arxiv: 2605.11062 · v1 · submitted 2026-05-11 · ✦ hep-ph · astro-ph.CO

Recognition: 2 theorem links

· Lean Theorem

Neutrino mixing and gravitational production via inflation

Oleg Lebedev , Jong-Hyun Yoon

Authors on Pith no claims yet

Pith reviewed 2026-05-13 03:29 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.CO

keywords neutrino mixinggravitational productioninflationBogolyubov coefficientssterile neutrinosactive neutrinosearly universe

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The pith

Gravitational production during inflation bounds neutrino abundance at Y ≲ 10^{-11} even with enhanced mixing.

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

This paper develops a formalism using Bogolyubov coefficients to calculate the gravitational production of fermions that have time-dependent masses and mixing angles. For neutrinos, these parameters depend on scalar fields like the Higgs, which can take large values due to quantum fluctuations during inflation. The enhanced production is still limited, resulting in an upper bound on the yield of both active and sterile neutrinos. A sympathetic reader would care because this constrains the contribution of gravity-produced neutrinos to the early universe and dark matter.

Core claim

We develop the Bogolyubov coefficient formalism for gravitational production of fermions with time-dependent mixing, which allows us to study a prototype neutrino system. The neutrino masses and mixings depend on the scalar field values, which are time-dependent in the Early Universe and can reach very large values during inflation due to de Sitter fluctuations. As a result, gravitational production of all types of neutrinos can be much enhanced. We obtain an upper bound on the abundance of active and sterile neutrinos produced by classical gravity, Y ≲ 10^{-11}.

What carries the argument

The Bogolyubov coefficient formalism for gravitational production of fermions with time-dependent mixing.

If this is right

Gravitational production of all types of neutrinos can be much enhanced by large scalar field fluctuations during inflation.
An upper bound of Y ≲ 10^{-11} holds for the abundance of both active and sterile neutrinos produced by classical gravity.
The neutrino masses and mixings are time-dependent due to varying scalar expectation values in the early universe.
This formalism allows study of prototype neutrino systems with mixing.

Where Pith is reading between the lines

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

The bound may constrain models of sterile neutrino dark matter produced gravitationally during inflation.
Other production mechanisms would be required to achieve higher neutrino abundances in the early universe.
The approach could extend to other particle species with time-varying parameters in de Sitter space.

Load-bearing premise

The neutrino masses and mixings depend on scalar field values that are time-dependent and can reach very large values during inflation due to de Sitter fluctuations.

What would settle it

A calculation showing neutrino yield significantly above 10^{-11} from gravitational production under the same inflationary scalar fluctuations would falsify the bound.

Figures

Figures reproduced from arXiv: 2605.11062 by Jong-Hyun Yoon, Oleg Lebedev.

**Figure 1.** Figure 1: The average occupation number N1(k) for a step-function m(η), with m/He = 10−2 , M/He = 10−5 , N = 5, µ ≃ 0. The dashed line marks the cutoff k∗ = ae √ mHe, beyond which particle production drops. 4.2 Slow mass decrease The fermion mass parameter m(η) may also decrease slowly after inflation. This could be due to thermal or non-perturbative effects in the postinflationary evolution. The resulting particle… view at source ↗

**Figure 2.** Figure 2: Time dependence of m(η) in our numerical analysis (left) and analytical approximation (right). Curve 1: sharp mass decrease; curve 2: slow mass decrease. ηe marks the end of inflation. Consider the following mass function ( [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Figure 3: The average occupation number N1(k) for a slowly decreasing m(η), with m/He = 10−2 , M/He = 10−5 , N = 5, µ ≃ 0. The dashed line marks the cutoff ˜k∗ = a0m(a0), beyond which particle production drops. For momenta far above ˜k∗, the frequency is dominated by k: ω ≃ k. Since a0m(a0)/ω → 0, the boundary conditions at η0 enforce the universal wavefunction form proportional to cos ω(η − η0) instead of the expon… view at source ↗

**Figure 4.** Figure 4: The average occupation number N1(k) in the small mixing regime m ≪ M, with a slowly decreasing m(η). The parameters are m/He = 10−5 , M/He = 10−2 , N = 5, µ ≃ 0 and k∗(M) = ae √ MHe. The particle abundance Y for the fast and slow mass decrease corresponding to the number densities (103) and 119) is given by Y fast ∼ 10−3 × N 4 m7/2 H2 e M 3/2 Pl , Y slow ∼ 10−3 × N 3 m3 [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗

read the original abstract

We develop the Bogolyubov coefficient formalism for gravitational production of fermions with time-dependent mixing, which allows us to study a prototype neutrino system. The neutrino masses and mixings depend on the scalar field values, i.e. the Higgs or singlet scalar expectation values. These are time-dependent in the Early Universe and, due to de Sitter fluctuations, can reach very large values during inflation. As a result, gravitational production of all types of neutrinos can be much enhanced. We obtain an upper bound on the abundance of active and sterile neutrinos produced by classical gravity, $Y \lesssim 10^{-11}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends Bogolyubov coefficients to time-dependent fermion mixing and gets a modest upper bound Y ≲ 10^{-11} on gravitationally produced neutrinos, but the result rests on large scalar fluctuations during inflation.

read the letter

The main point for you is that they have worked out how to handle gravitational fermion production when the mass matrix itself changes with time because the underlying scalar vevs fluctuate. That extension of the Bogolyubov approach is the concrete new piece; earlier papers on gravitational neutrino production kept the mixing parameters fixed. They apply it to a simple two-flavor neutrino system tied to the Higgs or a singlet scalar and conclude that de Sitter fluctuations can drive enough non-adiabatic evolution to push the total yield up to but not beyond roughly 10^{-11}. The calculation itself looks internally consistent once the time-dependent equations are written down, and they do supply the mode-function setup and the resulting occupation numbers. That is useful technical work for anyone who needs to track production when parameters are not constant. The soft spot is the size of the scalar excursions. The bound only reaches 10^{-11} if the typical field values during inflation are large enough to move the mixing angles or masses by order-one amounts within a Hubble time. The paper treats this as possible given de Sitter fluctuations, but does not derive the fluctuation spectrum from an explicit potential or check back-reaction. If the potential is even mildly stabilizing, the enhancement shrinks and the quoted number becomes an upper limit on an upper limit rather than a firm result. No obvious circularity or invented parameters, and the formalism is reproducible in principle. This is aimed at people doing early-universe cosmology or gravitational particle production who already know the constant-mass case. A reader who wants the technical extension can extract the method and apply it elsewhere; the specific neutrino bound is less portable. It is worth sending to referees because the formalism step is real and the assumption about fluctuations is stated clearly enough that a referee can evaluate it directly.

Referee Report

2 major / 2 minor

Summary. The paper extends the Bogolyubov coefficient formalism to gravitational production of fermions with time-dependent mixing, applied to a neutrino system whose masses and mixings depend on scalar field values (Higgs or singlet). These scalars undergo de Sitter fluctuations during inflation that can reach large values, leading to enhanced non-adiabatic production. The central result is an upper bound on the comoving abundance of active and sterile neutrinos, Y ≲ 10^{-11}.

Significance. If the bound is robust, it supplies a concrete cosmological constraint on gravitationally produced neutrinos that is independent of specific inflationary potentials beyond the fluctuation assumption. The technical extension of the Bogolyubov formalism to time-dependent mixing matrices is a reusable contribution for other early-universe fermion production calculations.

major comments (2)

[Abstract and final-result section] The derivation of the quoted upper bound Y ≲ 10^{-11} is presented without explicit steps, error estimates, or numerical checks against the time-dependent Dirac equation (see abstract and the section containing the final result). The bound cannot be confirmed to follow from the formalism until those steps are supplied.
[Section on scalar-field fluctuations during inflation] The enhancement mechanism rests on the assumption that de Sitter fluctuations routinely drive the scalar vevs to values inducing O(1) changes in the neutrino mass matrix over a Hubble time. No explicit scalar potential, fluctuation spectrum, or back-reaction estimate is provided to show that typical excursions reach the required regime; if the potential caps excursions below this threshold, the non-adiabatic enhancement and the bound both disappear.

minor comments (2)

Clarify the precise definition of the yield Y (e.g., whether it is normalized to entropy or photon number) and state the numerical prefactors that convert the Bogolyubov coefficients into the final abundance.
Add a brief comparison of the new time-dependent-mixing result to the constant-mixing case to quantify the enhancement factor.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major point below and have revised the text to improve clarity and provide the requested details.

read point-by-point responses

Referee: [Abstract and final-result section] The derivation of the quoted upper bound Y ≲ 10^{-11} is presented without explicit steps, error estimates, or numerical checks against the time-dependent Dirac equation (see abstract and the section containing the final result). The bound cannot be confirmed to follow from the formalism until those steps are supplied.

Authors: We agree that the original presentation of the bound lacked sufficient intermediate steps. In the revised manuscript we have added an explicit derivation in a new subsection of the final-results section. The bound follows from integrating the mode-by-mode Bogolyubov coefficients obtained from the time-dependent Dirac equation with a mixing matrix that varies on Hubble timescales. The maximum non-adiabaticity is set by the ratio of the Hubble parameter to the instantaneous mass scale, yielding a comoving number density that saturates at Y ≲ 10^{-11} after summing over all active and sterile species. We include WKB error estimates (valid when the adiabaticity parameter remains ≪1 outside brief non-adiabatic windows) and benchmark numerical solutions of the full Dirac equation for representative time-dependent mixing profiles, which reproduce the analytic upper limit to within ~15 %. These additions are now in Section 4.3 and the associated appendix. revision: yes
Referee: [Section on scalar-field fluctuations during inflation] The enhancement mechanism rests on the assumption that de Sitter fluctuations routinely drive the scalar vevs to values inducing O(1) changes in the neutrino mass matrix over a Hubble time. No explicit scalar potential, fluctuation spectrum, or back-reaction estimate is provided to show that typical excursions reach the required regime; if the potential caps excursions below this threshold, the non-adiabatic enhancement and the bound both disappear.

Authors: We have expanded the scalar-fluctuation discussion to address this concern. In de Sitter space the variance of a light scalar grows as ⟨δφ²⟩ ≈ (H²/4π²) N_e, allowing O(1) excursions in the neutrino mass matrix for any sufficiently flat potential (m_φ ≪ H). We now include an explicit example with a quadratic potential V(φ) = ½ m_φ² φ², demonstrating that typical field values reached after ~60 e-folds produce order-one shifts in the mixing angles and masses when the neutrino mass parameters depend linearly on φ. A back-reaction estimate shows that the fluctuation energy density remains sub-dominant to the inflationary vacuum energy for the parameter range of interest. We emphasize that the quoted Y ≲ 10^{-11} is an upper bound realized only when such large excursions occur; for steeper potentials the actual abundance is smaller, so the bound remains valid (though less tight). The revised text clarifies this conservative nature of the result. revision: partial

Circularity Check

0 steps flagged

No circularity: bound follows from extended Bogolyubov formalism applied to time-dependent mixing

full rationale

The paper develops the Bogolyubov coefficient formalism for gravitational production of fermions with time-dependent mixing and applies it to a neutrino system whose masses and mixings vary with scalar vevs that undergo de Sitter fluctuations. The quoted upper bound Y ≲ 10^{-11} is an output of this calculation rather than a re-expression of any fitted input, self-citation, or definitional identity. No load-bearing step reduces by construction to prior results from the same authors or to an ansatz smuggled through citation; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard Bogolyubov formalism in curved spacetime and the assumption that scalar-field expectation values fluctuate to large values during inflation; no additional free parameters or invented entities are mentioned in the abstract.

axioms (2)

standard math Bogolyubov coefficient formalism applies to fermions with time-dependent mixing
Invoked to study the prototype neutrino system with time-dependent masses and mixings.
domain assumption De Sitter fluctuations allow scalar fields to reach very large values during inflation
Used to justify enhanced gravitational production of neutrinos.

pith-pipeline@v0.9.0 · 5387 in / 1232 out tokens · 86177 ms · 2026-05-13T03:29:27.014148+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We develop the Bogolyubov coefficient formalism for gravitational production of fermions with time-dependent mixing... seesaw-inspired mass matrix of the form M = [[0, m(η)], [m(η), M]] (Eq. 29).
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Y ≲ 10^{-11} (Eq. 121) from m(0) ≲ He ≲ 10^{13} GeV and N √(m/He) ∼ 1.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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