 Uniform-density curvature perturbation

The uniform-density curvature perturbation is the spatial scalar curvature measured on density-defined hypersurfaces, with an explicitly chosen sign convention. If the spatial scalar potentials are $C,E$, one convention is $\zeta =-C+{\nabla }^{2}E/3+\mathcal{H}\delta \rho /{\bar{\rho }}^{\prime }$. Its two time-shift terms cancel under a cosmological gauge transformation. The density derivative must be nonzero. This is a different slicing from the comoving curvature perturbation, although the variables are related for regular adiabatic long-wavelength modes, with signs depending on convention.