Metric and Stress-Energy Perturbations

  In this section, we discuss the representation of the perturbations for the cosmological fluids and the geometry of space time. We start by defining the basis in which we shall expand such perturbations and their representation under various gauge choices.

We assume that background is described by an FRW metric $g_{\mu\nu} = a^2 \gamma_{\mu\nu}$ with scale factor a(t) and constant comoving curvature $K = - H_0^2(1-\Omega_{\rm tot})$ in the spatial metric $\gamma_{ij}$.Here greek indices run from to 3 while latin indices run over the spatial part of the metric: i,j=1,2,3. It is often convenient to represent the metric in spherical coordinates where

$$\gamma_{ij} dx^i dx^j = \vert K\vert^{-1} \left[ d\chi^2 + \sin_K^2\chi ( d \theta^2 + \sin^2\theta\, d\phi^2 ) \right]\,,$$ (1)

with

$$\sin_K(\chi) = \cases { \sinh(\chi)\,, & $K<0\,,$\space \cr \sin(\chi)\,, & $K\gt\,,$\space \cr}$$ (2)

where the flat-limit expressions are regained as $K \rightarrow 0$ from above or below. The component corresponding to conformal time

$$x^0 \equiv \eta = \int {dt\over a(t)}$$ (3)

is $\gamma_{00}=-1$.

Small perturbations $h_{\mu\nu}$ around this FRW metric

$$g_{\mu\nu} = a^2(\gamma_{\mu\nu} + h_{\mu\nu})\,,$$ (4)

can be decomposed into scalar (m=0, compressional), vector ($m=\pm 1$, vortical) and tensor ($m=\pm 2$, gravitational wave) components from their transformation properties under spatial rotations [6,1].