Cosmological Perturbation Theory, 26.4.2011 version

18 ADIABATIC AND ISOCURVATURE PERTURBATIONS IN A SIMPLIFIED UNIVERSE45When solving for perturbations it turns out to be more convenient to use y (or log y) as timecoordinate instead of η. Inverting Eq. (18.7), we have that(√ ) √ 1 + y − 1η = 1 + y − 1 η 3 = √ η eq , (18.13)2 − 1andH =√ 1 + yy2η 3=√ 1 + yyH eq√2. (18.14)18.2 PerturbationsIn terms of the component perturbations the total perturbations are nowδ =v =and the relative entropy perturbation is11 + y δ r + y1 + y δ m (18.15)44 + 3y v r + 3y4 + 3y v m (18.16)S ≡ S mr = δ m − 3 4 δ r . (18.17)From the pair (18.15,18.17) we can solve δ m and δ r in terms of δ and S:δ m = 3 + 3y4 + 3y δ + 44 + 3y S (18.18)δ r = 4 + 4y4 + 3y δ − 4y4 + 3y S . (18.19)Likewise we can express v m and v r in terms of the total and relative velocity perturbations, vand v m − v r :v m = v + 44 + 3y (v m − v r ) (18.20)v r = v − 3y4 + 3y (v m − v r ). (18.21)We can now also relate the total entropy perturbation S to S:(1S =3(1 + w)δρ¯ρ − 1 )δpyc 2 = ... =s ¯ρ 4 + 3y S = 1 3 (1 − 3c2 s)S . (18.22)The Bardeen equation (15.40) becomes nowH −2 δ ′′ C + ( 1 − 6w + 3c 2 s)H −1 δ ′ C − 3 (2 1 + 8w − 6c2s − 3w 2) δ C( ) k 2[δC − (1 + w)(1 − 3c 2 sH)S]= −c 2 s= −c 2 s( kH) 2 (δ C −y )1 + y S . (18.23)We get the entropy evolution equation by derivation Eq. (17.25),S ′ = −k(v m − v r ) ⇒ S ′′ = −k(v m ′ − v′ r ). (18.24)