Nonequilibrium Corrections to the Spectra of Massless Neutrinos in the Early Universe

A.D. Dolgov; D.V. Semikoz; S.H. Hansen

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Nonequilibrium Corrections to the Spectra of Massless Neutrinos in the Early Universe

A.D. Dolgov, S.H. Hansen, D.V. Semikoz

High Energy Physics - Phenomenology · Submitted 13 Mar 1997 · 10 references

Pith's verdict: A careful 1997 numerical study whose qualitative result has held up, but whose headline numbers are quoted at the top of their own convergence range.

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arXiv IDhep-ph/9703315 (version 1)

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CitationA.D. Dolgov, D.V. Semikoz, S.H. Hansen. "Nonequilibrium Corrections to the Spectra of Massless Neutrinos in the Early Universe." arXiv:hep-ph/9703315 (1997). https://pith.science/paper/STAPTYPJ

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Editor's note

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Worth your time if you care about N_eff, but read Table 3 before you quote the abstract back at anyone. What is genuinely new relative to Hannestad–Madsen (ref. [7]): the factor of two for identical-particle νν scattering, the inclusion of photon cooling through the full energy-conservation equation (6) rather than assuming entropy conservation in the electromagnetic sector, and the ΔY = 1.4e-4 number, which [7] did not compute. The photon-cooling point is not cosmetic — Fig. 2 shows it moves the energy-density correction by about 20%, and the explanation for why (the distortion is dominated by x ~ 1 where ΔT is still small, so a small shift in Tγ gets relatively amplified) is correct and clearly laid out. The validation is unusually good for the era: two formulations of the collision integral (D-functions versus explicit phase space in Appendix B), the identities (44) and (45) as an algebraic check on D1, D2, D3, two solution schemes, grid checks in both momentum and time, and a Bulirsch–Stoer cross-check. That is real evidence, not decoration. Where it is soft is the convergence bookkeeping. For νμ, Table 3 runs 0.25% at 100 momentum points, 0.33% at 200, and 0.40% in the δ-formulation, and the abstract quotes 0.40%. Section 3.3 calls the 100-versus-200 difference small; a 32% shift on doubling the grid is not small. Same pattern for νe: 0.79, 0.83, 0.94. The text says the full-distribution results are systematically smaller by about 10%, but for νμ the gap is 18 to 37%, so the stated size of the discrepancy understates it for the species whose number is smallest. And ΔY = 1.4e-4 is quoted bare, with no error bar, despite that spread. None of this threatens the sign or the rough magnitude — the later N_eff = 3.044 work confirms the picture — but the abstract's two numbers are the paper's product, and they sit at the top of their own range. One correction to the reader's report: the labelled weakest assumption, exact electromagnetic-plasma equilibrium, is actually well defended, since Section 1 gives the Γ >> H argument and Fig. 2 quantifies the sensitivity. The convergence issue is the better target. Send it out to a specialist in neutrino decoupling or BBN; this is not a broad-interest paper.

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Major objections (3)

[Table 3, Section 3.3, abstract:s2] Table 3 shows the νμ(τ) energy-density correction at 0.25-0.26% (100 momentum points), 0.33% (200 points, full distribution), and 0.40% (200 points, δ-formulation), while the abstract quotes 0.4%. That is a factor-1.6 spread across nominally equivalent runs, and the νe entry moves 0.79% → 0.83% → 0.94%. Section 3.3 states that 'the difference in the results is small, which means that a 100 points grid is a good approximation', but a 32% shift in the νμ number on doubling the grid is not small, and the text does not reconcile this with its own table. Section 3.3 also says the full-distribution results are 'systematically smaller by about 10%'; for νμ the gap is 18% (0.33 vs 0.40) and 37% (0.25 vs 0.40). The paper attributes the full-versus-δ difference to 'finite precision' without diagnosing which run is closer to the continuum limit, so there is no stated basis for preferring 0.40% over 0.33%. The qualitative effect and its sign are secure; the quoted precision is not. Please add an explicit convergence/error estimate and state which run is adopted and why.
[Section 3.4, Fig. 2, abstract:s3] The ΔY = 1.4×10^-4 is quoted without an uncertainty, although it is derived from the same δρν and δ(y) that Table 3 shows to vary by 20-30% between formulations. Fig. 2 further shows that switching between entropy-conserving and energy-conserving treatments of the photon sector changes δρν by about 20%. The paper adopts the energy-conserving treatment, which is the better choice, but the residual sensitivity to the assumption of exact thermal equilibrium and zero chemical potential for the e±γ plasma (Section 2) is not quantified. Please propagate the spread in δρν and δ(y) into an uncertainty on ΔY, and state explicitly what fraction of that uncertainty comes from the photon-sector modeling.
[Section 3.4, Fig. 5] The BBN modification uses δ_i(p,T_ν) 'fitted as functions of momentum and temperature', and Fig. 5 states that all curves are well approximated by δ = A y (y - B), but the coefficients A and B are not given for either species or for either statistics. Without them the ΔY = 1.4×10^-4 cannot be reproduced from the text alone, and the fit form cannot be checked against the plotted curves. Please tabulate A and B (with the fit range in y) for νe and νμ(τ), FD and MB.

Minor objections (5)

[Eq. (14)] As printed, Eq. (14) gives (T_eff - T_0)/T_0 = (1 - e^{-y}) δ/y. A direct expansion of T_eff = E/ln[(1-f)/f] with f = f_eq(1+δ) gives (1 + e^{-y}) δ/y, which reduces to 2δ/y at y→0; the printed factor vanishes there. Please check the sign/factor and the definition of T_0 relative to T_ν = 1/a.
[Eq. (46)] Eq. (46) is dimensionally inconsistent as printed: the right-hand side has dimensions of E^{-1} rather than E, and the 1/E1 that appears in the intermediate expression two lines above is missing. Eq. (50) retains its 1/E1. Please restore the factor and check the π-power (1/(3π^3) vs 1/(9π^2)) between the two.
[Section 4, ref. [7]] Section 4 attributes the residual differences with ref. [7] to 'different numerical accuracies'. Given the spread in Table 3, this attribution would be more convincing if the paper stated its own estimated accuracy on the same footing (e.g., the 100-vs-200 point difference for the quantity being compared).
[Section 2, Tables 1 and 2] The paper states that it takes a 'twice larger contribution' for νaνa → νaνa than ref. [7] because of identical particles in the initial state, and the symmetrization factor S is described in one sentence. Since this is a direct disagreement with a published result, please spell out the convention (the 1/2! for identical pairs and the factor 2 for identical initial particles) and, if possible, show the explicit phase-space count for the identical-particle case.
[Section 2] Section 2 says the two solution methods give results that 'are very close (see the next Section and Table 3)', but Table 3 shows 0.83% vs 0.94% for νe and 0.33% vs 0.40% for νμ. Please soften or quantify this statement.

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The paper

The paper claims

The paper argues that residual weak interactions with hotter electrons and positrons distort the cosmic neutrino spectra, raising the energy density of electron neutrinos by about 0.9% and of muon and tau neutrinos by about 0.4%, which in t

The reading

The paper asks whether cosmic neutrinos really keep a perfect thermal spectrum once they decouple, and argues that they do not: residual annihilation of hotter electrons and positrons into neutrino pairs, plus elastic neutrino-electron scattering, pushes the neutrino spectra out of equilibrium. Solving the coupled kinetic equations numerically, the authors find energy-density corrections of about 0.9% for electron neutrinos and 0.4% for muon and tau neutrinos. The effect matters because neutrino energy density and the weak reaction rates set the neutron-to-proton ratio before primordial nucleosynthesis; the authors propagate their distorted spectra into a standard nucleosynthesis code and find a $1.4$ times $10^{-4}$ increase in $^4$He. The paper also emphasizes that the answer is sensitive to how the photon-electron plasma cools, not just to the neutrino collision integrals.

What carries the argument

The machinery is a set of coupled integro-differential kinetic equations for the neutrino distribution functions $f_{nu_e}(x,y)$ and $f_{nu_mu}=f_{nu_tau}$, together with a covariant energy-conservation equation that determines the photon temperature $T_gamma(x)$. The collision integral is reduced from nine to two momentum integrations by using the four-fermion interaction, integrating out the momentum delta function and angular variables, and expressing the remaining angular dependence through the functions $D_1$, $D_2$, and $D_3$. The temperature difference between photons and neutrinos is the driving force, while elastic neutrino-neutrino scattering smooths the distortion. The authors solve the system both for the full distributions and for small deviations from equilibrium, and check the numerical result against Boltzmann-statistics runs and an independent treatment of the direct reaction terms.

Extended reading notes

Core claim

The central claim is that cosmic neutrinos do not retain an exactly equilibrium spectrum after weak decoupling. Because the electromagnetic plasma is heated by $e^+e^-$ annihilation, it stays slightly hotter than the neutrinos, and the residual reactions $e^+e^-$ to neutrino-antineutrino pairs and elastic neutrino-electron scattering drive the neutrino spectra away from equilibrium. The authors report relative energy-density corrections of approximately 0.9% for electron neutrinos and 0.4% for muon and tau neutrinos, with the electron neutrino more strongly affected because it has charged-current couplings to electrons. Feeding these corrected spectra into a nucleosynthesis calculation raises the primordial $^4$He abundance by $1.4$ times $10^{-4}$ and changes $^3$He and $^7$Li by amounts of order $5$ times $10^{-4}$ and $-6$ times $10^{-4}$, respectively.

Load-bearing premise

The calculation assumes that the photon-electron-positron plasma stays in exact thermal equilibrium with a single temperature and zero chemical potential, so the entire distortion is driven by the one temperature difference between that plasma and the neutrinos.

Editorial extensions

If this is right

The neutrino energy density is slightly higher than the equilibrium value, so the expansion rate of the universe around neutrino decoupling and nucleosynthesis is changed at the sub-percent level.
Primordial $^4$He increases by about $1.4$ times $10^{-4}$, with smaller relative changes in $^3$He and $^7$Li, so precision nucleosynthesis comparisons should include the distorted neutrino spectra.
The spectral correction is momentum-dependent and largest in the high-energy tail, so weak reaction rates need the full distortion $\delta(y)$, not only a shifted effective neutrino temperature.
Treating the electromagnetic plasma as entropy-conserving rather than including the energy transfer to neutrinos underestimates the distortion by roughly 20%, so the photon cooling law is part of the leading calculation.

Reading between the lines

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

If the correction is real, it shifts the effective number of relativistic neutrino species at nucleosynthesis by a small amount; disentangling that from other new-physics contributions would require matching the full momentum-dependent distortion rather than a single temperature shift.
The same mechanism should operate for any light relic that decouples while a hotter annihilating species is still present, so a spectral distortion of this shape may be a generic signature of imperfect decoupling.
Because the paper finds a 20% sensitivity to the photon cooling treatment, evolving the electromagnetic plasma kinetically alongside the neutrinos would be the natural next step beyond the paper's exact-equilibrium assumption for photons and electrons.

Original abstract

Distortion of the equilibrium spectra of cosmic neutrinos due to interaction with hotter electrons and positrons in the primeval cosmic plasma is considered. The set of integro-differential kinetic equations for neutrinos is accurately numerically solved. The relative corrections to neutrino energy densities are approximately 0.9% for $\nu_e$ and 0.4% for $\nu_\mu$ and $\nu_\tau$. This effect results in $1.4 \cdot 10^{-4}$ increase in the primordial $^4 He$ abundance.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.29 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.27 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Distortion of the equilibrium spectra of cosmic neutrinos due to interaction with hotter electrons and positrons in the primeval cosmic plasma is considered. The set of integro-differential kinetic equations for neutrinos is accurately numerically solved. The relative corrections to neutrino energy densities are approximately 0.9% for $\nu_e$ and 0.4% for $\nu_\mu$ and $\nu_\tau$. This effect results in $1.4 \cdot 10^{-4}$ increase in the primordial $^4 He$ abundance.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Distortion of the equilibrium spectra of cosmic neutrinos due to interaction with hotter electrons and positrons in the primeval cosmic plasma is considered.	background 0.51	0.44	0.07	asserted 0.81
s1	The set of integro-differential kinetic equations for neutrinos is accurately numerically solved.	method 0.99	0.92	0.05	asserted 1.00
s2	The relative corrections to neutrino energy densities are approximately 0.9% for $\nu_e$ and 0.4% for $\nu_\mu$ and $\nu_\tau$.	result 1.00	0.91	0.52	asserted 0.97
s3	This effect results in $1.4 \cdot 10^{-4}$ increase in the primordial $^4 He$ abundance.	result 0.73	0.85	0.44	asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.85 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The ratio Tγ/Tν as a function of the dimensionless ”time” x = ma for the cases of Fermi-Dirac and Maxwell-Boltzmann statistics. The value x = 1 corresponds to Tν = 1MeV

**Figure 2.** The relative distortion of the total neutrino energy density δρ/ρeq (for the definition see eq. (11)) as a function of x = ma for the case of ”entropy conservation” (upper curve) and energy conservation according to eq. (6) (lower curve). In the first case we neglect the energy exchange between neutrinos and e ± which is equivalent to the requirement of entropy conservation in the electromagnetic plasma

**Figure 3.** The ”time” evolution of the correction to the neutrino distribution functions δj = (fνj − f eq ν )/f eq ν for several fixed values of momentum y = 3, 5, 7. In Fig.3a and 3b we present respectively the results for electronic neutrinos, and for muonic (tau) ones

**Figure 4.** The evolution of the non-equilibrium corrections to the distribution functions for the momentum value y = 5 for the electronic (solid curve) and muonic (tau) (dotted curve) neutrinos in the cases of FD and MB statistics

**Figure 5.** The distortion of the neutrino spectra δj = (fνj − f eq ν )/f eq ν as functions of the dimensionless momentum y at the final ”time” x = 60. The dashed lines a and c correspond to Maxwell-Boltzmann statistics, while the solid lines b and d correspond to Fermi-Dirac statistics. The upper curves a and b are for electronic neutrinos, while the lower curves c and d are for muonic (tau) neutrinos. All the curves can be well approximated by a second order polynomial in y like δ = Ay(y − B). 25

Figures reproduced from arXiv:hep-ph/9703315 by the authors.

References

Works cited (10)

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[1]

S. Weinberg, Gravitation and Cosmology, Wiley, New York, 1972

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D.A. Dicus, E.W. Kolb, A.M. Gleeson, E.C.G. Sudarshan, V.L. Teplits, a nd M.S. Turner, Phys. Rev. D 26 (1982) 2694

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N.C. Raha and B. Mitra, Phys. Rev. D 44 (1991) 393

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S. Dodelson and M.S. Turner, Phys. Rev. D 46 (1992) 3372

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D.V. Semikoz and I.I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995); D.V. Semikoz and I.I. Tkachev, hep-ph/9507306, to appear in Phys.Rev.D

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W.H.Press et al., “Numerical Recipes in C: the art of scientiﬁc compu ting”, Cambridge U. Press, 1992

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[10]

L. Kawano, Fermilab-Pub-92/04-A (1992). 24 Figure Captions: Fig. 1 The ratio Tγ/Tν as a function of the dimensionless ”time” x = ma for the cases of Fermi-Dirac and Maxwell-Boltzmann statistics. The value x = 1 corresponds to Tν = 1M eV . Fig. 2 The relative distortion of the total neutrino energy density δρ/ρeq (for the deﬁnition see eq. (11)) as a funct...

View on Pith 1992

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73 later papers cite it

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1 later paper by other authors is listed below: 1 used its method or tool.

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Obtained its D_l functions as in this paper from the following integrals.

Relic neutrino decoupling with flavour oscillations revisited (2016) · 1.00 · paraphrased: the licence does not allow quoting

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69 arXiv papers cite this in 112 sentences

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Contests it 1 paper

We use squared-matrix elements for the collisional processes for active neutrinos consistent e.g. with the results of Table 3 of [ref] , modulo our different definition of $|M_fi|^2$, while we find once again that the results reported e.g. in [ref] are a factor two larger than the correct ones.

Massive sterile neutrinos in the early universe: From thermal decoupling to cosmological constraints · Equations of motion · contest 0.90

Builds on it 1 paper

This is an extension of the comprehensive study of this physics in the zero lepton-number case with the code [ref] , and in other works [ref] .

Lepton asymmetry, neutrino spectral distortions, and big bang nucleosynthesis · Introduction · 0.98

Uses its method or tool 10 papers

We employ a collision integral, $C_j$, originally derived in Refs. [ref] and modified in Ref. [ref] df_j(ϵ)/dt= C_j({f_k}), where the collision integral for $f_j(ϵ)$ is a functional of the entire set of occupation numbers ${f_k}$.

Insights into neutrino decoupling gleaned from considerations of the role of electron mass · Full Neutrino Transport and Nonequilibrium Energy Distributions · 0.94
Fortunately, by applying a technique developed in Refs. [ref] [For practical use, we refer the readers to Appendix A of Ref. [ref] and Appendix D in Ref. [ref] .], we can drastically reduce it to a 3-dimensional integral.

Dirac neutrinos and $N_{{\rm eff}}$ · Boltzmann equation · 1.00
As in previous analyses of neutrino decoupling in the early universe [ref] , in our calculations the collision terms for the diagonal components $ϱ_αα$ (the distribution functions of flavour neutrinos $f_ν_α$) are solved numerically after they are analytically reduced to two-dimensional integrals, following the process described in [ref] .

Relic neutrino decoupling with flavour oscillations revisited · Boltzmann equations · 0.97

Uses its data or values 12 papers

The current best-fit value from the latest Planck 2018+BAO data [ref] , $N_eff=2.99 ±0.17$, is in agreement with the SM prediction, $N_eff(SM) = 3.046$ [ref] .

Hot Axions and the $H_0$ tension · Introduction · 0.96
Within the SM, the prediction taking into account the non-instantaneous neutrino decoupling is $N_eff^SM = 3.044$ [ref] , whereas the presence of GW implies [equation] Since $ρ_GW∝a^-4$ and $ρ_γ∝T^4$, the above relation can be utilized to put a constraint on the GW energy density redshifted to today via [equation] where, as before, $Ω_γ^(0) h^2≃2.47×10^-5$.

Measuring Inflaton Couplings via Primordial Gravitational Waves · $$ · 0.94
Within the SM, taking the non-instantaneous neutrino decoupling into account, one finds $N_eff^SM = 3.044$ [ref] , while the presence of GW results in a modification [equation] The above relation can be utilized to put a constraint on the GW energy density red-shifted to today via [ref] [equation] which leads to $^(0) h^2≃5.62×10^-6$ where, $Ω_γ^(0) h^2≃2.47×10^-5$ is the relic photon abundance at the present epoch while $f=k/(2πa_0)$ is the present day frequency of the physical wave number $k$.

Testing Axionic Dark Matter during Gravitational Reheating · Inflationary gravitational wave · 0.92

Compares with it 3 papers

The results of the present work are compared with those of [ref] .

Restrictions on the lifetime of sterile neutrinos from primordial nucleosynthesis · Standard Model BBN · 1.00
[This is explicitly shown in their Eq. (D.1), where their factor two larger value in $|M|^2$ is compensated by the further $1/g$ in their integral prefactor, $g=2$ being the spin multiplicity.] In relation instead to the Ref. [ref] , we find systematically lower matrix elements compared to their Tables 1, 2, by a factor 2.

Massive sterile neutrinos in the early universe: From thermal decoupling to cosmological constraints · Heavy sterile neutrino model · 0.95
Expression [ref] precisely matches the one obtained from Equation B.2 in [ref] .

Neutrino decoupling beyond the Standard Model: CMB constraints on the Dark Matter mass with a fast and precise $N_{\rm eff}$ evaluation · Neutrino collision terms in the SM in the Maxwell-Boltzmann approximation · 0.99

Cites it as background 14 papers

A precision calculation of $N_eff^SM$ requires one to solve the process of neutrino decoupling in the early Universe and this has been a subject of intense study for more than 40 years now, see [ref] for references devoted to this problem, see [ref] for reviews and [ref] for a book.

Neff in the Standard Model at NLO is 3.043 · Introduction · 1.00
However, the overlap between neutrino decoupling and $e^±$ annihilations, known as incomplete neutrino decoupling, leads to slightly non thermal neutrino spectra, and to an increased neutrino energy density (both typically of order 1%), which is usually described by an effective number of thermalised neutrinos $$ departing from $3$ [ref] .

Neutrino decoupling including flavour oscillations and primordial nucleosynthesis · Introduction · 0.82
First, several studies solved the Boltzmann equations for neutrino distribution functions [ref] .

A review of neutrino decoupling from the early universe to the current universe · Introduction · 0.93

Discussion

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