and Cosmology

7. Cosmology II: Inhomogeneities in the Universe
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different from that of the mean Universe. If the initial
density is sufficiently large, the expansion of the sphere
will come to a halt, i.e., R(t) will reach a maximum;
after this, the sphere will recollapse.
If t max is the time of maximum expansion, then the
sphere will, theoretically, collapse to a single point at
time t coll = 2t max . The relation t coll = 2t max follows from
the time reversal symmetry of the equation of motion:
the time to the maximum expansion is equal to the
time from that point back to complete collapse. 5 The
question of whether the expansion of the sphere will
come to a halt depends on the density contrast δ(t i )
or δ 0 – compare the discussion of the expansion of the
Universe in Sect. 4.3.1 – and on the model for the cosmic
background.
Special Case: The Einstein–de Sitter Model. In the
special case of Ω m = 1andΩ Λ = 0, this behavior can
easily be quantified analytically; we thus treat this case
separately. In this cosmological model, any sphere with
δ 0 > 0 is a “closed universe” and will therefore recollapse
at some time. For the collapse to take place before
t 1 , δ(t i ) or δ 0 needs to exceed a threshold value. For instance,
for a collapse at t coll ≤ t 0 , a linearly extrapolated
overdensity of
δ 0 ≥ δ c = 3 20 (12π)2/3 ≃ 1.69 (7.32)
is required. More generally, one finds that δ 0 ≥ δ c (1+ z)
is needed for the collapse to occur before redshift z.
Violent Relaxation and Virial Equilibrium. Of
course, the sphere will not really collapse to a single
point. This would only be the case if the sphere was perfectly
homogeneous and if the particles in the sphere
moved along perfectly radial orbits. In reality, smallscale
density and gravitational fluctuations will exist
within such a sphere. These then lead to deviations of
the particles’ tracks from perfectly radial orbits, an effect
that is more important the higher the density of
the sphere becomes. The particles will scatter on these
fluctuations in the gravitational field and will virialize;
this process of violent relaxation has already been described
in Sect. 6.2.6 and occurs on short time-scales –
5 For the same reason that it takes a stone thrown up into the air the
same time to reach its peak altitude as to fall back to the ground from
there
roughly the dynamical time-scale, i.e., the time it takes
the particles to fully cross the sphere. In this case, the
virialization is essentially complete at t coll . After that,
the sphere will be in virial equilibrium, and its average
density will be 6
〈ρ〉 = (1 + δ vir ) ρ(t coll ),
where (1 + δ vir ) ≃ 178Ω −0.6
m . (7.33)
This relation forms the basis for the statement that the
virialized region, e.g., of a cluster, is a sphere with an
average density ∼ 200 times the critical density ρ cr of
the Universe at the epoch of collapse. Another conclusion
from this consideration is that a massive galaxy
cluster with a virial radius of 1.5 h −1 Mpc must have
formed from the collapse of a region that originally had
a comoving radius of about six times this size, roughly
10 h −1 Mpc. Such a virialized mass concentration of
dark matter is called a dark matter halo.
Up to now, we have considered the collapse of a homogeneous
sphere. From the above arguments one can
easily convince oneself that the model is still valid if
the sphere has a radial density gradient, e.g., if the density
decreases outwards. In this case, the initial density
contrast will also decrease as a function of radius. The
inner regions of such a sphere will then collapse faster
than the outer ones; a halo of lower mass will form first,
and only later, when the outer regions have also collapsed,
will a halo with higher mass form. From this it
follows that halos of low initial mass will grow in mass
by further accretion of matter.
The spherical collapse model is a simple model for
the non-linear evolution of a density perturbation in the
Universe. Despite being simplistic, it represents the fundamental
principles of gravitational collapse and yields
approximate relations, e.g., for the collapse time and
mean density inside the virialized region, as they are
found from numerical simulations.
6 This result is obtained from conservation of energy and from the
virial theorem. The total energy E tot of the sphere is a constant. At
the time of maximum expansion, it is given solely by the gravitational
binding energy of the system since then the expansion velocity, and
thus the kinetic energy, vanishes. On the other hand, the virial theorem
implies that in virial equilibrium E kin =−E pot /2, and by combining
this with the conservation of energy E tot = E kin + E pot one is then
able to compute E pot in equilibrium and hence the radius and density
of the collapsed sphere.