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It appears to be common in the discussion of perturbative FRW cosmologies to choose a gauge using hypersurfaces for special values of some quantity, like surfaces of constant density $\rho$, constant inflaton field $\phi$, or zero spatial curvature $\Psi = 0$.

What guarantees that this foliates spacetime?

It seems clear that in general there may be local density spikes that appear and disappear, so the constant density surface isn't space-like. Further, I don't think monotonicity of these quantities (perhaps because we're assuming small perturbations?) is a sufficient condition to guarantee foliation because monotonicity itself doesn't appear to be a gauge-invariant condition. (If the density $\rho$ were to be monotonically decreasing with respect to some choice of time coordinate $t$, there is another set of coordinates $x$ and $t$ for which it is not.)

Alternatively, am I wrong in thinking that this is expected to define a foliation in general? Maybe the only thing that matters is that you can define local gauge-invariant quantities (e.g., spatial curvature on constant-density hypersurfaces $-\zeta = \Psi + (H/\dot{\bar{\rho}})\delta \rho$), and it's not necessary that this defines a preferred coordinate system.

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