= Slow-roll normalization for exponential inflation
{title2=$\phi=\frac{2m_{\rm pl}}A\log[C(t+b)],\quad C^2=\frac{A^4V_0}{96\pi m_{\rm pl}^2}$}
For $V=V_0e^{-A\phi/m_{\rm pl}}$, neglecting acceleration of the field and <kinetic energy> gives $3H\dot\phi=AV/m_{\rm pl}$ and $H^2=8\pi V/(3m_{\rm pl}^2)$. Integration yields the displayed field with $a\propto(t+b)^{16\pi/A^2}$. The exact flat scalar-dominated solution has the same power but $C^2=A^4V_0/[2m_{\rm pl}^2(48\pi-A^2)]$. The two normalizations agree at leading slow-roll order. Inflation requires $A^2<16\pi$; the stronger slow-roll conditions require $A^2/(8\pi)\ll1$.
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