Lattice Simulations of Inflation
  The scalar field theory of cosmological inflation constitutes nowadays one of
the preferred scenarios for the physics of the early universe. In this paper we
aim at studying the inflationary universe making use of a numerical lattice
simulation. Various lattice codes have been written in the last decades and
have been extensively used for understating the reheating phase of the
universe, but they have never been used to study the inflationary phase itself
far from the end of inflation (i.e. about 50 e-folds before the end of
inflation). In this paper we use a lattice simulation to reproduce the
well-known results of some simple models of single-field inflation,
particularly for the scalar field perturbation. The main model that we consider
is the standard slow-roll inflation with an harmonic potential for the inflaton
field. We explore the technical aspects that need to be accounted for in order
to reproduce with precision the nearly scale invariant power spectrum of
inflaton perturbations. We also consider the case of a step potential, and show
that the simulation is able to correctly reproduce the oscillatory features in
the power spectrum of this model. Even if a lattice simulation is not needed in
these cases, that are well within the regime of validity of linear perturbation
theory, this sets the basis to future work on using lattice simulations to
study more complicated models of inflation.
