Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 14B (Cosmology)

During matter domination,The perturbation equation becomesSubstitution of the power law ${\delta }_k\propto t^{\alpha }$ giveswhose roots are $\alpha =2/3$ and $\alpha =-1$. Hence the cosmic-time matter density modes give

Normalize $a(t_0)=1$ and put $k_0=2\pi /(c{t}_0)$. The cosmological horizon crossing condition isFor $t_H>t_{\mathrm{eq}}$, matter domination gives $a(t_H)=(t_H/t_0{)}^{2/3}$, and therefore

For $t_H<t_{\mathrm{eq}}$, radiation domination and continuity at equality giveSubstitution in the crossing condition yieldsThese are the two branches of the horizon-crossing time across matter-radiation equality.

Neglect the decaying mode from part a. A mode that enters during matter domination grows in amplitude byso its variance grows by $(t_0/t_H{)}^{4/3}$. For $k<k_{\mathrm{eq}}$, part b gives

For $k>k_{\mathrm{eq}}$, the mode enters during radiation domination and is assumed not to grow significantly until equality. It then grows by $a(t_0)/a_{\mathrm{eq}}$. Since the equality mode satisfiesthe post-equality variance growth isTherefore the broken matter power spectrum from horizon entry is