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.
- c ? 2011 RAS, MNRAS 000, ??–?? r6A..
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
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
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
give us an insight on key physical processes that took place in the
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-
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.
Primordial non-Gaussianity is commonly parametrized by the non-
linear parameter fNLand may be written as follows (Lo Verde et al.
Bζ(k1,k2,k3) = (2π)4fNL
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,
= (2π)3δD(k123) Bζ(k1,k2,k3),
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arXiv:1109.6778v1 [astro-ph.CO] 30 Sep 2011
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
The WMAP 7-year estimate for flocal
G(x) − ?ζ2
is (Komatsu et al. 2011)
The bispectrum for this shape can be derived using Eq. (4) and it is
given by (Lo Verde et al. 2008; Komatsu 2010)
= 32 ± 21(68%C.L.) .
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)
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)
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),
δR(k,z) = D(z)W(k,R)M(k)T(k)ζ (k,z),
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
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
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? = fNL
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,
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
3, and above a certain threshold value, δc,
dM(z, M) = −2ρ
dM(M,z) = −
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A new signature of primordial non-Gaussianities3
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
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,
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),
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,
dz(z, M > Mlim) =dV
where dV/dz(z) is the comoving volume element which is given by
with H (z) = H0
dark energy equation of state parameter w is assumed to be con-
dM(M,z) = −
M?2∝ fNLis the skewness of the smoothed
(z, M, fNL) =dnST
dnPS/dM(z, M, fNL)
dnPS/dM(z, M, fNL= 0).
dM(z, M, fNL),
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
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.
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4A. M. M. Trindade et al.
(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).
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,
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
z,wef f(z), fNL= 0
z,wef f(z) = −1, fNL? 0
dz = NNG
|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,
?2|? W (k,V)|2T2(k)
where k = |k|,? W (k,V) is the Fourier transform of the rectangular
δr at the comoving distance r,
In Eq. (21), bNG
ple, given by,
volume V = rΘxrΘyδr, with angular area ∆Ω = Θx× Θyand depth
? W (k,V) = sinc
ef fis the effective non-Gaussian bias of the sam-
ef f(z,k) =
MlimdM bNG(z, M,k)dnNG
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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
b(z, M,k) = bG(z, M) +
bG(z, M) − 1
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,
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-
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
MR= M(k)T (k)W (k,R), R is the radius corresponding to
themass M andα = k2+y2+2kyµ.Notethatthepreviousexpression
DM, being σ2
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
cluster abundance. This is the result of there being a wef fthat max-
imizes the cluster number density at each redshift. The appearance
ence of primordial non-Gaussianities.
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
Avelino, P. P. & Viana, P. T. P. 2000, MNRAS, 314, 354
Baldauf, T., Seljak, U., Senatore, L., & Zaldarriaga, M. 2011, ArXiv e-
Bardeen, J. M., Bond, J. R., Kaiser, N., & Szalay, A. S. 1986, ApJ, 304,
Carlstrom, J. E., Ade, P. A. R., Aird, K. A., et al. 2009, ArXiv e-prints
Chiu, W. A., Ostriker, J. P., & Strauss, M. A. 1998, ApJ, 494, 479
Creminelli, P. 2003, J. Cosmology Astropart. Phys., 10, 3
Fedeli, C., Moscardini, L., & Matarrese, S. 2009, MNRAS, 397, 1125
Fosalba, P., Gazta˜ naga, E., & Elizalde, E. 2000, in Astronomical Society
of the Pacific Conference Series, Vol. 200, Clustering at High Redshift,
ed. A. Mazure, O. Le F` evre, & V. Le Brun, 408–+
Komatsu, E. 2010, Classical and Quantum Gravity, 27, 124010
Komatsu, E., Dunkley, J., Nolta, M. R., et al. 2009, ApJS, 180, 330
Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18
Komatsu, E. & Spergel, D. N. 2001, Phys. Rev. D, 63, 063002
Lo Verde, M., Miller, A., Shandera, S., & Verde, L. 2008, J. Cosmology
Astropart. Phys., 4, 14
Lyth, D. H. & Rodr´ ıguez, Y. 2005, Physical Review Letters, 95, 121302
Maldacena, J. 2003, Journal of High Energy Physics, 5, 13
Matarrese, S. & Verde, L. 2008, ApJ, 677, L77
Matarrese, S., Verde, L., & Jimenez, R. 2000, ApJ, 541, 10
Melin, J.-B., Bartlett, J. G., & Delabrouille, J. 2006, A&A, 459, 341
Press, W. H. & Schechter, P. 1974, ApJ, 187, 425
Robertson, B. E. 2010, ApJ, 716, L229
Robinson, J. & Baker, J. E. 2000, MNRAS, 311, 781
Robinson, J., Gawiser, E., & Silk, J. 2000, ApJ, 532, 1
Salopek, D. S. & Bond, J. R. 1990, Phys. Rev. D, 42, 3936
Seery, D. & Lidsey, J. E. 2005, J. Cosmology Astropart. Phys., 6, 3
Sefusatti, E. & Komatsu, E. 2007, Phys. Rev. D, 76, 083004
Sheth, R. K., Mo, H. J., & Tormen, G. 2001, MNRAS, 323, 1
Slosar, A., Hirata, C., Seljak, U., Ho, S., & Padmanabhan, N. 2008, J.
Cosmology Astropart. Phys., 8, 31
Sugiyama, N. 1995, ApJS, 100, 281
Wagner, C., Verde, L., & Boubekeur, L. 2010, J. Cosmology Astropart.
Phys., 10, 22
c ? 2011 RAS, MNRAS 000, ??–??
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, ??–??
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, ??–??