Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 3 / d

The background remains matter dominated, so $H=2/(3t)$ and $4\pi G{\bar{\rho }}_m=2/(3t^2)$. For the reduced gravitational clustering strength, substitute ${\Delta }_m\propto t^s$ into the modified equation:The growing root is the matter-era growth with reduced gravitational clustering strengthIt satisfies $0<s(f)<2/3$ for $0<f<1$. Between $t_f$ and the onset $t_{\Lambda }$ of dark-energy domination, standard growth supplies $(t_{\Lambda }/t_f{)}^{2/3}$ whereas the modified model supplies $(t_{\Lambda }/t_f{)}^{s(f)}$. Subsequent growth freezes in both, so