Cosmological Perturbation Theory, 26.4.2011 version

2 THE BACKGROUND UNIVERSE 3so that the background metric isThat is,whereandds 2 = ḡ µν dx µ dx ν = a 2 (η) [ −dη 2 + δ ij dx i dx j] = a 2 (η)(−dη 2 + dx 2 + dy 2 + dz 2 ). (2.6)ḡ µν = a 2 (η)η µν ⇒ ḡ µν = a −2 (η)η µν . (2.7)Using the conformal time, the Friedmann equations are (exercise)( aH 2 ′) 2= = 8πGa 3 ¯ρa2 (2.8)H ′ = − 4πG3 (¯ρ + 3¯p)a2 , (2.9)′≡ddη = a d dt= a( )˙ (2.10)H ≡ a′= aH = ȧ (2.11)ais the conformal, or comoving, Hubble parameter. Note that( aH ′ ′) ′ (= = a′′ a′) 2a a − = (aȧ)˙− ȧ 2 = aä = a 2äaa = a2 (Ḣ + H2 ). (2.12)The energy continuity equationbecomes just˙¯ρ = −3H(¯ρ + ¯p) (2.13)¯ρ ′ = −3H(¯ρ + ¯p) ≡ −3H(1 + w)¯ρ . (2.14)For later convenience we define the equation-of-state parameterw ≡ ¯p¯ρ(2.15)and the “speed of sound squared” 4c 2 s ≡ ˙¯p˙¯ρ ≡ ¯p′¯ρ ′ . (2.16)These two quantities always refer to the background values.From the Friedmann equations (2.8,2.9) and the continuity equation (2.14) one easily derivesadditional useful background relations, likeandH ′ = − 1 2 (1 + 3w)H2 , (2.17)w ′1 + w = 3H(w − c2 s), (2.18)¯p ′ = w¯ρ ′ + w ′¯ρ = −3H(1 + w)c 2 s ¯ρ. (2.19)Eq. (2.17) shows that w = − 1 3 corresponds to constant comoving Hubble length H−1 = const.For w < − 1 3 the comoving Hubble length shrinks with time (“inflation”), whereas for w > −1 3it grows with time (“normal” expansion). When w = const., we have c 2 s = w.4 This turns out to be the speed of sound if our ρ and p describe ordinary fluid. Even if they do not, wenevertheless define this quantity, although the name is then misleading.