= Mészáros equation
{c}
{title2=$\delta_{yy}+\frac{2+3y}{2y(1+y)}\delta_y-\frac3{2y(1+y)}\delta=0$}
In a flat universe containing <pressureless matter> and smooth <radiation in cosmology>, put $y=a/a_{\rm eq}$ 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 $y'^2=C(1+y)$ and $y''=C/2$. The subhorizon <cosmological Poisson equation> gives $\delta''+\mathcal H\delta'=3C\delta/(2y)$. 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.
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