Inflaton Effective Potential from Photons for an arbitrary first slow-roll parameter.

We present the contribution that photons make to the effective potential of a charged inflaton for inflationary geometries with an arbitrary first slow roll parameter ϵ. We find a small, nonlocal contribution and a numerically larger, local part. The local part involves the first and second derivatives of ϵ, 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 ϵ=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.