Draft version October 10, 2014Preprint typeset using LATEX style emulateapj v. 11/10/09

A CMB GIBBS SAMPLER FOR LOCALIZED SECONDARY ANISOTROPIES

Philip Bull1,2, Ingunn K. Wehus3,2, Hans Kristian Eriksen1, Pedro G. Ferreira2, Unni Fuskeland1, KrzysztofM. Gorski3,4, Jeffrey B. Jewell3,

Draft version October 10, 2014

ABSTRACT

As well as primary fluctuations, CMB temperature maps contain a wealth of additional informa-tion in the form of secondary anisotropies. Secondary effects that can be identified with individualobjects, such as the thermal and kinetic Sunyaev-Zel’dovich (SZ) effects due to galaxy clusters, aredifficult to unambiguously disentangle from foreground contamination and the primary CMB, whichcurrently inhibits their use as precision cosmological probes. We develop a Bayesian formalism forrigorously characterising anisotropies that are localised on the sky, taking the TSZ and KSZ effects asan example. Using a Gibbs sampling scheme, we are able to efficiently sample from the joint poste-rior distribution for a multi-component model of the sky with many thousands of correlated physicalparameters. The posterior can then be exactly marginalised to estimate properties of the secondaryanisotropies, fully taking into account degeneracies with the other signals in the CMB map. We showthat this method is computationally tractable using a simple implementation based on the existingCommander component separation code, and also discuss how other types of secondary anisotropycan be accommodated within our framework.Subject headings: cosmic microwave background — cosmology: observations — methods: statistical

1. INTRODUCTION

Observations of the temperature anisotropies of theCosmic Microwave Background (CMB) radiation havebeen instrumental in providing high-precision measure-ments of important cosmological quantities such as theage, geometry, and energy content of the Universe. Butwhile the task of characterising the primary anisotropiesmay seem essentially complete – as of the Planck 2013data release, measurements of the temperature autospec-trum of the CMB are cosmic variance dominated for mul-tipoles ` . 1500 (Planck Collaboration 2013c) – a wealthof information remains to be picked out of CMB temper-ature maps in the form of secondary anisotropies.

Secondary anisotropies are distortions of the primaryCMB signal due to inhomogeneities between the surfaceof last scattering and the observer (Aghanim et al. 2008).Gravitational effects such as weak lensing and the inte-grated Sachs-Wolfe (ISW) effect have been strongly de-tected (Das et al. 2011; Planck Collaboration 2013b,d;Hernandez-Monteagudo et al. 2013), as have scatter-ing phenomena such as the thermal Sunyaev-Zel’dovich(TSZ) effect (Birkinshaw et al. 1984; Shirokoff et al. 2011;Wilson et al. 2012; Planck Collaboration 2013e,f). Thekinetic Sunyaev-Zel’dovich (KSZ) effect, caused by theDoppler boosting of CMB photons scattered off ionisedgas travelling with a bulk peculiar velocity, has also re-cently been detected (Hand et al. 2012; Sayers et al.2013). These effects variously probe the spatial and tem-poral variations of the gravitational potential, the density

p.j.bull@astro.uio.no1 Institute of Theoretical Astrophysics, University of Oslo,

P.O. Box 1029 Blindern, N-0315 Oslo, Norway2 Astrophysics, University of Oxford, DWB, Keble Road, Ox-

ford OX1 3RH, UK3 Jet Propulsion Laboratory, California Institute of Technol-

ogy, Pasadena, CA 91109, USA4 Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478

Warszawa, Poland

field, and the peculiar velocity field on large scales, andcan therefore furnish tests of dark energy, modified grav-ity, and even the inflationary epoch by constraining thegeometry, expansion and growth history of the Universe.

The secondary anisotropies are generally dominatedby foregrounds and the primary CMB, so detecting andcharacterising them is a delicate process, even with mod-ern high-resolution data. Detections of secondaries canbe roughly divided into two categories: local, whereanisotropies in a given direction can be identified as be-ing caused by a particular astrophysical object; and non-local, where a secondary signal due to the combinationof many objects is detected statistically across a largerregion of the sky.

A number of different methods have been used tomeasure non-local signals: fitting models to the angu-lar power spectrum (Fowler et al. 2010; Shirokoff et al.2011); using some other statistical property of the CMBmap, such as higher-order (non-Gaussian) moments, toseparate off a given signal (Pierpaoli et al. 2005; Wilsonet al. 2012; Munshi et al. 2013); stacking the signal frommany directions to average down all but the target signal(Granett et al. 2008; Diego & Partridge 2009; Komatsuet al. 2011); and cross-correlating the CMB map withtracers (e.g. galaxies) that are uncorrelated with all butthe target signal (Fosalba et al. 2003; Afshordi 2004; Gi-annantonio et al. 2006; Ho et al. 2008; Hand et al. 2012;Sherwin et al. 2012; Hernandez-Monteagudo et al. 2014).These all allow small secondary anisotropy signals to bepicked out by essentially combining the signal from allavailable pixels into a single statistical quantity.

Detecting localised signals is often a more difficultprospect. Because astrophysical objects typically sub-tend small angles on the sky, only a limited numberof pixels are available to provide information about agiven object, making it harder to attain a high enoughsignal-to-noise ratio to get a definitive detection. Thelimited amount of information also makes it harder to

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disentangle other signals from the secondary anisotropy,especially if they have similar frequency spectra orshapes/angular sizes, and there are fewer options foraveraging-down contaminating signals.

Previous approaches to this problem have tended torely on a combination of frequency information and an-gular filters matched to the size/shape of the secondaryanisotropy to try and pick-out its signal, while rejectingor at least averaging down other signals as much as pos-sible (Aghanim et al. 2001; Herranz et al. 2002, 2005;Forni & Aghanim 2005; Schafer & Bartelmann 2007;Feroz et al. 2009; Mak et al. 2011; Atrio-Barandela et al.2012; Carvalho et al. 2012). Such methods may either beblind, applying filters of a range of sizes over the entiremap, or can apply parameter estimation techniques tofit parametrised models to the objects, given prior infor-mation on their positions and/or sizes. These methodswork well if the secondary has a distinctive spectrum andmulti-frequency data are available, but this is not alwaysthe case – the KSZ effect has the same flat spectrumas the primary CMB, for example, and only a few CMBexperiments have more than one or two frequency bands.

Angular information is also valuable, but most filter-ing and model fitting techniques are unable to blindlydistinguish signals with similar angular structures. Assuch, the estimated signal for an individual object willretain some level of residual contamination from otherfluctuating components. One example of this is the con-tamination of the KSZ signal by the primary CMB –primary anisotropies on the arcminute scales character-istic of galaxy clusters cannot be fully removed by a filter,and so will either bias the estimated signal, or must betreated as an effective source of noise, significantly in-creasing the statistical errors (Aghanim et al. 2001; Her-ranz et al. 2005; Feroz et al. 2009; Carvalho et al. 2012).

In this paper we describe a novel method for charac-terising localised secondary anisotropies, based on apply-ing Bayesian inference to a physical parametric model ofall relevant signals on the sky. We use the Gibbs sam-pling technique to efficiently reconstruct the joint pos-terior distribution of the full sky model, which typicallyinvolves many hundreds of thousands of parameters forrealistic datasets. With the posterior in hand, we canthen marginalise over all other parameters to producestatistically-robust, unbiased estimates of the propertiesof the secondary anisotropies.

Importantly, because all of the signals that contributeto the CMB map are explicitly modelled, degeneracieswith local fluctuations in other signals can be fully takeninto account. Instead of being treated as random noise,the fluctuations are reconstructed from the data, allow-ing them to be cleanly separated from the secondary sig-nal in a statistical manner. For the primary CMB, thisis equivalent to performing a constrained Gaussian reali-sation of the anisotropies behind the cluster. The uncer-tainties associated with this procedure are automaticallypropagated in full by the Gibbs sampling scheme.

The paper is organised as follows. In Section 2 weoutline a general Gibbs sampling scheme for estimat-ing localised signals in the presence of primary CMBanisotropies, various types of foreground emission, andnoise. We then specialise to a couple of example sec-ondary anisotropies: the TSZ effect (Section 3), and theKSZ effect (Section 4), and demonstrate a simple proof-

of-concept implementation of the SZ Gibbs scheme inSection 5. Examples of other localised signals that canbe accommodated by our framework are discussed in Sec-tion 6, and we conclude in Section 7.

2. GIBBS SAMPLING OF LOCALISED SIGNALS

Gibbs sampling is a popular Monte Carlo techniquefor performing Bayesian inference on complex paramet-ric models. In this section, we outline a Gibbs samplingscheme for the joint estimation of CMB anisotropies,galactic foregrounds, and spatially-localised signals frommulti-frequency full-sky data. This is based on the CMBanalysis framework previously described by Jewell et al.(2004); Wandelt et al. (2004); Eriksen et al. (2004, 2008),which allows for straightforward marginalisation overboth CMB and foreground signals by sampling from thejoint posterior distribution of all components.

2.1. Data model

We begin by defining a data model for an observationof the sky at a given frequency,

d(ν) = B(ν)

Ncomp∑i=1

Gi(ν)Tiai + n(ν). (1)

In this expression, d is a vector of observed values, dp(ν),for each pixel p and frequency ν, and n denotes instru-mental noise. The signal components are broken downinto a set of unknown stochastic amplitudes (ai), anamplitude-to-sky projection operator (Ti), a frequency-dependent mixing operator (Gi), and an instrumentalbeam convolution operator (B). Note that the index iruns over both signal types (CMB, foregrounds, SZ am-plitudes etc.) and individual components within eachsignal type (different SZ clusters, CMB harmonics etc.).

Next, we specify the statistical properties of ai andnp(ν). In this paper we assume the signal amplitudesand noise to be Gaussian, having covariance matrices Sand N respectively. One is often interested in estimatingthe signal covariance matrix from the data, so in generalS is unknown and must be jointly estimated with therest of the model parameters. The basic structure ofS can often be specified a priori, and parametrised interms of a relatively small number of free parameters.For example, an isotropic, Gaussian CMB componentwill have a diagonal signal covariance matrix with theCMB angular power spectrum coefficients C` along thediagonal. The noise covariance matrix will be assumedto be completely known, although in principle it couldalso be specified using some parametric model.

To complete the data model, we must define an inven-tory of relevant signal components and specify the prop-erties of G and T for each. Depending on the component,these can also be modelled parametrically, with free pa-rameters to be estimated from the data; for example,the frequency mixing matrix of a galactic synchrotroncomponent might take the form of a power-law spectrumwith an unknown spectral index. We will make no as-sumptions about the statistical distributions followed bythese additional parameters for the time being; as willsoon become apparent, they are not generally Gaussian.

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2.2. Posterior mapping by Gibbs sampling

Once the data model has been defined, the remainingproblem is to map out the full joint posterior distribu-tion P (a,S,G,T|d). The posterior is unlikely to take theform of a known analytic distribution that can be sam-pled from directly, suggesting the use of a Markov ChainMonte Carlo (MCMC) method to obtain samples. Owingto the extremely high dimensionality of the problem (e.g.there are (`max + 1)2 parameters for the CMB compo-nent alone), popular MCMC techniques like Metropolis-Hastings and nested sampling are unsuitable, as theytend to scale poorly with dimension (Allison & Dunk-ley 2014). Maximum likelihood techniques are a possiblealternative, although these are by nature approximateand thus fail to fully propagate statistical uncertainty.

Instead, we will make use of the Gibbs sampling algo-rithm (which is technically a special case of Metropolis-Hastings). While it may not be possible to sample fromthe joint posterior directly, it is often the case that itcan be broken down into a set of conditional distribu-tions that are tractable. One can show that iterativelysampling from the conditionals (Fig. 1) results in a setof samples that eventually converges to the joint poste-rior. In other words, by breaking the sampling prob-lem up into a series of comparatively simpler steps, wecan reconstruct the full posterior distribution withoutrecourse to approximations or any other “lossy” proce-dures. This holds even for extremely high-dimensionalparameter spaces if there are high-dimensional joint con-ditionals that can be evaluated efficiently (for example,if most parameters can be drawn from a multivariateGaussian distribution).

As shown in Fig. 1, Gibbs samplers explore the param-eter space using a series of orthogonal sub-steps. This isinefficient for parameters that are strongly correlated, forwhich the optimum strategy would be to explore alongthe degeneracy direction. As such, care must be takento either avoid parametrisations with degenerate param-eters, or to write down joint conditionals for strongly cor-related parameters that can be evaluated directly. Oth-erwise, the chain will spend a long time slowly explor-ing the strongly correlated subspace, and the resultingMCMC chain will have a long correlation length, result-ing in fewer independent samples.

For our problem, the Gibbs scheme is

ai+1 ← P (a|Si,Gi,Ti,d) (2)

Si+1 ← P (S|ai+1,Gi,Ti,d) (3)

Gi+1 ← P (G|ai+1,Si+1,Ti,d) (4)

Ti+1 ← P (T|ai+1,Si+1,Gi+1,d). (5)

One can of course further subdivide the conditional sam-pling steps using Bayes’ Theorem and other basic statis-tical relations if necessary. In general, though, it is moreefficient to simultaneously sample as many parameters aspossible in each step, in order to reduce the correlationlength of the chain.

For the remainder of this section, we will show howeach of these Gibbs steps can be sampled in practise.For a more detailed discussion of the general properties ofGibbs sampling techniques, see Gelfand & Smith (1990);Casella & George (1992).

(ai, bi)

(ai, bi+1)

(ai+1, bi+1)

(ai+1, bi+2)

ab

Fig. 1.— Illustration of the iterative sampling procedure thatforms the basis of Gibbs sampling. The algorithm alternately sam-ples from the conditional distributions of the various parameters –P (a|bi), then P (b|ai+1) and so on. Each sub-step is in an orthog-onal direction in parameter space.

Joint amplitude sampling— Under the assumption thatthe likelihood is Gaussian, Eq. (2) reduces to a sin-gle multivariate Gaussian distribution for all a, meaningthat the amplitude parameters for all components (po-tentially hundreds of thousands of them) can be sampledsimultaneously. Simplifying the notation for the signal tos = U · a, with U = BGT, one can see this by writing

P (a|d,S,G,T)∝P (d|a,S,G,T)P (a|S,G,T)

∝ e− 12 (d−U·a)TN−1(d−U·a) · e− 1

2aTS−1a

∝ e−12 (a−d)T (S−1+UTN−1U)(a−d). (6)

The distribution has covariance(S−1 + UTN−1U

)−1

and (Wiener-filtered) mean

d =(S−1 + UTN−1U

)−1UTN−1d. (7)

We have suppressed sums over frequency here; see Erik-sen et al. (2008) for a derivation of the above in the fullmulti-frequency case.

Sampling from this distribution is conceptuallystraightforward: one first generates a pair of vectors ofN (0, 1) random variables, (ω0, ω1), and then solves thelinear system Ma = b for a, where

M=S−1 + UTN−1U (8)

b=UTN−1d + S−12ω0 + (UTN−1U)

12ω1. (9)

In practise, solving this system is a significant compu-tational challenge, owing to its high dimensionality andtypically poor conditioning of the matrix operator M.We discuss this more in Section 5.2; see also Eriksenet al. (2008) for a detailed discussion of this problem.

A useful feature of the joint amplitude sampling stepis marked by the presence of prior-dependent (S) termsin Eqs. (8) and (9). These ensure that the solution is de-fined even in regions where the data have been masked.Solving the linear system therefore amounts to drawinga constrained realization of the amplitudes that is sta-tistically consistent with the available data and otherparameters of the data model. The availability of ‘un-cut’ amplitude map samples simplifies subsequent Gibbs

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steps that may rely on spherical harmonic analysis, suchas those involving angular power spectrum estimation.

Signal covariance— Eq. (3) can be written as

P (S|d,G,a)∝P (d|S,G,a)P (S|G,a) ∝ P (S|G,a)

=P (a|G,S) · P (S|G)

P (a|G)∝ P (a|G,S)

=e−

12aTS−1a√|S|

. (10)

This is an inverse Wishart distribution for S, assumingflat priors for S and a. Prior knowledge of the form ofS can be used to further simplify Eq. (10) to one of thespecial cases of the inverse Wishart distribution.

Notice that the dependence on the data, d, hasdropped out of Eq. (10). This is because we are nowconditioning on a, which contains all of the informationnecessary to estimate S.

General parameters— For Eqs. (4) and (5), a simpleapplication of Bayes’ Theorem yields

P (G|d,a,S,T)∝P (d|G,a,S,T)P (G|a,S,T)

∝ e− 12 (d−U·a)TN−1(d−U·a) · P (G)(11)

P (T|d,a,S,G)∝P (d|T,a,S,G)P (T|a,S,G)

∝ e− 12 (d−U·a)TN−1(d−U·a) · P (T).(12)

In this very general notation, the above equations donot tell us how to sample from these distributions, andthe priors for G and T are left general. This is becausethe form of these terms depends on the particular modelchosen for each component. In the following sectionswe will consider specific examples (summarised in Table5.2), for which the above equations simplify significantly.

2.3. CMB component

The CMB signal, s, is a statistically isotropic field thatcan be expanded in spherical harmonics, Y`m(n) (wheren is a unit vector direction in the sky), such that thesignal in an individual pixel p is

sp =∑`m

a`mY`m(np). (13)

We identify the coefficients a`m and spherical harmonicoperator Y`m(np) with the amplitudes (a) and projectionoperator (T) for this component respectively. The signalcovariance (S) is given in harmonic space by

〈a`ma`′m′〉 ≡ S = C`δ``′δmm′ ,

which reduces Eq. (10) to a set of inverse gamma dis-tributions, independent for each `. The mixing operator(G) is the identity, since the CMB frequency spectrumis very close to blackbody, and so has a flat spectrum inbrightness temperature.

2.4. Extended foreground components and offsetestimation

Galactic synchrotron, free-free, thermal dust emission,and other extended foregrounds typically have complexspatial structures that do not follow simple statisti-cal distributions like the CMB. As such, it is critical

to include in the analysis some frequency channels forwhich these signals dominate – say, below 30 GHz forsynchrotron/free-free or above 353 GHz for dust. It isthen straightforward to reconstruct these componentspixel-by-pixel, although a notable exception is spinningdust, which does not dominate at any frequency (PlanckCollaboration 2014), and is consequently subject to con-siderable degeneracies.

For each component one must write down an explicitparametrisation of G(ν), based on some small number ofparameters, θp, per pixel. For example, synchrotron isoften modelled in terms of a power law in brightness tem-perature, fsynch(ν;βs) = νβs , while thermal dust is welldescribed by a modified blackbody with free emissivityindex and temperature. These parameters may then besampled using Eq. (11), which reduces to an effective χ2

mapping of the respective parameters.In addition to foreground parameters, there are signifi-

cant uncertainties in the absolute offset and dipole termsof a given CMB map. These degrees of freedom are easilydescribed in terms of four full-sky templates: one full-skyconstant and three orthogonal dipole modes, each withan unconstrained overall linear amplitude. The appropri-ate sampling algorithm in this case is the usual Gaussiangiven in Eq. (6), with G(ν, ν′) = δνν′ , S

−1 = 0, and Tlisting the four monopole and dipole templates.

2.5. Spatially-localised components

As well as separating foregrounds and other effectsfrom the primary CMB, we are also interested in de-tecting and characterising secondary anisotropies. Wewill concentrate on the thermal and kinetic Sunyaev-Zel’dovich effects due to galaxy clusters in subsequentsections, but for now the discussion is kept general.

Unlike extended foreground components, which aretypically modelled as large coherent structures covering asizeable portion of the sky, many secondary anisotropiesare associated with discrete objects, and are thereforestrongly localised. The spatial distribution of the sec-ondary anisotropy is best captured by specifying a col-lection of spatial templates of limited size, each centredaround the location of an individual object. Each lo-calised template will have a separate amplitude associ-ated with it, although the amplitudes may be correlatedbetween objects. The shapes and frequency spectra ofthe templates may vary from object to object too, so toaccount for this one can define parametric spatial andspectral profiles with parameters that can be tuned (orsampled) for each object individually.

To define the model for this type of component, onemust also specify the number and positions of its con-stituent objects, and basic information on the templatefor each object, such as its angular size. This requiresa source of prior information, typically in the form of acatalogue of objects. For SZ clusters, for example, onecould use a catalogue of ‘candidate’ clusters from a blindSZ detection algorithm (Appendix A), or one of a num-ber of X-ray cluster catalogues. Clearly, the specificationof the component will only be as complete (and accurate)as the catalogue used, leading to issues with missing orduplicated objects, position errors, and the like. We willreturn to these problems later.

An object with index j, centred on direction nj , has

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Component Type Spectrum G(ν) Spatial dependence T Covariance S Amplitude a

CMB anisotropies 1 Y`m(p) C`δ``′δmm′ a`m

Spatial template (e.g. monopole/dipole) δνν′ Pre-defined template T (p) ∞ Aν′

Pixel-based foreground f(ν;θp) δpp′ ∞ Ap′

Thermal SZ (clusters) x ex−1ex+1

− 4 TTSZ(p, nj ,θj) ∞ Aj

Kinetic SZ (clusters) 1 TKSZ(p, nj ,θj) See Eq. (29) vlosj

TABLE 1Signal component types, defined by their spectral and spatial dependence.

the projection operator

Tj = Ti(p, nj ,θTj ), (14)

where Ti is a parametric spatial profile shared by all ob-jects of this type of component, and θTj are the profileparameters for the individual object. For localised sig-nals, Ti will typically be zero beyond some given angulardistance from nj , although this is not compulsory.

Similarly, the frequency mixing operator is given by

Gj = fi(ν,θGj ), (15)

where fi is a shared parametric spectral function andθGj are the spectral parameters for an individual object.Each object has a single overall amplitude, aj . Correla-tions between amplitudes are specified by a single signalcovariance matrix for all objects, i.e.

Si = 〈ajak〉. (16)

In the next two sections, we will consider the TSZ andKSZ effects for galaxy clusters as two specific examplesof localised components.

3. THERMAL SZ FROM GALAXY CLUSTERS

The thermal SZ effect is caused by the Compton scat-tering of CMB photons by hot gas in the intergalacticmedium (Sunyaev & Zeldovich 1972). The CMB gainsenergy from the gas, effectively leading to a shift in itsspectrum along the affected line of sight. This is man-ifested as an apparent decrement in the CMB temper-ature at low frequencies (ν . 217 GHz), and an incre-ment at higher frequencies, which distinguishes TSZ fromthe flat-spectrum primary CMB signal. As free electronsdominate the scattering, the magnitude of the shift de-pends primarily on the integrated electron pressure alongthe line of sight. In galaxy clusters, the thermal pressurecan be related to the cluster size and mass, and so theTSZ effect can be used as a way of probing a cluster’sphysical properties. This is useful for understanding howstructure forms, as well as providing constraints on cos-mological parameters such as the normalisation of thematter power spectrum (Battye & Weller 2003). An-other useful property of the TSZ effect is that the surfacebrightness is constant as a function of redshift (Sunyaev& Zeldovich 1972; Rephaeli 1995), making it possible todetect clusters out to high redshift (z & 1).

We will assume here that a catalogue of positions, an-gular sizes, and redshifts of clusters has already beencompiled from a previous blind survey, so that our taskis to accurately characterise the clusters’ properties. Thisis particularly critical for the most massive clusters, as

some cosmological tests are extremely sensitive to the lo-cation of the high-mass cut-off of the cluster mass func-tion (Matarrese et al. 2000). Note that while blind de-tection algorithms are capable of providing some infor-mation on cluster properties, they often rely on simpli-fied or approximate treatments of issues such as resid-ual foreground contamination, statistical error propaga-tion, overlap between clusters, and so on, so it is im-portant to perform a more specialised characterisationpost-detection.

3.1. Model definition

Following the discussion in Section 2.5, we begin bydefining a parametric spatial template for the clusterTSZ signal. The fractional temperature change due tothe thermal SZ effect along a line of sight is given by(Sunyaev & Zeldovich 1972)

∆T

T= f(ν)y(n) (17)

y(n) =σTmec2

∫Pe(n, l)dl, (18)

where Pe is the electron pressure, l is a distance alongthe line of sight, and f(ν) is the frequency spectrum,

f(ν) = xex − 1

ex + 1− 4; x = hν/kBTCMB, (19)

which can immediately be identified with the mixing op-erator, G. In our model, we adopt the ‘universal’ pres-sure profile of Arnaud et al. (2010),

Pe(x) = P500(z)P0(M500/M?)

αp(x)

(c500x)γ [1 + (c500x)α

]β−γα

, (20)

where x = r/R500 and M? = 3× 1014h−1M�. The pres-sure at radius R500 in a gravity-only self-similar model is(Nagai et al. 2007)

P500(z) = 1.65× 10−3h(z)83 (M500/M?)

23h2 keVcm−3,

and the running of the mass scaling with radius iswell-fit by αp(x) = 0.22 (1 − 8x2/(1 + 8x3)). Thenormalisation and shape parameters of the univer-sal profile have best-fit values of [P0, c500, γ, α, β] =

[8.403h−32 , 1.177, 0.3081, 1.0510, 5.4905], calibrated from

the REXCESS sample of 33 local X-ray clusters at smallradii (Bohringer et al. 2007) and hydrodynamic simula-tions at large radii.

The pressure profile is fully specified once the char-acteristic mass, radius, and redshift of the cluster are

6

10-2 10-1 100 101

θ/θ500

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

⟨ ∆T/T⟩

TSZ (150 GHz)TSZ (220 GHz)TSZ (350 GHz)KSZ (400 km/s)

Fig. 2.— Temperature fluctuation due to thermal SZ(black/grey) and kinetic SZ (red) at 150, 220, and 350 GHz, aver-aged over a circular top-hat aperture of radius θ. The lines shownare for a ‘typical’ SZ cluster at z = 0.1, with M500 = 1014M�,R500 = 1 Mpc, and line of sight velocity v · n = 400 kms−1. Thegrey band illustrates the approximate range of temperature fluctu-ations for the primary CMB. We have defined θ500 = R500/DA(z).

given. The TSZ projection operator for a cluster j, withpressure profile centred about the direction n, is then

TTSZj (p) = y(p, nj ,θ

Tj ), (21)

where the full set of parameters for the spatial profile isθT = {M500, R500, z, P0, c500, γ, α, β}. The TSZ profilefor a typical cluster is shown in Fig. 2, as a function offrequency and effective beam size.

For a sufficiently realistic model of the cluster spatialprofile, there should be no need for a separate amplitudedegree of freedom, aj , because Eqs. (17) and (20) definea complete mapping between the magnitude of the TSZsignal, y, and physical cluster parameters such as M500

and R500. We introduce aj in our model for a couple ofreasons, however. First of all, it is advantageous to beable to keep the cluster shapes fixed for some applica-tions, since the spatial templates are time-consuming tocompute. From Eq. (20), one can see that there is analmost one-to-one correspondence between aj and M500,the parameter of most cosmological interest. The statis-tics of aj are also a good proxy for detection significance.By holding the shape parameters fixed, but allowing aj tovary, we can therefore get good estimates of these quan-tities with considerably reduced computational expense.

Secondly, despite being one of the most accurate mod-els available, the universal profile is still a simplification.Arnaud et al. (2010) show the universal pressure profileto fit all clusters in the REXCESS sample to better than25% at radii greater than 0.2R500, but at smaller radiithey find the dispersion to be much greater, deviatingfrom the universal relation by up to a factor of four de-pending on how morphologically disturbed the cluster is.Allowing aj to vary could at least help to reduce biasesin other components of the data model due to this sort ofmodelling error (although one must be careful in choos-ing which profile parameters are also allowed to vary, assome are strongly degenerate with aj).

3.2. Amplitude sampling

Sampling of the {aj} parameters proceeds jointly withall other amplitude degrees of freedom, as described inSection 2.2, but it is instructive to explicitly write outthe linear system for just the CMB and a localised TSZcomponent, which we will now do.

From Section 2.2, the data model for a single frequencymay be written as dν = Uν · a + nν , where

a= (aCMB, aTSZ) (22)

Uν =Bν(1 · Y , f(ν)TTSZ) (23)

are symbolic block vectors of the amplitudes and (beamconvolved) mixing/projection operators for each compo-nent. The TSZ amplitudes are given by the block vectoraTSZ = (a1, a2, · · · , aN ), and the spatial templates byTTSZ = T = (T1, T2, · · ·TN ), where Tj = T (p, nj ,θj).

In this notation, the linear operator (8) can be written

as M = S−1 +∑ν N−1ν , with

S−1 =

(S−1

CMB

S−1TSZ

)N−1ν =

(BTν N

−1ν Bν BTν N

−1ν BνTf(ν)

f(ν)(BνT)TN−1ν Bν f(ν)(BνT)TN−1

ν BνTf(ν)

).

We will set S−1TSZ = 0 in the rest of this paper, but include

it here for the sake of generality. The bottom-right blockof the inverse noise operator contains the TSZ-TSZ term

(N−1ν,22)jk = f(ν)(BνTj)

TN−1ν BνTkf(ν). (24)

For j 6= k, this accounts for any overlap between clus-ters, ensuring that neighbouring clusters do not bias oneanother. Finally, the right-hand side of the linear system(9) can be written as

b =

S− 1

2

CMBω0 +∑ν B

Tν

[N−1ν dν +N

− 12

ν ων

]S− 1

2

TSZω1 +∑ν f(ν)(BνT)T

[N−1ν dν +N

− 12

ν ων

] ,

where ω are randomly-drawn white noise maps.In principle there is a physical prior on the TSZ am-

plitudes: aj ≥ 0. This violates our assumption that allamplitude parameters are Gaussian, without which wewould be unable to simultaneously sample large num-bers of amplitudes efficiently. To resolve this conflict, wechoose a looser interpretation of aj , treating it as a ‘diag-nostic parameter’ that quantifies detection significance,and is allowed to go negative.

3.3. Shape sampling

The TSZ frequency spectrum is completely fixed, sothere are no spectral parameters to sample. This leavesonly the shape parameters, θT , defined above. Unfortu-nately, the conditional distribution for θT is difficult tosample from analytically due to the dependence of (21)on a numerical line of sight integration, and the complex-ity of the pressure profile itself. As such, we fall back onthe Metropolis-Hastings algorithm to sample from (12).This is tractable owing to the reasonably small number ofshape parameters for each cluster, although sampling isrelatively slow because each proposal requires the clustertemplate to be recalculated for a new set of parameters.

7

The parameters should strictly only be sampled one clus-ter at a time to preserve the Gibbs scheme, which disal-lows parallelisation of the sampling algorithm.

A couple of approximations can be made to speed upcomputations. The first involves assuming that all clus-ters share the same shape parameters. In this case, thecluster profile recalculation can be parallelised effectively,and only one set of shape parameters need be sampledper Gibbs iteration. The shape parameters then repre-sent some ‘average’ profile for the ensemble of clusters.

The second involves approximating the likelihood foreach cluster to be independent of the shape parametersof any other clusters, in which case sampling for eachcluster can happen in parallel. This is a good approxi-mation unless clusters overlap. A possible refinement ofthis method would be to sample in parallel for all clus-ters except those that overlap by more than a pre-definedamount, for which sampling would instead happen se-quentially.

3.4. Catalogues and prior information

While recent high-resolution, high-sensitivity CMB ex-periments have greatly increased the number of clustersdetected using the SZ effect, the majority are detectedonly with comparatively low SNR, or are barely resolved.The CMB data alone are therefore insufficient to stronglyconstrain the physical properties of most clusters, and wemust look to other datasets to provide additional infor-mation. Fortunately, extensive cluster catalogues basedon X-ray and galaxy redshift surveys are available (e.g.Koester et al. (2007); Piffaretti et al. (2011)), that can beused to put priors on the cluster parameters θT . Prior in-formation is naturally incorporated into the Gibbs sam-pling procedure through the P (T) term in (12).

Most important from our perspective are the redshift,characteristic mass (M500), and radius (R500) of the clus-ters. Without some prior information on these parame-ters, the shape sampling method of Section 3.3 will beaffected by strong degeneracies. Composite X-ray cat-alogues such as MCXC (Piffaretti et al. 2011) providegood estimates of these parameters for ∼ 1800 clusters,although it should be noted that the different selectionfunctions for SZ and X-ray surveys mean that not all SZclusters are present in the catalogue. Also, the parame-ters are typically estimated using scaling relations thatare subject to systematic uncertainties in calibration, andwhich may disagree between SZ and X-ray observations(Planck Collaboration 2011).

A further requirement for the catalogues used by ourGibbs scheme is that they are free from duplicate entries.Eq. (24) shows why this is the case – if cluster k isactually a duplicate of j, there will be a 100% overlapbetween them, leading to large off-diagonal entries in theTSZ-TSZ block of the linear operator. This can causethe system to become degenerate, leading to ill-definedsolutions.

4. KINETIC SZ AND PECULIAR VELOCITIES

The kinetic SZ effect is also caused by the Comptonscattering of CMB photons, but this time it is the co-herent (bulk) motions of the scattering electrons withrespect to the CMB that imprint the signal, which is ef-fectively just a Doppler shift. This has a flat spectrum

and is therefore not so readily distinguished from theprimary CMB as the thermal SZ effect.

One can use the KSZ effect to probe the cosmologi-cal peculiar velocity field on large scales. This encodes agreat deal of information about the growth of structurein the Universe, and can be used to constrain dark en-ergy and modifications to General Relativity out to highredshift (Bhattacharya & Kosowsky 2008a; Kosowsky& Bhattacharya 2009; Keisler & Schmidt 2013). TheKSZ effect is also sensitive to other phenomena thatwould cause a CMB dipole to be seen in the clusterrest frame, for example in inhomogeneous cosmologicalmodels that violate the Copernican Principle (Goodman1995; Garcia-Bellido & Haugbølle 2008; Bull et al. 2012).

Because of the relative weakness of the signal andits lack of a distinctive spectral signature, the KSZ ef-fect is susceptible to various systematic errors that canseverely bias peculiar velocity measurements (Aghanimet al. 2001; Bhattacharya & Kosowsky 2008b). The mea-sured velocity is also degenerate with the optical depthof the cluster (Sehgal et al. 2005), so some way of in-dependently determining this must be found. Our pro-posed approach is well-suited to addressing these diffi-cult problems: through careful modelling, judicious useof prior information, and rigorous propagation of errors,one can break degeneracies and mitigate biases, even forextremely weak signals.

4.1. Model definition

For a cluster with bulk peculiar velocity v, the frac-tional temperature change due to the KSZ effect is (Sun-yaev & Zeldovich 1980)

∆T

T=−(v · n/c) τ(n) (25)

τ(n) =

∫σTne(n, l)dl. (26)

The shape of the cluster’s KSZ emission is governed bythe electron number density, ne. We already have a well-motived parametric form for the electron pressure profile(Eq. 20), so rather than choosing ne independently, weuse the ideal gas law ne ≈ Pe/kBTe and a ‘universal’temperature profile (Loken et al. 2002),

T (r) = 11.2

(R500h

Mpc

)2(1 + 0.75

r

R500

)−1.6

keV, (27)

to define an ne that is also a function of the TSZ shapeparameters defined in Section 3.1. By taking the TSZand KSZ profiles to be governed by the same set of pa-rameters, we explicitly take into account their commondependence on the physical properties of the cluster; in-formation gleaned from the stronger TSZ signal helpsbreak the degeneracy between the peculiar velocity andoptical depth. The projection operator for the KSZ com-ponent is then

TKSZj (p) = −τ(p, nj ,θ

Tj )/c, (28)

and the amplitude parameter is the bulk velocity pro-jected along the line of sight to the cluster, aj = vj · nj .Since the KSZ effect is just a temperature change alongthe line of sight, it has a flat spectral dependence, soG = 1.

8

4.2. Velocity correlations

In contrast with the TSZ case, we will not neglect thesignal covariance here. The peculiar velocities of clustersare correlated over large scales, with a covariance thatcan be calculated from linear cosmological perturbationtheory. While individual cluster velocities are of littleintrinsic interest, velocity correlations can provide infor-mation on the growth of structure, matter power spec-trum, and other cosmological parameters (Bhattacharya& Kosowsky 2007; Macaulay et al. 2012).

The KSZ velocity covariance matrix is given by (Gorski1988; Macaulay et al. 2012)

(SKSZ)jk =

∫k2dk

2π2Pvv(k)Fjk(k) + σ2

∗δjk, (29)

where the first term on the right hand side is the line ofsight velocity correlation from linear theory, 〈(vj ·nj)(vk ·nk)〉, and the second term is a nonlinear velocity disper-sion modelled as an uncorrelated noise term with vari-ance σ2

∗. The window function Fjk depends on the ori-entation of the pair of clusters (j, k) with respect to theobserver. Different (but equivalent) expressions for Fjkare given by Dodelson (2003) and Ma et al. (2011). Thevelocity power spectrum is related to the matter powerspectrum by

Pvv(k) =κ(zj)κ(zk)P (k, z = 0)/k2 (30)

κ(z) =H(z)f(z)/(1 + z)D(z), (31)

where D(z) is the linear growth factor normalized tounity today and f(z) = d logD/d log a is the growth rate.

4.3. Velocity covariance matrix amplitude

Given a suitable choice of parametrisation, the cos-mological functions that enter (30) can be constraineddirectly through the Gibbs sampling procedure. As asimple illustration, consider a parametrisation that hasonly the overall amplitude of the matter power spectrum,APS, as a free parameter. Replacing P (k) 7→ APSPfid(k)in (30), where Pfid(k) is a fiducial power spectrum, wecan rewrite (3) as

APS←P (APS|G,a,T,d) ∝ P (APS|aKSZ) (32)

∝ exp

(−1

2aT

KSZS−1KSZaKSZ

)/√|SKSZ| . (33)

If we set σ∗ = 0, the signal covariance matrix is simplyproportional to the fiducial linear theory velocity covari-ance matrix, SKSZ = APSSfid, and the pdf reduces to theinverse gamma distribution,

Γ−1(A;α, β)∝ exp(−β/A) / Aα+1 (34)

∝ exp

(−1

2aTS−1

fid a / APS

)/(AN/2PS

√|Sfid|

),

where N is the number of clusters and α = N/2 − 1.Efficient direct sampling algorithms exist for this distri-bution (e.g. Eriksen et al. 2004, and references therein).We do not know of a direct sampler for the general case,where σ∗ 6= 0, but an alternative method is discussed inSection 5.4.

5. GIBBS SAMPLER IMPLEMENTATION

Implementing a numerical code to efficiently carry outthe sampling procedure described in previous sections istractable but challenging. In this Section we discuss thecomputational difficulties associated with the proposedGibbs sampling scheme, and suggest solutions for eachof them. Our discussion is partially based on a simpleproof-of-concept implementation for the TSZ effect builton top of the Commander CMB component separationcode (Eriksen et al. 2004, 2008). Commander alreadyimplements a subset of the proposed Gibbs scheme, andcan be extended in a relatively modular fashion to ac-commodate a localised TSZ component. It is optimisedfor lower-resolution full-sky analyses, however, and lacksa suitable solver for the high-resolution analysis neededfor TSZ clusters. We therefore use this code as a simpletestbed, and defer full implementation to a later work.

5.1. Full Gibbs scheme for SZ

Combining results from the previous sections, a suit-able Gibbs scheme for the localised TSZ and KSZ signalsfrom galaxy clusters is

ai+1 ← P (a|Ci`, AiPS,θiFG,θ

iSZ,d) (35)

Ci+1` ← P (C`|ai+1

CMB) (36)

Ai+1PS ← P (APS|ai+1

KSZ) (37)

θi+1FG ← P (θFG|ai+1,θiFG,θ

iSZ,d) (38)

θi+1SZ ← P (θSZ|ai+1,θi+1

FG ,θiSZ,d), (39)

where a = (aCMB,aFG,aTSZ,aKSZ) are the amplitudeparameters, θFG are the foreground spectral parameters,and the cluster shape parameters θSZ are those of theuniversal pressure profile defined in Section 3.1.

The Commander code already includes (35), (36) and(38). The amplitude sampling step (35) must be gener-alised to include the new SZ components, and steps (37)and (39) need to be implemented from scratch. We focusonly on (35) here, deferring detailed implementation ofthe other steps for later work.

5.2. Constrained realisation solver

The computational complexity of the Gibbs scheme isentirely dominated by the joint amplitude sampling step(35), which involves solving a large linear system to drawa constrained realisation of all of the amplitude param-eters – potentially millions of them. For realistic CMBdata, with millions of multi-frequency pixels, inhomoge-neous noise, and masked regions, there is a wide spreadin the signal-to-noise ratio per pixel. The eigenvalues ofthe linear operator (8) therefore have a large dynamicrange, making the system poorly-conditioned. This,combined with its high dimensionality, results in unac-ceptably slow convergence for most linear solvers. With-out a computationally-efficient global amplitude sam-pling step, the Gibbs scheme is intractable, so this issueis of central importance.

There are a number of ways to speed-up the solutionof the linear system. Commander uses a preconditionedconjugate gradient (PCG) solver, which works by mul-tiplying both sides of the system by a preconditioningmatrix, Q, and then solving the resulting modified sys-tem, QMa = Qb. If one can design a preconditioner

9

4 2 0 2 4Standard score (σ)

0.0

0.1

0.2

0.3

0.4

0.5pd

f

Expected GaussianBest-fit GaussianTSZ (300 clusters)

Fig. 3.— Distribution of standard scores, (〈ai〉 − ai,in)/σi, forthe TSZ amplitudes of 300 simulated clusters. The mean and stan-dard deviation for each cluster are calculated from a Gibbs chainwith 575 samples, and the input amplitudes are those used in thesimulation.

such that Q≈M−1, the resulting modified system willbe well-conditioned, and if both Q and QM can be eval-uated quickly, it can be solved much faster. An efficientpreconditioner for the joint amplitude sampling problemwas described in Eriksen et al. (2004, 2008), and hasbeen shown to work well on masked, full-sky, low-noise,multi-frequency foreground-contaminated CMB data upto ` ' 200 (Planck Collaboration 2013a), which is suffi-cient for foreground component separation.

Higher resolution methods are needed to sample SZamplitudes, however, as a typical cluster at z & 0.1 sub-tends only a few arcminutes. Increasing the angular res-olution by even a factor of 2-4 results in a considerablehardening of the problem, as the number of pixels re-quired increases as the square. The noise is also higher atsmall scales, further contributing to the poor condition-ing of the system. The result is that substantially moresophisticated solvers are required to make the problemtractable. One such method is the multi-level algorithmdescribed by Seljebotn et al. (2014). This is capableof rapidly solving the amplitude sampling system up to` ≈ 2000, but has yet to be extended to multi-frequencydata with more than just the CMB plus noise.

An alternative is to reduce the complexity of the prob-lem by working in the flat-sky limit. This is suitable forexperiments such as ACT and SPT, which cover onlya few thousand square degrees; so while their angularresolution is higher, the total number of pixels is typ-ically smaller. Importantly, in the flat-sky limit onealso benefits from being able to use Fast Fourier Trans-forms (FFTs) instead of the slower and more cumber-some spherical harmonic transforms.

To demonstrate the amplitude sampler, we apply theexisting Commander PCG solver to a basic full-sky sim-ulation for three frequency channels (143, 217, and 353GHz), each of which has a primary CMB component,TSZ signals for 300 clusters, uncorrelated white noise,and instrumental beam effects. No foreground contam-ination or masks are included. A Gaussian CMB reali-sation is drawn using the Planck best-fit angular powerspectrum (Planck Collaboration 2013c). The beam is

chosen to be a uniform 40 arcmin. across all bands,and the noise covariance for each channel is obtained bysmoothing the noise maps for the corresponding PlanckHFI channels to the same resolution. To compensate forthe comparatively low resolution of the simulation, werescale the angular sizes of all clusters by a factor of 7,roughly corresponding to the ratio between the widthof our chosen beam (40′) and the Planck HFI beams(∼ 5′ − 7′). Correspondingly, we scale the amplitudesof the clusters by 1/72 in order to preserve their inte-grated flux, and thus the signal-to-noise ratio per cluster.The clusters are chosen to have the angular distributionand physical properties of the entries with the largestθ500 from the Planck SZ catalogue (Planck Collabora-tion 2013f), except Virgo and Coma which are too largeafter rescaling.

Fig. 3 shows the distribution of cluster amplitudes re-covered by running the modified Commander Gibbs sam-pler over the simulations. The standard scores (i.e. therecovered TSZ amplitude minus the amplitude input intothe simulation, weighted by the standard deviation esti-mated from the Gibbs chain) are consistent with the unitGaussian distribution. This is what one would expect ifthe recovered amplitudes are Gaussian-distributed andunbiased, and the standard deviation has been estimatedcorrectly (in other words, that the statistical uncertaintyhas been propagated correctly).

5.3. Spatial template calculation

The SZ spatial templates for each cluster are calcu-lated according to Eqs. (17) and (25), both of whichrequire line of sight integrations. This can be computa-tionally intensive for high-resolution data, especially asthe templates must be recalculated several times duringthe shape-sampling Gibbs step (39). Simple numericaltechniques like spline interpolation of the radial clusterprofile and integrating for many clusters in parallel canreadily be used to speed-up the process, however.

Once calculated, the templates must be convolved withthe instrumental beam in each band. While the convolu-tion would be fastest in spherical harmonic domain, thistends to introduce ringing artefacts, biasing the beam-convolved template by several percent (Fig. 4), whichin turn biases the SZ amplitudes. A much more accu-rate method is to perform the convolution directly on afiner grid in pixel space (i.e. each pixel is subdividedinto 4 or 16 sub-pixels, the pixel-space convolution iscalculated, and then the result is averaged back onto thecoarser grid). This is considerably more expensive thanthe spherical harmonic method, so the beam-convolvedtemplates should be cached if possible. Fortunately, theconvolution can be done independently per cluster, perband, making it easy to parallelise.

5.4. Velocity covariance matrix amplitude

Current CMB experiments lack the sensitivity to de-tect the KSZ effect from significant numbers of individualgalaxy clusters, so only statistical detections (e.g. Handet al. (2012)) will be possible for the foreseeable future.Our formalism suggests a natural quantity to use as astatistic: the velocity covariance, defined in Section 4.2.

To investigate the properties of this statistic, we definea simplified Gibbs scheme based on steps (35) and (37),

10

0 1 2 3 4 5 6 7 8 9 10Radius (θ/θ500)

0.90

0.95

1.00

1.05

1.10

TSH/T

pix

Fig. 4.— Ratio of TSZ templates for an example cluster (θ500 '30′) after beam convolution in spherical harmonic space and pixelspace (Nside = 1024). The ratio is shown as a function of anglefrom the center of the cluster, normalised to θ500, and the beamFWHM (θFWHM ' 14′) is shown as a dotted vertical line. Ringingartefacts are clearly visible.

where only the cluster KSZ amplitudes and the velocitycovariance matrix amplitude are free parameters,

a← P (a|APS,d) (40)

APS ← P (APS|aKSZ). (41)

We simulate velocity data, d, for a range of signal-to-noise ratios by drawing Gaussian realisations of veloci-ties with covariance SKSZ, and adding white noise withcovariance N = (SNR)−2 × diag(SKSZ), where SNR isan assumed signal-to-noise ratio per cluster, equal for allclusters. The velocity covariance matrix is calculated forthe 813 clusters with confirmed redshifts in the PlanckSZ catalogue (Planck Collaboration 2013f); a subset ofthese is taken when fewer clusters are needed. For thesake of simplicity, the non-linear velocity dispersion, σ∗,is set to zero in these simulations; the qualitative picturestays the same for non-zero σ∗, however.

One way of quantifying a detection of the KSZ effectis to use the amplitude parametrisation of the velocitycovariance matrix that was discussed in Sect. 4.3. Onemight expect that a constraint on APS that is inconsis-tent with zero at some confidence level would count asa detection, but the velocity covariance matrix must re-main positive definite, and so APS is always greater thanzero with 100% confidence. Furthermore, the distribu-tion of APS will have finite variance even for perfect,noise-free observations – a given (finite) set of correlatedpeculiar velocities is always consistent with having beendrawn from a finite range of distributions with differentAPS. Both of these effects can be seen in Fig. 5, wherethe marginal distribution of APS is plotted from simu-lations for 300 clusters, with varying SNR. As the noisedecreases, the marginal distribution rapidly approachesthe ideal distribution for the given SKSZ. The width ofthe estimated distribution, approximated by the stan-dard deviation, is shown in Fig. 6 as a function of bothSNR and the number of clusters.

A better way of quantifying detection significance istherefore to compare the estimated marginal distribution

0.0 0.5 1.0 1.5 2.0APS

0

1

2

3

4

5

6

pdf

SNR = 0.5SNR = 1SNR = 2SNR = 10

Fig. 5.— Marginal distributions of APS from simulations withdifferent signal-to-noise ratios per cluster (for 300 clusters, 5000samples). As the SNR increases, the marginal distribution con-verges towards the theoretical inverse gamma distribution for thegiven velocity covariance matrix (grey).

of APS with what would be expected in the ideal case. Auseful measure of the closeness of two distributions is theKullback-Liebler divergence (Kullback & Leibler 1951),also known as the relative entropy or information gain.For a pair of normalised discrete (binned) distributions,pi and qi, this is

∆S =∑i

pi log(pi/qi), (42)

where the sum is over bins. Since ∆S is not invariantunder qi ↔ pi, we specify that qi is the ideal referencedistribution here. As p approaches q, ∆S → 0, so smaller∆S denotes a stronger detection. Note that there is nodefinitive value of ∆S corresponding to a null detection,though; any constraint on the velocities, however weak,provides information on APS, and thus reduces ∆S.

In reality, σ∗ 6= 0, and so Eq. (33) cannot be reducedto the inverse gamma distribution. One can sample fromthe more general distribution using a simple inversionsampling algorithm: (1) Evaluate Eq. (33) over a gridof APS values; (2) integrate the result to find the cu-mulative distribution function (cdf); (3) draw from theuniform distribution, u ← U[0, 1]; and finally (4) drawAPS by evaluating the (spline interpolated) inverse cdfat u, i.e., APS ← cdf−1(u). This method is sufficientlyaccurate as long as the APS grid is dense enough, andis also relatively efficient; while each evaluation of (33)requires an expensive matrix inversion and determinantevaluation, these are fast enough for matrices with a fewthousand clusters or smaller, and the evaluation over thegrid can be performed in parallel.

Another problem that arises is the long correlationlength of Gibbs chains in the low SNR limit. In this case,the sampler spends most of its time exploring the jointprior, P (aKSZ, APS), and without good data to constrainthe KSZ amplitudes, APS is degenerate with an overallscaling of all aKSZ. Because the Gibbs scheme alternatelysamples from the conditional distributions, it is unableto move directly along the degeneracy direction, and so

11

0 100 200 300 400 500 600 700 800No. clusters

10-1

100

101

SN

R /

clu

ster

1.00

0.50

0.10

0.0

5

Fig. 6.— Standard deviation of the velocity covariance matrixamplitude, σ(APS), estimated from simulations, as a function ofthe number of clusters in the catalogue and the detection SNR foreach cluster.

exploration of the joint prior is slow, hence the highly cor-related samples. This issue can be overcome by adaptingthe low signal-to-noise CMB sampling algorithm of Jew-ell et al. (2009). This alternates between (1) samplingfrom the standard conditional distributions, and (2) anMCMC step based on a deterministic rescaling of theamplitudes. The MCMC step allows large jumps in APS

in the noise-dominated regime, significantly speeding-upexploration of the joint prior space, but reduces to thestandard sampling method in the signal-dominated case.

6. APPLICATION TO OTHER LOCALISEDSIGNALS

The sampling scheme proposed in Section 2 is suit-able for a variety of other types of localised secondaryanisotropy besides the TSZ and KSZ effects. The onlysignificant modification that is needed is to choose a dif-ferent parametric form for the spatial profile and fre-quency dependence, and to source a different catalogueof positions and (optionally) other properties of the tar-get objects. In the remainder of this section, we brieflyoutline some other types of secondary anisotropy thatmay benefit from the careful statistical treatment pro-vided by our method.

6.1. Integrated Sachs-Wolfe effect

The integrated Sachs-Wolfe (ISW) effect is a temper-ature change in the CMB caused by the decay of grav-itational potentials along the line of sight as dark en-ergy begins to dominate. The expected amplitude of theISW effect is predicted to be small compared with theprimary anisotropies, and so it is generally necessary tocross-correlate CMB sky maps with tracers of large-scalestructure in order to pick out the signal.

While interesting as an independent confirmation ofthe existence of dark energy, there have also been recentclaims of a detection of the ISW effect in the direction ofsupervoids that is anomalously large compared to ΛCDMpredictions (Granett et al. 2008; Papai et al. 2011; Na-dathur et al. 2012; Flender et al. 2013). The ISW effectdue to a large supervoid could potentially also explainthe CMB cold spot (Kovacs et al. 2014).

6.2. Topological defects

Phase transitions associated with spontaneous symme-try breaking in the early universe should give rise to topo-logical defects – the inhomogeneous boundaries betweenregions of different vacuum states (Durrer 1999). De-fects leave discontinuities and other characteristic non-Gaussian patterns in the CMB, with shapes that dependon the type of underlying symmetry that was broken.Most theories predict that only a few defects can be ex-pected to be visible in the CMB, and so have been in-voked as possible explanations of rare anomalies such asthe cold spot (Cruz et al. 2005).

There have been a number of searches for evidenceof defects in the CMB, with varying degrees of success(Jeong & Smoot 2005; Cruz et al. 2008; Feeney et al.2012). In order to claim that a detected anomaly is infact the result of a topological defect, a number of otherpotential explanations must first be ruled out; for ex-ample, a claimed detection of a cosmic texture may alsobe plausibly explained by the existence of an interven-ing cluster or void. The measured temperature profileof the anomaly can be used as part of a Bayesian modelcomparison to try and distinguish between the variousoptions (Cruz et al. 2008).

6.3. Signatures of pre-inflationary physics

A number of proposed models of pre-inflationaryphysics suggest processes that can imprint patterns intothe CMB that are not expected in the standard (Gaus-sian and isotropic) picture. These patterns typically takethe form of circles, concentric rings, or other simple ge-ometric shapes in CMB temperature or its variance, su-perimposed on anisotropies that are otherwise well de-scribed by a Gaussian random field. Examples of mod-els which predict such effects include (Wehus & Eriksen2011): bubble collisions in multiverse scenarios (Feeneyet al. 2013); cyclic cosmologies where primordial blackholes collide (Gurzadyan & Penrose 2010); and modelsin which massive particles exist before inflation (Fialkovet al. 2010). A closed spacetime topology would also givesimilar effects (Cornish et al. 1998).

Such theories normally predict a characteristic shapefor the patterns, and often impose other constraints suchas pairing of the shapes, or fixed concentricities. Thismakes it possible to construct well-defined spatial profilesfor the expected signal.

7. DISCUSSION

Secondary anisotropies of the CMB are a rich sourceof cosmological information – if they can be detectedand characterised accurately. In this paper, we have de-scribed a Bayesian method to rigorously and reliably dis-entangle secondary signals in CMB temperature mapsfrom other effects while simultaneously providing accu-rate estimates of statistical uncertainty. The basis ofthe method is a parametric physical model of the mi-crowave sky that includes primary and secondary CMBanisotropies, foreground contamination, and noise. Onecan then use a tailor-made Gibbs sampling scheme to ef-ficiently sample from the full joint posterior distributionfor the model, which may include many thousands, oreven millions, of parameters. After marginalising overeverything else, one is left with a consistent statistical

12

determination of the secondary signal. Though compu-tationally intensive, this method has the key advantagesof avoiding biases due to degeneracies with other signals(by modelling them), and correctly propagating uncer-tainties without relying on calibration against simula-tions or otherwise. The latter is particularly importantfor secondaries that are only marginally detected, whereinaccuracies in error estimates could make the differencebetween claiming a detection or not.

While our proposed method guarantees statistical self-consistency, its ability to accurately describe the actualdata depends on the suitability of the chosen model of thesky. As we discussed in Section 3, the sampling frame-work is extremely flexible, supporting the sampling ofcluster shape parameters and so on. It is therefore pos-sible to arbitrarily extend the sky model to be more re-alistic, incorporating effects such as non-sphericity andnon-thermal pressure support in galaxy clusters for ex-ample. Introducing additional Gibbs steps can lead toa significant increase in computational complexity, andgenerally increases uncertainties by requiring more pa-rameters to be marginalised, so these downsides must beweighed against any expected improvement in accuracy.

In Section 4, we gave an example of using our frame-work to directly estimate a cosmological statistic – inthis case, the velocity covariance. The advantage of thismethod is that the impact of foregrounds and the pri-mary CMB can be folded directly into the estimateduncertainty on the statistic, based solely on the avail-able data. The alternative is to calibrate the statisticoff simulations, which can be computationally intensiveand may lack some effects that are present in the realdata. Mapping-out the full joint posterior distribution,as we do, also has other advantages; one can marginaliseover the secondary signal to obtain rigorous estimatesof some other signal component. This is of particularinterest in cases where secondary anisotropies are bothcontaminants and interesting signals in their own right– for example, the contamination of the cosmic infraredbackground by the thermal SZ effect. Though successfulwhen applied to the CMB (Planck Collaboration 2013a),blind component separation methods are less likely tobe useful for this sort of problem, as they tend to mixtogether physical foregrounds, leaving only the primaryCMB behind after cleaning.

As discussed in Section 5, our Gibbs sampling method

is computationally intensive, and requires the use of someclever algorithms to speed it up. We demonstrated thetools necessary to make the method practical, but thenext step is to construct a full implementation. For thefull sky, the best option is most likely the multi-levelsolver of Seljebotn et al. (2014), but in the near term aflat sky version is a more straightforward prospect. Fulldetails of a flat-sky implementation, including full nu-merical validation of the method, are deferred to a forth-coming paper (Louis & Bull 2014).

The focus of this paper has been on secondaryanisotropies of the CMB, such as the TSZ and KSZ ef-fects, and the others listed in Section 6, but one couldalso consider extending our Gibbs sampling framework toadditional datasets, such as surveys of large scale struc-ture (LSS). We have already considered a basic way ofdoing this in our discussion of the SZ effect, where ex-ternal cluster catalogues were used to determine the po-sitions of galaxy clusters on CMB maps. A more so-phisticated ‘combined’ Gibbs scheme could also sampleparameters of the external cluster survey, such as detec-tion thresholds or selection functions, or indeed any otherquantity with statistical uncertainty attached. One couldeven define a scheme that sampled the density field re-constructed from galaxy redshift surveys and then cross-correlated it with the CMB. These possibilities are leftfor future work; we only wish to note here that properlyaccounting for correlated (physical/nuisance) parametersbetween disparate datasets is likely to become more cru-cial as cross-correlation analyses become more common,and that Gibbs sampling provides the flexibility to tacklethis problem.

Acknowledgements — We are grateful to G. Addison,R. Battye, T. Louis, E. Macaulay, and M. Schammel foruseful discussions. This project was supported by ERCStarting Grant StG2010-257080. PB acknowledges ad-ditional support from STFC, and hospitality from Cal-tech/JPL. IKW acknowledges support from ERC grant259505. PGF acknowledges support from Leverhulme,STFC, BIPAC and the Oxford Martin School. Part ofthe research was carried out at the Jet Propulsion Lab-oratory, California Institute of Technology, under a con-tract with NASA. Some of the results in this paper havebeen derived using the HEALPix software and analysispackage (Gorski et al. 2005).

APPENDIX

BLIND DETECTION OF THE TSZ EFFECT

The Gibbs sampling framework outlined in Section 2 can also be used to perform blind detections of clusters, usingonly the frequency dependence of the thermal SZ effect. The most suitable model for blind TSZ detection within ourframework is a pixel-based component (see Section 2.4) with a free amplitude and fixed TSZ spectrum (Eq. 19) ineach pixel. One could also impose a signal covariance matrix based on the TSZ angular power spectrum predictedfrom linear theory and simulations, but for experiments with multiple frequency channels there is generally enoughdata to make this unnecessary.

While the spectrum of the TSZ effect is rather distinctive, other foreground emission must nevertheless be includedin the data model as well. Otherwise, one runs the risk that a substantial fraction of the unmodelled componentscould be misidentified, causing significant contamination of the estimated TSZ signal. The precise definition of theseother components will depend on the frequency coverage of the experiment in question, but for the 70 – 350 GHzwindow, where the TSZ signal is largest, the most important contaminants are typically CO emission lines, free-free,and thermal dust (Planck Collaboration 2013a). Contamination by point sources is also an issue, and so either a pointsource mask or a reliable cross-matching procedure is required as well.

13

With a data model in hand, one can then use a variation on the Gibbs scheme of Section 2.2 to sample from the jointposterior. A map of TSZ amplitudes is produced with each Gibbs iteration by virtue of the first step in the scheme (Eq.2). Once the Gibbs chain has converged, each of these maps will be a sample from the marginal distribution for the TSZcomponent – that is, the CMB, galactic foregrounds, and other model parameters are automatically marginalised inthese maps. This procedure directly propagates the uncertainty associated with the component separation procedurein full, so there is no need to perform simulations to estimate detection significance and the like.

The TSZ marginal maps can then be processed using an existing source finder or filtering technique to try andidentify clusters (e.g. Herranz et al. (2005)). Estimates of the noise are given by the standard deviation from the chainfor each pixel, and an effective signal map is given by the mean.

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