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REVIEW 2 major objections 4 minor 57 references

Freezing-in Cannibals with Low-reheating Temperature

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Slow reheating plus 2-to-3 cannibal self-interactions can reopen freeze-in dark matter parameter space that instantaneous-reheating calculations exclude, bringing MeV-scale dark matter into HL-LHC and FCC reach.

desk verdict A solid and genuinely new combination of non-instantaneous reheating with cannibal self-interactions in freeze-in; the main caveat is the unvalidated Maxwell-Boltzmann closure that shifts contours but does not break the qualitative conclusion. read the letter →

arxiv 2506.09155 v1 pith:ODSNJWVF submitted 2025-06-10 hep-ph hep-th

Nicolás Bernal , Esau Cervantes , Kuldeep Deka , Andrzej Hryczuk This is my paper
classification hep-phhep-th PACS 95.35.+d98.80.Cq
keywords freeze-indarkmattercannibal2-to-3number-changingprocesses3-to-2cannibalizationnon-instantaneousreheatinglowtemperatureZ3scalarHiggsportal
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

Freeze-in dark matter is usually computed assuming the universe reheated instantly and that dark-sector particles barely interact with each other. This paper removes both assumptions at once: it follows a complex scalar dark matter candidate, stabilized by a $\mathbb{Z}_3$ symmetry, through a reheating phase that can end at temperatures as low as a few MeV, with $2\to 3$ and $3\to 2$ 'cannibal' self-interactions active inside the dark sector. The central claim is that these two effects reinforce each other: the prolonged temperature evolution of the SM bath changes when dark matter is produced, while the $2\to 3$ processes convert kinetic energy into extra particles and raise the final abundance. As a result, the Higgs portal coupling needed to match the observed relic density drops, and parameter regions that standard instantaneous-reheating or no-self-interaction calculations exclude become viable. A sympathetic reader should care because this puts light, MeV-scale dark matter into territory that the HL-LHC and a future FCC could test through invisible Higgs decays.

What carries the argument

The engine of the calculation is a two-moment closure of the Boltzmann equation for the dark sector. The paper assumes the DM distribution stays in a Maxwell-Boltzmann form with a single effective dark temperature, $f(E,T') = (n/n_{\rm eq})\exp(-E/T')$, so the full phase-space evolution is replaced by two ordinary differential equations: one for the comoving number density $N=a^3 n$ and one for the dark temperature $T'$, sourced by the zeroth and second moments $C_0$ and $C_2$ of the collision operator. These moments collect the Higgs-portal production terms, the elastic-scattering kinetic-equilibration terms, and the $2\leftrightarrow 3$ cannibal collision integrals, while the background is set by separate equations for the inflaton and radiation energy densities with the Hubble rate $H$ depending on both. Keeping $T'\neq T$ is what lets the calculation see the cannibal phase, where the dark sector cools as $2\to 3$ splits particles and later heats above the SM bath when $3\to 2$ annihilations enforce chemical equilibrium.

What would settle it

Recompute the benchmark $m=10\,\mathrm{MeV}$, $T_{\rm rh}=1\,\mathrm{GeV}$, $\lambda_{\rm hs}=1.8\times 10^{-7}$, $\lambda_s=0.01$ by solving the full unintegrated Boltzmann equation for $f(p,t)$ without the Maxwell-Boltzmann Ansatz; if the resulting relic abundance differs from the two-moment result by more than the factor by which the $2\to 3$ processes are claimed to boost the yield, the central claim is not quantitatively established.

Watch

Extended reading notes

Core claim

In the $\mathbb{Z}_3$-symmetric complex-scalar model with cubic self-couplings, the DM abundance is set by three competing engines: Higgs decay and SM annihilation feed particles through the Higgs portal $\lambda_{hs}$; $2\to 3$ reactions ($S^*S\to SSS$) convert kinetic energy into extra particles, boosting the yield; and $3\to 2$ reactions enforce a cannibalistic chemical equilibrium that later depletes and heats the dark sector. The paper's discovery is quantitative: solving the coupled number-density and dark-temperature equations together with an inflaton-to-radiation reheating background shows that these engines turn on and off at different scale factors depending on $T_{\rm rh}$, and that their combined effect can change the required $\lambda_{hs}$ by orders of magnitude. Concretely, regions of the $[m,\lambda_{hs}]$ plane that are excluded when reheating is taken as instantaneous or when $\lambda_s=g_s=0$ become consistent with the observed relic abundance once low reheating temperatures and $2\to 3$ cannibalism are included, especially for DM masses of order a few MeV.

Load-bearing premise

The load-bearing premise is that a single-temperature Maxwell-Boltzmann distribution with the Ansatz $f(E,T')=(n/n_{\rm eq})e^{-E/T'}$, together with the approximation $(1\pm f)\simeq 1$, captures the energy dependence of the collision integrals well enough; if the early freeze-in distribution is not thermalized to that shape, the predicted shifts in the viable parameter space are not quantitatively reliable.

Editorial extensions

If this is right

  • Lower reheating temperatures demand larger portal couplings to compensate for entropy dilution, so the required $\lambda_{hs}$ rises sharply as $T_{\rm rh}$ drops toward the BBN bound.
  • Where $2\to 3$ self-interactions are active, the final DM abundance can be boosted by more than an order of magnitude, allowing a smaller $\lambda_{hs}$ than freeze-in with negligible self-interactions.
  • For a light DM mass around $4$ MeV, a large enough self-coupling $\lambda_s$ moves the relic-density contour below the current LHC invisible-Higgs limit and into the projected HL-LHC and FCC reach.
  • In the opposite regime of large $\lambda_{hs}$ and $\lambda_s\to 1$, efficient $3\to 2$ reactions can deplete an overproduced dark sector, making the final yield nearly insensitive to $\lambda_s$.
  • The combined constraints from BBN, the Bullet cluster, LHC invisible Higgs decays, and direct detection leave a window of light DM with low $T_{\rm rh}$ that only the cannibal-plus-reheating treatment opens.

Reading between the lines

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

  • Trusting the quantitative opening of parameter space requires the single-temperature Ansatz to hold during the early dilute phase; if the true distribution from Higgs decay is narrower than the Maxwell-Boltzmann form, the $2\to 3$ boost could be over- or under-estimated, so a full phase-space solution for the benchmark cases would be the natural check.
  • The same two-moment machinery should apply to other cannibal dark sectors, such as $\mathbb{Z}_2$-stabilized models with $4\to 2$ processes, suggesting the qualitative interaction between low reheating temperatures and number-changing self-interactions is a general feature, not specific to $\mathbb{Z}_3$.
  • A sharp, cosmology-independent test of the newly opened region is the invisible Higgs branching ratio: if FCC-era measurements exclude $\lambda_{hs}$ at the values the contours require for $m\sim$ MeV and $T_{\rm rh}\sim 6$ MeV, that part of the claim would be closed experimentally.
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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 studies freeze-in production of a complex scalar dark matter candidate with a Z3 symmetry and cubic self-interactions, which allow 2-to-3 and 3-to-2 cannibal processes, in a cosmological background with non-instantaneous reheating ending at low reheating temperature T_rh. The authors solve coupled Boltzmann equations for the dark-sector number density and effective temperature, using a Maxwell-Boltzmann ansatz with a single temperature T', together with background equations for inflaton and radiation energy densities. They find that the interplay between prolonged reheating and dark-sector cannibalization can enhance the DM yield, permitting smaller Higgs portal couplings and opening new viable parameter regions for light DM (m ~ MeV) that would be excluded in the standard instantaneous-reheating or no-self-interaction treatments. The paper presents two detailed benchmarks, parameter scans in m, T_rh, lambda_hs, and lambda_s, and discusses constraints from BBN, the Bullet Cluster, Higgs invisible decays, direct detection, and future collider sensitivities.

Significance. If the central result survives scrutiny, this is a useful and timely extension of freeze-in dark matter studies. The paper's main strength is that it computes relic abundances forward from the model parameters rather than fitting them, and it explicitly tracks the non-instantaneous reheating epoch, which is often approximated away. The two-moment formalism in Section 3 is clearly laid out, and the benchmarks in Figs. 2 and 3 provide a helpful physical picture of the different dynamical regimes. The claimed new viable regions, especially the low-lambda_hs region for light DM in Fig. 6, are potentially testable at HL-LHC and FCC. The principal weakness is that the Maxwell-Boltzmann closure for the dark-sector distribution is assumed rather than validated in the early, non-thermal freeze-in phase, and this assumption directly affects the magnitude of the cannibal boost that drives the central new-viable-region claim.

major comments (2)
  1. [Section 3, Eq. (3.5)] The Maxwell-Boltzmann closure f(E,T') = (n/n_eq) exp(-E/T') is assumed throughout, including the early freeze-in phase when the dark sector is populated by h -> S* S decays and has a narrow, non-thermal spectrum. The 2<->3 collision integrals in Eqs. (4.10)-(4.11) involve products of three distribution functions, so the cannibal boost that shifts the lambda_hs contours in Figs. 4 and 6 depends on the shape of f, not only on n and T'. The only justification offered in Section 3 is the generic statement that efficient cannibal processes require efficient elastic self-scattering, which drives the distribution to equilibrium; no quantitative check is given for the epoch when the 2->3 rate first becomes comparable to H. This is load-bearing because the paper's main new claim, that self-interactions open otherwise excluded low-lambda_hs regions for light DM, relies on the size of this boost. Please validate the ansatz, for example by comparing against a full momentum-resolved solution for one benchmark in each regime (dark freeze-out after reheating and during reheating), or by testing a broader class of distribution shapes and showing that the resulting relic abundance is stable.
  2. [Section 4.2.2 and Appendix A, Eq. (4.9)] The fermion elastic scattering contribution to the second moment, Eq. (4.9), is derived under a set of relativistic approximations in Appendix A: the CM-frame replacements E_i ~ |p_i|, the approximation sqrt(z^2-1) << 3z, and the neglect of the momentum dependence of the t-channel Higgs propagator. These approximations are reasonable for the stated mass hierarchy, but the paper does not quantify their accuracy for the low-temperature regime where T' is comparable to or below the DM mass, which is precisely where kinetic decoupling occurs and where the temperature evolution matters for the final abundance. Since the abundance in the low-T_rh scenarios is sensitive to the competition between Hubble expansion and kinetic-equilibration rates, please provide an estimate of the error introduced by these approximations, or cross-check Eq. (4.9) against a direct numerical evaluation for representative parameters.
minor comments (4)
  1. [Footnote 4] Footnote 4 states that for lambda_s = 10^-2 the 2->3 interactions 'never impact the DM production significantly,' but Section 5.1 and Fig. 2 show the 2->3 rate exceeding the Hubble rate and producing a noticeable drop in the dark-sector temperature. These statements appear contradictory and should be reconciled or clarified.
  2. [Section 3, Eq. (3.11)] The expectation value <p^4/E^3> in Eq. (3.11) is not defined explicitly. Please define it as an integral against the distribution function and, if it is evaluated with the ansatz of Eq. (3.5), state the resulting closed-form expression or note that it is computed numerically.
  3. [Appendix A, Eq. (A.5)] The transformation displayed for (p_1^2/E_1)_cm in Eq. (A.5) is difficult to follow: the variable z is introduced without definition, and the expression contains a '-' sign whose origin is not explained. Please clarify the notation and the steps leading from the CM-frame expression to the final angular integral.
  4. [Section 4.2.2] The text assumes an instantaneous QCD phase transition at T_QCD = 200 MeV, but it does not describe how the effective degrees of freedom g_* and g_*s are implemented in the background equations across the transition. Please state whether tabulated temperature-dependent values are used and how the discontinuity is treated.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the relic abundance is computed forward from microphysical rates; the Maxwell-Boltzmann closure and borrowed collision integrals are assumptions and derivations, not fitted to the target.

full rationale

The paper's central claim is that non-instantaneous reheating combined with 2-to-3 cannibal self-interactions shifts and opens viable freeze-in parameter space. The derivation is forward: given the model parameters (lambda_hs, lambda_s, m, T_rh), the coupled Boltzmann equations for the DM number density and temperature are solved and the final relic abundance is compared to observation. No parameter is fitted to the observed abundance; instead, contours satisfying the relic condition are traced in parameter space. This is not circular. The main assumptions are the Maxwell-Boltzmann closure in Eq. (3.5) and the two-moment truncation, but these are explicit modeling assumptions rather than quantities defined in terms of the final result. The approximation (1 +/- f) ~ 1 is also stated and is a known simplification, not a hidden fit. The paper does rely on previous work by overlapping authors, notably Ref. [25] (Cervantes and Hryczuk) for the 2-to-3 self-interaction collision integrals and for some production and elastic-scattering rates, and Ref. [49] (Bernal) for the boost of freeze-in through thermalization. These are self-citations, but they are parameter-free derivations with stated assumptions and do not contain the target relic abundance as an input. Invoking them is standard scientific practice and does not make the present calculation circular. The unvalidated shape of the dark-sector distribution during early freeze-in, highlighted by the skeptic, is a robustness and correctness concern about the moment closure, not a circularity: even if the Ansatz were wrong, the paper would be inaccurate rather than tautological. The new viable regions, such as the m = 4 MeV, T_rh = 6 MeV contour in Fig. 6, emerge from the dynamics of the solved equations rather than from a self-imposed normalization. Thus no step reduces, by construction or by definition, to its own inputs.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities; it scans known model parameters. Quantitative results depend on the modeling assumptions listed, most critically the Maxwell-Boltzmann closure and the perturbative reheating background.

free parameters (5)
  • Higgs portal coupling λ_hs = scanned (e.g. 1.8×10^-7 for m=10 MeV, T_rh=1 GeV)
    Central coupling controlling DM production; scanned to match relic density across benchmarks and contours.
  • Quartic self-coupling λ_s = scanned (e.g. 10^-2 in benchmarks, up to O(1))
    Controls cannibal 2↔3 and elastic self-interaction rates; chosen by hand across values.
  • Trilinear self-coupling g_s = fixed via k = g_s^2/(3 λ_s μ_s^2) = 2
    Not scanned independently; ratio k=2 chosen to make 3↔2 dominate over 4↔2 and to reduce parameter space.
  • Reheating temperature T_rh = scanned from ~4 MeV to 150 GeV
    Defines the non-instantaneous reheating history; central to the paper's new scenarios.
  • Dark matter mass m = scanned, approx 1 MeV to tens of GeV
    Mass of complex scalar S; selection of light masses is the focus of new viable regions.
assumptions (5)
  • domain assumption The reheating period is described by perturbative inflaton decay with 100% branching to SM states and immediate thermalization (Eqs. 2.1, 2.2).
    Background used for all DM production; non-perturbative preheating or delayed thermalization would change the SM bath temperature history.
  • domain assumption SM and DM distributions are Maxwell-Boltzmann and quantum statistics factors (1±f) are set to 1 (Section 3, Eq. 3.5 and footnote 2).
    Enables the two-moment closure; not valid for very dilute, non-thermal DM and for relativistic species.
  • domain assumption The Z3-symmetric complex scalar with potential Eq. (4.2) is the full dark sector, with k=2 ensuring 3↔2 dominance over 4↔2.
    Model choice inherited from Refs [25,43,44]; the results are specific to this setup.
  • domain assumption Initial condition is set at a_EWPT with T=150 GeV, with inflationary scale ensuring T_max >> 150 GeV so all SM species remain in equilibrium; production before EWPT is neglected.
    Neglect of earlier production is stated to be small, but not quantified in this paper.
  • domain assumption QCD phase transition is instantaneous at T_QCD = 200 MeV, with hadronic production treated via a Breit-Wigner using Γ_hadrons from Ref. [54].
    Affects the C0 and C2 rates for SM annihilations in the 100 MeV to 200 MeV range.

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Cite this review

Pith. "Pith review of Freezing-in Cannibals with Low-reheating Temperature." pith.science (2026). https://pith.science/paper/ODSNJWVF

@misc{pith2026250609155,
  author       = {Pith},
  title        = {Pith review of: Freezing-in Cannibals with Low-reheating Temperature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ODSNJWVF}},
  note         = {Machine review of arXiv:2506.09155}
}
abstract

The freeze-in mechanism provides a compelling framework for dark matter (DM) production, particularly suited to scenarios involving feeble interactions with the Standard Model (SM). In this work, we highlight a possible interplay of a non-instantaneous reheating phase and dark sector self-interactions, specifically $2 \to 3$ and $3 \to 2$ cannibalization processes. As an example we study the freeze-in production of a complex scalar DM candidate stabilized by a $\mathbb{Z}_3$ symmetry permitting cubic self-couplings, enabling number-changing interactions that drive internal thermalization and significantly modify the dark sector number density and temperature evolution. We numerically solve the coupled Boltzmann equations for the DM number density and temperature alongside the evolving SM bath, accurately capturing the dynamics of a prolonged reheating epoch. Our analysis reveals a rich and distinctive phenomenology arising from the interplay between the Universe's thermal history, Higgs portal mediated production, and cannibalistic self-interactions. Compared to scenarios with instantaneous reheating or negligible self-interactions, our framework opens new viable regions in parameter space, particularly for light DM, potentially within reach of future probes.

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