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In Synchronous gauge in cosmology with coordinates comoving with cold dark matter, the linearized Einstein field equations and matter conservation give the displayed density equation. Primes denote conformal time derivatives. The radiation term contains its pressure contribution and cannot be dropped on superhorizon scales merely because radiation oscillates after cosmological horizon entry.

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.

Define the matter power spectrum by the Fourier transform conventionStatistical isotropy then givesFor the growing mode found above, write $P_{\delta }(k,\tau )={\tau }^4{P}_A(k)$. At cosmological horizon entry, ${\tau }_H\propto k^{-1}$, so the variance contribution per logarithmic interval is proportional to $k^3{k}^{-4}{P}_A(k)=P_A(k)/k$. Making it independent of $k$ requires $P_A(k)\propto k$. HenceThis is the Harrison-Zeldovich spectrum. Scale invariance applies at horizon entry, not to the unprocessed density variance on a common late-time slice. In the notation ${\delta }_{\mathbf{k}}\propto k^{n/2}{\tau }^2$, it means $n=1$, with random Fourier coefficients supplying the ensemble statistics.

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.