= Quadratic-inflation expansion from Planck density
{title2=$N\simeq4\pi(m_P/m)^2$}
For $V=m^2\phi^2/2$ and unreduced <Planck mass>, the <slow-roll e-fold count> is $N=2\pi(\phi_i^2-\phi_f^2)/m_P^2$. A formal initial potential $V_i\sim m_P^4$ gives $\phi_i^2\sim2m_P^4/m^2$, hence the displayed large-field estimate and expansion factor $e^N$. The final-field contribution is order one in $N$. Without specifying $m/m_P$ there is no unique numerical expansion factor. The extrapolation to Planck density is a classical estimate, rather than a controlled treatment of quantum-gravity corrections.
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