Logarithmic CDM growth after radiation-era horizon entry (${\delta }_C=C_1(k)+C_2(k)\log (k\tau ),C_2(k)\sim A_R(k)k^{-2}$)

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.

In an instantaneous-transition approximation, match the superhorizon adiabatic CDM mode in radiation domination and its first derivative to the radiation-era logarithmic solution at ${\tau }_h=1/k$. Its value is $A_R/k^2$ and its derivative is $2A_R/k$. The two matching equations give the displayed expression. Its coefficient two belongs to this approximation; a smooth radiation forcing through entry changes the constants while preserving the logarithmic CDM growth after radiation-era horizon entry scaling.