LATTICEEASY: A Program for Lattice Simulations of Scalar Fields in an Expanding Universe

Gary Felder; Igor Tkachev

REVIEW

LATTICEEASY: A Program for Lattice Simulations of Scalar Fields in an Expanding Universe

Gary Felder, Igor Tkachev

High Energy Physics - Phenomenology · Submitted 13 Nov 2000 · 11 references

Pith's verdict: A short, honest announcement of a tool that turned out to matter — the physics is standard, the validation is thin, and the paper says up front that the real documentation lives on the website.

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arXiv IDhep-ph/0011159 (version 1)

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CitationGary Felder, Igor Tkachev. "LATTICEEASY: A Program for Lattice Simulations of Scalar Fields in an Expanding Universe." arXiv:hep-ph/0011159 (2000). https://pith.science/paper/SCLNW7GG

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Editor's note

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Short version: this is a four-page software announcement, not a physics paper, and it should be read that way. LATTICEEASY solves the standard Euler-Lagrange equations for interacting scalars on a lattice with optional self-consistent Friedmann expansion, and the authors put the code on the web. That is the contribution, and it is a real one — the tool has been used for parametric resonance, gravitational wave production, defect formation, and thermalization, and the model-file design (a header the user edits to supply V and its derivatives) is the right call for a field where everyone's potential is different. The abstract's claim at [abstract:s0] is exactly what the paper delivers, no more and no less. What the paper does well: the algorithm description is clear and correct. Staggered leapfrog, fields and derivatives at offset times, rescalings chosen to kill the 3H f-dot term so the integrator stays stable — that is the right trick and they explain why. The initial conditions (quantum fluctuations in momentum space, Gaussian random phases, Fourier transformed to the grid) are stated precisely enough to reimplement. Section IV is candid about what is missing: metric perturbations, gauge fields, fermions. Where it is soft: validation. There is no convergence test, no dt-refinement study, no comparison to an analytic or independent numerical result. The paper states the leapfrog stability condition qualitatively (no first-derivative terms) but never gives the actual criterion, dt * omega_max < 2, and in preheating omega_max grows both through the oscillating effective mass and through k_max = pi/(a dx) as the box redshifts. A dt that is stable at t=0 can go unstable later, exactly in the backreaction epoch the code exists for. Related: Eq. (5) gives a UV-divergent variance on a lattice; the paper does not say how the grid cuts it off or why late-time spectra do not care. Both are cheap to check with diagnostics the code already prints (energy conservation, spectra), and neither appears in the paper. I would weight these as real but not disqualifying. The paper explicitly says it is a summary and points to the website for derivations, so some of this may be in the documentation — but the paper is what we are reviewing, and a tool paper that ships no benchmark is asking for trust it has not earned. The self-citation pattern in the references is usage evidence, not derivation, so it does not bother me. Who should read it: anyone doing lattice reheating, and anyone who needs a citable reference for the code. Worth a referee, with a request for a dt-stability criterion and one convergence demonstration.

Referee report

Significance

The contribution is a publicly released, documented, platform-independent solver for a class of problems (preheating, parametric resonance, tachyonic instability, defect formation, thermalization) that the authors correctly argue cannot be treated by linear or Hartree analysis once fluctuations become large. Releasing the code, with the equations and rescalings written down and a model-file interface that lets users add potentials without touching the integrator, is a genuine service to the field and is the kind of artifact that makes published reheating results reproducible. The paper is honest about scope: Section IV names the physics it does not yet cover, and Section I states plainly that quantum effects are neglected and why that is justified once preheating begins. These are strengths and I weigh them in the assessment. What the manuscript does not yet supply is any quantitative evidence that the integrator is accurate in the regime the abstract advertises, namely reheating.

Major objections (3)

[Section III.A and Section II (parameters.h)] The paper identifies the one stability condition it can remove analytically (the 3Hḟ term, eliminated by rescaling) but never states the condition it cannot remove. For fixed-step staggered leapfrog the requirement is dt·ω_max < 2 with ω_max² ≈ k_max² + ∂²V/∂f², and in preheating both terms move in the unsafe direction: the effective mass oscillates with amplitude set by the inflaton, and k_max = π/(a dx) grows as the comoving box is redshifted. A dt that is stable at initialization can therefore cross the stability boundary in the backreaction-dominated epoch that the program exists to simulate. Section II exposes the time step as a free user parameter with no stated criterion, and no dt-refinement study is reported anywhere in the manuscript. Because the abstract's claim that the program is 'particularly useful for the study of reheating' rests on this, I ask the authors to state an explicit time-step criterion (e.g. dt·ω_max ≲ 0.1 at the largest effective mass reached) and to include at least one refinement test showing a late-time spectrum or variance is unchanged when dt is halved.
[Section III.B, Eqs. (5)-(7)] Eq. (5) seeds each mode with <|f_k|²> = 1/(2ω_k), whose integral over the lattice Brillouin zone diverges quadratically in the UV, so the real-space variance of the initial fluctuations depends on grid spacing dx and on N. The manuscript does not say how this is regulated (a momentum cutoff at k_max, a subtracted variance, or nothing at all) nor why late-time spectra should be insensitive to that choice. Since the initial fluctuation amplitude feeds directly into the early growth of the resonance, this is a load-bearing omission for a code paper. A short paragraph stating the regularization used and one comparison at fixed physical box size with two different N (or two dx) would settle it.
[Section II, Figures 1 and 2] The manuscript contains no benchmark against an independent result. The two figures are illustrative only: Fig. 1 shows a spectrum rising and smoothing, Fig. 2 shows a two-dimensional histogram settling at |σ| = v, but neither is compared to linear-theory growth rates, to a Hartree computation, or to any analytic expectation, and neither states the run parameters used. The code already computes overall energy conservation as a built-in output (Section II), which makes a quantitative validation cheap: a plot of fractional energy non-conservation versus time for a representative reheating run, together with a comparison of the early parametric-resonance growth rate to the linearized result, would convert the paper's central claim from an assertion into evidence. I recognize the paper defers derivations and detail to the website, but a journal-version code paper should contain at least a summary of validation in the manuscript itself.

Minor objections (7)

[Eqs. (3)-(4) and the units statement in Section III] As typeset, Eq. (3) reads with a single power of a in the numerator, which is not dimensionally consistent with the stated convention; please verify that it is printed as ä/a = -(4π/3)(ρ + 3p). Relatedly, the units line in Section III is garbled in the preprint and should state unambiguously whether M_P = 1.22×10¹⁹ GeV is the Planck mass or the reduced Planck mass, since the 8π factors in Eqs. (3)-(4) depend on that choice.
[Eqs. (6)-(7)] Eq. (7) defines m² = ∂²V/∂f² without saying at what field value it is evaluated. Please state explicitly that it is evaluated at the homogeneous background value used to initialize the run, since for the two-field model of Eq. (1) the answer differs depending on whether φ or χ is meant.
[Figures 1 and 2] The axis labels in both figures are illegible in the submitted version (e.g. the vertical label of Fig. 1 and the 'ΣR/Slash1v' style labels of Fig. 2). Please supply clean labels and captions that state the model parameters, box size, grid size, and time of each snapshot.
[Section II (model files)] Only twofldm.h is named. A short list of the model files distributed with the code, and a one-line description of each potential, would help readers judge whether their model is already covered.
[Section I] The statement that quantum effects are 'exponentially small once preheating begins' is the paper's justification for the classical treatment. One sentence naming the regime in which this fails (e.g. at low occupation numbers or for light fields with large quantum pressure) would make the domain of validity explicit.
[References [10] and [11]] Both are listed as 'in preparation'. They are used only as evidence of prior use of the code, which is acceptable, but the text should not lean on them for any physics claim until they are available.
[Footnote 1 (Section II)] The claim that the code 'should work with any C++ compiler' is supported only by experience with gcc, and the footnote documents a gcc-specific crash. Please soften the portability claim or state which compilers and platforms have actually been tested.

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The paper

The paper claims

The authors release a lattice program that solves the classical field equations of an expanding universe, so that reheating after inflation can be followed past the point where linear and Hartree approximations break down.

The reading

This paper presents LATTICEEASY, a C++ program that evolves interacting scalar fields on a three-dimensional lattice in an expanding universe. The motivation is reheating: after inflation, fluctuations in the fields grow until linear analysis and the Hartree approximation stop being valid, and the authors argue the only remaining description is to solve the classical equations of motion directly on a grid. The aim is a reusable instrument rather than a new physical result, one in which a new potential is added by writing a single header file and in which the cosmic expansion can be switched off, fixed as a power law, or solved self-consistently from the fields' own energy density. If the program does what is claimed, groups studying preheating, defect formation, or thermalization can run nonlinear simulations without rebuilding the solver each time.

What carries the argument

The load-bearing mechanism is a staggered leapfrog integrator on a fixed time step, made applicable to an expanding universe by rescaled variables. Field values $f$ and their derivatives $dot{f}$ are stored half a step apart; derivatives advance the fields by a full step, then the fields give the second derivatives that advance the derivatives. The method is stable only for equations of the form $ddot{f} = ddot{f}(f)$ with no first-derivative term, so the program rescales field and spacetime variables to remove the $3Hdot{f}$ friction term, with a fixed relationship enforced between the field and time rescalings. Initial conditions are set in momentum space rather than configuration space: each mode gets a random phase and a Gaussian amplitude with dispersion $langle |f_k|^2 rangle = 1/(2omega_k)$ and $omega_k^2 = k^2 + m^2$, $m^2 = partial^2 V/partial f^2$, then Fourier transformed to the grid, with optional homogeneous field and derivative values added on top.

Extended reading notes

Core claim

The claim is a tool claim: that a single, portable C++ code can solve the Euler-Lagrange equations $ddot{f} + 3Hdot{f} + partial V/partial f = 0$ for an arbitrary collection of interacting scalar fields on a lattice, optionally coupled to the Friedmann equations $ddot{a} = -frac{4pi a}{3}(rho + 3p)$ and $(dot{a}/a)^2 = frac{8pi}{3}rho$ so that expansion is computed from the simulated fields themselves. The authors report that every potential-specific piece of physics lives in a model file, a header the program reads in, which must supply the potential and its first and second derivatives; a second file holds run parameters such as grid size, time step, random seed, and which outputs to produce. Built-in outputs cover field means and variances, spectra, energy-density components, energy conservation, and one- and two-dimensional field histograms, with a hook for model-specific output. The evidence offered is that the program has already been used in the authors' published work on parametric resonance, gravitational-wave production, phase transitions and topological defects, thermalization, and disordered chiral condensates, and that it is now publicly available with full documentation.

Load-bearing premise

The fixed-step staggered leapfrog, once the fields and time are rescaled to remove the expansion friction term, stays stable and converged in the strongly nonlinear, high-occupancy regimes where users actually apply it; the paper offers no convergence test, time-step criterion, or resolution study to back this.

Editorial extensions

If this is right

Groups with a new inflationary potential can run nonlinear lattice simulations by writing one header file, rather than writing a solver from scratch.
Because expansion can be solved self-consistently from the simulated fields, reheating can be followed through the transition from inflaton domination to a radiation-dominated universe within a single run.
Built-in spectra, energy-component breakdowns, and energy-conservation output give a direct numerical check on whether a run is resolving the dynamics it claims to resolve.
The same code covers applications beyond reheating, including phase transitions, topological defect formation, and the classical limit of nonequilibrium quantum field theory.
The authors' stated plans — coupling field and metric perturbations, and adding vector fields for gauge models — mark out the boundary of what this version can currently be trusted to compute.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A tool paper of this kind is only as good as its convergence behaviour, and the natural next step for a user community is a published resolution study: same model, several time steps and lattice sizes, comparing late-time spectra and energy conservation.
Because the initial conditions are classical realizations of quantum vacuum fluctuations, the code's domain of validity is set by when occupation numbers become large; a reader could test this directly by running the same model with and without the initial fluctuation spectrum and checking where results diverge.
The reference list doubles as a map of what the method has been applied to and what it has not, so the gaps there are a reasonable guide to which cosmological models remain unexamined by lattice methods.

Original abstract

We describe a C++ program that we have written and made available for calculating the evolution of interacting scalar fields in an expanding universe. The program is particularly useful for the study of reheating and thermalization after inflation. The program and its full documentation are available on the Web at http://physics.stanford.edu/gfelder/latticeeasy/ . In this paper we provide a brief overview of what the program does and what it is useful for.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,522 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.17 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.04 · mentions code or data availability 0.98.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We describe a C++ program that we have written and made available for calculating the evolution of interacting scalar fields in an expanding universe. The program is particularly useful for the study of reheating and thermalization after inflation. The program and its full documentation are available on the Web at http://physics.stanford.edu/gfelder/latticeeasy/ . In this paper we provide a brief overview of what the program does and what it is useful for.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	We describe a C++ program that we have written and made available for calculating the evolution of interacting scalar fields in an expanding universe.	contribution 0.96	0.92	0.19	asserted 1.00
s1	The program is particularly useful for the study of reheating and thermalization after inflation.	contribution 0.71	0.17	0.10	asserted 0.93
s2	The program and its full documentation are available on the Web at http://physics.stanford.edu/gfelder/latticeeasy/ .	contribution 0.85	0.53	0.03	asserted 1.00
s3	In this paper we provide a brief overview of what the program does and what it is useful for.	contribution 0.89	0.71	0.08	asserted 0.93

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.91.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. The power spectrum of the field

**Figure 2.** FIG. 2. Two-dimensional histogram of the fields

Figures reproduced from arXiv:hep-ph/0011159 by the authors.

References

Works cited (11)

[1]

S. Khlebnikov and I. Tkachev, Phys. Rev. Lett. 77, 219 (1996), [hep-ph 9603378]

View on Pith 1996
[2]

S. Khlebnikov and I. Tkachev, Phys. Rev. Lett. 79, 1607 (1997), [hep-ph 9610477]

View on Pith 1997
[3]

S. Khlebnikov and I. Tkachev, Phys. Rev. D 56, 653 (1997), [hep-ph 9701423]

View on Pith 1997
[4]

L. Kofman, A. Linde, and A. A. Starobinsky, Phys. Rev. D 56, 3258 (1997), [hep-ph 9704452]

View on Pith 1997
[5]

E. W. Kolb, A. Riotto, I. Tkachev, Phys. Rev. D 56, 6133 (1997), [astro-ph 9703119]

View on Pith 1997
[6]

E. W. Kolb, A. Riotto, I. Tkachev, Phys. Lett. B423, 348 (1998), [hep-ph 9801306]

View on Pith 1998
[7]

S. Khlebnikov, L. Kofman, A. Linde, I. Tkachev, Phys. Rev. Lett. 81, 2012 (1998), [hep-ph 9804425]

View on Pith 1998
[8]

I. Tkachev, S. Khlebnikov, L. Kofman, A. Linde, Phys. Lett. B440, 262 (1998), [hep-ph 9805209]

View on Pith 1998
[9]

G. Felder, L. Kofman, A. Linde, I. Tkachev, JHEP 0008 010 (2000), [hep-ph 0004024]

View on Pith 2000
[10]

G. Felder, L. Kofman, in preparation

View on Pith
[11]

G. Felder, P. Greene, J. Garcia-Bellido, L. Kofman, A. Linde, and I. Tkachev, in preparation. 4

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Self-resonance after inflation: oscillons, transients and radiation domination
astro-ph.CO 2017-10 conditional novelty 5.0 of 10

For inflaton potentials with a non-quadratic minimum, self-resonance fragments the post-inflationary condensate and drives the equation of state to 1/3 (radiation domination) for any flattening scale M, while a quadra...
Parametric Resonance in the Early Universe - A Fitting Analysis
astro-ph.CO 2016-09 conditional novelty 5.0 of 10

Lattice simulations of parametric resonance across preheating and spectator scenarios yield power-law fits for the backreaction time, the true decay time, and the late-time energy fractions, replacing the misleading l...
Postinflationary vacuum instability and Higgs-inflaton couplings
hep-ph 2016-08 conditional novelty 5.0 of 10

Resonant Higgs production during preheating destabilizes the electroweak vacuum for quartic Higgs-inflaton couplings above about 3e-8 and trilinear couplings above about 1e8 GeV, leaving a narrow viable window.
Gravitational waves from oscillons after inflation
astro-ph.CO 2016-07 reject novelty 5.0 of 10

The originally reported large oscillon-induced gravitational-wave peak in asymmetric preheating is a numerical artifact; corrected simulations show a subdominant peak for the example model.
Gravitational Wave Production At The End Of Inflation
astro-ph 2006-12 conditional novelty 5.0 of 10

Preheating after inflation generates a stochastic gravitational wave background whose amplitude is independent of the inflationary energy scale while its peak frequency scales inversely with it, reaching Omega_gw h^2 ...
Tachyonic Instability and Dynamics of Spontaneous Symmetry Breaking
hep-th 2001-06 accept novelty 5.0 of 10

Spontaneous symmetry breaking from a tachyonic (negative mass squared) potential completes within one oscillation of the field distribution, because the instability converts potential energy into colliding classical w...
PSpectCosmo: A Pseudo-Spectral Code for Cosmological Dynamics Spanning Inflation and Reheating
astro-ph.CO 2024-11 reject novelty 3.0 of 10

PSpectCosmo applies a pseudo-spectral method to inflationary lattice simulations and claims to resolve the divergent vacuum energy density by tracking a quantum-to-classical ratio R.
Inflationary Cosmology after Planck 2013
hep-th 2014-02 conditional novelty 3.0 of 10

A broad class of inflationary models, with non-minimal coupling to gravity or a conformal boundary in field space, all predict the same spectral tilt and tensor fraction, landing on the region Planck 2013 prefers.
EuCAPT White Paper: Opportunities and Challenges for Theoretical Astroparticle Physics in the Next Decade
astro-ph.HE 2021-10 unverdicted novelty 2.0 of 10

A 135-author European roadmap identifies ten open problem areas in theoretical astroparticle physics and argues that progress requires cross-subfield, multi-messenger coordination.
The First Three Seconds: a Review of Possible Expansion Histories of the Early Universe
astro-ph.CO 2020-06 conditional novelty 2.0 of 10

The review argues that radiation domination before Big Bang Nucleosynthesis is an assumption rather than a measurement, and catalogues the causes, consequences, and current constraints on nonstandard expansion histori...
Gravitational waves from inflation
astro-ph.CO 2016-05 accept novelty 2.0 of 10

A survey of inflationary gravitational-wave production mechanisms, their predicted power spectra, and how the tensor-to-scalar ratio and tensor tilt can discriminate among models.
Nonperturbative Dynamics Of Reheating After Inflation: A Review
hep-ph 2014-10 accept novelty 2.0 of 10

The review synthesizes the nonperturbative physics of post-inflation reheating, from parametric preheating through nonlinear fragmentation and thermalization, and surveys models and observables.
Primordial magnetogenesis
astro-ph.CO 2010-07 accept novelty 2.0 of 10

Standard inflation dilutes magnetic seeds to roughly 10^-50 G, so viable primordial fields require modified electromagnetism, non-flat cosmology, or non-standard physics.
f(R) theories
gr-qc 2010-02 accept novelty 2.0 of 10

f(R) gravity can be built to be cosmologically viable and consistent with solar-system tests, and it leaves measurable signatures in the growth of structure and in the inflationary tensor-to-scalar ratio.
Reheating in Inflationary Cosmology: Theory and Applications
hep-th 2010-01 accept novelty 2.0 of 10

Reheating proceeds through a rapid preheating stage of exponential particle production, followed by a longer thermalization phase that sets the initial conditions for the hot early universe.
Inflationary Cosmology
hep-th 2007-05 conditional novelty 2.0 of 10

A 25-year review of inflation arguing that chaotic initial conditions, eternal self-reproduction, and the string landscape form the only currently viable early-universe framework, with CMB data now matching its predictions.
Inflation Dynamics and Reheating
astro-ph 2005-07 accept novelty 2.0 of 10

A comprehensive review of multi-field inflation, reheating and preheating, and the curvaton and modulated-reheating alternatives for generating cosmic density perturbations, organized around a two-by-two transfer matr...
The Gravitational-Wave Physics
gr-qc 2017-03 conditional novelty 1.0 of 10

The review surveys three gravitational-wave source classes (inflation and preheating, first-order phase transitions, compact binaries) and the use of binary mergers as standard sirens for cosmology.
Lectures on Reheating after Inflation
astro-ph.CO 2019-07 unverdicted

Lecture notes providing a generic introduction to reheating after inflation, covering its theoretical, phenomenological, and observational aspects.

How later papers used it

116 later papers cite it

Evidence: substantial. 115 of the citing papers share no author with it, and 70 of those carry a confident label from Pith's citation classifier.

Pith read 28 of the 149 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

23 later papers by other authors are listed below: 23 used its method or tool.

Uses its method or tool 23 of 67 papers with this label

Used the numerical code LATTICEEASY to compute the root mean square of the scalar spectator field in its model and to calculate the modified term in the potential.

Transient Acceleration after Non-minimal M-flation Preheating (2023) · 1.00 · paraphrased: the licence does not allow quoting
Performed lattice simulations with the parallel code ClusterEasy and computed the resulting GW spectrum with the method of another work, extended to account for the expansion of the universe.

Gravitational Waves from the Non-Perturbative Decay of Condensates along Supersymmetric Flat Directions (2009) · 0.96 · paraphrased: the licence does not allow quoting
Used LATTICEEASY to simulate the scalar field evolution of a field in its potential.

What can we learn from the stochastic gravitational wave background produced by oscillons? (2017) · 1.00 · paraphrased: the licence does not allow quoting
Performed a lattice calculation using LATTICEEASY, with fields and derivatives evaluated on an evenly spaced spatial grid in an expanding background, to solve the coupled differential equations including the spatial dependence of the Higgs field.

On the Cosmological Stability of the Higgs Instability (2022) · 1.00 · paraphrased: the licence does not allow quoting
Adapted the publicly available LATTICEEASY code, incorporating the evolution of an equation together with the scalar field equations into the staggered leapfrog integrator routine.

A Gravitational Wave Background from Reheating after Hybrid Inflation (2007) · 1.00 · paraphrased: the licence does not allow quoting
Studied the non-linear dynamics of the scalar field of the KKLT model using a modified version of Lattice-Easy to solve its equation on lattices.

Revisiting oscillon formation in KKLT scenario (2020) · 1.00 · paraphrased: the licence does not allow quoting
Used a modified version of LATTICEEASY to simulate multi-stream inflation.

Lattice Simulation of Multi-Stream Inflation (2021) · 1.00 · paraphrased: the licence does not allow quoting
Performed lattice simulations of its system using the publicly available numerical package LATTICEEASY to examine the behavior of inflaton and moduli fields during preheating and estimate moduli abundance from parametric resonance.

Non-thermal moduli production during preheating in $\alpha$-attractor inflation models (2023) · 1.00 · paraphrased: the licence does not allow quoting
Used LatticeEasy to simulate the evolution of a spectator field and its daughter field in a matter-dominated background.

A Cosmic Microscope for the Preheating Era (2020) · 1.00 · paraphrased: the licence does not allow quoting
Computed the field evolution using LATTICEEASY, which uses a staggered leapfrog integrator.

Gravitational Waves From the End of Inflation: Computational Strategies (2007) · 1.00 · paraphrased: the licence does not allow quoting
Implemented a Klein-Gordon finite-difference solver similar to LatticeEasy in the AMReX framework.

Post-inflationary structure formation boosted by parametric self-resonance (2023) · 1.00 · paraphrased: the licence does not allow quoting
Adapted the publicly available LATTICEEASY code, incorporating the evolution of an equation together with the scalar field equations into the staggered leapfrog integrator routine.

A new gravitational wave background from the Big Bang (2008) · 1.00 · paraphrased: the licence does not allow quoting
Set the spectra of quantum fluctuations in the lattice in a similar way as in Latticeeasy, imposing spectra for the Higgs field amplitude and derivatives in momentum space.

Higgs-curvature coupling and post-inflationary vacuum instability (2017) · 1.00 · paraphrased: the licence does not allow quoting
Solved the equations of motion and the Friedmann equation numerically using a parallelized version of LatticeEasy to study the non-linear dynamics.

The Equation of State and Duration to Radiation Domination After Inflation (2016) · 1.00 · paraphrased: the licence does not allow quoting
Used the publicly available code LATTICEEASY to study dark matter production during preheating.

Non-thermal production of Dark Matter after Inflation (2018) · 1.00 · paraphrased: the licence does not allow quoting
To solve Eqs. [ref], [ref], [ref] and [ref], we use the public LatticeEasy package [ref] , but with the integrator modified to the 4th order Runge-Kutta method.

Spillway Preheating (2021) · Model and approach · 1.00
Integrated its equations of motion numerically using the LatticeEasy code, which uses a staggered leapfrog integrator.

On the Suppression of Parametric Resonance and the Viability of Tachyonic Preheating after Multi-Field Inflation (2009) · 1.00 · paraphrased: the licence does not allow quoting
Employed the same second order algorithm used in this paper.

Particle Production near an AdS Crunch (2010) · 1.00 · paraphrased: the licence does not allow quoting
Confirmed its results by using LATTICEEASY.

Fate of Electroweak Vacuum during Preheating (2016) · 1.00 · paraphrased: the licence does not allow quoting
Confirmed its expected late-time yields with a lattice computation using LATTICEEASY, approximating the vector interaction by that of a scalar field with a quartic coupling.

Parametric Resonance Production of Ultralight Vector Dark Matter (2018) · 1.00 · paraphrased: the licence does not allow quoting
To solve this problem we have exploited a method adopted in Ref. [ref] to correct the staggered leapfrog for this particular equation.

The Effect of Non-minimally Coupled Scalar Field on Gravitational Waves from First-order Vacuum Phase Transitions (2022) · NUMERICAL METHODS AND THE POWER SPECTRA · 1.00
Used this paper to find numerical upper bounds on the portal couplings as a function of m_ϕ/M_p, using the condition that the instability timescale is longer than the parametric resonance regime lasts.

Stability, reheating and leptogenesis (2019) · 0.99 · paraphrased: the licence does not allow quoting
Performed its numerical simulations using a new lattice code containing modified elements of both LatticeEasy and DEFROST.

Preheating After Modular Inflation (2009) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

95 arXiv papers cite this in 115 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 33 sentences without a confident label

Uses its method or tool 58 papers

Due to quantum fluctuations of the scalar fields during preheating the effective potential acquires the following correction [ref] [equation] In our work, to calculate the dispersion of the spectator field, $⟨Ψ̃_i^2⟩$, we perform a lattice simulation by using the computational code LATTICEEASY [ref] .

Transient Acceleration after Non-minimal M-flation Preheating · Preheating in non-$𝕄$-flation · 0.92
We perform lattice simulations with the parallel code ClusterEasy [ref] and we compute the resulting GW spectrum with the method of [ref] , extended here to take into account the expansion of the universe.

Gravitational Waves from the Non-Perturbative Decay of Condensates along Supersymmetric Flat Directions · Introduction · 0.96
To properly solve the system of coupled differential equations, including the spatial dependence of the Higgs field, we have performed a lattice calculation making use of the program LATTICEEASY, where the fields and its derivatives are evaluated on a spatial grid with evenly spaced points, in an expanding background [ref] .

On the Cosmological Stability of the Higgs Instability · The dynamics in the interior of the Higgs bubbles · 1.00

Uses its data or values 2 papers

For example, for the $ϕ^4$ and $ϕ^2$ chaotic inflationary models, lattice simulations [ref] show that preheating can lead to GWs with frequencies of around $10^6 ∼10^8$ Hz and peak power of $Ω_gwh^2 ≈10^-9 ∼10^-11$ at present [ref] .

The Gravitational-Wave Physics · During Preheating · 0.88
In particular, the authors of Ref. [ref] determined that numerically, the EoS parameter exhibited a slow increase from a value of $ω_reh=0$ at the end of inflation to $ω_reh∼0.3$ during the reheating epoch.

Reconstructing inflation and reheating in the framework of a generalized $\mathcal{F}(H)$ Friedmann equation · Introduction · 0.94

Compares with it 1 paper

HLattice conserves energy with an accuracy of order $∼10^-8$, as compared to $10^-3-10^-5$, which has obtained using previous codes such as DEFROST [ref] or LATTICEASY [ref] .

Nongaussianity from Particle Production During Inflation · HLattice Simulations · 0.96

Cites it as support 1 paper

Being a strong out of equilibrium process (and also turbulent, according to theoretical and numerical studies [ref] ), the reheating period is therefore a suitable scenario for primordial magnetogenesis.

Primordial magnetogenesis · Magnetogenesis during reheating · 0.96

Cites it as background 8 papers

As for the quantum nature of the inflaton, it is described by the statistical uncertainty generated by semi-classical approximation [ref] .

Lattice Simulation of Multi-Stream Inflation · Evolution Equations · 0.87
Preheating is complicated by the importance of backreaction after the first few bursts of particle production which affects resonances; these effects can only be recovered using lattice simulations [ref] , which have only become feasible for more complicated scenarios in recent years.

On the Suppression of Parametric Resonance and the Viability of Tachyonic Preheating after Multi-Field Inflation · Introduction · 0.84
The application of numerical methods to the study of non-linear field dynamics in the early Universe has grown substantially, as reflected e.g. by the number of specialized packages developed in recent years [ref] , see also [ref] .

CosmoLattice 2.0 · Introduction · 0.99

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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