= Radiation-background Bessel basis for matter perturbations
{title2=$\delta_+=I_0(\sqrt{6a/a_{\rm eq}}),\quad\delta_-=K_0(\sqrt{6a/a_{\rm eq}})$}
In a radiation-dominated background with matter self-gravity retained, $\delta_{aa}+\delta_a/a-3\delta/(2aa_{\rm eq})=0$. A change to $x=\sqrt{6a/a_{\rm eq}}$ gives the <Modified Bessel differential equation> of order zero. The <Modified Bessel function of the first kind> increases as $1+3a/(2a_{\rm eq})+\cdots$; 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.
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