Cosmological Perturbation Theory, 26.4.2011 version

9 PERTURBATION IN THE ENERGY TENSOR 20It is also equal to the fluid velocity observed by a comoving (i.e., one whose x i = const.) observer,since the ratio of change in comoving coordinate dx i to change in conformal time dη equals theratio of the corresponding physical distance adx i to the change in cosmic time dt = adη.To express u µ and u ν in terms of v i , write them asu µ = ū µ + δu µ ≡ ( a −1 + δu 0 ,a −1 v 1 ,a −1 v 2 ,a −1 v 3)u ν = ū ν + δu ν ≡ (−a + δu 0 ,δu 1 ,δu 2 ,δu 3 ) . (9.9)These are related by u ν = g µν u ν and u µ u µ = −1. Usingwe getg µν = a 2 [ −1 − 2A −Bi−B i (1 − 2D)δ ij + 2E ij], (9.10)u 0 = g 0µ u µ= a 2 (−1 − 2A)(a −1 + δu 0 ) − δ ij a 2 B i a −1 v j= −a − a 2 δu 0 − 2aA (9.11)(where we dropped higher than 1 st order quantities, like B i v j ), from which followsδu 0 = −a 2 δu 0 − 2aA. (9.12)Likewiseδu i = u i = g iµ u µ = −aB i + av i . (9.13)We solve the remaining unknown, δu 0 fromu µ u µ = ... = −1 − 2aδu 0 − 2A = −1 ⇒ δu 0 = − 1 a A (9.14)Thus we have for the 4-velocityu µ = 1 a (1 − A,v i) and u µ = a(−1 − A,v i − B i ). (9.15)Inserting this into Eq. (9.5) we getT ν µ = ¯T ν µ + δT νµ [ −¯ρ 0=0 ¯pδ ji] [ −δρ (¯ρ + ¯p)(vi − B+i )−(¯ρ + ¯p)v i δpδji]. (9.16)There are 5 remaining degrees of freedom in the space part, δTj i , corresponding to perturbationsaway from the perfect fluid from. We write them as( )δTj i = δpδi j + Σ ij ≡ ¯p δp¯p + Π ij . (9.17)Here Σ ij and Π ij ≡ Σ ij /¯p are symmetric and traceless, which makes the separation intoandδp ≡ δT k k (9.18)Σ ij ≡ δT i j − 1 3 δi j δT k k (9.19)unique (the trace and the traceless part of δT i j ). Σ ij is called anisotropic stress (or anisotropicpressure). For a perfect fluid Σ ij = 0.