Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 41 / 2 / iv / Solution

After the decay products dominate the background, radiation domination gives ${\rho }_r\propto a^{-4}$. Neglecting curvature and vacuum terms, the Friedmann equation implies $\dot{a}/a=C/a^2$, so $a\propto (t-t_B{)}^{1/2}$ andThe displayed $1/(2t)$ form uses a cosmic-time origin with $t_B=0$. After a late transition the extrapolated radiation-era origin need not be the original Big Bang; this time shift does not change the growth law.

For the remaining pressureless matter, neglect its self-gravity to leading order in ${\rho }_m/{\rho }_r$. Its density contrast obeys $\ddot{\delta }+(t-t_B{)}^{-1}\dot{\delta }=0$. Using the conserved expansion-weighted derivative of a passive density perturbation givesThus logarithmic growth of matter perturbations during radiation domination replaces the static exponential growth and the matter-era $t^{2/3}$ growth. The expansion persists while its dominant radiation component supplies little sustained clustering in this pressureless test-component approximation. The residual matter can eventually become important again because ${\rho }_m/{\rho }_r\propto a$; the approximation is for the radiation-dominated interval. The equation is not a pressureless growth equation for the relativistic decay products themselves, whose pressure and, when a fluid description applies, radiation acoustic oscillations must be retained.