Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 4 / c / Solution

When $H\ll m_{\sigma }$ over an oscillation, the background equation reduces toTime averaging gives $⟨{\dot{\bar{\sigma }}}^2⟩=m_{\sigma }^2⟨{\bar{\sigma }}^2⟩$, henceThe oscillating quadratic scalar is therefore pressureless matter. Its continuity equation giveswith the slowly varying amplitude obeying $\Sigma \propto a^{-3/2}$.