 Mészáros equation

In a flat universe containing pressureless matter and smooth radiation in cosmology, put $y=a/a_{\mathrm{eq}}$ and $C=H_0^2{\Omega }_{m,0}^2/{\Omega }_{r,0}$. The Friedmann equation gives ${\mathcal{H}}^2=C(1+y)/y^2$, so $y^{\prime 2}=C(1+y)$ and $y^{\prime \prime }=C/2$. The subhorizon cosmological Poisson equation gives ${\delta }^{\prime \prime }+\mathcal{H}{\delta }^{\prime }=3C\delta /(2y)$. The chain rule produces the displayed equation. Smoothing radiation neglects its rapid acoustic forcing after horizon entry; it is not an exact treatment of all radiation perturbations.