and Cosmology

7.5 Non-Linear Structure Evolution
of baryons with photons: although matter dominates
the Universe for z < z eq , the density of baryons remains
smaller than that of radiation for a long time,
until after recombination begins. Since photons and
baryons interact with each other by photon scattering
on free electrons, which again are tightly coupled electromagnetically
to protons and helium nuclei, and since
radiation cannot fall into the potential wells of dark
matter, baryons are hindered from doing so as well.
Hence, the baryons are subject to radiation pressure.
For this reason, the density distribution of baryons is
initially much smoother than that of dark matter. Only
after recombination does the interaction of baryons with
photons cease to exist, and the baryons can fall into the
potential wells of dark matter, i.e., some time later the
distribution of baryons will closely resemble that of the
dark matter.
The linear theory of the evolution of density fluctuations
will break down at the latest when |δ|∼1;
the above equations for the power spectrum P(k, t) are
therefore valid only if the respective fluctuations are
small. However, very accurate fitting formulae now exist
for P(k, t) which are also valid in the non-linear
regime. For some cosmological models, the non-linear
power spectrum is displayed in Fig. 7.6.
7.5 Non-Linear Structure Evolution
Linear perturbation theory has a limited range of applicability;
in particular, the evolution of structures like
clusters of galaxies cannot be treated within the framework
of linear perturbation theory. One might imagine
that one can evolve the system of equations (7.2)–(7.4)
to higher orders in the small variables δ and |u|, and so
consider a non-linear perturbation theory. In fact, a quite
extensive literature exists on this topic in which such
calculations have been performed. It is worth mentioning,
though, that while this higher-order perturbation
theory indeed allows us to follow density fluctuations
to slightly larger values of |δ|, the achievements of this
theory do not, in general, justify the large mathematical
effort. In addition, the fluid approximation is no longer
valid if gravitationally bound systems form because, as
mentioned earlier, multiple steams of matter will occur
in this case.
However, for some interesting limiting cases, analytical
descriptions exist which are able to represent
the non-linear evolution of the mass distribution in the
Universe. We shall now investigate a special and very
important case of such a non-linear model. In general,
studying the non-linear structure evolution requires
the use of numerical methods. Therefore, we will also
discuss some aspects of such numerical simulations.
7.5.1 Model of Spherical Collapse
We consider a spherical region in an expanding Universe,
with its density ρ(t) enhanced compared to the
mean cosmic density ρ(t),
ρ(t) = [1 + δ(t)] ρ(t), (7.30)
where we use the density contrast δ as defined in (7.1).
For reasons of simplicity we assume that the density
within the sphere is homogeneous although, as we will
later see, this is not really a restriction. The density
perturbation is assumed to be small for small t, so that
it will grow linearly at first, δ(t) ∝ D + (t), as long as
δ ≪ 1. If we consider a time t i which is sufficiently
early such that δ(t i ) ≪ 1, then δ(t i ) = δ 0 D + (t i ), where
δ 0 is the density contrast linearly extrapolated to the
present day. It should be mentioned once again that
δ 0 ̸= δ(t 0 ), because the latter is determined by the nonlinear
evolution.
Let R com be the initial comoving radius of the overdense
sphere; as long as δ ≪ 1, the comoving radius will
change only marginally. The mass within this sphere is
M = 4π 3 R3 com ρ 0 (1 + δ i ) ≈ 4π 3 R3 com ρ 0 , (7.31)
because the physical radius is R = aR com , and
ρ = ρ 0 /a 3 . This means that a unique relation exists between
the initial comoving radius and the mass of this
sphere, independent of the choice of t i and δ 0 , if only
we choose δ(t i ) = δ 0 D + (t i ) ≪ 1.
Due to the enhanced gravitational force, the sphere
will expand slightly more slowly than the Universe as
a whole, which again will lead to an increase in its density
contrast. This then decelerates the expansion rate
even further, relative to the cosmic expansion rate. Indeed,
the equations of motion for the radius of the sphere
are identical to the Friedmann equations for the cosmic
expansion, only with the sphere having an effective Ω m
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