testing cosmology with galaxy clusters, the cmb and galaxy clustering

July 4, 2013 SuperJEDI Mauritius

Testing cosmology with galaxy clusters, the CMB and galaxy clustering

David Rapetti DARK Fellow

Dark Cosmology Centre, Niels Bohr Institute University of Copenhagen

In collaboration with Steve Allen (KIPAC), Adam Mantz (KICP), Chris Blake

(Swinburne), David Parkinson (Queensland), Florian Beutler (LBNL), Sarah Shandera (Pennsylvania)


Combined constraints on growth and expansion: breaking degeneracies “A combined measurement of cosmic growth and expansion from clusters of galaxies, the CMB and galaxy clustering”,

MNRAS 2013 (arXiv:1205.4679) David Rapetti, Chris Blake, Steven Allen, Adam Mantz, David Parkinson, Florian Beutler


GR γ~0.55

Modeling linear, time-dependent departures from GR

Linear power spectrum

Variance of the density fluctuations

General Relativity Phenomenological parameterization

Growth rate Scale independent in the synchronous gauge

Number density of galaxy clusters


Modeling linear, time-dependent departures from GR

Having measurements of σ8(z) allows us to obtain f(z)

To measure g we need growth, f(z), and expansion, Ωm(z), measurements

4 D. Rapetti et al.

2.4 The Alcock-Paczynski e!ect and

redshift-space distortions

The Alcock-Paczynski test is a geometrical means of prob-ing the cosmological model by a comparison of the ob-served tangential and radial dimensions of objects whichare assumed to be isotropic in the correct choice of model.It can be applied to the 2-point statistics of galaxy clus-tering if the redshift space distortions, the principal addi-tional source of anisotropy, can be successfully modelled(Ballinger et al. 1996; Matsubara & Suto 1996; Matsubara2000; Seo & Eisenstein 2003; Simpson & Peacock 2010). Byequating radial and tangential physical scales, the AP testdetermines the observable F (z) = (1 + z)DA(z)H(z)/c,where DA(z) is the physical angular diameter distance andc is the speed of light.

In the model fit for F (z), the normalized growth rate,f!8(z), is determined simultaneously. Here f(z) is again thelogarithmic rate of change of the growth factor at redshiftz (see equation 2) and !8(z) = [D(z)/D(0)]!8. In B11,RSD were modelled using the fitting formulae provided byJennings et al. (2011) to determine the density-velocity andvelocity-velocity power spectra, marginalizing over a linearbias factor. Tests were performed to ensure that the resultswere not very sensitive to the model used for the non-linearRSD, the real-space power spectrum, or the range of scalesfitted.

For a low-redshift survey such as 6dFGS, the Alcock-Paczynski distortion is negligible (since distances inh!1 Mpc are approximately independent of the assumed cos-mological model)8. For 6dFGS, the growth rate measure-ment of Beutler et al. (2012) was obtained by again assum-ing the model of Jennings et al. (2011) to described non-linear RSD.

For the BOSS measurement of the RSD and AP ef-fect, the modeling of the matter density and velocity fieldswas performed by following the approach of Reid & White(2011). The latter uses perturbation theory to calculate thenon-linear redshift space clustering of halos in the quasilin-ear regime and the halo model framework to describe thegalaxy-halo relation. This model was tested against a largeset of galaxy catalogs from N-body simulations and onlyfit over those scales where the quasilinear velocity field wasthought to dominate the signal and the small-scale randomvelocities could be simply modeled and marginalized over.

For all the RSD and AP e!ect measurements employedin the paper (see Section 4.2), the parameters used to fit the2D galaxy power spectrum and galaxy correlation functiondata have negligible covariance with the parameters in thecurrent analysis. Also, the linear model as well as the non-linear corrections assumed in those analyses lie within theGR+"CDM paradigm tested here9.

8 Beutler et al. (2012) calculated the uncertainties in F (z) andshowed that they are unimportant.9 The non-linear modeling from Jennings et al. (2011) used inthe WiggleZ and 6dFGS analyses also encompasses a range ofquintessence dark energy models.

3 COVARIANCES

In this section, we describe the principle degeneracies be-tween our most relevant growth and expansion parameters,for each the observations used.

3.1 CMB anisotropies

From the normalization and tilt of the CMB temperatureanisotropy power spectrum, we can primarily constrain thescalar amplitude and spectral index of primordial fluctua-tions; from the position of its first peak, the mean energydensity of curvature and dark energy; and from the ampli-tudes of the second and third peaks, those of dark and bary-onic matter. These measurements provide strong constraintson the content of the background energy density and its lin-ear density fluctuations at high redshift. For a given valueof the growth index, ", these translate into tight constraintson the amplitude of the matter power spectrum today, !8. Amodel with faster perturbation growth, i.e. with a small ",implies large fluctuations today, i.e. large !8, and vice-versa.This provides a large, negative correlation between !8 and" (see Figure 1). At low redshift, the ISW e!ect of the CMBdata (see Section 2.3) constrains ", which is otherwise un-constrained by this data set.

3.2 Distribution of galaxies

From measurements of the shape of the galaxy power spec-trum and correlation function, we use constraints on theproduct f(z)!8(z) and on the quantity F (z), where the lat-ter are purely expansion history constraints, i.e. on #m(z).For this data set, both of these constraints are crucial tomeasure " = ln f(z)/ ln#m(z).10 However, having a con-straint on f(z)!8(z), instead of on f(z), yields a positive cor-relation between " and !8 (see Figure 1) as long as #m < 1.The faster the perturbations grow (small "), the smallerthe perturbation amplitude, !8, needs to be to provide thesame amount of anisotropy in the distribution of galaxies,f(z)!8(z). Note that the current uncertainty on the bias ofbaryonic matter limits the ability of using the normaliza-tion of the galaxy power spectrum to measure !8, and thusto break the degeneracy with ".

3.3 Cluster abundance and masses

For clusters, we have direct constraints on !8(z) and #m(z)from abundance, mass calibration and gas mass fractiondata (see Sections 2.2 and 4.1). !8(z) measurements pro-vide us with constraints not only on !8(z = 0), fromthe local cluster mass function, but also on the growthrate f(z) = !(1 + z)d ln!8(z)/dz, from which togetherwith those on #m(z), we can constrain ". The evolution of!8(z) = !8e

!g(z) depends on ", #m and w as follows

10 Note that without the AP e!ect constraints on "m(z), no rel-evant constraints on ! can be obtained from RSD measurementsalone. For the same reason, the addition of the BAO constraintson "m(z) improves significantly the measurement of ! for thecombination gal+BAO (see the right panel of Figure 1).

c! 2011 RAS, MNRAS 000, 1–13

4 D. Rapetti et al.









3 COVARIANCES










c! 2011 RAS, MNRAS 000, 1–13

4 D. Rapetti et al.









3 COVARIANCES










c! 2011 RAS, MNRAS 000, 1–13

Growth and expansion from clusters, the CMB and galaxies 5

g(z) =

! z

0

(1 + z!)"1 "p(z!)! 1#"!

p(z!)!dz! (8)

= (3w!)"1 ["(z)! "(0)] , (9)

where "(z) = [p(z)! 1]1"! p(z)! 2F1 [1, 1; 1 + !; p(z)], 2F1

is a hypergeometric function, p(z) = p0(1 + z)"3w andp0 = !m/(!m ! 1). In practice, a negative degeneracy be-tween #8 and ! exists due to the limited precision of clus-ter mass estimates, but it is notably smaller than those de-scribed above (see Figure 1). Within the precision of thedata, indistinguishable cluster mass functions can be pro-duced by a model with a small initial amplitude, #8, anda slow growth rate (large !), or one with a slightly largerinitial amplitude and faster growth.

For the !+wCDM model, the dependence of #8(z) onthe product w ! implies a negative correlation on the w, !plane (see Figure 2). Within the precision of the data, afast expansion history (small w) can be mimicked by a slowgrowth history (large !), and vice-versa.

4 DATA ANALYSIS

4.1 Galaxy cluster data

For clusters we use two experiments: growth of structure(M10a,b) and gas mass fraction (fgas; Allen et al. 2008)11.

Following the methods developed by M10a,b for thecluster growth analysis, we self-consistently and simultane-ously combine X-ray survey and follow-up observations toobtain the best constraints possible while accounting fullyfor selection biases. We employ the survey data to deter-mine cluster abundances and the follow-up data to calibratecluster masses from two observables, luminosity and tem-perature. For the survey data we employ three wide-areacluster samples drawn from RASS: the Bright Cluster Sam-ple in the northern sky (BCS; z < 0.3 and FX(0.1!2.4 keV)> 4.4"10"12 erg s"1 cm"2), the ROSAT-ESO Flux LimitedX-ray sample in the southern sky (REFLEX; z < 0.3 and FX

> 3.0"10"12 erg s"1 cm"2), and the Bright Massive ClusterSurvey with # 55 per cent sky coverage (Bright MACS; 0.3< z < 0.5 and FX > 2 " 10"12 erg s"1 cm"2). To keep sys-tematic uncertainties to a minimum, for all three sampleswe impose a lower luminosity cut of 2.5 " 1044h"2

70 erg s"1

(0.1! 2.4 keV) leaving a total of 78 clusters from BCS; 126clusters from REFLEX; and 34 clusters from Bright MACS.In total we use 238 clusters. For 94 of these clusters we usefollow-up observations from CXO or pointed observationsfrom ROSAT (M10b; distributed along the same redshiftrange of the survey data 0 < z < 0.5) to constrain simulta-neously the luminosity-mass (L–M) and temperature-mass(T–M) relations using the model from M10b (see a briefdescription in Section 4.1.1).

For the fgas analysis, we use the methods and data setof Allen et al. (2008) for 42 massive, hot (kT > 5 keV), dy-namically relaxed, X-ray luminous galaxy clusters spanningthe redshift range 0.05 < z < 1.1.

11 Note that the cluster growth analysis employs the fgas analysisto calibrate the masses for the scaling relations of Section 4.1.1using gas mass as a proxy for total mass (see details in M10a).

4.1.1 Scaling relations model

We model the L–M scaling relation as (M10b)

$$(m)% = %"m0 + %"m

1 m+ %"m2 log10(1 + z) , (10)

with a log-normal intrinsic scatter at a given mass of

#"m(z) = #"m(1 + #!"mz) , (11)

where $ & log10[L500E(z)"1/1044 erg s"1] and m &log10[E(z)M500/10

15 M#]. The subscript 500 refers to quan-tities measured within radius r500, at which the mean, en-closed density is 500 times the critical density of the Universeat redshift z. We model the T–M scaling relation $t(m)%,where t & log10 (kT500/ keV), and its scatter #tm(z) us-ing the same equations 10 and 11 but with the parameters%tm0 , %tm

1 , %tm2 , #tm and #!

tm instead of those with index $.When %"m

2 = 0 and %tm2 = 0 we have “self-similar” evolution

of the L–M and T–M relations respectively (Kaiser 1986;Bryan & Norman 1998)12. #!

"m = 0 and #!tm = 0 correspond

to scaling relations with non-evolving scatter.M10b showed that current data do not require depar-

tures from self-similar evolution and constant scatter. R10demonstrated that ! correlates weakly with departures fromself-similarity and constant scatter in the L–M relation andnegligibly for those in the T–M relation. Here we thereforeassume self-similar evolution and constant scatter for bothrelations (%"m

2 = #!"m = %tm

2 = #!tm = 0).

4.2 Galaxy clustering data

For WiggleZ, a series of growth and expansion analyses haverecently been released, and here we build on one in particu-lar: the joint analysis of the AP e"ect and growth of struc-ture presented by B11, which contains four redshift bins ofwidth #z = 0.2, spanning the redshift range 0.1 < z < 0.9.The WiggleZ survey at the Australian Astronomical Obser-vatory was designed to extend the study of large-scale struc-ture over large cosmic volumes to higher redshifts z > 0.5,complementing SDSS observations at lower redshifts. Thesurvey, which began in August 2006, completed observationsin January 2011 and has obtained of order 200,000 redshiftsfor UV-bright emission-line galaxies covering of order 1000square degree of equatorial sky.

For the WiggleZ analysis we fit our cosmological mod-els to the joint measurements of RSD and AP distortionpresented by B11. For this, we use the constraints ob-tained by B11 as a bivariate Gaussian likelihood for f #8(z)and F (z), including the large correlations between them.From B11, we have four bins with e"ective redshifts z =(0.22, 0.41, 0.60, 0.78) and f #8(z) = (0.53 ± 0.14, 0.40 ±0.13, 0.37 ± 0.08, 0.49 ± 0.12), F (z) = (0.28 ± 0.04, 0.44 ±0.07, 0.68 ± 0.06, 0.97 ± 0.12) and correlation coe$cientsr = (0.83, 0.94, 0.89, 0.84).

For the 6dFGS analysis we use the growth rate of struc-ture measurement obtained by Beutler et al. (2012). The6dFGS is a combined redshift and peculiar velocity survey

12 Self-similar evolution is entirely determined by the E(z) fac-tors in the definitions of !, t and m.

c! 2011 RAS, MNRAS 000, 1–13


g(z) =

! z

0

(1 + z!)"1 "p(z!)! 1#"!

p(z!)!dz! (8)

= (3w!)"1 ["(z)! "(0)] , (9)

where "(z) = [p(z)! 1]1"! p(z)! 2F1 [1, 1; 1 + !; p(z)], 2F1



4 DATA ANALYSIS





70 erg s"1






$$(m)% = %"m0 + %"m

1 m+ %"m2 log10(1 + z) , (10)


#"m(z) = #"m(1 + #!"mz) , (11)



1 , %tm2 , #tm and #!







2 = #!"m = %tm

2 = #!tm = 0).






c! 2011 RAS, MNRAS 000, 1–13


g(z) =

! z

0

(1 + z!)"1 "p(z!)! 1#"!

p(z!)!dz! (8)

= (3w!)"1 ["(z)! "(0)] , (9)

where "(z) = [p(z)! 1]1"! p(z)! 2F1 [1, 1; 1 + !; p(z)], 2F1



4 DATA ANALYSIS





70 erg s"1






$$(m)% = %"m0 + %"m

1 m+ %"m2 log10(1 + z) , (10)


#"m(z) = #"m(1 + #!"mz) , (11)



1 , %tm2 , #tm and #!







2 = #!"m = %tm

2 = #!tm = 0).






c! 2011 RAS, MNRAS 000, 1–13


g(z) =

! z

0

(1 + z!)"1 "p(z!)! 1#"!

p(z!)!dz! (8)

= (3w!)"1 ["(z)! "(0)] , (9)

where "(z) = [p(z)! 1]1"! p(z)! 2F1 [1, 1; 1 + !; p(z)], 2F1



4 DATA ANALYSIS





70 erg s"1






$$(m)% = %"m0 + %"m

1 m+ %"m2 log10(1 + z) , (10)


#"m(z) = #"m(1 + #!"mz) , (11)



1 , %tm2 , #tm and #!







2 = #!"m = %tm

2 = #!tm = 0).






c! 2011 RAS, MNRAS 000, 1–13


g(z) =

! z

0

(1 + z!)"1 "p(z!)! 1#"!

p(z!)!dz! (8)

= (3w!)"1 ["(z)! "(0)] , (9)

where "(z) = [p(z)! 1]1"! p(z)! 2F1 [1, 1; 1 + !; p(z)], 2F1



4 DATA ANALYSIS





70 erg s"1






$$(m)% = %"m0 + %"m

1 m+ %"m2 log10(1 + z) , (10)


#"m(z) = #"m(1 + #!"mz) , (11)



1 , %tm2 , #tm and #!







2 = #!"m = %tm

2 = #!tm = 0).






c! 2011 RAS, MNRAS 000, 1–13


g(z) =

! z

0

(1 + z!)"1 "p(z!)! 1#"!

p(z!)!dz! (8)

= (3w!)"1 ["(z)! "(0)] , (9)

where "(z) = [p(z)! 1]1"! p(z)! 2F1 [1, 1; 1 + !; p(z)], 2F1



4 DATA ANALYSIS





70 erg s"1






$$(m)% = %"m0 + %"m

1 m+ %"m2 log10(1 + z) , (10)


#"m(z) = #"m(1 + #!"mz) , (11)



1 , %tm2 , #tm and #!







2 = #!"m = %tm

2 = #!tm = 0).






c! 2011 RAS, MNRAS 000, 1–13

Rapetti et al 13


Flat ΛCDM + growth index γ Rapetti et al 13

clusters (XLF+fgas): BCS+REFLEX+MACS CMB (ISW): WMAP galaxies (RSD+AP): WiggleZ+6dFGS+BOSS Gold: clusters+CMB+galaxies (+BAO+SNIa+SH0ES)

! = 0.616± 0.061" 8 = 0.791± 0.019!m = 0.277± 0.011H0 = 70.2±1.0


Flat ΛCDM + γ: full pdf’s

Gold, solid line: clusters+CMB (ISW)+galaxies Red, dashed line: clusters Blue, dotted line: CMB (ISW) Green, long-dashed line: galaxies

Rapetti et al 13


Gold, solid line: clusters+CMB (ISW)+galaxies Red, dashed line: clusters Blue, dotted line: CMB (ISW) Green, long-dashed line: galaxies

Flat ΛCDM + γ: full pdf’s Rapetti et al 13


Flat ΛCDM + growth index γ Rapetti et al 13

clusters (XLF+fgas): BCS+REFLEX+MACS CMB (ISW): WMAP galaxies (RSD+AP): WiggleZ+6dFGS+BOSS For General Relativity γ~0.55 Magenta: clusters+galaxies Purple: clusters+CMB Turquoise: CMB+galaxies Gold: clusters+CMB+galaxies


Rapetti et al 13

For General Relativity γ~0.55 Magenta: clusters+galaxies Purple: clusters+CMB Turquoise: CMB+galaxies Gold: clusters+CMB+galaxies Platinum: clusters+CMB+galaxies+BAO (Reid et al 12; Percival et al 10)+SNIa (Suzuki et al 12) +SH0ES (Riess et al 11)

Flat wCDM + growth index γ: growth plane


Rapetti et al 13

Flat wCDM + growth index γ: expansion planes

Platinum: clusters + CMB + galaxies + BAO (Reid et al 12; Percival et al 10) + SNIa (Suzuki et al 12) + SH0ES (Riess et al 11)


Rapetti et al 13

For General Relativity γ~0.55 Magenta: clusters+galaxies Purple: clusters+CMB Turquoise: CMB+galaxies Gold: clusters+CMB+galaxies

Flat wCDM + growth index γ: expansion+growth


Rapetti et al 13

Flat wCDM + growth index γ: expansion+growth

For General Relativity γ~0.55 For ΛCDM w=-1 Gold: clusters+CMB+galaxies Platinum: clusters+CMB+galaxies+BAO+SNIa+SH0ES

! = 0.604± 0.078" 8 = 0.789± 0.019w = !0.967!0.053

+0.054

"m = 0.278!0.011+0.012

H0 = 70.0±1.3


Flat wCDM + γ: full pdf’s

Red, dashed line: clusters; Purple, dotted line: clusters+CMB; Gold, solid line: clusters+CMB+galaxies; Platinum, long-dashed line: all

Rapetti et al 13


Beyond ΛCDM: Primordial non-Gaussianity “X-ray cluster constraints on non-Gaussianity”,

arXiv:1304.1216 Sarah Shandera, Adam Mantz, David Rapetti, Steven Allen


Testing the Gaussianity of the primordial fluctuations

1.  When cumulants beyond skewness (correlations beyond the bispectrum) are important, we can only properly describe the non-Gaussianity with a one-parameter model if we can use this parameter to specify the amplitude of all the correlations.

2.  We assume two different ways to scale higher moments with the skewness based on particle physics models of inflation.

3.  Cluster counts probe smaller scales (0.1-0.5h/Mpc) than the CMB and the galaxy bias.

4.  Cluster counts are sensitive to any non-Gaussianity and to higher order moments of the probability distribution of primordial fluctuations.



– so much so that pursuing limits on non-Gaussianity down to the minimal levels expected

from single field slow-roll inflation is an important task. Galaxy clusters form from rare

primordial over-densities, with more massive and higher redshift clusters tracing rarer initial

fluctuations, and so their abundance is sensitive to non-Gaussianity in the primordial inho-

mogeneities. Constraints from cluster number counts are complementary to other probes of

non-Gaussianity in three ways: they probe smaller scales than the CMB or galaxy bias do

(cluster constraints are at k ≈ 0.1–0.5h/Mpc), they are sensitive to any non-Gaussianity

(including any shape for the primordial bispectrum), and they are sensitive to higher order

cumulants of the probability distribution of the primordial inhomogeneities.

The non-Gaussian statistics of the primordial perturbations are completely described by

the set of correlation functions �Φ(�k1)Φ(�k2) . . .Φ(�kn)�c, where Φ(k) is the Bardeen potential

in momentum space, and n runs from 3 to infinity. The subscript c stands for ‘connected’ andpicks out the parts of the correlations that vanish for a Gaussian field. Clearly, a single pa-

rameter cannot describe this series of functions. However, if the field is weakly non-Gaussian,

the three-point function is likely to generate the strongest observational signal, so many non-

Gaussian models are classified by naming the configuration of the three-point function and

its amplitude. The Wilkinson Microwave Ansiotropy Prove (WMAP) satellite, for example,

reports constraints on the parameters f localNL , f equil

NL , forthNL , where the labels ‘local’, ‘equilat-

eral’, and ‘orthogonal’ refer to specific, scale-independent functional forms assumed for the

three-point correlation. Translation invariance requires that the three-point correlation has a

factor δ3D(�k1+�k2+�k3) (a Dirac delta function), similar to the term δ3D(

�k1+�k2) in the power

spectrum. The three-point correlation is then just a function of two independent momenta

and is called the bispectrum.

The local, equilateral and orthogonal bispectra are shown in Eq.(2.3) below. Inter-

estingly, though, object number counts are not sensitive to the details of the bispectrum’s

momentum dependence. Instead, only the integrated moments of the smoothed density fluc-

tuations δR are relevant. For example,

�δ3R� =�

d3�k1(2π)3

�d3�k2(2π)3

�d3�k3(2π)3

M(k1, R, z)M(k2, R, z)M(k3, R, z)�Φ(�k1)Φ(�k2)Φ(�k3)�c(1.1)

where the terms M(ki, R, z) contain a window function, the corresponding factors from the

Poisson equation, the transfer function and the growth factor converting the linear perturba-

tion in the gravitational potential to the smoothed density perturbation. (The full expressions

for these quantities can be found in Appendix A.) We characterize the non-Gaussianity by

the dimensionless ratios of the cumulants of the density field

Mn,R =�δnR�c�δ2R�n/2

(1.2)

which are by construction redshift independent and nearly independent of the smoothing

scale, R, if the primordial bispectrum is scale independent.1

When cumulants beyond the skewness (correlations beyond the bispectrum) are relevant,

a one-parameter model is only useful if we can use it to specify the amplitude of all thecorrelations. In this paper we use M3 and a choice for how higher moments scale with M3

to describe non-Gaussian fluctuations. The scalings we consider are motivated by particle

1Scale independence means that the bispectrum (e.g., those in Eq.(2.3)) contains no length scale otherthan the factors k−1

i in the P (ki) terms.

– 2 –

– so much so that pursuing limits on non-Gaussianity down to the minimal levels expected

from single field slow-roll inflation is an important task. Galaxy clusters form from rare

primordial over-densities, with more massive and higher redshift clusters tracing rarer initial

fluctuations, and so their abundance is sensitive to non-Gaussianity in the primordial inho-

mogeneities. Constraints from cluster number counts are complementary to other probes of

non-Gaussianity in three ways: they probe smaller scales than the CMB or galaxy bias do

(cluster constraints are at k ≈ 0.1–0.5h/Mpc), they are sensitive to any non-Gaussianity

(including any shape for the primordial bispectrum), and they are sensitive to higher order

cumulants of the probability distribution of the primordial inhomogeneities.

The non-Gaussian statistics of the primordial perturbations are completely described by

the set of correlation functions �Φ(�k1)Φ(�k2) . . .Φ(�kn)�c, where Φ(k) is the Bardeen potential

in momentum space, and n runs from 3 to infinity. The subscript c stands for ‘connected’ andpicks out the parts of the correlations that vanish for a Gaussian field. Clearly, a single pa-

rameter cannot describe this series of functions. However, if the field is weakly non-Gaussian,

the three-point function is likely to generate the strongest observational signal, so many non-

Gaussian models are classified by naming the configuration of the three-point function and

its amplitude. The Wilkinson Microwave Ansiotropy Prove (WMAP) satellite, for example,

reports constraints on the parameters f localNL , f equil

NL , forthNL , where the labels ‘local’, ‘equilat-

eral’, and ‘orthogonal’ refer to specific, scale-independent functional forms assumed for the

three-point correlation. Translation invariance requires that the three-point correlation has a

factor δ3D(�k1+�k2+�k3) (a Dirac delta function), similar to the term δ3D(

�k1+�k2) in the power

spectrum. The three-point correlation is then just a function of two independent momenta

and is called the bispectrum.

The local, equilateral and orthogonal bispectra are shown in Eq.(2.3) below. Inter-

estingly, though, object number counts are not sensitive to the details of the bispectrum’s

momentum dependence. Instead, only the integrated moments of the smoothed density fluc-

tuations δR are relevant. For example,

�δ3R� =�

d3�k1(2π)3

�d3�k2(2π)3

�d3�k3(2π)3

M(k1, R, z)M(k2, R, z)M(k3, R, z)�Φ(�k1)Φ(�k2)Φ(�k3)�c(1.1)

where the terms M(ki, R, z) contain a window function, the corresponding factors from the

Poisson equation, the transfer function and the growth factor converting the linear perturba-

tion in the gravitational potential to the smoothed density perturbation. (The full expressions

for these quantities can be found in Appendix A.) We characterize the non-Gaussianity by

the dimensionless ratios of the cumulants of the density field

Mn,R =�δnR�c�δ2R�n/2

(1.2)

which are by construction redshift independent and nearly independent of the smoothing

scale, R, if the primordial bispectrum is scale independent.1

When cumulants beyond the skewness (correlations beyond the bispectrum) are relevant,

a one-parameter model is only useful if we can use it to specify the amplitude of all thecorrelations. In this paper we use M3 and a choice for how higher moments scale with M3

to describe non-Gaussian fluctuations. The scalings we consider are motivated by particle

1Scale independence means that the bispectrum (e.g., those in Eq.(2.3)) contains no length scale otherthan the factors k−1

i in the P (ki) terms.

– 2 –

2 The effect of Primordial non-Gaussianity on object number counts

Our basic tool is a series expansion for the ratio of the non-Gaussian mass function tothe Gaussian one. The expansion we use is based on a Press-Schechter model for haloformation applied to non-Gaussian probability distributions for the primordial fluctuations.A detailed derivation of the non-Gaussian mass function we use is given in Appendix Aand was developed in [30–32]. The weakly non-Gaussian probability distributions that themass function is based on are asymptotic expansions that deviate substantially from theactual probability density function (PDF) for sufficiently rare fluctuations. Fortunately, ourcosmology is already sufficiently constrained to determine that the clusters in our sampledo not lie in that regime. However, the clusters are sufficiently rare that truncating theexpansion below at a single term (the skewness) is not sufficient to test the full range ofmodels that are only as skewed as current CMB constraints allow.

We add non-Gaussianity to the cosmology by considering a mass function of the form�

dn

dM

�

NG

=

�dn

dM

�

T,M300

�nNG

nG

��Edgeworth

�(2.1)

where the first term on the right hand side is the Gaussian mass function of Tinker et al.[33] for clusters identified as spheres containing a mean density 300 times that of the meanmatter density of the Universe, 300 ρ̄m(z). The ratio of the non-Gaussian mass function tothe Gaussian one will be given as a series expansion, defined below. This factor will bea function of mass, redshift, and parameters that characterize the amplitude of the non-Gaussianity, which we define next.

2.1 Parametrizing the level of non-Gaussianity

Since object number counts are not sensitive to the details of the momentum space corre-lations, we consider the dimensionless, connected moments (the cumulants, divided by theappropriate power of the amplitude of fluctuations) of the density fluctuations smoothed ona given scale R, as defined in Eq.(1.2). Most constraints on non-Gaussianity have so far beenreported for a parameter that measures the size of the three-point correlation in momentumspace, or bispectrum. This is an extremely useful first statistic because this correlation shouldbe exactly zero if the fluctuations were exactly Gaussian. However, because the bispectrumis a function of two momenta, the non-Gaussian parameters most often quoted assume ashape for the bispectrum.

A generic homogeneous and isotropic bispectrum for the potential Φ can be written as�Φ(�k1)Φ(�k2)Φ(�k3)

�

c= (2π)3δ3D(�k1 + �k2 + �k3) B(k1, k2, k3) (2.2)

where the function B(k1, k2, k3) determines the shape. Bispectra are colloquially named bythe (triangle) configuration of the three momentum vectors that are most strongly correlated.To interpret our constraints on M3 in terms of familiar bispectra, we consider the templatesfor ‘local’, ‘equilateral’ and ‘orthogonal’ bispectra:

Blocal = 2f localNL (P (k1)P (k2) + P (k1)P (k3) + P (k2)P (k3)) (2.3)

Bequil = 6f equilNL [−P (k1)P (k2) + 2 perm.− 2(P (k1)P (k2)P (k3))

2/3

+P (k1)1/3P (k2)

2/3P (k3) + 5 perm.]

Borth = 6forthNL [−3P (k1)P (k2) + 2 perm.− 8(P (k1)P (k2)P (k3))

2/3

+3P (k1)1/3P (k2)

2/3P (k3) + 5 perm.]

– 4 –

Hierarchical scaling (local)

Feeder scaling (two field model)

Non-Gaussian mass function

Dimensionless ratios of the cumulants of the density field




dn

dM

�

NG

=

�dn

dM

�

T,M300

�nNG

nG

��Edgeworth

�(2.1)





�

c= (2π)3δ3D(�k1 + �k2 + �k3) B(k1, k2, k3) (2.2)




2/3

+P (k1)1/3P (k2)

2/3P (k3) + 5 perm.]


2/3

+3P (k1)1/3P (k2)

2/3P (k3) + 5 perm.]

– 4 –

Integrated moments of the smoothed density fluctuations

Generic homogeneous and isotropic bispectrum of the potential Object number counts are sensitive to the value of the total skewness and to the scaling ofhigher moments, rather than any details of the momentum space correlations.

In addition to the dependence on a parameter like fNL, the cumulants also have nu-merical coefficients that typically have to do with combinatorics. For example, beginningwith Eq.(1.3), the bispectrum contains three terms linear in f local

NL , each with two equivalentways to take the expectation value of pairs of fields ΦG. We will the choose the constantsof proportionality equal to combinatoric factors for the moments that are generated in thelocal ansatz and a simple two-field extension that gives feeder scaling:4

Hierarchical Mhn = n! 2n−3

�Mh

36

�n−2(2.7)

Feeder Mfn = (n− 1)! 2n−1

�Mf

38

�n/3. (2.8)

For a given scaling of the moments, we can determine a series expansion for the probabilitydistribution and for the mass function that can be consistently truncated at some order inthe moments.

For the single parameter scenarios, we report constraints in terms of the scaling as-sumed and the parameter M3, which can be compared with other constraints on particularbispectrum shapes using Table 1.

2.2 The mass function in terms of M3 and the scaling of higher moments

We will assume the non-Gaussian factor in the mass function of Eq.(2.1) takes the followingform:

nNG

nG

��Edgeworth

≈ 1 +F h,f�1 (M)

F �0(M)

+F h,f�2 (M)

F �0(M)

+ . . . (2.9)

Each term in the series is normalized by the Press-Schechter Gaussian term, F �0(M) =

(e−ν2c /2/√2π)(dσ/dM)(νc/σ), where νc = δc/σ, δc = 1.686 is the collapse threshold, and σ =

σ(M) is the variance in density fluctuations smoothed on the appropriate scale (Eq.(A.4)).Although the first term, F h �

1 (M) or F f �1 (M), is proportional to M3 regardless of how the

higher moments scale, the exact form of all higher order terms depends on the choice ofscaling. For the hierarchical and feeder scaling, F h �

n (M) and F f �n (M) are given in Eq.(A.14)

of the Appendix. Truncating this series after the first term is clearly unphysical since noprobability distribution with only a non-zero skewness can be positive everywhere. Althoughfor some objects (low mass, low redshift) this truncation does not cause a significant error,for rarer fluctuations it does. Keeping higher terms in the series is therefore important. Howsignificant these terms are in the context of cluster constraints depends on the mass and red-shift of the objects as well as the amplitude and scaling of the non-Gaussianity considered.In Section 5, we show several examples to illustrate how relevant the higher terms are as afunction of mass, redshift, skewness and scaling. Although this mass function has been shownto agree reasonably well with simulations, it does not come from a first principles derivation.In Section 5 we also contrast it to the Dalal et al mass function from simulations of the localansatz [20].

4The local ansatz was given in Eq.(1.3) with f2NL�ΦG(x)

2� � 1 to ensure weak non-Gaussianity. Themoments generated have the hierarchical scaling with I = fNL. To obtain representative combinatorics forthe feeder scaling, we use a scenario where one Gaussian field and one subdominant but highly non-Gaussianfield contribute to the inhomogeneities in the gravitational potential: Φ(x) = φG+σG+f̃NL[σG(x)

2−�σG(x)2�],

with f̃NLP1/2σ � 1. In that case I = f̃NLPσ/P1/2

Φ .

– 6 –

Object number counts are sensitive to the value of the total skewness and to the scaling ofhigher moments, rather than any details of the momentum space correlations.




�Mh

36

�n−2(2.7)


�Mf

38

�n/3. (2.8)





nNG

nG

��Edgeworth

≈ 1 +F h,f�1 (M)

F �0(M)

+F h,f�2 (M)

F �0(M)

+ . . . (2.9)










2−�σG(x)2�],


Φ .

– 6 –

physics models of inflation, and our constraints on the total dimensionless skewness can

always be re-written in terms of a particular bispectrum using Eq. (1.1) and Eq. (1.2).

Most previous work on the utility of cluster counts to constrain non-Gaussianity has

focused on the local ansatz [18, 19], where one assumes that the non-Gaussian field Φ(x) is

a simple, local transformation of a Gaussian field ΦG(x):

Φ(x) = ΦG(x) + f localNL [ΦG(x)

2− �ΦG(x)

2�]. (1.3)

In this useful model, f localNL is the single parameter that all correlation functions depend on,

and the cumulants scale2

as (f localNL )

n−2. Non-Gaussianity of the local type has a bispectrum

that most strongly correlates Fourier modes of very different wavelengths. This particular

mode coupling generates strong signals in other large scale structure observables – most no-

tably introducing a scale dependence in the bias of any biased tracer of the underlying dark

matter distribution [11, 14, 20]. For this reason, papers that have analyzed the potential for

future surveys to constrain non-Gaussianity have largely focused on non-Gaussianity captured

by the local ansatz only, and on the superior constraints from the bias compared to number

counts. The bias may allow us to probe ∆f localNL ∼ O(1) in the near future [21, 22], although

this optimism is still subject to a full understanding of the relevant systematics [14, 23, 24].

Regardless, the motivation for looking at cluster number counts to constrain a scale-invariant,

local ansatz is certainly weak. However, single field models of inflation cannot produce large

local non-Gaussianity over a wide range of scales [25]. Since non-Gaussianity that does not

strongly correlate modes of very different wavelength is not particularly detectable in the

galaxy bias [13, 26, 27], that measure alone tests only a subset of viable inflation models.

Furthermore, number count constraints are sensitive to higher order correlation functions.

As we will demonstrate here, number counts can distinguish between scenarios with indis-

tinguishable bispectra that nonetheless are generated by qualitatively different inflationary

physics. That is, a model may have a local-shape bispectrum but higher order moments that

do not scale as those from the standard local ansatz do (e.g., [28]). In this case, number

counts or other measurements sensitive to higher moments will provide complementary in-

formation to the halo bias constraints. For the rest of the paper, we will assume that f localNL

is a parameter measured from the bispectrum alone (see Eq. (2.3) below), which does not

imply the entire series of correlations that Eq. (1.3) generates, unless we specify the local

ansatz as our model. This use of f localNL is more in line with how it is observationally defined

(e.g., in analyses of the CMB and halo bias).

This paper provides the first constraint on primordial non-Gaussianity from X-ray de-

tected clusters, and the first large scale structure constraint that can be usefully applied to

any model for primordial non-Gaussianity. In the next Section, we define our parametriza-

tion of the effects of primordial non-Gaussianity on cluster abundance, using a semi-analytic

non-Gaussian mass function in terms of a single new parameter. In Section 3 we discuss sev-

2This scaling can be seen from taking the Fourier transform of Eq. (1.3) and computing the connectedcorrelation functions. In order for the correlations of the non-Gaussian field to be non-zero they must beproportional to an even number of Gaussian fields. The connected piece is proportional to a delta functionrequiring the vector sum of all momenta to be zero, so it comes from the term in Φn with no more thantwo factors of the linear term from the right hand side of Eq. (1.3). Each contracted pair of Gaussian fieldscontributes a delta function in momentum and each quadratic (fNL) term contains a convolution integral.The result can have any number of convolution integrals remaining but only one delta function. In the weaklynon-Gaussian case the term with exactly two factors of the linear term dominates, so an n-point function hasn−2 factors of the quadratic term and is proportional to (fNL)n−2. In addition, there are 2(n−2)+2 = 2n−2factors of the Gaussian field which give n− 2 factors of the power spectrum amplitude.

– 3 –



Hierarchical scaling (local)

Feeder scaling (two field model)




dn

dM

�

NG

=

�dn

dM

�

T,M300

�nNG

nG

��Edgeworth

�(2.1)





�

c= (2π)3δ3D(�k1 + �k2 + �k3) B(k1, k2, k3) (2.2)




2/3

+P (k1)1/3P (k2)

2/3P (k3) + 5 perm.]


2/3

+3P (k1)1/3P (k2)

2/3P (k3) + 5 perm.]

– 4 –

Generic homogeneous and isotropic bispectrum of the potential





�Mh

36

�n−2(2.7)


�Mf

38

�n/3. (2.8)





nNG

nG

��Edgeworth

≈ 1 +F h,f�1 (M)

F �0(M)

+F h,f�2 (M)

F �0(M)

+ . . . (2.9)










2−�σG(x)2�],


Φ .

– 6 –





�Mh

36

�n−2(2.7)


�Mf

38

�n/3. (2.8)





nNG

nG

��Edgeworth

≈ 1 +F h,f�1 (M)

F �0(M)

+F h,f�2 (M)

F �0(M)

+ . . . (2.9)










2−�σG(x)2�],


Φ .

– 6 –

physics models of inflation, and our constraints on the total dimensionless skewness can

always be re-written in terms of a particular bispectrum using Eq. (1.1) and Eq. (1.2).

Most previous work on the utility of cluster counts to constrain non-Gaussianity has

focused on the local ansatz [18, 19], where one assumes that the non-Gaussian field Φ(x) is

a simple, local transformation of a Gaussian field ΦG(x):

Φ(x) = ΦG(x) + f localNL [ΦG(x)

2− �ΦG(x)

2�]. (1.3)

In this useful model, f localNL is the single parameter that all correlation functions depend on,

and the cumulants scale2

as (f localNL )

n−2. Non-Gaussianity of the local type has a bispectrum

that most strongly correlates Fourier modes of very different wavelengths. This particular

mode coupling generates strong signals in other large scale structure observables – most no-

tably introducing a scale dependence in the bias of any biased tracer of the underlying dark

matter distribution [11, 14, 20]. For this reason, papers that have analyzed the potential for

future surveys to constrain non-Gaussianity have largely focused on non-Gaussianity captured

by the local ansatz only, and on the superior constraints from the bias compared to number

counts. The bias may allow us to probe ∆f localNL ∼ O(1) in the near future [21, 22], although

this optimism is still subject to a full understanding of the relevant systematics [14, 23, 24].

Regardless, the motivation for looking at cluster number counts to constrain a scale-invariant,

local ansatz is certainly weak. However, single field models of inflation cannot produce large

local non-Gaussianity over a wide range of scales [25]. Since non-Gaussianity that does not

strongly correlate modes of very different wavelength is not particularly detectable in the

galaxy bias [13, 26, 27], that measure alone tests only a subset of viable inflation models.

Furthermore, number count constraints are sensitive to higher order correlation functions.

As we will demonstrate here, number counts can distinguish between scenarios with indis-

tinguishable bispectra that nonetheless are generated by qualitatively different inflationary

physics. That is, a model may have a local-shape bispectrum but higher order moments that

do not scale as those from the standard local ansatz do (e.g., [28]). In this case, number

counts or other measurements sensitive to higher moments will provide complementary in-

formation to the halo bias constraints. For the rest of the paper, we will assume that f localNL

is a parameter measured from the bispectrum alone (see Eq. (2.3) below), which does not

imply the entire series of correlations that Eq. (1.3) generates, unless we specify the local

ansatz as our model. This use of f localNL is more in line with how it is observationally defined

(e.g., in analyses of the CMB and halo bias).

This paper provides the first constraint on primordial non-Gaussianity from X-ray de-

tected clusters, and the first large scale structure constraint that can be usefully applied to

any model for primordial non-Gaussianity. In the next Section, we define our parametriza-

tion of the effects of primordial non-Gaussianity on cluster abundance, using a semi-analytic

non-Gaussian mass function in terms of a single new parameter. In Section 3 we discuss sev-

2This scaling can be seen from taking the Fourier transform of Eq. (1.3) and computing the connectedcorrelation functions. In order for the correlations of the non-Gaussian field to be non-zero they must beproportional to an even number of Gaussian fields. The connected piece is proportional to a delta functionrequiring the vector sum of all momenta to be zero, so it comes from the term in Φn with no more thantwo factors of the linear term from the right hand side of Eq. (1.3). Each contracted pair of Gaussian fieldscontributes a delta function in momentum and each quadratic (fNL) term contains a convolution integral.The result can have any number of convolution integrals remaining but only one delta function. In the weaklynon-Gaussian case the term with exactly two factors of the linear term dominates, so an n-point function hasn−2 factors of the quadratic term and is proportional to (fNL)n−2. In addition, there are 2(n−2)+2 = 2n−2factors of the Gaussian field which give n− 2 factors of the power spectrum amplitude.

– 3 –


�

c= (2π)3δ3

D(�k1 + �k2 + �k3) B(k1, k2, k3) (2.2)

where the function B(k1, k2, k3) determines the shape. Bispectra are colloquially named bythe (triangle) configuration of the three momentum vectors that are most strongly correlated.To interpret our constraints onM3 in terms of familiar bispectra, we consider the templatesfor ‘local’, ‘equilateral’ and ‘orthogonal’ bispectra:

Blocal = 2flocalNL (P (k1)P (k2) + P (k1)P (k3) + P (k2)P (k3)) (2.3)

Bequil = 6fequilNL [−P (k1)P (k2) + 2 perm.− 2(P (k1)P (k2)P (k3))2/3

+P (k1)1/3P (k2)2/3

P (k3) + 5 perm.]Borth = 6f

orthNL [−3P (k1)P (k2) + 2 perm.− 8(P (k1)P (k2)P (k3))2/3

+3P (k1)1/3P (k2)2/3

P (k3) + 5 perm.]

where the power spectrum, P (k), is defined from the two-point correlation function by

�Φ(�k1)Φ(�k2)

�= (2π)3δ3

D(�k1 + �k2)P (k1) ≡ (2π)3δ3D(�k1 + �k2)2π

2 ∆2Φ(k0)k

31

�k1

k0

�ns−1

(2.4)

where ∆Φ(k0) is the RMS amplitude of fluctuations at a pivot point k0 and any runningof that amplitude with scale is parametrized with the spectral index ns. In the best fitcosmology from the seven-year WMAP data, baryon acoustic oscillations and Hubble pa-rameter measurements, the pivot point is k0 = 0.002 Mpc−1, the spectral index is a constantns = 0.967 [34, 35], and the amplitude is such that σ8 = 0.81. Observationally, the parameterf

localNL is typically measured by looking for a bispectrum of the form given in the first line of

Eq. (2.3), which has weaker implications than the definition of all the correlation functions asin Eq. (1.3). The best current limits on the amplitudes of the bispectra in Eq. (2.3) come fromthe Planck Satellite maps of the CMB [10], which limit f

localNL = 2.7± 5.8, f

equilNL = −42± 75,

and forthNL = −25± 39 at the 68.3% confidence level.Table 1 shows the result of computing M3 as defined in Eq. (1.2) using the local,

equilateral, and orthogonal templates and the WMAP7 best fit cosmology in Eq. (1.1). Wehave smoothed on a scale corresponding to 1014

h−1

M⊙ halos (h is related to the Hubbleparameter today, H0 = 100h km/s/Mpc).

Table 1. The conversions between the parameter M3 and the amplitudes of particular bispectrafNL. These numbers assume the WMAP7 best fit cosmology and change by at most a few percent ifthe best fit cosmologies from our analysis are used instead.

Shape M3

Local 0.00031 flocalNL

Equilateral 0.000086 fequilNL

Orthogonal −0.000062 forthNL

For some non-Gaussian scenarios (notably the local ansatz and typical single field mod-els) the parameter M3 is interchangeable with the f

localNL , f

equilNL , or f

orthNL as a description

– 5 –







�Mh

36

�n−2(2.7)


�Mf

38

�n/3. (2.8)





nNG

nG

��Edgeworth

≈ 1 +F h,f�1 (M)

F �0(M)

+F h,f�2 (M)

F �0(M)

+ . . . (2.9)










2−�σG(x)2�],


Φ .

– 6 –





�Mh

36

�n−2(2.7)


�Mf

38

�n/3. (2.8)





nNG

nG

��Edgeworth

≈ 1 +F h,f�1 (M)

F �0(M)

+F h,f�2 (M)

F �0(M)

+ . . . (2.9)










2−�σG(x)2�],


Φ .

– 6 –





�Mh

36

�n−2(2.7)


�Mf

38

�n/3. (2.8)





nNG

nG

��Edgeworth

≈ 1 +F h,f�1 (M)

F �0(M)

+F h,f�2 (M)

F �0(M)

+ . . . (2.9)










2−�σG(x)2�],


Φ .

– 6 –

The sets {km}h are again non-negative integer solutions to k1 + 2k2 + · · · + sks = s and

r = k1+k2+ · · ·+km, but the sets {km}f are solutions to 3k1+4k2+ · · ·+(s+2)ks = s+2.

Now for either scaling, truncating the series at some finite s in the sums above keeps all terms

up to the same order in M3: Ms3 for hierarchical scalings and Ms/3

3 for feeder scalings.

To write the mass function we will need derivatives of all the terms in the expansion

with respect to mass (or smoothing scale). In general, the derivatives can be found using the

relationship for the Hermite polynomials:

νHn(ν)−dHn(ν)

dν= Hn+1(ν) . (A.12)

The ratio of the non-Gaussian Edgeworth mass function to the Gaussian has the same struc-

tural form for either scaling:

nNG

nG

��Edgeworth

≈ 1 +F

h,f�1 (M)

F�0(M)

+F

h,f�2 (M)

F�0(M)

+ . . . (A.13)

with the derivatives of each term F�s = dFs/dM for s ≥ 1:

Fh �s (ν) = F

�0

�

{km}h

�Hs+2r

s�

m=1

1

km!

�Mm+2,R

(m+ 2)!

�km

(A.14)

+Hs+2r−1σ

ν

d

dσ

�s�

m=1

1

km!

�Mm+2,R

(m+ 2)!

�km��

Ff �s (ν) = F

�0

�

{km}f

�Hs+2

s�

m=1

1

km!

�Mm+2,R

(m+ 2)!

�km

+Hs+1σ

ν

d

dσ

�s�

m=1

1

km!

�Mm+2,R

(m+ 2)!

�km��

where the {km} again satisfy the relationships given below Eq.(A.11) and we have used

F�0 =

e− ν2c

2

√2π

dσ

dM

νc

σ. (A.15)

We have shared computer code to evaluate series for these two scalings at http://www.slac.

stanford.edu/~amantz/work/nongauss13/.

A.1 Other non-Gaussian mass functions

The mass function above has been shown to agree relatively well with N -body simulations

of the local ansatz [45, 55], although there is some disagreement in the literature about just

how well it agrees [67]. Some work has found evidence of a better fit if one shifts the collapse

threshold [68], but our data are not sensitive to that level of detail. Ideally, one would like

to use mass functions explicitly calibrated on simulations. To that end, Dalal et al. [20]

proposed the following

dn

dM=

�dMG

dn

dMG

dP

dM(MG) (A.16)

– 20 –

Press-Schechter normalization

Edgeworth expansion

Hierarchical scaling

Feeder scaling


Shandera et al 13

Gaussian distribution Μ3=0 Purple: clusters Gold: clusters+CMB

Flat ΛCDM + beyond skewness: hierarchical

−0.2 −0.1 0.0 0.1 0.2

0.7

0.8

0.9

1.0

1.1

M3

σ8

●●

clustersclusters+CMB

●

●

●●●

●

Hierarchical model

−600 −300 0 300 600fNL local

103M3 = !1!28+24

! 8 = 0.81!0.03+0.02

fNLlocal = !3!91

+78


Shandera et al 13


Flat ΛCDM + beyond skewness: feeder

−0.04 −0.02 0.00 0.02 0.04

0.7

0.8

0.9

1.0

1.1

M3

σ8

●●


●

●

●●●

●

Feeder model

−100 −50 0 50 100fNL local

103M3 = !1!28+24

! 8 = 0.81!0.03+0.02

fNLlocal = !14!21

+22


Shandera et al 13


Flat ΛCDM + beyond skewness: hierarchical

−0.2 −0.1 0.0 0.1 0.2

1.1

1.2

1.3

1.4

1.5

1.6

M3

β lm

●●


●

●

●●●

●

Hierarchical model

−600 −300 0 300 600fNL local

103M3 = !1!28+24

!lm =1.33!0.08+0.07

fNLlocal = !3!91

+78


Shandera et al 13


Flat ΛCDM + beyond skewness: feeder

−0.04 −0.02 0.00 0.02 0.04

1.1

1.2

1.3

1.4

1.5

1.6

M3

β lm

●●


●

●

●●●

●

Feeder model

−100 −50 0 50 100fNL local

103M3 = !1!28+24

!lm =1.32!0.05+0.06

fNLlocal = !14!21

+22


Flat ΛCDM + beyond skewness: redshift

−0.2 −0.1 0.0 0.1 0.2 0.3

0.6

0.7

0.8

0.9

1.0

1.1

M3

σ8

●●

clusters (all)clusters (z < 0.3)

●

●

●●●

●

Hierarchical model

−600 −300 0 300 600 900fNL local

Shandera et al 13

−0.04 −0.02 0.00 0.02 0.040.

60.

70.

80.

91.

01.

1M3

σ8

●●


●

●

●●●

●

Feeder model

−150 −100 −50 0 50 100 150fNL local


Shandera et al 13

� � � � � � � � � � ��

� � � � � � � � � � ��

� � � � � � � � � � ��

� � � � � ��

13.5 13.75 14. 14.25 14.5 14.75 15. 15.25 15.51.0

1.1

1.2

1.3

1.4z�0, fNL�30

�dndM

�NG�

dndM

�G

z = 0 , f localNL = 30

LMSV, skew onlyLMSV, hierarchical

LMSV, feeder

Dalal et al.

Log10(M/M⊙ h−1)

� � � � � � ��

��

� � � � � ��

��

� � � � � ��

�

� � � ��

�

13.5 13.75 14. 14.25 14.5 14.75 15. 15.25 15.51.0

1.1

1.2

1.3

1.4z�1, fNL�30

�dndM

�NG�

dndM

�G

z = 1.0 , f localNL = 30

Log10(M/M⊙ h−1)

� � � � � � � � � � ��

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Shandera et al 13

Table 3. The constraints on the skewness can be converted to constraints on the amplitude of any

bispectrum. The shape of the bispectrum is independent of the scaling, although the usual local

ansatz corresponds to a local-shape bispectrum with hierarchical moments.

Scaling Data Local Bispectrum Equil. Bispectrum Orthog. Bispectrum

h CL −73+129−113 −271

+482−422 346

+538−615

h CL+CMB −3+78−91 −12

+289−338 15

+430−369

f CL −28+35−13 −106

+134−48 130

+60−164

f CL+CMB −14+22−21 −52

+85−79 63

+97−104

skew-only CL −29+532−78 −105

+1916−280 146

+389−2658

skew-only CL+CMB −9+234−65 −35

+841−234 48

+324−1167

q

Density

0 0.2 0.4 0.6 0.8 1

0.0

0.5

1.0

1.5 Hierarchical

Feeder

Figure 4. Posterior probability density of the q parameter for hierarchical and feeder scalings. In

our parametrization, small values of q boost higher order non-Gaussian moments relative to M3 and

q = 1 corresponds to the one-parameter models with either scaling. The clusters+CMB data disfavor

very small values of q, consistent with the overall preference for Gaussian initial fluctuations, although

only the feeder scaling shows a clear (if modest) preference for q = 1. At 95.4% confidence, the lower

limits on q are respectively q > 0.10 and q > 0.18 for the hierarchical and feeder scenarios.

q > 0.10 and q > 0.18 with the hierarchical and feeder scalings, assuming a uniform prior

from zero to one.

For ease of comparison to the literature (Section 5), we also obtained constraints on

M3 keeping only the first term in the non-Gaussian mass function, proportional to the

skewness. As shown in Table 2, the resulting error bars are larger than when we include more

terms in the series, reflecting the fact that those terms generically increase the deviation of

the mass function from the Gaussian one at higher masses and higher redshifts (again, see

Figure 6). For example, keeping only the first term, our constraints from cluster and CMB

data correspond to f localNL = −9

+234−78 (Table 3). When including all relevant terms we find

f localNL = −3

+78−91 for the hierarchical scaling and f local

NL = −14+22−21 for the feeder scaling.

In light of the favorable comparison of our results to those obtained previously (see Sec-

– 12 –

−0.2 −0.1 0.0 0.1 0.2 0.3

0.6

0.7

0.8

0.9

1.0

1.1

M3

!8

!!


!

!

!!!

!

Hierarchical model

−600 −300 0 300 600 900fNL local

−0.04 −0.02 0.00 0.02 0.04

0.6

0.7

0.8

0.9

1.0

1.1

M3

!8

!!


!

!

!!!

!

Feeder model

−150 −100 −50 0 50 100 150fNL local

Figure 5. As in Figure 2, but comparing constraints obtained from the full cluster data set with

those from only clusters at redshifts z < 0.3.

tion 5), we briefly investigate what characteristics of our data set influence the results. There

is a practical limitation, however, since any attempt to reduce the overall size of the sample,

the number of clusters with mass estimates from follow-up data, or the mass/redshift ranges

covered, necessarily impacts constraints on the full set of cosmological and scaling relation

parameters. Consequently, we confine ourselves to a single, limited, but informative compar-

ison by asking how our constraints change when data at z ≥ 0.3 are excluded. In detail, this

low-redshift sample contains 203 clusters, of which 61 have follow-up data, compared to 237

and 94 for the full data set. As shown in Figure 5 for single-parameter non-Gaussian models

using the full hierarchical and feeder scalings, the constraining power of this low-redshift data

set is significantly reduced. Results for the low-redshift clusters only are shown in Table 2.

5 Comparison with the literature

Previous forecasts for constraints on non-Gaussianity from cluster counts have been done by

Pillepich et al. [7] for the eROSITA X-ray mission, Sartoris et al. [43] for future X-ray surveys

resembling the Wide Field X-ray Telescope concept, Oguri [44] for a variety of future optical

surveys, Cunha et al. [6] for optically selected clusters in the Dark Energy Survey (DES),

and Mak and Pierpaoli [8] for future surveys using the Sunyaev-Zel’dovich effect. There have

been three previous cluster constraints on non-Gaussianity: two based on clusters detected

in the SPT survey, by Benson et al. [15], who find f localNL = −192±310, and Williamson et al.

[16], who report f localNL = 20 ± 450; and one based on the SDSS maxBCG cluster catalogue,

by Mana et al. [17], who have f localNL = 282± 317.8

The existing forecasts and constraints use a variety of prescriptions for the non-Gaussian

mass function (listed in Table 4). These mass functions are given in Eq.(A.19) in the Ap-

pendix and are plotted in Figure 6. The levels of non-Gaussianity shown are M3 = 0.009

8This result corresponds to their analysis of only cluster number counts, without including either the clusterpower spectrum or CMB data.

– 13 –

Benson et al 13 (SPT+CMB)

−0.2 −0.1 0.0 0.1 0.2 0.3

0.6

0.7

0.8

0.9

1.0

1.1

M3

!8

!!


!

!

!!!

!

Hierarchical model

−600 −300 0 300 600 900fNL local

−0.04 −0.02 0.00 0.02 0.04

0.6

0.7

0.8

0.9

1.0

1.1

M3

!8

!!


!

!

!!!

!

Feeder model

−150 −100 −50 0 50 100 150fNL local



























– 13 –

Williamson et al 11 (SPT+CMB)

−0.2 −0.1 0.0 0.1 0.2 0.30.

60.

70.

80.

91.

01.

1M3

!8

!!


!

!

!!!

!

Hierarchical model

−600 −300 0 300 600 900fNL local

−0.04 −0.02 0.00 0.02 0.04

0.6

0.7

0.8

0.9

1.0

1.1

M3

!8

!!


!

!

!!!

!

Feeder model

−150 −100 −50 0 50 100 150fNL local



























– 13 –

Mana et al 13 (MaXBCG)

and M3 = 0.031, which correspond to the local model bispectrum with f localNL = 30 and 100,

respectively. Notice that, below about 1015 M⊙, the Dalal et al. mass function [20] deviatesa little less from the Gaussian for a given value of fNL than the LoVerde et al. mass function[30] (hereafter LMSV). However, some authors have found the LMSV mass function agreesbetter with simulation results if a reduced collapse threshold, δc ∼ 1.5, is used. If that ad-justment is made, the Dalal et al mass function would deviate more from the Gaussian thanLMSV; see [45] for a comparison of all these cases. Since the Dalal et al. mass function wascalibrated on simulations of the local ansatz, in principle it should include information abouthigher moments. This technique, though, has only been tried against one set of simulationsand only for non-Gaussianity of the local type. A more precisely calibrated, more generalnon-Gaussian mass function will be important for any future analysis of non-Gaussianitywith clusters.

Table 4. Gaussian mass functions and non-Gaussian extensions used in the literature. The non-Gaussian mass functions are either the first order semi-analytic expression from LoVerde et al. [30](LMSV) or the mass function calibrated on N-body simulations of the local ansatz by Dalal et al.[20]. All non-Gaussian mass functions also make use of a Gaussian mass function such as those fit bySheth and Tormen [46], Warren et al. [47], Jenkins et al. [48] or Tinker et al. [33, 49].

Author Mass Function usedBenson [15] Jenkins + fDalal

Cunha [6] Jenkins + fDalal

Mak [8] Tinker + fLMSV,skew only

Mana [17] Tinker + fLMSV,skew only

Oguri [44] Warren +fLMSV,skew only

Pillepich [7] Tinker + fLMSV,skew only

Sartoris [43] Sheth-Tormen + fLMSV,skew only

Williamson [16] Jenkins + fDalal

This work Tinker + fLMSV,many terms

Apart from the non-Gaussian mass function, these forecasts and analyses differ fromone another and from ours in two principal ways: the form and complexity assumed for themass–observable relation and its intrinsic scatter, and priors on the associated parameters.The most pessimistic forecasts in the literature find marginalized one sigma errors on f local

NLaround O(103) (e.g, some cases analyzed in [6, 7]). Those results assume that the scalingrelations will be constrained solely through self-calibration [50] rather than with estimatesof cluster masses, which can significantly boost the constraining power [51]. In addition,some forecasts assume significant photometric redshift errors [7]. As outlined in Section 4,all of the clusters in our sample have spectroscopic redshifts and for nearly half we also havefollow-up X-ray data that significantly improve the mass determinations.

Among the SPT results, Benson et al. use a smaller area of the survey than Williamsonet al., but have an improved mass calibration and extend their sample to lower SZ detectionsignificance (i.e. lower mass). In comparison, our cluster data set is significantly larger thaneither of the SPT cluster samples, contains more massive clusters (although at lower red-shifts), has a larger intrinsic scatter in the mass–observable relation (although the parametersof the scaling relation are better constrained), and uses a more straightforward mass calibra-

– 14 –
















– 14 –
















– 14 –
















– 14 –

testing cosmology with galaxy clusters, the cmb and galaxy clustering

Technology

pz 2f1

z measurements

constraints evant constraints

growth rate fz

bao constraints

direct constraints

ln fz ln mz

local cluster mass function