 Newtonian-gauge matter density from a constant gravitational potential

In matter domination with the constant growing potential, the full Newtonian-gauge energy constraint gives the stated density contrast. The second term dominates inside the Hubble radius and grows as $a$, because ${\mathcal{H}}^2\propto a^{-1}$. Outside the Hubble radius, the constant $-2\Phi$ term dominates and is gauge dependent. The comoving density combination ${\Delta }_c={\delta }_c+3\mathcal{H}{\theta }_c/k^2$ cancels it for ${\theta }_c=2k^2\Phi /(3\mathcal{H})$, leaving the cosmological Poisson equation relation. Matter-spectrum power laws must specify whether they refer to this comoving density or to subhorizon Newtonian-gauge density.