 Uniform-density curvature perturbation

The uniform-density curvature perturbation is a dimensionless spatial-curvature amplitude evaluated on hypersurfaces where the density perturbation vanishes. Fix the scalar spatial metric conventionHere primes denote conformal time, $\mathcal{H}=a^{\prime }/a$ and ${\bar{\rho }}^{\prime }\ne 0$. Under the passive cosmological gauge transformation $\tilde{\eta }=\eta +T$, $\tilde{\Phi }=\Phi +\mathcal{H}T$ and $\delta \tilde{\rho }=\delta \rho -{\bar{\rho }}^{\prime }T$, so the two changes cancel in $\zeta$. Choosing $T=\delta \rho /{\bar{\rho }}^{\prime }$ reaches uniform-density slicing and gives $\zeta =-{\Phi }_{\delta \rho =0}$.

Using background energy conservation, the same convention isIt agrees with the uniform-density curvature sign test. For a spatially flat background, the intrinsic three-dimensional Ricci perturbation on these slices is $\delta {}^{(3)}R=-4a^{-2}{\nabla }^2\zeta$: $\zeta$ is its scalar amplitude, rather than the Ricci scalar itself. The full-Hessian $E$ term is a spatial-coordinate contribution and cancels from that Ricci scalar.

An overall reversal of the definition of $\zeta$ is also used, including in the explicitly signed superhorizon conservation of uniform-density curvature equation. Signs must follow the declared metric and curvature definition together. Uniform-density slicing also differs from comoving gauge in cosmology; their regular adiabatic large-scale relation needs additional dynamical assumptions.^[ref]