Quartic-inflation expansion from Planck density ($N\sim 2\pi /\sqrt{\lambda }$)

With unreduced Planck mass $M_P$ and initial potential-dominated density $\lambda {\varphi }_i^4/4\sim M_P^4$, one has ${\varphi }_i^2\sim 2M_P^2/\sqrt{\lambda }$. The slow-roll e-fold count is $N\simeq \pi ({\varphi }_i^2-{\varphi }_e^2)/M_P^2$. Thus small positive $\lambda$ produces a very large expansion factor $e^N$ in this classical quartic model. The order-one endpoint correction is negligible compared with $2\pi /\sqrt{\lambda }$; initial kinetic energy, spatial curvature and the model's validity at trans-Planckian field values are independent assumptions.