Nonlinear curvature covector (${\zeta }_i=-{\partial }_i\log a+(H/\Pi ){\partial }_i\varphi$)

With negligible shear, a minimally coupled scalar field serving as a clock, and $\Pi \ne 0$, the Hamiltonian constraint and momentum constraint imply $\dot{H}=-4\pi GN{\Pi }^2$ and ${\partial }_{i}H=-4\pi G\Pi {\partial }_i\varphi$. Differentiating the local Friedmann relation yields $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$ and ${\partial }_i\Pi =-(3H+V_{,\varphi }/\Pi ){\partial }_i\varphi$. 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.