In previous sections, we have discussed the main elements of cluster formation in the standard CDM cosmology. Although this model is very successful in explaining a wide variety of observations, some of its key assumptions and ingredients are not yet fully tested. This provides motivation to explore different assumptions and alternative models.

As discussed in Section 3.7, the halo
mass function for a
Gaussian random field is uniquely specified by the peak height
=
_{c} /
(*R*, *z*),
where *R* is the filtering scale
corresponding to the cluster mass scale *M*. For sufficiently large
mass, that is rare peaks with
≫ 1, the mass function
becomes exponentially sensitive to the value of
. At the same time, the
mass function also determines the halo bias (see
Section 3.8). Again, for
≫ 1
and Gaussian perturbations, the bias function scales as
*b*() ~
^{2} /
_{c} =
/
(*R*,
*z*). Therefore, the cluster 2-point
correlation function can be written as
_{cl}(*r*)
= ^{2}(_{R}(*r*)
/ _{R}^{2}), where
_{R}(*r*) is the
correlation function of the smoothed fluctuation field (see
Section 3.1). Once the peak height
is constrained by requiring a model to predict the observed cluster
abundance, the value of the cluster correlation function at a single
scale *r* provides a
measurement of the shape of the power spectrum through the ratio of
the clustering strength at the scale *r* and at the cluster
characteristic scale *R*. These predictions are only valid under two
assumptions, namely Gaussianity of primordial density perturbations
and scale independence of the linear growth function *D*(*z*), as
predicted by the standard theory of gravity. Therefore, the combination of
number counts and large-scale clustering studies offers a powerful means
to constrain the possible violation of either one of these two
assumptions that hold for the
CDM model.

In this section, we briefly review the specifics of cluster formation in models with non-Gaussian initial density field and with non-standard gravity, the most frequently discussed modifications to the standard structure formation paradigm.

**5.1. Mass function and bias of clusters in
non-Gaussian models**

One of the key assumptions of the standard model of structure
formation is that initial density perturbations are described by a
Gaussian random field (see Section 3.1).
The simplest single-field, slow-roll inflation models predict nearly
Gaussian initial density fields. However, deviations from Gaussianity
are expected in a broad range of inflation models that violate
slow-roll approximation, and have multiple fields, or modified kinetic
terms (see
Bartolo et
al. 2004
for a review). Given that there is
no single preferred inflation model, we do not know which specific
form of non-Gaussianity is possibly realized in nature. Deviations
from Gaussianity are parameterized using a heuristic functional
form. One of the simplest and most common choices for such a form is the
local non-Gaussian potential given by
_{NG}(**x**) =
-(_{g}(**x**) +
*f*_{NL}[_{g}(**x**)^{2} -
<_{g}^{2}>]),
where _{NG} is
the usual Newtonian potential,
_{g}
is the Gaussian random field with zero mean,
and the parameter *f*_{NL} = const controls the degree and
nature of non-Gaussianity (e.g.,
Salopek
& Bond 1990,
Matarrese,
Verde & Jimenez 2000,
Komatsu
& Spergel 2001).
The simplest inflation models predict *f*_{NL}
10^{-2} (e.g.,
Maldacena
2003),
but a number of models that predict much
larger degree of non-Gaussianity exist as well
(Bartolo et
al. 2004).
The current CMB constraint on scale-independent non-Gaussianity is
*f*_{NL} = 30 ± 20 at the 68% confidence level (e.g.,
Komatsu 2010)
and there is thus still room for existence
of sizable deviations from Gaussianity.

The non-Gaussian fields with *f*_{NL} < 0 have a PDF of the
potential field that is skewed toward positive values and the
abundance of peaks that seed the collapse of halos is reduced compared
to Gaussian initial conditions. Conversely, the PDF of the potential
field in models with *f*_{NL} > 0 has negative skewness,
and hence an increased number of potential minima (density peaks). This
would result in an enhanced abundance of rare objects, such as massive
distant clusters, relative to the Gaussian case (see, e.g., figure 1 in
Dalal et
al. 2008
for an illustration of the effect of *f*_{NL} on the
large-scale structure that forms). The suppression or
enhancement of abundance of halos increases with increasing peak height.

The mass functions resulting from non-Gaussian initial
conditions have been studied both analytically (e.g.,
Chiu, Ostriker
& Strauss 1998,
Matarrese,
Verde & Jimenez 2000,
Lo Verde et
al. 2008,
Afshordi
& Tolley 2008)
and using cosmological simulations
(Grossi et
al. 2007,
Dalal et
al. 2008,
Lo Verde et
al. 2008,
LoVerde
& Smith 2011,
Wagner &
Verde 2012).
These studies showed that accurate formulae for the halo abundance from the
initial linear density field exist for the non-Gaussian models as
well. The general result is that the fractional change in the abundance
of the rarest peaks is of order unity for the initial fields with
|*f*_{NL}| ~ 100. The abundance of clusters is thus only
mildly sensitive to deviations of Gaussianity within the currently
constrained limits
(Scoccimarro, Sefusatti & Zaldarriaga 2004,
Sefusatti
et al. 2007,
Sartoris et
al. 2010,
Cunha, Huterer
& Doré 2010).
In contrast, primordial non-Gaussianity may also leave an imprint in the
spatial distribution of clusters in the form of a scale-dependence of
large-scale linear bias.

As was discovered by
Dalal et
al. (2008)
and confirmed in subsequent analytical
(Matarrese
& Verde 2008,
McDonald 2008,
Afshordi
& Tolley 2008,
Taruya,
Koyama & Matsubara 2008,
Slosar et
al. 2008)
and numerical studies
(Desjacques,
Seljak & Iliev 2009,
Pillepich,
Porciani & Hahn 2010,
Grossi et
al. 2009,
Shandera,
Dalal & Huterer 2011),
the linear bias of collapsed objects in the models with *local*
non-Gaussianity can be described as a function of wavenumber *k* by
*b*_{NG} = *b*_{g}
+ *f*_{NL} × const / *k*^{2}, where
*b*_{g} is the linear bias in the corresponding
cosmological model with the Gaussian initial conditions discussed in
Section 3.8. This scale dependence
arises because in the
non-Gaussian models the large-scale modes that boost the abundance of
peaks are correlated with the peaks themselves, which enhances (or
suppresses) the peak amplitudes by a factor proportional to
*f*_{NL}
*f*_{NL} /
*k*^{2}. Because this effect of
modulation increases with increasing peak height,
=
_{c} /
(*M*, *z*),
the scale-dependence of bias increases
with increasing halo mass. This unique signature can be used as a
powerful constraint on deviations from Gaussianity (at least for
models with *local* non-Gaussianity) in large samples of clusters
in which the power spectrum or correlation function can be measured on
large scales.

**5.2. Formation of clusters in modified gravity
models**

Recently, there has been a renewed interest in modifications to the standard GR theory of gravity (e.g., see Capozziello & de Laurentis 2011, Durrer & Maartens 2008, Silvestri & Trodden 2009) for recent reviews). These models have implications not only for cosmic expansion, but also for the evolution of density perturbations and, therefore, for the formation of galaxy clusters.

For instance, in the class of the *f*(*R*) models, cosmic
acceleration arises from a modification of gravity law given by the
addition of a general function *f*(*R*) of the Ricci curvature
scalar *R* in the Einstein-Hilbert action (see, e.g.,
Sotiriou
& Faraoni 2010,
Jain &
Khoury 2010
for recent reviews). Such modifications result
in enhancements of gravitational forces on scales relevant for
structure formation in such a way that the resulting linear
perturbation growth rate *D* becomes scale dependent; whereas on very
large scales gravity behaves similarly to GR gravity, on smaller
scales it is enhanced compared to GR and the rate of structure
formation is thereby also enhanced. The nonlinear halo collapse and
growth are also faster in *f*(*R*) models, which leads to
enhanced abundance of massive clusters
(Schmidt et
al. 2009,
Ferraro,
Schmidt & Hu 2011,
Zhao, Li &
Koyama 2011)
compared to the
predictions of the models with GR gravity and identical cosmological
parameters. Likewise, the peaks collapsing by a given *z* have lower
peak height in the modified
gravity models compared to the peak
height in the standard gravity model. This results in the reduced bias
of clusters of a given mass compared to the standard model.
Furthermore, the scale dependence of the linear growth also induces a
scale dependence of bias, thus offering another route to detect
modifications of gravity
(Parfrey,
Hui & Sheth 2011).
Qualitatively similar effects on cluster abundance and bias are
expected in the braneworld-modified gravity models based on higher
dimensions, such as the Dvali-Gabadadze-Porrati (DGP,
Dvali, Gabadadze &
Porrati 2000)
gravity model
(Schäfer & Koyama 2008,
Khoury &
Wyman 2009,
Schmidt 2009,
Schmidt, Hu
& Lima 2010)
and its successors with similar LSS phenomenology consistent with current
observational constraints, such as models of ghost-free massive
gravity
(de Rham,
Gabadadze & Tolley 2011,
D'Amico et
al. 2011).

A general consequence of modifying gravity is that the Birkhoff
theorem no longer holds, which does not allow a straightforward
extension of the spherical collapse model described in
Section 3.2 to a generic model of modified
gravity. Nevertheless, numerical calculations of spherical collapse
have been presented for a number of specific models (e.g.,
Schäfer & Koyama 2008,
Schmidt et
al. 2009,
Schmidt, Hu
& Lima 2010,
Martino,
Stabenau & Sheth 2009).
For both the *f*(*R*) and the DGP classes of models, the
results of simulations obtained so far suggest that halo mass function
and bias can still be described by the universal functions of peak
height, in which the threshold for collapse and the linear growth rate
are modified appropriately from their standard model values
(Schmidt et
al. 2009,
Schmidt, Hu
& Lima 2010).
This implies that it should be
possible to calibrate mass function and bias of halos in the modified
gravity models with the accuracy comparable to that in the standard
structure formation models.