Long-wavelength approximation in cosmology ($k/(aH)\ll 1$)

When physical wavelengths greatly exceed the Hubble radius, spatial derivatives are small compared with temporal expansion scales. A cosmological gradient expansion then approximates neighbouring regions as locally homogeneous universes, while retaining finite perturbation amplitudes. This does not alone discard frozen tensor cosmological perturbations.

With vectors and independent tensors neglected, finite scalar cosmological perturbations of the local spatial scale can be encoded by $h_{ij}=a_{\mathrm{bg}}^2{e}^{2\zeta }{\delta }_{ij}$. Scalar lapse and shift potentials complete its 3+1 decomposition of spacetime. The exact shift-square term is $h^{ij}{N}_i{N}_j$, independently of the parametrization of the scalar shift.

The traceless mixed extrinsic curvature of a spatial hypersurface can have leading solution ${\tilde{K}}^i{}_j=C^i{}_j(\mathbf{x})a_{\mathrm{loc}}^{-3}$. Cosmic inflation suppresses this anisotropic expansion. An anisotropic spatial shape can nevertheless freeze to a nonzero constant, so shear damping does not prove the absence of tensor cosmological perturbations.