 Monomial-inflation curvature amplitude

For $V=C_p(\Phi /M_{\mathrm{Pl}}{)}^p$ and the leading endpoint estimate ${ϵ}_V=1$, the slow-roll approximation gives ${\Phi }_N^2/M_{\mathrm{Pl}}^2=2pN+p^2/2$ and ${ϵ}_V=p/(4N+p)$. Thus ${\mathcal{P}}_{\mathcal{R}}=C_p(2pN+p^2/2{)}^{p/2}(4N+p)/(24{\pi }^2{M}_{\mathrm{Pl}}^{4}p)$ and the scalar spectral index obeys $n_s-1=-2(p+2)/(4N+p)$. The coefficient sets the amplitude, whereas the monomial power and exit e-fold count set the leading tilt. Changing the power without specifying the coefficient leaves the overall normalization undetermined.