 Existence conditions for exponential-potential scaling

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 $V_0>0$.