The development of gravitational instabilities - RESCEU

A new Euler equation (up to second order)
1
H ˙θ (2) +(2+ Ḣ
H 2 )θ(2) + 3 2 Ω m(1+ɛ(k))δ m (2) = −β(k 1 , k 2 )θ 2 −[S Eul. (k 1 , k 2 )+S Intr. (k 1 , k 2 )] (δ m (1) ) 2
S Eul. (k 1 , k 2 )= (k 2.k)
k 2 1
a 2 m 2 ( ¯φ)
k2
2 S(k 1 )η(k 2 )
S Intr. (k 1 , k 2 )= a2 m 2 ( ¯φ)
k2
2 S(k)˜η(k 2 )+ a2 m 2 ( ¯φ)
k1
2
a 2 m 2 ( ¯φ)
k2
2 S(k 1 )S(k 2 )µ(k)
H 2
η(k) =S(k)
m 2 ( ¯φ)
d(β eff (φ))
k( ¯φ)dφ , ˜η(k) =S(k) H 2 d(A(φ)β eff (φ))
m 2 ( ¯φ) k( ¯φ)dφ
(negligible in→∞ limit
µ(k) = S(k)
3Ω m
H 2
m 4 ( ¯φ)
d 3 V eff
2M 2 Pl dϕ3
(negligible in→0 limit