= Solution
In a flat <matter-dominated universe>, $a\propto t^{2/3}$, $H=2/(3t)$ and $4\pi G\bar\rho_c=2/(3t^2)$. Substitution of $\delta_k\propto t^\beta$ gives $\beta(\beta-1)+(4/3)\beta-2/3=0$, whose roots are $2/3$ and $-1$. Hence
$$
\boxed{\delta_k(t)=A(k)(t/t_{\rm eq})^{2/3}+B(k)(t/t_{\rm eq})^{-1}.}
$$
During <radiation domination>, $a\propto t^{1/2}$ while $\bar\rho_c/\bar\rho_{\rm total}\propto a\ll1$. The matter self-gravity term is suppressed relative to $H^2$, leaving approximately $\ddot\delta_k+t^{-1}\dot\delta_k=0$, with solutions constant and $\log t$. Thus there is no growing power-law mode comparable to the later $t^{2/3}$ growth. The elementary estimate in part (ii) neglects this slow <logarithmic growth of matter perturbations during radiation domination>.
Back to article page
