Uniform-density curvature perturbation ($\zeta$)

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.

For spatial metric $a^2(1-2\Phi ){\delta }_{ij}$, the combination $\zeta =-\Phi +\delta \rho /[3(\bar{\rho }+\bar{P})]$ is invariant under a time displacement: $\Phi$ shifts by $\bar{N}HT$ and $\delta \rho$ by $3\bar{N}H(\bar{\rho }+\bar{P})T$. The opposite overall sign is also valid. The sum $\Phi +\delta \rho /[3(\bar{\rho }+\bar{P})]$ is not invariant. This is a useful convention check before applying superhorizon conservation of uniform-density curvature.

With the convention $\zeta =\Phi -\delta \rho /[3(\bar{\rho }+\bar{P})]$, energy conservation gives ${\zeta }^{\prime }=\mathcal{H}\delta P_{\mathrm{nad}}/(\bar{\rho }+\bar{P})+(\nabla \cdot \mathbf{v})/3$. The non-adiabatic pressure perturbation vanishes for adiabatic cosmological perturbations, and regular velocity gradients are suppressed on superhorizon scales. Curvature is then conserved to leading gradient order, allowing primordial initial conditions to be propagated between cosmological eras.