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arxiv: 2606.03395 · v1 · pith:ESPTXBEInew · submitted 2026-06-02 · 🌌 astro-ph.CO

Self-interacting neutrinos in cosmological perturbation theory -- integrating the collision kernel

Jakob K. Mogensen , Steen Hannestad , Thomas Tram This is my paper

Pith reviewed 2026-06-28 08:53 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords self-interacting neutrinosBoltzmann hierarchycollision kernelmultipole projectionYukawa potentialrecurrence relationcosmological perturbationsanalytic integration
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The pith

The multipole integrals for neutrino-neutrino scattering admit an exact rational-plus-π² representation for every ℓ.

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

The paper derives an exact analytic expression for the multipole integral that sets the coefficients α_ℓ in the Boltzmann hierarchy for self-interacting neutrinos. It expresses the integration kernel as angular derivatives of the Yukawa potential, moves the derivatives onto the Legendre polynomials, and reduces the momentum integrals to a base family governed by a first-order recurrence. This produces closed-form expressions consisting of rational numbers and multiples of π², together with recurrence relations and an approximation for numerical codes. The result replaces numerical integration of the collision term with exact arithmetic in cosmological perturbation calculations.

Core claim

The collision kernel for neutrino-neutrino scattering mediated by a light scalar is projected onto momentum-averaged multipoles by evaluating the multipole integral for α_ℓ. This integral is computed exactly by writing the kernel as angular derivatives of the Yukawa potential e^{-P/2}/P, shifting the derivatives onto Legendre polynomials, and showing that the remaining momentum integrals belong to a single base family obeying a first-order recurrence. The procedure yields an exact rational-plus-π² representation for every multipole together with recurrence relations, a closed form for the base integral, and an asymptotically constrained approximation suitable for Boltzmann codes.

What carries the argument

The multipole integral for the coefficients α_ℓ, reduced via angular derivatives of the Yukawa potential e^{-P/2}/P and a first-order recurrence on a base momentum integral.

If this is right

  • The coefficients α_ℓ admit exact expressions as rational numbers plus multiples of π² for any multipole order ℓ.
  • A single base integral plus recurrence relations suffice to generate all higher multipoles without repeated integration.
  • Boltzmann solvers can implement the collision term using exact rational arithmetic rather than numerical quadrature.
  • The method provides both the exact values and a compact approximation constrained by asymptotic behavior.

Where Pith is reading between the lines

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

  • Cosmological constraints on neutrino self-interactions can be computed more rapidly and accurately with these exact coefficients.
  • Similar techniques based on derivative shifts and recurrences may apply to other scattering processes with Yukawa mediators in early-universe calculations.
  • The exact form clarifies the analytic structure of the collision term and may reveal symmetries or simplifications not visible numerically.

Load-bearing premise

The integration kernel for neutrino-neutrino scattering can be expressed as angular derivatives of the Yukawa potential and the remaining momentum integrals reduce to a single base family obeying a first-order recurrence.

What would settle it

Numerical quadrature of the multipole integral for a chosen ℓ and P yielding a value that differs from the rational-plus-π² expression obtained via the recurrence.

read the original abstract

Cosmological constraints on self-interacting neutrinos require a Boltzmann hierarchy in which the collision term is projected onto momentum-averaged multipoles. We revisit the collision kernel for neutrino-neutrino scattering mediated by a light scalar and derive an exact analytic expression for the multipole integral that determines the coefficients $\alpha_\ell$. The key idea is to express the integration kernel as angular derivatives of the Yukawa-potential $\frac{\mathrm{e}^{-P/2}}{P}$, move the derivatives onto Legendre polynomials, and reduce the remaining momentum integrals to a single base family obeying a first-order recurrence. This gives an exact rational-plus-$\pi^2$ representation for every multipole, together with a compact implementation based on exact rational arithmetic. We provide the recurrence relations, a closed form for the base integral, and an asymptotically constrained approximation suitable for Boltzmann codes such as CLASS. Our numerical implementation is publicly available in the Jupyter notebook IntegralComputation.ipynb.

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.

Referee Report

1 major / 2 minor

Summary. The paper derives an exact analytic expression for the multipole integrals determining the coefficients α_ℓ in the collision term for neutrino-neutrino scattering mediated by a light scalar. The central method expresses the angular dependence of the integration kernel as derivatives of the Yukawa potential e^{-P/2}/P, transfers those derivatives onto Legendre polynomials, and reduces the remaining momentum integrals to a single base family obeying a first-order recurrence, yielding exact rational-plus-π² representations for every multipole together with recurrence relations, a closed-form base integral, an asymptotically constrained approximation, and a public Jupyter notebook implementation.

Significance. If the derivation holds, the result supplies a parameter-free, numerically stable analytic route to the collision coefficients that can be directly inserted into Boltzmann solvers such as CLASS. This removes the need for on-the-fly numerical quadrature of the kernel and supplies falsifiable, machine-reproducible expressions (via the supplied notebook) for all multipoles, which is a concrete advance for precision cosmology of self-interacting neutrinos.

major comments (1)
  1. [abstract / key-idea paragraph] The key step (abstract and the paragraph describing the method) asserts that the full angular dependence of the neutrino-neutrino collision kernel equals angular derivatives acting on e^{-P/2}/P after the energy integrals are performed. No explicit term-by-term identity is supplied showing that residual angular structure from the matrix element, statistics factors, or momentum mapping is absent; because this representation is load-bearing for the subsequent Legendre differentiation and single-base-family recurrence, an intermediate derivation confirming exact equality is required.
minor comments (2)
  1. [Implementation section] The public notebook IntegralComputation.ipynb is a strength; however, the manuscript should state the exact version of the notebook and the floating-point precision used when comparing the recurrence output to direct numerical quadrature of the original kernel.
  2. [Notation] Notation for the momentum variable P and the angular variable should be defined once at first use and kept consistent between the Yukawa potential and the Legendre expansion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the work's significance, and constructive comment. We address the major comment below.

read point-by-point responses

  1. Referee: [abstract / key-idea paragraph] The key step (abstract and the paragraph describing the method) asserts that the full angular dependence of the neutrino-neutrino collision kernel equals angular derivatives acting on e^{-P/2}/P after the energy integrals are performed. No explicit term-by-term identity is supplied showing that residual angular structure from the matrix element, statistics factors, or momentum mapping is absent; because this representation is load-bearing for the subsequent Legendre differentiation and single-base-family recurrence, an intermediate derivation confirming exact equality is required.

    Authors: We agree that the manuscript would benefit from an explicit term-by-term verification of this central identity. While the overall reduction is outlined in the text, we did not supply a detailed expansion confirming the absence of residual angular contributions from the matrix element, statistics factors, and momentum mapping after energy integration. In the revised version we will insert this intermediate derivation (as an expanded subsection or appendix) to demonstrate the exact equality, thereby justifying the subsequent transfer of derivatives onto Legendre polynomials and the reduction to a single base family. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper presents a direct mathematical reduction: the collision kernel is expressed as angular derivatives of the Yukawa potential e^{-P/2}/P, derivatives are moved onto Legendre polynomials, and momentum integrals are reduced to a single base family with a first-order recurrence, yielding exact rational-plus-π² forms. No self-citations, fitted parameters, or self-referential definitions appear in the provided abstract or description. The central claim is a self-contained analytic derivation from the kernel, independent of external benchmarks or prior author results, so the derivation does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on standard properties of Legendre polynomials, angular integration, and the functional form of the Yukawa potential; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The collision kernel for neutrino-neutrino scattering mediated by a light scalar admits an exact representation as angular derivatives of the Yukawa potential e^{-P/2}/P.
    Invoked in the key idea paragraph of the abstract as the starting point for moving derivatives onto Legendre polynomials.
  • domain assumption The momentum integrals that remain after the angular derivatives are applied belong to a single base family that obeys a first-order recurrence relation.
    Stated in the abstract as the step that reduces all multipoles to a closed rational-plus-π² form.

pith-pipeline@v0.9.1-grok · 5691 in / 1539 out tokens · 21128 ms · 2026-06-28T08:53:14.731116+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

29 extracted references · 1 canonical work pages

  1. [1]

    NuFit-6.0: updated global analysis of three-flavor neutrino oscillations,

    I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J. P. Pinheiro, and T. Schwetz, “NuFit-6.0: updated global analysis of three-flavor neutrino oscillations,”JHEP12 (2024) 216,arXiv:2410.05380 [hep-ph]

    Pith/arXiv arXiv 2024

  2. [2]

    Neutrino Mass and New Physics,

    R. N. Mohapatra and A. Y. Smirnov, “Neutrino Mass and New Physics,”Ann. Rev. Nucl. Part. Sci.56(2006) 569–628,arXiv:hep-ph/0603118

    Pith/arXiv arXiv 2006

  3. [3]

    Neutrino self-interactions: A white paper,

    J. M. Berrymanet al., “Neutrino self-interactions: A white paper,”Phys. Dark Univ.42 (2023) 101267,arXiv:2203.01955 [hep-ph]

    arXiv 2023

  4. [4]

    Structure formation with strongly interacting neutrinos - Implications for the cosmological neutrino mass bound,

    S. Hannestad, “Structure formation with strongly interacting neutrinos - Implications for the cosmological neutrino mass bound,”JCAP02(2005) 011,arXiv:astro-ph/0411475

    Pith/arXiv arXiv 2005

  5. [5]

    Cosmological signatures of interacting neutrinos,

    N. F. Bell, E. Pierpaoli, and K. Sigurdson, “Cosmological signatures of interacting neutrinos,” Phys. Rev. D73(2006) 063523,arXiv:astro-ph/0511410

    Pith/arXiv arXiv 2006

  6. [6]

    Constraining Models of Neutrino Mass and Neutrino Interactions with the Planck Satellite,

    A. Friedland, K. M. Zurek, and S. Bashinsky, “Constraining Models of Neutrino Mass and Neutrino Interactions with the Planck Satellite,”arXiv:0704.3271 [astro-ph]

    Pith/arXiv arXiv

  7. [7]

    Are There Real Goldstone Bosons Associated with Broken Lepton Number?,

    Y. Chikashige, R. N. Mohapatra, and R. D. Peccei, “Are There Real Goldstone Bosons Associated with Broken Lepton Number?,”Phys. Lett. B98(1981) 265–268

    1981

  8. [8]

    Left-Handed Neutrino Mass Scale and Spontaneously Broken Lepton Number,

    G. B. Gelmini and M. Roncadelli, “Left-Handed Neutrino Mass Scale and Spontaneously Broken Lepton Number,”Phys. Lett. B99(1981) 411–415

    1981

  9. [9]

    Neutrinoless universe,

    J. F. Beacom, N. F. Bell, and S. Dodelson, “Neutrinoless universe,”Phys. Rev. Lett.93(2004) 121302,arXiv:astro-ph/0404585

    Pith/arXiv arXiv 2004

  10. [10]

    Updated constraints on non-standard neutrino interactions from Planck,

    M. Archidiacono and S. Hannestad, “Updated constraints on non-standard neutrino interactions from Planck,”JCAP07(2014) 046,arXiv:1311.3873 [astro-ph.CO]

    Pith/arXiv arXiv 2014

  11. [11]

    Limits on Neutrino-Neutrino Scattering in the Early Universe,

    F.-Y. Cyr-Racine and K. Sigurdson, “Limits on Neutrino-Neutrino Scattering in the Early Universe,”Phys. Rev. D90(2014) no. 12, 123533,arXiv:1306.1536 [astro-ph.CO]

    Pith/arXiv arXiv 2014

  12. [12]

    Neutrino puzzle: Anomalies, interactions, and cosmological tensions,

    C. D. Kreisch, F.-Y. Cyr-Racine, and O. Doré, “Neutrino puzzle: Anomalies, interactions, and cosmological tensions,”Phys. Rev. D101(2020) no. 12, 123505,arXiv:1902.00534 [astro-ph.CO]

    arXiv 2020

  13. [13]

    Updated constraints on massive neutrino self-interactions from cosmology in light of theH0 tension,

    S. Roy Choudhury, S. Hannestad, and T. Tram, “Updated constraints on massive neutrino self-interactions from cosmology in light of theH0 tension,”JCAP03(2021) 084, arXiv:2012.07519 [astro-ph.CO]

    arXiv 2021

  14. [14]

    Neutrino self-interactions in post-reionization era: Lyman-α, 21-cm and cross-spectra,

    S. Pal and S. Pal, “Neutrino self-interactions in post-reionization era: Lyman-α, 21-cm and cross-spectra,”arXiv:2604.15287 [astro-ph.CO]

    Pith/arXiv arXiv

  15. [15]

    Towards a complete scheme of cosmological neutrino self-interactions: Collision term for a wide range of mediator masses,

    I. Pérez-Castro, J. De-Santiago, G. Garcia-Arroyo, J. Venzor, and A. Pérez-Lorenzana, “Towards a complete scheme of cosmological neutrino self-interactions: Collision term for a wide range of mediator masses,”arXiv:2602.12477 [hep-ph]

    Pith/arXiv arXiv

  16. [16]

    Directly probing neutrino interactions through CMB phase shift measurements,

    G. Montefalcone, S. Ghosh, K. K. Boddy, D. W. R. Ho, and Y. Tsai, “Directly probing neutrino interactions through CMB phase shift measurements,”Phys. Rev. D113(2026) no. 2, 023540,arXiv:2509.20363 [astro-ph.CO]. – 26 –

    arXiv 2026

  17. [17]

    Self-interacting neutrinos in light of recent CMB and LSS data,

    A. Poudou, T. Simon, T. Montandon, E. M. Teixeira, and V. Poulin, “Self-interacting neutrinos in light of recent CMB and LSS data,”Phys. Rev. D112(2025) no. 10, 103535, arXiv:2503.10485 [astro-ph.CO]

    arXiv 2025

  18. [18]

    Strong constraints on a simple self-interacting neutrino cosmology,

    D. Camarena and F.-Y. Cyr-Racine, “Strong constraints on a simple self-interacting neutrino cosmology,”Phys. Rev. D111(2025) no. 2, 023504,arXiv:2403.05496 [astro-ph.CO]

    arXiv 2025

  19. [19]

    Selfinteracting warm dark matter,

    S. Hannestad and R. J. Scherrer, “Selfinteracting warm dark matter,”Phys. Rev. D62(2000) 043522,arXiv:astro-ph/0003046

    Pith/arXiv arXiv 2000

  20. [20]

    Boltzmann hierarchy for interacting neutrinos I: formalism,

    I. M. Oldengott, C. Rampf, and Y. Y. Y. Wong, “Boltzmann hierarchy for interacting neutrinos I: formalism,”JCAP04(2015) 016,arXiv:1409.1577 [astro-ph.CO]

    Pith/arXiv arXiv 2015

  21. [21]

    Interacting neutrinos in cosmology: exact description and constraints,

    I. M. Oldengott, T. Tram, C. Rampf, and Y. Y. Y. Wong, “Interacting neutrinos in cosmology: exact description and constraints,”JCAP11(2017) 027,arXiv:1706.02123 [astro-ph.CO]

    Pith/arXiv arXiv 2017

  22. [22]

    Cosmological perturbation theory in the synchronous and conformal Newtonian gauges,

    C.-P. Ma and E. Bertschinger, “Cosmological perturbation theory in the synchronous and conformal Newtonian gauges,”Astrophys. J.455(1995) 7–25,arXiv:astro-ph/9506072

    Pith/arXiv arXiv 1995

  23. [23]

    Dodelson and F

    S. Dodelson and F. Schmidt,Modern Cosmology. Academic Press, 2020

    2020

  24. [24]

    The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes,

    D. Blas, J. Lesgourgues, and T. Tram, “The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes,”JCAP07(2011) 034,arXiv:1104.2933 [astro-ph.CO]

    Pith/arXiv arXiv 2011

  25. [25]

    I. S. Gradshte˘ ın and I. M. Ryzhik,Table of Integrals, Series, and Products. Academic Press, San Diego, California, USA, fifth edition ed., 1994

    1994

  26. [26]

    Range-Separated Exchange Functionals with Slater-Type Functions,

    M. Seth and T. Ziegler, “Range-Separated Exchange Functionals with Slater-Type Functions,” Journal of Chemical Theory and Computation8(2012) no. 3, 901–907. https://pubs.acs.org/doi/10.1021/ct300006h

    work page doi:10.1021/ct300006h 2012

  27. [27]

    Weaker yet again: mass spectrum-consistent cosmological constraints on the neutrino lifetime,

    J. Z. Chen, I. M. Oldengott, G. Pierobon, and Y. Y. Y. Wong, “Weaker yet again: mass spectrum-consistent cosmological constraints on the neutrino lifetime,”Eur. Phys. J. C82 (2022) no. 7, 640,arXiv:2203.09075 [hep-ph]

    arXiv 2022

  28. [28]

    HYPERDIRE, HYPERgeometric functions DIfferential REduction: MATHEMATICA-based packages for differential reduction of generalized hypergeometric functionspFp−1,F 1,F2,F3,F4,

    V. V. Bytev, M. Y. Kalmykov, and B. A. Kniehl, “HYPERDIRE, HYPERgeometric functions DIfferential REduction: MATHEMATICA-based packages for differential reduction of generalized hypergeometric functionspFp−1,F 1,F2,F3,F4,”Comput. Phys. Commun.184 (2013) 2332–2342,arXiv:1105.3565 [math-ph]

    Pith/arXiv arXiv 2013

  29. [29]

    The Computation of Classical Constants,

    D. V. Chudnovsky and G. V. Chudnovsky, “The Computation of Classical Constants,” Proceedings of the National Academy of Sciences86(1989) no. 21, 8178–8182. – 27 –

    1989