Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 3 / b / Solution

For an approximately scale-invariant primordial spectrum, the late cosmological density power spectrum behaves schematically asModes with $k\ll k_{\mathrm{eq}}$ enter the horizon after matter-radiation equality and retain $T(k)\simeq 1$, so $P(k)\sim k^{n_s}$. Modes with $k\gg k_{\mathrm{eq}}$ enter during radiation domination and grow only logarithmically until equality. Their transfer function behaves as $T(k)\sim \log (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$, giving a strongly falling small-scale spectrum. The result is a turnover near the matter-radiation equality scale $k_{\mathrm{eq}}$.