Matching the CDM growing mode at horizon entry (${\delta }_C=\frac{A_R}{k^2}[1+2\log (k\tau )](\tau \ge k^{-1})$)

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.