Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 9E / b / Solution

The physical horizon size at conformal time $\tau$ is $a(\tau )c\tau$. Horizon crossing occurs when it equals the physical wavelength:and hence

At late times the decaying mode is neglected, so the growing solution in part (a) obeys $\delta \propto {\tau }^2\propto a$. For the equality-crossing mode this gives