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REVIEW 2 major objections 5 minor 66 references

When inflaton energy passes through long-lived seesaw neutrinos, the Universe cycles matter→radiation→matter→radiation and the Standard Model temperature cools with the unusual powers a^-1/4 and a^-3/8, reshaping dark-matter production.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 09:28 UTC pith:QS6AETAM

load-bearing objection Elegant two-step reheating formalism with clean scalings, but the load-bearing collisionless-RHN assumption is unchecked and a Pauli-blocking estimate calls the benchmark into question. the 2 major comments →

arxiv 2607.27325 v1 pith:QS6AETAM submitted 2026-07-29 hep-ph astro-ph.CO

Seesaw Cosmology

Nicol\'as Bernal , Chee Sheng Fong , \'Oscar Zapata This is my paper
classification hep-ph astro-ph.CO
keywords seesaw cosmologyright-handed neutrinosreheatingnonthermal distributionequation of statedark matter productionUV freeze-inpre-BBN thermal history
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Seesaw cosmology is the claim that reheating can proceed in two steps: the inflaton first decays into right-handed neutrinos (RHNs), and only then do those RHNs decay into Standard Model particles. Because the RHNs are produced relativistically and live long enough to dominate the energy budget, the expansion of the Universe oscillates between matter-like and radiation-like behavior—0, 1/3, 0, 1/3—instead of going directly to radiation. The key result is a distinctive temperature history: after an early plateau, T falls as a^-1/4 during relativistic-RHN domination and as a^-3/8 during nonrelativistic-RHN domination, before the standard a^-1. This matters because pre-BBN cosmology is almost unconstrained, and any process that depends on the temperature during reheating—dark-matter production, baryogenesis, gravitational-wave backgrounds—will shift accordingly. The paper shows concretely that direct dark-matter production from inflaton decays is enhanced by m_phi/(2m_N), while ultraviolet freeze-in acquires critical powers p=12 and p=20.

Core claim

The central discovery is that the full production-time dependence of the nonthermal RHN distribution cannot be replaced by a single average momentum: RHNs born at different times have different redshifted momenta, time-dilated decay rates, and equations of state. Retaining that distribution, the paper derives an integro-differential Boltzmann system and analytic solutions in four asymptotic regimes. The effective equation of state alternates as 0 -> 1/3 -> 0 -> 1/3, and the SM temperature follows T ∝ a^0, a^-1/4, a^-3/8, a^-1. In the hierarchical limit m_phi >> 2m_N, the final DM yield from inflaton decays is controlled by m_N and enhanced by m_phi/(2m_N); for UV freeze-in, the critical expo

What carries the argument

The central object is the production-time-dependent differential RHN number density δn_N(a,a') da' (Eq. 3.9), which tracks how many RHNs created between a' and a'+da' survive until a, including the redshift of their momentum p_N(a,a') = (a'/a)√(m_phi^2/4 - m_N^2) and the time-dilated decay rate Γ~_N = (m_N/E_N) Γ_N. Integrating this distribution over production times yields the total RHN number density, energy density, pressure, and SM radiation source; the relativistic-to-nonrelativistic transition emerges from the distribution rather than from any single-momentum approximation. This machinery produces the a^-1/4 and a^-3/8 temperature scalings and the p=12,20 freeze-in critical powers.

Load-bearing premise

The whole four-stage history assumes the right-handed neutrinos do not scatter, annihilate, or inverse-decay between production and decay: they are collisionless, so their nonthermal momentum distribution is only redshifted and time-dilated.

What would settle it

If one includes even a small elastic-scattering or number-changing interaction among RHNs (or inverse decays that repopulate the inflaton), the distribution in Eq. (3.9) no longer holds. A Boltzmann simulation with a nonzero scattering cross section would show whether the a^-1/4 and a^-3/8 scalings and the p=12,20 behavior persist; if RHNs thermalize before anr, the temperature history collapses to the characteristic-momentum/prompt-thermalization result and the claimed departures disappear.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If correct, the pre-BBN thermal history is not generic radiation domination but a four-stage sequence whose duration is set by seesaw parameters, making reheating more predictive because RHN decay widths are bounded by neutrino mass-squared differences.
  • Dark-matter yield from a subdominant inflaton decay channel is set by m_N rather than m_phi and can be boosted by m_phi/(2m_N) relative to standard reheating at fixed Trh and branching fraction.
  • UV freeze-in production is not dominated by the final reheating temperature for p ≥ 12: for p=12 there is a logarithmic enhancement, for p=20 a R^3 ln(R2) boost, and for p>20 a double power-law enhancement, so the RHN-dominated eras can dominate the DM abundance.
  • The maximum temperature Tmax depends on both Γ_phi and Γ_N, while Trh depends only on Γ_N; both are seeded by neutrino parameters, so the seesaw scenario links observable neutrino masses to the duration of each cosmological era.

Where Pith is reading between the lines

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

  • The collisionless assumption is likely to break when RHN number densities are high enough for elastic scattering or inverse decays; a natural testable extension is to include a finite thermalization rate and check whether the four-stage sequence survives when RHNs scatter even weakly.
  • The alternating equation of state 0→1/3→0→1/3 should leave a characteristic imprint on primordial gravitational-wave spectra, since the spectral tilt changes with each era; a dedicated computation could distinguish seesaw cosmology from single-momentum approximations.
  • The enhancement m_phi/(2m_N) suggests that in minimal seesaw scenarios with normal or inverted mass ordering, the dark-matter abundance and the duration of the RHN eras could be predicted from neutrino oscillation data plus the inflaton mass, defining target parameter regions for experiments.
  • The p=12 and p=20 critical powers single out operators of mass dimension D=8 and D=12 as having special freeze-in behavior; laboratory probes of such operators, combined with the measured DM abundance, could constrain the ratios R1 and R2 and hence the seesaw parameters.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes a reheating history in which the inflaton decays exclusively into long-lived right-handed neutrinos (RHNs), which subsequently decay into the Standard Model. The authors solve the coupled Boltzmann system while retaining the production-time dependence of the nonthermal RHN distribution, including the relativistic-to-nonrelativistic transition and the time-dilated decay rate of each production cohort. They find an alternating effective equation of state (inflaton matter, RHN radiation, RHN matter, SM radiation) and derive piecewise temperature scalings T(a) proportional to a^0, a^{-1/4}, a^{-3/8}, and a^{-1} in the four eras. They apply this thermal history to dark matter from direct inflaton decay, obtaining an enhancement of order m_phi/(2 m_N) at fixed reheating temperature and branching fraction, and to UV freeze-in, where they identify critical powers p=12 and p=20 that control how much of the DM is produced in the intermediate RHN-dominated eras. The benchmark numerical evolution in Fig. 1 is consistent with the analytic estimates.

Significance. If the underlying assumptions hold, this is a qualitatively new pre-BBN thermal history that connects type-I seesaw neutrino masses to the expansion and temperature evolution before nucleosynthesis. The paper contains a genuinely useful technical step: the exact source identity in Eq. (3.11), which removes momentum dependence from the SM radiation source, is elegant and correctly implemented. The analytic scalings are derived, not fitted, and the benchmark numerics confirm the four-stage sequence. The DM implications are concrete and falsifiable: the p=12 and p=20 critical powers give sharply different predictions from standard reheating and from the near-simultaneous prompt-thermalization treatment in Ref. [29]. The main weakness is that the entire production-time-resolved picture is conditional on the RHNs being collisionless from production to decay, and this condition is asserted but not quantitatively established.

major comments (2)
  1. [Section 3, Eq. (3.9) paragraph] The paper states that RHNs are 'collisionless between their production and decay, neglecting elastic scattering, number-changing reactions, and inverse inflaton decays.' This assumption is load-bearing: the production-time-dependent distribution delta n_N(a,a') in Eq. (3.9), the energy/pressure integrals in Eqs. (3.13)-(3.14), and the resulting temperature scalings in Eq. (1.3)/(4.14) all rely on it. If RHN scattering or inverse processes are efficient enough to redistribute momenta or change the RHN number density, the distribution collapses toward the prompt-thermalization/characteristic-momentum treatment of Ref. [29] and the claimed departures disappear. The paper does not estimate any relevant rate (e.g., NN->lH, lH->N, or pair annihilation) relative to H in the region where the four-stage sequence is claimed. For the benchmark parameters of Fig. 1, the minimal Casas-Ibarra Yukawa c
  2. [Section 3, paragraph after Eq. (3.9)] The text claims that inverse inflaton decay (production of inflatons from two RHNs) is 'always kinematically blocked.' This is stronger than what follows from the stated assumptions. For two RHNs produced from different inflaton decays, the invariant mass s of the pair is not fixed; after redshift, a continuum of pair energies is present, and pairs with s = m_phi^2 are kinematically allowed. The total 3-momentum of such a pair is generally nonzero, so the final inflaton has nonzero momentum, but the process is not forbidden. Whether it is negligible is a dynamical question controlled by the small inflaton-RHN coupling and the phase-space density of resonant pairs. The statement as written is not justified and should either be corrected or replaced by a quantitative bound on the rate of NN -> phi relative to H.
minor comments (5)
  1. [Section 2, Eq. (2.2)] The phrase 'with n′ referring to the N_i fulfilling...' is awkwardly typeset; please clarify that the sum is over RHN species with M_i < m_phi/2.
  2. [Table 1] The header 'T able 1' contains a formatting typo; should be 'Table 1'.
  3. [Section 4, transition at a_nr] The definition of a_nr in Eq. (4.9) uses the highest-momentum cohort produced at a_phi. This is a clean asymptotic criterion, but before a_nr some older cohorts are already nonrelativistic and after a_nr the highest-momentum cohort is not instantly cold. The analytic scalings are therefore only strictly valid in the limit R2 >> 1. The paper already requires well-separated regimes in Section 4, but it would be helpful to state this explicitly near Eq. (4.9) and to quantify the residual correction.
  4. [Section 5.2] In Eq. (5.20) and the surrounding text, g_star and g_star s should be evaluated at T_rh and taken constant across the eras; this is a standard approximation but should be stated once for clarity.
  5. [Note added] The note comparing Eq. (4.16) with Ref. [29] is useful. It would strengthen the paper to state explicitly which predictions beyond T_rh differ from the prompt-thermalization treatment and to emphasize that the difference is controlled by the collisionless assumption.

Circularity Check

0 steps flagged

No significant circularity: the central temperature history and dark-matter results are derived from the Boltzmann equations, not fitted, and self-citations are non-load-bearing.

full rationale

The paper's central claims are derived, not assumed. The four-stage equation of state and the T(a) scalings in Eq. (1.3)/(4.14) follow from solving the Boltzmann system (3.1)-(3.3) with the production-time-resolved RHN distribution (3.9), energy and pressure integrals (3.13)-(3.14), and the asymptotic solutions (4.1)-(4.16). The plateau T ∝ a^0, then a^{-1/4}, then a^{-3/8}, then a^{-1}, are analytic consequences of these equations; they are not inputs. Similarly, the dark-matter enhancement factor m_phi/(2 m_N) in Eq. (5.10) is obtained by comparing the derived yield (5.8) with the standard yield (5.9), and the critical freeze-in powers p=12 and p=20 emerge from the integrals (5.15)-(5.19) using the derived T(a) scalings. None of these quantities is fitted to reproduce an outcome; the benchmark parameters in Fig. 1 are explicitly illustrative and chosen to exhibit well-separated asymptotic regimes. Self-citations, e.g., [68] for UV freeze-in, appear in literature/build-block contexts but are not load-bearing: the relevant integrals are derived in the paper, and the plateau and matter-domination scalings are traced to independent works [25,52]. The Note added states that Eq. (4.16) agrees with the independent near-simultaneous paper [29], providing external consistency. The main caveat, the assumption that RHNs are 'collisionless between their production and decay' (Section 3, after Eq. 3.9), is an explicit physical assumption rather than a circular reduction; its unquantified robustness is a scientific limitation, but it does not make the derivation equivalent to its inputs. Overall, no circular step meeting the required evidence threshold is present.

Axiom & Free-Parameter Ledger

6 free parameters · 9 axioms · 0 invented entities

The central contributions are analytic scalings that depend on standard cosmology, type-I seesaw relations, and the collisionless/nonthermal assumptions. The benchmark numerical example uses four free cosmological parameters; no parameter was fitted to a target observable. The heaviest burden is the collisionless nonthermal RHN assumption. No new particle or force is introduced.

free parameters (6)
  • H_I (Hubble scale at beginning of reheating) = 1e9 GeV in benchmark
    Sets the overall normalization of the example evolution; the analytic scalings are independent of its value. Chosen below the BICEP/Keck bound.
  • m_phi (inflaton mass) = 3.3e9 GeV in benchmark
    Sets m_phi >> 2 m_N so RHNs are born relativistic; required for the four-stage history.
  • m_N (RHN mass) = 1.7e6 GeV in benchmark
    Together with m_phi sets R2 and the duration of the relativistic-RHN era; chosen heavy enough for on-shell decay.
  • Gamma_phi (inflaton decay width to RHNs) = 2.5e6 GeV in benchmark
    Sets the duration of inflaton domination; must satisfy H_I >> Gamma_phi >> Gamma_N.
  • Gamma_N (RHN decay width) = 3.1e-5 GeV in benchmark
    Set at the n=2 NO lower bound from neutrino oscillation data (Eq. 2.8); controls T_rh and the RHN-dominated eras. The paper says the minimal cosmology has four free parameters.
  • DM application parameters (p, Lambda, m_chi, N_chi, Br) = varied: p=6..30, Lambda > T_max, etc.
    Scanned/illustrative in Sec. 5; the parametric enhancements such as m_phi/(2 m_N) and R1/R2 boosts are independent of their numerical values.
axioms (9)
  • domain assumption The dominant energy density after inflation is a real scalar singlet in a quadratic potential, oscillating around zero, with pressureless equation of state (w=0).
    Used in Secs. 2-3; sets inflaton-domination scaling rho_phi ~ a^-3 and decay kinematics p_N ~ m_phi/2.
  • ad hoc to paper Perturbative inflaton decay to RHN pairs is the only reheating channel; nonadiabatic fermion production and inflaton-induced mass corrections are negligible.
    Footnote 1 in Sec. 2 explicitly leaves the complementary regime for future work.
  • domain assumption RHNs are collisionless once produced; elastic scattering, number-changing reactions, and inverse inflaton decays are neglected.
    Sec. 3 after Eq. (3.9); this is the weakest assumption. If absent, the nonthermal distribution thermalizes and the scalings change.
  • domain assumption SM decay products thermalize on timescales much shorter than H^-1, allowing a well-defined SM temperature.
    Sec. 3; needed for the T(a) scalings and for the freeze-in rate gamma(T).
  • domain assumption The heavy-RHN zero-temperature decay width Gamma_N = (y y^dag)_ii M_i/(8 pi) with the Casas-Ibarra seesaw relation, so Gamma_N is tied to measured neutrino mass splittings.
    Eqs. (2.3)-(2.9); links the reheating parameters to neutrino data.
  • ad hoc to paper The inflaton decays predominantly into one RHN species; other RHNs build the light-neutrino spectrum but are negligibly populated.
    Sec. 2; a single nonthermal population is tracked. Multi-RHN extension is left for future work.
  • domain assumption Parameter hierarchy H_I >> Gamma_phi >> Gamma_N and m_phi >> 2 m_N holds, so the four asymptotic stages are well separated.
    Sec. 4 opening; needed for the piecewise analytic solutions to be accurate.
  • domain assumption For UV freeze-in, the effective operator description with gamma_chi(T) = T^p / Lambda^(p-4) is valid only for T_max < Lambda.
    Sec. 5.2; the paper states this consistency condition explicitly.
  • domain assumption Dark matter is stable, out of equilibrium, and its annihilation/decay terms are negligible during production.
    Sec. 5 opening; simplifies Eq. (5.1) to a pure source equation.

pith-pipeline@v1.3.0-daily-deepseek · 16305 in / 28479 out tokens · 234833 ms · 2026-08-01T09:28:22.144748+00:00 · methodology

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read the original abstract

We study perturbative reheating in which the inflaton transfers its energy to the Standard Model through right-handed neutrinos (RHNs) responsible for light-neutrino masses via the type-I seesaw mechanism. We refer to the resulting nonstandard thermal history as $seesaw$ $cosmology$. When produced relativistically and sufficiently long lived, the RHNs generate a characteristic sequence of inflaton, relativistic-RHN, nonrelativistic-RHN, and Standard Model radiation domination. We solve the Boltzmann system while retaining the production-time dependence of the nonthermal RHN distribution and its relativistic-to-nonrelativistic transition. The Standard Model temperature rapidly approaches a plateau during inflaton domination and subsequently scales as $a^{-1/4}$ and $a^{-3/8}$ during relativistic- and nonrelativistic-RHN domination, respectively. We investigate the implications of seesaw cosmology for dark-matter production. Direct production through inflaton decays can be enhanced relative to conventional reheating by a factor of order $m_\phi/(2m_N)$, while ultraviolet freeze-in exhibits the critical temperature powers $p = 12$ and $20$, leading to potentially large contributions before the final radiation-dominated era. Seesaw cosmology therefore connects neutrino-mass generation, the pre-BBN thermal history and phenomena such as dark-matter production and baryogenesis.

discussion (0)

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