# A new signature of primordial non-Gaussianities from the abundance of galaxy clusters

**ABSTRACT** The evolution with time of the abundance of galaxy clusters is very sensitive

to the statistical properties of the primordial density perturbations. It can

thus be used to probe small deviations from Gaussianity in the initial

conditions. The characterization of such deviations would help distinguish

between different inflationary scenarios, and provide us with information on

physical processes which took place in the early Universe. We have found that

when the information contained in the galaxy cluster counts is used to

reconstruct the dark energy equation of state as a function of redshift,

assuming erroneously that no primordial non-Gaussianities exist, an apparent

evolution with time in the effective dark energy equation of state

arises,characterized by the appearance of a clear discontinuity.

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**ABSTRACT:**The redshift dependence of the abundance of galaxy clusters is very sensitive to the statistical properties of primordial density perturbations. It can thus be used to probe small deviations from Gaussian initial conditions. Such deviations constitute a very important signature of many inflationary scenarios, and are thus expected to provide crucial information on physical processes which took place in the very early Universe. We have determined the biases which may be introduced in the estimation of cosmological parameters by wrongly assuming the absence of primordial non-Gaussianities. Although we find that the estimation of the present-day dark energy density using cluster counts is not very sensitive to the non-Gaussian properties of the density field, we show that the biases can be considerably larger in the estimation of the dark energy equation of state parameter $w$ and of the amplitude of the primordial density perturbations. Our results suggest that a significant level of non-Gaussianity at cluster scales may be able to reconcile the constraint on the amplitude of the primordial perturbations obtained using galaxy cluster number counts from the Planck Sunyaev-Zeldovich Catalog with that obtained from the primary Cosmic Microwave Background anisotropies measured by the Planck satellite.Monthly Notices of the Royal Astronomical Society 04/2013; 435(1). · 5.52 Impact Factor

Page 1

Mon. Not. R. Astron. Soc. 000, ??–?? (2011)Printed 3 October 2011(MN LATEX style file v2.2)

A new signature of primordial non-Gaussianities from the

abundance of galaxy clusters

A. M. M. Trindade1,2?, P P. Avelino1,2and P. T. P. Viana1,2

1Centro de Asrof´ ısica da Universidade do Porto, Rua das Estrelas 687, 4150-762 Porto, Portugal

2Departamento de F´ ısica e Astronomia da Faculdade de Ciˆ encias da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

Accepted 2011 ???. Received 2011 ???; in original form 2011 September 29

ABSTRACT

The evolution with time of the abundance of galaxy clusters is very sensitive to the statistical

properties of the primordial density perturbations. It can thus be used to probe small devia-

tions from Gaussianity in the initial conditions. The characterization of such deviations would

help distinguish between different inflationary scenarios, and provide us with information on

physical processes which took place in the early Universe. We have found that when the infor-

mation contained in the galaxy cluster counts is used to reconstruct the dark energy equation

of state as a function of redshift, assuming erroneously that no primordial non-Gaussianities

exist, an apparent evolution with time in the effective dark energy equation of state arises,

characterized by the appearance of a clear discontinuity.

Key words: galaxies: clusters – surveys

1INTRODUCTION

One of the most fundamental predictions of the simplest standard,

single field, slow-roll inflationary cosmology, is that the primordial

density fluctuations, that seeded the formation of the large-scale

structure we see today, were nearly Gaussian distributed (see e.g.

Creminelli 2003; Maldacena 2003; Lyth & Rodr´ ıguez 2005; Seery

& Lidsey 2005; Sefusatti & Komatsu 2007. Such prediction seems

to be in good agreement with current observations of the cosmic

microwave background anisotropies (e.g. Slosar et al. 2008) and

large-scale structure (e.g. Komatsu et al. 2011). Nevertheless, a

significant, potentially observable level of non-Gaussianity may be

produced in some inflationary models where any of the conditions

that give rise to the standard single-field, slow-roll inflation fail.

The detection of primordial non-Gaussianities would decrease

considerablythenumberofviableinflationarymodels,anditwould

give us an insight on key physical processes that took place in the

earlyUniverse.Suchdetectioncouldbeachievedthroughthestatis-

tical characterization of the properties of the large-scale structure,

namely the bispectrum and/or trispectrum of the galaxy distribution

(e.g. Sefusatti & Komatsu 2007; Matarrese & Verde 2008), or the

determination of the evolution with time of the abundance of mas-

sive collapsed objects such as galaxy clusters (see e.g. Matarrese

et al. 2000; Robinson & Baker 2000). These form at high peaks of

the density field δ(x) = δρ/ρ and their number density as a function

of redshift depends on the growth of structure, thus being sensitive

to the dynamics and energy content of the Universe and to the sta-

tistical properties of the primordial density fluctuations.

In this work, we address the issue of how primordial non-

?E-mail: Arlindo.Trindade@astro.up.pt

Gaussianities may affect the determination of the effective dark en-

ergy equation of state w using the evolution with time of the galaxy

cluster abundance. Throughout, and unless stated otherwise, we

consider our fiducial cosmological model to be a flat ΛCDM model

with WMAP 7-year cosmological parameters (WMAP+BAO+H0)

(Komatsu et al. 2011), namely, a Hubble constant, H0, equal to

100hkms−1Mpc−1with h = 0.704, fractional densities of matter

and baryons today of Ωm = 0.272, Ωbh2= 0.023 respectively,

a scalar spectral index, ns, equal to 0.963, and we normalize the

power spectrum so that σ8= 0.809.

2PRIMORDIAL NON-GAUSSIANITY

Primordial non-Gaussianity is commonly parametrized by the non-

linear parameter fNLand may be written as follows (Lo Verde et al.

2008),

Bζ(k1,k2,k3) = (2π)4fNL

P2

(k1k2k3)3A(k1,k2,k3),

ζ(K)

(1)

where K = k1+k2+k3,ζ istheprimordialcurvatureperturbationand

A is an auxiliar function that contains the shape of the bispectrum,

Bζ, of ζ. Further, Pζ∝ kns−1is the dimensionless power-spectrum,

while ki = |ki| with kibeing the wave vectors. In Fourier space,

the functions Pζand Bζare defined by means of the two and three-

point correlation functions,

?

?

ζk1ζk2

?

= (2π)3δD(k12)(2π)3Pζ(k1)

4πk3

1

,

(2)

ζk1ζk2ζk3

?

= (2π)3δD(k123) Bζ(k1,k2,k3),

(3)

c ? 2011 RAS

arXiv:1109.6778v1 [astro-ph.CO] 30 Sep 2011

Page 2

2A. M. M. Trindade et al.

where kij...n≡ ki+ kj+ ... + kn. Note that fNLmay or may not be

scale-dependent. In this work only the later case is considered.

The bispectrum of ζ is the lowest order statistics sensitive

to non-Gaussian features. Depending on the underlying physical

mechanism responsible for generating non-Gaussianities, different

triangular configurations (shapes) will arise. There are broadly four

classes of triangular shapes or, equivalently, four different bispec-

trum parametrizations can be found in the literature: Local, Equilat-

eral, Folded and Orthogonal. Here we shall only consider the first

two. They are defined as follows:

- Local shape: It arises from multi-field, inhomogeneous re-

heating, curvaton and ekpyrotic models. Mathematically, the Local

shape can be characterized by a simple Taylor expansion around

the Gaussian curvature perturbation, ζG, (Salopek & Bond 1990;

Komatsu & Spergel 2001; Lo Verde et al. 2008),

ζ (x) = ζG(x) +3

5flocal

The WMAP 7-year estimate for flocal

NL

?

ζ2

G(x) − ?ζ2

G(x)?

?

.

(4)

NL

is (Komatsu et al. 2011)

flocal

NL

The bispectrum for this shape can be derived using Eq. (4) and it is

given by (Lo Verde et al. 2008; Komatsu 2010)

3

10K−2(ns−1)?

k3

= 32 ± 21(68%C.L.) .

(5)

Alocal=

k3

1(k2k3)ns−1+

3(k1k2)ns−1?

2(k1k3)ns−1+ k3

.

(6)

This quantity is maximized for the so-called squeezed triangle con-

figuration, i.e. k3? k2≈ k1.

- Equilateral shape: It is characteristic of inflationary models

where scalar fields have a non-canonical kinetic term (for example

Dirac-Born-Infield inflation, see Alishahiha et al. 2004). The math-

ematical expression for the Equilateral shape is (Lo Verde et al.

2008; Komatsu 2010)

9

10K−2(ns−1)?

−2(k1k2k3)1+2(ns−1)/3+ k2+(ns−1)/3

This quantity reaches a maximum at k1 ≈ k2 ≈ k3. Constraints

from WMAP 7-year set the level of non-Gaussianity for this shape

at (Komatsu et al. 2011)

fequi

Aequi=

−k3

1(k2k3)ns−1+ perm.

1

k1+2(ns−1)/3

2

kns−1

3

+ perm.

?

.

(7)

NL= 26 ± 140(68%C.L.) .

As mentioned before, the abundance of rare objects such as

galaxy clusters holds relevant information that can be used to probe

the initial conditions. For such information to be of use, the statis-

tics of the density perturbation, δR, smoothed on a scale R, have

to be characterized. However, we have previously defined the non-

Gaussianity in the primordial curvature perturbation, ζ, rather than

in the smoothed linear density field. The relation between ζ and the

linear perturbation to the matter density today smoothed on a scale

R is given by (Lo Verde et al. 2008),

(8)

δR(k,z) = D(z)W(k,R)M(k)T(k)ζ (k,z),

where

M(k) =2

5

Ωm

H2

0

with D(z) being the linear growth factor (Komatsu et al. 2009),

T (k) is the transfer function adopted from Bardeen et al. 1986, and

W (k,R) is the smoothing top-hat window. We use the shape param-

eter given by Sugiyama 1995, Γ = Ωmhexp

(9)

1c2

k2,

(10)

?

−Ωb

?

1 +

√2h/Ωm

??

.

Using Eq. (9) and the definitions given in Eqs. (2) and (3), one

may compute the variance

?

?∞

and the three-point function for the smoothed density field (Lo

Verde et al. 2008)

?

?

with ζi≡ ζ (ki), Fi≡ W (ki,R)M(ki)T (ki), K = k1+ k2+ k3and

k12=

k2

Eqs. (11) and (12) will be of special relevance in the next section,

since in order to incorporate non-Gaussian initial conditions in the

prediction of rare objects, one has to derive a non-Gaussian prob-

ability density function (PDF) for the smoothed density field, δR.

This can be done by using a mathematical procedure that enables

us to construct the PDF from its comulants (see Lo Verde et al.

2008; Matarrese & Verde 2008 for details).

σ2(R) = δ2

R

=

d3k1

(2π)3

dk

kF2(k)Pζ(k) ,

?

d3k2

(2π)3F1F2?ζ1ζ2? =

=

0

(11)

?δ3

d3k1

(2π)3

R? = fNL

?

d3k1

(2π)3

?

d3k2

(2π)3

?

Pζ(K)

d3k2

(2π)3F1F2F3?ζ1ζ2ζ3?

?2 A(k1,k2,k12)

=

d3k2

(2π)3F1F2F12(2π)4?

(k1k2k12)3

, (12)

?

1+ k2

2+ 2k1· k2(hereadotrepresentsthescalarproduct).

3 THE DARK ENERGY EQUATION OF STATE FROM

THE GALAXY CLUSTER ABUNDANCE

3.1 Halo Mass Function

The comoving number density of virialized halos per unit of vol-

ume at a given redshift z, dn/dM (z, M), with a mass, M, in the

range [M, M + dM] is called the mass function. In the presence of

non-Gaussian initial conditions, the expression for the mass func-

tion has been derived using extensions of the Press-Schechter (PS)

formalism (Press & Schechter 1974). This formalism asserts that

the fraction of matter ending up in objects of mass M is propor-

tional to the probability that the density fluctuations smoothed on

the scale R = (3M/4πρ)

can be written as,

??∞

where ρ, σM, δcand P(ν, M) are respectively, the comoving mass

density, the r.m.s of mass fluctuation in spheres of radius R, the

critical overdensity in the spherical collapse model and the PDF

of the smoothed density field. Here we have incorporated the red-

shift dependence of the threshold for the collapse as δc(z) =

1.686D(0)D−1(z). For Gaussian initial conditions the mass func-

tion acquires the following form,

?

πσM

The cosmological parameters enter Eq. (14) essentially through the

variance and the linear growth factor, as well as, through the critical

density contrast δc(z).

There are several prescriptions to change Eq. (14) in the pres-

ence of non-Gaussian initial conditions (Chiu et al. 1998; Robinson

et al. 2000; Avelino & Viana 2000; Fosalba et al. 2000; Matarrese

et al. 2000). Here we will adopt that which has been proposed by

Lo Verde et al. 2008, and which has been shown to provide a good

1

3, and above a certain threshold value, δc,

dn

dM(z, M) = −2ρ

M

d

dM

δc(z)/σM

dνP(ν, M)

?

(13)

dn

dM(M,z) = −

2¯ ρ

M2

δc(z)

dln σM

dln Me−δ2c(z)/(2σ2

M).

(14)

c ? 2011 RAS, MNRAS 000, ??–??

Page 3

A new signature of primordial non-Gaussianities3

0.0

0

0.51.0

z

(a)

1.52.0

10

20

30

40

50

dN/dz/deg2

Gaussian

Local,fNL= −100

Local,fNL= −50

Local,fNL= 50

Local,fNL= 100

Local,fNL= 350

0.0

0

0.51.0

z

(b)

1.52.0

5

10

15

20

25

30

35

40

45

dN/dz/deg2

Gaussian

Equi.,fNL= −100

Equi.,fNL= −50

Equi.,fNL= 50

Equi.,fNL= 100

Equi.,fNL= 350

(c)

Figure 1. The number of galaxy cluster per unit of redshift per square degree with mass M > Mlim= 2 × 1014h−1M?(considering w = −1) and different

levels of non-Gaussianity, fNL= (−100,−50,50,100,350), for Local, panel (a), and Equilateral parametrizations (b) panel. In panel (c) is shown the effect

that a change on the constant dark energy equation of state parameter , w, has on number of clusters per unit of redshift per square degree (with Gaussian initial

conditions. fNL= 0).

fit to results from N-body simulations (see Wagner et al. 2010 and

references therein),

?

π

?δ4

where S3M = ?δ3

density field. If fNL= 0, then S3M= 0 and Eq. (15) reduces to the

Gaussian mass function.

Numerical simulations have shown that the PS form of the

mass function under-predicts the abundance of high-mass objects

and over-predicts low-mass ones. Therefore, to be in better agree-

ment with the results of numerical simulations, we follow Lo

Verde et al. 2008 and model the departures from Gaussianity us-

ing the mass function suggested by Sheth et al. 2001, with the non-

Gaussian mass function becoming,

dnNG

dMdM

The Seth-Tormen mass function has been calibrated using nu-

merical simulations with Gaussian initial conditions and it is given

by (Sheth et al. 2001),

?

dln σM

dln Me−aδ2c(z)/(2σ2

with a = 0.707, A = 0.322184 and p = 0.3.

Given Eqs. (16) and (17), one may now compute the number

of cluster per unit of redshift above a certain mass threshold, Mlim,

dN

dz(z, M > Mlim) =dV

Mlim

where dV/dz(z) is the comoving volume element which is given by

??z

with H (z) = H0

dark energy equation of state parameter w is assumed to be con-

stant.

dn

dM(M,z) = −

2

ρ

M2e−δ2c(z)/2σ2

− 2δ2

σ2

M

M

?dlnσM

+1

6

dln M

dS3M

dln MσM

?δc(z)

σM

+S3MσM

6

×

c(z)

σ4

M

c(z)

− 1

??

?δ2

c(z)

σ2

M

− 1

??

, (15)

M?/?δ2

M?2∝ fNLis the skewness of the smoothed

(z, M, fNL) =dnST

dnPS/dM(z, M, fNL)

dnPS/dM(z, M, fNL= 0).

(16)

dnST

dM

(z, M)

=

−

2a

πA

?

1 +

?aδc

σ2

?−p?

M)

¯ ρ

M2

δc(z)

σM

×

(17)

dz(z)

?∞

dMdn

dM(z, M, fNL),

(18)

dV

dz(z) =

4π

H (z)

0

dx

H (x)

?2

,

(19)

?

Ωm(1 + z)3+ (1 − Ωm)(1 + z)3(1+w)?1/2, where the

Figures 1a and 1b illustrate the impact of the primordial non-

Gaussianities on the number of galaxy clusters with mass threshold

M > Mlim = 2 × 1014h−1M?and w = −1, showing the strong

effect that fNL ? 0 has on the observed number of clusters as a

function of redshift. Figure 1c shows how the number of clusters

is affected when the dark energy equation of state parameter w is

changed, always assuming Gaussian initial conditions, for the same

mass threshold. The effect of fNLand w on the abundance of galaxy

clusters is quite different. On one hand, increasing w above w = −1

flattens the slope of the cluster abundance above z ≈ 0.5, which

translates in an increase in the number of high-z clusters. On the

other hand, changes in fNLmodify the cluster abundance more uni-

formly in redshift. The difference in behaviour occurs because w

affects both the volume factor and the mass function, while fNL

changes only the tail of the distribution of the density fluctuation,

thus modifying just the mass function.

3.2Estimation of wef f

Figures 1a, 1b and 1c, suggest that the redshift evolution of the

number density of galaxy clusters in non-Gaussian models could

be wrongly taken to be the result of an effective dark energy equa-

tion of state different from the real one, under the assumption of

Gaussian initial conditions. In order to test this hypothesis, we have

generated the redshift evolution of the cluster number density for

different non-Gaussian initial conditions in bins of redshift with

width ∆z = 0.05 up to redshift 2, assuming a mass threshold of

Mlim = 2.0 × 1014h−1M?and a sky area of 4000 square degrees.

These survey characteristics were motivated by the expected sen-

sitivity and sky coverage of the South Pole Telescope (SPT) sur-

vey (Melin et al. 2006; Carlstrom et al. 2009). We have considered

fLocal

NL

respectively for the Local and Equilateral bispectrum parametriza-

tions given in section 2.

Having generated a mock redshift distribution of the number

density of galaxy clusters with non-Gaussian initial conditions, we

then computed an effective dark energy equation of state, wef f, us-

ing Eq. (18) with fNL= 0, that mimics the distribution of the num-

ber of galaxy clusters in the presence of non-Gaussian initial con-

= (−100,−30,30,100) and fEquil.

NL

= (−330,−100,100,330),

c ? 2011 RAS, MNRAS 000, ??–??

Page 4

4A. M. M. Trindade et al.

0.40.50.60.70.80.9

z

−8

−7

−6

−5

−4

−3

−2

−1

w∗(z)

(a)(b) Local (c) Equilateral

Figure 2. In figure 2a it is plotted the wef fas a function of redshift that maximizes the abundance of clusters (with Gaussian initial conditions). The vertical

dotted line corresponds to the redshift z∗∼ 0.75, where w∗= −1. In figures 2b and 2c are plotted the reconstructed effective dark energy equation of state,

wef ffor different values of fNLand both Local and Equilateral parametrizations, respectively. The vertical dotted line is the same as in 2a).

Model

σ8,Gauss

fNL

-100-3030100

Local0.7971 0.80540.81260.8209

(a)

Model

σ8,Gauss

fNL

-330 -100100330

Equilateral0.80070.80650.81150.8173

(b)

Table 1. The computed σ8obtained by fixing the present-day non-Gaussian

number density of galaxy clusters for the Local parametrization (panel 1a)

and Equilateral parametrization (panel 1b).

ditions at the i-th redshift bin, by solving the following equation,

?zi+∆z/2

?zi+∆z/2

Thus, the effective dark energy equation of state, wef f, is defined

as being the the equation of state which is constant throughout the

history of the universe and that reproduces the number of clusters

of the mock non-Gaussian catalogue in i-th redshift bin.

The present-day number density of galaxy clusters was as-

sumed to be that which is obtained in the fiducial model (with non-

Gaussian initial conditions and σ8= 0.809). The normalization of

the power spectrum, ie. the value of σ8, for each of the other cos-

mological models considered here is then obtained by demanding

that such number is recovered. Table 1 shows the computed σ8for

different values of fNLand Local and Equilateral parametrizations.

Figures 2b and 2c show the reconstructed dark energy equa-

tion of state, wef f, as a function of the redshift for different values

of fNLand the Local and Equilateral parametrizations. At low red-

shift, our reconstructed wef f is very close to our fiducial w = −1.

On the other hand, at high redshift, our computed wef f deviates

from −1, with this effect being more evident for higher values of

NG

i

=

zi−∆z/2

dN

dz

dN

dz

?

?

z,wef f(z), fNL= 0

?

dz =

=

zi−∆z/2

z,wef f(z) = −1, fNL? 0

?

dz = NNG

i

(20)

|fNL| in both parametrizations. However, the most interesting fea-

ture in Figures 2b and 2c is the presence of a redshift interval where

wef fis undefined, which widens with increasing fNL. This happens

in a redshift range centered at the redshift z∗∼ 0.75, at which the

value of wef f, which we will call w∗, that maximizes the cluster

abundance is equal to −1 (see Figure 2a). In this interval, the prod-

uct of the comoving volume with the integral of the mass function,

when assuming Gaussian initial conditions, is always smaller than

the same quantity for non-Gaussian initial conditions, i.e. fNL? 0.

If our fiducial cosmological model had a different value for w, then

the discontinuity would appear at the redshift at which w = w∗.

3.3Estimation of wef fwith observational uncertainties

The observational estimation of the number density of galaxy clus-

ters is affected by two sources of statistical uncertainty: the shot-

noise, or Poisson uncertainty, and the cosmic variance. The statis-

tical uncertainty associated with the former increases, for example,

with the cluster mass threshold, as clusters then become more rare,

while the statistical uncertainty associated with the later increases,

for example, as the cosmic volume surveyed gets smaller.

Let us first generalize the calculation of the cosmic variance

of the dark matter halo distribution given in (Robertson 2010) to

the case of non-Gaussian initial conditions, for a sample of objects

at a given redshift z in a volume V and mass threshold Mlim,

?

M2(k)(2π)3Pζ(k)

4πk3

σ2

cv(z,V)

=

D2(z)

dk

(2π)3

?

bNG

ef f(z,k)

?2|? W (k,V)|2T2(k)

×

,

(21)

where k = |k|,? W (k,V) is the Fourier transform of the rectangular

δr at the comoving distance r,

?kxrΘx

In Eq. (21), bNG

ple, given by,

?+∞

volume V = rΘxrΘyδr, with angular area ∆Ω = Θx× Θyand depth

? W (k,V) = sinc

2

?

sinc

?kyrΘy

2

?

sinc

?kzδr

2

?

.

(22)

ef fis the effective non-Gaussian bias of the sam-

bNG

ef f(z,k) =

MlimdM bNG(z, M,k)dnNG

?+∞

dM(z, M)

MlimdMdnNG

dM(z, M)

.

(23)

c ? 2011 RAS, MNRAS 000, ??–??

Page 5

A new signature of primordial non-Gaussianities5

In the presence of primordial non-Gaussianities, the linear bias pa-

rameter bNGis scale-dependent (Matarrese & Verde 2008) and may

be written as,

?

where bGis the linear bias parameter of dark halos of fixed mass

M for gaussian initial conditions. Here we adopt the formula sug-

gested by Sheth, Mo and Tormen (2001), but see also Sheth et al.

2001 and Fedeli et al. 2009, for bG,

bG(z, M) = 1 +aδ2

δc(z)

b(z, M,k) = bG(z, M) +

bG(z, M) − 1

?

δc(z)ΓR(k) ,

(24)

c(z)/σ2

M− 1

+

2p/δc(z)

?

1 +

aδ2

c(z)/σ2

M

?2,

(25)

with a = 0.75 and p = 0.3.

The scale dependence of the linear bias in the presence of non-

Gaussian initial conditions is encapsulated in the term ΓRin Eq.

(24), and has the form, derived in Matarrese & Verde 2008 and

Fedeli et al. 2009,

?+∞

where?

only holds up to the Hubble radius, given that at larger scales the

non-Gaussian bias is expected to remain constant (Baldauf et al.

2011). It is straightforward to show that in the case of Gaussian

initial conditions Eq. (21) reduces to σ2

variance of the dark matter distribution.

Our objective in this section is to investigate if the lifting of

the degeneracy between fNLand wef f, found in the previous sec-

tion, still holds if the statistical observational uncertainty is taken

into account. wef fis estimated in similar fashion as in the previous

section, but we introduce two sources of statistical uncertainty at

the 1σ level in each redshift bin, the cosmic variance in the distri-

butionofgalaxyclustercounts(N), N2σ2

we add, as usual, in quadrature.

Figures 3 and 4 show the 1σ statistical uncertainty in the re-

constructed wef f, as a function of redshift, when the cosmic vari-

ance plus shot-noise is taken into account. As we can see, for the

SPT reference survey, the statistical observational uncertainty does

not make the discontinuity disappear for values of fNLin the order

of ±100 and ±330 for the Local and Equilateral parametrizations

respectively. But, as expected, with a smaller sky coverage, it be-

comes increasingly difficult to detect the discontinuity with high

statistical confidence.

ΓR(k)

=

1

8π2σ2

?+1

M?

dµ?

MR

0

dyy2?

?4πk3

MR(y)

×

−1

MR

?√α

(2π)3

Bζ

?

Pζ(k)

y,√α,k

?

,

(26)

MR= M(k)T (k)W (k,R), R is the radius corresponding to

themass M andα = k2+y2+2kyµ.Notethatthepreviousexpression

cv= b2σ2

DM, being σ2

DMthe

cvandshot-noise, N,which

4CONCLUSION

In this work, we investigated whether the presence of primordial

non-Gaussianities has an impact on the estimation of the effec-

tive dark energy equation of state, when one uses the abundance of

galaxy clusters as a tool to probe different cosmological scenarios.

We computed the effective dark energy equation of state, wef fper

redshift bin, assuming Gaussian initial conditions, that is capable

of reproducing the galaxy cluster counts expected in several non-

Gaussian models, thus constructing a correspondence fNL ?→ wef f

for each redshift bin. The most important result of this work is the

discovery of a redshift interval where no value for the effective dark

energyequationofstateiscapableofreproducingthenon-Gaussian

cluster abundance. This is the result of there being a wef fthat max-

imizes the cluster number density at each redshift. The appearance

ofsuchadiscontinuitythusconstitutesanewdiagnosticofthepres-

ence of primordial non-Gaussianities.

5 ACKNOWLEDGEMENTS

We thank Ant´ onio da Silva for useful discussions during the preparation

of this paper and Cosimo Fedeli for hints on the numerical calculation of

the non-Gaussian bias. This work was funded by FCT (Portugal) through

contract PTDC/CTE-AST/64711/2006.

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Page 6

6A. M. M. Trindade et al.

Figure 3. In this figure is plotted the reconstructed effective dark energy equation of state for the Local parametrization, when we consider cosmic variance +

shot-noise as source of observational uncertainty.

c ? 2011 RAS, MNRAS 000, ??–??

Page 7

A new signature of primordial non-Gaussianities7

Figure 4. In this figure is plotted the reconstructed effective dark energy equation of state for the Equilateral parametrization, when we consider cosmic

variance + shot-noise as source of observational uncertainty.

c ? 2011 RAS, MNRAS 000, ??–??

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