Cosmological perturbation theory

Definition

Let $a(\eta) > 0$ be the scale factor of a spatially flat Friedmann–Lemaître–Robertson–Walker metric in conformal time $\eta$. The background metric is $$\bar g = a(\eta)^2 \left( -d\eta^2 + \delta_{ij}\, dx^i dx^j \right),$$ with indices $i, j \in \set{1, 2, 3}$. Let $\bar T$ be a background stress–energy tensor that solves the Einstein field equations with $\bar g$. Linear cosmological perturbation theory studies metrics $g = \bar g + \delta g$ and stress–energy tensors $T = \bar T + \delta T$ with small $\delta g$ and $\delta T$. It requires the Einstein field equations to hold to first order in $\delta g$ and $\delta T$. The perturbed metric is written as $$ds^2 = a^2 \left[ -(1 + 2A)\, d\eta^2 + 2 B_i\, dx^i d\eta + \left( \delta_{ij} + h_{ij} \right) dx^i dx^j \right],$$ where $A$, $B_i$ and $h_{ij} = h_{ji}$ are functions of $(\eta, x)$ treated to first order. They are split into scalar, vector and tensor parts with respect to rotations of the spatial slices:

1. $B_i = \partial_i B + S_i$ with $\partial_i S_i = 0$.
2. $h_{ij} = -2\psi\, \delta_{ij} + 2\, \partial_i \partial_j E + \partial_i F_j + \partial_j F_i + h^{T}_{ij}$, with summation over repeated indices. The conditions are $\partial_i F_i = 0$, $h^{T}_{ii} = 0$ and $\partial_i h^{T}_{ij} = 0$.

Here $A, B, \psi, E$ are scalar perturbations, $S_i, F_i$ are vector perturbations, and $h^{T}_{ij}$ is the tensor perturbation. A change of coordinates $x^\mu \mapsto x^\mu + \xi^\mu$ with small $\xi$ changes $\delta g$ by the Lie derivative $\mathcal{L}_\xi \bar g$, up to a sign fixed by convention. It changes $\delta T$ by $\mathcal{L}_\xi \bar T$ in the same way. This freedom is called the gauge freedom.