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Ultra-Relativistic Freeze-Out During Reheating

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Ultra-relativistic freeze-out during reheating can produce the observed dark matter abundance across thirteen orders of magnitude in mass while keeping the relic cold enough for structure formation.

desk verdict Solid new UFO mechanism, but the light-DM coldness claim assumes away the kinetic decoupling epoch — the sub-eV boundary needs a real calculation. read the letter →

arxiv 2505.04703 v1 pith:TORUBSIO submitted 2025-05-07 hep-ph astro-ph.CO

Stephen E. Henrich , Mathieu Gross , Yann Mambrini , Keith A. Olive This is my paper · ORCID
classification hep-phastro-ph.CO
keywords ultra-relativisticfreeze-outreheatingdarkmatterrelicdensityfreeze-ininflatondecaycoldLyman-alphaconstraintheavymediator
topics Dark Matter
open problems 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

Most production mechanisms either assume dark matter is cold at decoupling (WIMP) or barely interacts (FIMP), and relativistic freeze-out is normally rejected because it gives hot eV-scale relics. The paper argues that if decoupling happens during reheating, this rejection fails: the inflaton keeps pouring radiation into the bath after dark matter freezes out, diluting the dark matter relative to the radiation and letting it cool while the bath is replenished. The result is that ultra-relativistic freeze-out during reheating can match the observed relic density over a huge parameter space, with dark matter masses from $10^{-7}$ GeV to $10^{6}$ GeV and interaction scales from $10^{3}$ to $10^{14}$ GeV, while remaining cold by structure formation. Because the couplings sit between those of WIMPs and FIMPs, the mechanism is also a natural target for detection experiments.

What carries the argument

The machinery is the Boltzmann equation for the comoving number density $Y_\chi = n_\chi a^3$ after decoupling, $dY_\chi/da = a^2\langle\sigma v\rangle n_{\rm eq}^2/H(a)$, integrated from the freeze-out scale $a_{\rm FO}$ to $a_{\rm RH}$ during reheating. The temperature evolution is controlled by the inflaton potential exponent $k$, giving $T \propto a^{-3(k-1)/(2(k+2))}$ and $H(T) \propto T^{2k/(k-1)}$ for $k<7$; this makes the source term grow with $a$ even after $\Gamma < H$, so dark matter production continues in freeze-in fashion while the bath is replenished by inflaton decay. Critical exponents $n_c$ and $n_*$ separate whether UFO is possible and whether the post-freeze-out production is UV- or IR-dominated, and the analytic relic formulas for $k=2,n=2$ and $k=4,n=2$ are the paper's concrete outputs.

What would settle it

Compute the elastic scattering rate $\Gamma_{\rm el} = n_R\langle\sigma v\rangle$ below $T_{\rm FO}$ for the $k=2,n=2$ benchmark and evolve the dark matter momentum distribution with kinetic decoupling treated separately; if $\Gamma_{\rm el}$ remains above $H$ until $T \approx T_{\rm FO}/100$, the Lyman-$\alpha$ mass bound rises well above the quoted 5 keV and the sub-eV dark matter benchmark is excluded.

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Extended reading notes

Core claim

The paper's central claim is that dark matter which reaches thermal equilibrium during reheating and freezes out while still relativistic—ultra-relativistic freeze-out, or UFO—is a complete, self-consistent production mechanism. After freeze-out the annihilation term drops out, but the production term $n_{\rm eq}^2\langle\sigma v\rangle$ keeps feeding dark matter; simultaneously the inflaton continues to decay, increasing the comoving radiation density. The net effect is a relative dilution $n_\chi/n_R \propto (T_{\rm FO}/T_{\rm RH})^3$ for UV-dominated cases, which breaks the classical relation between dark matter mass and relic density and lets masses far above the eV scale satisfy $\Omega_\chi h^2=0.12$. The paper derives the conditions under which UFO is possible, the freeze-out temperature, and the resulting relic abundance, and it shows that the dark matter is cold enough at structure formation to dodge Lyman-$\alpha$ and $N_{\rm eff}$ constraints, distinguishing UV-UFO, whose abundance is sensitive to $T_{\rm FO}$, from IR-UFO, whose abundance is sensitive to $m_\chi$ and $T_{\rm RH}$ but not to $T_{\rm FO}$.

Load-bearing premise

The calculation lets dark matter free-stream from the instant of chemical freeze-out, assuming kinetic decoupling happens at the same temperature; if elastic scattering with the bath continues below $T_{\rm FO}$, the relic is warmer and the paper's coldness and $N_{\rm eff}$ conclusions tighten.

Editorial extensions

If this is right

  • The standard ceiling $m_\chi \lesssim 100$ eV for relativistic freeze-out is replaced by a much wider band: with reheating-era dilution, masses from $10^{-7}$ GeV to $10^6$ GeV can give the observed relic density.
  • Heavy-vector or heavy-scalar mediator models with intermediate couplings will generically pass through WIMP-like freeze-out, UFO, and freeze-in as $\Lambda$ and $T_{\rm RH}$ vary, so UFO is not a separate model class but a contiguous regime.
  • UFO couplings lie between typical WIMP and FIMP values, giving the dark sector a realistic chance of being seen in direct and indirect searches, unlike FIMPs.
  • UV-UFO and IR-UFO relics have different fingerprints: a UV abundance depends on the freeze-out temperature, while an IR abundance depends on $m_\chi$ and $T_{\rm RH}$, so measuring the dark matter mass and interaction scale could identify which regime operated.
  • Because dark matter equilibrates before freeze-out, the final abundance does not depend on its production history during or after inflation, removing the initial-condition problem of freeze-in.

Reading between the lines

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

  • Beyond the paper: if kinetic decoupling is delayed below the chemical freeze-out temperature, the dark matter temperature evolution used in Sec. IV C underestimates the velocity dispersion; treating elastic scattering separately would tighten the Lyman-$\alpha$ bounds and may remove the paper's sub-eV examples.
  • Beyond the paper: the same dilution logic should apply to semi-relativistic freeze-out, so the boundary between UFO and WIMP-like freeze-out is likely a smooth ramp rather than a sharp line; the paper's analytic boundaries could be tested with full numerical Boltzmann solvers.
  • Beyond the paper: the model-independent parametrization $\langle\sigma v\rangle = T^n/\Lambda^{n+2}$ suggests UFO should be implemented in concrete portal models and checked against future cosmological surveys of the small-scale matter power spectrum, which are more sensitive than the simple 5 keV velocity criterion.
  • Beyond the paper: a late enough freeze-out would leave a small thermal velocity in the dark matter today; the absence of any cutoff in the small-scale power spectrum would push $T_{\rm FO}/T_{\rm RH}$ to large values and favor the UV-dominated regime.
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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 / 5 minor

Summary. The paper systematically studies ultra-relativistic freeze-out (UFO) of dark matter during the inflaton-dominated reheating epoch. It derives the conditions for UFO, the freeze-out temperature (Eqs. 35, 37), constraints on the reheating temperature, and the resulting relic abundance from the Boltzmann equation with post-freeze-out production (Secs. III and IV). The authors distinguish UV-dominated from IR-dominated out-of-equilibrium production and show that the continued production of radiation during reheating dilutes the dark matter relative to the bath, relaxing the classic eV-mass bound. They quote a broad allowed parameter region: m_chi from about 10^-7 GeV to 10^6 GeV, T_RH from 10^-2 to 10^15 GeV, and Lambda from 10^3 to 10^14 GeV. They also analyze the dark-matter temperature at structure formation and the Neff constraint (Sec. IV C) to argue that the relic can be cold.

Significance. If the result holds, the paper opens a substantial new region of dark-matter parameter space, bridging the WIMP and FIMP paradigms. The analytic derivations are detailed and are compared with numerical Boltzmann solutions in representative cases (Figs. 4 and 7); the authors explicitly acknowledge the O(30) deviation in the m_chi > T_RH analytic estimate. The UV/IR distinction is a useful organizing principle. However, the coldness analysis in Sec. IV C relies on an implicit identification of chemical freeze-out with kinetic decoupling that is not justified and affects the advertised light-cold-DM boundary. The higher-mass UFO windows (above roughly a keV) are not affected by this flaw, so the central mechanism remains viable, but the abstract's mass range and the 'sub-eV cold DM' example require revision after a proper treatment of kinetic decoupling.

major comments (2)
  1. [Sec. IV C 1, Eq. (83)] The derivation of the DM temperature at structure formation assumes that the DM free-streams from the chemical freeze-out temperature T_FO onward. This is not justified, because the elastic scattering rate of DM with the SM bath, n_R <sigma v>, exceeds the Hubble rate at T_FO by about (g_R/g_chi)(3k/(k+2)), which is roughly 160 for g_R=106.75, g_chi=1, and k=2. For k=2 during reheating H ~ T^4 while the elastic rate ~ T^5, so kinetic decoupling occurs at T_kd ~ T_FO/160 rather than at T_FO. Between T_FO and T_kd the DM momentum distribution is held at the bath temperature, which cools as a^{-3/8} (for k=2) rather than a^{-1}; at later times the momentum is larger than the free-streaming estimate by (T_FO/T_kd)^{5/3}. For the quoted benchmark T_FO=10^5 GeV and T_RH=100 GeV in Sec. IV C 2, the Lyman-alpha bound in Eq. (87) rises from about 50 eV to roughly 0.2 keV, so the statement that 'even sub-eV DM particles would be cold' is not supported as written. The abstract's lower boundary m_chi ~ 10^-7 GeV ~ 100 eV is therefore at least partially compromised. The authors should compute T_kd from n_R <sigma v> = H and evolve the DM temperature from that point, or restrict the coldness claim to masses above roughly the keV scale. The higher-mass windows are likely unaffected, but this is a load-bearing correction for the paper's light-cold-DM claim.
  2. [Sec. IV C 1, Eq. (83)] The same assumption of simultaneous chemical and kinetic decoupling enters the Neff bound. If kinetic decoupling occurs at T_kd < T_FO, the DM temperature at BBN is larger than the free-streamed value used in Eq. (83), because the DM tracks the bath temperature during the intervening reheating period. The resulting Delta Neff can exceed the quoted bound in parts of parameter space where T_RH is not much smaller than T_kd, so the statement that the Neff constraint is 'always satisfied' in Sec. IV C 1 needs to be re-examined after the kinetic-decoupling calculation is included. This is a direct consequence of the same premise, but it is separately relevant for the BBN-compatibility claim.
minor comments (5)
  1. [Sec. IV C 2, Eq. (87)] The numerical coefficient 5 keV in Eq. (87) depends on the specific Lyman-alpha bound v_chi < 2x10^-4 at T ~ 1 eV and on the chosen degrees of freedom. The authors should specify the conversion and the assumed g_* values so the reader can track the provenance of this coefficient.
  2. [Sec. IV A 3, Eq. (68) and Fig. 7] The analytic estimate for the m_chi > T_RH regime is a factor of ~30 below the numerical solution, as the authors note. Because the abstract and the parameter-space figures rely on these analytic formulas, the text should explicitly state that the analytic results in this regime are approximate at the O(30) level, even if the origin of the discrepancy is understood.
  3. [Sec. IV A 2, Eq. (58)] The quantity n_eq^chi(a_RH) in Eq. (58) is not defined in the text; it would be clearer to write n_eq(a_RH) and define the convention (the equilibrium number density of a massless scalar with g_chi degrees of freedom).
  4. [Sec. III B, after Eq. (35)] The identification Lambda = M/g in Eq. (36) is made via Eq. (5), but the low-energy matching of Lambda to the mediator mass and coupling is stated somewhat abruptly. A one-sentence reminder of the matching in the low-temperature limit would improve readability.
  5. [Secs. IV A 3 and IV B] The cases with k > 7 are repeatedly left as 'an exercise for the motivated reader.' Since the paper already derives T_FO and the T_RH constraints for k > 7, a few formulae showing the resulting relic abundance for the k=8 example would make the paper self-contained for readers interested in those potentials.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the UFO relic abundance and parameter space are derived from the Boltzmann equation with independently specified reheating dynamics and a cross-section ansatz; Omega_chih^2=0.12 is applied as a contour, not as a fitted input.

full rationale

The claimed derivation is not circular. The freeze-out temperature is the solution of Gamma(TFO)=3k/(k+2)H(TFO) with Gamma=n_chi<sigma v> (Eqs. 7, 21, 35), where the cross-section ansatz (Eq. 3) and the reheating Hubble rate (Eqs. 28-29) are stated inputs, not functions of the final relic density. The post-freeze-out number density is then obtained by integrating the Boltzmann equation (Eqs. 51-56) from the equilibrium abundance at TFO, with the continuing production term computed from the same <sigma v> and the bath density; the final conversion to Omega_chih^2 (Eq. 69) is a standard kinematic factor. Nowhere is Omega_chih^2=0.12 used to define Lambda, TRH, or m_chi; the figures impose it as a contour after the abundance formula, which is why the resulting iso-density curves have nontrivial shapes (Eqs. 70-72). The UV/IR UFO split is derived by comparing the two integration limits in Eq. (56), and the critical exponent n*=(10-2k)/(k-1) is the same threshold the authors derived in their earlier freeze-in work; because that threshold is independently derivable from the same scaling and is not used to force the UFO relic density, citing it is not load-bearing. The self-citations [4-6] supply the reheating temperature-scale-factor relations and the freeze-in solution; these are parameter-free published results whose assumptions do not include the UFO abundance or the coldness claim, so they are independent support rather than circular premises. The Sec. IV C estimate of DM coldness uses the assumption T'=TFO(aFO/a) (Eq. 77), i.e. chemical and kinetic decoupling at the same instant; if elastic scattering keeps DM coupled longer the Lyman-alpha bound tightens, but this is a physical approximation that can be checked externally, not a step that makes Eq. (87) true by construction. Overall, the central claims are self-contained against the Boltzmann dynamics and standard cosmology, and no prediction reduces to a fitted parameter or to a self-citation by definition.

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

The central claim rests on the reheating dynamics (inflaton power-law k, thermalized SM bath), the cross-section ansatz, and the freeze-out criteria. The most fragile item is the implicit assumption of simultaneous chemical and kinetic decoupling used for the coldness calculation, which is not supported by the paper's equations. No new particles or forces are introduced; the scan parameters (TRH, Lambda, m_chi, k, n) are model inputs, not fitted constants.

free parameters (6)
  • TRH (reheating temperature)
    Treated as a free parameter scanned over 10^-2 to 10^15 GeV; sets the end of reheating and the normalization of H(T) and Tmax (Sec. III A).
  • Lambda (BSM interaction scale)
    Effective scale in <sigma v> = T^n / Lambda^(n+2), scanned over 10^3 to 10^14 GeV; not fitted to data, it labels the interaction strength.
  • m_chi (DM mass)
    Dark matter mass scanned over 10^-7 to 10^6 GeV; constrained by Omega_chi h^2 = 0.12 and structure formation.
  • k (inflaton potential exponent) = 2,4,6,8,10,12
    Equation of state parameter w = (k-2)/(k+2) during reheating; shown for representative values. k>4 may be affected by fragmentation, acknowledged in Sec. I.
  • n (cross-section temperature exponent) = 2,4, etc.
    Defines the interaction type in <sigma v> proportional to T^n; specific values correspond to heavy mediator (n=2) or higher-dimensional operators.
  • Dimensionless coupling g = 1 in figures
    Set to O(1) for the parameter-space plots; the paper notes smaller couplings shift the FI/UFO boundary (Sec. V limitations).
assumptions (6)
  • domain assumption The inflaton potential is V(phi) = lambda M_P^4 (phi/M_P)^k with a fixed k during reheating, and the solutions for rho_phi(a) and rho_R(a) from refs. [4,5] apply.
    This parameterizes the expansion history and underpins all H(T) relations (Sec. I and Eqs. 22-23).
  • domain assumption The SM decay products form a fully thermalized bath with a well-defined temperature T at all times during reheating.
    Used implicitly in all T-a relations and in the equilibrium freeze-out picture; no thermalization timescale is modeled.
  • domain assumption The DM-SM interaction is captured by <sigma v> = T^n / Lambda^(n+2) in the relativistic regime, with no dependence on m_chi.
    Central parametrization (Eqs. 3-5); the m_chi-dependent terms are neglected, acknowledged as a limitation near the UFO/WIMP boundary (Sec. V).
  • ad hoc to paper The DM kinetically decouples at the same moment it chemically freezes out, so its momentum redshifts as a^-1 from TFO.
    Used in Sec. IV C (Eq. 77) for the coldness and Neff constraints; not justified in the text and likely violated for these couplings.
  • domain assumption Equilibrium is achieved when Gamma = (3/2)(1+w)H (Eq. 7), and freeze-out occurs when Gamma = H with the equilibrium density n_eq used at TFO.
    Standard freeze-out criterion; the O(1) factor choice shifts TFO slightly but not the qualitative results.
  • standard math The Boltzmann equation for DM with only the production term after freeze-out (Eq. 51) is valid.
    Standard freeze-in approximation applied after UFO; validated by numerics in Figs. 4 and 7.

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Cite this review

Pith. "Pith review of Ultra-Relativistic Freeze-Out During Reheating." pith.science (2026). https://pith.science/paper/TORUBSIO

@misc{pith2026250504703,
  author       = {Pith},
  title        = {Pith review of: Ultra-Relativistic Freeze-Out During Reheating},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TORUBSIO}},
  note         = {Machine review of arXiv:2505.04703}
}
abstract

We perform a thorough investigation of (ultra)relativistic freeze-out (UFO) during reheating. While the standard WIMP (non-relativistic freeze-out) and FIMP (freeze-in) paradigms have been explored in detail during the reheating epoch, UFO has not been systematically studied, despite the fact that it is operative in a broad region of parameter space. Although dark matter (DM) is ``hot" at the time of relativistic freeze-out, we show that it can easily undergo enough cooling by the time of structure formation to be compatible with $\Lambda$CDM. Unlike standard WIMP-like freeze-out, there can be significant out-of-equilibrium DM production after UFO, similar to the freeze-in mechanism. However, unlike freeze-in, UFO can accommodate much stronger couplings. The UFO parameter space consistent with $\Omega_{\chi}h^2=0.12$ is quite large, with DM masses spanning about 13 orders of magnitude ($10^{-7} \text{ GeV} \lesssim m_{\chi} \lesssim 10^{6}$ GeV), reheating temperatures spanning 17 orders of magnitude ($10^{-2} \text{ GeV} \lesssim T_{\rm RH} \lesssim 10^{15} \text{ GeV}$) and Beyond the Standard Model (BSM) effective interaction scales spanning 11 orders of magnitude ($10^{3} \text{ GeV} \lesssim \Lambda \lesssim 10^{14}\text{ GeV}$). Interestingly, the most suitable range of couplings for UFO lies precisely between the typical couplings for WIMPs and FIMPs, rendering UFO quite attractive from the standpoint of detection. Particle physics models that are easily amenable to UFO include heavy vector or scalar portal interactions, along with nonrenormalizable effective interactions. Finally, we show there is a distinction between UV UFO and IR UFO, where the relic abundance for the former is sensitive to the freeze-out temperature, while the abundance for the latter is sensitive to the DM mass and the reheating temperature but insensitive to the freeze-out temperature.

Figures

Figures reproduced from arXiv: 2505.04703 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Λ [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Λ [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

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Forward citations

Cited by 2 Pith papers

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Reference graph

Works this paper leans on

49 extracted references · 15 canonical work pages · cited by 2 Pith papers

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    Note that for k >4, the evolution of the inflaton is subject to the effects of fragmentation [13, 14] which cause the equation of state to evolve to k = 4

    The overall scale of the potential, determined by λ, is fixed by the normalization of the cosmic microwave background (CMB) anisotropy spectrum and depends on k [4]. Note that for k >4, the evolution of the inflaton is subject to the effects of fragmentation [13, 14] which cause the equation of state to evolve to k = 4. These effects can be avoided and hi...

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    Minimal conditions for equilibrium We first consider the condition that the dark compo- nent must be in equilibrium at early times. As discussed 7 in Section III A, the minimal condition for equilibrium to be reached at some time during reheating is Γ( Tmax) > 3k k+2H(Tmax) for M > Tmax or Γ(M) > 3k k+2H(M) for M <Tmax . These constraints can be translate...

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    To do this, we can saturate the bound in Eq

    Conditions for TFO > TRH and TFO > mχ Next, we turn to the condition that TFO > TRH, namely the requirement that relativistic freeze-out oc- curs before reheating rather than after reheating. To do this, we can saturate the bound in Eq. (34), namely TFO =TRH so that TRH < 3k k + 2 π2 gχζ(3) rα 3 Λn+2 MP 1 n+1 , (47) valid for all k. Note that this upper l...

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    However, inflaton decay continues to steadily increase the co-moving number density of SM particles during reheat- ing

    Generalities When UFO occurs, particles in the Standard Model ra- diation bath do not produce DM particles at a rate com- mensurate with the Hubble expansion (Γ( T ) < H(T )). However, inflaton decay continues to steadily increase the co-moving number density of SM particles during reheat- ing. This leads the DM number density nχ to drop rel- ative to the...

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    radiation dominated

    mχ < TRH Starting with k < 7, we can use Eq. (23) to obtain T (a) and Eq. (22) with H =√ρϕ/3MP , so that dYχ da ∝a 3n+26−8k−3kn 2k+4 . (54) 10 This is easily integrated from aFO to some desired scale a after freeze-out Yχ(a) =YFO + g2 χζ(3)2 π4 r 3 α Tn+4 RH MP Λn+2 a (3k−3)(n+6)−6k 2k+4 RH × 2k + 4 3n− 3nk− 6k + 30 h a (3−3k)(n+6)+12k+12 2k+4 − a (3−3k)(...

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    Indeed, when T < mχ, the exponentially suppressed Boltzmann factor significantly reduces the number of targets in the neq×neq⟨σv⟩ pro- duction term

    mχ > TRH If the dark matter mass is higher than the reheating temperature, the IR freeze-in production stops at a tem- perature greater than TRH. Indeed, when T < mχ, the exponentially suppressed Boltzmann factor significantly reduces the number of targets in the neq×neq⟨σv⟩ pro- duction term. In this case, we should integrate Eq. (52) from aend to am, wh...

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    We start with UV dominated production

    Dark radiation constraint from Neff We will again consider the cases of UV and IR pro- duction separately, since the temperature dependence of the abundance after freeze-out is distinct. We start with UV dominated production. After freeze-out, the DM is relativistic and the co-moving number density is approx- imately constant. Therefore, after freeze-out,...

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    Structure formation constraint The free–streaming length of a dark matter candi- date which was in thermal equilibrium with the standard model bath depends on its mass and on the ratio of its decoupling temperature TFO to the reheating tempera- ture. Relativistic (or fast) particles will have long free streaming lengths and erase structure on small scales...

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