Cosmological Perturbation Theory, 26.4.2011 version

18 ADIABATIC AND ISOCURVATURE PERTURBATIONS IN A SIMPLIFIED UNIVERSE46Using the v N ievolution equations (17.22), this becomes (Exercise)S ′′ = Hk(v m − v r ) + Hkv r + 1 4 k2 δ N r . (18.25)Here the δ N r is converted to the comoving gauge using the total fluid velocity (see Eq. 17.17),so that Eq. (18.24) becomesS ′′δ N r = δ C r − 3H(1 + w r)k −1 v N (18.26)( ) 4= Hk (v m − v r ) + 14 + 3y4 k2 δr C . (18.27)Replacing v m − v r by −k −1 S ′ we get (Exercise) the Kodama-Sasaki equationH −2 S ′′ + 4 ( ) k 2 ( 1 + y4 + 3y H−1 S ′ =H 4 + 3y δC − y )4 + 3y S . (18.28)orH −2 S ′′ + 3c 2 s H−1 S ′ = 1 ( ) k 2 ( )13 H 1 + w δ − (1 − 3c2 s )S ,where δ ≡ δ C .The two equations (18.23) and (18.28) form a pair of ordinary differential equations, fromwhich we can solve the evolution of the perturbations δ C ⃗ k(η) and S ⃗k (η). Since the coefficientfunctions of these equations can more easily be expressed in terms of the scale factor y, it may bemore convenient to use y as the time coordinate instead of η. The time derivatives are convertedwith (Exercise)H −1 f ′ = y dfdyand the equations becomey 2 d2 δdy 2 + 3 2 (1 − 5w + 2c2 s)y dδandH −2 f ′′ = y 2d2 fdy 2 + 1 df2(1 − 3w)ydy( kdy − 3 2 (1 + 8w − 6c2 s − 3w 2 )δ = −Hy 2d2 Sdy 2 + 1 2 (1 − 3w + 6c2 s)y dS ( k=dy H(18.29)) 2 (c 2 s δ −y )1 + y S (18.30)) 2 ( 1 + y4 + 3y δC − y )4 + 3y S .For solving the other perturbation quantities, we collect here the relevant equations:Φ = − 3 ( ) H 2δ (18.31)2 k( )v N 2 k (H=−1 Φ ′ + Φ )3(1 + w) H( ) Hδ N = δ − 3 (1 + w)v NkR= −Φ −2 (Φ ′ + HΦ )3(1 + w)HWhen judging which quantities are negligible at superhorizon scales (k ≪ H), one has to exrcisesome care, and not just look blindly at powers of k/H in equations which contain different perturbationquantities. From Eq. (18.31a) one sees that at superhorizon scales, a small comovingδ can still be important and cause a large gravitational potential perturbation Φ. Eq. (15.38),( ) δH −1 R ′ = −c 2 s1 + w − 3S ,