 Radiation-background Bessel basis for matter perturbations

In a radiation-dominated background with matter self-gravity retained, ${\delta }_{aa}+{\delta }_a/a-3\delta /(2a{a}_{\mathrm{eq}})=0$. A change to $x=\sqrt{6a/a_{\mathrm{eq}}}$ gives the Modified Bessel differential equation of order zero. The Modified Bessel function of the first kind increases as $1+3a/(2a_{\mathrm{eq}})+\cdots$; the Modified Bessel function of the second kind decreases with a leading negative logarithm. The early solution space has the constant/logarithmic behaviors, not the $a$ growth of matter domination. This background approximation must not be extrapolated through equality.