= Solution
Insert
$$
a=(t/t_0)^\gamma,\qquad \phi=\phi_0+\alpha\log t.
$$
Matching powers of $t$ requires $\lambda\alpha=2$, so $\alpha=2/\lambda$. Writing $\overline V=V_0e^{-\lambda\phi_0}$, the scalar equation and Friedmann equation give
$$
\lambda\overline V=\alpha(3\gamma-1),
\qquad
\boxed{\gamma=\frac{16\pi G}{\lambda^2}}.
$$
Thus
$$
\overline V=\frac{2(3\gamma-1)}{\lambda^2},
\qquad
\phi_0=\lambda^{-1}\log(V_0/\overline V).
$$
Existence with positive potential requires $\gamma>1/3$, or $\lambda^2<48\pi G$. Accelerated expansion requires $\gamma>1$, hence
$$
\boxed{\lambda^2<16\pi G}.
$$
Back to article page
