= Cosmological contraction with a free homogeneous scalar
{title2=$a(\tau)=(-\tau)^{1/2},\quad\phi_0=\sqrt{3/2}\log(-\tau)$}
In units $8\pi G=1$, a free homogeneous scalar has $\rho=P=\dot\phi_0^2/2$ and therefore a stiff equation of state. Its equation gives $a^2\phi_0'=\text{constant}$, while the <Friedmann equation> gives $3\mathcal H^2=\phi_0'^2/2$. The displayed contracting branch satisfies both equations for negative <conformal time>. Constant normalizations and an additive scalar shift can be restored.
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