cosmology – Ellis et al.Evaluating scalars at a constant redshift for a geodetic observer.

S

O

aa

z

zwIzwSIS

zwSzwdzwSI

1

;),,1(),,(

);~,,()~,,(~),,(

0

0

002

0

LC average of flux for any space-time amounts to the area of the 2-sphere! (JM subleading correction drops out)

Exact Result - Flux

Anisotropies always “mimic” acceleration!

GLC MetricFLRW

τ can be mapped to the time coordinate in the synchronous gauge of arbitrary space-time.

IBD et al. ’12

Averaged dL at Constant RedshiftWe write the GLC exact results in terms of 2nd order in standard cosmological perturbation theory (SPT) in the Poisson Gauge.Novelty: In principle, exact treatment of the geodesic equations and the averaging hyper-surface for any spacetime and any DE model, as long as the geodesic equation is unchanged.Previous attempts are limited to perturbations about FLRW and had to solve order by order: Vanderveld et al. 2007 – post Newtonian, Barausse et al. 2005, Kolb et al. 2006 – SG superhorizon, Pyne at al 2005, Bonvin et al. 2006 1st order, Wang 1998, 2000 flux analysisRebuttal to Kolb et al.: Hirata et al., Geshnizjani et al.

The Perturbed QuantitiesEFE gives Poisson eq. that

connects the density contrast and the gravitational potential:

Both the area distance and the measure of integration are expressed in terms of the gravitational potential and its derivatives. Vector and tensor pert. do not contribute.

Statistical PropertiesIn principle, we can now calculate <dL>(z) to first order in the gravitational potential ~ void model.In order not to resort to a specific realization we need LC+statistical/ensemble average. If perturbations come from primordial Gaussian fluc. (inflation)

In the flux - Doppler terms ~k2. Flux is optimal.~k3 contributions – lensing, dominates at large redshift, z>0.3, appears in all other f(dL) and in variance/dispersion.

Interpretation & Analysis

No Divergences

!

Linear PS of the grav. potential

“Doppler2” term in Flux (CDM)

∞

1 Mpc-1

0.1 Mpc-1

↵

Lensing2 Term – Not in Flux (CDM)

∞

1 Mpc-1

0.1 Mpc-1

Interpretation & AnalysisSuperhorizon scales are subdominant.

At small enough scales, the transfer function wins.At intermediate scales, the phase space factor competes with the initial small amplitude.In principle the upper limit can be infinite. In practice, up to what scale do we trust our spectrum? Linear treatment k<0.1-1 Mpc-1 and non-linear treatment k<30h Mpc-1.

Fractional corrections to the Flux and dL, ΛCDM,

kUV=10h, 30h Mpc-1

Distance Modulus Average and Dispersion, at z>0.5, Lensing Dominates!

Lensing DispersionHolz &Linder 2005, Hui &Green 2006

Us ~0.056zLight~0.055zDark~0.05zH&L~0.093z

Hubble ParameterTension between CMB value and SNIa.The exact same f_d, f_Φ, only at z<<1. Current analysis does not take them into account. Calibration? better fit?All observables are biased at a percent level, but flux is still optimal.“Fitting” the Union2.1 distance modulus 0.01<z<0.1 gives an increase of 2% in H0

Closer Look at the Lensing Integrand

Lensing DispersionCurrent data up to z~1.3Most Conservative Approach: Constrains late time power spectrum and/or numerical simulations.

h(k,z) incorporates the dependence on the EOS parameter w(z) etc.The dispersion can be used to constrain EOS, primordial power spectrum etc.

Primordial Power Spectrum

Many inflationary models predict enhanced spectra (Particle production, features, several inflationary epochs etc.)Even with PLANCK, Ly-alpha – measurements only up to k=1 Mpc-1

Lensing DispersionCurrent data has SN up to z~1.3Statistical Analysis: dispersion<0.12 in mag. (March et al. 2011)Extended halofit, Inoue & Takahashi 2012, up to k=320h Mpc-1

Þ Can constrain the amplitude on scales 1<k<30h to roughly 10-7 and better.

ConclusionsNo divergences. Useful for light-like signals.Flux is the optimal observable. Different bias or “subtraction” mechanisms, in order to extract cosmological parameters. The effect is several orders of magnitude larger than the naïve expectations due to the large phase space factor.Irreducible Scatter - The dispersion is large ~ 2-10% ΛCDM, of the critical density depending on the spectrum. Scatter is mostly from the LC average. It gives theoretical explanation to part of the scatter of SN measurements.The effect is too SMALL AND has the WRONG z dependence to simulate observable CC!

Open Issues/Future Prospects

1. Using the lensing dispersion to constrain the power spectrum.

2. The effect on the Hubble parameter determination.3. Matching the effect to other probes: CMB, LSS4. Applying LC averaging to cosmic shear, BAO, kSZ, strong

lensing etc.5. Other applications, averaging of EFE ….Many open

theoretical and pheno. problems.

Constraints from Spectral Distortions (deviations from

BB spectrum)Chluba, Erickcek, IBD 2012:

kmax=1 Mpc-1

Summary & ConclusionsApplication of light cone averaging formalism to the dL-z relation.

INHOMOGENEITIES CANNOT FAKE DARK ENERGY AT THE OBSERVABLE LEVEL!The effect of averaging can, in principle, be distinguished from the homogeneous contribution of CC/DE.

Variance gives theoretical explanation to the intrinsic scatter of SN measurements.

Open Issues/Future Prospects

1. Estimates of the non-linear regime, especially 0.1 Mpc-1<k< 1 Mpc-1.

2. Cosmological parameter analysis.3. Matching the effect to other probes: CMB, LSS4. Applying LC averaging to BAO, kSZ and many

more.5. Other applications, averaging of EFE ….Many

open theoretical and pheno. problems.

GLC to FLRW NG 1st Order

LC Calculation and LCDM

Pure FLRW:

Transform from the GLC metric to the longitudinal gaugePerturbed:

Measure of Integration

The deviation from the unperturbed measure:

For future reference:Subhorizon fluc. H0<k. Superhorizon fluc. are subdominant.No UV (k∞) or IR (z 0, k 0) divergences.

<dL>(z), <f(dL)>(z)

A= redshift, V= light-cone coordinate

Averaged dL at Constant RedshiftEthrington’s Reciprocity Law, for any spacetime:

SPT:

Measure of

Integration

Fluctuations in scalar

Functions of dLStandard pert. theory: the gravitational potential, density contrast etc. are gaussian random variables.Overbars and {…} denote ensemble average, <..> denote LC average.

Averages of different functions of scalars receive different contributions.

Statistical Properties

In principle, we can now calculate <dL>(z) to first order in the gravitational potential ~ void model.In order not to resort to a specific realization we need LC+statistical/ensemble average. If perturbations come from primordial Gaussian fluc. (inflation)

BR of Statistical and LC AveragingThe mean of a scalar:

=> Effects are second order, but we have the full backreaction of the inhomogeneities of the metric at this order!The variance to leading order:

Dominant Terms

Doppler effect due to the perturbation of the geodesic.

The Lensing Term:

Universe Composition Today

CC/DE becomes relevant only at z~1, Coincidence Problem?Based on CMB, LSS and SNIa observations.

Do perturbations

alter the picture???

Interpretation & AnalysisdL is a stochastic observable – mean, dispersion, skewness...

In the flux - The dominant contribution are Doppler terms ~k2

Any other function of dL gets also k3 contributions – lensing contribution, dominates at large redshift, z>0.3In principle the upper limit can be infinite. In practice, up to what scale do we trust our spectrum? Linear treatment k<0.1-1 Mpc-1 and non-linear treatment k<30h Mpc-1

No Divergences!

Results and LessonsUnlike volume averages: No divergencesThe contribution from inhomogeneities is several orders of magnitude larger than the naïve expectations due to the large phase space factor.The size of the contribution (fd,fΦ) is strongly dependent on the quantity whose average is considered. Our approach is useful whenever dealing with information carried by light-like signals travelling along our past light-cone.

GLC to FLRW NG 1st Order

LC Calculation and LCDM

Pure FLRW:

Perturbed:

Comparing by defining an effective redshift and averaging at constant redshift and w=w0

Non-Trivial AveragesOff-Center LTBAnisotropic Models (Except Kantowski-Sachs)More general metrics.Perturbed FLRW

Application: calculating the averaged luminosity – distance redshift relation

Past attempts: Vanderveld et al. – post Newtonian, Barausse et al., Kolb et al. – SG superhorizon, Pyne at al.,