Gauge-field production during axion inflation in the gradient expansion formalism

We study the explosive production of gauge fields during axion inflation in a novel gradient expansion formalism that describes the time evolution of a set of bilinear electromagnetic functions in position space. Based on this formalism, we are able to simultaneously account for two important effects that have thus far been mostly treated in isolation: (i) the backreaction of the produced gauge fields on the evolution of the inflaton field and (ii) the Schwinger pair production of charged particles in the strong gauge-field background. This allows us to show that the suppression of the gauge-field production due to the Schwinger effect can prevent the backreaction in scenarios in which it would otherwise be relevant. Moreover, we point out that the induced current, $\boldsymbol{J} = \sigma \boldsymbol{E}$, also dampens the Bunch--Davies vacuum fluctuations deep inside the Hubble horizon. We describe this suppression by a new parameter $\Delta$ that is related to the time integral over the conductivity $\sigma$ and which hence renders the description of the entire system inherently nonlocal in time. Finally, we demonstrate how our formalism can be used to construct highly accurate solutions for the mode functions of the gauge field in Fourier space, which serves as a starting point for a wealth of further phenomenological applications, including the phenomenology of primordial perturbations and baryogenesis.