Gradient evolution identity for comoving curvature ($-A{\mathcal{R}}^{\prime }=\mathcal{H}{c}_a^2{\nabla }^2\varphi$)

For adiabatic scalar perturbations without anisotropic stress, define $A=4\pi G{a}^2(\bar{\rho }+\bar{P})={\mathcal{H}}^2-{\mathcal{H}}^{\prime }$ and $c_a^2={\bar{P}}^{\prime }/{\bar{\rho }}^{\prime }$. In conformal Newtonian gauge, $\mathcal{R}=-\varphi -\mathcal{H}({\varphi }^{\prime }+\mathcal{H}\varphi )/A$. Background conservation gives $A^{\prime }/A=-\mathcal{H}(1+3c_a^2)$. Differentiating the expression for $\mathcal{R}$ and using the scalar-potential equation gives the displayed identity. For regular adiabatic modes with nonzero background enthalpy, evolution is suppressed on scales $k\ll \mathcal{H}$.