Relative to massless-neutrino growth after $z_\nu$,
$$
\frac{\delta_{cb}}{\delta_0}
=\left(\frac{a}{a_\nu}\right)^{-3f_\nu/5}
\simeq1-\frac35f_\nu\log\frac{1+z_\nu}{1+z}.
$$
The total contrast has the additional factor $1-f_\nu$. Squaring its amplitude to obtain the <cosmological density power spectrum> gives
$$
\boxed{
P_{f_\nu}(k,z)\simeq
\left[1-2f_\nu-\frac65f_\nu
\log\frac{1+z_\nu}{1+z}\right]P_0(k,z)}.
$$
The two terms respectively encode the unclustered neutrino fraction and the accumulated slowing of cold-matter growth.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-310/3/d/solution