Master Thesis

5.2 InitialConditions
According to [25] the equations 5.36 and 5.37 are simplified corresponding to the stage of the universe far before recombination
and re-parameterized to equations, that depend on x= kτ, instead ofτ. The differential equation can be written
d
in matrix notation
d ln x U= AU, were A is a matrix and U=� Dgc, x−1Vc, Dgγ, x−1Vγ, Dg b, x−1Vb, Dgν, x−1Vν, x−2 �
Πν is the
vector of perturbation variables. Further considerations yield, that the first order approximation U(x)= ��
�λ x
has four non-decaying modes U i which remain constant (λ i= 0).
A further distinction of the modes can be made by using the multi-component entropy perturbationΓ=Γ int+Γ rel.
The internal entropy perturbationsΓ int= �
αΓα of the matter species(α) are obviously assumed to vanish in the com-
mon literature. Assuming that the energy momentum transfer Q (α)µ of the matter species can be neglected the relative
entropy perturbation can be written as ([27], [3]):
Γ rel
with S αβ =
= 1�
2p
α,β
D α
1+ω α
(ρ α+p α)·(ρ β+ p β)
ρ+p
− D β
1+ω β
(c 2
α− c2
β )Sαβ i c i
x 0
i U i
(5.41)
(5.42)
, wereαandβ are matter species indices. From this one concludes, that two matter species are adiabatic, if S αβ= 0 is
satisfied. As a result the density perturbations have the specific fraction 3D ν= 3D γ= 4D c= 4D b if they are adiabatic.
32 5.2 Initial Conditions