and Cosmology

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7. Cosmology II: Inhomogeneities in the Universe
only few stars and are therefore not visible. One consequence
For t = t 0 , the function f(Ω m ) can be expressed
where we defined the function
f(Ω m ) :=
a(t) dD +
D + (t) da = d log D where we used the previously defined function f(Ω m ).
+
d log a . (7.44) This Poisson equation for ψ can be solved, and by computing
the gradient the peculiar velocity field can be
of this explanation is that the low-mass satellite
galaxies that are seen in our Local Group should be
dominated by dark matter. Given the faintness and low
surface brightness of these galaxies, obtaining kinematical
information for them is very difficult and requires
large telescopes for spectroscopy of individual stars in
by a very simple and very accurate approximation,
f(Ω m ) ≈ Ωm 0.6 . This was first discovered for the case
where Ω Λ = 0, but it was later found that a cosmological
constant has only a marginal effect on this relation.
Introducing corrections arising from Ω Λ , one obtains
the slightly more accurate approximation
these objects. The results of such investigations indicate
that the dwarf galaxies in the Local Group are indeed f ≈ Ωm
0.6 (
Λ
1 + Ω )
m
.
dark matter dominated, with a mass-to-light ratio of
70 2
(7.45)
∼ 100 in Solar units. However, this conclusion is based
on the assumption that the stars in these systems are in
From the smallness of the last term, one can see that the
correction for Λ is marginal indeed, because of which
dynamical equilibrium, an assumption which is difficult one sets f = Ωm
0.6 in most cases.
to test.
On the other hand, g(x) is the gradient of the gravitational
potential, g ∝−∇φ. This implies that u(x)
is a gradient field, i.e., a scalar function ψ(x) exists
7.6 Peculiar Velocities
such that u =∇ψ, where the gradient is taken with
respect to the comoving spatial coordinate x. Therefore,
As mentioned on several occasions before, cosmic
sources do not exactly follow the Hubble expansion, but
have an additional peculiar velocity. Deviations from
∇·g ∝−∇ 2 φ ∝−δ, so that also ∇·u ∝−δ; here,
the Poisson equation (7.10) has been utilized. Taken
together, these results yield for today
the Hubble flow are caused by local gravitational fields,
and such fields are in turn generated by local density ∇·u(x) =−H 0 Ωm 0.6 0(x). (7.46)
fluctuations. These inevitably lead to an acceleration,
which affects the matter and generates peculiar velocities.
In numerical simulations, the peculiar velocities
We would like to derive this result in somewhat more
detail and begin with the linearized form of Eq. (7.8),
of individual particles are followed in the computations
∂δ
automatically. In this brief section, we will investigate
∂t + 1 ∇·u = 0 ,
a
the large-scale peculiar velocities as they are derived
(7.47)
from linear perturbation theory.
Since the spatial dependence of the density contrast δ
is constant in time, δ(x, t) = δ 0 (x) D + (t) (see Eq. 7.14),
where the gradient is, here and in the following, always
taken with respect to comoving coordinates. The fact
that δ(x, t) factorizes (see Eq. 7.14) immediately yields
the acceleration vector g has a constant direction in
the framework of linear perturbation theory. Hence, one ∂δ
obtains the peculiar velocity in the form
∂t = Ḋ+ δ.
D +
∫
u(x) ∼ dt g(x, t),
Combining this equation with (7.47) and, as above,
defining u =∇ψ leads to
i.e., parallel to g(x). Quantitatively, we obtain for today,
thus for t = t 0 , a relation between the velocity and ∇ 2 ψ =∇·u =−a Ḋ+ δ =−a ȧ 1 dD +
acceleration field:
D + D + da δ
2
=−aH(a) f(Ω m )δ≈−aH(a)Ωm 0.6 u(x) = f(Ω m ) g(x), (7.43)
3H 0 Ω m (7.48)