= Newtonian-gauge matter density from a constant gravitational potential
{c}
{title2=$\delta_c=-2\Phi-2k^2\Phi/(3\mathcal H^2)$}
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.
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