Inflaton effective potential from photons for general ε

Abstract

We accurately approximate the contribution that photons make to the effective potential of a charged inflaton for inflationary geometries with an arbitrary first slow roll parameter $\epsilon$. We find a small, nonlocal contribution and a numerically larger, local part. The local part involves first and second derivatives of $\epsilon$, coming exclusively from the constrained part of the electromagnetic field which carries the long range interaction. This causes the effective potential induced by electromagnetism to respond more strongly to geometrical evolution than for either scalars, which have no derivatives, or spin one half particles, which have only one derivative. For $\epsilon = 0$ our final result agrees with that of Allen on de Sitter background, while the flat space limit agrees with the classic result of Coleman and Weinberg.