Comoving-gauge density contrast ($\Delta =\delta -3\mathcal{H}(1+w)v$)

For a perfect fluid with $P=w\rho$, in Newtonian gauge in cosmology the density perturbation on fluid-comoving slices is the displayed combination of the fractional density perturbation and velocity potential. The Einstein energy and momentum constraints combine into ${\nabla }^2\varphi =4\pi G{a}^2\bar{\rho }\Delta$ when scalar anisotropic stress vanishes. This gives a relativistic Poisson constraint valid beyond the subhorizon limit.