Solution
= Solution
Since $dt=da/\dot a$ on the collapsing branch, the time for $a$ to decrease from $a_0$ to zero is
$$
\boxed{
\tau
=\sqrt{\frac{3\rho}{2p_0}}
\int_0^{a_0}
\frac{da}{\sqrt{a_0^3/a^3-1}}
}.
$$
Equivalently, scaling $a=a_0x$ gives
$$
\boxed{\tau
=a_0\sqrt{\frac{3\rho}{2p_0}}
\int_0^1\frac{x^{3/2}}{\sqrt{1-x^3}}\,dx.}
$$