and Cosmology

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7. Cosmology II: Inhomogeneities in the Universe
It is convenient to consider the problem in comoving Accordingly, the gravitational potential Φ is written as
∂δ
∂t + 1 of the linearized form of equation (7.11) for the pressure-free case,
a ∇·[(1 + δ) u] = 0 . (7.8) ∂u/∂t + Hu =−a −1 ∇φ. Finally, the Laplacian of φ is replaced by
the Poisson equation (7.10).
coordinates; hence we define, as in (4.4),
r = a(t) x .
Φ(r, t) = 2π 3 Gρ(t)|r|2 + φ(x, t) ; (7.9)
In a homogeneous cosmos, x is a constant for every
matter particle, and its spatial position r changes only
due to the Hubble expansion. Likewise, the velocity
the first term is the Newtonian potential for a homogeneous
density field, and φ satisfies the Poisson equation
for the density inhomogeneities,
field is written in the form
v(r, t) = ȧ ( r
)
a r + u a , t ∇ 2 φ(x, t) = 4πGa 2 (t)ρ(t)δ(x, t)
, (7.5)
= 3H2 0 Ω m
δ(x, t), (7.10)
where u(x, t) is a function of the comoving coordinate x.
2a(t)
In (7.5), the first term represents the homogeneous Hubble
expansion, whereas the second term describes the
where in the last step we used ρ ∝ a −3 and the definition
deviations from this homogeneous expansion. For this
of the density parameter Ω m . Then, the Euler equation
reason, u is called the peculiar velocity.
(7.3) becomes
∂u
Transforming the Fluid Equations to Comoving Coordinates.
We will now show how the above equations
∂t + u ·∇
a
u + ȧ
a u =− 1
ρ a ∇ P − 1 ∇φ, (7.11)
a
read in comoving coordinates. For this, we first note where (4.13) has been utilized.
that the partial derivative ∂/∂t in (7.2) means a time
derivative at fixed r. If the equations are to be written in Linearization. In the homogeneous case, δ ≡ 0, u ≡ 0,
comoving coordinates, this partial time derivative needs φ ≡ 0, ρ = ρ, and (7.7) then implies ˙ρ + 3Hρ = 0,
to be transformed into one where x is kept fixed. For which also follows immediately from (4.17) in the case
example,
of P = 0. Now we will look for approximate solutions
( ) ( )
∂ ∂ ( r )
of the above set of equations which describe only small
ρ(r, t) = ρ x
∂t r ∂t r a , t deviations from this homogeneous solution. For this
( ) ∂
= ρ x (x, t) − ȧ
reason, in these equations we only consider first-order
∂t x a x ·∇ xρ x (x, t),
terms in the small parameters δ and u, i.e., we disregard
terms that contain uδ or are quadratic in the velocity u.
(7.6) After this linearization, we can eliminate the peculiar
where ∇ x is the gradient with respect to comoving
coordinates, and where we define the function equations 2 and then obtain a second-order differential
velocity u and the gravitational potential φ from the
ρ x (x, t) ≡ ρ(ax, t). Note that ρ x (x, t) and ρ(x, t) both equation for the density contrast δ,
describe the same physical density field, but that ρ
∂ 2 δ
and ρ x are different mathematical functions of their
∂t + 2ȧ ∂δ
arguments. After these transformations, (7.2) becomes
= 4πGρδ .
a ∂t
(7.12)
∂ρ
∂t + 3ȧ a ρ + 1 It is remarkable that neither does this equation contain
∇·(ρu) = 0 , (7.7) derivatives with respect to spatial coordinates, nor do
a
the coefficients in the equation depend on x. Therefore,
where from now on all spatial derivatives are to be
(7.12) has solutions of the form
considered with respect to x. For notational simplicity
we from now on set ρ ≡ ρ x and δ ≡ δ(x, t), and note that
the partial time derivative is to be understood to mean at
δ(x, t) = D(t) ˜δ(x),
fixed x. Writing ρ = ρ(1 + δ) and using ρ ∝ a −3 , (7.7) For this, the linearized form of (7.8), ∂δ/∂t + a −1 ∇·u = 0, is differentiated
with respect to time and combined with the reads in comoving coordinates
divergence