Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 3

During matter domination, $a\propto t^{2/3}$, $H=2/(3t)$ and $4\pi G{\bar{\rho }}_m=2/(3t^2)$. Trying ${\delta }_m=t^p$ giveswith roots $p=2/3,-1$. The growing linear cosmological density perturbation is therefore

After all species are nonrelativistic, each background density scales as $a^{-3}$, sois constant. Since $\delta {\rho }_i={\bar{\rho }}_i{\delta }_i$,

Neutrino free streaming gives ${\delta }_{\nu }\simeq 0$ on the stated scales, so ${\delta }_m=(1-f_{\nu }){\delta }_{cb}$. With derivatives with respect to $N=\log a$, the growth equation becomesFor ${\delta }_{cb}\propto a^p$,Henceto first order.

Relative to massless-neutrino growth after $z_{\nu }$,The total contrast has the additional factor $1-f_{\nu }$. Squaring its amplitude to obtain the cosmological density power spectrum givesThe two terms respectively encode the unclustered neutrino fraction and the accumulated slowing of cold-matter growth.