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Aspects of Gravitational Portals and Freeze-in during Reheating

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read During reheating, freeze-in from the gravitationally produced radiation bath can dominate over the conventional decay bath and alone set the dark matter relic density when $m_{\chi}>T_{\rm RH}$.

desk verdict A well-motivated extension of freeze-in during reheating with a real error in Eq. (16) that undermines the printed numbers; worth refereeing after a simple fix. read the letter →

arxiv 2412.13288 v1 pith:NHVAROHT submitted 2024-12-17 hep-ph astro-ph.CO

Stephen E. Henrich , Yann Mambrini , Keith A. Olive This is my paper · ORCID
classification hep-phastro-ph.CO
keywords gravitationalportalfreeze-inreheatingdarkmatterrelicdensityinflatoncondensateradiationbathT-modelinflationBSMscale
topics Dark Matter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks how dark matter is produced in the first moments after inflation, while the inflaton condensate is still dominating the universe and reheating it. It shows that three production routes must be tracked together: direct gravitational scattering of the inflaton, freeze-in from the radiation bath produced by inflaton decay, and freeze-in from a second, often neglected radiation bath that the inflaton itself creates through gravitational scattering. The central result is that this gravitational bath can be the dominant source of freeze-in dark matter and can by itself account for the observed relic density, provided the dark matter mass exceeds the reheating temperature and certain $k$- and $n$-dependent conditions are met. The paper also derives, for each freeze-in interaction, an upper limit on the new-physics scale $\Lambda$ above which direct gravitational production from the inflaton beats thermal freeze-in. A complete calculation of the dark matter relic abundance during reheating therefore has to include the gravitational bath, not just the decay bath.

What carries the argument

The central object is the gravitational bath: Standard Model radiation produced by graviton exchange from the oscillating inflaton condensate, $\phi\phi\to h_{\mu\nu}\to$ SM fields, whose energy density redshifts as $\rho^h_R\propto a^{-4}$ and peaks at a temperature $T^h_{\max}\simeq 1.0\times10^{12}$ GeV for $k=2$. It is compared with the decay bath produced by inflaton decay. The argument is carried by closed-form Boltzmann-equation solutions parameterized by $k$, the power of the inflaton potential about its minimum, and $n$, the temperature power of the freeze-in rate $R_{\chi}\propto T^{n+6}/\Lambda^{n+2}$, together with the crossing temperature $T_{\times}$ at which the two bath densities are equal; for $m_{\chi}>T_{\times}$, freeze-in is fed by the gravitational bath even for interactions whose production is otherwise dominated by late, low-temperature times. The analytic results hinge on critical values $n_c^h=-6/(k+2)$ and $n_c^y=(10-2k)/(k-1)$ that separate production dominated by early, high-temperature times from production dominated by late, low-temperature times.

What would settle it

Identify a stable scalar dark matter candidate in a T-model reheating scenario with $k=4$ and $m_{\chi}\gtrsim128$ GeV: the paper predicts direct gravitational inflaton scattering alone overproduces such particles, so a confirmed stable scalar at that mass would contradict the framework. Alternatively, in a parameter region where the gravitational bath should dominate, for example $m_{\chi}>T_{\rm RH}$ with $\Lambda\sim10^{10}$-$10^{12}$ GeV, a precision measurement of the dark matter relic density that matches only the decay-bath prediction would falsify the claim that the gravitational bath is a required third production source.

Watch

Extended reading notes

Core claim

The paper's central claim is that a complete account of dark matter production during reheating must include three sources, and that the newly emphasized one, freeze-in from the gravitational bath, can dominate. For a general inflaton potential $V(\phi)\propto\phi^k$ about its minimum and a general freeze-in rate $R_{\chi}\propto T^{n+6}/\Lambda^{n+2}$, the paper derives analytic relic densities for all three sources for arbitrary $k$ and $n$ and compares them. It finds that when $m_{\chi}>T_{\rm RH}$, freeze-in from the gravitational bath can exceed the decay-bath contribution and saturate $\Omega_{\chi}h^2=0.12$, subject to $k$- and $n$-dependent constraints; for $k=4$ and $n=2$, for instance, the gravitational bath provides the dominant source at reheating temperatures $T_{\rm RH}\lesssim 60$ GeV. Conversely, for each interaction there is an $m_{\chi}$- and $T_{\rm RH}$-dependent upper bound on $\Lambda$ above which direct gravitational production from the inflaton condensate exceeds ordinary freeze-in, so freeze-in studies using high BSM scales must be re-examined.

Load-bearing premise

The calculation assumes that the gravitationally produced Standard Model radiation thermalizes instantaneously into a bath with a well-defined temperature $T_h$, so that freeze-in rates written as functions of temperature apply throughout reheating; if thermalization is delayed or incomplete, the early production history and all comparisons between the gravitational and decay baths change.

Editorial extensions

If this is right

  • A complete freeze-in relic calculation during reheating must include the gravitational bath as a third source; for $m_{\chi}>T_{\rm RH}$ it can dominate and alone give $\Omega_{\chi}h^2=0.12$ under $k$- and $n$-dependent conditions.
  • For every freeze-in interaction considered, there is an $m_{\chi}$- and $T_{\rm RH}$-dependent upper limit on the BSM scale $\Lambda$ above which direct gravitational production from the inflaton exceeds thermal freeze-in, so freeze-in studies with high $\Lambda$ must include that source.
  • For $k=4$, stable scalar dark matter heavier than about 128 GeV, and fermionic dark matter above about $1.8\times10^9$ GeV, are overproduced by direct gravitational inflaton scattering regardless of reheating temperature.
  • When $\Lambda<T^h_{\max}$, the effective interaction index $n$ jumps by 4 once the bath cools below $\Lambda$; for the gravitational bath this makes production peak near $T\sim\Lambda$ rather than at $T_{\max}$ or $T_{\rm RH}$, a regime that is unique to the gravitational bath.
  • For $k=6$, freeze-in from the gravitational bath never exceeds the decay bath for any allowed reheating temperature above the big-bang nucleosynthesis limit of about 4 MeV, so whether the gravitational bath matters at all is controlled by the inflaton potential's $k$.

Reading between the lines

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

  • The same gravitational bath should source any very weakly coupled relic produced at temperatures near $T^h_{\max}\sim10^{12}$ GeV, including gravitinos, axions, or a baryon asymmetry; those channels would inherit the same competition between direct and bath production.
  • The derived upper limits on $\Lambda$ imply that freeze-in parameter scans assuming $\Lambda\gtrsim10^{10}$ GeV should be rerun with direct gravitational production included, and experimental bounds on mediator masses may need to be reinterpreted as bounds on where freeze-in applies at all.
  • If thermalization of the gravitational bath is not instantaneous, the effective crossing temperature $T_{\times}$ and the $m_{\chi}>T_{\rm RH}$ dominance region would shift; a dedicated treatment of delayed thermalization could sharpen or relax the conclusions for $k=2$ and $k=4$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper analyzes dark-matter production during reheating from three sources: direct gravitational scattering of the inflaton condensate, freeze-in from the radiation bath produced by inflaton decay, and freeze-in from the radiation bath produced by gravitational scattering of the inflaton. The freeze-in rates are parameterized as R_chi proportional to T^{n+6}/Lambda^{n+2}, and the inflaton potential near its minimum is taken as V(phi) proportional to phi^k. The authors derive analytic relic-density expressions for each source and for the regimes m_chi < T_RH, m_chi > T_RH, and Lambda below the maximal bath temperature, and they compare the three production mechanisms in the (m_chi, T_RH) plane for k = 2, 4, 6 and several values of n. The central claims are that freeze-in from the gravitationally produced bath can exceed that from the conventional decay bath for m_chi > T_RH, and that for each interaction there is a BSM scale Lambda above which direct gravitational production from the inflaton dominates thermal freeze-in.

Significance. If the quantitative results are correct, the paper makes a useful and timely contribution: it identifies an often-neglected third source of FIMP dark matter during reheating and shows that it can be the dominant source in a non-negligible part of parameter space. The analytic framework is a strength: the appendix collects the n- and k-dependent relic-density formulas, and the structure of the Boltzmann-equation solutions is transparent enough for independent checks. The predicted Omega_chi h^2 contours and the Lambda upper limits are falsifiable and directly relevant for model-building. The main caveat is that the numerical realization of the headline claim currently rests on an internally inconsistent set of formulas, so the quantitative support must be re-established after correction.

major comments (2)
  1. [Section II.A, Eq. (16)] The printed expression for a_hmax/a_end is the reciprocal of the value implied by Eq. (15). Setting d ln rho_R^h / d ln a = 0 in Eq. (15) gives a_hmax/a_end = [(6k-3)/(2k+4)]^{(k+2)/(8k-14)}, whereas Eq. (16) prints the inverse base. For k = 2 this gives 0.79 instead of 1.27, so the printed peak occurs before a_end even though Eq. (15) vanishes at a_end and rises afterwards. Because Eq. (49) explicitly instructs the reader to substitute Eq. (16) into the scale-factor ratios, the printed forms of Eqs. (46), (53), (54), and the k = 2 result (47) inherit a suppression factor [(8/9)/(9/8)]^{9/2} ≈ 1/13 for k = 2. At the same time, Eqs. (17), (57), and (A.3) already use the correct ratio, so the paper is internally inconsistent rather than uniformly mis-scaled. The authors should correct Eq. (16), re-derive every expression that substitutes it, and re-evaluate the affected figures and parameter-space statements.
  2. [Section III, numerical implementation] Because the corrected and printed forms of the gravitational-bath density differ by an order of magnitude in the k = 2 regime, the numerical results in Section III need to be recomputed or explicitly mapped to the corrected expressions. For example, Eq. (47) is the k = 2 gravitational-bath relic density for n > n_h^c; if evaluated using Eq. (49) with the printed Eq. (16), the resulting Omega_h values are suppressed by about a factor of 13. However, Eq. (57) already contains (9/8)^9, which corresponds to the corrected peak ratio. It is therefore unclear which expressions were used to generate each panel in Figs. 10-12 and 15-19. The authors should state the formula used for each figure and, where Eq. (16) was used, correct the curves and the associated discussion of gravitational-bath dominance and upper limits on Lambda.
minor comments (4)
  1. [Section II.A] The instantaneous-thermalization assumption is stated in the paragraph on the decay bath, but the gravitational bath is also treated as a thermal bath with temperature T_h in the freeze-in rate R_chi(T); please state explicitly that the same assumption is made for the gravitational bath and comment on its regime of validity.
  2. [Figure captions] In Figs. 2 and 3, the quantities y_max and T_RH^max are used in the captions but are only defined in the surrounding text; please add definitions in the captions for readability.
  3. [Appendix A.12] The heading of Eq. (A.12) says "For the decay bath with T_h^max > Lambda > m_chi > T_RH," but the decay bath's maximum temperature is T_y^max; the notation should be corrected.
  4. [Equation (52)] The bracket in the second line of Eq. (52) is missing a closing parenthesis; please check the parentheses and the displayed formula.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the freeze-in comparisons are derived from published, parameter-free bath solutions, and the observed relic density enters only as a final normalization target.

full rationale

The central derivation is self-contained in the sense required by the circularity test. The relic density for each source is obtained by integrating the Boltzmann equation (31) with explicit production rates and the inflaton-dominated Hubble rate (32); the observed relic density Ωh² = 0.12 is introduced only after the integrals are evaluated, through the conversion factor in Eq. (39), so it is a target rather than an input to the production calculation. The gravitational-bath density quoted in Eq. (15) and the decay-bath density in Eq. (19) are taken from Refs. [31] and [9,10], respectively, but they are re-displayed in the text as explicit analytic solutions of the Boltzmann equations (10) with stated model assumptions; they are parameter-free given the potential and coupling inputs and do not encode the paper's relic-density result. The freeze-in rate Rχ ∝ T^{n+6}/Λ^{n+2} is an explicitly stated parametrization of interaction types, not a function fitted to the relic density. The derived Λ upper limits and the T× crossing comparisons are equalities between independently obtained density expressions, not fitted parameters renamed as predictions. Self-citations are numerous and some are load-bearing as inputs, but they are published, externally checkable calculations whose assumptions do not include the target abundance; under the reviewing rules they therefore count as real evidence and do not raise the circularity score. The stated limitations of the paper, such as instantaneous thermalization and the restriction to k = 2, 4, 6, are physical assumptions and scope choices, not circular steps. The arithmetic consistency concern about Eq. (16) raised in the skeptic note is a quantitative correctness issue, not a circularity issue, and does not alter this verdict.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The derivation is otherwise self-contained given standard cosmology. The gravitational-bath production rate and N = 4 scalar content are imported from Ref. [31]; the freeze-in rate parameterization is assumed; CMB normalization sets λ. No new particles, mediators, forces, or conserved quantities are introduced.

free parameters (3)
  • λ (inflaton potential normalization) = 2.5e-11 (k=2), 3.3e-12 (k=4), 4.6e-13 (k=6)
    Fixed from the CMB amplitude As via Eq. (9); sets ρend, mϕ, and T_hmax. Input from Planck, not derived here.
  • y (inflaton decay coupling) = 1e-7 base value; TRH varied independently in most constraints
    Sets Γϕ and the decay-bath temperature; the paper scans TRH rather than fitting y.
  • Scan axes: TRH, mχ, Λ, n, k, σ
    These are free model inputs over which relic-density contours are drawn; they are not fitted to make the derivation work.
assumptions (6)
  • domain assumption Hubble expansion during DM production is dominated by the oscillating inflaton condensate (Eq. 32).
    Used to integrate the Boltzmann equation in Eq. (31); breaks down if radiation or dark matter backreaction dominates before reheating.
  • domain assumption The gravitationally produced Standard Model radiation thermalizes instantaneously into a bath with temperature T_h.
    Stated in Section II A; without it the gravitational-bath freeze-in rates Rχ(T) are not well defined.
  • domain assumption The DM-SM interaction rate is exactly Rχ ∝ T^{n+6}/Λ^{n+2} with constant n over the integration range, switching sharply at T = Λ in Section II E.
    Parameterization in Eq. (44) and matching in Section II E; neglects mediator widths, threshold factors, and resonant production.
  • domain assumption Gravitational production of the SM bath involves only N = 4 real scalars; conformal invariance forbids massless fermions and gauge bosons.
    Used in Eq. (10) from Ref. [31]; if other SM species are produced, T_hmax and the bath density change.
  • domain assumption Inflaton decay products thermalize instantaneously and the decay bath follows Eqs. (19) and (5).
    Standard assumption cited to Refs. [53-62]; affects the decay-bath comparison.
  • standard math Freeze-in Boltzmann equations with no DM backreaction and no dark matter annihilation (Eqs. 29-33).
    Standard freeze-in formalism; valid when the DM-SM interaction is too weak for thermal equilibrium.

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Cite this review

Pith. "Pith review of Aspects of Gravitational Portals and Freeze-in during Reheating." pith.science (2026). https://pith.science/paper/NHVAROHT

@misc{pith2026241213288,
  author       = {Pith},
  title        = {Pith review of: Aspects of Gravitational Portals and Freeze-in during Reheating},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NHVAROHT}},
  note         = {Machine review of arXiv:2412.13288}
}
abstract

We conduct a systematic investigation of freeze-in during reheating while taking care to include both direct and indirect production of dark matter (DM) via gravitational portals and inflaton decay. Direct production of DM can occur via gravitational scattering of the inflaton, while indirect production occurs through scattering in the Standard Model radiation bath. We consider two main contributions to the radiation bath during reheating. The first, which may dominate at the onset of the reheating process, is produced via gravitational scattering of the inflaton. The second (and more standard contribution) comes from inflaton decay. We consider a broad class of DM production rates parameterized as $R_{\chi} \propto T^{n+6}/\Lambda^{n+2}$, and inflaton potentials with a power-law form $V(\phi) \propto \phi^{k}$ about the minimum. We find the relic density produced by freeze-in for each contribution to the Standard Model bath for arbitrary $k$ and $n$, and compare these with the DM density produced gravitationally by inflaton scattering. We find that freeze-in production from the gravitationally-produced radiation bath can exceed that of the conventional decay bath and account for the observed relic density provided that $m_{\chi} > T_{\rm RH}$, with additional $k$- and $n$-dependent constraints. For each freeze-in interaction considered, we also find $m_{\chi}$- and $T_{\rm RH}$-dependent limits on the BSM scale, $\Lambda$, for which gravitational production will exceed ordinary freeze-in production.

Figures

Figures reproduced from arXiv: 2412.13288 by the authors.

Figure 1
Figure 1. Since ρ h R ∝ a −4 , and ρ y R ∝ a − 6k−6 k+2 eventually the decay bath begins to dominate. On the other hand as discussed in more detail below, for k > 7, as both bath redshift as a −4 , reheating is completely determined by ρ h R if T y max < T h max. FIG. 2. The value of the decay coupling ymax, such that the peak value of the the energy densities of the two radiation baths are equal, for three values of k = 2, 4… view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Contributions of each of the three freeze-in sources [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: We notice that a region is still excluded because of an overabundance due to gravitational scattering of the inflaton, above the solid line, for larger masses mχ than found in the scalar case. The limit follows TRH ∝ m−3 χ , as expected from Eq. (43). All points below …
Figure 7
Figure 7. Figure 7: FIG. 7. As in Fig [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Relic density summary plot for dark matter produc [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Contributions to the (fermionic) dark matter relic [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 13
Figure 13. Figure 13: FIG. 13. As in Fig [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 12
Figure 12. Figure 12: FIG. 12. A zoom of the boxed region in Fig. [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Contributions to the (fermionic) dark matter relic [PITH_FULL_IMAGE:figures/full_fig_p016_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Relic density summary plot for fermionic dark mat [PITH_FULL_IMAGE:figures/full_fig_p016_16.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Contributions to the (fermionic) dark matter relic [PITH_FULL_IMAGE:figures/full_fig_p016_18.png]
Figure 20
Figure 20. Figure 20: FIG. 20. DM relic density summary plot for [PITH_FULL_IMAGE:figures/full_fig_p017_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. DM Relic density summary plot for [PITH_FULL_IMAGE:figures/full_fig_p018_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. As in Fig [PITH_FULL_IMAGE:figures/full_fig_p018_22.png]

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Forward citations

Cited by 2 Pith papers

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  1. Seesaw reheating

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.