Adiabatic radiation-era cold-dark-matter transfer solution (${\delta }_C=KF(x),x=k\tau /\sqrt{3}$)

In a radiation-dominated background with comoving cold dark matter, neglect its contribution to gravity but retain the radiation perturbation. The regular adiabatic initial conditions select $F(x)={\int }_0^x(1-\cos t)dt/t+\sin x/x-(1-\cos x)/x^2-1/2$. It obeys $x^2{F}^{(4)}+5x{F}^{(3)}+x^2{F}^{\prime \prime }+x{F}^{\prime }=0$, with $F\sim x^2/8$ at zero and $F\sim \ln x+\gamma -1/2$ at infinity. The accompanying radiation contrast is $K[-2\cos x+4\sin x/x-4(1-\cos x)/x^2]/3$. This exact leading-radiation solution explains the transition from common superhorizon growth to acoustic radiation oscillations and logarithmic matter growth.