 Solution

For conserved nonrelativistic dark matter and radiation, ${\rho }_{\mathrm{DM}}\propto a^{-3}$ and ${\rho }_{\gamma }\propto a^{-4}$, so $y={\rho }_{\mathrm{DM}}/{\rho }_{\gamma }=a/a_{\mathrm{eq}}$ obeys $\dot{y}=yH$. Denote differentiation with respect to $y$ by a prime. ThenThe supplied Friedmann equations implySubstitution in the linear growth equation and division by $y^2{H}^2$ give the Mészáros equationFor $D_1=1+3y/2$, $D_1^{\prime }=3/2$ and $D_1^{\prime \prime }=0$, making the two remaining terms cancel exactly. For reduction of order, put $\delta =f{D}_1$ and $P=(2+3y)/[2y(1+y)]=1/y+1/[2(1+y)]$. The equation reduces to $f^{\prime \prime }+(2D_1^{\prime }/D_1+P)f^{\prime }=0$, whenceUsing the supplied antiderivative and reversing an irrelevant overall sign gives the independent solutionIts Wronskian is $D_1{D}_2^{\prime }-D_1^{\prime }{D}_2=-1/[y\sqrt{1+y}]\ne 0$. The asymptotic expansions areThus subhorizon dark matter during radiation domination has constant and logarithmic modes, rather than the superhorizon $a^2$ growth of b. A combination with negative $A_2$ supplies positive logarithmic growth. During matter domination, the growing mode is proportional to $a$, agreeing with b, while the second mode decays as $a^{-3/2}$. These results neglect subhorizon radiation clustering; the Mészáros equation is the approximation stated in the question, not a full photon–matter perturbation system through horizon entry.