Primordial non-Gaussianity from the covariance of galaxy cluster counts
PPrimordial non-Gaussianity from the covariance of galaxy cluster counts
Carlos Cunha, Dragan Huterer, and Olivier Dor´e
2, 3 Department of Physics, University of Michigan, 450 Church St, Ann Arbor, MI 48109-1040 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 California Institute of Technology, Pasadena, CA 91125 (Dated: October 29, 2018)It has recently been proposed that the large-scale bias of dark matter halos depends sensitivelyon primordial non-Gaussianity of the local form. In this paper we point out that the strong scaledependence of the non-Gaussian halo bias imprints a distinct signature on the covariance of clustercounts. We find that using the full covariance of cluster counts results in improvements on constraintson the non-Gaussian parameter f NL of three (one) orders of magnitude relative to cluster counts(counts + clustering variance) constraints alone. We forecast f NL constraints for the upcomingDark Energy Survey in the presence of uncertainties in the mass-observable relation, halo bias, andphotometric redshifts. We find that the DES can yield constraints on non-Gaussianity of σ ( f NL ) ∼ I. INTRODUCTION
Primordial non-Gaussianity provides cosmology one ofthe precious few connections between primordial physicsand the present-day universe. Standard inflationary the-ory with a single-field, slowly rolling scalar field, predictsthat the spatial distribution of structures in the universetoday is very nearly Gaussian random (e.g. [1–5]; for anexcellent recent review, see [6]). Departures from Gaus-sianity, barring contamination from systematic errors orlate-time non-Gaussianity due to secondary processes,can therefore be interpreted as violation of this “vanilla”inflationary assumption. Constraining or detecting pri-mordial non-Gaussianity is therefore an important andbasic test of the cosmological model.Constraints on primordial non-Gaussianity have beentraditionally obtained from observations of the cosmicmicrowave background, as nonzero non-Gaussianity gen-erates a non-zero three-point correlation function (orits Fourier transform, the bispectrum) of density fluc-tuations [7–13]. Increasingly sophisticated algorithmshave been developed to constrain non-Gaussianity, [14–18] and, to the extent that it can be measured, Gaussian-ity has so far been confirmed [19–21]. For example, themost recent constraints from the Wilkinson MicrowaveAnisotropy Probe (WMAP) indicate f NL ≈ ±
21 (1 σ ;[22]), where the exact constraints depend somewhat onthe choice of the statistical estimator applied to the data,the CMB map used, and details of the foreground sub-traction. Here f NL is the parameter describing non-Gaussianity in the widely studied “local” model, wherethe non-Gaussian potential Φ NG is defined byΦ NG ( x ) = Φ G ( x ) + f NL (Φ ( x ) − (cid:104) Φ (cid:105) ) , (1)and where Φ G is the Gaussian potential. Correspond-ing constraints can be obtained on other classes of non-Gaussian models. For example, for “equilateral” modelswhere most power comes from equilateral triangle con-figurations, f eqNL = 26 ±
140 (1 σ ; [22]). The CMB is not the only cosmological probe to be sen-sitive to the presence of primordial non-Gaussianity. Ithas been known for a relatively long time that the abun-dance of dark matter halos [23–29] (or voids [30, 31]) issensitive to the presence of primordial non-Gaussianity.This dependence is easy to understand: halos populatethe high tail of the probability density distribution ofstructures in the universe, and the shape of this distri-bution is sensitive to departures from Gaussianity. How-ever, while the halo abundance is rather powerful in con-straining models that are non-Gaussian in the density(rather than the potential) [32], for the popular modelsof the local type (cf. Eq. (1)) the abundance is much lessconstraining than the CMB anisotropy and not compet-itive with the CMB constraints (e.g. [33, 34]).Some of us [35] have recently shown that the cluster-ing of dark matter halos is very sensitive to primordialnon-Gaussianity of the local type. This exciting devel-opment paves way to using the large-scale structure toprobe primordial non-Gaussianity nearly three orders ofmagnitude more accurately than using the abundance ofhalos. Dalal et al. [35] found, analytically and numeri-cally, that the bias of dark matter halos acquires strongscale dependence b ( k ) = b + f NL ( b − δ c m H a g ( a ) T ( k ) c k , (2)where b is the usual Gaussian bias (on large scales,where it is constant), δ c ≈ .
686 is the collapse thresh-old, a is the scale factor, Ω M is the matter fraction rel-ative to critical, H is the Hubble constant, k is thewavenumber, T ( k ) is the transfer function, and g ( a ) isthe growth suppression factor . This result has been con- The usual linear growth D ( a ), normalized to be equal to a inthe matter-dominated epoch, is related to the suppression factor g ( a ) via D ( a ) = ag ( a ). a r X i v : . [ a s t r o - ph . C O ] J un firmed by other researchers using a variety of methods,including the peak-background split [36–38], perturba-tion theory [39–41], and numerical (N-body) simulations[42–44]. Astrophysical measurements of the scale depen-dence of the large-scale bias, using galaxy and quasarclustering as well as the cross-correlation between thegalaxy density and CMB anisotropy, have recently beenused to impose constraints on f NL already comparableto those from the cosmic microwave background (CMB)anisotropy [36, 38], giving f NL = 28 ±
23 (1 σ ), with somedependence on the assumptions made in the analysis [38].In the future, constraints on f NL are expected to be oforder a few [35, 45, 46]. The sensitivity of the large-scalebias to other models of primordial non-Gaussianity hasnot been investigated yet (though see preliminary analy-ses in [47, 48]). Clustering of galaxy clusters, in particular, can verystrongly constrain primordial non-Gaussianity. Clustershave an advantage of being large, relatively simple ob-jects that are easy to find using either optical or X-ray light, or else from their Sunyaev-Zeldovich signature.Clusters already provide interesting constraints on darkenergy [49, 50] and they hold promise for precision mea-surements of cosmological and dark energy parameters(e.g. [51]). Since clusters are massive and hence signif-icantly biased objects, their counts (via the mass func-tion) and clustering (via the mass function and bias) areboth sensitive to primordial non-Gaussianity. Recently,Oguri [52] has argued that the variance of cluster counts(i.e. scatter measured in each cell individually), in com-bination with the cluster counts, leads to interesting im-provements on f NL constraints relative to the counts-onlycase.In this paper we point out that including the covari-ance of cluster counts in angle and redshift leads tovery significant further improvements in the cluster con-straints on local primordial non-Gaussianity. The prin-cipal reason for the improvement is simple: covariance isdetermined by the cluster power spectrum, which is pro-portional to the halo bias squared. At large scales, thenon-Gaussian contribution to the halo bias dominates (cf.Eq. (2)), and this results in a strong f NL signal in the co-variance. Furthermore, we explore the sensitivity of theconstraints to various assumptions about statistical andsystematic errors in modeling the cluster mass-observablerelation, as well as the presence of other cosmological pa-rameters. We find that the bulk of the information aboutlocal non-Gaussianity comes from the far-separation co-variances of cluster counts-in-cells.This paper is organized as follows. In Sec. II, we de-scribe the methodology that we use to obtain constraintsfrom both counts and clustering of galaxy clusters. InSec. III we describe our fiducial assumptions about thecosmological model and data as well as solutions to var-ious challenges in calculating the constraints. In Sec. IVwe describe the forecasted constraints on f NL from theDark Energy Survey. We discuss our results in Sec. V,and conclude in Sec. VI. II. METHODOLOGY
We address the following problem: how well can thecosmological parameters be recovered using counts ofgalaxy clusters in pixels distributed in angle and radiuson the sky? We largely follow the formalism of Hu andCohn [53] and Lima and Hu [54].Assume that clusters are counted in square pixels offixed angular size θ pix , corresponding to comoving size L pix ( z ) = θ pix r ( z ), where r is the comoving distance.The clusters are also binned in the mass-observable (i.ethe observable proxy for cluster mass), with intervals[ M α obs , M α +1obs ] where α refers to a specific mass-observablebin. The number density of clusters at a given redshift z with observable in the range M α obs ≤ M obs ≤ M α +1obs isgiven by¯ n α ( z ) ≡ (cid:90) M α +1obs M α obs dM obs M obs (cid:90) dMM d ¯ nd ln M p ( M obs | M ) (3)where p ( M obs | M ) is the observable-mass relation (ex-plained in Appendix A) and d ¯ n/d ln M is the mass func-tion. Uncertainties in the redshifts distort the volumeelement; we fully take into account the photometric red-shift uncertainties following [55]; details are shown in Ap-pendix B.We adopt the mass function from Dalal et al. [35] whoused N-body simulations to parametrize the shift in massof a typical halo in the presence of non-Gaussianity. Themass shift, M G → M , is adequately described by a Gaus-sian with mean and variance respectively given by (cid:28) MM G (cid:29) − . × − f NL σ σ ( M G , z ) − (4)var (cid:18) MM G (cid:19) = 1 . × − ( f NL σ ) . σ ( M G , z ) − , (5)where σ ( M, z ) is the amplitude of mass fluctuations onmass scale M and at redshift z . The final non-Gaussianmass function is given by [35] dndM = (cid:90) dM G dndM G dPdM ( M G ) , (6)where dP/dM ( M G ) is the probability distribution that aGaussian halo of mass M G maps to a non-Gaussian haloof mass M , and is given by the Gaussian with the meanand variance given in Eqs. (4) and (5). For dn/dM G , weadopt the Jenkins mass function [56].On large scales, the number counts of clusters m ( x )trace the linear density perturbation δ ( x ) m i ( M α , x ) ≡ m iα = ¯ m i (1 + b ( M α , z ) δ ( x )) (7)where i refers to the pixel (i.e. its angular and radialcoordinates), and α indicates the mass bin. The spatialcovariance of cluster counts is [57] S αij = (cid:104) ( m iα − ¯ m iα )( m jα − ¯ m jα ) (cid:105) ≡ ¯ m iα ¯ m jα ξ αij , (8)where ξ αij is the pixel real-space correlation function ξ αij ≡ (cid:90) d k (2 π ) | W i ( k ) W j ( k ) | cos( k x ∆ x ij ) × cos( k y ∆ y ij ) cos( k z ∆ z ij ) b iα b jα P ( k ; z ) . (9)If i and j come from different redshift bins, the geometricmean of the two redshifts is adopted for z . In the limit r ij (cid:29) L pix , ξ αij → ξ ( r ij ), where the latter quantity isthe standard two-point correlation function in real space.∆ x ij = L pix n xij is the physical separation between i and j in the x direction (transverse to the line of sight), and n xij is the number of pixels separating them; ∆ y ij isdefined equivalently. Finally, the window function W isthe Fourier transform of the square pixel in the presenceof redshift errors W ( k ) i = exp (cid:32) − σ z,i k z H i (cid:33) × (10) j ( k x L pix / j ( k y L pix / j ( k z ∆ z/ H i ) , where the index i refers to the redshift bin, σ z,i is the red-shift scatter at the radial distance corresponding to the i th pixel, and H i is the Hubble parameter. The photo-zbias is implicit in the ∆ z ij term in Eq. (9).The expression for the full Fisher matrix for galaxycluster counts and their covariance is quite complicated(see [53]), but a reasonable approximation is given by [58] F µν = ¯ m t,µ C − ¯ m ,ν + 12 Tr[ C − S ,µ C − S ,ν ] , (11)where the first term encodes information from clustercounts, and the second from the covariance. Here µ and ν are indices that refer to both cosmological and nui-sance parameters (including f NL ). The cluster countshave been arranged as the vector ¯ m . S = { S αij } is thesample covariance matrix from Eq. (8), and C ≡ N + S is the total covariance. N ij = ¯ m i δ ij is the (shot) noisematrix. The derivative with respect to f NL can be com-puted analytically, using the fact that P ( k, z ) ∝ b ( k, z )and Eq. (2). III. FIDUCIAL ASSUMPTIONS ANDCALCULATIONAL CHALLENGES
We implement the procedure outlined above for opti-cally selected clusters. In our fiducial setup we divide the In the linear regime, the correlation between pixels i and j con-tains the product of the growth factors corresponding to z i and z j . Therefore, the corresponding power spectrum, P ( k, z ) inEq. (9), should use the growth function equal to the geometricmean of the two growth functions. Instead, we effectively usethe growth function which is evaluated at the redshift equal tothe geometric mean of the two redshifts z i and z j . Results areinsensitive to this approximation, specially because most of theinformation comes from relatively close redshift pairs. sky into the 11 ×
11 field of pixels of 41 .
32 sq. deg. each,for a total of 5,000 sq. deg. which matches expectationsfor the Dark Energy Survey (DES). The facing surfaceof each pixel is a square with a side L pix ( z ) = θ pix r ( z )(see Sec. II). Each pixel has redshift depth ∆ z = 0 . M th = 10 . h − M (cid:12) and also bin in mass, using 5 massbins of width ∆ ln M th = 0 .
2, with the exception ofthe highest-mass bin, which we extend to infinity. Us-ing smaller bins in angle or redshift yields better results,up to the point where the covariance matrix becomesdominated by shot-noise (which occurs for bins with areaaround 0.1-1 sq. deg.). For very large number of pixels,the Gaussian approximation used to define the covari-ance used in our Fisher matrix would break down. In ourfiducial case we have about 1 . × clusters subdividedinto 3 ,
025 pixels, so that we are well within the Gaussianregime. In addition, results for large angular pixels areless sensitive to systematics due to non-linear physics orangular mask uncertainties. In Sec. IV we consider de-partures from the fiducial assumption, namely variationsin the mass-threshold, maximum redshift range and pixelarea.We assume fiducial cosmological parameters based onthe fifth year data release of the Wilkinson MicrowaveAnisotropy Probe [59]. Thus, we set the baryon den-sity, Ω b h = 0 . m h =0 . k =0 . − , δ ζ = 4 . × − , the tilt, n = 0 . τ = 0 . DE = 0 . w = −
1. In this cosmology, σ = 0 . when calculating themarginalized constraints on parameters.To study systematic errors in cluster cosmology, weadd a generous set of nuisance parameters described inAppendix A (see also Cunha [61] and Cunha et al. [51]),with 10 nuisance parameters describing the bias and vari-ance in the mass-observable relation and 3 parametersdescribing uncertainty in the halo bias ( a c , p c , and δ c ,cf. Eq. (B5)). The assumption of 3 nuisance parametersdescribing the Gaussian halo bias is somewhat ad hoc but conservative since for a given mass function the halobias can be predicted to roughly 10% accuracy in therange of scales we are interested [62]. We fix the photo-z scatter to 0 .
02 everywhere except in Sec. IV D wherewe consider the effects of including 10 additional nui-sance parameters describing photometric redshift errors.In this exploratory paper we do not consider models fornon-Gaussianity other than the one from Eq. (1), or ob-servational systematic errors (e.g. atmospheric blurringor completeness variations across the sky). The study of W. Hu, private communication
FIG. 1:
Left panel : Sensitivity of the variance of cluster counts to non-Gaussianity. The black lines shows the variance S ii , theshort-dashed red line shows the (auto)correlation function ξ αii , and the long-dashed blue line shows the squared mean counts.Note that S ii = ( ¯ m ) ξ αii We assumed a pixel with area 40 sq. deg. and radial redshift extent ∆ z = 0 .
2, centered at z = 0 . Rightpanel : Sensitivity of the covariance of cluster counts to non-Gaussianity. We show the off-diagonal elements of the clusteringmatrix, normalized by the variance of f NL = 0 case ( S Gauss ii ) as a function of angle between the i th and j th pixel. We use thesame pixelization as for the left panel.We show the Gaussian case ( f NL = 0), and four non-Gaussian models ( f NL = ±
20 and f NL = ± these effects is left for future work.Evaluating the expression for the Fisher matrix withthe signal matrix of this size is clearly challenging: thetotal size of the matrix S (see Eq. (9)) is N × N , where N = N pixels × N mass × N redshift = 121 × × , ∼ elements of the matrix, each of which involvesthe numerical computation of a rapidly oscillating tripleintegral; see Eqs. (8) and (9). Unlike previous workswhich studied constraints on dark energy [53, 55, 58],we cannot ignore the off-diagonal elements (i.e. the pixel co variance) of the matrix S since those elements, whilebeing very small for the Gaussian case, become signifi-cant for f NL (cid:54) = 0 (see the right panel of Fig. 1) due tothe f k − dependence scaling of the power spectrumas k →
0. To reduce the size of the covariance matrixwe assume that the information from the different massbins is independent, so that we can estimate the Fishermatrix for each mass bin separately and then add themin the end. The scatter in the mass-observable relationcan generate correlations between mass bins at a givenpixel. In addition, as Seljak [63] and McDonald and Sel-jak [64] noticed, correlating the halos of different massesat large separations would lead to improved constraintsin our analysis, making our assumption conservative.
A. Regularization of the covariance
As Wands and Slosar [65] pointed out, the two-pointcorrelation function for biased tracers of structure has aninfrared divergence if f NL is not zero. However, the mea-sured correlation function from any survey is of coursefinite because one cannot measure variance of the den-sity field on scales larger than the survey. To that effect,Wands and Slosar [65] suggest regularizing the correla-tion function ξ ( r ) by subtracting from it the varianceof the density field evaluated at the scale of the sur-vey. However, Cunha and Slosar (private communica-tion) found out that the regularization of Wands andSlosar [65] contains a typo; the correct regularizationterm is given byΣ ( R ) ≡ (cid:90) d k (2 π ) | W R ( k ) | b iα b jα P ( k ; z ) , (12)where we use the mass bin α and redshift bin i corre-sponding to those of the correlation function ξ αij fromwhich this is being subtracted. If i and j come from dif-ferent redshift bins, the geometric mean of the two red-shifts is taken. The difference from what is presented inWands and Slosar [65] is that our expression has | W R ( k ) | instead of | W R ( k ) | (cf. Eqs. 47, 49 and 50 in [65]). Us-ing the above expression, the observed 2-pt correlationat a given survey volume has the desirable property thatit integrates to zero over the survey volume.We approximate the window function | W R ( k ) | as theFourier transform of a spherical top-hat, and adopt R =2 h − Gpc as the linear dimension of our survey. For themain analysis in this paper, the effects of the divergenceare not significant, since all of our results (except in Sec.V) assume zero fiducial f NL , and the analytic expressionfor the derivative dS ij /df NL is weakly sensitive to the in-tegration boundary. The divergence of the two-point cor-relation does affect the covariance for non-zero f NL andfor pixel separation greater than a few hundred Mpc. Weuse the lower boundary of integration k min = 10 − , andcheck that results are stable vis-a-vis variations in thisvalue, or whether the regularization mentioned above hasbeen applied or not. For Fig. 1 and the results in Sec. V,we do apply the corrected Wands-Slosar regularizationprescription (cf. Eq. 12).Besides its impact on the regularization, the choice ofsurvey geometry is important since the distribution ofpixel-pixel separations depends on the geometry. We as-sume that the survey itself has square shape (and im-plicitly work in a flat-sky approximation), and assumea 11 ×
11 field of square-shaped pixels for each redshiftbin. To populate the covariance matrix, we precompute S ij as a function of pixel separation for integer values ofthe separation along a row of pixels in Eq. (9) — thatis, we set ∆ y ij = 0 and vary ∆ x ij at each redshift. Weuse linear interpolation to estimate the covariance forpixels whose physical separation, in units of ∆ x i ( i +1) , isnon-integer. We find that the effects of disregarding thepixel orientation are negligible (by changing the orien-tation of bins and finding little change in the results).Pre-computation of the covariance matrix elements as afunction of pixel separation greatly reduces the numberof covariance terms we need to calculate.As the right panel of Fig. 1 shows, in the Gaussian casethe off-diagonal terms of S ij fall off very fast. We findthat covariance terms for pixels in different redshifts tobe negligible, because we use broad redshift bins. Hence,we only calculate covariance between different redshiftbins when estimating the derivative of the covariancewith respect to f NL . To save time, for the results shownin IV we only calculate terms in adjacent redshift bins.We checked that including larger redshift separations im-proves unmarginalized constraints by about 30%. But in-cluding the regularization removes most of the improve-ment (for fiducial f NL = 0). To calculate the derivativesof the covariance with respect to f NL , we use the fact thatthe derivative of the bias with respect to f NL is analyticso that dξ αij df NL ≡ (cid:90) d k (2 π ) | W ( k ) | cos( k x ∆ x ij ) cos( k y ∆ y ij ) × cos( k z ∆ z ij ) d ( b iα b jα ) df NL P ( k ; z ) . (13)In calculating dS ij /df NL , we only keep the dominantterm, which is the one with derivative with respect to ξ αij . That is, we assume that dS αij df NL (cid:39) ¯ m iα ¯ m jα dξ αij df NL . (14)The terms we ignore correspond to the sensitivity of clus-ter counts to non-Gaussianity, and they would only en-hance the impact of f NL , though slightly, as will be shownin the following sections. In a real survey one actuallyhas to calculate the covariance at non-zero values of f NL for which our approach of evaluating the derivative ana-lytically at f NL = 0 would be insufficiently general. Forthis sensitivity study, however, the analytic derivative isperfectly acceptable. We examine the sensitivity to theconstraints around different fiducial values of f NL in Sec.V. IV. RESULTS
Our results are presented as follows. First we discussthe sensitivity of cluster counts and clustering of countsto f NL , and examine unmarginalized constraints on f NL .Second, we examine the degeneracies with cosmologicalparameters and nuisance parameters due to modeling un-certainties in the observable-mass relation and in the halobias. Third and last, we look at the impact of photomet-ric redshift uncertainties. A. Sensitivity of cluster covariance
The effect of non-Gaussianity on clustering is a combi-nation of several effects, which can be identified fromEq. (8). The dominant effect is due to the explicitmodification of the halo bias (Eq. (2)) which affects ξ αij (cf. Eq. (9)) In addition, non-Gaussianity affects themass function, which affects the mean cluster counts (cf.Eqs. 3 and B1), and the average cluster linear bias (cf.Eq. (B7)). The left panel of Fig. 1 shows the depen-dence of the different terms that make up the clusteringcovariance S ij , as a function of f NL . For this sensitivityplot, we assume a 40 sq. deg. pixel with redshift thickness∆ z = 0 . z = 0 . M th = 10 . h − M (cid:12) , and show only the diagonal ele-ments i = j for clarity. The relation between the func-tions plotted in this figure is S ij = ¯ m ξ αij . It is apparentfrom the figure that ξ αij encodes most of the dependenceof the clustering signal on f NL , and that the clusteringcovariance ( S ij , or ξ αij ) is much more sensitive to f NL thanthe mean counts ¯ m . As mentioned previously, we neglectthe implicit mass function dependence of f NL when cal-culating the covariance. Including it would only enhancethe impact of f NL , albeit slightly.In the right panel of Fig. 1 we plot the absolute valueof the clustering covariance as a function of angular sep-aration between the centroids of two pixels. For refer-ence, at z = 0 .
5, a one-degree separation corresponds
FIG. 2: 1 − σ uncertainties in the parameter f NL as a function of the maximum angular separation between pixel centroids inthe covariance matrix. The left panel shows the unmarginalized constraints while the right panel shows marginalized constraintsassuming Planck priors and fixed halo-bias and observable-mass nuisance parameters. Zero separation indicates the case ofpure variances (as considered by Oguri [52]). The maximum angular separation between pixels for a 5 ,
000 sq. deg. surveydivided into 41.3 sq. deg pixels is about 90 degrees (or 10 √ f NL , but disregarding the covariance between different redshift bins. The blueshort dashed line corresponds to constraints derived using only cluster counts. The red dashed line shows the constraints whenonly the clustering of clusters is used, and the solid black line shows the combined constraints from counts and clustering. to about 23 . h − Mpc. For f NL = 0, the clustering co-variance is large and positive at small separations, butbecomes negative at intermediate pixel separations ( ∼ ∼ h − Mpc at z = 0 . ξ ( r ) (see e.g. Ref. [66]). The effect of nonzero f NL depends on its sign as well as on the scale. For posi-tive f NL , the covariance increases monotonically with f NL roughly up to the scale of the survey. Beyond that scale( ∼ ◦ in our example), the covariance reverses its trendwith f NL and becomes negative due to the integral con-straint imposed by the regularization. For negative f NL ,the dependence of the covariance S ij on f NL is more com-plicated because the total bias becomes negative at largeenough scales; thus, for f NL < | f NL | only on scales ( (cid:46) ◦ in the rightpanel of Fig. 1) for which the bias correction – secondterm in Eq. (2) – is subdominant. Note that Fig. 1 hidesthe fact that the number of pixels at a given separationincreases with separation: the number of off-diagonal el-ements in the covariance is much bigger than the numberof diagonal elements, and this gives a “’geometric boost”to the covariance. B. Unmarginalized constraints from clustering andcounts
Both panels of Fig. 2 show f NL constraints as a func-tion of the maximum pixel separation allowed in the co-variance (cf. Eq. 8) used to generate the Fisher matrixconstraints (cf. Eq. 11).In the left panel of Fig. 2 we see that the cluster countsyield better unmarginalized constraints than the variance of cluster counts alone; however, once the covariances (i.e.off-diagonal terms of the signal matrix S ij ) are included,the clustering information rapidly beats that from thecounts. In Table I we show the unmarginalized f NL con-straints for a variety of survey expectations. Changes inthe constraints improve in the direction expected: thelower the mass-threshold and the higher the maximumredshift, the better. This Table also shows that decreas-ing the angular area of the pixels to 12 . results insubstantial ( O (50%)) improvements. The improvementwith decreasing pixel size, for f NL constraints, does nothappen if we consider only the variance in counts. Forother parameters, that are sensitive to small scale in-formation, such as Ω DE and w , the smaller pixels dotranslate into better constraints even if only the samplevariance is used. Further refinements of the pixelizationleads to improvement up to the regime of shot-noise dom-ination, (which occurs for pixels of ∼ . − ). Un-marginalized constraints are of order 10 − in this regime, Unmarginalized error σ ( f NL )Assumption Number Counts Covariance Both
Fiducial 1 . × pix 1 . × z max = 0 . . ×
13 2.3 z max = 1 . . × M th = 10 . . × M th = 10 . . ×
10 2.3
TABLE I: Unmarginalized constraints on f NL . The fiducialcase assumes no nuisance parameters, 5 bins in mass and red-shift each, and other assumptions as in the text. Variationsin the assumptions are shown in the first column, followedby the total number of clusters in the 5 ,
000 deg survey weassumed, while cluster counts, covariance, and combined pro-jected 1- σ constraints on f NL are given in the following threecolumns. though observational systematics are likely to dominatestatistical errors of this size. C. Degeneracies with cosmological and nuisanceparameters.
In the right panel of Fig. 2 we show the marginalized constraints on f NL assuming Planck priors and fixed nui-sance parameters (both halo bias and mass-observable).We see that the change in the constraints from combinedcounts and clustering is even more remarkable than theunmarginalized constraints shown in the right panel. Thefull clustering covariance yields about one order of mag-nitude better constraints than if only the variance is used.As we shall see, this fractional improvement remains evenwhen we include nuisance parameters.Tables II and III show f NL constraints using the vari-ance of cluster counts, and the full covariance, respec-tively. The results assumed Planck priors on the cos-mological parameters, 10 nuisance parameters describingthe mass-observable relation and 3 nuisance parametersdescribing uncertainties in the Gaussian halo bias.Comparing the last columns of Tables II and III, wesee that the counts+covariance combination yields aboutan order of magnitude improvement over simply usingcounts+variance. For the counts+variance, the uncer- The slight degradation in f NL constraints from counts seen in theright panel is real, and is due to adding the (positive) covariancematrix elements to the counts noise; see the first term on theRHS of Eq. (11). Using the full covariance therefore yields veryslightly worse constraints. tainties in the halo bias parameters are the main sourceof degradation to f NL constraints. Without the infor-mation from large separations provided by the full co-variance, the Fisher matrix cannot disentangle the ef-fects due to the Gaussian bias from the f NL contribu-tion. When the full covariance is used (cf. Table III), theerrors in the mass-observable relation are the dominantsource of degradation. Marginalizing over all nuisanceparameters, assuming flat priors, yields a degradation of ∼ σ ( f NL ). This is not large, considering we added13 nuisance parameters, but not negligible either. Evenmodest prior information can improve the marginalizedconstraints significantly.There are two principal reasons for the strong improve-ment of errors when the covariance is added:1. The strong scale dependence of the bias as a func-tion implies that most signal comes from the co-variances, since the covariances have longer leverarms in k than the variance alone (and are muchmore sensitive than counts which only depend onnon-Gaussianity via the mass function);2. The signature of f NL in the covariance is unique, asno other cosmological parameter leads to a similareffect — therefore, the degeneracy with other cos-mological parameters is very small, as first notedby [35].Comparing the f NL constraints for the full covariance forfixed nuisance parameters (Table III) to the unmarginal-ized constraints (Table I), we see that degeneracies withcosmological parameters only result in a small degrada-tion of f NL constraints (from 1.7 to 1.8).Tables II and III also show the constraints obtainedusing counts alone, or (co)variance by itself. The in-formation about f NL from the counts is very degener-ate with the cosmological and nuisance parameters. The“ ∞ ” symbols indicate that the Fisher matrix could notbe inverted, i.e., that particular technique was unableto simultaneously constrain all of the parameters. Fromthe last row of both tables, we see that cluster countsare effective at constraining the cosmological parametersand mass-observable relation (from the mass binning)whereas the (co)variance constrains mainly the nuisanceparameters and f NL .Marginalization degrades the counts + covariance f NL constraints roughly independently of the different surveyassumptions, so one can use Table I to infer marginalizedconstraints. For example, from Table I, we see that us-ing 12 . pixels yields about 60% better constraints.The full marginalized constraints are also improved bya similar factors so that, for example, σ ( f NL ) ∼ . . marginalized over the 13 nuisance parameters(compared to σ ( f NL ) = 6 . pixels). Marginalized errors - Variance only
Nuisance parameters Counts Variance Counts+VarianceHalo bias M obs σ (Ω DE ) σ ( w ) σ ( f NL ) σ (Ω DE ) σ ( w ) σ ( f NL ) σ (Ω DE ) σ ( w ) σ ( f NL )Marginalized Marginalized ∞ ∞ ∞ ∞ ∞ ∞ Known Marginalized 0.095 0.32 . × ∞ ∞ ∞ Marginalized Known ∞ ∞ ∞ Known Known 0.0046 0.021 TABLE II: Marginalized constraints on f NL and dark energy with cluster counts, variance of the counts, and the two combined.The fiducial case assumes 5 bins in mass and redshift each with a mass threshold M th = 10 . , maximum redshift z max = 1 . ∞ indicate that the method was unable to constrain the parameters. Marginalized errors - Full Covariance
Nuisance parameters Counts Covariance Counts+CovarianceHalo bias M obs σ (Ω DE ) σ ( w ) σ ( f NL ) σ (Ω DE ) σ ( w ) σ ( f NL ) σ (Ω DE ) σ ( w ) σ ( f NL )Marginalized Marginalized ∞ ∞ ∞ ∞ ∞ ∞ Known Marginalized 0.097 0.33 . × Marginalized Known ∞ ∞ ∞
Known Known 0.0051 0.023 TABLE III: Marginalized constraints on f NL and dark energy with cluster counts, covariance of the counts, and the twocombined. The fiducial case assumes 5 bins in mass and redshift each with a mass threshold M th = 10 . , maximum redshift z max = 1 .
0, and other assumptions as in the text. Assumptions about the nuisance parameters are varied, and are shown inthe first two columns. Entries with ∞ indicate that the method was unable to constrain the parameters. D. Photometric redshift errors
To study the effects of photometric redshift errors, weadd 10 nuisance parameters to the analysis, namely twoparameters — one each describing the photo-z scatterand bias — in each of the five redshift bins. The resultsare summarized in Table IV.If either the halo bias or the mass-observable nui-sance parameters are fixed, then the degradation fromthe inclusion of photo-z’s is not very damaging. In otherwords, the additional correlations between either photo-zand halo bias parameters, or between photo-z and mass-observable parameters, do not cause substantial addi-tional degradation to f NL constraints (relative to the casewhere only the photo-z parameters are unknown).However when all 23 nuisance parameters (10 for thephoto-z’s, 10 for the mass-observable relation, and 3 forhalo bias) are left free, one cannot simultaneously con-strain dark energy and f NL , and the constraints on bothdrastically degrade. We traced the biggest source ofdegradation to the redshift evolution parameters in themass-observable relation and to the photo-z bias nui-sance parameters. Simply adding a 33% prior to the one parameter describing the evolution of the bias in P ( M obs | M ) (parameter a in Eq. (A3)) was enough toreclaim respectable accuracy, with σ ( f NL ) = 18 . .
01 with all other parameters free, then σ ( f NL ) = 7 . ∼
15% worse than when photo-z parame-ters are fixed . For a survey such as the DES, these re-quirements should be relatively easy to satisfy, given thatspectroscopic samples of 10 -10 galaxies will be avail-able to calibrate the photometric redshift errors (see e.g.Eqs. (19) and (20) in Hearin et al. [67]). Unlike f NL , the dark energy constraints are sensitive to both biasand scatter of the photo-z’s. For a prior uncertainty in the photo-z bias of 0 .
01 per bin, the photo-z scatter needs to be known to0 . per bin to achieve small ( (cid:46) σ (Ω DE )and σ ( w ) relative to the case of perfectly known photo-z errors. The effects of photo-z uncertaintiesNuisance parametersHalo bias M obs σ (Ω DE ) σ ( w ) σ ( f NL )Known Known 0.016 0.041 Marginalized Known 0.021 0.053
Known Marginalized 0.11 0.36
Marginalized a Marginalized a a a a TABLE IV: Effect of photometric redshift uncertainties onthe marginalized constraints on f NL . The fiducial case as-sumes 5 bins in mass and redshift each with a mass-threshold M th = 10 . and maximum redshift z max = 1 .
0, and otherassumptions as in the text. Variations are in the first twocolumns, while cluster, covariance, and combined projected 1- σ constraints on f NL are given in the following three columns.In the bottom row, superscript a signals that a Fisher matrixprior of F a ,a = 10 is added to the nuisance parameter a defined in Eq. (A3), which describes the redshift evolution ofthe bias in the mass-observable relation. V. DISCUSSIONA. Choice of the fiducial model
In our fiducial approach we estimated errors in f NL around f NL = 0. However, it is a slightly different matterto estimate the detectability of non-Gaussianity, whichrequires estimating the signal-to-noise at which a non-zero fiducial value of f NL can be differentiated from zero .The detectability is independent of the fiducial value ifthe observable quantity is linear in the parameter(s); thisis clearly not the case here since the clustering signal isa quadratic function of the bias, which itself dependslinearly on f NL .Fig. 3 shows the fiducial unmarginalized constraints on f NL as a function of its fiducial value. Unlike in the re-sults shown previously, here we calculate all elements ofthe covariance matrix and its derivative with respect to f NL (which is why the constraints for f NL = 0 shown inthe plot are slightly better than what is shown in TableI). The figure shows tightest constraints for | f NL | (cid:39)
10— more than 4 times stronger than those for our fiducialassumption of f NL = 0. The “witch’s hat” shape shownin Fig. 3 can be understood by examining the secondterm on the RHS of Eq. (11) that contains the Fisherinformation from the covariance of cluster counts. The f NL constraints are set by the competition between the Arguably the best approach might be to use the Bayesian modelselection techniques and, for a range of f NL values, test if thehypothesis f NL = 0 can be rejected. We do not pursue such anapproach in this paper. signal, represented by the derivative of the covariancewith respect to f NL , S ,µ , and the noise, given by the to-tal covariance, C . These two quantities vary with f NL at different rates; the total covariance depends (roughly)quadratically on f NL whereas S ,µ only has a linear de-pendence. In addition, the matrix elements of S ,µ and C have different sensitivity to f NL at each angular separa-tion, and it is the relative importance of the off-diagonalmatrix elements relative to the diagonal elements thatsets the shape of the curve in Fig. 3.For very small values of | f NL | ( (cid:28) C . This can beseen in the f NL = 0 curves in the right panel of Fig. 1and in the right panel of Fig. 3. Note that the plots hidethe fact that the number of pixels at a given separationincreases with separation: the number of off-diagonal el-ements in the covariance is much bigger than the numberof diagonal elements, and this gives a “’geometric boost”to the covariance.For large values of | f NL | ( (cid:29) f NL = ±
100 curves inthe right panel of Fig. 1). Therefore, the constraints on f NL now worsen with the increasing value of | f NL | , albeitslowly.Finally, in the intermediate range of | f NL | ∼
10, the off-diagonal elements of C are small relative to the diagonaland near-diagonal elements. For example, the right panelof Fig. 1 shows that, for f NL = 20, the far-separation co-variances are much smaller than the variances. Howeverthe derivatives of the sample covariance, d S /df NL , areonly moderately smaller for the off-diagonal pixels thanfor the diagonal ones (e.g. a factor of ∼ f NL = 20;see the right panel of Fig. 3). Therefore, it is at theseintermediate values of | f NL | ∼
10 that we find the bestsignal-to-noise, and best constraints on f NL .In summary, the dependence on the fiducial value of f NL can be understood rather simply. For small f NL , thelarge-scale covariances do not add much signal. For large f NL the covariances add too much noise. At intermedi-ate f NL , the signal-to-noise relation is “just right”. Wecaution that the shape of the curve in Fig. 3 depends onthe volume (and geometry) of the survey as well as inthe number density of sources. The width of the pixelsaffect the width of the central part of the “hat” slightly.Smaller bins tend to shift the minima to smaller valuesof | f NL | . We conclude that the power of a DES-like clus-ter surveys to rule out the Gaussian hypothesis may beeven greater than indicated in Tables in this paper, sincethe error at f NL (cid:54) = 0 nearly always smaller than thatfor f NL = 0. This is another exciting development, butwarrants further investigation, and in particular a moredetailed study of the dependencies on the overall sur-vey volume and selection. In this initial study we simplyadopt the conservative errors, and show the f NL = 0 re-sults everywhere except in Fig. 3.0 FIG. 3: Left panel: Unmarginalized 1 − σ constraints on f NL as a function of the fiducial value of this parameter, assumingfive redshift and five mass bins. The “witch’s hat” shape can be explained from the competition between the derivative of thecovariance with respect to f NL , and the total covariance at the fiducial f NL ; see text. Right panel: Derivative of the signalmatrix elements S ij with respect to f NL as a function of angular separation between pixels i and j , for f NL = − , − , , z = 0 .
5, a separation of 1 degree corresponds to about 23 h − Mpc.
B. Clusters vs. galaxies
It is useful to compare cluster constraints obtainedhere with the expected constraints from a similar, DES-type, galaxy survey. Forecasts of constraints on primor-dial non-Gaussianity from galaxy clustering were studiedrecently [35, 45, 74] using the Fisher matrix and a sim-ple, Feldman-Kaiser-Peacock (FKP [68]) estimator thatcounts modes of P ( k ) and combines them with the sur-vey volume and its galaxy density. Perhaps counterintu-itively, our constraints are a factor of a few better thanthose from galaxies estimated previously. We now ex-plain the origin of this apparent discrepancy.Both clusters and galaxies probe the power spectrumof dark matter halos (and thus the halo bias). However,there are some important differences • Clusters additionally probe the mass function,which determines the counts, and also weakly af-fects the bias b ( M, z ); see Eqs. (B6) and (B7); • The number density of galaxies may be significantlyhigher, depending on how they and the clusters areselected. However, as mentioned in Sec. III, thelarger size of galaxy samples may not bring muchadditional information, since the constraints on f NL benefit from very large-scale halo separations, andnot from intra-halo correlations; • Clusters reside in more massive halos than galaxies,and thus have a higher bias. The higher the bias,the stronger is the correlation (cf. Eq. 9); • With regards to systematics, clusters can natu-rally be binned by the mass-observable, which helpsbreak degeneracies with nuisance parameters. Thisallows utilization of the cross-correlation betweendifferent mass bins to reduce the impact of samplevariance (e.g. [63, 64]), which we do not exploit inthis paper. • Large spectroscopic samples of galaxies are ex-pected in the near future, whereas clusters willrely on photometric redshifts; therefore, galaxy red-shifts are likely to be more accurate than clusterredshifts;Given all these differences, it is difficult to predictwhether clusters or galaxies will give a stronger con-straints on primordial non-Gaussianity without a directcalculation. We have verified that the FKP estimatorof galaxy constraints on f NL indeed gives a weaker re-sult, and is in rough agreement with previous estimatesin [35, 45, 46].However, as discussed in Tegmark et al. [69], theFKP estimator is only optimal and lossless on scalesmuch smaller than the linear size of the survey. Sincegood constraints on f NL benefit from precisely the large-wavelength modes, it is not surprising that the FKP es-timator for galaxies indicates worse constraints than ourpixel-based estimator for clusters. We have additionallyverified that constraints on the constant part of the bias, b (see Eq. (2)), or the dark energy equation of state w , which do not benefit as much from large-wavelength1modes, are comparable when estimated from the pixel-based formalism (from this paper) and the FKP approachassuming the same survey volume and number density ofobjects. C. Comparison to previous work
Numerous papers have studied the power of clustercounts alone to probe primordial non-Gaussianity (e.g.[26, 32–35]). To the extent that such constraints are gen-erally weak due to degeneracies, and strongly depend onthe priors and nuisance parameters varied, our results(see the “counts” columns in Table III) are in broadagreement with these studies.A more interesting comparison can be made withthe recent work of Oguri [52] who studied thecounts+variance case of clusters, corresponding to re-sults in our Table II. The main difference between thetwo studies is that we additionally considered the co-variance of cluster counts, and found that it leads to ahuge further improvement in the constraints. However,even for the counts+variance case only, our results differsubstantially, and we forecast a much weaker constrainton non-Gaussianity than Ref. [52]. For example, we get σ ( f NL ) ∼ σ ( f NL ) ∼ . Thesediscrepancies could probably be explained by a num-ber of other differences in the analyses: mass functions(Ref. [52] uses the LoVerde et al. [70] mass function withanalytic fit for skewness, while we use Dalal et al. massfunction from Eqs. (4)-(6)); cosmological parameter pri-ors (Ref. [52] uses the diagonal priors on some parameterswhile we use the full, off-diagonal Planck prior Fisher ma-trix), etc. We have not attempted to reproduce resultsfrom Ref. [52] using the assumptions made in that paper. D. Issues for future study
There are a number of effects that remain to be studiedin detail, but are beyond the scope of this preliminaryanalysis. We now list them here: • Fisher matrix approximation: in this paper we haveassumed the fiducial value of f NL = 0 and calcu-lated the errors on f NL by taking the derivatives ofobservables with respect to this parameter. This“Fisher error” will be a good approximation tothe true error if the error itself is small. There-fore, at least in the cases where the f NL error istight, we expect the Fisher approximation is a good Ref. [52] assumes only two mass-observable nuisance parameters. one, though this should eventually be checked withMarkov chain Monte Carlo methods. • Calculational issues:
The computation of the clus-ter covariance is time consuming, particularly forsmall but non-zero values of f NL . In this work wehave largely avoided this issue by using the Fishermatrix approximation and taking analytic deriva-tives around f NL = 0 (and a few other values),which enabled us to only evaluate the covarianceat the fiducial Gaussian model. With real data,however, a full exploration of parameter space willbe necessary, which might be sufficiently time con-suming to warrant analysis using a smaller set ofobservable parameters. For example, one could re-sort to using larger pixels and a coarser binningin redshift, or perhaps using no pixels at all. Onecould also explore speeding up the covariance cal-culations with various mathematical tricks. • Mass function: we have assumed the Dalal et al.[35] mass function which has been calibrated fromnumerical simulations and simply shifts the massof halos with non-Gaussianity. A number of al-ternative mass functions have recently been pro-posed in the literature and studied numerically[42, 70]. While the agreement in the relevant quan-tity n NG ( M, z ) /n G ( M, z ) is becoming good, thereis still no uniform agreement in the communityabout the convergence. The overall constraints areexpected to be robust given that most of the ef-fect of non-Gaussianity comes from the bias scalingas f NL k − and not the mass function. Neverthe-less, we expect constraints in this paper to be onthe conservative side: given that the Dalal et al.mass function predicts a smaller effect due to non-Gaussianity than some of the other popular func-tions, use of these other mass functions would onlyincrease the effects due to non-Gaussianity and thusimprove the error bars on f NL . • Corrections to the bias formula:
While the de-pendence of bias on f NL is established to followEq. (2) both analytically and numerically, it couldbe that there are second-order corrections to thebias formula. These have been discussed in the lit-erature; for example, it appears that a small con-stant offset in bias is warranted by the simulationsand some analytical results [41, 43, 44]. Studyof these higher-order corrections is very importantbut, given that there is no convergence in the com-munity on this issue as of yet, we leave their inclu-sion for future work. • Relativistic corrections and gauge dependence:
Wands and Slosar [65] have shown that, to first-order, the scale-dependent bias does not receive rel-ativistic corrections at large scales, using a spheri-cal collapse model. However, other authors have2shown that higher-order corrections in the mat-ter perturbations can produce non-Gaussianity (seee.g. [71, 72]). How the higher-order correction prop-agate to the halo bias is yet to be understood indetail. • Observational systematics:
In this paper we havemodeled the systematic uncertainties in under-standing of the Gaussian bias b ( M, z ) and the re-lation between cluster mass and its observationalproxy by introducing nuisance parameters that de-scribe uncertainty in these relations. However, wehave not attempted to model observational uncer-tainties, such as variations in atmospheric seeingor photometric calibration. Clearly, knowledge ofsuch uncertainties over large angular scales will beimportant if measurements of non-Gaussianity arenot to be substantially degraded. We leave thestudy of observational systematics for future work.
VI. CONCLUSIONS
In this paper we studied how well primordial non-Gaussianity of the local type can be probed with galaxyclusters. We took into account cluster number counts, aswell as the full covariance of cluster counts-in-cells. Weallowed generous uncertainties in the knowledge of thecluster mass-observable relation, the photometric red-shifts, and the Gaussian halo bias (we did not considersystematics due to uncertainties in angular selection,which may be important.) As we discuss at length inSec. III, the Fisher matrix calculation is computationallychallenging, and we resorted to a number of conservativeapproximations, the most important of which is usingvery large pixels. Since angular selection issues are ex-pected to be most significant at small angular scales, ourpixel choices partly justify neglecting angular uncertain-ties.We found that most information on primordial non-Gaussianity comes from the previously neglected covari-ance of counts. The covariance links cluster overdensitiesacross large distances, and thus benefits the constraintson primordial non-Gaussianity of the local type. Thereason is easy to understand: the non-Gaussian param-eter f NL enters through the term proportional to k − inthe bias, and correlates cluster counts in bins separatedby hundreds of megaparsecs. Other cosmological param-eters do not lead to these far-separation correlations incluster counts (see the right panel of Fig. 1). Correlationsof cluster counts across vast spatial distances of hundredsof megaparsecs therefore represent a smoking-gun signa-ture of primordial non-Gaussianity of the local type.The combination of counts and clustering is particu-larly effective at breaking degeneracies of f NL with cos-mological and nuisance parameters, since the two statis-tical probes complement each other very well. While ourfull set of 23 freely varying nuisance parameters can de- grade f NL constraints by factors of a few, even modestprior uncertainties on some of them break degeneraciesand restore the accuracy in f NL . For example, the biasin each photo-z bin needs to be known to 0.01 to keep f NL constraints within 15% of their values for the case ofperfectly known photo-z’s.We investigated the sensitivity of our results to thechoice of fiducial value of f NL and found that the uncer-tainty in f NL at f NL (cid:54) = 0 is smaller than that for f NL = 0.In other words, a non-zero small value of f NL may evenbe more sensitively differentiated from the f NL = 0 casethan indicated in our Tables; the reason for this is ex-plained in Sec. V A.Our forecasts indicate very strong constraints on pri-mordial non-Gaussianity, which is perhaps surprising.However, closer inspection reveals a number of effectsthat help clusters achieve these numbers; we discuss thesein Sec. V B. In particular, we use the pixel-based estima-tor, which is well suited for extracting signal from verylarge scales. Previous error forecasts of non-Gaussianityfrom galaxy clustering used the suboptimal FKP esti-mator; dark-energy studies that did use the pixel-basedestimator only considered variance of cluster counts.To achieve the full potential of forecasted constraintsdiscussed here, a few more issues need to be carefullystudied. Particularly important are theoretical uncer-tainties in linking dark matter halos to observed clustersof galaxies, and observational systematics across largeangular scales. While constraints on primordial non-Gaussianity have improved two orders of magnitude be-tween COBE [73] and WMAP [22], another one or eventwo orders of magnitude improvement may be possiblewith upcoming surveys of large-scale structure, especiallyif they include both dark matter halo counts and theirclustering covariance. Acknowledgements
We are extremely grateful to Neal Dalal for contribut-ing crucially to this project at its early stages. We thankWayne Hu for pointing out the reference [69] to us, andAnˇze Slosar and Adam Becker for useful discussions. Wealso thank the Aspen Center for Physics, where this workstarted, for hospitality. CC and DH are supported by theDOE OJI grant under contract DE-FG02-95ER40899.DH is additionally supported by NSF under contractAST-0807564, and NASA under contract NNX09AC89G.Part of the research described in this paper was carriedout at the Jet Propulsion Laboratory, California Insti-tute of Technology, under a contract with the NationalAeronautics and Space Administration.3
Appendix A: Parametrization of mass-observablerelation
We assume a log-normal form for the probability ofmeasuring an observable signal, denoted M obs , given truemass M , p ( M obs | M ) = 1 √ πσ ln M exp (cid:2) − x ( M obs ) (cid:3) , (A1)where x ( M obs ) ≡ ln M obs − ln M − ln M bias ( M obs , z ) √ σ ln M ( M obs , z ) . (A2)For the optical survey, the mass threshold of the ob-servable is set to M th = 10 . h − M (cid:12) and the redshiftlimit is z = 1, corresponding to the projected sensitiv-ity of the Dark Energy Survey. Different studies suggesta wide range of scatter for optical observables, rangingfrom a constant σ ln M = 0 . . < σ ln M < . ∼ . P ( M | M obs ), where M was determined using weaklensing and M obs was an optical richness estimate. Wechoose a fiducial mass scatter of σ ln M = 0 . M bias ( M obs , z ) = ln M bias0 + a ln(1 + z )+ a (ln M obs − ln M pivot ) , (A3) σ M ( M obs , z ) = σ + (cid:88) i =1 b i z i + (cid:88) i =1 c i (ln M obs − ln M pivot ) i . (A4)We set M pivot = 10 h − M (cid:12) . In all, we have 10 nuisanceparameters for the optical mass errors (ln M bias0 , a , a , σ , b i , c i ).There are few, if any, constraints on the number of pa-rameters necessary to realistically describe the evolutionof the variance and bias with mass. Ref. [54] shows that acubic evolution of the mass-scatter with redshift capturesmost of the residual uncertainty when the redshift evolu-tion is completely free (as assumed in the Dark EnergyTask Force (DETF) report [78]). While generous, thisparametrization assumes a lognormal distribution of themass-observable relation that may fail for low-masses (seee.g. [79]). However, [51] show that more complex distri-butions do not degrade results substantially ( ∼ − Appendix B: Photometric redshift errors andGaussian halo bias
Uncertainties in the redshifts distort the volume el-ement. Assuming photometric techniques are used todetermine the redshifts of the clusters (hereafter photo-z’s), and a perfect angular selection the mean number ofclusters in a photo-z bin z p i ≤ z p ≤ z p i +1 is¯ m α,i = (cid:90) z p i +1 z p i dz p (cid:90) dV ¯ n α W th i (Ω) p ( z p | z ) (B1)where W th i (Ω) is an angular top hat window function. Weparametrize the probability of measuring a photometricredshift, z p , given the true cluster redshift z as [55] p ( z p | z ) = 1 (cid:112) πσ z exp (cid:2) − y ( z p ) (cid:3) (B2)where y ( z p ) ≡ z p − z − z bias (cid:112) σ z , (B3) z bias is the photometric redshift bias and σ z is the scatterin the photo-z’s.On large scales, the number counts of clusters m ( x )trace the linear density perturbation δ ( x ) m i ( M α , x ) ≡ m iα = ¯ m i (1 + b ( M α , z ) δ ( x )) (B4)where M α denotes a bin in mass and i refers to the pixelon the sky defined by its angular location and redshift.The (Gaussian) halo bias may be very roughly approxi-mated by [80] b ( M ; z ) = 1 + a c δ c /σ − δ c + 2 p c δ c [1 + ( aδ c /σ ) p c ] (B5)with a c = 0 . p c = 0 .
3, and δ c = 1 .
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