Uniform-density curvature sign test ($\zeta =-\Phi +\delta \rho /[3(\bar{\rho }+\bar{P})]$)

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.