= Solution
The physical horizon size at conformal time $\tau$ is $a(\tau)c\tau$. Horizon crossing occurs when it equals the physical wavelength:
$$
\frac{2\pi a(\tau_H)}k=a(\tau_H)c\tau_H,
$$
and hence
$$
\boxed{\tau_H=\frac{2\pi}{kc}}.
$$
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
$$
\boxed{
D_{\rm eq}
=\frac{\delta(\mathbf k_{\rm eq},\tau_0)}
{\delta(\mathbf k_{\rm eq},\tau_{\rm eq})}
=\left(\frac{\tau_0}{\tau_{\rm eq}}\right)^2
=\frac{a(\tau_0)}{a(\tau_{\rm eq})}}.
$$
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