Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 15E / i / 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 $-1$. HenceDuring radiation domination, $a\propto t^{1/2}$ while ${\bar{\rho }}_c/{\bar{\rho }}_{\mathrm{total}}\propto a\ll 1$. 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.