Most cosmological observables arise from perturbations around a homogeneous background. To compare theory with data, one must evolve metric and matter fluctuations consistently and separate physical perturbations from coordinate artifacts.

Scalar perturbations of the FLRW metric can be written using potentials such as $\Phi$ and $\Psi$ in the Newtonian gauge. For standard matter with negligible anisotropic stress, one often has $\Phi=\Psi$.

The comoving curvature perturbation $\mathcal R$ and the curvature perturbation on uniform-density hypersurfaces $\zeta$ are central gauge-invariant variables. On superhorizon scales and for adiabatic perturbations, $\zeta$ is conserved to good approximation.

In the subhorizon, pressureless limit, density perturbations satisfy $$\ddot\delta + 2H\dot\delta - 4\pi G\rho_m\delta = 0.$$ This growth equation is modified by massive neutrinos, clustering dark energy, or modified gravity.