Slow-roll normalization for exponential inflation ($\varphi =\frac{2m_{\mathrm{pl}}}{A}\log [C(t+b)],C^2=\frac{A^4{V}_0}{96\pi m_{\mathrm{pl}}^2}$)

For $V=V_0{e}^{-A\varphi /m_{\mathrm{pl}}}$, neglecting acceleration of the field and kinetic energy gives $3H\dot{\varphi }=AV/m_{\mathrm{pl}}$ and $H^2=8\pi V/(3m_{\mathrm{pl}}^2)$. 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 $C^2=A^4{V}_0/[2m_{\mathrm{pl}}^2(48\pi -A^2)]$. The two normalizations agree at leading slow-roll order. Inflation requires $A^2<16\pi$; the stronger slow-roll conditions require $A^2/(8\pi )\ll 1$.