= Nonlinear curvature covector
{title2=$\zeta_i=-\partial_i\log a+(H/\Pi)\partial_i\phi$}
= Separate-universe curvature covector
{synonym}
With negligible shear, a <minimally coupled scalar field> serving as a clock, and $\Pi\ne0$, the <Hamiltonian constraint> and <momentum constraint> imply $\dot H=-4\pi GN\Pi^2$ and $\partial_iH=-4\pi G\Pi\partial_i\phi$. Differentiating the local Friedmann relation yields $\dot\Pi=-N(3H\Pi+V_{,\phi})$ and $\partial_i\Pi=-(3H+V_{,\phi}/\Pi)\partial_i\phi$. Inserting these identities into $\dot\zeta_i$ cancels all terms, including gradients of the <lapse function>. The conserved covector becomes the gradient of a <uniform-density curvature perturbation> at linear adiabatic order. Dropping the shear constraint is essential; this is not a proof of conservation in an arbitrary non-attractor phase.
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