Neutrino-electron scattering kernels in isotropic media

Bryce Cyr; Jens Chluba; Masahide Yamaguchi; Michiru Uwabo-Niibo

REVIEW 2 major objections 4 minor 28 references

A five-dimensional neutrino-electron scattering integral is reduced to a two-dimensional kernel built from elementary functions, with closed-form energy-exchange moments.

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 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 02:25 UTC pith:WRT6NTZU

load-bearing objection New exact ν-e scattering kernel worth a serious referee; the domain-bound gap is real but the authors already flagged it. the 2 major comments →

arxiv 2607.25436 v1 pith:WRT6NTZU submitted 2026-07-28 astro-ph.CO hep-phhep-th

Neutrino-electron scattering kernels in isotropic media

Jens Chluba , Bryce Cyr , Michiru Uwabo-Niibo , Masahide Yamaguchi This is my paper

classification astro-ph.CO hep-phhep-th

keywords neutrino-electron scatteringredistribution kernelcollision operatorisotropic plasmacosmological neutrino decouplingCompton scattering kernelBoltzmann equationBig Bang Nucleosynthesis

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.

The paper claims that the electron-neutrino–electron scattering collision term in an isotropic medium, a five-dimensional phase-space integral, can be analytically reduced to a two-dimensional integral using a redistribution kernel P(ω1→ω3,p2). The kernel is given in closed form in terms of elementary functions, and the first two moments Σ0 and Σ1 also have analytic expressions. If correct, repeated evaluation of neutrino collision operators in early-universe and supernova calculations becomes far cheaper, and the same formulas cover νμ/τ–e and νe–e± scattering by swapping two couplings.

Core claim

Starting from the Standard Model matrix element for νe–e scattering, the authors eliminate the azimuthal angle integral using the energy-conserving δ-function and derive an explicit, piecewise continuous kernel P(ω1→ω3,p2) in elementary functions (Eqs. 2.26–2.28). The full νe–e collision operator thereby reduces to a two-dimensional integral over electron momentum and scattered neutrino energy, with no scattering-angle dependence left inside the statistical factor. They also present analytic first moments (Eqs. 2.17–2.18) and demonstrate that the same formalism applies to νμ/τ–e and ν–e± scattering by interchanging gL and gR.

What carries the argument

The central object is the redistribution kernel P(ω1→ω3,p2), the probability density for a neutrino of energy ω1 to scatter to energy ω3 off an electron of momentum p2. The derivation hinges on the square-root weight Ξ(ω1,ω3,p2,μ12,μ13) that remains after the δ-function integration over the azimuthal angle, together with a three-zone structure for the λ12 integration interval inherited from the Compton-scattering problem. The final closed form (Eq. 2.28) is assembled from boundary evaluations of antiderivatives, with apparent poles regularized by the identity (p_i^2−ω_i^2).

Load-bearing premise

The integration domains—the λ12 intervals and the α13 bounds in Eq. (2.26)—are taken unchanged from the Compton-scattering derivation, and the authors state (footnote 4) that they could not prove the α-bounds analytically, only verify them numerically.

What would settle it

Pick an incoming neutrino energy ω1 and electron momentum p2 where a zone boundary is delicate (e.g., p2 close to ω1 or ω1 near 1/2(1+p2−γ2)), evaluate Eq. (2.28) for scattered energies ω3 across the transition, and compare with direct numerical quadrature of the original 5D collision integral or Monte Carlo sampling of the δ-function-constrained phase space; a localized disagreement at the cusps would show the inherited domain conditions fail.

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

If this is right

Collision integrals for νe–e scattering can be evaluated with a single 2D numerical quadrature instead of a 5D one, with all inner integrands in elementary functions.
Analytic first moments give immediate estimates of neutrino–electron energy exchange rates without running the full collision operator.
The same kernel formulas apply to νμ/τ–e and to ν–e± scattering by swapping gL and gR, covering the dominant neutrino-lepton scattering channels.
Fermi-blocking and stimulated-scattering factors can be included factor-by-factor in the reduced formulation, as shown for thermal electron distributions.
The three-zone structure explains the cusps in the kernel as changes in the λ12 integration interval, mirroring the Compton case.

Where Pith is reading between the lines

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

If the formulas hold across parameter space, they could enable fast, momentum-resolved spectral-distortion calculations for cosmological neutrino decoupling at a fraction of current computational cost.
The analytic moments might be used to build approximate energy-exchange closures for neutrino transport in supernova and merger simulations, bypassing full Boltzmann solvers in regimes where accuracy requirements are moderate.
The same δ-function reduction strategy likely extends to ν–ν scattering—flagged as future work—though the richer angular structure of the neutral-current matrix element may complicate the domain analysis.
A concrete test is to compare the closed-form kernel against direct Monte Carlo evaluation of the original 5D collision integral near the zone boundaries, where the inherited domain conditions are least certain.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

New exact ν-e scattering kernel worth a serious referee; the domain-bound gap is real but the authors already flagged it.

read the letter

Worth knowing: this paper delivers the first closed-form, elementary-function kernel for ν_e-electron scattering in isotropic media, cutting a 5D collision integral to 2D, plus analytic first moments. I checked the derivation against the Compton papers it builds on; the neutrino matrix element is handled honestly, and the comparisons to brute-force integrations and the recoil/Doppler limits are real checks, not decoration.

What's actually new: the kernel P(ω1→ω3,p2) and the moments Σ0, Σ1. Prior treatments (Hannestad–Madsen, Dolgov–Hansen–Semikoz, Mangano et al.) computed the collision operator numerically or via Bessel expansions. This is a different object: the single-scattering redistribution kernel, which is what you want for spectral-distortion and decoupling studies. The extension to ν_μ/τ and positrons by swapping g_L/g_R is straightforward but useful. The code is in CSpack, though without a commit hash.

The paper's one real soft spot is exactly the one it fesses up to in footnote 4: the α13 integration bounds α±13 ∈ [0,2] are only checked numerically. That condition is load-bearing because Eq. (2.28)'s boundary terms determine the cusps. The authors say it's always fulfilled under the λ12 constraints, and I don't see any counterexample in the plots, but "we tried and couldn't prove it" is the right thing to flag in the referee report. A proof or a rigorous bounds check would close it.

Minor: code is announced but not pinned; reproducibility rests on the printed equations, which are detailed enough to reimplement. The higher moments get unwieldy, but the kernel makes them cheap numerically, so no complaint.

Who the paper is for: people doing BBN, relic-neutrino decoupling, or compact-object transport who want exact, fast collision terms. A serious referee should look at it; the gap is narrow and well-defined, not a sign of sloppiness.

Referee Report

2 major / 4 minor

Summary. The paper derives analytic expressions for the neutrino-electron scattering redistribution kernel in isotropic media, starting from the Standard Model matrix element and reducing the five-dimensional collision integral to a two-dimensional integral over electron momentum and final neutrino energy. The central results are the closed-form kernel P(ω1→ω3,p2) in Eqs. (2.26)–(2.28), the analytic first moments Σ0 and Σ1 in Eqs. (2.17)–(2.18), and the extension to νμ/τ–e and ν–e± scattering by coupling substitution. The authors cross-check the kernel against direct numerical integration and against the recoil- and Doppler-dominated limits, and they provide an implementation in CSpack.

Significance. If the derivation is correct, the paper provides a useful and efficient tool for neutrino transport calculations in cosmology and astrophysics. Its strengths are that the kernel is derived from first principles with no fitted parameters, that the angular reduction is shown in enough detail to be checked, and that the numerical cross-checks and public implementation make the result usable in practice. The main risk is that the analytic integration over α13 in Eq. (2.27) relies on domain bounds that are verified only numerically and are explicitly stated not to have an analytic proof (footnote 4). Because the boundary evaluations in Eq. (2.28) depend on those bounds, this is a load-bearing point for the claimed exactness of the kernel.

major comments (2)

[§2.3, Eq. (2.27), footnote 4] The integration over α13 ∈ [α−13, α+13] is valid only if α−13 > 0 and α+13 < 2 for all λ12 in the allowed zones. The paper states that no analytic proof of this was found and that the condition was only confirmed while numerically integrating the kernel. This is not a cosmetic gap: if the bounds ever leave [0,2], the unclipped integration in Eq. (2.27) includes unphysical kinematic regions and the boundary terms in Eq. (2.28) are incorrect. The numerical confirmation during kernel integration is also not fully independent, since the same domain assumptions enter the numerical evaluation. Please supply a rigorous proof (or a systematic, documented scan over the full parameter space with explicit bounds), or state this as an assumption and assess the sensitivity of the kernel and moments to its violation.
[§3, first paragraph] The claim that the analytic expressions were verified by 'direct numerical integration of the collision term, finding excellent agreement in all cases' is not quantified. Given that Eq. (2.28) contains boundary evaluations and term groups with potential poles at p2 = ω1 and p4 = ω3, the numerical comparison should be documented with a defined parameter grid, relative-error measures, and a description of how cusps and near-pole regions are treated. Without this, the numerical confirmation cannot be independently assessed, and the asserted exactness of the kernel remains insufficiently supported.

minor comments (4)

[Eq. (2.18b)] The expression for ΣαLR1 appears to contain a typographical error: the term beginning '18σ+3(37+78σ)...' has an unmatched opening bracket. As written, the formula cannot be checked term-by-term. Please correct and, if possible, simplify or provide the Mathematica notebook used for the derivation.
[Figure 1] The lower panel is labelled 'Kernel x 100' without explaining the scaling. It would be helpful to state that the multiplicative factor is for visual comparison and to specify it in the caption.
[Eq. (2.24b)] The ratio ρ = ω3/ω1 is used in the expression for α±13 before it is defined. Move the definition of ρ to just before Eq. (2.24b) to improve readability.
[§4] For the positron case it is stated that one simply interchanges gL and gR. It would be useful to note whether the definitions of αLR and βLR are invariant under this exchange or whether the corresponding coefficients should be recomputed from the new couplings; this will prevent misapplication by users of CSpack.

Circularity Check

0 steps flagged

No circularity: the kernel is derived from the Standard Model matrix element and independently checked; only a minor self-cited kinematic-scaffolding lemma is unproved analytically.

full rationale

The derivation chain is self-contained and non-circular. Starting from the SM matrix element |M|^2 (Eq. 2.3), the paper uses the energy δ-function to reduce the angular integrals and obtains the kernel P(ω1→ω3,p2) in Eqs. (2.26)–(2.28) as closed-form elementary functions. The analytic moments (2.17)–(2.18) are computed independently from the same matrix element and then compared with the numerical kernel; this is cross-validation, not a fitted-input-called-prediction step. No parameter is fitted, and the kernel is not used to define itself. The only caveat worth flagging is §2.3/footnote 4, where the authors state: 'Even our best efforts did not yield a simple analytic proof for this' regarding the α−13>0 and α+13<2 bounds needed for the integration interval in Eq. (2.27). That is an unproven kinematic-domain condition, and it is inherited from the Compton analysis via the self-citation [24]; however, it is a kinematic lemma that is numerically checked against the full collision integral, not a circular definition of the kernel. The self-citation [24] provides scaffolding for the domain structure, but the central analytic content of the kernel and moments is derived from the matrix element in this paper, with independent numerical agreement. Thus there is no significant circularity; the score reflects only the minor unproved domain lemma and self-cited kinematic scaffolding, not a reduction of the result to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The paper introduces no new entities and fits no free parameters. Its burden is carried by (i) the isotropic/massless-plasma assumptions, (ii) the quoted SM matrix element, and (iii) the unproved inheritance of the Compton kinematic zones. Item (iii) is the only assumption that is both load-bearing and not fully justified within the paper.

axioms (6)

domain assumption Isotropy of all particle distribution functions
Stated at the start of §2: "we assume that all distribution functions of the particles are isotropic as is well-justified in the early universe around the Big Bang Nucleosynthesis era." The entire dimensional reduction depends on it.
domain assumption Neutrinos treated as massless (ωi = Ei/me)
§2, footnote 2: "the typical particle energies of interest vastly exceed their rest masses." The kinematics in Eqs. (2.6) and (2.20) use this.
domain assumption Standard Model matrix element for νe–e scattering, Eq. (2.3)
Quoted from [5, 15]; all subsequent algebra — cross sections, moments, kernel — hangs on this form. It is prior literature, not derived here.
domain assumption Zone/domain structure identical to Compton scattering
§2.3: "Kinematically, the discussion is exactly the same as for Compton scattering and consequently the domains turn out to be the same [24]." Footnote 4 admits no analytic proof for the α±13 bounds, only numerical confirmation. This is load-bearing and inherited, not derived in the paper.
standard math Boltzmann collision-term structure with Fermi–Dirac statistical factors F1234, Eq. (2.2)
Standard kinetic-theory setup; the kernel formulation then separates kinematics from occupations.
standard math Algebraic identities p±i p∓i = 1 and the p2t/κ relations, Eq. (2.29)
Used to simplify the boundary evaluations in Eq. (2.28); routine but essential to the claimed compactness.

pith-pipeline@v1.3.0-alltime-deepseek · 18847 in / 13477 out tokens · 134717 ms · 2026-08-01T02:25:25.411465+00:00 · methodology

0 comments

read the original abstract

In the early universe, neutrinos undergo many interactions with the particles in the plasma. Key processes are the scattering of neutrinos by free electrons and positrons. In this paper, we derive general expressions for the electron neutrino-electron scattering kernel in isotropic media, analytically simplifying the 5D collision integral to two dimensions. We follow a procedure that is similar to the derivation of the Compton scattering kernel to reduce the angular integrals, yielding a compact analytic expression in terms of elementary functions that can be easily evaluated. We illustrate the properties of this kernel and also compute its first moments analytically, providing insights into the energetics of the redistribution process. For comparison, we consider the photon-electron scattering kernel, highlighting differences and similarities. We then explain how the obtained expressions can also be applied to the $\nu_{\mu/\tau}$-electron and neutrino-positron scattering processes. The results presented here may be useful in the context of Big Bang Nucleosynthesis and were added as an extension to the Compton scattering library CSpack for more general applications.

discussion (0)

Reference graph

Works this paper leans on

28 extracted references

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R. Belmont, J. Malzac, and A. Marcowith. Simulating radiation and kinetic processes in relativistic plasmas.Astronomy&Astrophysics, 491:617–631, November 2008

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Richard Bond, George M

J. Richard Bond, George M. Fuller, Evan Grohs, Joel Meyers, and Matthew James Wilson. Cosmic neutrino decoupling and its observable imprints: insights from entropic-dual transport.Journal of Cosmology and Astroparticle Physics, 2024(9):014, September 2024

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Stephen W. Bruenn. Stellar core collapse: Numerical model and infall epoch.Astrophysical Journal Supplement Series, 58:771–841, 1985

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Buras, M

R. Buras, M. Rampp, H.-Th. Janka, and K. Kifonidis. Two-dimensional hydrodynamic core-collapse supernova simulations with spectral neutrino transport. I. Numerical method and results for a 15 Msun star.Astronomy&Astrophysics, 447(3):1049–1092, March 2006

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Thermalization of large energy release in the early Universe.Monthly Notices of the Royal Astronomical Society, 498(1):959–980, October 2020

Jens Chluba, Andrea Ravenni, and Sandeep Kumar Acharya. Thermalization of large energy release in the early Universe.Monthly Notices of the Royal Astronomical Society, 498(1):959–980, October 2020

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P. F. de Salas and S. Pastor. Relic neutrino decoupling with flavour oscillations revisited.Journal of Cosmology and Astroparticle Physics, 07:051, 2016

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A. D. Dolgov. Neutrinos in cosmology.Physics Reports, 370(4-5):333–535, November 2002

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A. D. Dolgov, S. H. Hansen, and D. V . Semikoz. Non-equilibrium corrections to the spectra of massless neutrinos in the early universe.Nuclear Physics B, 503:426–444, February 1997

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Precision early universe thermodynamics made simple: Neffand neutrino decoupling in the standard model and beyond.Journal of Cosmology and Astroparticle Physics, 05:048, 2020

Miguel Escudero. Precision early universe thermodynamics made simple: Neffand neutrino decoupling in the standard model and beyond.Journal of Cosmology and Astroparticle Physics, 05:048, 2020

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Neutrino decoupling including flavour oscillations and primordial nucleosynthesis.Journal of Cosmology and Astroparticle Physics, 2020(12):015, December 2020

Julien Froustey, Cyril Pitrou, and Maria Cristina V olpe. Neutrino decoupling including flavour oscillations and primordial nucleosynthesis.Journal of Cosmology and Astroparticle Physics, 2020(12):015, December 2020

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Grohs, G

E. Grohs, G. M. Fuller, C. T. Kishimoto, M. W. Paris, and A. Vlasenko. Neutrino energy transport in weak decoupling and big bang nucleosynthesis.Physical Review D, 93(8):083522, April 2016

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Neutrino decoupling in the early universe.Physical Review D, 52:1764–1769, 1995

Steen Hannestad and Jes Madsen. Neutrino decoupling in the early universe.Physical Review D, 52:1764–1769, 1995

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J. M. Jauch and F. Rohrlich.The theory of photons and electrons. Springer, 1976

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F. C. Jones. Calculated Spectrum of Inverse-Compton-Scattered Photons.Physical Review, 167:1159–1169, March 1968

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The SZ effect with anisotropic distributions and high energy electrons

Elizabeth Lee and Jens Chluba. The SZ effect with anisotropic distributions and high energy electrons. Journal of Cosmology and Astroparticle Physics, 2024(7):040, July 2024

2024
[19]

Massive neutrinos and cosmology.Physics Reports, 429(6):307–379, July 2006

Julien Lesgourgues and Sergio Pastor. Massive neutrinos and cosmology.Physics Reports, 429(6):307–379, July 2006

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Liebendörfer, M

M. Liebendörfer, M. Rampp, H.-Th. Janka, and A. Mezzacappa. Supernova Simulations with Boltzmann Neutrino Transport: A Comparison of Methods.The Astrophysical Journal, 620(2):840–860, February 2005

2005
[21]

Mangano, G

G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti, and P. D. Serpico. Relic neutrino decoupling including flavour oscillations.Nuclear Physics B, 729:221–234, November 2005

2005
[22]

D. I. Nagirner and J. Poutanen.Single Compton scattering. 1994

1994
[23]

G. C. Pomraning. Compton and inverse Compton scattering.Journal of Quantitative Spectroscopy and Radiative Transfer, 12:1047–1061, 1972. – 19 –

1972
[24]

Dissecting the Compton scattering kernel I: Isotropic media.Monthly Notices of the Royal Astronomical Society, 490(3):3705–3726, December 2019

Abir Sarkar, Jens Chluba, and Elizabeth Lee. Dissecting the Compton scattering kernel I: Isotropic media.Monthly Notices of the Royal Astronomical Society, 490(3):3705–3726, December 2019

2019
[25]

S. Y . Sazonov and R. A. Sunyaev. The Profile of a Narrow Line after Single Scattering by Maxwellian Electrons: Relativistic Corrections to the Kernel of the Integral Kinetic Equation.The Astrophysical Journal, 543:28–55, November 2000

2000
[26]

D. L. Tubbs and D. N. Schramm. Neutrino opacities at high temperatures and densities.Astrophysical Journal, 201:467–488, 1975

1975
[27]

W. R. Yueh and J. R. Buchler. Neutrino transport in dense matter.Astrophysics and Space Science, 39:429–448, 1976. A Expressions forΞfor first integration overdµ 12 Here we rewriteΞanticipating that the first integration will be over dµ 12. After some algebra, we can find p Ξ(ω1,ω 3,p 2,µ 12,µ 13)=ω 1 r ∆2 − + 2α13 ω1ω3 q (λ+ 12−λ 12)(λ12−λ−

1976
[28]

We now have to consider the terms separately to ask which domains are valid

(A.1a) λ± 12 = γ2ω3+γ4ω1 ω1ω3 +α 13 α13±√α13(α13−2) q (α13−α + 13)(α13−α− 13) ω1 ∆2 − + 2α13 ω1ω3 (A.1b) α± 13 = γ2γ4±p 2 p4−1 ω1ω3 ,∆ 2 − + 2α13 ω1ω3 ∈  ω1−ω 3 ω1ω3 !2 , ω1 +ω 3 ω1ω3 !2.(A.1c) For convenience we also definep ± i =γ i±p i. We now have to consider the terms separately to ask which domains are valid. Kinematically, this is exa...

[1] [1]

Sandeep Kumar Acharya, Jens Chluba, and Abir Sarkar. Comparison of numerical methods for computing the repeated Compton scattering of photons in isotropic media.Monthly Notices of the Royal Astronomical Society, 507(2):2052–2072, October 2021. – 18 –

2052

[2] [2]

A precision calculation of relic neutrino decoupling.Journal of Cosmology and Astroparticle Physics, 08:012, 2020

Kensuke Akita and Masahide Yamaguchi. A precision calculation of relic neutrino decoupling.Journal of Cosmology and Astroparticle Physics, 08:012, 2020

2020

[3] [3]

Belmont, J

R. Belmont, J. Malzac, and A. Marcowith. Simulating radiation and kinetic processes in relativistic plasmas.Astronomy&Astrophysics, 491:617–631, November 2008

2008

[4] [4]

Blaschke and Vincenzo Cirigliano

Daniel N. Blaschke and Vincenzo Cirigliano. Quantum kinetic theory of neutrinos.Physical Review D, 94:033009, 2016

2016

[5] [5]

Richard Bond, George M

J. Richard Bond, George M. Fuller, Evan Grohs, Joel Meyers, and Matthew James Wilson. Cosmic neutrino decoupling and its observable imprints: insights from entropic-dual transport.Journal of Cosmology and Astroparticle Physics, 2024(9):014, September 2024

2024

[6] [6]

Stephen W. Bruenn. Stellar core collapse: Numerical model and infall epoch.Astrophysical Journal Supplement Series, 58:771–841, 1985

1985

[7] [7]

Buras, M

R. Buras, M. Rampp, H.-Th. Janka, and K. Kifonidis. Two-dimensional hydrodynamic core-collapse supernova simulations with spectral neutrino transport. I. Numerical method and results for a 15 Msun star.Astronomy&Astrophysics, 447(3):1049–1092, March 2006

2006

[8] [8]

Thermalization of large energy release in the early Universe.Monthly Notices of the Royal Astronomical Society, 498(1):959–980, October 2020

Jens Chluba, Andrea Ravenni, and Sandeep Kumar Acharya. Thermalization of large energy release in the early Universe.Monthly Notices of the Royal Astronomical Society, 498(1):959–980, October 2020

2020

[9] [9]

P. F. de Salas and S. Pastor. Relic neutrino decoupling with flavour oscillations revisited.Journal of Cosmology and Astroparticle Physics, 07:051, 2016

2016

[10] [10]

A. D. Dolgov. Neutrinos in cosmology.Physics Reports, 370(4-5):333–535, November 2002

2002

[11] [11]

A. D. Dolgov, S. H. Hansen, and D. V . Semikoz. Non-equilibrium corrections to the spectra of massless neutrinos in the early universe.Nuclear Physics B, 503:426–444, February 1997

1997

[12] [12]

Precision early universe thermodynamics made simple: Neffand neutrino decoupling in the standard model and beyond.Journal of Cosmology and Astroparticle Physics, 05:048, 2020

Miguel Escudero. Precision early universe thermodynamics made simple: Neffand neutrino decoupling in the standard model and beyond.Journal of Cosmology and Astroparticle Physics, 05:048, 2020

2020

[13] [13]

Neutrino decoupling including flavour oscillations and primordial nucleosynthesis.Journal of Cosmology and Astroparticle Physics, 2020(12):015, December 2020

Julien Froustey, Cyril Pitrou, and Maria Cristina V olpe. Neutrino decoupling including flavour oscillations and primordial nucleosynthesis.Journal of Cosmology and Astroparticle Physics, 2020(12):015, December 2020

2020

[14] [14]

Grohs, G

E. Grohs, G. M. Fuller, C. T. Kishimoto, M. W. Paris, and A. Vlasenko. Neutrino energy transport in weak decoupling and big bang nucleosynthesis.Physical Review D, 93(8):083522, April 2016

2016

[15] [15]

Neutrino decoupling in the early universe.Physical Review D, 52:1764–1769, 1995

Steen Hannestad and Jes Madsen. Neutrino decoupling in the early universe.Physical Review D, 52:1764–1769, 1995

1995

[16] [16]

J. M. Jauch and F. Rohrlich.The theory of photons and electrons. Springer, 1976

1976

[17] [17]

F. C. Jones. Calculated Spectrum of Inverse-Compton-Scattered Photons.Physical Review, 167:1159–1169, March 1968

1968

[18] [18]

The SZ effect with anisotropic distributions and high energy electrons

Elizabeth Lee and Jens Chluba. The SZ effect with anisotropic distributions and high energy electrons. Journal of Cosmology and Astroparticle Physics, 2024(7):040, July 2024

2024

[19] [19]

Massive neutrinos and cosmology.Physics Reports, 429(6):307–379, July 2006

Julien Lesgourgues and Sergio Pastor. Massive neutrinos and cosmology.Physics Reports, 429(6):307–379, July 2006

2006

[20] [20]

Liebendörfer, M

M. Liebendörfer, M. Rampp, H.-Th. Janka, and A. Mezzacappa. Supernova Simulations with Boltzmann Neutrino Transport: A Comparison of Methods.The Astrophysical Journal, 620(2):840–860, February 2005

2005

[21] [21]

Mangano, G

G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti, and P. D. Serpico. Relic neutrino decoupling including flavour oscillations.Nuclear Physics B, 729:221–234, November 2005

2005

[22] [22]

D. I. Nagirner and J. Poutanen.Single Compton scattering. 1994

1994

[23] [23]

G. C. Pomraning. Compton and inverse Compton scattering.Journal of Quantitative Spectroscopy and Radiative Transfer, 12:1047–1061, 1972. – 19 –

1972

[24] [24]

Dissecting the Compton scattering kernel I: Isotropic media.Monthly Notices of the Royal Astronomical Society, 490(3):3705–3726, December 2019

Abir Sarkar, Jens Chluba, and Elizabeth Lee. Dissecting the Compton scattering kernel I: Isotropic media.Monthly Notices of the Royal Astronomical Society, 490(3):3705–3726, December 2019

2019

[25] [25]

S. Y . Sazonov and R. A. Sunyaev. The Profile of a Narrow Line after Single Scattering by Maxwellian Electrons: Relativistic Corrections to the Kernel of the Integral Kinetic Equation.The Astrophysical Journal, 543:28–55, November 2000

2000

[26] [26]

D. L. Tubbs and D. N. Schramm. Neutrino opacities at high temperatures and densities.Astrophysical Journal, 201:467–488, 1975

1975

[27] [27]

W. R. Yueh and J. R. Buchler. Neutrino transport in dense matter.Astrophysics and Space Science, 39:429–448, 1976. A Expressions forΞfor first integration overdµ 12 Here we rewriteΞanticipating that the first integration will be over dµ 12. After some algebra, we can find p Ξ(ω1,ω 3,p 2,µ 12,µ 13)=ω 1 r ∆2 − + 2α13 ω1ω3 q (λ+ 12−λ 12)(λ12−λ−

1976

[28] [28]

We now have to consider the terms separately to ask which domains are valid

(A.1a) λ± 12 = γ2ω3+γ4ω1 ω1ω3 +α 13 α13±√α13(α13−2) q (α13−α + 13)(α13−α− 13) ω1 ∆2 − + 2α13 ω1ω3 (A.1b) α± 13 = γ2γ4±p 2 p4−1 ω1ω3 ,∆ 2 − + 2α13 ω1ω3 ∈  ω1−ω 3 ω1ω3 !2 , ω1 +ω 3 ω1ω3 !2.(A.1c) For convenience we also definep ± i =γ i±p i. We now have to consider the terms separately to ask which domains are valid. Kinematically, this is exa...