The development of gravitational instabilities - RESCEU

The closure theory
Valageas P., A&A, 2007
Taruya, Hiramatsu, ApJ 2008, 2009
It makes use of the unequal time power spectra
and of a non-linear propagator.
〈 〉
δΦa (k, η)
δΦ b (k ′ , η ′ )
〈Φ a (k, η)Φ b (k ′ , η ′ )〉 = (2π) 3 δ Dirac (k − k ′ ) R ab (k, η, η ′ )
= G ab (k, η, η ′ )δ Dirac (k − k ′ )
Then evolution equations for those quantities are derived using the Direct-Interaction (DI)
approximation in which one separates the field expression in a DI part and a Non-DI part. At
leading order in Non-DI >> DI, one gets a set of closed equations,
∂
∂η R ab(k, η, η ′ )+Ω ac R cb (k, η, η ′ )=
∫ η
0
∫ η
0
dη ′′ M as (k, η, η ′′ )R bs (k, η ′ , η ′′ )+
dη ′′ N al (k, η, η ′′ )G bl (k, η ′ , η ′′ )
∂
∂η G ab(k, η, η ′ )+Ω ac G cb (k, η, η ′ )=
∫ η
0
dη ′′ M as (k, η, η ′′ )G bs (k, η ′ , η ′′ )
M as (k, η, η ′′ )=
∫
4 d 3 k ′ γ apq (k − k ′ , k ′ )γ lrs (k ′ − k, k)
×G ql (k ′ , η, η ′′ )R pr (|k − k ′ |, η, η ′′ )
N al (k, η, η ′′ )=
∫
2 d 3 k ′ γ apq (k − k ′ , k ′ )γ lrs (k ′ − k, k)
×R qs (k ′ , η, η ′′ )R pr (|k − k ′ |, η, η ′′ )
These equations can more rigorously be derived in a large N
expansion.