Here we introduce a few concepts and analytic results from the theory of structure formation which underly the use of galaxy clusters as cosmological probes. These provide us with the vocabulary which pervades the literature on analytic and numerical models of galaxy cluster evolution. Material in this section has been derived from three primary sources: Padmanabhan (1993) [12] for the spherical top-hat model for nonlinear collapse, Dodelson (2003) [4] for Press-Schechter theory, and Bryan & Norman (1998) [13] for virial scaling relations.
In the linear regime, both super- and sub-horizon scale perturbations grow
as t2/3 in the matter-dominated era. This means that
after recombination,
the linear power spectrum retains its shape while its amplitude grows as
t4/3 before the onset of cosmic acceleration
(Fig. 6a). When
2(k)
for a
given k approaches unity linear theory no longer applies, and some other
method must be used to determine the fluctuation's growth. In general,
numerical simulations are required to model the nonlinear phase of growth
because in the nonlinear regime, the modes do not grow independently.
Mode-mode coupling modifies both the shape and amplitude of the power
spectrum over the range of wavenumbers that have gone nonlinear.
At any given time, there is a critical wavenumber which we shall call the
nonlinear wavenumber knl which determines which portion of
the spectrum has evolved into the nonlinear regime. Modes with k <
knl are said to
be linear, while those for which k > knl are nonlinear
(Fig. 6b).
Conventionally, one defines the nonlinear wavenumber such that
(knl,
z) = 1. From this one can derive a
nonlinear mass scale Mnl(z) =
(4
/ 3)
(z)
(2
/
knl)3.
A more useful and rigorous definition of the nonlinear mass scale comes
from evaluating the amplitude of mass fluctuations within spheres or
radius R at epoch z. The enclosed mass is M =
(4
/ 3)
(z) R3. The mean
square mass fluctuations (variance) is
![]() |
(13) |
where W is the Fourier transform of the top-hat window function
![]() |
(14) |
If we approximate P(k) locally with a power-law
P(k,z) = D2(z)
km, where D is the linear growth factor, then
2(M)
D2 R-(3 + m)
D2 M-(3 + m)/3. From this we see
that the RMS fluctuations are a
decreasing function of M. At very small mass scales, m
-3, and
the fluctuations asymptote to a constant value. We now define the nonlinear
mass scale by setting
(Mnl) = 1. We get that
([17])
![]() |
(15) |
For m > -3, the smallest mass scales become nonlinear first. This is the origin of hierarchical ("bottom-up") structure formation (Fig. 6b).
We now ask what happens when a spherical volume of mass M and radius R
exceeds the nonlinear mass scale. The simplest analytic model of the
nonlinear evolution of a discrete perturbation is called the spherical
top-hat model. In it, one imagines as spherical perturbation of radius
R and some constant overdensity
=
3M / 4
R3 in an Einstein-de Sitter
(EdS) universe. By Birkhoff's theorem the equation of motion for R is
![]() |
(16) |
whereas the background universe expands according to Eq. 6
![]() |
(17) |
Comparing these two equations, we see that the perturbation evolves like a universe of a different mean density, but with the same initial expansion rate. Integrating Eq. 16 once with respect to time gives us the first integral of motion:
![]() |
(18) |
where E is the total energy of the perturbation. If E < 0, the perturbation is bound, and obeys
![]() |
(19) |
where Rm and tm are the radius and
time of "turnaround". At turnaround
(as
), the fluctuation reaches its
maximum proper radius (see Fig. 7). As
t
2tm, R
0, and we say the
fluctuation has collapsed.
A detailed analysis of the evolution of the top-hat perturbation is
given in Padmanabhan (1993, Ch. 8) for general
m.
Here we merely quote results for an EdS universe.
The mean linear overdensity at turnaround; i.e., the value one
would predict from the linear growth formula
~
t2/3, is 1.063. The actual overdensity at
turnaround using the nonlinear model is 4.6. This illustrates that
nonlinear
effects set in well before the amplitude of a linear fluctuation reaches
unity. As R
0, the
nonlinear overdensity becomes infinite.
However, the linear overdensity at t = 2tm is
only 1.686. As the fluctuation
collapses, other physical processes (pressure, shocks, violent relation)
become important which
establish a gravitationally bound object in virial equilibrium before
infinite density is reached. Within the framework of the spherical top-hat
model, we say virialization has occurred when the kinetic and gravitational
energies satisfy virial equilibrium: |U| = 2K. It is easy to
show from conservation of energy that this occurs when R =
Rm / 2; in other
words, when the fluctuation has collapsed to half its turnaround
radius. The nonlinear overdensity at virialization
c
is not infinite since the radius is finite. For an EdS universe,
c =
18
2
180. Fitting
formulae for non-EdS models are provided in the next section.
The spherical top-hat model can be scaled to perturbations of arbitrary
mass. Using virial equilibrium arguments, we can predict various physical
properties of the virialized object. The ones that interest us most are
those that relate to the observable properties of gas in galaxy clusters,
such as temperature, X-ray luminosity, and SZ intensity change. Kaiser
[14]
first derived virial scaling relations for clusters in an EdS
universe. Here we generalize the derivation to non-EdS models of
interest. In order to compute these scaling laws, we must assume some
model for the distribution of matter as a function of radius within the
virialized object. A top-hat distribution
with a density
=
c
(z)
is not useful because it is
not in mechanical equilibrium. More appropriate is the isothermal,
self-gravitating, equilibrium sphere for the collisionless matter, whose
density profile is related to the one-dimensional velocity dispersion
[15]
![]() |
(20) |
If we define the virial radius rvir to be the radius of a
spherical volume within which the mean density is
c times
the critical density at that redshift (M =
4
rvir3
crit
c / 3),
then there is a relation between the virial mass M and
:
![]() |
(21) |
Here we have introduced a normalization factor
f
which will be used to
match the normailization from simulations. The redshift dependent Hubble
parameter can be written as H(z) = 100hE(z)
km s-1 with the function E2(z) =
m (1 +
z)3 +
k (1 +
z)2 +
,
where the
's have
been previously defined.
The value of
c is
taken from the spherical top-hat
model, and is 18
2
for the critical EdS model, but has a
dependence on cosmology through the parameter
(z) =
m (1 +
z)3 / E2(z).
Bryan and Norman (1998)
provided fitting formulae for
c for the
critical for
both open universe models and flat, lambda-dominated models
![]() |
(22) |
where x = (z) - 1.
If the distribution of the baryonic gas is also isothermal, we can define a
ratio of the "temperature" of the collisionless material
(T =
µ mp
2 / k)
to the gas temperature:
![]() |
(23) |
Given equations (22) and (23), the relation between temperature and mass is then
![]() |
(24) |
where in the last expression we have added the normalization factor
fT and set
= 1.
The scaling behavior for the object's X-ray luminosity is easily
computed by
assuming bolometric bremsstrahlung emission and ignoring the temperature
dependence of the Gaunt factor: Lbol
2
T1/2 dV
Mb
T1/2. where Mb is the
baryonic mass of the cluster. This is infinite for an isothermal density
distribution, since
is
singular. Observationally and
computationally, it is found that the baryon distribution rolls over to a
constant density core at small radius. A procedure is described in
Bryan and Norman (1998)
which yields a finite luminosity:
![]() |
(25) |
Eliminating M in favor of T in Eq. 25 we get
![]() |
(26) |
The scaling of the SZ "luminosity" is likewise easily computed. If we
define LSZ as the integrated SZ intensity change:
LSZ =
dA
ne
T( kT /
(mec2))dl
Mb T, then
![]() |
(27) |
We note that cosmology enters these relations only with the combination of
parameters h2
c
E2, which comes from the relation between the
cluster's mass and the mean density of the universe at redshift z. The
redshift variation comes mostly from E(z), which is equal to (1 +
z)3/2 for an EdS universe.
3.4. Statistics of hierarchical clustering: Press-Schechter theory
Now that we have a simple model for the nonlinear evolution of a spherical
density fluctuation and its observable properties as a function of its
virial mass, we would like to estimate the number of virialized objects of
mass M as a function of redshift given the matter power spectrum. This is
the key to using surveys of galaxy clusters as cosmological probes. While
large scale numerical simulations can and have been used for this purpose
(see below), we review a powerful analytic approach by Press and Schechter
[16]
which turns out to be remarkably close to the numerical results. The basic
idea is to imagine smoothing the cosmological density field at any epoch z
on a scale R such that the mass scale of virialized objects of interest
satisfies M = (4 / 3)
(z)R3. Because the density
field (both smoothed and unsmoothed) is a Gaussian random field, the
probability that the mean overdensity in spheres of radius R exceeds a
critical overdensity
c is
![]() |
(28) |
where (R,
z) is the RMS density
variation in spheres of radius R as discussed above.
Press and Schechter suggested that this probability be identified with the
fraction of particles which are part of a nonlinear lump with mass
exceeding M if we take
c = 1.686,
the linear
overdensity at virialization. This assumption has been tested against
numerical simulations and found to be quite good
[9]
(however, see below). The fraction of the
volume collapsed into objects with mass between M and M +
dM is given by
(dp / dM)dM. Multiply this by the average number
density of such objects
m
/M to get the number density of collapsed
objects between M and M + dM:
![]() |
(29) |
The minus sign appears here because p is a decreasing function of M. Carrying out the derivative using the fact that dM / dR = 3M / R, we get
![]() |
(30) |
The term is square brackets is related to the logarithmic slope of the
power spectrum, which on the mass scale of galaxy clusters is close to
unity. Eq. 30 is called the halo mass function, and it has the
form of a power law multiplied by an exponential. To make this more
explicit, approximate the power spectrum on scales of interest as a
power law as we have done above. Substituting the
scaling relations for
in Eq. 30 one gets the result
[17]
![]() |
(31) |
Here, Mnl(z) is the nonlinear mass scale. To be
more consistent with
the spherical top-hat model, it satisfies the relation
(Mnl, z) =
c ; i.e., those
fluctuations in the smoothed density field that
have reached the linear overdensity for which the spherical top-hat model
predicts virialization.
3.5. Validating the Halo Mass Function using N-body Simulations
Below we discuss how one goes about numerically simulating the nonlinear
evolution of the density fluctuations described by the
CDM power
spectrum. Here we simply mention two works which made detailed
comparisons of the PS formula with halo populations found in dark matter
N-body simulations. The first is by Jenkins et al. (2001)
[74]
who analyzed the results of the "Hubble Volume" simulation - a
simulation of dark matter clustering carried
out in a cubic volume 3 Gpc/h on a side with 10243 dark
matter particles (Fig. 8).
This yields a dark matter particle mass of 2.2 × 1012
M
,
implying that a galaxy cluster halo would typically contain 100-1000
particles. The relatively poor mass resolution is offset by the very
large volume, which permits exploring the cluster mass function across a
broad range of masses including the very high mass end.
Fig. 8 shows a slice through
the simulation volume on which the dark matter density field is plotted.
Jenkins et al. (2001) identified dark matter halos
using the friends-of-friends algorithm
[52]
and found that while the PS formula gives a good approximation
to the numerical data, it underpredicts the number of rare, massive
objects, and overpredicts the number of "typical" objects as shown in
the right panel of Fig. 8.
![]() |
Figure 8. The Hubble Volume Simulation [76]. Left: slice through the dark matter density field. Yellow/orange peaks correspond to galaxy clusters. Right: Halo multiplicity function, as measured from the simulation (solid lines), with Press-Schechter prediction superposed (dashed line). From [74]. |
Warren et al. (2006) [75] were interested in testing the validity of the PS formula over a wider range of mass scales than can be obtained from a single simulation. They simulated 16 boxes of different physical size but the same number of DM particles (10243) nested in such a way that together they define a composite halo mass function covering 5 orders of magnitude in mass scale. They derived a fitting formula for the composite halo mass function for the WMAP3 concordance cosmological parameters by assuming a parameterized form for the halo multiplicity function of:
![]() |
(32) |
where the multiplicity function
f() is related to
the mass function n(M) via
![]() |
(33) |
where A, a, b and c come from the fitting procedure, and are documented in [75].
To illustrate one application of this formula, Fig. 9 shows the cluster halo mass function computed using the Warren et al. (2006) fit for three different redshifts for the cosmological parameters adopted in the simulation shown in Fig. 1. Overplotted on the semi-analytic predictions are the halo mass functions obtained from the simulation itself. The departure of the simulation from the predictions at the low mass end are due to finite resolution effects discussed in Sec. 4 below.
![]() |
Figure 9. Cumulative number of galaxy clusters exceeding mass m for three redshifts. Solid lines: z = 0.1, short-dashed lines z = 1, long-dashed lines z = 2. The thick lines are from the simulation shown in Fig. 1, while the thin lines are predictions using the Warren [75] fitting function. Note the rapid redshift evolution of the number of massive clusters. The departure of the simulation from the predictions at the low mass end are due to finite resolution effects. From [77]. |
3.6. Application to galaxy clusters
The aerial density of galaxy clusters can be calculated by multiplying
the redshift-dependent halo mass function using the Warren fitting
formula with the redshift-dependent differential volume element for one
square degree on the sky. Fig. 10a
shows the result varying the amplitude of the matter power spectrum
8 holding all
other cosmological
parameters fixed, while Fig. 10b shows the
result varying the matter density
m holding
all other cosmological parameters fixed. In
general we see a rapid rise in the number of clusters with increasing
redshift due to the increasing volume element. However, the space
density of clusters declines rapidly with redshift (see
Fig. 9), and thus the aerial density peaks at
z
1 and then
declines rapidly toward higher redshift.
![]() |
![]() |
Figure 10. Predicted aerial
density of galaxy clusters for the WMAP3 concordance cosmological
model. Left: varying the amplitude of the matter power spectrum
|
From these curves and the virial scaling relations given above it is easy to predict the expected number of clusters of a given X-ray temperature, X-ray luminosity, or SZ luminosity as a function of redshift [18, 13], bearing in mind that real clusters may not perfectly obey the virial scaling relations. In fact they don't as discussed in my 2004 Varenna lectures [69] and by Borgani in these proceedings.