Constraints on gravity and dark energy from the pairwise kinematic Sunyaev-Zeldovich effect
Eva-Maria Mueller, Francesco de Bernardis, Rachel Bean, Michael Niemack
CConstraints on gravity and dark energy from the pairwise kinematicSunyaev-Zeldovich effect
Eva-Maria Mueller , , Francesco de Bernardis , Rachel Bean , Michael Niemack . Department of Physics, Cornell University, Ithaca, NY 14853, USA, Department of Astronomy, Cornell University, Ithaca, NY 14853, USA.
We calculate the constraints on dark energy and cosmic modifications to gravity achievable withupcoming cosmic microwave background (CMB) surveys sensitive to the Sunyaev-Zeldovich (SZ)effects. The analysis focuses on using the mean pairwise velocity of clusters as observed through thekinematic SZ effect (kSZ), an approach based on the same methods used for the first detection ofthe kSZ effect, and includes a detailed derivation and discussion of this statistic’s covariance undera variety of different survey assumptions.The potential of current, Stage II, and upcoming, Stage III and Stage IV, CMB observations areconsidered, in combination with contemporaneous spectroscopic and photometric galaxy observa-tions. A detailed assessment is made of the sensitivity to the assumed statistical and systematicuncertainties in the optical depth determination, the magnitude and uncertainty in the minimumdetectable mass, and the importance of pairwise velocity correlations at small separations, wherenon-linear effects can start to arise.In combination with Stage III constraints on the expansion history, such as those projected by theDark Energy Task Force, we forecast 5% and 2% for fractional errors on the growth factor, γ , forStage III and Stage IV surveys respectively, and 2% constraints on the growth rate, f g , for a Stage IVsurvey for 0 . < z < .
6. The results suggest that kSZ measurements of cluster peculiar velocities,obtained from cross-correlation with upcoming spectroscopic galaxy surveys, could provide robusttests of dark energy and theories of gravity on cosmic scales.
I. INTRODUCTION
The accelerating expansion of the universe continuesto be one of the most puzzling problems in cosmology.The background evolution of the universe is constrainedby measurements of the cosmic microwave background(CMB) (e.g. [1–4]), baryon acoustic oscillations (BAO)in the galaxy two point correlation function (e.g. [5–8]),as well as type 1a supernovae (SN) (e.g. [9, 10]).There is significant interest in differentiating betweenalternative explanations of cosmic acceleration by ex-tending beyond the expansion history to dark energy’simpact on the growth of structure (see [11–13] for re-views). This approach is key if a modification of gravityon astrophysical scales is responsible for cosmic accel-eration. Large scale structure observations provide twocomplementary probes of the properties of gravity: thebending of light due to a gravitational potential and theeffect of gravity on the motions of non-relativistic ob-jects. The latter manifests as the peculiar velocities ofgalaxies imprinted in redshift space distortions (RSD)in the galaxy correlation function [14] as well as clus-ter motions as observed through the kinematic Sunyaev-Zel’dovich (kSZ) effect [15]. Upcoming surveys such asthe Dark Energy Survey (DES) [16], HyperSuprimeCam(HSC) [87], the Large Synopic Survey Telescope (LSST)[17] and the Euclid [18] and WFIRST [19] space tele-scopes, will provide gravitational lensing surveys out toredshift 2, and beyond. Concurrently spectroscopic sur-veys such as eBOSS [20], DESI [21] and spectroscopyfrom Euclid and WFIRST, will provide both BAO andredshift space clustering measurements over overlappingepochs and survey areas. Each of those probes, though having the potential to constrain gravity, are affectedby systematic effects. Cosmological measurements us-ing weak gravitational lensing (WL) will require precisephotometric redshift and point spread function calibra-tions along with characterization of intrinsic alignmentcontamination of shear correlations, e.g. [22], that canbias and dilute dark energy constraints [23–25]. Accuratemodeling of redshift space clustering into the non-linearregime requires precise descriptions of the galaxy cluster-ing correlations beyond the Kaiser formula [26]. Clustersare high density environments that are highly affected bythe underlying theory of gravity. The peculiar velocitiesof clusters provide an alternative, complementary mea-surement of the cosmological gravitational potential fieldthat has different systematic uncertainties. Consideringthese as part of a multiple tracer approach will providethe clearest picture of gravity’s properties.Cluster motions leave a secondary imprint in the CMBknown as the kSZ effect [15], the process of CMB pho-tons passing through a cluster and being Doppler shifteddue to the cluster’s peculiar velocity relative to the CMBrest frame. This provides a potentially powerful comple-mentary measurement of gravity’s influence on cosmicstructure to the peculiar motions of individual galaxies[27–34]. Despite its potential, the kSZ has been hardto measure; the signal is small when compared to thethermal SZ effect and emission from dusty galaxies, anddoesn’t have a distinct frequency dependence. Observa-tional efforts to constrain the cluster peculiar velocitieshave come from multi-band photometry in combinationwith X-ray spectra [35–39] and spectroscopy around thethermal SZ null frequency [40]. Recent work extractedthe kSZ signature from individual clusters by combin- a r X i v : . [ a s t r o - ph . C O ] A ug ing sub-mm, X-ray and sub-arcminute resolution CMBdata to respectively remove dusty galaxy emission, esti-mate electron density and fit thermal and kinematic SZtemplates [41]. Data from the WMAP and
Planck satel-lites have been used to place upper limits on the bulkflows and statistical variation in cluster peculiar veloc-ities [42, 43], while South Polar Telescope (SPT) data[44, 45] and Atacama Cosmology Telescope (ACT) data[46] have been used to place limits on the kSZ signal fromthe epoch of reionization.Multi-band methods do not yet provide a practical ap-proach to extract the kSZ signal from thousands of clus-ters as desired for large scale cosmological correlations.Cross-correlating arcminute resolution CMB maps withcluster positions and redshifts determined by a spectro-scopic large scale structure survey can enable extractionof the pairwise kSZ signal [32, 33, 47]. Indeed, the firstdetection of the kSZ effect in the CMB spectrum wasmade by combining CMB measurements from the ACT[48] with the SDSS BOSS spectroscopic survey [20] tomeasure the mean pairwise momentum of clusters, usingluminous red galaxies as a tracer for clusters [49]. Thepairwise approach for extracting the kSZ signal measuresthe difference in peculiar velocities of nearby clusters as afunction of the comoving distance between the clusters.This approach minimizes contributions from the CMB,thermal SZ, and foregrounds, which can be treated asapproximately constant on these scales, and by averag-ing over many clusters pairs any effects independent ofthe separation will cancel. CMB surveys such as ACT-Pol [50], SPTPol [51], Advanced ACTPol [52], SPT-3G[53], and a next-generation, so-called Stage IV CMB sur-vey [54] in combination with overlapping galaxy surveys,such as those described above, can improve upon this de-tection and enable the use of mean pairwise velocities asa cosmological probe.In this paper, we study the constraints on dark energyand cosmic modifications to gravity expected from an-alyzing the mean pairwise velocity of clusters observedthrough the kSZ effect by upcoming CMB observationsin combination with spectroscopic large scale structureredshift surveys. In section II the analytical formalismused to construct statistics and associated covariances forcluster velocity correlations is summarized. The analysisapproach and findings are presented in section III, withconclusions and implications for future work discussed insection IV. A detailed derivation of key results in II ispresented in Appendix A.
II. FORMALISM
We consider the mean pairwise velocity of clusters de-rived from the kSZ effect as a probe for dark energymodels and modifications to general relativity. SectionII A outlines how the growth of structure can be usedto constrain modified gravity, II B summarizes the halomodel approach to analytically calculate the mean pair- wise velocity of clusters, and II C presents the formalismto estimate the covariance matrix of the mean pairwisevelocity. In sections II D and II E we discuss the fiducialcosmological model and survey assumptions.
A. Cosmic structure and modified gravity
Even though on large scales the universe appears ho-mogenous and isotropic, initial local matter overdensi-ties form galaxies and galaxy clusters and evolve into thelarge scale structure of the universe. The growth of thesestructures depends on the underlying physical theory andcan therefore be used to constrain cosmological models.According to linear theory, the matter over-density, δ m , is related to the velocity of dark matter particles,˙ δ m ∝ v m , which connects the time evolution of the per-turbations to the dark matter velocity. Any tracer ofthe underlying dark matter velocity distribution can beused to constrain cosmology and in particular modifiedgravity models. In a variety of modified gravity scenariosthe evolution of the density perturbations can be quitedifferent from standard gravity even though the back-ground expansion of the universe is undistinguishablefrom a Λ CDM universe (for example [55–57]). The lin-ear perturbation equations have a solution of the form δ m ( (cid:126)x, t ) = D a ( t ) δ ( x ), factorizing the spatial and tempo-ral dependency, with D a being the growth factor. Wecan define the growth rate at a given scale factor, a , as f g ( a ) ≡ d ln D a d ln a (1)to parametrize the growth of structure. The growth rateis well approximated by f g ( a ) ≈ Ω m ( a ) γ with the growthindex γ (cid:39) .
55 for standard gravity [58, 59]. Pairwise ve-locity statistics can be used to constrain the cosmologicalmodel of the universe and the underlying theory of grav-ity [60, 61].
B. Motion of clusters as a probe of cosmology
We analytically model the expected large scale motionof clusters under cosmological gravitational interactionsby considering the properties of dark matter particles, inlinear theory, and then using a halo model to infer thevelocity statistics of gravitationally bound halos, whichwe use as proxies for galaxy clusters.Following the formalism outlined in [62], we assumelinear theory to describe the mean pairwise streamingvelocity, v , between two dark matter particles, at posi-tions r i and r j , in terms of their comoving separation r = | r i − r j | , v ( r ) = − f g ( a ) H ( a ) ar ¯ ξ ( r, a )1 + ξ ( r, a ) (2)
20 40 60 80 100 120 140 160 180 r [ Mpc/h ]0.80.91.01.11.2 V ( r ) / V ( r ) fid V ( r ) [ k m / s ] z = 0 . w = − . γ = 0 . w = − . γ = 0 . w = − . γ = 0 . w = − . γ = 0 . w = − . γ = 0 .
44 20 40 60 80 100 120 140 160 180 r [ Mpc/h ]0.60.81.01.21.4 V ( r ) / V ( r ) fid V ( r ) [ k m / s ] z = 0 . M min = 2 . × M (cid:12) M min = 1 . × M (cid:12) M min = 1 . × M (cid:12) M min = 0 . × M (cid:12) M min = 1 . × M (cid:12) FIG. 1: [Left panel] The mean pairwise cluster velocity, V , for different values of the dark energy equation of state parameter, w , and the modified gravity parameter, γ , at z = 0 .
15 and assuming a minimum cluster mass of M min = 1 × M (cid:12) . A morenegative w leads to an increase in V . A decreased value of γ increases the growth rate and therefore increases V whereas ahigher value of γ has the opposite effect. The same fractional change in γ has a greater effect on the amplitude of V thanchanging w . [Right panel] The mean pairwise cluster velocity, V , for different minimum mass cut-offs at redshift of z = 0 . M min changes the shape as well as the amplitude of V . Higher M min leads to an increase in mean pairwisevelocity since the more massive clusters tend to have higher streaming velocities. [Lower panels] Ratio of the mean pairwisevelocity, V , for the different scenarios to that for the fiducial model, V fid . where ξ is the dark matter 2-point correlation functionand ¯ ξ the volume averaged correlation function, respec-tively defined as, ξ ( r, a ) = 12 π (cid:90) dkk j ( kr ) P ( k, a ) , (3)¯ ξ ( r, a ) = 3 r (cid:90) r dr (cid:48) r (cid:48) ξ ( r, a ) , (4)with P ( k, a ) being the dark matter power spectrum and j ( x ) = sin( x ) /x is the zeroth order spherical Bessel func-tion.The properties of dark matter halos of mass M , relativeto the dark matter distribution, can be modeled using ahalo bias b ( M, z ) = 1 + δ − σ ( M, a = 1) σ ( M, a = 1) δ crit D a , (5)where M ( r ) = 4 πR ¯ ρ/
3, ¯ ρ is the average cosmologicalmatter density, the critical overdensity is taken to havethe standard ΛCDM value of δ crit ≈ . σ ( m, a ) = 12 π (cid:90) ∞ dkk P ( k, a ) W ( kR ( M )) . (6) Surveys will generally include cluster halos over a rangeof masses above some limiting mass threshold, M min . Toanalyze the mass statistics we consider a mass averaged cluster pairwise velocity statistic, V , for pairs of clustersseparated by a comoving distance rV ( r, a ) = − H ( a ) af g ( a ) r ¯ ξ h ( r, a )1 + ξ h ( r, a ) , (7)which has an analogous expression to that in (2) [62, 63],with ξ h ( r, a ) = 12 π (cid:90) dkk j ( kr ) P lin ( k, a ) b (2) h ( k ) , (8)¯ ξ h ( r, a ) = 3 r (cid:90) r dr (cid:48) r (cid:48) ξ ( r, a ) b (1) h ( k ) . (9)The mass-averaged halo bias moments, b ( q ) h , are given by b ( q ) h = (cid:82) M max M min dM M n ( M ) b q ( M ) W [ kR ( M )] (cid:82) M max M min dM M n ( M ) W [ kR ( M )] (10)where n ( M ) is the number density of halos of mass M ,given by the Jenkins mass function, and the top-hat win-dow function W ( x ) = 3(sin x − x cos x ) /x .In Figure 1 we show the mean pairwise velocity, V asa function of cluster separation r for a number of cosmo-logical models at z = 0 .
15 for a survey with limiting mass M min = 10 M (cid:12) , assuming all other survey specificationsare fixed (left panel) and for various assumptions on thelimiting mass (right panel). The figure suggests that, aswith other linear growth rate related statistics, the equa-tion of state, w , and growth exponent, γ , have degen-erate effects on the pairwise velocity amplitude, throughtheir effects on the growth factor, and do not alter theshape of the function. However, as indicated in sectionIII D, the redshift dependence of these parameters helpsto break the degeneracy. To be more specific, the ampli-tude of V as a function of z is different for variations in γ compared to w . Increasing the minimum cluster massshifts the peak of the pairwise velocity function to largerscales (on scales below 60 Mpc) and boosts the overallamplitude on scales larger than this, because the largerclusters have a larger streaming velocity. C. Covariance matrix
Measurements of cluster velocities are subject to anumber of statistical and systematic uncertainties. First,discreteness effects need to be taken into consideration; asmooth continuous field is typically assumed to underly adiscrete distribution of local objects, which leads to shotnoise. For a large sample size the shot noise should be ap-proximately Gaussian resulting in an error proportionalto 1 /N [64], where N is the number of objects in the sam-ple. If the number of objects (e.g. clusters) in the sampleis not sufficiently large, the Gaussian limit breaks down,and an additional non-Gaussian contribution to the shotnoise can become relevant [65].Second, as in any cosmological survey, the measure-ment will be subject to cosmic variance due to the finitesize of the sample. Third, in addition to the statisticalerrors we include a velocity measurement error [63] toaccount for the accuracy of the measurements and theuncertainty in the optical depth of the clusters. The to-tal covariance for the mean pairwise velocity is thereforea combination of cosmic variance, shot noise, and thevelocity measurement error: C total V ( r, r (cid:48) ) = C cosmic V ( r, r (cid:48) ) + C shot V ( r, r (cid:48) )+ C measurement V ( r, r (cid:48) ) . (11)A detailed derivation of the covariance terms can befound in Appendix A. We summarize the results here.Defining an estimator for the mean pairwise velocity,ˆ V , enables the covariance matrix to be calculated using C V ( r, r (cid:48) ) = (cid:104) ˆ V ( r ) ˆ V ( r (cid:48) ) (cid:105) − (cid:104) ˆ V ( r ) (cid:105)(cid:104) ˆ V ( r (cid:48) ) (cid:105) (12)where (cid:104) ... (cid:105) is the volume average. For analyzing a surveywe include binning, as observations will be combined not at just one radius r but in bins of width ∆ r ,ˆ V ( r ) → V ∆ ( r ) = 1 V bin (cid:90) r +∆ r/ r − ∆ r/ ˜ r d ˜ r (cid:90) d Ω ˆ V (˜ r ) , (13)assuming spherical symmetry and where a ∆ subscriptindicates binned quantities over bins of size ∆ r .The covariance between the mean pairwise velocitiesof two cluster pairs, with the two pairs separated by r and r (cid:48) and using bin width ∆ r , can be expressed as C V ( r, r (cid:48) ) = 4 π V s ( a ) (cid:18) H ( a ) a ξ h ( r, a ) (cid:19) f g ( a ) × (cid:34)(cid:90) dk (cid:18) P lin ( k, a ) b (1) h ( k ) + 1 n ( a ) (cid:19) W ∆ ( kr ) W ∆ ( kr (cid:48) )+ ∆ rV ∆ ( r (cid:48) ) (cid:90) dkk P lin ( k, a ) b (1) h ( k ) n ( a ) W ∆ ( kr ) (cid:35) , (14)where V s ( a ) is the survey volume, and W ∆ ( kr ) = 3 R ˜ W ( kR min ) − R ˜ W ( kR max ) R − R (15)˜ W ( x ) = 2 cos( x ) + x sin( x ) x . (16)The first term in (14) is the Gaussian contributionto the covariance, which includes both cosmic variance( ∝ P ) and shot noise ( ∝ /n ). The second term is an ad-ditional contribution that is often neglected, which arisesif the Gaussian limit breaks down; we refer to this termas ‘Poisson’ shot noise as in [65]. While we find it issubdominant in comparison to the Gaussian terms for amass cut-off M ≤ × M (cid:12) (see Figure 3), it can beimportant for surveys with smaller cluster number densi-ties. The purely Gaussian shot noise contribution on theother hand is not insignificant and should be included.We include a contribution to the covariance due to theuncertainty in measuring the velocity given by [63], C measurement V ( r, r (cid:48) ) = 2 σ v N pair δ r,r (cid:48) (17)where σ v is the measurement error discussed in more de-tail in section III D, and N pair is the number of pairs ineach separation bin given by N pair ( r, a ) = n ( a ) V s ( a )2 × (cid:0) V ∆ ( r ) n ( a ) + 4 πr n ( a ) ξ h ( r, a )∆ r (cid:1) . (18)As shown in Figure 2, the number of cluster pairs in-creases rapidly with decreased minimum mass. The mea-surement error term in the covaraince is proportional to1 /N pairs and will increase quickly with an increasing num-ber of bins since the number of cluster pairs directly de-pends on the size of the r -bin.Figure 3 shows the diagonal elements of the covariancematrix for the different covariance components, for a bin
20 40 60 80 100 120 140 160 180 r/ ( Mpc/h )10 N p a i r ( r ) M min = 2 × M (cid:12) M min = 1 × M (cid:12) M min = 6 × M (cid:12) FIG. 2: Number of cluster pairs, N pair ( r ), versus separationin bins of ∆ r = 2 Mpc/h for different mass cutoffs at redshift0 . < z < .
2. This assumes a Jenkins mass function and a6000 square degree survey. width of ∆ r = 2 Mpc / h. As cluster separation increases,the covariance becomes dominated by cosmic variance,while at smaller separations (cid:46)
40 Mpc / h, the contribu-tions from each of the terms becomes comparable. Asa result of the multiple contributions to the covariancematrix, and their respective sensitivities to bin size andcluster separation, the total covariance matrix slightlydepends upon the number of bins. The measurement er-ror and shot noise can be reduced by choosing a coarserbinning with the trade-off of decreased resolution and lossof information. On the other hand, the fractional contri-bution of the cosmic variance will increase as the size ofthe bins increases. Once the cosmic variance dominatesnothing can be gained from a coarser binning. A verycoarse binning marginally reduces the constraints, e.g.using ∆ r = 20 Mpc / h lowers the FoM by 30% comparedto ∆ r = 2 Mpc / h, however, any bin size smaller than∆ r = 5 Mpc / h leads to equivalent results. Throughoutthe analysis we assume a bin size of ∆ r = 2 Mpc / h.Off-diagonal covariances between cluster pairs of differ-ent separations are important. Figure 4 shows the covari-ance contributions from cosmic variance and shot noiseand indicates the comparative importance of off-diagonalterms. The off-diagonal contributions have a notable ef-fect on the Fisher matrix amplitudes as a function of sep-aration, giving rise to the differences between the left andright panels in Figure 3. The right panel shows the effecton the Fisher matrix of changing key model assumptions,the minimum detectable cluster mass and the mean pair-wise velocity uncertainty. Altering the mass limit has alarger effect than comparable changes to the the measure-ment error because the number of clusters and clusterpairs strongly depend on the limiting mass (see Figure 2), changing shot noise as well as the measurement errorcontribution to the covariance significantly. D. Cosmological Model
For our analysis we consider constraints on 9 cosmo-logical parameters: p = { Ω b h , Ω m h , Ω k , Ω Λ , w , w a , n s , ln A s , γ } (19)where Ω b , Ω m , Ω k and Ω Λ are the dimensionless baryon,matter, curvature and dark energy densities respectively, h is the Hubble constant in units of 100 km/s/Mpc, w and w a are the dark energy equation of state parameters,such that the equation of state is w ( a ) = w + (1 − a ) w a , γ is the growth rate exponent, such that f g = Ω m ( a ) γ ,and n s and A s are the spectral index and normalizationof the primordial spectrum of curvature perturbations.Throughout this paper we assume a fiducial modelthat is a ΛCDM cosmological model with parame-ters consistent with those adopted in [18]: Ω b h =0 . , Ω m h = 0 . , Ω k = 0 , Ω Λ = 0 . , w = − . , w a = 0 , n s = 1 , ln(10 A s ) = 3 . p µ and p ν , from (19), is given by F µν = N z (cid:88) i N r (cid:88) p,q ∂V ( r p , z i ) ∂p µ Cov − i ( r p , r q , z i ) ∂V ( r p , z i ) ∂p ν , (20)where Cov ( r p , r q , z i ) is the covariance matrix betweentwo clusters pairs as defined in II C, including a redshiftbin with mid-point z i and the clusters in each pair havingcomoving separations of r p and r q . N z and N r are thenumber of redshift and spatial separation bins, respec-tively.We quote results in terms of the Dark Energy Figuresof Merit (FoM) [66] defined asFoM = det (cid:2) ( F − ) (cid:3) − / w ,w a (21)FoM GR = det (cid:2) ( F − ) (cid:3) − / w ,w a . (22)( F − ) w ,w a is the 2 × γ . Thisprocedure is equivalent to marginalizing over the 7 pa-rameters (for MG) or 6 parameters (for GR) of the modelconsidered.Throughout this paper we consider results in combina-tion with either simply a Planck-like CMB prior or a DarkEnergy Taskforce (DETF) [66] prior that includes CMB,SN, and non-kSZ related LSS constraints on the back-ground cosmological and dark energy parameters. Wedo not include a prior on the modified gravity parame-ters unless stated otherwise. For the Planck-like CMBsurvey, we consider complementary constraints on thecosmological parameters from the temperature ( T ) andpolarization ( E ) measurements up to l = 3000 as sum-marized in Table I [88].
20 40 60 80 100 120 140 160 180r [Mpc/h]0 . . . . . . . p C o v V ( r ) ( r , r ) / V ( r ) Cosmic varianceGaussian shotMeasurement errorPoisson shot 20 40 60 80 100 120 140 160 180r [Mpc/h]0 . . . . . / (cid:0) V ( r ) p F V ( r ) ( r , r ) (cid:1) M min = 2 × M (cid:12) , σ refv M min = 1 × M (cid:12) , σ refv M min = 5 × M (cid:12) , σ refv M min = 1 × M (cid:12) , σ refv × M min = 1 × M (cid:12) , σ refv / FIG. 3: [Left panel] The relative error on the mean pairwise velocity of clusters at redshift 0 . < z < . r = 2 Mpc / h assuming a Stage III like survey (see Table II). The Poisson shot noise is sub-dominant compared to theother terms, the Gaussian shot noise term however cannot be neglected. [Right panel] One over the diagonal terms of the totalFisher matrix relative to the mean pairwise velocity for varying the minimum mass and the measurement error. The effect ofthe minimum mass on the fisher matrix is more prevailing than the dependency on the measurement error.
20 40 60 80 100 120 140 160 180r [Mpc/h]20406080100120140160180 r [ M p c / h ] C cosmic V ( r ) [( km/s ) ] 4080120160200240280320 20 40 60 80 100 120 140 160 180r [Mpc/h]20406080100120140160180 r [ M p c / h ] C shot V ( r ) [( km/s ) ] 102030405060708090 FIG. 4: 2D contour plots of the cosmic variance [left panel] and the shot noise term [right panel] at redshift 0 . < z < . r = 2 Mpc / h, a lower mass limit of M min = 1 × M (cid:12) , and a sky coverage of 6000 squaredegrees. Note that both terms have notable non-zero off-diagonal terms that affect the total inverse covariance used in theFisher analysis, and that while the cosmic variance values are larger, the Gaussian noise term should not be neglected as it canhave a significant effect, particularly for small separations. E. Survey Specifications
We forecast cosmological constraints for three differ-ent combinations of surveys: 1) a current (Stage II)CMB survey, such as ACTPol [50], combined with agalaxy sample that includes spectroscopic redshifts, suchas SDSS BOSS [20], 2) a near-term (Stage III) survey,such as Advanced ACTPol [52], also combined with SDSSBOSS, and 3) a longer-term (Stage IV) survey, such as CMB-S4 [54], combined with a next generation spectro-scopic survey, such as DESI [21].The mean cluster pairwise velocity can be measuredby cross-correlating the kSZ signal with cluster positionsand redshifts. For the cluster sample, we assume that aspectroscopic survey provides redshifts to luminous redgalaxies (LRGs) over an overlapping area with the CMBsurvey. Recent studies show that the kSZ signal can beextracted from the CMB maps using LRGs of the BOSS
Frequency (GHz)100 143 217 f sky θ FWHM (arcmin) 10.7 8.0 5.5 σ T ( µ K) 5.4 6.0 13.1 σ E ( µ K) - 11.4 26.7TABLE I: CMB survey specifications, for the sky coverage, f sky , beam size, θ FWHM , and noise levels per pixel for thetemperature and polarization detections at 3 frequencies, fora Planck-like survey. Survey StageSurvey Parameters II III IVCMB ∆ T instr ( µK arcmin) 20 7 1Galaxy z min z max z bins, N z M min (10 M (cid:12) ) 1 1 0 . T inst , along withthe assumed optical large scale structure survey redshift range z min < z < z max , redshift binning, and minimum detectablecluster mass, M min are shown. We consider an effective skycoverage by estimating the degree of overlap between the re-spective CMB and optically selected cluster datasets. survey as a proxy for clusters [67]. Using LRGs creates alarge, precise positioned sample of tracers to extract thekSZ correlation.However, there are several factors that need to be con-sidered in using LRGs as cluster tracers. LRGs are notperfect tracers of a cluster’s center, with perhaps 40% ofbright LRGs and 70% of faint LRGs off-centered, satel-lite galaxies [68] that may be related to cluster mergers[69]. The imprecise match between LRGs and clusterscould lead to detrimental misalignments, such as tryingto extract the kSZ signal from positions that are not as-sociated with clusters or an incomplete cluster catalog ifspectroscopic measurements of an LRG near the clustercenter were not obtained. The theoretical mean clusterpairwise velocity is an observable averaged over all clusterpairs assuming a complete sample above a limiting min-imum mass. While Hand et al. [67] optimize the angularsize of the CMB sub-map used in the stacking approachto minimize the overall covariance, this does not ensurethat the cluster sample obtained from the LRGs is com-plete. Further studies are needed to quantify the effectsof using LRGs as cluster tracers and ensure that no biasis introduced in the analysis before this approach can beused for cosmological constraints. Another issue is thatthe uncertainty in the minimum mass of the cluster sam-ple associated with the LRGs is difficult to estimate, al-though, the minimum mass uncertainty could be treatedas an additional nuisance parameter in the analysis. To acknowledge these issues in our forecasts, we as-sume a scenario that aims to maximize cluster complete-ness and purity, with a well defined cluster mass cut-off, rather than cluster number density. We select a sur-vey area that has photometric and spectroscopic galaxycatalogs and overlapping CMB kSZ data. Specifically,we consider BOSS and a DESI-like survey, for whichwe expect photometric catalogs to exist over the surveyarea. We note that Euclid spectroscopic and imagingsurveys, and LSST imaging with overlapping WFIRSTimaging and spectroscopy would also provide future valu-able datasets at higher redshifts. The uncertainties inthe cosmological parameters evolve as the square rootof the sky coverage. Requiring spectroscopic redshifts,e.g. from BOSS, limits the survey area, but providesconfidence that the comoving cluster separation can beaccurately calculated as in [49]. Photometric informationallows cluster detection, and mass estimates, using algo-rithms, such as the friends-of-friends, as used in redMaP-Per [70], to maximize the completeness and purity of thecluster sample, with the drawback of a limited number ofclusters and a volume-limited catalog. A study of usingonly photometric information to extract the kSZ signalcan be found in [71].The survey specification assumed in our analysis forthe CMB and large scale structure Stage II, III and IVsurveys are given in Table II. We assume a BOSS-likespectroscopic survey for Stage II and III and a DESI-like Stage IV survey with redshift ranges that are deter-mined by the redshift coverage of the LRG sample andassume joint photometric survey data. We assume StageII and Stage III have access to the same or compara-ble LRG surveys so retain the same limiting mass, butdo slightly increase CMB overlap with these data dueto the larger survey area planned for Advanced ACT-Pol [52]. For Stage IV we assume a deeper LRG surveythat provides lower minimum mass, higher z , and largeroverlap. Our minimum mass assumptions are conserva-tive, and will likely be improved upon at each respectivestage. As an example, the LSST survey projects thatthe minimum detectable cluster mass at z ∼ . ∼ × M (cid:12) after a single visit image inall bands, and be better than 10 M (cid:12) in all bands inthe complete ten-year survey [17]. Similarly the SDSS-derived MaxBCG Catalog already achieves 90% purityand >
85% completeness for clusters of masses exceeding10 M (cid:12) [72].The measurement error for the radial peculiar velocity, v , of a cluster is a combination of the instrumental sen-sitivity as well as the uncertainty in the optical depth, τ ,for each cluster as the kSZ signal is proportional to τ asfollows [15], ∆ T kSZ T CMB = − vc τ, (23)where T CMB is the temperature of the CMB. We estimatethe total measurement error by adding those two sources
Redshift bin0 .
15 0 .
25 0 .
35 0 .
45 0 . τ τ / ¯ τ ) σ τ (km/s) 120 σ instr Stage II 290 440 540 - -(km/s) Stage III 100 150 190 - -Stage IV 15 22 27 34 42 σ v Stage II 310 460 560 - -(km/s) Stage III 160 200 230 - -Stage IV 120 120 120 120 130TABLE III: The assumed individual contribution from instru-ment sensitivity, σ instr , and uncertainty in τ , σ τ . The valuesof τ and fractional uncertainty in τ , (∆ τ / ¯ τ ) , are estimatedfrom simulations assuming a convolution over a 1.3 (cid:48) beam. σ v is the total measurement uncertainty for the reference case. of uncertainty in quadrature as σ v = (cid:113) σ + σ τ . (24)The accuracy of the instrument is given by σ instr = ∆ T instr ∆ T kSZ × v = ∆ T pixel / (cid:112) N pixel τ v/cT CMB × v (25)where ∆ T pixel is the sensitivity of the instrument perpixel and N pixel being the number of pixels of a clus-ter. We assume that an average size cluster will have N pixel ≈ σ τ = ∆ ττ × v. (26)Assumed uncertainties contributing to the measure-ment error are summarized in Table III. We use the scat-ter in the optical depth, | ∆ τ / ¯ τ | , and the mean value of τ from simulations [73] [89], to obtain an indicative es-timate for the intrinsic dispersion in τ averaged over allcluster masses. For the fiducial analysis we do not includeany further dispersion arising from potential additionalmeasurement accuracy in determining τ . Section III Dincludes a discussion of the impact of additional factorsaffecting the measurement error on the cosmological con-straints. III. ANALYSIS
Section III A summarizes and compares the results ofeach survey. The effect of modeling assumptions on theminimum detectable cluster mass, the minimum clusterseparation considered, the measurement error, and thedark energy model are discussed in sections III B-III E. − . − . − . − . − . . . w − . . . . . . γ kSZ + CMB prior . < z < . . < z < . FIG. 5: 2D projected likelihoods for the w − γ parameterspace, showing the 68% and 95% confidence levels for StageIV-like survey are shown for two well-separated spectroscopicredshift bins, 0 . < z < . . < z < . . < z < . w and γ and improves the kSZ driven constraints onthe growth history. A. Potential kSZ constraints on dark energy andmodified gravity
In this section we discuss the potential of upcomingkSZ surveys to constrain dark energy and modified grav-ity parameters. Figure 1 shows both the equation ofstate, driving the expansion history, and γ , that mod-ifies the growth history of density perturbations, havequalitatively similar effects on the pairwise velocity func-tion through their effect on the linear growth factor. Forcluster measurements in each individual redshift bin thiscreates a degeneracy between the equation of state and γ parameters. As shown in Figure 5, the use of multipleredshift bins allows the differences in the evolution of thegrowth rate for the dark energy and modified gravity pa-rameters to be distinguished. The constraints on w and γ from the low and high redshift bins are markedly or-thogonal; in combination this complementarity tightensthe constraints, in particular on the growth factor. InFigure 6 we present the 2D marginalized constraints inthe w − γ parameter plane for kSZ in combination with aPlanck-like CMB and DETF priors on all parameters ex-cluding γ , including CMB, BAO, weak lensing and super-novae measurements for a combination of Stage III like − . − . − . − . − . − . − . − . − . w . . . . . . . . γ + CMB+ DETF Stage IIIStage IVStage IIIStage IIDETF Stage III FIG. 6: 2D projected likelihoods for w − γ parameterspace, showing the 68% confidence levels for Stage II (red),III (green) and IV (blue)-like surveys when combined withPlanck-like CMB priors only (dashed) and DETF stage IIIGR priors (solid, excluding DETF constraints on γ ) [66]. Forcomparison, the projected DETF Stage III constraints alone(including γ ), that includes CMB, SN and non-kSZ relatedLSS constraints, are shown (black solid line). surveys [66]. The suite of Stage III DETF-motivated ob-servables would provide stronger constraints on the equa-tion of state, through the addition of geometric measure-ments that constrain the expansion history. These breakthe degeneracy between w and γ from kSZ and CMBmeasurements alone. With the addition of a CMB prioron the data, however, the data can constrain γ to 10%,8% and 5% respective in the Stage II through IV surveyspecifications. The kSZ is a less powerful tool for con-straining the dark energy equation of state. A Stage IV-like survey can achieve figure of merits of FoM GR = 61with a CMB prior (which has FoM GR = 1 .
15 alone),and FoM = 292 with DETF Stage III data included(FoM GR = 116).Complementary, contemporaneous constraints frombaryonic acoustic oscillations and type Ia supernovaewill provide significantly tighter constraints on the back-ground expansion history and the equation of state. Ifwe include the impact of a DETF Stage III prior on allparameters, excluding the growth factor, the degeneracybetween the equation of state and growth factor is sig-nificantly reduced and the projected constraints on γ areimproved, with fractional errors of 5% and 2% for StageIII and Stage IV surveys. Table IV summarizes the darkenergy figure of merit (FoM), assuming modified gravity (9 parameters, marginalizing over γ ), General Relativity(GR) (8 parameters, fixing γ ) and a flat, General Rela-tivity cosmology (7 parameters, fixing γ and Ω k ), as wellas the 1 σ constraints of w , w a (marginalizing over γ )and γ for a Stage II, Stage III and Stage IV like survey, asspecified in Table II.These results could provide valuable complementaryconstraints to those on γ from spectroscopic galaxyclustering surveys. Projections include constraints of∆ γ/γ (cid:39)
5% from measurements at z > .
65 using OII fora DESI-like survey [74], and comparable from a Euclid-like Hα survey, for which [75] projected ∆ γ/γ = 4%(assuming a luminosity function [76] that has since beenrevised downwards to lower Hα number counts [77, 78]). B. Dependence on minimum mass of the galaxycluster sample
For cluster abundance measurements knowing the pre-cision with which the minimum mass is known is impor-tant. To assess the degree of precision required for thepairwise measurements we consider the impact on thecosmological constraints of marginalizing over the mini-mum mass, with a 15% prior on M min . The middle panelof Table IV shows the effects of this marginalization: theconstraints are loosened only slightly compared to thefiducial case, that has no marginalization over the min-imum mass. This implies that a precise knowledge ofthe minimum mass is not crucial to achieve cosmologicalconstraints. An explanation for the comparative insen-sitivity of the dark energy constraints to uncertaintiesin M min , can be understood with reference to Figure 1.While varying dark energy parameters and M min bothchange the large scale pairwise velocity amplitude theminimum mass also changes the shape of the pairwisevelocity function. This means that uncertainties in theminimum mass can be discerned from those in dark en-ergy, and do not translate into a comparable degradationof constraints on w or γ .The measurement uncertainty on the mean pairwisevelocity decreases with the number of clusters used forthe cross-correlation. The upper panels of Figure 7presents the dependence of the FoM and ∆ γ/γ con-straints on the assumed minimum observed mass. The in-creased number density of clusters and cluster pairs aris-ing from a lower mass bound, below ∼ M (cid:12) , signifi-cantly improves the statistical uncertainties in the pair-wise velocity. For our analysis we integrated over a Jenk-ins mass function using the minimum observed mass asour lower limit. As the number density of clusters dropsoff quickly for higher masses the constraints deterioratestrongly for a minimum mass above M > × M (cid:12) .Assuming that the complications discussed in sectionII E, in determining LRG centrality and cluster mass es-timates, can be controlled, in principle one could achievemuch higher number densities and a smaller minimummass. This would increase the number of pairs in the0 Fiducial assumptions + Uncertainty in M min + Lower M min Stage II Stage III Stage IV Stage II Stage III Stage IV Stage II Stage III Stage IV+CMB FoM MG GR flat
37 57 128 29 39 94 52 68 206 σ ( w ) 0.72 0.68 0.33 0.73 0.69 0.33 0.63 0.55 0.18 σ ( w a ) 2.6 2.5 1.2 2.6 2.5 1.2 2.3 2.0 0.6∆ γ/γ MG
131 152 273FoM GR
133 156 292FoM flat
181 213 405 σ ( w ) 0.10 0.08 0.06 σ ( w a ) 0.29 0.26 0.21∆ γ/γ γ ) from the DETF Stage III survey.For reference, the Planck-like Fisher matrix alone has FoM GR = 1 .
15 and DETF has FoM GR = 116. [Central columns] Resultsin which the impact of an uncertainty in the exact minimum mass of the cluster sample, M min , is included by marginalizingover M min as a nuisance parameter with a 15% prior imposed. [Right columns] Results for a more optimistic mass cut-off of M min = 4 × M (cid:12) for Stage II and III and M min = 1 × M (cid:12) for Stage IV with marginalization over M min with a 15%prior imposed as well. Constraints as a function of M min are also shown at the top of Figure 7. cluster sample. The window functions for lower masshalos would include additional information in the massaveraged statistics from the power spectrum at smallerscales that would lead to tighter constraints on the cos-mological parameters. The right columns of Table IVshow the results assuming a more optimistic mass cut-offthan the reference case, M min = 4 × M (cid:12) for StageII and III and M min = 1 × M (cid:12) for Stage IV. To ac-count for the uncertainty in mass we marginalize over theminimum mass assuming a 15% prior. The GR figures ofmerit, with a CMB prior, are improved from FoM GR = 8and FoM GR = 14 for Stage II and III to FoM GR = 12and FoM GR = 20, compared to the reference scenarios,and by a factor of 1 . γ reduces to ∆ γ/γ = 0 .
07, 0 .
06, and 0 .
02 for Stage II, IIIand IV.
C. Dependence on the non-linearity cut-off
As shown in Figure 3, the inverse covariance rises atlower cluster separations so that the inclusion of clusterpairs at small separation can have a potentially signif-icant effect on improving the dark energy constraints.Simulation show a deviation from the predicted theoret-ical mean pairwise velocity, however, starting at separa-tions of r <
45 Mpc / h [63] so that the non-linear correc-tions to the cluster motion needs to be considered. Equa-tion (2) has two major deficiencies: It relies on lineartheory to model the underlying dark matter distribution[60, 62] and it assumes a linear, scale-independent bias[62]. The former leads to a discrepancy of the dark matterpairwise velocity with linear theory at non-linear scalesaround r ≤
10 Mpc / h, the latter introduces deviationsat even larger scales. It is worthwhile, therefore, to as- sess how accurately the mean pairwise velocity of clusterscan be modeled in the transition to the non-linear regimeand how the cosmological constraints depend upon theassumed limiting minimum mass.In Figure 7 we highlight the sensitivity of the figures ofmerit and uncertainty in the modified gravity parameter γ to the assumptions about the smallest cluster separa-tions to be included in the analysis, parametrized here by r min . For a Stage IV like survey including all scales up to r = 5 Mpc / h more than doubles the FoM compared toan analysis with separations above 50 Mpc / h excludedand halves the uncertainty on γ .In this work we chose a moderate approach cutting offour analysis in the mildly non-linear regime using a min-imum separation of r min = 20 Mpc /h . On-going work onusing an perturbative approach to model non-linearities[79] and improved N-body simulations suggests that theformalism will be improved in the near future to fullyexploit the mildly non-linear regime. D. Dependence on the measurement error
Central to utilizing the kSZ for cosmology, is the abilityto measure the pairwise momentum accurately, and thenin turn extract the pairwise velocity, from the momen-tum, through being able to determine the cluster opticaldepths. In this section we investigate in more detail thesensitivity of the constraints to these important effects.As described in section II E, the measurement error ofa given cluster is given by the combination in quadra-ture of the instrument noise and the uncertainty in theoptical depth of the cluster. In the fiducial analysis weinclude an uncertainty in the measurement of τ based onthe intrinsic dispersion in the optical depth observed in1 M min [ M (cid:12) ]050100150 F i g u r e s o f m e r i t kSZ+CMB priorsGRMG Stage IVStage IIIStage II 10 M min [ M (cid:12) ]0 . . . . . . . . . . ∆ γ / γ kSZ+CMB priorsStage IVStage IIIStage II0 10 20 30 40 50 r min [ Mpc/h ]020406080100 F i g u r e s o f m e r i t kSZ+CMB priorsGRMG Stage IVStage IIIStage II 0 10 20 30 40 50 r min [ Mpc/h ]0 . . . . . . . . . ∆ γ / γ kSZ+CMB priorsStage IVStage IIIStage II FIG. 7: [Upper panel] The impact of the assumed minimum mass of the cluster sample, M min , on the dark energy figures ofmerit (FoM) and uncertainty on the growth factor, ∆ γ/γ , for the Stage II (red), Stage III (green) and Stage IV (blue) referencesurvey specifications (as given in Table II) with a Planck-like CMB prior on all parameters except γ . FoM plots show resultsassuming standard general relativity (‘GR’, solid lines) and when the growth factor is marginalized over (‘MG’, dashed lines).[Lower panel] The impact of including observations on small scales, denoted by the minimum separation r min . While includingsmaller-scale observations below ∼
20 Mpc /h would appear to improve both the FoM and ∆ γ/γ , as discussed in the text, wenote that caution must be used in including these scales, with the potential for additional theoretical uncertainties, not includedhere, as non-linear effects become important. cluster simulations, averaged over all masses. While thisdoesn’t include the measurement error in estimating theoptical depth, it also does not include additional infor-mation in the mass dependence of the optical depth thatcould reduce the intrinsic dispersion estimator throughthe creation of a fitting function. Possible ways to esti-mate τ beyond the scope of this paper include combiningthermal SZ and X-ray observations to break the electrontemperature-optical depth degeneracy that will partiallyaffect even multi-frequency arcminute resolution obser-vations [80]. This technique relies on theoretical assump-tions and modeling to connect the electron temperatureto the X-ray temperature, that need more detailed test-ing against simulations. A polarization sensitive stage IVCMB survey may be able to measure τ by stacking clus- ters to extract the polarization signal introduced by thescattering, which depends directly on the optical depth(see e.g. [81]).To understand the impact of greater uncertainty inthe determination of τ on the cosmological constraints,we consider two potential forms of uncertainties, shownin Figure 8. The first is the effect of increased statisticaldispersion, σ τ in the optical depths of the cluster sam-ple and the second is a systematic offset in the τ . Forthe latter, we introduce a nuisance parameter, b τ ( z ), ineach redshift bin that scales the amplitude of the meanpairwise velocity, ˆ V ( z ) = b τ ( z ) V ( z ), and consider its ef-fect on cosmological constraints when marginalizing over b τ ( z ). Additionally we consider a constant, redshift in-dependent nuisance parameter, b τ , that scales the ampli-2 σ τ [ km/s ]01020304050607080 F i g u r e s o f m e r i t kSZ+CMB priorsGRMG Stage IVStage IIIStage II 0 200 400 600 800 1000 σ τ [ km/s ]0 . . . . . . . . . ∆ γ / γ kSZ+CMB priorsStage IVStage IIIStage II10 − − − ∆ b priorτ F i g u r e s o f m e r i t kSZ+CMB priors b τ ( z ) b τ Stage IVStage IIIStage II 10 − − − ∆ b priorτ . . . . . . ∆ γ / γ kSZ+CMB priors b τ ( z ) b τ Stage IVStage IIIStage II
FIG. 8: The impact of modeling assumptions in the determination of τ for each cluster on the dark energy FoM [left panels] andfractional constraints on the growth factor [right panels] for Stage II (red), Stage III (green) and Stage IV (blue) surveys. Theupper panels show the effect of increasing a statistical dispersion in the τ measurement, σ τ , in the pairwise velocity covariance.The upper left shows the FoM assuming the growth rate is determined by GR (solid lines) and marginalizing over a freelyvarying γ (‘MG’, dashed lines), which corresponds to the ∆ γ/γ constraints in the upper right panel. The lower panels showthe effect of a prior on a systematic offset in the τ value, parameterized by a multiplicative bias in each redshift bin (solidlines), b τ ( z ), and a redshift independent multiplicative bias (dashed lines), b τ . A detailed discussion of the relative sensitivitiesis provided in the text. tude across all clusters. For clarity, when studying theimpact of b τ we remove the σ τ contribution to the co-variance and purely parameterize the uncertainty in τ through a prior on b τ .Table II shows that for a near-term Stage II survey thenoise will be dominated by the instrument accuracy, fora more sensitive Stage III both components become com-parable, and for a Stage IV survey the velocity accuracymay be limited by the accuracy of τ . This is reflectedin the top panels of Figure 8 in which varying the am-plitude of σ τ between 0 and 1000 km/s only minimallychanges the constraints on the dark energy FoM and theconstraints on γ for Stage II and III.For Stage II and Stage III surveys, conclusions for theeffect of b τ on w and w a are similar to those for σ τ . The constraints on these dark energy parameters are princi-pally determined by the Planck-like prior, independent ofthe kSZ constraints, and uncorrelated with b τ . For theStage IV survey the kSZ constraints provide additionalconstraints on the equation of state, increasing their cor-relation with b τ , and the prior has a more pronouncedeffect on improving the FoM once below ∆ b τ (cid:46) − .Equivalently Stage II and III are not affected by the as-sumptions on the τ bias model; marginalizing over theamplitude in each redshift bin yields similar results to in-troducing a constant bias factor across all redshifts. ForStage IV slightly larger FoM are achieved for a redshiftindependent b τ model without imposing any prior.For the growth parameter, which is predominantly con-strained by the kSZ data, the model assumptions on the τ γ = γ + γ a (1 − a )Stage II Stage III Stage IVFoM MG
130 151 269 σ ( w ) 0.10 0.08 0.06 σ ( w a ) 0.29 0.26 0.21∆ γ /γ σ ( γ a ) 0.68 0.51 0.14TABLE V: A summary of the dark energy FoM and 1 σ marginalized constraints on for the dark energy parametersin the γ − γ a parametrization for Stage II, III and IV sce-narios in combination with a DETF prior on all parametersexcept γ and γ a . nuisance parameter are more important. Marginalizingover the amplitude in each redshift bin, b τ ( z ), withoutimposing any prior doubles the uncertainty in γ com-pared to a constant, redshift independent nuisance pa-rameter b τ . The difference between this behavior andthe FoM constraints (shown in Figure 8 lower panels) in-dicates that the redshift dependence of the FoM versus γ helps to break the degeneracy between them. A prioron the bias ∆ b τ (cid:46) − leads to a factor of 5 to 10 im-provement in the parameter constraints for the redshiftdependent b τ ( z ) model and a factor of 3 to 4 improve-ment for a constant b τ . For the Stage IV survey we findthat the multiple-redshift bins and improved covariancereduce the degeneracy between the τ bias parameter and γ , so that the systematic bias and the growth parametercan be constrained simultaneously by the data, and theprior has less effect.Beyond uncertainties in τ , the measurement uncer-tainty also depends on the peculiar velocity of the cluster,see (24). Even though the peculiar velocities of clustersare in principle distributed over a range of velocities, herefor simplicity we assume a rms velocity of v = 300 km/sthat corresponds to the peak velocity of the distributionfound in simulations [82] for all clusters to calculate thetotal measurement error. Fortunately the peak veloc-ity does not strongly depend on the mass of the cluster[82]. While future observations will reduce the velocitymeasurement uncertainty, there is an irreducible error ofaround σ v = (50 − E. Dependency on the Modified Gravityparametrization
In the previous sections we parametrized modifiedgravity models using one extra parameter γ that is as-sumed constant across all redshifts. Not all modifiedgravity models are well represented by such a simple pa-rameterization. Some models are better fit by a moregeneral parametrization that allows for a monotonic red-shift dependence in γ , γ ( a ) = γ + (1 − a ) γ a [85], equiva-lent to the dark energy w − w a model. Table V summa- .
35 0 .
40 0 .
45 0 .
50 0 .
55 0 .
60 0 .
65 0 .
70 0 . γ − . − . − . − . . . . . . γ a + DETF Stage III+ CMBStage IVStage IIIStage II FIG. 9: Marginalized constraints on the γ − γ a parame-ter space showing the 68% confidence contours for the pair-wise velocity constraints in combination with a Planck-like(dashed) or DETF Stage III (solid) prior on all parametersexcept γ and γ a . The fiducial model assumes GR with γ = 0 .
55 and γ a = 0. rizes the figures of merit as well as the 1 σ constraints on { w , w a , γ , γ a } . Introducing an additional extra param-eter loosens the constraints on the parameters with theadvantage of imposing a smaller theoretical prior on mod-ified gravity. Figure 9 shows the 1 σ and 2 σ constraintson γ − γ a for Stage II, III and IV.An even more general approach is to directly constrainthe growth rate in redshift bins as a ‘model-independent’way. This approach is particularly applicable for spec-troscopic galaxy surveys which can isolate peculiar ve-locity data, and hence the growth rate, in precise red-shift bins. In Figure 10 we present the forecasts for thisparametrization. We find, in combination with the StageIII DETF constraints on the equation of state, an uncer-tainty in f g of less than 2% at z ∼ . − . σ over a comparable redshift range to that consideredin our analysis [21]. DESI, along with Euclid [75] andWFIRST [19] will also provide complementary spectro-scopic constraints on the growth rate at higher redshifts,1 < z < . . . . . . . . . . ∆ f g / f g + DETF Stage III+ CMB Stage IVStage IIIStage II FIG. 10: Expected fractional 1 σ errors on the growth rate, f g , in each redshift bin for Stage II (red), Stage III (green)and Stage IV (blue) when combined with a Planck-like CMB(dashed) or DETF Stage III (solid) prior on all parametersexcept f g ( z ). IV. CONCLUSIONS
Recent analyses have demonstrated that the kSZ canbe successfully extracted from sub arcminute resolutionCMB maps by cross-correlating them with cluster posi-tions and redshift from spectroscopic large scale structuresurveys. In this paper we have considered the potentialto apply this technique, in light of planned CMB andLSS surveys with greater sensitivity and larger areas, toconstrain dark energy and modifications to gravity oncosmic scales using the mean pairwise velocity of clustersas an observable. We have extended the model presentedin [63] to account for the dependence on the binning incluster separations, shot noise, and potential contribu-tions to the total covariance matrix due to small numberdensities, that we show are significant, despite being fre-quently neglected, and provided a detailed derivation ofthe covariance components.The projected constraints are intimately related to notonly the quality of future data, determined by the in-strumental precision, but also to the modeling of the un-certainties in transforming the kSZ observations into ve-locity estimates that constrain the large scale structuregrowth history. We investigate a range of uncertainties,using reasonable assumptions based on simulations andprojected survey capabilities. We also study the sen-sitivity to assumptions by varying theoretical priors to understand and estimate the robustness of the results.We included a study of the effect of survey assumptionson the minimum detectable cluster mass and the mini-mum cluster separation that could be included, in lightof the influence of non-linear effects in the cluster mo-tions/correlations.The mean pairwise velocity is modeled assuming lin-ear theory for the underlying matter distribution as wellas the halo bias. Variations in the equation of state andthe growth rate affect the linear growth factor in simi-lar ways, that leads to degenerate effects on the pairwisevelocity amplitude. However, the different redshift de-pendence of these effects helps to break the degeneracy,and constraints on the expansion history, such as thosefrom Type 1a supernovae, BAO and CMB geometric con-straints, break it further, allowing growth information tobe extracted from the kSZ.The cluster sample’s minimum mass has a significantimpact on the predicted constraints. A smaller minimummass leads to an increase of the number of clusters in thecatalog (assuming the catalog is nearly complete) and sig-nificantly reduces the errors on all cosmological parame-ters. Assuming an optimistic mass cut-off for the upcom-ing cluster catalogs leads to an improvement on the fig-ures of merit (including CMB priors) from FoM GR = 61to FoM GR = 110 and a reduction of the 1 σ uncertaintyof the modified gravity parameter γ from 5% to 2% fora Stage IV survey compared to our fiducial assumption.In contrast, the uncertainty in the exact minimummass had only a mild impact on the dark energy andmodified gravity constraints. This was understood interms of the additional effect of the minimum mass onthe shape, as a function of cluster pair separation, as wellas amplitude, of the pairwise velocity. Marginalizing overthe minimum mass, while imposing a 15% prior in ouranalysis, in a scenario in which the covariance remainsunchanged, reduces the FoM for a Stage IV survey fromFoM GR = 61 to FoM GR = 43 and marginally loosens theconstrains on γ since the mean pairwise velocity is onlyweakly dependent on the assumed mass cut-off. In com-parison to the abundance of clusters as a cosmologicalprobe, the mean pairwise velocity of clusters appears tobe more robust to uncertainties in the mass calibration.Considering pairwise correlations down to cluster sep-arations of r = 5Mpc / h doubles the FoM compared toan analysis that excludes all scales below r = 50Mpc / h.While extending the analysis to smaller separations couldsignificantly improve the constraints, including scales inthe non-linear regime without accurate modeling couldalso potentially bias the constraints and introduce moresystematic uncertainties.Improved constraints on τ in clusters are critical foraccurate extraction of cluster streaming velocities fromkSZ measurements. We studied the impact of uncertain-ties in the τ measurement by considering constraints aswe varied the level of statistical uncertainty in individualcluster τ measurements and, separately, the effect of asystematic offset in the τ determinations. The later was5parameterized by a multiplicative bias parameter in eachredshift bin, b τ ( z ), as well as a constant, redshift inde-pendent bias, b τ . We found that the effect of σ τ on thedark energy FoM was minimal reflecting that the princi-pal constraints come from the external CMB or DETFprior. For Stage IV, in particular, the dispersion in τ doeshave a notable impact on the growth factor constraintsas the instrument contribution to the measurement er-ror and shot noise contributions have decreased. For thesystematic offset in τ , we found that the prior on b τ or b τ ( z ) had the biggest impact for Stage II and Stage IIIsurveys for which significant degeneracies exist betweenthe τ bias and γ . Though, a ∼
10% prior on the ampli-tude of τ enables these surveys to provide competitiveconstraints. For a Stage IV survey and a b τ bias model,the redshift bins and reduced covariance allowed both b τ and γ to be extracted from the data without the need fora prior on τ .In addition to a minimal model to modify gravity, inwhich a modification to the growth rate is parameterizedby a single parameter, γ , we also predict constraints formore general modified gravity parametrizations. We usea γ parametrization that monotonically varies with thescale factor, and a model independent approach of mea-suring the growth rate as a function of redshift, f g ( z ),directly. We forecast ∼ −
8% 1 σ errors on f g for StageII, 4% −
6% for Stage III and ∼
2% constraints for StageIV when combined with a Stage III DETF constraints onthe expansion history.Potential improvements in the covariance could includetaking advantage of multi-frequency information avail-able in upcoming surveys (e.g. [52, 53]) to improve thekSZ signal extraction and reduce the measurement error.Larger LRG catalogs could also be used, such as in thefirst kSZ detection; however, this increases uncertainty in the minimum mass of the cluster sample. Similarly,with the improvements in cluster photometric redshiftuncertainties that are coming from improved algorithmsand spectroscopic training sets, it may be feasible to usephotometric surveys, without spectroscopic follow up, tosignificantly enlarge the cluster sample. This will degradethe redshift accuracy, and therefore the measurements ofthe cluster separation, particularly on small scales; how-ever, the larger sample size will help compensate andmight even improve the constraining power.Measurements of the kSZ effect provide complemen-tary constraints on the growth of structure to weak lens-ing and redshift space distortion measurements by pro-viding measurements on larger physical scales and usinga highly complementary, and more massive, tracer of thecosmological gravitational field, that is not dependentupon a characterization of galaxy bias. Having a vari-ety of cosmological probes of dark energy and modifiedgravity with different systematics is going to be vital forreducing systematic effects and biases in parameter es-timation and determining the properties of dark energyand gravity in a variety of epochs and regimes.
Acknowledgments
The authors would like to thank Nicholas Battaglia,Joanna Dunkley, Kira Hicks, Arthur Kosowsky, ThomasLoredo, Eduardo Rozo and David Spergel for useful in-puts and discussions on pairwise statistics, survey capa-bilities and astrophysical uncertainties, and comments onthe manuscript.The work of EMM and RB is supported by NASA ATPgrants NNX11AI95G and NNX14AH53G, NASA ROSESgrant 12-EUCLID12- 0004, NSF CAREER grant 0844825and DoE grant de-sc0011838.
Appendix A: Derivation of the mean pairwise velocity covariance
In this section we provide a detailed derivation of the Gaussian and non-Gaussian contributions to the covariancematrix given in equation (14) in the main text. The covariance matrix specified in (12) in terms of the volume averageof the estimator, ˆ V of the pairwise velocity V , C V ( r, r (cid:48) ) = (cid:104) ˆ V ( r ) ˆ V ( r (cid:48) ) (cid:105) − (cid:104) ˆ V ( r ) (cid:105)(cid:104) ˆ V ( r (cid:48) ) (cid:105) . (A1)Let’s first consider a covariance between the pairwise cluster velocities of two cluster pairs, each with respectiveseparations r and r (cid:48) , we will then incorporate the effect of including finite bin sizes in the cluster separations. Usingthe expression for the mean pairwise cluster velocity, given in (2), the covariance of V can be written as C V ( r, a, r (cid:48) , a (cid:48) ) = 11 + ξ h ( r, a ) 23 rH ( a ) af g ( a ) 11 + ξ h ( r (cid:48) , a (cid:48) ) 23 r (cid:48) H ( a (cid:48) ) a (cid:48) f g ( a (cid:48) ) (A2) × (cid:104) (cid:104) ˆ¯ ξ h ( r ) ˆ¯ ξ h ( r (cid:48) ) (cid:105) − (cid:104) ˆ¯ ξ h ( r ) (cid:105)(cid:104) ˆ¯ ξ h ( r (cid:48) ) (cid:105) (cid:105) . (A3)For simplicity in the following derivation, we drop the subscript “h” (denoting halo) from the correlation function, ξ h and mass average correlation function, ¯ ξ h , denoting them respectively by ξ and ¯ ξ . Similarly we use P ( k, a ) todenote the halo linear dark matter power spectrum, given in full by P dmlin ( k, a ) b (2) h ( k ), and the cluster number density n cl is denoted n .6We define an estimator of the volume averaged correlation function ¯ ξ equivalently to the estimator of the correlationfunction ξ as ˆ¯ ξ ( (cid:126)r ) = 1 V ( (cid:126)r ) (cid:90) (cid:126)r d r V ( (cid:126)r ) (cid:90) d xW ( (cid:126)x ) (cid:90) d x (cid:48) W ( (cid:126)x (cid:48) ) δ ( (cid:126)x ) δ ( (cid:126)x (cid:48) ) δ (3) D ( (cid:126)x − (cid:126)x (cid:48) − (cid:126)r ) (A4)= 1 V ( (cid:126)r ) (cid:90) (cid:126)r d r (cid:90) d k (2 π ) (cid:90) d k (2 π ) δ (cid:126)k δ ∗ (cid:126)k e i(cid:126)k (cid:126)r h ( (cid:126)k − (cid:126)k , (cid:126)r ) (A5)where h ( (cid:126)k, (cid:126)r ) = 1 V ( (cid:126)r ) (cid:90) d xe i(cid:126)k(cid:126)r W ( (cid:126)x ) W ( (cid:126)x + (cid:126)r ) . (A6)The covariance matrix at a given redshift (dropping the subscript a) becomes C ¯ ξ ( (cid:126)r, (cid:126)r (cid:48) ) = 1 V ( (cid:126)r ) (cid:90) (cid:126)r d r V ( (cid:126)r (cid:48) ) (cid:90) (cid:126)r (cid:48) d r (cid:48) (cid:90) d k (2 π ) (cid:90) d k (2 π ) e i(cid:126)k (cid:126)r h ( (cid:126)k − (cid:126)k , (cid:126)r ) × (cid:90) d k (cid:48) (2 π ) (cid:90) d k (cid:48) (2 π ) e i(cid:126)k (cid:48) (cid:126)r (cid:48) h ( (cid:126)k (cid:48) − (cid:126)k (cid:48) , (cid:126)r (cid:48) ) × (cid:104) (cid:104) δ (cid:126)k δ ∗ (cid:126)k δ (cid:126)k (cid:48) δ ∗ (cid:126)k (cid:48) (cid:105) − (cid:104) δ (cid:126)k δ ∗ (cid:126)k (cid:105)(cid:104) δ (cid:126)k (cid:48) δ ∗ (cid:126)k (cid:48) (cid:105) (cid:105) . (A7)The expectation value of the four δ (cid:48) s including noise is (cid:104) (cid:104) δ (cid:126)k δ ∗ (cid:126)k δ (cid:126)k (cid:48) δ ∗ (cid:126)k (cid:48) (cid:105) − (cid:104) δ (cid:126)k δ ∗ (cid:126)k (cid:105)(cid:104) δ (cid:126)k (cid:48) δ ∗ (cid:126)k (cid:48) (cid:105) (cid:105) (A8)= (2 π ) δ (3) D ( (cid:126)k + (cid:126)k (cid:48) ) (cid:18) P ( (cid:126)k ) + 1 n (cid:19) (2 π ) δ (3) D ( (cid:126)k + (cid:126)k (cid:48) ) (cid:18) P ( (cid:126)k ) + 1 n (cid:19) (A9)+ (2 π ) δ (3) D ( (cid:126)k − (cid:126)k (cid:48) ) (cid:18) P ( (cid:126)k ) + 1 n (cid:19) (2 π ) δ (3) D ( (cid:126)k − (cid:126)k (cid:48) ) (cid:18) P ( (cid:126)k ) + 1 n (cid:19) (A10)+ (2 π ) δ (3) D ( (cid:126)k − (cid:126)k + (cid:126)k (cid:48) − (cid:126)k (cid:48) ) T full ( (cid:126)k, (cid:126)k , (cid:126)k (cid:48) , (cid:126)k (cid:48) ) . (A11)Evaluating the first two terms of the above equation leads to the Gaussian contribution of the covariance matrix C ξ ( (cid:126)r, (cid:126)r (cid:48) ) = 1 V ( (cid:126)r ) (cid:90) (cid:126)r d r V ( (cid:126)r (cid:48) ) (cid:90) (cid:126)r (cid:48) d r (cid:48) (cid:90) d k (2 π ) (cid:90) d k (2 π ) (cid:18) P ( (cid:126)k ) + 1 n (cid:19) (cid:18) P ( (cid:126)k ) + 1 n (cid:19) (A12) × h ( (cid:126)k − (cid:126)k , (cid:126)r ) h ∗ ( (cid:126)k (cid:48) − (cid:126)k (cid:48) , (cid:126)r (cid:48) ) (cid:16) e i(cid:126)k (cid:126)r + i(cid:126)k(cid:126)r (cid:48) + e i(cid:126)k (cid:126)r − i(cid:126)k (cid:126)r (cid:48) (cid:17) . (A13)Using the approximation (cid:90) d k (2 π ) h ( (cid:126)k, (cid:126)r ) h ∗ ( (cid:126)k, (cid:126)r (cid:48) ) = 1 V s ( a ) δ aa (cid:48) (A14)the Gaussian terms become C ¯ ξ ( (cid:126)r, (cid:126)r (cid:48) ) = 1 V ( (cid:126)r ) (cid:90) (cid:126)r d r V ( (cid:126)r (cid:48) ) (cid:90) (cid:126)r (cid:48) d r (cid:48) V s (cid:90) d k (2 π ) (cid:32) P ( (cid:126)k ) + 2 P ( (cid:126)k ) n + 1 n (cid:33) (cid:104) e i(cid:126)k ( (cid:126)r + (cid:126)r (cid:48) ) + e i(cid:126)k ( (cid:126)r − (cid:126)r (cid:48) ) (cid:105) (A15)= 1 π V s ( a ) 1 V ( r ) V ( r (cid:48) ) (cid:90) k dk (cid:90) r πr dr (cid:90) r (cid:48) πr (cid:48) dr (cid:48) (cid:18) sin( kr ) kr (cid:19) (cid:18) sin( kr (cid:48) ) kr (cid:48) (cid:19) P ( k ) (A16)= 9 π rr (cid:48) V s ( a ) (cid:90) dk (cid:18) P ( k ) + 2 P ( k ) n + 1 n (cid:19) j ( kr ) j ( kr (cid:48) ) . (A17)The remaining non-Gaussian terms come from the trispectrum [86] T full ( (cid:126)k, (cid:126)k , (cid:126)k (cid:48) , (cid:126)k (cid:48) ) = 1 n (cid:104) P ( (cid:126)k − (cid:126)k + (cid:126)k (cid:48) ) + P ( (cid:126)k + (cid:126)k (cid:48) − (cid:126)k (cid:48) ) + P ( (cid:126)k − (cid:126)k (cid:48) − (cid:126)k ) + P ( (cid:126)k (cid:48) − (cid:126)k − (cid:126)k (cid:48) ) (cid:105) + 1 n (cid:104) P ( (cid:126)k − (cid:126)k ) + P ( (cid:126)k + (cid:126)k (cid:48) ) + P ( (cid:126)k − (cid:126)k (cid:48) ) (cid:105) + 1 n (A18)7dropping all the terms proportional to the bispectrum, and four point functions, assuming a Gaussian density distri-bution.Evaluating the first term of (A18) leads to a non-zero contribution only for a separation with r = 0, proportionalto (cid:104) ξ ( (cid:126)r ) δ (3) D ( (cid:126)r (cid:48) ) + 2 ξ ( (cid:126)r (cid:48) ) δ (3) D ( (cid:126)r ) (cid:105) and is therefore not relevant for this work. Similarly, the last term leads to (cid:90) d k (2 π ) (cid:90) d k (2 π ) e i(cid:126)k (cid:126)r h ( (cid:126)k − (cid:126)k , (cid:126)r ) (cid:90) d k (cid:48) (2 π ) (cid:90) d k (cid:48) (2 π ) e i(cid:126)k (cid:48) (cid:126)r (cid:48) h ( (cid:126)k (cid:48) − (cid:126)k (cid:48) , (cid:126)r (cid:48) ) 1 n × (2 π ) δ (3) D ( (cid:126)k − (cid:126)k + (cid:126)k (cid:48) − (cid:126)k (cid:48) )= 1 n V s δ (3) D ( (cid:126)r ) δ (3) D ( (cid:126)r (cid:48) ) . (A19)The only non-zero term, proportional to 1 /n , gives rise to a non-Gaussian contribution to the covariance, we willdenote as ’Poisson’ shot noise term, and can be evaluated using (cid:90) d k (2 π ) (cid:90) d k (2 π ) e i(cid:126)k (cid:126)r h ( (cid:126)k − (cid:126)k , (cid:126)r ) (cid:90) d k (cid:48) (2 π ) (cid:90) d k (cid:48) (2 π ) e i(cid:126)k (cid:48) (cid:126)r (cid:48) h ( (cid:126)k (cid:48) − (cid:126)k (cid:48) , (cid:126)r (cid:48) ) × (2 π ) δ (3) D ( (cid:126)k − (cid:126)k + (cid:126)k (cid:48) − (cid:126)k (cid:48) ) × (cid:18) n (cid:104) P ( (cid:126)k − (cid:126)k ) + P ( (cid:126)k + (cid:126)k (cid:48) ) + P ( (cid:126)k − (cid:126)k (cid:48) ) (cid:105)(cid:19) (A20)= (cid:90) d k (2 π ) (cid:90) d k (2 π ) e i(cid:126)k (cid:126)r h ( (cid:126)k − (cid:126)k , (cid:126)r ) (cid:90) d k (cid:48) (2 π ) e i ( (cid:126)k − (cid:126)k + (cid:126)k (cid:48) ) (cid:126)r (cid:48) h ( (cid:126)k − (cid:126)k, (cid:126)r (cid:48) ) × (cid:18) n (cid:104) P ( (cid:126)k − (cid:126)k ) + P ( (cid:126)k + (cid:126)k (cid:48) ) + P ( (cid:126)k − (cid:126)k (cid:48) ) (cid:105)(cid:19) (A21)= 1 n V s (cid:90) d k (2 π ) (cid:90) d k (cid:48) (2 π ) e i(cid:126)k(cid:126)r e i(cid:126)k (cid:48) (cid:126)r (cid:48) (cid:104) P ( (cid:126)k + (cid:126)k (cid:48) ) + P ( (cid:126)k − (cid:126)k (cid:48) ) (cid:105) (A22)= 1 n V s (cid:90) d k (2 π ) (cid:90) d k (cid:48) (2 π ) e i(cid:126)k (cid:48) (cid:126)r e i ( (cid:126)k − (cid:126)k (cid:48) ) (cid:126)r (cid:48) P ( (cid:126)k ) + 1 n V s (cid:90) d k (2 π ) (cid:90) d k (cid:48) (2 π ) e i ( (cid:126)k + (cid:126)k (cid:48) ) (cid:126)r e i(cid:126)k (cid:48) (cid:126)r (cid:48) P ( k ) (A23)= 1 n V s (cid:90) d k (2 π ) (cid:90) d k (cid:48) (2 π ) e i(cid:126)k (cid:48) ( (cid:126)r − (cid:126)r (cid:48) ) e i(cid:126)k(cid:126)r (cid:48) P ( (cid:126)k ) + 1 n V s (cid:90) d k (2 π ) (cid:90) d k (cid:48) (2 π ) e i(cid:126)k(cid:126)r e i(cid:126)k (cid:48) ( (cid:126)r + (cid:126)r (cid:48) ) P ( k ) (A24)= 1 n V s ( δ (3) D ( (cid:126)r − (cid:126)r (cid:48) ) ξ ( (cid:126)r ) + δ (3) D ( (cid:126)r + (cid:126)r (cid:48) ) ξ ( (cid:126)r )) (A25)dropping the term proportional to P (0). Using spherical symmetry δ D ( (cid:126)r − (cid:126)r (cid:48) ) ξ ( (cid:126)r (cid:48) ) + δ D ( (cid:126)r (cid:48) − (cid:126)r ) ξ ( (cid:126)r ) = δ D ( r − r (cid:48) )4 πr ξ ( r (cid:48) ) + δ D ( r (cid:48) − r )4 πr (cid:48) ξ ( r ) (A26)and volume averaging over r and r (cid:48) leads to1 V ( r ) 1 V ( r (cid:48) ) (cid:90) r (cid:48) (cid:90) r (cid:18) δ D (˜ r − ˜ r (cid:48) )4 π ˜ r ξ (˜ r (cid:48) ) + δ D (˜ r (cid:48) − ˜ r )4 π ˜ r (cid:48) ξ (˜ r ) (cid:19) π ˜ r π ˜ r (cid:48) d ˜ rd ˜ r (cid:48) (A27)= V ( r (cid:48) ) ¯ ξ ( r ) if r (cid:48) > r V ( r ) ¯ ξ ( r (cid:48) ) if r (cid:48) < r V ( r (cid:48) ) ¯ ξ ( r ) + V ( r ) ¯ ξ ( r (cid:48) ) if r (cid:48) = r (A28)(A29)where we have used the integral expression for the Dirac delta function (cid:90) r δ D ( (cid:126) ˜ r − (cid:126) ˜ r (cid:48) ) d r = (cid:90) r (cid:90) − π ˜ r d ˜ rdµ (cid:90) dk (2 π ) (cid:90) − dµ (cid:48) πk e ikµ ˜ r − ikµ (cid:48) r (cid:48) (A30)= 2 π (cid:90) dkkj ( kr ) j ( kr (cid:48) ) r (A31)8to calculate the integrals as (cid:90) r (cid:90) r (cid:48) δ D ( (cid:126) ˜ r − (cid:126) ˜ r (cid:48) ) ξ ( (cid:126) ˜ r (cid:48) ) d ˜ rd ˜ r (cid:48) = (cid:90) r (cid:48) π (cid:90) dkkj ( kr ) j ( k ˜ r (cid:48) ) r π (cid:90) dk (cid:48) k (cid:48) j ( k (cid:48) ˜ r (cid:48) ) P ( k (cid:48) )4 π ˜ r (cid:48) d ˜ r (cid:48) = 2 π (cid:90) dk (cid:48) P ( k (cid:48) ) j ( rk (cid:48) ) k (cid:48) r if r (cid:48) > rj ( r (cid:48) k (cid:48) ) k (cid:48) r (cid:48) if r (cid:48) < r j ( rk (cid:48) ) k (cid:48) r + j ( r (cid:48) k (cid:48) ) k (cid:48) r (cid:48) if r (cid:48) = r (A32)(A33)The total covariance for the pairwise velocity correlation of cluster pairs of exact separation r and r (cid:48) can thereforebe written as C V ( r, a, r (cid:48) , a (cid:48) ) = C Gaus.V ( r, a, r (cid:48) , a (cid:48) ) + C P oiss.V ( r, a, r (cid:48) , a (cid:48) ) (A34)with C Gaus.V ( r, a, r (cid:48) , a (cid:48) ) = 4 π V s ( a ) H ( a ) a ξ ( r, a ) H ( a (cid:48) ) a (cid:48) ξ ( r (cid:48) , a (cid:48) ) f g ( a ) f g ( a (cid:48) ) δ aa (cid:48) (cid:90) dk (cid:18) P ( k, a ) + 1 n ( a ) (cid:19) (cid:18) P ( k, a (cid:48) ) + 1 n ( a (cid:48) ) (cid:19) × j ( kr ) j ( kr (cid:48) ) (A35) C P oiss.V ( r, a, r (cid:48) , a (cid:48) ) = 1 V s ( a ) 13 π H ( a ) a ξ ( r, a ) H ( a (cid:48) ) a (cid:48) ξ ( r (cid:48) , a (cid:48) ) f g ( a ) f g ( a (cid:48) ) δ aa (cid:48) (cid:40) rn ( a ) r (cid:48) ¯ ξ ( r, a ) if r (cid:48) ≥ r r (cid:48) n ( a (cid:48) ) r ¯ ξ ( r (cid:48) , a (cid:48) ) if r (cid:48) < r (A36)Now we consider the statistics calculated by binning cluster separations in a bin of width ∆ r . In this case thepairwise velocity estimate is averaged over cluster pairs with separations within the finite bin,ˆ V ( r ) → V bin (cid:90) r +∆ r/ r − ∆ r/ ˜ r d ˜ r (cid:90) d Ω ˆ V (˜ r ) (A37)where we again assume spherical symmetry. Volume averaging over a bin of size ∆ r = R max − R min yields j ( kr ) → R i, max − R i, min (cid:90) R i, max R i, min r j ( kr ) dr. (A38)Using that (cid:90) R i, max R i, min r j ( kr ) dr = R i, min ˜ W ( kR i, min ) − R i, max ˜ W ( kR i, max ) (A39)(A40)with ˜ W ( x ) = 2 cos( x ) + x sin( x ) x . (A41)Binning in r translates into replacing the Bessel function with a function related to the bin limits, j ( kr ) → R i, max − R i, min (cid:16) R i, min ˜ W ( kR i, min ) − R i, max ˜ W ( kR i, max ) (cid:17) ≡ W ∆ ( kr ) . (A42)Rewriting the volume averaged correlation function in terms of the power spectrum,2 rr (cid:48) ¯ ξ ( r ) = 3 r (cid:48) π (cid:90) dkkP ( k ) j ( kr ) , (A43)and with 1 r → π ∆ rV ∆ ( r ) , (A44)9the full, angle-averaged covariance for the mean pairwise velocity, excluding measurement error, is given by the sumof a Gaussian cosmic variance and shot noise component plus a Poisson component, C V ( r, a, r (cid:48) , a (cid:48) ) = C Gaus .V ( r, a, r (cid:48) , a (cid:48) ) + C Poiss .V ( r, a, r (cid:48) , a (cid:48) ) (A45)with C Gaus .V ( r, a, r (cid:48) , a (cid:48) ) = 4 π V s ( a ) H ( a ) a ξ ( r, a ) H ( a (cid:48) ) a (cid:48) ξ ( r (cid:48) , a (cid:48) ) f g ( a ) f g ( a (cid:48) ) δ aa (cid:48) (cid:90) dk (cid:18) P ( k, a ) + 1 n ( a ) (cid:19) (cid:18) P ( k, a (cid:48) ) + 1 n ( a (cid:48) ) (cid:19) × W ∆ ( kr ) W ∆ ( kr (cid:48) ) (A46) C Poiss .V ( r, a, r (cid:48) , a (cid:48) ) = 4 π V s ( a ) H ( a ) a ξ ( r, a ) H ( a (cid:48) ) a (cid:48) ξ ( r (cid:48) , a (cid:48) ) f g ( a ) f g ( a (cid:48) ) δ aa (cid:48) (cid:40) ∆ r (cid:48) n ( a ) V ∆ ( r (cid:48) ) (cid:82) dkkP ( k, a ) W ∆ ( kr ) if r (cid:48) ≥ r ∆ rn ( a (cid:48) ) V ∆ ( r ) (cid:82) dkkP ( k, a (cid:48) ) W ∆ ( kr (cid:48) ) if r (cid:48) < r. 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