Lattice Simulations of Abelian Gauge Fields coupled to Axion during
  Inflation
  We use a non-linear lattice simulation to study a model of axion inflation
where the inflaton is coupled to a U(1) gauge field through Chern-Simons
interaction. This kind of models have already been studied with a lattice
simulation in the context of reheating. In this work, we focus on the deep
inflationary phase and discuss the new aspects that need to be considered in
order to simulate gauge fields in this regime. Our main result is reproducing
with precision the growth of the gauge field on the lattice induced by the
rolling of the axion on its potential, thus recovering the results of linear
perturbation theory for this model. In order to do so, we study in detail how
the spatial discretization, through the choice of the spatial derivatives on
the lattice, influences the dynamics of the gauge field. We find that the
evolution of the gauge field is highly sensitive to the choice of the spatial
discretization scheme. Nevertheless, we are able to identify a discretization
scheme for which the growth of the gauge field on the lattice reproduces the
one of continuous space with good precision.
