and Cosmology

7.5 Non-Linear Structure Evolution
7.5.2 Number Density of Dark Matter Halos
291
Press–Schechter Model. The model of spherical collapse
allows us to approximately compute the number
density of dark matter halos as a function of their mass
and redshift; this model is called the Press–Schechter
model.
We consider a field of density fluctuations δ 0 (x), featuring
fluctuations on all scales according to the power
spectrum P 0 (k). Assume that we smooth this field with
a comoving smoothing length R, by convolving it with
a filter function of this scale. In our example of the waves
on a lake, we could examine a picture of its surface taken
through a pane of milk-glass, by which all the contours
on small scales would be blurred. Then, let δ R (x) be
the smoothed density field, linearly extrapolated to the
present day. This field does not contain any fluctuations
on scales R, because these have been smoothed out.
Each maximum in δ R (x) corresponds to a peak with
characteristic scale R and, according to (7.31), each
of these maxima corresponds to a mass peak of mass
M ∼ (4πR 3 /3)ρ 0 . If the amplitude δ R of the density
peak is sufficiently large, a sphere of (comoving) radius
R around the peak will decouple from the linear
growth of density fluctuations and will begin to grow
non-linearly. Its expansion will come to a halt, and then
it will recollapse. This process is similar to that in the
spherical collapse model and can be described approximately
by this model. The density contrast required
for the collapse, δ R ≥ δ min , can be computed for any
cosmological model and for any redshift.
If the statistical properties of δ 0 (x) are Gaussian –
which is expected for a variety of reasons – the statistical
properties of the fluctuation field δ 0 are completely
defined by the power spectrum P(k). Then the number
density of density maxima with δ R ≥ δ min can be
computed, and hence the (comoving) number density
n(M, z) of relaxed dark matter halos in the Universe as
a function of mass M and redshift z can be determined.
The Mass Spectrum. The most important results of
the Press–Schechter model are easily explained (see
Fig. 7.7). The number density of halos of mass M
depends of course on the amplitude of the density fluctuation
δ 0 – i.e., on the normalization of the power
spectrum P 0 (k). Hence, the normalization of P 0 (k)
Fig. 7.7. Number density of dark matter halos with mass > M,
computed from the Press–Schechter model. The comoving
number density is shown for three different redshifts, z = 0
(upper curves), z = 0.33, and z = 0.5 (lower curves), for three
different cosmological models: an Einstein–de Sitter model
(solid lines), a low-density open model with Ω m = 0.3 and
Ω Λ = 0 (dotted lines), and a flat universe of low density with
Ω m = 1 − Ω Λ = 0.3 (dashed lines). The normalization of the
density fluctuation field has been chosen such that the number
density of halos with M > 10 14 h −1 M ⊙ at z = 0 in all models
agrees with the local number density of galaxy clusters. Note
the dramatic redshift evolution in the EdS model
can be determined by comparing the prediction of the
Press–Schechter model with the observed number density
of galaxy clusters, as we will discuss further in
Sect. 8.2.1 below. The corresponding result is called the
“cluster-normalized power spectrum”.
Furthermore, we find that n(M, z) is a decreasing
function of halo mass M. This follows immediately
from the previous argument, since a larger M requires
a larger smoothing length R, together with the fact that
the number density of mass peaks of a given amplitude
δ min decreases with increasing smoothing length.
For large M, n(M, z) decreases exponentially because
sufficiently high peaks become very rare for large
smoothing lengths. Therefore, very few clusters with
mass 2 × 10 15 M ⊙ exist. From Fig. 7.7, we can see
that the number density of clusters with M 10 15 M ⊙
today is about 10 −7 Mpc −3 , so the average separation
between two such clusters is larger than 100 Mpc, which