pith. machine review for the scientific record. sign in

arxiv: 2604.23224 · v1 · submitted 2026-04-25 · 🌌 astro-ph.CO · gr-qc· hep-ph

Recognition: unknown

Parametric Resonance in φ⁴ Preheating: An Exact Numerical Study

Hrisikesh Thakur , Malay K. Nandy
Authors on Pith no claims yet

Pith reviewed 2026-05-08 07:30 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-ph
keywords parametric resonancepreheatingphi^4 inflationnumerical simulationmode functionsoccupation numberschaotic inflationparticle production
0
0 comments X

The pith

Exact numerical integration of the coupled equations in φ⁴ preheating reveals resonance patterns that differ from those given by approximate analytical methods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper carries out a fully numerical solution of the inflaton field and the mode equations for produced particles in the φ⁴ chaotic inflation model, without relying on the usual linear or perturbative approximations. In the weak-coupling limit the short-wavelength modes quickly settle to steady oscillations while their occupation numbers saturate, whereas long-wavelength modes continue to grow and enter a nonlinear regime. Stronger coupling drives the system into stochastic behavior, with short modes showing intermittent bursts visible as staircase steps in the occupation number and long modes displaying steadier but fluctuating growth. These detailed, coupling-dependent structures matter because preheating determines how efficiently the inflaton energy converts into a hot plasma, setting the initial conditions for the radiation-dominated era.

Core claim

By evolving the complete set of nonlinear, coupled differential equations for the homogeneous inflaton and the inhomogeneous mode functions, the study finds that parametric resonance in the quartic model produces resonance bands whose detailed evolution depends strongly on the coupling strength; weak coupling yields rapid saturation for short modes and gradual growth for long modes, while rising coupling introduces stochasticity and step-like occupation-number growth for short wavelengths.

What carries the argument

The numerical solution of the coupled dynamical equations governing the homogeneous inflaton condensate and the mode functions of the daughter scalar field.

If this is right

  • Short-wavelength modes in weak coupling rapidly reach constant-amplitude oscillations with saturating occupation numbers.
  • Long-wavelength modes in weak coupling exhibit continued gradual growth that transitions into nonlinear oscillations.
  • Increasing the coupling strength drives the system toward stochastic dynamics with intermittent particle production.
  • In strong coupling, short modes display staircase evolution in occupation number while long modes grow more monotonically with small fluctuations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reported deviations from analytical predictions imply that earlier estimates of particle spectra or reheating efficiency in quartic models may require adjustment when full nonlinear evolution is included.
  • The emergence of stochastic behavior at strong coupling suggests that ensemble-averaged quantities rather than single-mode trajectories may be needed to predict observable consequences.
  • The distinction between short- and long-wavelength mode evolution points to a possible scale-dependent filtering effect that could shape the final power spectrum of produced particles.

Load-bearing premise

The numerical scheme and chosen resolution are assumed to capture all relevant nonlinear backreaction and mode-mode couplings without introducing artifacts.

What would settle it

Re-running the evolution at substantially higher momentum-space resolution and confirming whether the reported saturation, gradual growth, staircase steps, and stochastic regimes remain unchanged.

Figures

Figures reproduced from arXiv: 2604.23224 by Hrisikesh Thakur, Malay K. Nandy.

Figure 1
Figure 1. Figure 1: Damped oscillation of the inflaton field view at source ↗
Figure 2
Figure 2. Figure 2: Left panels: Time evolution of the mode function view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the evolution of occupation number view at source ↗
Figure 4
Figure 4. Figure 4: Left panels: Time evolution of the mode function view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the evolution of occupation number view at source ↗
Figure 6
Figure 6. Figure 6: Left panels: Time evolution of the mode function view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the evolution of occupation number view at source ↗
Figure 8
Figure 8. Figure 8: Left panels: Time evolution of the mode function view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the evolution of occupation number view at source ↗
Figure 10
Figure 10. Figure 10: Left panels: Time evolution of the mode function view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the evolution of occupation number view at source ↗
read the original abstract

Preheating after inflation proceeds through parametric resonance, leading to efficient particle production in scalar field models. In this work, we investigate the structure of parametric resonance in the $\phi^4$ chaotic inflationary model during the preheating phase by performing a fully numerical analysis of the coupled dynamical equations governing the inflaton field and the mode function of the produced particles, thereby avoiding the approximations commonly employed in earlier studies. Our results reveal resonance patterns that differ significantly from those obtained with approximate analytical treatments. In the weak coupling regime, short-wavelength modes rapidly settle into oscillations with nearly constant amplitude, while the corresponding occupation numbers approach saturation. However, the long-wavelength modes exhibit gradual amplitude growth, with occupation numbers transitioning into a non-linear oscillatory regime. As the coupling strength increases, the dynamics becomes increasingly nonlinear, leading to the emergence of stochastic behavior. In the strong coupling regime, short-wavelength modes display a step-like (staircase) evolution in the occupation number, indicative of intermittent bursts of particle production. However, the long-wavelength modes exhibit a more gradual, monotonic growth with small superimposed fluctuations. These findings highlight the rich, coupling-dependent, structure of parametric resonance in the quartic inflationary model and underscore the importance of exact numerical treatment in accurately capturing preheating dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper performs a direct numerical integration of the coupled inflaton and mode-function equations in the φ⁴ chaotic inflation model during preheating, avoiding common analytical approximations. It reports coupling-dependent resonance structures: in the weak-coupling regime short-wavelength modes rapidly saturate in amplitude while long-wavelength modes grow gradually with nonlinear occupation-number oscillations; in the strong-coupling regime short modes exhibit staircase-like occupation-number growth and long modes show monotonic increase with fluctuations, accompanied by the emergence of stochastic behavior.

Significance. If the numerical results prove robust, the work would usefully illustrate the limitations of approximate analytic treatments for capturing nonlinear back-reaction and mode-mode coupling in preheating, thereby refining predictions for particle-production efficiency. The approach receives credit for deriving results from the standard field equations without fitted parameters or self-referential definitions.

major comments (2)
  1. [Abstract and Results] The central claim that resonance patterns 'differ significantly' from approximate analytical treatments is load-bearing yet unsupported by quantitative comparisons, error bars, or direct overlays with prior results. No such metrics appear in the results or discussion sections, leaving the magnitude and robustness of the reported differences unverified.
  2. [Numerical Methods and Results] The reported staircase evolution, stochasticity, and saturation behaviors rest on the assumption that the chosen discretization fully captures nonlinear back-reaction without artifacts. No convergence tests with respect to lattice/mode count, time-stepping method, or k-space cutoff are presented, which directly undermines in the physical origin of these features.
minor comments (2)
  1. [Abstract] The abstract introduces 'stochastic behavior' without stating the diagnostic used to distinguish it from regular nonlinear oscillations.
  2. [Introduction] Notation for the coupling constant and the definition of occupation number should be introduced explicitly at the first appearance rather than assumed from prior literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive feedback. We appreciate the positive assessment of the numerical approach and its potential to illustrate limitations of analytic approximations. We address the two major comments below and commit to revisions that directly respond to the concerns raised.

read point-by-point responses

  1. Referee: [Abstract and Results] The central claim that resonance patterns 'differ significantly' from approximate analytical treatments is load-bearing yet unsupported by quantitative comparisons, error bars, or direct overlays with prior results. No such metrics appear in the results or discussion sections, leaving the magnitude and robustness of the reported differences unverified.

    Authors: We acknowledge that the manuscript presents the differences primarily through qualitative descriptions of the observed behaviors (rapid saturation of short-wavelength modes with gradual long-wavelength growth in the weak regime, and staircase-like stochastic features in the strong regime). While these descriptions are drawn from the exact numerical integration of the coupled equations, we agree that the claim would be strengthened by quantitative support. In the revised manuscript we will add direct comparisons, including overlays of our numerical occupation-number curves against representative analytic predictions from the literature, together with quantitative measures of deviation (e.g., relative differences in growth rates and saturation levels). These additions will be placed in the results section and will quantify the magnitude of the reported differences. revision: yes

  2. Referee: [Numerical Methods and Results] The reported staircase evolution, stochasticity, and saturation behaviors rest on the assumption that the chosen discretization fully captures nonlinear back-reaction without artifacts. No convergence tests with respect to lattice/mode count, time-stepping method, or k-space cutoff are presented, which directly undermines in the physical origin of these features.

    Authors: We agree that explicit convergence tests are necessary to establish that the reported features arise from the physics rather than from the numerical discretization. Although our integration employs the standard field equations without additional approximations, the original manuscript did not include systematic tests of lattice size, mode number, time-step accuracy, or k-space cutoff. In the revised version we will add a dedicated subsection (or appendix) presenting convergence studies across these parameters. The tests will demonstrate that the staircase evolution, stochastic fluctuations, and saturation behaviors remain robust under refinement, thereby confirming their physical origin. revision: yes

Circularity Check

0 steps flagged

No circularity: results are direct outputs of numerical integration of standard equations

full rationale

The paper conducts a fully numerical solution of the coupled inflaton and mode-function equations for the φ⁴ preheating model, explicitly avoiding analytical approximations. All reported resonance patterns, amplitude evolutions, occupation-number behaviors (saturation, staircase steps, stochasticity), and coupling-dependent distinctions are generated as simulation outputs rather than being presupposed, fitted, or defined in terms of the target quantities. No self-citations are invoked as load-bearing uniqueness theorems, no parameters are tuned to reproduce specific outcomes, and no ansatz or renaming of known results occurs. The derivation chain therefore remains self-contained against external benchmarks (the standard Klein-Gordon and mode equations), yielding a circularity score of 0.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The study rests on the standard equations of motion for a scalar field with quartic potential plus a coupled fluctuation mode in an expanding background; no new entities are introduced and the only free parameter is the coupling strength, which is scanned rather than fitted to data.

free parameters (1)
  • coupling strength
    Varied across weak-to-strong regimes to map different dynamical behaviors; not fitted to external data.
axioms (2)
  • standard math The inflaton and fluctuation modes obey the Klein-Gordon equation in a Friedmann-Robertson-Walker background with the φ⁴ potential.
    Standard relativistic field theory for scalar fields during preheating.
  • domain assumption Initial conditions are set at the end of inflation with small fluctuations.
    Conventional setup for preheating studies.

pith-pipeline@v0.9.0 · 5528 in / 1466 out tokens · 107727 ms · 2026-05-08T07:30:51.303019+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

44 extracted references · 42 canonical work pages · 2 internal anchors

  1. [1]

    Alan H. Guth. Inflationary universe: A possible solution to the horizon and flatness problems.Phys. Rev. D, 23:347–356, Jan 1981. URL:https://link.aps.org/doi/10.1103/PhysRevD.23.347, doi:10.1103/PhysRevD.23.347

    work page doi:10.1103/physrevd.23.347 1981

  2. [2]

    Starobinsky, Phys

    A.A. Starobinsky. A new type of isotropic cosmological models without singularity.Physics Letters B, 91(1):99–102, 1980. URL:https://www.sciencedirect.com/science/article/pii/ 037026938090670X,doi:10.1016/0370-2693(80)90670-X

    work page doi:10.1016/0370-2693(80)90670-x 1980

  3. [3]

    D. Kazanas. Dynamics of the universe and spontaneous symmetry breaking.Astrophysical Journal, 241:L59–L63, October 1980.doi:10.1086/183361

    work page doi:10.1086/183361 1980

  4. [4]

    Cosmological baryon-number domain structure and the first order phase transition of a vacuum.Physics Letters B, 99(1):66–70, 1981

    Katsuhiko Sato. Cosmological baryon-number domain structure and the first order phase transition of a vacuum.Physics Letters B, 99(1):66–70, 1981. URL:https://www.sciencedirect.com/ science/article/pii/0370269381908054,doi:10.1016/0370-2693(81)90805-4

    work page doi:10.1016/0370-2693(81)90805-4 1981

  5. [5]

    A. D. Linde. A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems.Physics Letters B, 108(6):389–393, February 1982.doi:10.1016/0370-2693(82)91219-9

    work page doi:10.1016/0370-2693(82)91219-9 1982

  6. [6]

    Albrecht and P

    Andreas Albrecht and Paul J. Steinhardt. Cosmology for grand unified theories with radiatively induced symmetry breaking.Phys. Rev. Lett., 48:1220–1223, Apr 1982. URL:https://link.aps. org/doi/10.1103/PhysRevLett.48.1220,doi:10.1103/PhysRevLett.48.1220

    work page doi:10.1103/physrevlett.48.1220 1982

  7. [7]

    A.D. Linde. Chaotic inflation.Physics Letters B, 129(3):177–181, 1983. URL:https://www. sciencedirect.com/science/article/pii/0370269383908377,doi:10.1016/0370-2693(83) 90837-7

    work page doi:10.1016/0370-2693(83 1983

  8. [8]

    Hawking and I.L

    S.W. Hawking and I.L. Moss. Supercooled phase transitions in the very early universe.Physics Letters B, 110(1):35–38, 1982. URL:https://www.sciencedirect.com/science/article/pii/ 0370269382909467,doi:10.1016/0370-2693(82)90946-7

    work page doi:10.1016/0370-2693(82)90946-7 1982

  9. [9]

    A Prescription for Successful New Inflation

    Paul J. Steinhardt and Michael S. Turner. Prescription for successful new inflation.Phys. Rev. D, 29:2162–2171, May 1984. URL:https://link.aps.org/doi/10.1103/PhysRevD.29.2162,doi: 10.1103/PhysRevD.29.2162

    work page doi:10.1103/physrevd.29.2162 1984

  10. [10]

    Liddle and David H

    Andrew R. Liddle and David H. Lyth. Cobe, gravitational waves, inflation and extended infla- tion.Physics Letters B, 291(4):391–398, 1992. URL:https://www.sciencedirect.com/science/ article/pii/037026939291393N,doi:10.1016/0370-2693(92)91393-N

    work page doi:10.1016/0370-2693(92)91393-n 1992

  11. [11]

    Liddle, Paul Parsons, and John D

    Andrew R. Liddle, Paul Parsons, and John D. Barrow. Formalizing the slow-roll approximation in inflation.Phys. Rev. D, 50:7222–7232, Dec 1994. URL:https://link.aps.org/doi/10.1103/ PhysRevD.50.7222,doi:10.1103/PhysRevD.50.7222

    work page doi:10.1103/physrevd.50.7222 1994

  12. [12]

    Coughlan and G.G

    G.D. Coughlan and G.G. Ross. Initial conditions for inflation.Physics Letters B, 157(2):151– 156, 1985. URL:https://www.sciencedirect.com/science/article/pii/0370269385915369, doi:10.1016/0370-2693(85)91536-9. 19

    work page doi:10.1016/0370-2693(85)91536-9 1985

  13. [13]

    A.D. Linde. Initial conditions for inflation.Physics Letters B, 162(4):281–286, 1985. URL:https://www.sciencedirect.com/science/article/pii/0370269385909232,doi:10. 1016/0370-2693(85)90923-2

    work page arXiv 1985

  14. [14]

    Brandenberger

    Andreas Albrecht and Robert H. Brandenberger. Realization of new inflation.Phys. Rev. D, 31:1225–1231, Mar 1985. URL:https://link.aps.org/doi/10.1103/PhysRevD.31.1225,doi: 10.1103/PhysRevD.31.1225

    work page doi:10.1103/physrevd.31.1225 1985

  15. [15]

    Brandenberger, and Richard A

    Andreas Albrecht, Robert H. Brandenberger, and Richard A. Matzner. Numerical analysis of inflation.Phys. Rev. D, 32:1280–1289, Sep 1985. URL:https://link.aps.org/doi/10.1103/ PhysRevD.32.1280,doi:10.1103/PhysRevD.32.1280

    work page doi:10.1103/physrevd.32.1280 1985

  16. [16]

    Inflation with generalized initial conditions.Phys

    Andreas Albrecht, Robert Brandenberger, and Richard Matzner. Inflation with generalized initial conditions.Phys. Rev. D, 35:429–434, Jan 1987. URL:https://link.aps.org/doi/10.1103/ PhysRevD.35.429,doi:10.1103/PhysRevD.35.429

    work page doi:10.1103/physrevd.35.429 1987

  17. [17]

    Particlephysicsandinflationarycosmology,

    Andrei Linde. Particle physics and inflationary cosmology, 2005. URL:https://arxiv.org/abs/ hep-th/0503203,arXiv:hep-th/0503203

    work page arXiv 2005

  18. [18]

    Springer Berlin Heidelberg, Berlin, Heidelberg, 2007.doi:10.1007/978-3-540-74353-8_1

    Andrei Linde.Inflationary Cosmology, pages 1–54. Springer Berlin Heidelberg, Berlin, Heidelberg, 2007.doi:10.1007/978-3-540-74353-8_1

    work page doi:10.1007/978-3-540-74353-8_1 2007

  19. [19]

    Keith A. Olive. Inflation.Physics Reports, 190(6):307–403, 1990. URL:https://www. sciencedirect.com/science/article/pii/037015739090144Q,doi:10.1016/0370-1573(90) 90144-Q

    work page doi:10.1016/0370-1573(90 1990

  20. [20]

    Inflation,

    DANIEL BAUMANN.INFLATION, pages 523–686. URL:https://www.worldscientific.com/ doi/abs/10.1142/9789814327183_0010,arXiv:https://www.worldscientific.com/doi/pdf/ 10.1142/9789814327183_0010,doi:10.1142/9789814327183_0010

    work page doi:10.1142/9789814327183_0010

  21. [21]

    Physics of the Dark Universe , author =

    J´ erˆ ome Martin, Christophe Ringeval, and Vincent Vennin. Encyclopædia inflationaris.Physics of the Dark Universe, 5-6:75–235, 2014. Hunt for Dark Matter. URL:https://www.sciencedirect. com/science/article/pii/S2212686414000053,doi:10.1016/j.dark.2014.01.003

    work page doi:10.1016/j.dark.2014.01.003 2014

  22. [22]

    Matzner, and James R

    Hannu Kurki-Suonio, Joan Centrella, Richard A. Matzner, and James R. Wilson. Inflation from inhomogeneous initial data in a one-dimensional back-reacting cosmology.Phys. Rev. D, 35:435– 448, Jan 1987. URL:https://link.aps.org/doi/10.1103/PhysRevD.35.435,doi:10.1103/ PhysRevD.35.435

    work page doi:10.1103/physrevd.35.435 1987

  23. [23]

    Goldwirth and Tsvi Piran

    Dalia S. Goldwirth and Tsvi Piran. Inhomogeneity and the onset of inflation.Phys. Rev. Lett., 64:2852–2855, Jun 1990. URL:https://link.aps.org/doi/10.1103/PhysRevLett.64.2852, doi:10.1103/PhysRevLett.64.2852

    work page doi:10.1103/physrevlett.64.2852 1990

  24. [24]

    Initial conditions for inflation.Physics Reports, 214(4):223– 292, 1992

    Dalia S Goldwirth and Tsvi Piran. Initial conditions for inflation.Physics Reports, 214(4):223– 292, 1992. URL:https://www.sciencedirect.com/science/article/pii/0370157392900739, doi:10.1016/0370-1573(92)90073-9. 20

    work page doi:10.1016/0370-1573(92)90073-9 1992

  25. [25]

    Hannu Kurki-Suonio, Pablo Laguna, and Richard A. Matzner. Inhomogeneous inflation: Numerical evolution.Phys. Rev. D, 48:3611–3624, Oct 1993. URL:https://link.aps.org/doi/10.1103/ PhysRevD.48.3611,doi:10.1103/PhysRevD.48.3611

    work page doi:10.1103/physrevd.48.3611 1993

  26. [26]

    Onset of inflation in inhomogeneous cosmology.Phys

    Osamu Iguchi and Hideki Ishihara. Onset of inflation in inhomogeneous cosmology.Phys. Rev. D, 56:3216–3224, Sep 1997. URL:https://link.aps.org/doi/10.1103/PhysRevD.56.3216,doi: 10.1103/PhysRevD.56.3216

    work page doi:10.1103/physrevd.56.3216 1997

  27. [27]

    Causality and cosmic inflation.Phys

    Tanmay Vachaspati and Mark Trodden. Causality and cosmic inflation.Phys. Rev. D, 61:023502, Dec 1999. URL:https://link.aps.org/doi/10.1103/PhysRevD.61.023502,doi:10.1103/ PhysRevD.61.023502

    work page doi:10.1103/physrevd.61.023502 1999

  28. [28]

    Price, and Javier Rasero

    Richard Easther, Layne C. Price, and Javier Rasero. Inflating an inhomogeneous universe.Journal of Cosmology and Astroparticle Physics, 2014(08):041, aug 2014. URL:https://dx.doi.org/10. 1088/1475-7516/2014/08/041,doi:10.1088/1475-7516/2014/08/041

    work page doi:10.1088/1475-7516/2014/08/041 2014

  29. [29]

    East, Matthew Kleban, Andrei Linde, and Leonardo Senatore

    William E. East, Matthew Kleban, Andrei Linde, and Leonardo Senatore. Beginning inflation in an inhomogeneous universe.Journal of Cosmology and Astroparticle Physics, 2016(09):010, sep

    2016

  30. [30]

    URL:https://dx.doi.org/10.1088/1475-7516/2016/09/010,doi:10.1088/1475-7516/ 2016/09/010

    work page doi:10.1088/1475-7516/2016/09/010 2016

  31. [31]

    Towards the th eory of reheating after inflation,

    Lev Kofman, Andrei Linde, and Alexei A. Starobinsky. Towards the theory of reheating after inflation.Phys. Rev. D, 56:3258–3295, Sep 1997. URL:https://link.aps.org/doi/10.1103/ PhysRevD.56.3258,doi:10.1103/PhysRevD.56.3258

    work page doi:10.1103/physrevd.56.3258 1997

  32. [32]

    Greene, Lev Kofman, Andrei Linde, and Alexei A

    Patrick B. Greene, Lev Kofman, Andrei Linde, and Alexei A. Starobinsky. Structure of resonance in preheating after inflation.Phys. Rev. D, 56:6175–6192, Nov 1997. URL:https://link.aps. org/doi/10.1103/PhysRevD.56.6175,doi:10.1103/PhysRevD.56.6175

    work page doi:10.1103/physrevd.56.6175 1997

  33. [33]

    David I. Kaiser. Resonance structure for preheating with massless fields.Phys. Rev. D, 57:702– 711, Jan 1998. URL:https://link.aps.org/doi/10.1103/PhysRevD.57.702,doi:10.1103/ PhysRevD.57.702

    work page doi:10.1103/physrevd.57.702 1998

  34. [34]

    David I. Kaiser. Preheating in an expanding universe: Analytic results for the massless case.Phys. Rev. D, 56:706–716, Jul 1997. URL:https://link.aps.org/doi/10.1103/PhysRevD.56.706, doi:10.1103/PhysRevD.56.706

    work page doi:10.1103/physrevd.56.706 1997

  35. [35]

    Calzetta and B

    E. Calzetta and B. L. Hu. Quantum fluctuations, decoherence of the mean field, and structure formation in the early universe.Phys. Rev. D, 52:6770–6788, Dec 1995. URL:https://link.aps. org/doi/10.1103/PhysRevD.52.6770,doi:10.1103/PhysRevD.52.6770

    work page doi:10.1103/physrevd.52.6770 1995

  36. [36]

    Bassett, Fabrizio Tamburini, David I

    Bruce A. Bassett, Fabrizio Tamburini, David I. Kaiser, and Roy Maartens. Metric pre- heating and limitations of linearized gravity.Nuclear Physics B, 561(1):188–240, 1999. URL:https://www.sciencedirect.com/science/article/pii/S0550321399004952,doi:10. 1016/S0550-3213(99)00495-2

    1999

  37. [37]

    Gravity, parametric resonance, and chaotic inflation.Phys

    Richard Easther and Matthew Parry. Gravity, parametric resonance, and chaotic inflation.Phys. Rev. D, 62:103503, Oct 2000. URL:https://link.aps.org/doi/10.1103/PhysRevD.62.103503, doi:10.1103/PhysRevD.62.103503. 21

    work page doi:10.1103/physrevd.62.103503 2000

  38. [38]

    Chaotic dynamics in preheating after inflation.Classical and Quantum Gravity, 23(2):353, dec 2005

    Yoshida Jin and Shinji Tsujikawa. Chaotic dynamics in preheating after inflation.Classical and Quantum Gravity, 23(2):353, dec 2005. URL:https://dx.doi.org/10.1088/0264-9381/23/2/ 006,doi:10.1088/0264-9381/23/2/006

    work page doi:10.1088/0264-9381/23/2/ 2005

  39. [39]

    D. I. Podolsky and A. A. Starobinsky. Chaotic reheating, 2002. URL:https://arxiv.org/abs/ astro-ph/0204327,arXiv:astro-ph/0204327

    work page internal anchor Pith review arXiv 2002

  40. [40]

    Evolution of nonlinear fluctuations in preheating after infla- tion.Classical and Quantum Gravity, 23(2):511, dec 2005

    Yasusada Nambu and Yohei Araki. Evolution of nonlinear fluctuations in preheating after infla- tion.Classical and Quantum Gravity, 23(2):511, dec 2005. URL:https://dx.doi.org/10.1088/ 0264-9381/23/2/015,doi:10.1088/0264-9381/23/2/015

    work page doi:10.1088/0264-9381/23/2/015 2005

  41. [41]

    Analysis of the evolution of curvature perturbations in preheating by usingδnformalism.Classical and Quantum Gravity, 24(6):1615, mar 2007

    Teruaki Suyama and Shuichiro Yokoyama. Analysis of the evolution of curvature perturbations in preheating by usingδnformalism.Classical and Quantum Gravity, 24(6):1615, mar 2007. URL: https://dx.doi.org/10.1088/0264-9381/24/6/015,doi:10.1088/0264-9381/24/6/015

    work page doi:10.1088/0264-9381/24/6/015 2007

  42. [42]

    Collapse of small-scale density per- turbations during preheating in single field inflation.Journal of Cosmology and Astroparticle Physics, 2010(09):034, sep 2010

    Karsten Jedamzik, Martin Lemoine, and J´ erˆ ome Martin. Collapse of small-scale density per- turbations during preheating in single field inflation.Journal of Cosmology and Astroparticle Physics, 2010(09):034, sep 2010. URL:https://dx.doi.org/10.1088/1475-7516/2010/09/034, doi:10.1088/1475-7516/2010/09/034

    work page doi:10.1088/1475-7516/2010/09/034 2010

  43. [43]

    Planck 2018 results. X. Constraints on inflation

    Y. Akrami et al. Planck 2018 results - X. constraints on inflation.A&A, 641:A10, 2020.doi: 10.1051/0004-6361/201833887

    work page internal anchor Pith review doi:10.1051/0004-6361/201833887 2018

  44. [44]

    P. A. R. Ade et al. Joint analysis of BICEP2/Keck array and Planck data.Phys. Rev. Lett., 114:101301, Mar 2015. URL:https://link.aps.org/doi/10.1103/PhysRevLett.114.101301, doi:10.1103/PhysRevLett.114.101301. 22

    work page doi:10.1103/physrevlett.114.101301 2015