 Solution

Work in conformal Newtonian gauge, where $\mathcal{R}=-\varphi +\mathcal{H}v$. SetThe momentum equation gives $v=-({\varphi }^{\prime }+\mathcal{H}\varphi )/A$, henceThis assumes nonzero background enthalpy, as required for the matter comoving slicing. Background energy conservation gives ${\bar{\rho }}^{\prime }=-3\mathcal{H}(\bar{\rho }+\bar{P})$ and thus

The density constraint gives $4\pi G{a}^2\delta \rho ={\nabla }^2\varphi -3\mathcal{H}({\varphi }^{\prime }+\mathcal{H}\varphi )$. Insert this into the pressure equation to obtainDifferentiating the expression for $\mathcal{R}$, using the formula for $A^{\prime }/A$ and then $A={\mathcal{H}}^2-{\mathcal{H}}^{\prime }$, groups the terms asThe preceding equation proves the gradient evolution identity for comoving curvature:For a Fourier mode, ${\nabla }^2\varphi =-k^2\varphi$. On super-Hubble scales, $k/\mathcal{H}\ll 1$, its evolution is gradient-suppressed. With the stated regular adiabatic relation between $\mathcal{R}$ and $\varphi$, $\mathcal{R}$ is constant at leading long-wavelength order. At finite wavelength there are corrections; the conclusion is not an exact claim that the Laplacian vanishes.