Exponential-potential scalar-dominated power-law solution ($a\propto t^{2/{\lambda }^2},\phi =(2M_{\mathrm{Pl}}/\lambda )\log (t/t_0)$)

For $V=V_0{e}^{-\lambda \phi /M_{\mathrm{Pl}}}$, $\lambda \ne 0$, the displayed ansatz satisfies the flat Friedmann equation and scalar field equation of motion when $V_0{t}_0^2=2M_{\mathrm{Pl}}^2(6-{\lambda }^2)/{\lambda }^4$. Its equation-of-state parameter is $w=-1+{\lambda }^2/3$ and its deceleration parameter is $q={\lambda }^2/2-1$. It gives exponential-potential power-law inflation only for $0<{\lambda }^2<2$; the same algebraic family includes decelerating expansion outside that range when its potential normalization is real.

For $V_0\ne 0$, real nonzero $\lambda$, and a finite positive logarithmic time scale $t_0$, the exponential-potential scalar-dominated power-law solution requires the displayed strict sign condition. Thus positive $V_0$ permits $0<{\lambda }^2<6$, while negative $V_0$ permits ${\lambda }^2>6$. The boundary ${\lambda }^2=6$ cannot occur with nonzero $V_0$ in this ansatz. Setting $\lambda =0$ gives a constant potential and a separate branch, including constant-field de Sitter spacetime for $V_0>0$.