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After cosmological horizon entry during radiation domination, neglect averaged radiation forcing and the small matter self-gravity term. The CDM density equation in a matter-radiation universe becomes $(\tau {\delta }_C^{\prime }{)}^{\prime }=0$, giving the displayed logarithmic solution. Entry at ${\tau }_h\sim k^{-1}$ matches a superhorizon amplitude of order $A_R/k^2$, fixing the coefficient scale. Detailed radiation forcing near entry determines its order-one numerical factor. This is the Mészáros effect for the initially adiabatic mode.

During matter domination in a flat FLRW metric, the cosmological continuity equation gives $\rho \propto a^{-3}$, and the Friedmann equation gives $a\propto {\tau }^2$ and $8\pi G\rho a^2=12/{\tau }^2$. The CDM density equation in a matter-radiation universe, with radiation neglected, is therefore ${\delta }^{\prime \prime }+2{\delta }^{\prime }/\tau -6\delta /{\tau }^2=0$. Its Euler-Cauchy equation powers satisfy $(p-2)(p+3)=0$, proving the displayed general solution. The growing density contrast is proportional to the scale factor, while the decaying mode is proportional to $a^{-3/2}$. This all-scale form uses the CDM-comoving synchronous gauge; other density slices require their corresponding gauge transformation.

In deep radiation domination, $a\propto \tau$ and $a^{\prime }/a=1/\tau$. Neglect ${\Omega }_c{\delta }_c$ against the radiation gravity source, so the supplied CDM density equation in a matter-radiation universe becomesOn superhorizon scales omit the $k^2{\delta }_r/3$ pressure-gradient term. Under adiabatic initial conditions, ${\delta }_r=(4/3){\delta }_c$ then givesThe Euler-Cauchy equation characteristic polynomial is $p^2-4$, soThese are the general growing and decaying adiabatic solutions in this approximation. A regular primordial growing mode discards the divergent ${\tau }^{-2}$ solution. Although its conformal-time growth is again quadratic, now ${\delta }_c\propto a^2$, rather than ${\delta }_c\propto a$ in matter domination.

If “general” is taken without imposing adiabaticity, both independent relative-entropy integration constants must also be retained. The gradient-free radiation equation givesSubstitution into the matter equation producesThis is the general solution of the leading supplied coupled system, described by the superhorizon radiation-era entropy integration constants. Setting both $S_0$ and $S_1$ to zero recovers the adiabatic result. Regularity and the full initial Einstein field equations constraints can further restrict which of these leading solutions represents a prescribed physical initial mode.

On scales well inside the horizon, radiation pressure drives oscillations at frequency approximately $k/\sqrt{3}$. Their amplitude is bounded at leading order, and their gravitational forcing of a slowly varying matter contrast can be averaged away. With this approximation, as well as negligible matter self-gravity,hence the logarithmic CDM growth after radiation-era horizon entry isThis is the Mészáros effect. It does not mean that ${\delta }_r$ itself vanishes. For a rapidly oscillating source $R_{\gamma }\cos (\omega \tau +\phi )$, the matter equation has a particular responseIts leading second derivative reproduces $3{\delta }_r/{\tau }^2$; the remaining terms are smaller by powers of $(k\tau {)}^{-1}$. Thus the suppressed cold-matter forcing by rapid radiation oscillations is only order ${\delta }_r/(k\tau {)}^2$ and does not change the secular logarithm. Radiation forcing near horizon entry determines $C_0,C_{\log }$ and cannot be averaged away there. Neither this smoothing approximation nor the neglect of matter gravity remains uniformly valid near equality.

During radiation domination, the cosmological continuity equation gives ${\rho }_R\propto a^{-4}$. The flat Friedmann equation in conformal time is $(a^{\prime }/a{)}^2=8\pi G{a}^2{\rho }_R/3$. Hence $a^{\prime 2}$ is constant, and choosing the Big Bang at $\tau =0$ yieldsFor the CDM density equation in a matter-radiation universe, neglect ${\rho }_C{\delta }_C$ while retaining ${\delta }_C$ itself. The adiabatic initial conditions give ${\delta }_R=4{\delta }_C/3$, soThis Euler-Cauchy equation has powers ${\tau }^p$ with $p(p-1)+p-4=0$, namely $p=2,-2$. Selecting the growing superhorizon adiabatic CDM mode in radiation domination gives

After cosmological horizon entry in radiation domination, the oscillating radiation source averages away. Neglecting both that source and CDM self-gravity in the CDM density equation in a matter-radiation universe givesso ${\delta }_C=C_1(k)+C_2(k)\log \tau$. The superhorizon adiabatic CDM mode in radiation domination has amplitude $A_R(k){\tau }_h^2\sim A_R(k)k^{-2}$ at entry ${\tau }_h\sim k^{-1}$, fixing this scale for the subsequent logarithmic CDM growth after radiation-era horizon entry.

More explicitly, in a sharp-transition approximation with ${\tau }_h=1/k$, continuity of ${\delta }_C$ and ${\delta }_C^{\prime }$ givesThe matching the CDM growing mode at horizon entry therefore yields the late subhorizon scalingHere $\simeq$ suppresses order-one coefficients and the nonlogarithmic term, as in the question. A smooth treatment of the radiation forcing through entry changes those coefficients; the factor two is exact only for the sharp matching model just used. This logarithmic growth, rather than growth proportional to $a$, is the Mészáros effect.

During matter domination, ${\rho }_C\propto a^{-3}$. The flat Friedmann equation in conformal time implies $a^{\prime 2}\propto a$, and henceWith radiation neglected, the CDM density equation in a matter-radiation universe reduces toFor this Euler-Cauchy equation, ${\delta }_C\propto {\tau }^p$ requires $p(p-1)+2p-6=(p-2)(p+3)=0$. Thus the matter-era growing and decaying density modes are ${\tau }^2$ and ${\tau }^{-3}$, respectively, andThere is no explicit $k$ in this pressureless equation, so the result holds on all scales within the stated CDM-comoving synchronous gauge description. The amplitude can still depend on $k$ through its earlier evolution; the result does not equate density contrasts on different time slicings.

During radiation domination, $a\propto \tau$ and $8\pi G{\rho }_R{a}^2=3/{\tau }^2$. The adiabatic initial conditions reduce the CDM density equation in a matter-radiation universe to ${\delta }_C^{\prime \prime }+{\delta }_C^{\prime }/\tau -4{\delta }_C/{\tau }^2=0$. Its Euler-Cauchy equation powers are two and minus two. The growing solution is the displayed one, in the stated CDM-comoving synchronous gauge.