= Gradient evolution identity for comoving curvature
{title2=$-A\mathcal R'=\mathcal Hc_a^2\nabla^2\phi$}
For adiabatic scalar perturbations without anisotropic stress, define $A=4\pi Ga^2(\bar\rho+\bar P)=\mathcal H^2-\mathcal H'$ and $c_a^2=\bar P'/\bar\rho'$. In <conformal Newtonian gauge>, $\mathcal R=-\phi-\mathcal H(\phi'+\mathcal H\phi)/A$. Background conservation gives $A'/A=-\mathcal H(1+3c_a^2)$. 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 $k\ll\mathcal H$.
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