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 $\sigma$ 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.
