 Linear cosmological perturbation theory

Linear cosmological perturbation theory describes small departures from a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker metric. Introduce a bookkeeping parameter $ϵ$ and expand the metric tensor and stress-energy tensor asSubstituting into the Einstein field equations separates the background equations from linear equations for the perturbations; products of two first-order perturbations are discarded. Smallness of $h_{ab}$ is assessed relative to the background metric tensor, not by its determinant alone.

Spatial homogeneity lets one analyze each Fourier mode separately. The fields and particle species at a given wavevector can remain coupled, but different wavevectors do not exchange amplitude through the linear equations. Products retained in nonlinear cosmological perturbation theory instead generate convolutions between modes. The linear density equations help connect initially small fluctuations to the growth of cosmic structure.^[ref]

A cosmological gauge transformation changes how the perturbed and background spacetimes are identified. For example, under $\tilde{\eta }=\eta +T$, a density perturbation transforms aswhere the prime is a conformal time derivative. A nonzero coordinate-dependent density perturbation can therefore represent a time relabelling of unchanged matter. One fixes a gauge or forms invariant combinations before relating perturbations to observations.

Tensor cosmological perturbations describe gravitational-wave degrees of freedom, while density and velocity perturbations describe other responses. A weak gravitational potential does not guarantee a small density contrast; collapsed structures can require a nonlinear matter treatment even when their metric potentials remain small.