 Shear-inclusive long-wavelength scalar equations

For zero shift vector, negative-expansion extrinsic curvature $K=-3H$ and $S^i{}_j={\tilde{K}}^i{}_j$, retain the leading spatial-gradient equations of a minimally coupled scalar. They give $\dot{\varphi }=N\Pi$, $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$, $\dot{H}=-4\pi GN{\Pi }^2-N{S}^i{}_j{S}^j{}_i/2$, and ${\dot{S}}^i{}_j=-3NH{S}^i{}_j$. The first-gradient momentum constraint is ${\partial }_{i}H=-4\pi G\Pi {\partial }_i\varphi -{\nabla }_j{S}^j{}_i/2$. Shear decays as $a^{-3}$ during expansion. Its neglect after inflationary damping yields the equations underlying conservation of the nonlinear curvature covector.