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DEFROST: A New Code for Simulating Preheating after Inflation
Andrei V. Frolov
TL;DR
Preheating requires numerical treatment because instability drives violent nonlinear transfer from a homogeneous inflaton to other fields. The paper develops DEFROST, an optimized scalar-field simulation code with visualization and applies it to two-field models. The simulations reproduce prior results and find persistent inhomogeneous states with nearly universal lognormal total-energy-density distributions rather than obvious thermalization.
Problem
Nonlinear preheating dynamics are difficult to study because full gravitational evolution is complex and expensive, while instability rapidly creates inhomogeneous field configurations.
Method
DEFROST numerically evolves scalar fields in an expanding universe using optimized finite-difference methods, computes perturbative gravitational observables, and supports three-dimensional visualization.
Results
The code reproduces previously published two-field preheating results and finds persistent highly inhomogeneous states whose total-energy-density distribution is nearly stationary and lognormal across tested models.
Takeaways & Limitations
The simulations show no obvious thermalization on the reported timescales, and the authors suggest the persistent state may reflect scalar-field turbulence.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
Preheating transfers inflaton energy into other fields through dynamical instabilities, requiring nonlinear numerical simulations. DEFROST targets this problem and reveals persistent, highly inhomogeneous late-time states with apparently universal lognormal energy-density distributions.
- Inflation leaves energy in a nearly homogeneous, low-entropy inflaton condensate that must decay into other matter-field excitations after inflation.
- Parametric resonance and, in hybrid models, tachyonic instability can drive efficient inflaton decay, while nonlinear evolution requires numerical simulation.
- Preheating studies address phenomena including topological defects, particle production, primordial black holes, magnetic fields, and gravitational waves.
- DEFROST is a new code designed to improve preheating-simulation accuracy and performance while supporting visualization and analysis of complex dynamics.
- After transient instability and bubble breakup, simulations produce a persistent clumpy state whose total-energy-density distribution appears universally lognormal across two-field models.
II. EQUATIONS OF MOTION
The paper simplifies preheating dynamics by evolving coupled scalar fields on a homogeneous expanding background and treating inhomogeneous gravity with linear perturbation theory. This avoids the expense and numerical difficulty of solving the full Einstein system while retaining gravitational observables.
- The baseline model contains N minimally coupled scalar fields interacting through a nonlinear potential, with energy density and pressure derived from the stress-energy tensor.
- Full 3+1-dimensional Einstein solvers are complex, expensive, and potentially numerically unstable, motivating a reduced treatment.
- Gravitational backreaction from field inhomogeneities is approximately 10^-3 in the presented simulations, so scalar evolution is treated in a homogeneous flat Friedmann-Robertson-Walker spacetime.
- The code evolves scalar fields while calculating inhomogeneous gravitational fields with linear perturbation theory and ignoring metric-perturbation backreaction on scalar evolution.
- The Hubble length L ≡ 1/H is evolved instead of H because L ∝ t has vanishing higher derivatives for constant equation of state, reducing second-order truncation error.
- Scalar perturbations are computed through gravitational potentials that are nondynamical Poisson-equation solutions, with Φ and Ψ equal only when anisotropic stress is absent.
III. PDE SOLVER IMPLEMENTATION
DEFROST uses second-order finite differences and a leapfrog solver on a three-dimensional grid, with isotropic spatial discretization and conservative energy calculations. Its implementation combines optimized memory and numerical-analysis choices to support efficient, robust simulation and output.
- The scalar-field equations are solved with a second-order accurate finite-difference scheme based on a leapfrog algorithm.
- Discretization of spatial differential operators: Using all 26 neighbors in a 3 × 3 × 3 cube enables a second-order accurate, fourth-order isotropic Laplacian that reduces directional artifacts.
- Discretization of spatial differential operators: DEFROST defaults to isotropic discretization C because it provides the best accuracy and stability, despite higher computational cost.
- Discretization of spatial differential operators: The gradient-square discretization is constructed conservatively so that discretized energy remains conserved during long simulations.
- The time step must satisfy the Courant stability condition and resolve the fastest oscillating field.
- Memory layout cycles previous, current, and next time-slice indices instead of copying large data blocks, while boundary layers support boundary conditions and MPI transition.
- The analysis pipeline estimates power spectra with FFT-based antialiasing and derives probability densities from a partially sorted cumulative distribution.
IV. INITIAL CONDITIONS
DEFROST initializes simulations at the end of inflation with homogeneous components and Gaussian field fluctuations, using finite-box methods designed to preserve accurate spectra and spatial correlations. The simulation box covers relevant cosmological and instability scales, while the random-field generator addresses finite-grid infrared errors.
- Field fluctuations: Quantum fluctuations are initialized as Gaussian random fields because they trigger preheating instability and become effectively classical after horizon exit or large occupation growth.DEFROST uses an effective-mass approximation for expanding-universe fluctuations.
- Simulation setup: Homogeneous-field initial conditions are supplied externally, while their handoff time is selected case by case near the end of inflation.The choice reflects the model trajectory and the limited spatial dynamic range available to the simulation.
- Simulation setup: Simulations start at the end of inflation, with comoving box size ℓ = 10/m selected to cover the relevant length scales.The box is chosen to include horizon, inflaton-mass, and unstable-band scales.
- Random-field generation: Naively discretizing the 1/(2ω) spectrum causes finite-grid infrared power loss and incorrect real-space two-point correlations.The problem arises because only a few long-wavelength modes fit inside the simulation box.
- Random-field generation: DEFROST instead realizes the Gaussian field by convolving white noise with a spherically symmetric, regularized kernel evaluated numerically and applied on the three-dimensional grid.The kernel is regularized with a Gaussian cutoff below the Nyquist frequency, and the convolution uses a discrete FFT.
- Implementation caveat: LATTICEEASY’s random-number implementation can generate correlated, non-Gaussian numbers, although reported results have not appeared strongly affected so far.The issue matters especially for studies of non-Gaussianity.
V. CHAOTIC INFLATION AND BROAD PARAMETRIC RESONANCE
The selected model contains a massive inflaton and a massless decay field coupled through the inflaton’s oscillations. Sufficiently strong coupling produces broad parametric resonance, whose instability bands are described using the Mathieu equation and Floquet theory.
- Model: The model uses two scalar fields: a massive inflaton φ and a massless decay product ψ interacting through a quartic coupling.The inflaton is initially overdamped, then oscillates with decreasing amplitude as it rolls toward smaller field values.
- Model: Oscillations of the inflaton modulate the decay field’s effective mass, producing a Fourier-mode equation for ψ_k.The coupling transfers the inflaton’s time dependence into the decay-field mode evolution.
- Broad parametric resonance: Strong instability occurs when the coupling is sufficiently large, because periodic mass modulation drives parametric resonance.Neglecting expansion and Hubble drag reduces the mode equation to the Mathieu equation.
- Broad parametric resonance: Floquet theory gives Mathieu solutions of the form e^µηP(η), with positive Re µ indicating exponential instability in selected parameter bands.The stability structure is represented by the dependence of Re µ on the dimensionless parameters A and q.
VI. NUMERICAL RESULTS
DEFROST reproduces preheating expansion results while achieving high numerical accuracy, then reveals instability-driven structure that settles into a long-lived, highly inhomogeneous state with lognormal energy-density statistics.
- Numerical validation: DEFROST reproduces the expansion-history results while satisfying the flat-model constraint to 10^-7 without accumulating error over time.The reported error mainly reflects neglected second-order density corrections from initial field fluctuations.
- Instability and field evolution: Instability begins near t = 100/m, exponentially amplifying decay-field fluctuations until nonlinear interactions limit their growth and backreaction makes the inflaton strongly inhomogeneous.The decay field draws energy from the inflaton zero mode, reducing its coherent oscillation amplitude without eliminating the homogeneous component.
- Density and gravitational structure: Total energy density becomes highly inhomogeneous, with peak densities exceeding ten times the average, then rapidly approaches a nearly stationary non-Gaussian distribution.The gravitational potential remains small despite these overdensities, reaching a maximum well depth of only 2 · 10^-3.
- Spatial evolution: Parametric resonance first produces a foam-like network of overdense bubble walls, which repulsive interactions then break into localized blobs that persist with little change.The resulting state is long-lived and statistically simple, but distinct from thermal equilibrium.
- Structure growth: The comoving density correlation length drops from ≳100/m to about 10^-1/m as structure forms, while gravitational-potential correlations continue growing toward larger scales.The long-time asymptotic behavior of the gravitational-potential correlation length is not reached within the reported simulation.
- Late-time behavior: The simulations do not reach thermalization, instead remaining in a simple, inhomogeneous, long-lived state during the reported evolution.The paper notes that thermalization may require a much longer time and that an intermediate scaling regime may exist.
VII. CONCLUSIONS
DEFROST enables faster and more accurate preheating simulations while revealing a persistent, highly inhomogeneous state with nearly universal lognormal energy-density distributions. The simulations also show unexpectedly growing small-scale gravitational-potential structure, linking late preheating phenomenology to large-scale structure formation despite different scales and physics.
- DEFROST improves preheating simulations through higher accuracy, faster performance, and full support for 3D visualization.The code is small, fast, easy to modify, and designed for visualization and analysis.
- The code reproduces published two-field preheating results and extends analysis to energy-density and scalar-potential evolution.These quantities had not previously been examined closely in the reported models.
- The evolving fields quickly settle into a simple, highly inhomogeneous state whose total energy-density distribution is nearly stationary and lognormal across tested two-field models.The stationarity holds apart from overall dilution caused by expansion.
- The simulations show no obvious thermalization even after 212/m, corresponding to five efolds after inflation for the massive-inflaton case.Field-value distributions and other correlators may continue evolving while the energy-density distribution remains in the reported state.
- Small-scale gravitational-potential structure can grow faster than the comoving box expands, although the mechanism remains unclear.Because gravitational interactions were neglected, scalar-field interactions are the only identified source of this growth in the simulations.
- The late-stage pattern of growing structure and lognormal density resembles large-scale structure formation, but occurs on much smaller scales and through different physics.The authors suggest analytical methods from large-scale structure might be applicable.