Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 310 / 3 / a / Solution

For radiation in cosmology, ${\bar{P}}_r={\bar{\rho }}_r/3$ and $\delta P_r=\delta {\rho }_r/3$, so both terms proportional to the equation-of-state difference vanish. During matter domination the gravitational potential is constant, ${\Phi }^{\prime }=0$. Writing ${\theta }_r=\nabla \cdot {\mathbf{v}}_r$, the continuity and Euler equations reduce toDifferentiate the first equation and substitute the second:Thus the radiation perturbation obeys the forced acoustic equation