and Cosmology

7. Cosmology II: Inhomogeneities in the Universe
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needs to be introduced. T(k) is called the transfer function;
it can be computed for any cosmological model if
the matter content of the Universe is specified. In particular,
T(k) depends on the nature of dark matter. One
distinguishes between cold dark matter (CDM) and hot
dark matter (HDM). These two kinds of dark matter
differ in the thermal velocities of their constituents at
time t eq , when radiation and matter had equal density.
The particles of CDM were non-relativistic at this time,
whereas those of HDM had velocities of order c. If
dark matter consists of weakly interacting elementary
particles, the difference between CDM and HDM depends
on the mass m of the particles. Assuming that the
“temperature” of the dark matter particles is close to
the temperature of the Universe, then a particle mass m
satisfying the relation
mc 2 ≫ k B T(t eq ) ≃ k B × 2.73 K (1 + z eq )
= k B × 2.73 K × 23 900 Ω m h 2 ∼ 6Ω m h 2 eV
indicates CDM, whereas HDM is characterized by the
opposite inequality, i.e., mc 2 ≪ k B T(t eq ); for instance,
neutrinos belong to HDM. The important distinction between
HDM and CDM follows from the considerations
below.
If density fluctuations become too large on a certain
scale, linear perturbation theory breaks down and (7.25)
is no longer valid. Then the true current power spectrum
P(k, t 0 ) will deviate from P 0 (k). Nevertheless, in this
case it is still useful to examine P 0 (k) – it is then called
the linearly extrapolated power spectrum.
7.4.2 Growth of Density Perturbations
Within the framework of linear Newtonian perturbation
theory in the “cosmic fluid”, δ(x, t) = D + (t)δ 0 (x) applies.
Modifications to this behavior are necessary for
several reasons:
• If dark matter consists of relativistic particles, these
are not gravitationally bound in the potential well of
a density concentration. In this case, they are able to
move freely and to escape from the potential well,
which in the end leads to its dissolution if these particles
dominate the matter overdensity. From this, it
follows immediately that for HDM small-scale density
perturbations cannot form. For CDM this effect
of free-steaming does not occur.
• At redshifts z z eq , radiation dominates the density
of the Universe. Since the expansion law a(t) is then
distinctly different from that in the matter-dominated
phase, the growth rate for density fluctuations will
also change.
• As discussed in Sect. 4.5.2, a horizon exists with
comoving scale r H,com (t). Physical interactions can
take place only on scales smaller than r H,com (t). For
fluctuations of length-scales L ∼ 2π/k r H,com (t),
Newtonian perturbation theory will cease to be valid,
and one needs to apply linear perturbation theory in
the framework of the General Relativity.
CDM and HDM. The first of the above points immediately
implies that a clear difference must exist between
HDM and CDM models as regards structure formation
and evolution. In HDM models, small-scale fluctuations
are washed out by free-streaming of relativistic
particles, i.e., the power spectrum is completely suppressed
for large k, which is expressed by the transfer
function T(k) decreasing exponentially for large k. In
the context of such a theory, very large structures will
form first, and galaxies can form only later by fragmentation
of large structures. However, this formation
scenario is in clear contradiction with observations. For
example, we observe galaxies and QSOs at z ∼ 6sothat
small-scale structure is already present at times when
the Universe had less than 10% of its current age. In
addition, the observed correlation function of galaxies,
both in the local Universe (see Fig. 7.4) and at higher
redshift, is incompatible with cosmological models in
which the dark matter is composed mainly of HDM.
Hot dark matter leads to structure formation that
does not agree with observation. Therefore we
can exclude HDM as the dominant constituent of
dark matter. For this reason, it is now commonly
assumed that the dark matter is “cold”. The achievements
of the CDM scenario in the comparison
between model predictions and observations fully
justify this assumption.
We shall elaborate on the last statement in quite some
detail in Chap. 8.
In linear perturbation theory, fluctuations grow on all
scales, or for all wave numbers, independent of each