Cosmological Perturbation Theory, 26.4.2011 version

9 PERTURBATION IN THE ENERGY TENSOR 19Note the background (written first) and perturbation parts in all these quantities. Since thebackground ¯R µ ν and Ḡµ ν are diagonal, the off-diagonals contain just the perturbation, and wehaveR 0 i = G0 i = δR0 i = δG0 i . (8.17)9 Perturbation in the Energy TensorConsider then the energy tensor 15 .The background energy tensor is necessarily of the perfect fluid form 16¯T µν = (¯ρ + ¯p)ū µ ū ν + ¯pḡ µν¯T µ ν = (¯ρ + ¯p)ū µ ū ν + ¯pδ µ ν . (9.1)Because of homogeneity, ¯ρ = ¯ρ(η) and ¯p = ¯p(η). Because of isotropy, the fluid is at rest,ū i = 0 ⇒ ū µ = (ū 0 ,0,0,0) in the background universe. Sinceū µ ū µ = ḡ µν ū µ ū ν = a 2 η µν ū µ ū ν = −a 2 (ū 0 ) 2 = −1, (9.2)we haveū µ = 1 a (1, ⃗0) and ū µ = a(−1,⃗0). (9.3)The energy tensor of the perturbed universe isT µ ν = ¯T µ ν + δT µ ν . (9.4)Just like the metric perturbation, the energy tensor perturbation has 10 degrees of freedom, ofwhich 6 are physical and 4 are gauge. It can likewise be divided into scalar+vector+tensor, with4+4+2 degrees of freedom, of which 2+2+2 are physical. The perturbation can also be dividedinto perfect fluid + non-perfect, with 5+5 degrees of freedom.The perfect fluid degrees of freedom in δT µ ν are those which keep T µ ν in the perfect fluid formT µ ν = (ρ + p)u µ u ν + pδ µ ν . (9.5)Thus they can be taken as the density perturbation, pressure perturbation, and velocity perturbationρ = ¯ρ + δρ, p = ¯p + δp , and u i = ū i + δu i = δu i ≡ 1 a v i . (9.6)The δu 0 is not an independent degree of freedom, because of the constraint u µ u µ = −1. Weshall callv i ≡ au i (9.7)the velocity perturbation. It is equal to the coordinate velocity, since (to first order)dx idη = uiu 0 = uiū 0 = aui = v i . (9.8)15 This section could actually have been earlier. We do not specify a gauge here, and the restriction to scalarperturbations is done only in the end.16 The “imperfections” can only show up in the energy tensor if there is inhomogeneity or anisotropy. Whetheran observer would “feel” the ¯p as pressure is another matter, which depends on the interactions of the fluidparticles. But gravity only cares about the energy tensor.