Three Dimensional Lattice Dispersion Relations for Finite Difference Methods in Scalar Field Simulations

Abstract

We calculate the lattice dispersion relation for three dimensional simulations of scalar fields. We argue that the mode frequency of scalar fields on the lattice should not be treated as a function of the magnitude of its wavevector but rather of its wavevector decomposition in Fourier space. Furthermore, we calculate how the lattice dispersion relation differs depending on the way that spatial derivatives are discretized when using finite difference methods in configuration space. For applications that require the mode frequency as an average function of the magnitude of the wavevector, we show how to calculate the radially averaged lattice dispersion relation. Finally, we use the publicly available framework LATTICEEASY to show that wrong treatment of dispersion relations in simulations of preheating leads to an inaccurate description of parametric resonance, which results in incorrect calculations of particle number densities during thermalization after inflation.