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Abstract We consider reheating for a charged inflaton which is minimally coupled toelectromagnetism. The evolution of such an inflaton induces a time-dependentmass for the photon. We show how the massive photon propagator can beexpressed as a spatial Fourier mode sum involving three different sorts ofmode functions, just like the constant mass case. We develop accurateanalytic approximations for these mode functions, and use them to approximatethe effective force exerted on the inflaton 0-mode. This effective forceallows one to simply compute the evolution of the inflaton 0-mode and tofollow the progress of reheating. 

We include the single graviton loop contribution to the linearized Einstein equation. Explicit results are obtained for one loop corrections to the propagation of gravitational radiation. Although suppressed by a minuscule loop-counting parameter, these corrections are enhanced by the square of the number of inflationary e -foldings. One consequence is that perturbation theory breaks down for a very long epoch of primordial inflation. Another consequence is that the one loop correction to the tensor power spectrum might be observable, in the far future, after the full development of 21 cm cosmology. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 2’. 

We employ an unregulated computation of the graviton self-energy from gravitons on the de Sitter background to infer the renormalized result. This is used to quantum-correct the linearized Einstein equation. We solve this equation for the potentials that represent the gravitational response to a static, point mass. We find large spatial and temporal logarithmic corrections to the Newtonian potential and to the gravitational shift. Although suppressed by a minuscule loop-counting parameter, these corrections cause perturbation theory to break down at large distances and late times. Another interesting fact is that gravitons induce up to three large logarithms, whereas a loop of massless, minimally coupled scalars produces only a single large logarithm. This is in line with corrections to the graviton mode function: a loop of gravitons induces two large logarithms, whereas a scalar loop gives none. 

A bstract We consider single graviton loop corrections to the effective field equation of a massless, minimally coupled scalar on de Sitter background in the simplest gauge. We find a large temporal logarithm in the approach to freeze-in at late times, but no correction to the feeze-in amplitude. We also find a large spatial logarithm (at large distances) in the scalar potential generated by a point source, which can be explained using the renormalization group with one of the higher derivative counterterms regarded as a curvature-dependent field strength renormalization. We discuss how these results set the stage for a project to purge gauge dependence by including quantum gravitational corrections to the source which disturbs the effective field and to the observer who measures it. 

A bstract We consider two nonlinear sigma models on de Sitter background which involve the same derivative interactions as quantum gravity but without the gauge issue. The first model contains only a single field, which can be reduced to a free theory by a local field redefinition; the second contains two fields and cannot be so reduced. Loop corrections in both models produce large temporal and spatial logarithms which cause perturbation theory to break down at late times and large distances. Many of these logarithms derive from the “tail” part of the propagator and can be summed using a variant of Starobinsky’s stochastic formalism involving a curvature-dependent effective potential. The remaining logarithms derive from the ultraviolet and can be summed using a variant of the renormalization group based on a special class of curvature-dependent renormalizations. Explicit results are derived at 1-loop and 2-loop orders. 

A bstract We consider quantum gravitational corrections to Maxwell’s equations on flat space background. Although the vacuum polarization is highly gauge dependent, we explicitly show that this gauge dependence is canceled by contributions from the source which disturbs the effective field and the observer who measures it. Our final result is a gauge independent, real and causal effective field equation that can be used in the same way as the classical Maxwell equation.