 Solution

For a homogeneous canonical massless scalar field, the flat Friedmann equation and field equation in conformal time arePut $C=\sqrt{3/2}$. On the negative-time branch, $\mathcal{H}=1/(2\tau )$, ${\varphi }_0^{\prime }=C/\tau$ and ${\varphi }_0^{\prime \prime }=-C/{\tau }^2$. ThereforeBoth equations are satisfied. Since $a^{\prime }<0$, this is cosmological contraction with a free homogeneous scalar. Its energy density and pressure are equal, so it is a stiff fluid, and the scale factor tends to zero as $\tau \uparrow 0$.