Cosmological Perturbation Theory, 26.4.2011 version

21 EARLY RADIATION-DOMINATED ERA 57For the above photon+neutrino adiabatic growing mode solution, these becomeηδ ′ c + kηv c = 0 (21.32)ηδ ′ b + kηv b = 0ηv ′ c + v c − kηΦ = 0ηv ′ b + v b − kηΦ = an e σ T4ρ γ3ρ b(v − v b ).21.2.1 The Completely Adiabatic SolutionOne solution for Eq. (21.32) is the completely adiabatic solution:δ c = δ b = 3 4 δ = −3 2Φ = const. (21.33)v c = v b = v = 1 2 kηΦ .To check this, substitute Eq. (21.33) into Eq. (21.32). This givesfor the δ equations, and0 + 1 2 (kη)2 Φ = 012 kηΦ + 1 2kηΦ − kηΦ = 0,so the equations are indeed satisfied. The δ equations are satisfied to accuracy (kη) 2 ≪ 1, i.e.,in a Hubble time, δ i will deviate from its initial value − 3 2 Φ by about −1 2 (kη)2 Φ, a negligiblechange.21.2.2 Baryon and CDM Entropy PerturbationsThere are three independent entropy perturbations: the neutrino, baryon, and CDM entropyperturbations,S ν ≡ 3 4 (δ ν − δ γ ) S b ≡ δ b − 3 4 δ γ S c ≡ δ c − 3 4 δ γ . (21.34)Their evolution equations areS ′ ν = −k(v ν − v γ ) S ′ b = −k(v b − v γ ) S ′ c = −k(v c − v γ ). (21.35)The relative entropy perturbation stays constant unless there is a corresponding relative velocityperturbation.Entropy perturbations also tend to stay constant at superhorizon scales 27 , as( ) kH −1 S i ′ = − (v i − v γ ). (21.36)HAssume now the neutrino-adiabatic growing mode solution of Sec. 21.1. This assumes S ν = 0,but we may still have baryon and CDM entropy perturbations.The baryon and neutrino density perturbations areδ b = 3 4 δ + S b and δ c = 3 4 δ + S c . (21.37)27 This property is not as general as the constancy of R at superhorizon scales: The result (21.36) relies on twoassumptions: 1) no interaction terms in the component energy continuity equations, and 2) the component fluidshave a unique relation p i = p i(ρ i). Note also that this does not hold for the “total entropy perturbation” S.