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DEFROST: A New Code for Simulating Preheating after Inflation

Andrei V. Frolov

TL;DR

This work introduces DEFROST, a fast and accurate 3D code for simulating non-linear preheating after inflation in an expanding universe. It evolves multiple scalar fields on a 3D lattice using a leapfrog integrator with an isotropic, high-accuracy discretization and employs FFT-based methods to compute the gravitational potential while tracking the Hubble length to high precision. The main findings show that, after broad parametric resonance, the system does not thermalize but instead forms a long-lived, inhomogeneous state with a universal lognormal one-point distribution of total energy density, and the associated gravitational-potential structures grow over time, reminiscent of turbulence-like statistics. These results, together with DEFROST’s superior speed and visualization capabilities, open avenues for larger-scale, more detailed explorations of early-universe non-equilibrium dynamics.

Abstract

At the end of inflation, dynamical instability can rapidly deposit the energy of homogeneous cold inflaton into excitations of other fields. This process, known as preheating, is rather violent, inhomogeneous and non-linear, and has to be studied numerically. This paper presents a new code for simulating scalar field dynamics in expanding universe written for that purpose. Compared to available alternatives, it significantly improves both the speed and the accuracy of calculations, and is fully instrumented for 3D visualization. We reproduce previously published results on preheating in simple chaotic inflation models, and further investigate non-linear dynamics of the inflaton decay. Surprisingly, we find that the fields do not want to thermalize quite the way one would think. Instead of directly reaching equilibrium, the evolution appears to be stuck in a rather simple but quite inhomogeneous state. In particular, one-point distribution function of total energy density appears to be universal among various two-field preheating models, and is exceedingly well described by a lognormal distribution. It is tempting to attribute this state to scalar field turbulence.

DEFROST: A New Code for Simulating Preheating after Inflation

TL;DR

This work introduces DEFROST, a fast and accurate 3D code for simulating non-linear preheating after inflation in an expanding universe. It evolves multiple scalar fields on a 3D lattice using a leapfrog integrator with an isotropic, high-accuracy discretization and employs FFT-based methods to compute the gravitational potential while tracking the Hubble length to high precision. The main findings show that, after broad parametric resonance, the system does not thermalize but instead forms a long-lived, inhomogeneous state with a universal lognormal one-point distribution of total energy density, and the associated gravitational-potential structures grow over time, reminiscent of turbulence-like statistics. These results, together with DEFROST’s superior speed and visualization capabilities, open avenues for larger-scale, more detailed explorations of early-universe non-equilibrium dynamics.

Abstract

At the end of inflation, dynamical instability can rapidly deposit the energy of homogeneous cold inflaton into excitations of other fields. This process, known as preheating, is rather violent, inhomogeneous and non-linear, and has to be studied numerically. This paper presents a new code for simulating scalar field dynamics in expanding universe written for that purpose. Compared to available alternatives, it significantly improves both the speed and the accuracy of calculations, and is fully instrumented for 3D visualization. We reproduce previously published results on preheating in simple chaotic inflation models, and further investigate non-linear dynamics of the inflaton decay. Surprisingly, we find that the fields do not want to thermalize quite the way one would think. Instead of directly reaching equilibrium, the evolution appears to be stuck in a rather simple but quite inhomogeneous state. In particular, one-point distribution function of total energy density appears to be universal among various two-field preheating models, and is exceedingly well described by a lognormal distribution. It is tempting to attribute this state to scalar field turbulence.

Paper Structure

This paper contains 7 sections, 48 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Equations of Motion
  3. PDE Solver Implementation
  4. Initial Conditions
  5. Chaotic Inflation and Broad Parametric Resonance
  6. Numerical Results
  7. Conclusions

Figures (9)

  • Figure 1: Three-dimensional spatial discretization stencil.
  • Figure 2: Memory layout of the spatial grid and field samples.
  • Figure 3: Expansion history of the universe in chaotic inflation. Numerical simulation starts at the end of inflation ($\epsilon=-1$). The size of comoving simulation box is selected to cover the scales corresponding to horizon size (red), inflaton mass (thin black line), and the wave modes in unstable band (not shown).
  • Figure 4: Stability bands of Mathieu equation.
  • Figure 5: Expansion history (left) and average equation of state (right) during preheating.
  • ...and 4 more figures