 Solution

Neglecting the rapidly oscillating radiation source on subhorizon scales, stress-energy conservation and the trace of the Einstein field equations giveEquivalently in cosmic time, ${\delta }_{m,tt}+2H{\delta }_{m,t}-4\pi G{\rho }_m{\delta }_m=0$. Radiation still contributes to the background expansion, even when its perturbation source is averaged away.

Put $\eta =a/a_{\mathrm{eq}}$ and let ${\rho }_{\mathrm{eq}}$ be the density of either component at matter-radiation equality. Then ${\rho }_m={\rho }_{\mathrm{eq}}{\eta }^{-3}$ and ${\rho }_r={\rho }_{\mathrm{eq}}{\eta }^{-4}$. With $C=(8\pi G/3){\rho }_{\mathrm{eq}}{a}_{\mathrm{eq}}^2$, the Friedmann equation givesApply the chain rule to the density equation:Thus the Mészáros equation isSubstituting $D_{+}=1+3\eta /2$ gives $(2+3\eta )(3/2)-3(1+3\eta /2)=0$, sois indeed a solution. It is approximately constant during radiation domination and grows as $a$ during matter domination. The second independent solution iswhich behaves as $\log (4/\eta )-3$ at small $\eta$. Thus smoothing radiation does not eliminate every logarithmic solution; matching through actual horizon entry fixes the combination excited.

Modes with $k\ll k_{\mathrm{eq}}$ enter after equality and retain the $P(k)\propto k$ shape. Modes with $k\gg k_{\mathrm{eq}}$ enter during radiation domination: their horizon-entry spectrum is $P_H\propto k^{-3}$ and they gain little growth until equality. The late matter power spectrum therefore turns over near $k_{\mathrm{eq}}$ and has a large-scale $k$ branch and a small-scale $k^{-3}$ branch, with the more accurate radiation-era matching givingEquivalently the cold-dark-matter transfer function falls approximately as $(k_{\mathrm{eq}}/k{)}^2$, up to the logarithm. The turnover is the consequence of the Mészáros effect; it is not a change of the primordial scale-invariance condition.