Cosmological Perturbation Theory, 26.4.2011 version

5 SEPARATION INTO SCALAR, VECTOR, AND TENSOR PERTURBATIONS 102. and transformations in the space coordinates 7where X i′kx i′ = X i′ k xk , (5.1)is independent of time; which is the case we consider in this section.We had chosen the coordinates for our background, FRW(0), so that the 3-metric was Euclidean,g ij = a 2 δ ij , (5.2)and we want to keep this property. This leaves us rotations. The full transformation matricesare then [ ] [ ][ ]1 0 1 0X µ′ ρ = =and X µ 1 00 X i′ k0 R i′ ρ= ′k0 R i , (5.3)k ′where R i′ k is a rotation matrix8 , with the property R T R = I, or R i′ k Ri′ l = (RT R) kl = δ kl . ThusR T = R −1 so that R i′ k = Rk i ′ .This coordinate transformation in the background induces the corresponding transformation,into the perturbed spacetime. Here the metric isx µ′ = X µ′ρx ρ , (5.4)g µν = a 2 [ −1 − 2A −Bi−B i (1 − 2D)δ ij + 2E ij]= a 2 η µν + a 2 [ −2A −Bi−B i −2Dδ ij + 2E ij]. (5.5)Transforming the metric,we get for the different componentsg ρ ′ σ ′ = Xµ ρ ′ X ν σ ′g µν , (5.6)g 0 ′ 0 ′ = Xµ 0X ν ′ 0 ′g µν = X 0 0 ′X0 0 ′g 00 = g 00 = a 2 (−1 − 2A)g 0 ′ l ′ = Xµ 0X ν ′ l ′g µν = X 0 0 ′Xj lg ′ 0j = −a 2 R j( lB ′ j)g k ′ l ′ = Xi k ′Xj lg ′ ij = a 2 −2Dδ ij R i k ′Rj l+ 2E ′ ij R i k ′Rj( l)′= a 2 −2Dδ kl + 2E ij R i k ′Rj l, (5.7)′from which we identify the perturbations in the new coordinates,A ′D ′= A= DB l ′= R j l ′ B jE k ′ l ′ = Ri k ′Rj l ′ E ij . (5.8)Thus A and D transform as scalars under rotations in the background spacetime coordinates,B i transforms as a 3-vector, and E ij as a 3-d tensor. While staying in a fixed gauge, we can thusthink of them as scalar, vector, and tensor fields on the 3-d Euclidean background space. We7 In this section we use ′ to denote the other coordinate system. Do not confuse with ′ ≡ d/dη in the othersections.8 In this notation R i′ j and R i j ′ are two different matrices, corresponding to opposite rotations; the position ofthe ′ indicates which way we are rotating. We have put the first index upstairs to follow the Einstein summationconvention—but we could have written R i ′ j and R ij ′ just as well.