= Flatness bound on the initial field in exponential inflation
{title2=$\phi_i\le\phi_R-\frac{Am_{\rm pl}}{8\pi}\frac{\log(a_RH_R/a_0H_0)}{1-A^2/(16\pi)}$}
For positive slope parameter $A$, the exponential-potential field increases during <cosmic inflation>. The quantity $aH$ grows by $e^{(1-\epsilon_H)N}$ with constant $\epsilon_H=A^2/(16\pi)$ and $N=8\pi(\phi_R-\phi_i)/(Am_{\rm pl})$. Starting from order-one <curvature> and demanding order-one or smaller present <curvature> gives the displayed upper bound on the initial field. A specified tighter final <curvature> tolerance adds one-half its logarithm to the numerator. Thus sufficient <cosmic inflation> sets an upper, not a lower, initial-field bound for this sign of the potential slope. Instantaneous <reheating> and subsequent entropy conservation determine $a_RH_R/a_0H_0$.
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