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

TL;DR

This paper derives three-dimensional lattice dispersion relations for scalar field simulations, emphasizing the importance of wavevector decomposition and discretization methods, and demonstrates the impact on simulation accuracy in preheating scenarios.

Contribution

It introduces a detailed calculation of lattice dispersion relations considering wavevector decomposition and discretization effects, improving the accuracy of scalar field simulations.

Findings

Incorrect dispersion treatment causes inaccurate parametric resonance modeling.

Proper lattice dispersion relations are crucial for reliable particle number density calculations.

Using the LATTICEEASY framework, the study shows significant differences in simulation outcomes.

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.