Lattice formulation of axion inflation. Application to preheating
  We present a lattice formulation of an interaction ${\phi\over\Lambda}
F\tilde F$ between an axion and some $U(1)$ gauge sector with the following
properties: it reproduces the continuum theory up to $\mathcal{O}(dx_\mu^2)$
corrections, it preserves exact gauge invariance and shift symmetry on the
lattice, and it is suitable for self-consistent expansion of the Universe. The
lattice equations of motion can no longer be solved by explicit methods, but we
propose an implicit method to overcome this difficulty, which preserves the
relevant system constraints down to arbitrary (tunable) precision. As a first
application we study, in a comoving grid in $(3+1)$ dimensions, the last efolds
of axion-inflation with quadratic potential and the preheating stage following
afterwards. We fully account for the inhomogeneity and non-linearity of the
system, including the gauge field contribution to the expansion rate of the
Universe and its backreaction into the axion dynamics. We characterize in
detail, as a function of the coupling, the energy transfer from the axion to
the gauge field. Two coupling regimes are identified, sub- and super-critical,
depending on whether the final energy fraction stored in the gauge field is
below or above $\sim 50\%$ of the total energy. The Universe is very
efficiently reheated for super-critical couplings, rapidly entering in a
radiation dominated stage. Our results on preheating confirm previously
published results.
