Lattice simulations of inflation

Abstract

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.

Lattice Simulations of Abelian Gauge Fields coupled to Axion during Inflation

“…(2.7) but with k eff instead of k, and this is why we refer to it as the effective momentum. Indeed, the lattice solution is equivalent to the continuous one via the identification k eff ↔ k, in a similar way to what is shown in [40] for the single field case. As we will see, this identification turns out to be very useful when comparing the results of the linear theory with the lattice ones.…”

“…( 4.12). The extra normalization factor L 3/2 is introduced to correct for the finite volume of space [32,33,40], while the 1/∆x 3 factor takes into account the dimensionality of the lattice in the computation of the discrete Fourier transform [32,40]. Since the growth of the gauge field occurs only approximately at horizon crossing, a few modes will already be tachyonic at the beginning of the simulation.…”

“…This procedure consists in initiating every classical mode A ± (κ m ) as a Gaussian random number with standard deviation equal to u ± (κ m ). We use the same procedure as [40], to which we refer for further details. After this, the gauge field A( n) in real space is obtained from eq.…”

“…Various lattice codes have been developed in the last decades to study both scalar [30][31][32][33][34][35][36][37] and gauge field [38,39] models during the reheating epoch and the final e-folds of inflation. Recently, lattice techniques have also been generalized to the deep inflationary regime for scalar field models [40,41]. However, gauge fields have never been simulated in this regime.…”

“…In particular, we aim at reproducing with precision the production of gauge field fluctuations induced by the rolling inflaton through the Chern-Simons interaction. Extending the work of [40] for the single scalar field case, we are going to reproduce with a non linear lattice simulation the well-known results of linear perturbation theory for this model. In order to do so, we study in detail how the discretization scheme, through the choice of the spatial derivatives, influences the dynamics of the gauge field on the lattice.…”