= Monomial inflation with a prescribed endpoint
{title2=$\phi_N^2=\phi_e^2+2nNM_{\rm Pl}^2$}
For $V=\lambda\phi^n$ on $\phi>0$, <slow-roll approximation> gives $\epsilon\simeq n^2M_{\rm Pl}^2/(2\phi^2)$ and $\eta\simeq2nM_{\rm Pl}^2/\phi^2$. The <slow-roll e-fold count> integrates to $N\simeq(\phi_N^2-\phi_e^2)/(2nM_{\rm Pl}^2)$. For $n=2,3,4$, the loose order-one conditions $\epsilon\leq1$ and $|\eta|\leq1$ imply $\phi^2\geq2nM_{\rm Pl}^2$ in this approximation. Genuine slow roll requires a much stronger hierarchy. A prescribed endpoint $\phi_e^2=2nM_{\rm Pl}^2$ gives $\phi_N^2=2n(N+1)M_{\rm Pl}^2$ and
$$
n_s\simeq1-\frac{n+2}{2(N+1)},\qquad r\simeq\frac{4n}{N+1}.
$$
The actual end criterion for accelerated expansion is $\epsilon=1$, whose leading potential estimate instead gives $\phi_e^2\simeq n^2M_{\rm Pl}^2/2$. The prescribed endpoint is a convention, not an exact slow-roll or inflation boundary.
Back to article page
