= 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 $\epsilon$ and expand the <metric tensor> and <stress-energy tensor> as
$$
g_{ab}=\bar g_{ab}+\epsilon h_{ab}+O(\epsilon^2),\qquad
T_{ab}=\bar T_{ab}+\epsilon\delta T_{ab}+O(\epsilon^2).
$$
Substituting 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.<Cosmology by David Tong>[Chapter 3, section 3.1: density perturbations and linear growth]{ref}
A <cosmological gauge transformation> changes how the perturbed and background spacetimes are identified. For example, under $\widetilde\eta=\eta+T$, a density perturbation transforms as
$$
\widetilde{\delta\rho}=\delta\rho-\bar\rho'T,
$$
where 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.
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