Towards the Theory of Reheating After Inflation

Alexei Starobinsky; Andrei Linde; Lev Kofman

REVIEW

Towards the Theory of Reheating After Inflation

Lev Kofman, Andrei Linde, Alexei Starobinsky

High Energy Physics - Phenomenology · Submitted 30 Apr 1997 · 46 references

Pith's verdict: The paper that made preheating calculable: the parabolic-scattering recursion is sound and reproduces the numerics, with soft spots confined to the last oscillation or two and to an acknowledged μ calibration.

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CitationAlexei Starobinsky, Andrei Linde, Lev Kofman. "Towards the Theory of Reheating After Inflation." arXiv:hep-ph/9704452 (1997). https://pith.science/paper/WCW7NAI6

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Editor's note

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Bottom line: this is the paper that turned preheating from a numerical curiosity into something you can calculate, and the central machinery holds up. The part worth your time is Sec. VII. Near each zero of φ the mode equation reduces to d²X/dτ² + (κ²+τ²)X = 0, whose exact parabolic-cylinder reflection and transmission amplitudes give a clean recursion (Eq. 55) for n_k after each kick. That reproduces the Mathieu growth index in the Minkowski limit (Fig. 9) and the envelope in the expanding case (Fig. 10). The physical picture — phases decorrelate between kicks, the band structure dissolves, n_k does a random walk in log — is right, and it is why the later lattice work had something to check against. The factor-of-two agreement with Khlebnikov–Tkachev is real external support, though the authors are right to tell you not to lean on it too hard. Credit where due: the |φ| form of the backreaction potential (Eq. 93) is derived, not assumed, and the two-stage picture with the crossover at n_χ ≈ m²Φ/g is a genuine result. The paper is also unusually candid about its own weak points — Sec. VIII C says outright that the Hartree truncation drops a diagram of comparable size, and Sec. X B says the particle-like cross-section is invalid near φ=0. That kind of self-reporting is rare and should be rewarded. Where it is soft: the stress-test note calls the single effective μ ≈ 0.13 load-bearing, and it is, but the note overstates the consequence. t₁ ∝ 1/μ (Eq. 111), so a 30% error in μ moves t₁, Φ₁ and q₁ by roughly 30%, and the superheavy bound M ∝ q₁^{1/4} by under 10%. That is not exponential propagation. The fair criticism is narrower: μ is calibrated on g = 10⁻⁴–10⁻³ and then used at g = 10⁻¹, and the log-normal tail that actually dominates the k-integral is acknowledged as a 'natural selection effect' but never quantified. Both are real, both are bounded, and the authors flag the second themselves. The rescattering termination criterion (Eq. 138) is the weakest quantitative piece, resting on a cross-section the authors concede does not apply where production happens; it affects the last oscillation or two. Who should read it: anyone doing post-inflationary cosmology, and anyone who wants a model of how to do analytic work when the standard band-structure tools fail. Send it out.

Referee report

Significance

If the central picture holds, this is a foundational contribution. The reduction of the broad-resonance mode equation to successive parabolic scatterings (Sec. VII B–C) is an elegant and reusable tool: it reproduces the Mathieu growth index (Fig. 9) and matches the expanding-universe numerics (Fig. 10), and it does so without requiring the static band structure that fails here. The insistence that n_k, not ⟨χ²⟩, is the correct adiabatic invariant (Figs. 3 and 6) corrects a practice common in the preheating literature and is a genuinely useful methodological point. The two-stage division with the criterion n_χ = m²Φ/g (Eq. 95) is clean and yields a concrete threshold, g ≳ 3×10⁻⁴, for efficient preheating; the resulting ⟨χ²⟩ estimates agree with the lattice simulations of Khlebnikov and Tkachev to within a factor of two (Sec. X C), which is a meaningful check. The superheavy-particle bound of Eq. (144) is a falsifiable prediction, and the nonthermal symmetry-restoration scale of ~10¹⁶ GeV is a concrete cosmological consequence. The paper also states clearly what it does not settle, which is a strength rather than a weakness.

Major objections (3)

[Sec. IX A, Eqs. (109)–(114) and the table on p. 30] The quantitative results of the first stage rest on a single effective growth index μ ≈ 0.13. That number is not derived from the analytic recursion of Eq. (80); it is read off the numerical table of Sec. VI, which spans g = 0.9×10⁻⁴ to 10⁻³ and shows μ ranging from 0.10 to 0.18. The Sec. IX A table then quotes results for g = 10⁻³, 10⁻², and 10⁻¹, i.e. one to two decades beyond the calibration range. Because t₁ ∝ 1/μ (Eq. 111) and Φ₁ ∝ μ (Eq. 113), the quoted Φ₁, q₁, and the superheavy bound M ∝ q₁^{1/4} ∝ μ^{1/2} (Eq. 144) all shift linearly with the assumed μ. The authors acknowledge the stochastic sensitivity in the footnote on p. 29 and in Ref. [44], but the manuscript should state the calibration range explicitly and give the scaling of the final numbers with μ, so that a reader can rescale if a better determination of μ becomes available. As written, the extrapolation in g is unverified.
[Sec. VII E and Eq. (64)] The k-integrated yield is obtained by steepest descent with the estimate μ''_k ∼ 2μ/Δk², and the 'natural selection effect' — modes that happen to receive positive kicks more often than the 3:1 average dominate the total — is invoked qualitatively but not quantified. If the yield is controlled by the tail of the log-normal distribution of n_k rather than by its mean, then the effective μ governing n_χ is not the single-mode mean, and Eq. (64) is an order-of-magnitude estimate whose prefactor and effective exponent are uncertain. The paper should either bound this effect or state plainly that Eq. (64) is an estimate and that the downstream numbers inherit that uncertainty. This is load-bearing because every subsequent result in Sec. IX is built on Eq. (64).
[Sec. VIII C, Eq. (103), and Sec. IX B] The authors show that the second polarization operator Π²_φ of Fig. 12b is of the same order as the Hartree contribution Π¹_φ, so the Hartree approximation may be insufficient for the δφ spectrum, and they explicitly defer a complete calculation. This is an honest and important observation, but the second-stage estimates of Sec. IX B and the rescattering analysis of Sec. X rest on the Hartree effective mass m²_φ ≈ g²⟨χ²⟩. The manuscript should state which of its quantitative results survive if Π²_φ modifies the φ mass and which are conditional on the Hartree truncation. As it stands, the reader cannot tell whether the final ⟨χ²⟩ and n_χ estimates are robust or contingent.

Minor objections (6)

[Fig. 7] The resonance-band structure along the line A = 2q is central to the argument of Sec. VI, but the axis labels of Fig. 7 are difficult to read. The figure should be redrawn with legible tick labels and a clear indication of the q-intervals corresponding to the first band and the stability gap.
[Eq. (72)] The elimination of the phase ϑ between Eqs. (68) and (71) is described as 'easy'; a one-line intermediate step would let a reader verify the growth index without reconstructing the algebra.
[Sec. X C, comparison with Ref. [30]] The text notes that the definition of q₀ in Ref. [30] differs slightly from the present one and advises comparing at fixed g rather than fixed q₀. A short table giving the two conventions side by side would make the comparison unambiguous.
[Sec. VIII B versus Sec. X B] Section VIII B argues that the high-frequency modulation of g²⟨χ²⟩ does not much affect the evolution of φ(t), while Sec. X B notes that the same modulation may serve as an additional source of φ-particles. These statements are consistent (background versus fluctuations), but the text would be clearer if the distinction were made explicit at the point where the modulation is first introduced.
[Eq. (144) and the following paragraph] The relaxation of the superheavy-production condition from gmΦ ≳ 4M² to gmΦ ≳ 2M² is asserted without derivation. A brief justification, or a reference to where the factor is computed, would help.
[Sec. XII] The discussion states that for g ≪ 3×10⁻⁴ the total energy density of produced particles is exponentially small, and that even for large g the main energy remains in φ-particles. It would strengthen the summary to quote the explicit small parameter controlling this suppression, rather than only the qualitative statement.

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The paper

The paper claims

Reheating after inflation is not a slow one-particle-at-a-time decay but a stochastic resonance, and the authors claim it can be computed analytically even though each growing mode jumps across thousands of resonance bands within a single振荡

The reading

After inflation ends, the inflaton field oscillates about the minimum of its potential and dumps its energy into other fields. The paper asks how that transfer actually happens in the simplest model, a massive inflaton $m^2varphi^2/2$ coupled to a second scalar $chi$ through $g^2varphi^2chi^2$. The authors argue that the answer is not perturbation theory and not the textbook narrow-resonance picture either: because the oscillation amplitude and frequency change quickly as the universe expands, the phase of each $chi$ mode at the moment the inflaton crosses zero is effectively random, so the standard Mathieu-equation band structure does not apply. They call the resulting process stochastic resonance, show that it still amplifies occupation numbers exponentially, and build an analytic method that treats each zero-crossing as a scattering event off a parabolic potential. With that tool they follow the whole history, including the backreaction of the produced particles and their rescattering, and extract when preheating stops, how many particles and how much fluctuation it leaves behind, and whether it can make particles heavier than the inflaton itself.

What carries the argument

The load-bearing device is the reduction of the mode equation near each zero of the inflaton to a universal form. Writing $X_k = a^{3/2}chi_k$ removes the Hubble friction, and near $varphi(t_j)=0$ the frequency becomes $k^2/a^2 + g^2Phi^2 m^2 (t-t_j)^2$, so with $tau = k_*(t-t_j)$ and $kappa = k/(a k_*)$, where $k_* = sqrt{g m Phi}$, one gets $d^2X/dtau^2 + (kappa^2+tau^2)X = 0$. That equation has exact parabolic-cylinder solutions, hence exact reflection and transmission amplitudes $R_k$ and $D_k$ (Eqs. 51–52), which convert into a two-by-two recursion for the Bogoliubov coefficients across one kick. Iterating the recursion gives the change in occupation number per kick, $n_k^{j+1} = e^{-pi kappa^2} + (1+2e^{-pi kappa^2})n_k^j - 2 e^{-pi kappa^2/2}sqrt{1+e^{-pi kappa^2}}sqrt{n_k^j(1+n_k^j)}sinTheta_j$. The sine term is the interference between consecutive kicks; when the phases $Theta_j$ are random, the whole band structure dissolves and what remains is a broad, phase-insensitive amplification with a resonance width $k lesssim k_*/sqrt{pi}$. This is what lets the authors replace a Mathieu stability chart, which fails here, with a single growth index and then integrate over momenta.

Extended reading notes

Core claim

The central claim is that broad parametric resonance in an expanding universe with a quadratic inflaton potential is a stochastic process, and that it is nevertheless tractable analytically. In the usual treatment each mode sits in a fixed instability band of the Mathieu equation and grows at a fixed rate. Here, the authors point out, the parameter $q = g^2Phi^2/4m^2$ falls as $t^{-2}$, so within a single oscillation a mode sweeps through order $sqrt{q}$ different bands, and the phase accumulated between successive zero-crossings of $varphi$ changes by far more than $pi$. The kicks therefore arrive with uncorrelated phases, and the occupation number $n_k$ can fall as well as rise from one kick to the next — something impossible for classical particles in a fixed background. Averaged over the randomness, the growth is still exponential, with an effective index $mu$ of order $0.1$–$0.2$ (the authors quote $mu approx 0.13$). From this they derive the duration of the first stage of preheating, the amplitude $Phi_1$ and $q_1$ at its end, the criterion $g lesssim 3times10^{-4}$ for preheating to finish before backreaction matters, the growth of the effective inflaton mass as $g n_chi/|varphi|$ (a $|varphi|$ term in the potential, not $varphi^2$), and finally a condition $g m varphi Phi gtrsim 4M^2$ under which $chi$ particles of mass $M gg m$ are copiously produced.

Load-bearing premise

Every exponential estimate rests on replacing the wildly fluctuating, phase-dependent growth index of Eq. (81) by one effective constant, $mu approx 0.13$, and then integrating $n_chi propto e^{2mu m t}$ over the entire first stage.

Editorial extensions

If this is right

For $g lesssim 3times10^{-4}$ preheating ends while backreaction is still negligible, and the perturbative reheating calculation remains the correct description of the end of the process.
For larger $g$ a second stage follows in which the produced $chi$ particles raise the inflaton's effective frequency, shortening each oscillation and making particle production faster rather than slower, until $q sim 1/4$.
The variance of $chi$ can reach $sqrt{langle chi^2 rangle} sim 10^{16}$ GeV, orders of magnitude above any thermal value at the same energy density, which can restore symmetry nonthermally and seed topological defects.
Particles with mass $M gg m$ can be produced in quantity provided $g m varphi Phi gtrsim 4M^2$; producing $M = 10m$ requires $g gtrsim 0.065$, or about $0.036$ with the relaxed criterion.
At the end of resonance the $chi$ particles outnumber nothing: rescattering leaves roughly ten times more $varphi$ particles than $chi$ particles, so the reheating temperature is still set by the later perturbative decay of the $varphi$ remnants, not by preheating.

Reading between the lines

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

If $n_k$ really obeys a log-normal distribution, the volume-averaged energy density is dominated by rare patches with anomalously large occupation rather than by the mean, so preheating would be strongly inhomogeneous on sub-horizon scales; that would be a source of non-Gaussian density perturbations and possibly of primordial black holes, which the paper only gestures at.
The parabolic-kick formalism depends only on the behaviour of $varphi$ near its zero, so it should transfer directly to any potential where the inflaton crosses zero with a linear zero, giving a general map from $V(varphi)$ to a growth index without ever writing down a Mathieu equation.
Because $mu$ is a random variable, the honest object is a Fokker–Planck equation for the distribution of $ln n_k$; replacing it by a single mean $mu$ will systematically underestimate the upper tail, which would push the superheavy-particle mass bound of Eq. (144) upward.
Soft $varphi$ modes with $k ll k_*$ are indistinguishable from the coherent condensate over the timescales that matter, so 'number of inflaton particles' is not a clean observable during preheating; field variances, not occupation numbers, are probably the right bookkeeping.

Original abstract

Reheating after inflation occurs due to particle production by the oscillating inflaton field. In this paper we describe the perturbative approach to reheating, and then concentrate on effects beyond the perturbation theory. They are related to the stage of parametric resonance called preheating. It may occur in an expanding universe if the initial amplitude of oscillations of the inflaton field is large enough. We investigate a simple model of a massive inflaton field coupled to another scalar field X. Parametric resonance in this model is very broad. It occurs in a very unusual stochastic manner, which is different from the parametric resonance in the case when the expansion of the universe is neglected. Quantum fields interacting with the oscillating inflaton field experience a series of kicks which occur with phases uncorrelated to each other. We call this process stochastic resonance. We develop the theory of preheating taking into account the expansion of the universe and backreaction of produced particles, including the effects of rescattering. The process of preheating can be divided into several distinct stages. At the first stage the backreaction of created particles is not important. At the second stage backreaction increases the frequency of oscillations of the inflaton field, which makes the process even more efficient than before. Then the effects related to scattering of X-particles terminate the resonance. We calculate the density of X-particles and their quantum fluctuations with all backreaction effects taken into account. This allows us to find the range of masses and coupling constants for which one has efficient preheating. In particular, under certain conditions this process may produce particles with a mass much greater than the mass of the inflaton field.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Reheating after inflation occurs due to particle production by the oscillating inflaton field. In this paper we describe the perturbative approach to reheating, and then concentrate on effects beyond the perturbation theory. They are related to the stage of parametric resonance called preheating. It may occur in an expanding universe if the initial amplitude of oscillations of the inflaton field is large enough. We investigate a simple model of a massive inflaton field coupled to another scalar field X. Parametric resonance in this model is very broad. It occurs in a very unusual stochastic manner, which is different from the parametric resonance in the case when the expansion of the universe is neglected. Quantum fields interacting with the oscillating inflaton field experience a series of kicks which occur with phases uncorrelated to each other. We call this process stochastic resonance. We develop the theory of preheating taking into account the expansion of the universe and backreaction of produced particles, including the effects of rescattering. The process of preheating can be divided into several distinct stages. At the first stage the backreaction of created particles is not important. At the second stage backreaction increases the frequency of oscillations of the inflaton field, which makes the process even more efficient than before. Then the effects related to scattering of X-particles terminate the resonance. We calculate the density of X-particles and their quantum fluctuations with all backreaction effects taken into account. This allows us to find the range of masses and coupling constants for which one has efficient preheating. In particular, under certain conditions this process may produce particles with a mass much greater than the mass of the inflaton field.

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	Reheating after inflation occurs due to particle production by the oscillating inflaton field.	background 1.00	0.06	0.03	asserted 1.00
s1	In this paper we describe the perturbative approach to reheating, and then concentrate on effects beyond the perturbation theory.	contribution 0.88	0.76	0.04	asserted 0.80
s2	They are related to the stage of parametric resonance called preheating.	background 0.91	0.11	0.04	asserted 0.99
s3	It may occur in an expanding universe if the initial amplitude of oscillations of the inflaton field is large enough.	background 0.91	0.11	0.04	hedged possibility 1.00
s4	We investigate a simple model of a massive inflaton field coupled to another scalar field X.	method 0.85	0.76	0.02	asserted 0.97
s5	Parametric resonance in this model is very broad.	result 0.97	0.72	0.05	asserted 1.00
s6	It occurs in a very unusual stochastic manner, which is different from the parametric resonance in the case when the expansion of the universe is neglected.	result 0.96	0.75	0.09	asserted 1.00
s7	Quantum fields interacting with the oscillating inflaton field experience a series of kicks which occur with phases uncorrelated to each other.	result 0.71	0.53	0.06	asserted 1.00
s8	We call this process stochastic resonance.	interpretation 0.49	0.83	0.09	asserted 0.99
s9	We develop the theory of preheating taking into account the expansion of the universe and backreaction of produced particles, including the effects of rescattering.	contribution 0.78	0.94	0.12	asserted 0.99
s10	The process of preheating can be divided into several distinct stages.	result 0.51	0.50	0.05	asserted 1.00
s11	At the first stage the backreaction of created particles is not important.	result 0.73	0.44	0.03	asserted 1.00
s12	At the second stage backreaction increases the frequency of oscillations of the inflaton field, which makes the process even more efficient than before.	result 0.81	0.71	0.09	asserted 1.00
s13	Then the effects related to scattering of X-particles terminate the resonance.	result 0.87	0.42	0.07	asserted 1.00
s14	We calculate the density of X-particles and their quantum fluctuations with all backreaction effects taken into account.	contribution 0.51	0.96	0.07	asserted 1.00
s15	This allows us to find the range of masses and coupling constants for which one has efficient preheating.	result 0.83	0.91	0.17	asserted 0.99
s16	In particular, under certain conditions this process may produce particles with a mass much greater than the mass of the inflaton field.	result 0.98	0.82	0.26	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.95.

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arXiv listing abstract v2

not examined

full text
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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. Oscillations of the field

**Figure 2.** FIG. 2. Narrow parametric resonance for the field

**Figure 3.** In the semiclassical regime when the frequency ωk(t) is changing adiabatically, nk is a constant which coincides with an adiabatic invariant. To appreciate the usefulness of the introduction of the adiabatic invariant nk, one should compare the evolution of the modes χk with the evolution of the occupation numbers corresponding to each of these modes shown in Figs. 2 and 3. As we see, in the narrow resonance regime χk vigorously oscillates, whereas ln nk grows like a straight line. In the broad resonance regime the field amplification occurs near the points φ(t) = 0 where the proce

**Figure 4.** FIG. 4. Early stages of parametric resonance in the the

**Figure 5.** FIG. 5. The same process as in Fig. 4 during a longer

**Figure 7.** FIG. 7. The structure of the resonance bands for the

**Figure 8.** FIG. 8. The change of the comoving particle number

**Figure 9.** FIG. 9. The characteristic exponent

**Figure 10.** FIG. 10. The characteristic exponent

**Figure 12.** FIG. 12. Two diagrams for the polarization operator of

Showing 10 of the paper's 12 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9704452 by the authors.

References

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diﬀers slightly from ours, so it is better to compare our results for a given g rather than for a given q0. In particular, one should compare their results for q0 = 10 6 with our results for g = 10 −2: √⟨χ2⟩r ≈ 3 × 1015 GeV. This result agrees, to within a factor of 2, with the results of the lattice simulation of [30]. One should not overemphasize the si...

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In fact, the table in Sec. VI might suggest that the chang e of parameters by 10% can lead to exponentially large changes in the ﬁnal results. This is indeed the case for the processes if g < 3 × 10−4, where backreaction is not important. Meanwhile, for g ≫ 3 × 10−4 backreaction makes the process terminate at a time which is somewhat less sensitive to the...

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See all 260 Pith citations

How later papers used it

755 later papers cite it

Evidence: substantial. 748 of the citing papers share no author with it, and 110 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 1,465 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

11 later papers by other authors are listed below: 11 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 12 of 76 papers with this label · 1 by the same authors

Solved its equation numerically using the leading order adiabatic expansion with modes expressed as this paper's Ansatz.

Despicable Dark Relics: generated by gravity with unconstrained masses (2018) · 1.00 · paraphrased: the licence does not allow quoting
In order to find $a^3 n_k$ we will adopt the parametrization of Ref. [ref] ,

On the Role of Cosmological Gravitational Particle Production in Baryogenesis (2024) · CGPP for spin-0 particles · 1.00
Used this paper's method of successive scattering matrices from preheating scenarios to find an analytic expression for the number of dark photons.

Analytic study of dark photon and gravitational wave production from axion (2020) · 1.00 · paraphrased: the licence does not allow quoting
Returning to our discussion, in order to compute $⟨χ^2 ⟩$, we write the solution of equation [ref] in terms of Bogoliubov coefficients $α_k = α_k(t)$ and $β_k = β_k(t)$ [ref] , yielding

Preheated inflation (2025) · Background dynamics · 0.98
Following this paper, wrote down the equation of motion of the field.

Hidden Photon and Axion Dark Matter from Symmetry Breaking (2021) · 0.98 · paraphrased: the licence does not allow quoting
Following this paper, gave the occupation number in a given mode function.

Gravitational-Wave Mediated Preheating (2014) · 0.98 · paraphrased: the licence does not allow quoting
Following this paper's standard procedure, obtained the equation of motion of the curvaton field.

The running curvaton (2019) · 0.98 · paraphrased: the licence does not allow quoting
Computed the occupation number of each comoving momentum mode as given by this paper to validate its analytical estimate.

Shrinking the Warm Little Inflaton (2023) · 0.98 · paraphrased: the licence does not allow quoting
Based its appendix on ideas and methods found in Section VII B of this paper.

Irruption of massive particle species during inflation (2014) · 0.98 · paraphrased: the licence does not allow quoting
Following this paper, wrote an expression accounting for unknown accumulated phases of the Bogoliubov coefficients.

Preheating in the Standard Model with the Higgs-Inflaton coupled to gravity (2008) · 0.94 · paraphrased: the licence does not allow quoting
Presented the formulation of particle production by following this paper.

Moduli Oscillation Induced by Reheating (2018) · 0.89 · paraphrased: the licence does not allow quoting

By the same authors

Used this paper's approximation to solve the equation analytically and obtain the occupation number of created particles.

Cosmological Fluctuations from Infra-Red Cascading During Inflation (2009) (same authors) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

668 arXiv papers cite this in 1,296 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 · 734 sentences without a confident label

Contests it 2 papers

This led Kofman, Linde, and Starobinsky [ref] to propose that the residual field could act as dark matter, but in fact detailed calculations [ref] show the relic abundance is far too high under standard assumptions.

Triple unification of inflation, dark matter, and dark energy using a single field · Cosmological evolution · contest 0.75
For example, the equation should include fluctuations that are present in such systems with dissipation [ref] .

Inflationary phenomenology of quadratic gravity in the Palatini formulation · Perturbative reheating · contest 0.72

Builds on it 3 papers

This formula can be extended to the case where the $ψ$ particles have a bare mass, the more general result is then [ref]

Constraints on generating the primordial curvature perturbation and non-Gaussianity from instant preheating · Instant preheating · 1.00
(See also Refs. [ref] .) Within the single-field attractor, the coupled equations of motion in Eq. ([ref]) reduce to

Preheating after Multifield Inflation with Nonminimal Couplings, I: Covariant Formalism and Attractor Behavior · Mode Expansion and Adiabatic Parameters · 0.80
Although the quantum nature of the inflaton coherent-oscillation induced GPP has been known (e.g., see [ref] ), the present paper extends the previous ideas to graviton-mediated scattering, and to our knowledge is the first to articulate clearly and to compute analytically the quantum interference effects.

Quantum interference in gravitational particle production · Introduction · 0.84

Uses its method or tool 70 papers

The final particle number density, at large $$ values, can be computed using the equation [ref] [equation] Substituting again $ω_k_*^2(t)≃ω_k=0^2(t)$ in the above expression we can factor out $ω_k$, and write [equation]

Quintessential inflation and nonlinear effects of the tachyonic trap mechanism · The Strongly Non-Linear Regime · 0.93
However, we will show that implementing the backreaction under the Hartree approximation [ref] leads to $ϕ_max$ significantly smaller than $10^-11$.

Cosmology in theories with spontaneous scalarization of neutron stars · Introduction · 0.96
Ignoring backreaction onto the trajectory while $ϕ⃗$ is close to the ESP, we can compute the number density of $χ$ particles with wave-number $k$ after the encounter, $t>t_$, to [ref] [equation] where we incorporated the subsequent dilution for $t>t_$ due to the expansion of the universe.

A Terminal Velocity on the Landscape: Particle Production near Extra Species Loci in Higher Dimensions · Particle Production near ESPs and Computation of the Terminal Velocity · 0.85

Uses its data or values 3 papers

Typical value of $⟨χ^2⟩$ in any Hubble patch is of the order of $10^-14$ at the time of the start of preheating [ref] [ During inflation $χ$ field does not get any super-horizon perturbation.

Constraints on variations in inflaton decay rate from modulated preheating · Dynamics of fields · 0.96
The total potential energy is equally distributed among the $𝒩$ fields and sums up to $m^2 φ_0^2/2$, where $φ_0 = 0.193$ [ref] , in line with single-field models.

On the Suppression of Parametric Resonance and the Viability of Tachyonic Preheating after Multi-Field Inflation · Implementation of LatticeEasy · 0.84
Using expressions for $Φ(t_r)$ from [ref] , immediately after preheating ends $m_χ_i∼√(g_i) ×10^15$.

Symmetric and Asymmetric Reheating · Rescattering and other effects · 0.91

Compares with it 20 papers

Furthermore, the relative increase resulting from each zero-crossing is consistent with that obtained using a Bogoliubov transformation [ref] .

Preheating in Palatini Higgs inflation on the lattice · Lattice results · 0.97
Following this approach, the growth rate was estimated in Ref. [ref] to be $μ̃=qm_aπ/4$, which differs from $μ̃=q m_a/2$ predicted by NPR [ref] .

Particle decay as asymptotic narrow parametric resonance · Introduction · 0.88
This however leads only to minor differences as compared to the standard case [ref] as we will discuss below.

Parametric Decay of the Curvaton · The parametric resonance · 0.80

Cites it as support 11 papers

From work in the context of inflationary reheating [ref] we know that the damping of the oscillations of $ϕ$ due to the expansion of space do not destroy the resonance.

Parametric Resonance and Backreaction Effects in Magnetogenesis from Ultralight Dark Matter · Stability of the Narrow Band Resonance · 0.98
Since the time duration of preheating is small compared to the Hubble time scale, these assumptions are valid and self-consistent during the preheating process [ref] .

Effective Action Approach for Preheating · One-loop effective action · 0.99
Preheating with such small coupling is not efficient [ref] , i.e. reheating has to proceed via perturbative inflaton decay as we analyzed in Sec. [ref].

Small Field Polynomial Inflation: Reheating, Radiative Stability and Lower Bound · Radiative Corrections and Stability · 0.98

Cites it as background 343 papers

Studies of reheating in inflationary cosmology [ref] have taught us that in a dynamical space-time only the broad resonance regime with $q>1$ can lead to a significant enhancement of the generation of the $χ$ particles and of the corresponding fluctuations on super-Hubble scales.

Preheating a bouncing universe · Broad Resonance with Weak Coupling · 0.84
Particle production can occur whenever the effective mass $m_eff$ of $χ$ becomes light [ref] , given that the coupling constant is chosen appropriately.

Preheating after Multi-field Inflation · Introduction · 0.86
Given the shape of the potential ([ref]), very different (p)reheating mechanisms could in principle take place, from a tachyonic production in the region between the end of inflation and the inflection point [ref] , an instant preheating mechanism [ref] and a parametric resonance effect around the minimum of the potential [ref] .

Preheating in the Standard Model with the Higgs-Inflaton coupled to gravity · Reheating in the Standard Model of particle physics · 0.80

Discussion

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In fact, the table in Sec. VI might suggest that the chang e of parameters by 10% can lead to exponentially large changes in the ﬁnal results. This is indeed the case for the processes if g < 3 × 10−4, where backreaction is not important. Meanwhile, for g ≫ 3 × 10−4 backreaction makes the process terminate at a time which is somewhat less sensitive to the...

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