Neutrino decoupling including flavour oscillations and primordial nucleosynthesis

• Below ∼2 MeV, weak interactions decouple neutrinos from the electromagnetic plasma, forming the cosmic neutrino background.Source: §1 · p. 2
• Incomplete decoupling during e± annihilation produces nonthermal spectra and increases the neutrino energy density by typically about 1%, shifting Neff from 3.Source: §1 · p. 3
• Flavour oscillations modify the flavour dependence of spectral distortions, reducing electron-neutrino distortion and increasing distortions in other flavours relative to no oscillations.Source: §1 · p. 3
• Neutrino decoupling predictions provide initial conditions for later cosmological stages, including BBN, CMB formation, and structure formation.Source: §1 · pp. 3–4
• The study derives and solves full neutrino quantum kinetic equations, including complete matrix-structured collision terms, and develops an adiabatic approximation for zero chemical potential.Source: §1 · p. 4

• The BBGKY formalism generalizes Boltzmann evolution to neutrino density matrices containing masses, mixings, and quantum coherence.Source: §2 · p. 4, §2.1 · p. 5
• The one-body density matrix has diagonal entries corresponding to occupation numbers, while off-diagonal entries encode flavour coherence.Source: §2.1 · p. 5
• A hierarchy of reduced density matrices is truncated using a mean-field approximation and a molecular-chaos ansatz for two-body collision correlations.Source: §2.1 · p. 6
• The collision term has gain–loss plus Hermitian-conjugate structure and includes charged-lepton scattering, neutrino annihilation, and neutrino self-interactions.Source: §2.1 · p. 6, §2.2.3 · p. 9
• The early-universe setup assumes three active species in a homogeneous, isotropic plasma of electrons, positrons, and photons, neglecting baryons because their density is negligible.Source: §2.1 · p. 7, §2.2 · p. 7
• Unlike earlier treatments, the self-interaction collision contribution is derived with its full matrix structure rather than discarding or damping off-diagonal components.Source: §2.2.2 · p. 8, §2.2.3 · p. 9

• Rapid oscillations can be averaged because previous results indicate that oscillations evolve much faster than the comparatively slow collision-driven evolution.Source: §3 · p. 11
• ATAO assumes adiabatic matter-basis evolution, averaged off-diagonal components, and collision rates and variations small compared with effective oscillation frequencies.Source: §3 · p. 12
• The adiabatic condition requires effective mixing-matrix elements to vary slowly relative to effective oscillation frequencies, including across possible MSW resonances.Source: §3 · p. 12
• Averaging over many oscillations washes out matter-basis non-diagonal components when collisions are not too strong, leaving an effective diagonal evolution.Source: §3 · p. 12, Adiabatic Transfer of Averaged… · p. 12
• The authors solve both the full QKEs and the ATAO approximation to assess the approximation's validity.Source: Adiabatic Transfer of Averaged… · p. 14

4.3 Neutrino temperature and spectra

• Including O(e^3) QED plasma corrections lowers Neff by approximately 0.001, from Neff ≃ 3.0444 without them to Neff ≃ 3.0434 with them in the no-oscillation case.These corrections account for the difference between this work's no-oscillation result and earlier values.Source: §4.3 · p. 17
• The ATAO approximation reproduces full-QKE frozen-out temperatures and Neff to 10^-6 while reducing computation time by at least a factor of 5.Combining ATAO with a direct Jacobian method reduces calculations from days on CPU clusters to a few hours on one CPU.Source: §4.1 · p. 16, §4.3 · p. 19
• Neff = 3.0440 with at least 10^-4 precision when flavour oscillations and O(e^3) QED plasma corrections are included.The authors identify QED corrections to collision rates as an out-of-scope ingredient that could lower Neff by 0.001.Source: §4.3 · p. 19

4.4 ATAO transfer functions

• Vacuum averaging suppresses electronic spectra and enhances the other flavours, but post-averaging no-oscillation results does not reproduce the oscillation-induced increase in Neff.The full treatment includes collision effects that the post-averaging construction misses.Source: §4.3 · p. 19, §4.4 · p. 21
• Neutrino decoupling is mostly sensitive to mixing angles, with little sensitivity to mass-squared differences and therefore to the neutrino mass hierarchy.The θ23 uncertainty changes Neff by only about 2 × 10^-5, beyond the stated accuracy goal, while normal- and inverted-hierarchy differences are extremely small below the MSW resonance.Source: §4.3 · p. 20, §4.4 · p. 22

Physical effects

• Incomplete decoupling affects BBN through nonthermal spectra, the photon-to-neutrino temperature ratio, weak-rate changes, and the clock effect from increased neutrino energy density.The higher expansion rate leaves less time for neutron decay and deuterium destruction.Source: §5 · p. 22

Implementation

• PRIMAT incorporates three neutrino-correction treatments—average temperature, effective νe temperature without distortions, and the full distorted νe spectrum—to compare light-element abundances with and without oscillations.All treatments share the same Neff, so their clock-effect contributions are identical.Source: §5 · p. 23

Flavour oscillations

• Using the true νe temperature delays weak freeze-out because Tνe exceeds the average temperature, while distorted spectra modify detailed balance in the opposite direction.The freeze-out shift is larger without oscillations because the effective νe temperature differs more strongly from the average temperature.Source: §5 · p. 24
• With oscillations, the final freeze-out neutron fraction is higher than in the average-temperature implementation, although the corresponding curves nearly coincide without oscillations.The result reflects the interplay between reduced νe heating and spectral-distortion corrections.Source: §5 · p. 24

Accuracy and scope

• The average-temperature implementation is less accurate with oscillations but still reproduces the light-element abundances to a relative precision of a few 10^-4, beyond experimental uncertainties.Its simplicity allows the neutrino information to be represented by one effective temperature parameter.Source: §5 · pp. 24–25

Quantum kinetic equations

• The BBGKY-like derivation yields neutrino QKEs with mean-field terms through O(1/m_W,Z^2) and collision terms retaining their full matrix structure.These equations govern neutrino evolution during weak decoupling with flavour mixing.Source: §6 · p. 25

Approximate evolution

• The ATAO approximation averages matter-basis off-diagonal evolution when oscillation frequencies greatly exceed collision rates and the matter basis evolves adiabatically.These assumptions permit a substantially faster numerical treatment while preserving the relevant decoupling dynamics.Source: §6 · pp. 25–26

Primordial nucleosynthesis

• The calculated Neff and spectral distortions resolve the reported discrepancy between and and produce BBN results consistent with.The resulting abundance changes lie well beyond current experimental uncertainties but clarify the physical processes involved.Source: §6 · p. 26

• The collision term is obtained from the evolution equation for the correlated part of the two-body density matrix, requiring a corresponding split of the three-body density matrix.Source: A Derivation of the formal collision… · p. 26
• Under the molecular chaos ansatz, collisions create correlations that subsequently evolve freely without a mean-field background, simplifying the correlation equation.Source: A Derivation of the formal collision… · p. 26
• The decoupling calculation depends on the Fermi and Newton constants, with the Newton constant known to only 4 × 10−5 precision.Source: A Derivation of the formal collision… · p. 26
• The correlation equation can be solved from the initial condition C(t = 0) = 0 using an evolution operator.Source: A Derivation of the formal collision… · p. 27
• A clear separation between collision, density-matrix, and oscillation timescales permits extending the collision integral to infinite time while holding operators at their initial values.Source: A Derivation of the formal collision… · p. 27
• The energy-dependent exponential factors arise because the momentum-space density matrix satisfies ˆtϱ(p) = p ϱ(p).Source: A Derivation of the formal collision… · p. 27

• The relevant early-universe interactions involve neutrinos and antineutrinos scattering through weak processes with electrons, positrons, and other neutrinos.Source: B Interaction potential matrix elements · p. 27
• The Hamiltonian is separated into charged-current interactions, neutral-current interactions with the electron-positron background, and neutrino self-interactions.Source: B Interaction potential matrix elements · pp. 27–28
• At lowest order, gauge-boson exchange is represented by the usual four-Fermi effective theory, with momentum transfer specified for t- and s-channel processes.Source: B Interaction potential matrix elements · p. 28
• Neutral-current couplings use gL = −1/2 + sin2 θW and gR = sin2 θW, with sin2 θW ≃ 0.231, and are assembled into flavour-space matrices.Source: B Interaction potential matrix elements · p. 28
• Non-standard interactions can generalize the couplings to non-diagonal matrices.Source: B Interaction potential matrix elements · p. 28
• The interaction matrix elements required for the collision term are collected in Table 4 at lowest order in the gauge-boson propagator expansion.Source: B Interaction potential matrix elements · p. 28, Table 4 · p. 29

• The BBGKY formalism is applied explicitly to derive the neutrino-neutrino scattering contribution to the collision integral.Source: C Neutrino self-interactions collision… · p. 29
• Homogeneity identifies the momenta of corresponding density matrices, and the collision matrix receives two flavour-dependent contributions.Source: C Neutrino self-interactions collision… · p. 29
• When particles 1 and 3 or particles 1 and 4 share a flavour, distinct scattering amplitudes contribute to the collision matrix.Source: C Neutrino self-interactions collision… · pp. 29–30
• The matrix products combine into the expected gain − loss + h.c. structure, with exchange symmetry requiring twice the displayed result except for identical-flavour cases.Source: C Neutrino self-interactions collision… · p. 30
• The statistical factor contains matrix-valued occupation and blocking terms together with trace contributions.Source: C Neutrino self-interactions collision… · p. 30
• The collision integral is obtained from the correlation term after removing the momentum-conserving delta function and applying the stated normalization.Source: C Neutrino self-interactions collision… · p. 30

• The continuity equation for energy density and pressure is rewritten as an evolution equation for the dimensionless photon temperature z(x).Source: D Energy conservation and QED equation… · p. 30
• The temperature equation contains neutrino collision-integral traces and additional functions G1 and G2 of x/z.Source: D Energy conservation and QED equation… · p. 30
• The functions G1 and G2 encode finite-temperature QED corrections to the plasma equation of state and are computed order by order in α = e2/4π.Source: D Energy conservation and QED equation… · p. 31
• Although the QED expressions differ formally from previous literature, integration-by-parts identities show that the expressions are equivalent.Source: D Energy conservation and QED equation… · p. 31

Generalized definitions

• The formalism introduces global particle and antiparticle indices, with barred notation distinguishing antiparticle degrees of freedom and operators.The generalized operator c_I denotes either a particle or antiparticle operator.Source: E Quantum kinetic equations with… · p. 32
• Isotropy removes anomalous pairing densities, allowing separate neutrino and antineutrino density matrices with a transposed antineutrino convention.This convention preserves matching transformation properties between the two density matrices.Source: E Quantum kinetic equations with… · p. 32

Antineutrino QKE

• Compared with neutrinos, the antineutrino vacuum term acquires a minus sign from the reversed commutator, while the mean-field term does not.The result follows after incorporating the antiparticle kinetic and interaction coefficients into the antineutrino equation.Source: E Quantum kinetic equations with… · p. 33

Antiparticle interactions

• Antiparticle collision integrals follow from the neutrino expressions by exchanging ϱ with ¯ϱ and left- with right-handed couplings.The interaction conventions also require accounting for altered operator ordering and channel assignments.Source: E Quantum kinetic equations with… · pp. 32–33

Antineutrino QKE

• The resulting antineutrino QKE is derived but not solved numerically because the calculation assumes zero asymmetry; it is used to check numerical stability.The numerical evolution therefore focuses on the neutrino equation under the stated zero-asymmetry setup.Source: E Quantum kinetic equations with… · pp. 32–33

Factorization of the CP phase

• The CP phase can be factorized from the QKE evolution because muon and tau neutrinos have identical interactions, including in the collision term.The factorization applies to the vacuum, mean-field, and collision contributions after a unitary transformation.Source: F Effect of the CP violating phase · p. 34
• The transformed density matrix has the same evolution equation and initial condition as the δ = 0 system, establishing equality throughout the evolution.This yields the relation between density matrices with and without the CP phase.Source: F Effect of the CP violating phase · p. 34

Consequences for observables

• Neff is independent of the CP phase because the trace of the neutrino density matrix is unchanged.This conclusion follows directly from the density-matrix relation.Source: F Effect of the CP violating phase · p. 34
• With no initial chemical potentials, the CP phase has no effect on BBN because light-element abundances depend on Neff, zνe, and δgνe.Initial degeneracies would invalidate the coincident initial conditions and could allow CP-phase signatures in primordial abundances.Source: F Effect of the CP violating phase · p. 34

Quantitative spectral effect

• The CP phase affects only muon- and tau-neutrino distribution functions through a [cos(δ) − 1] dependence, with an expected difference of approximately 4.6 × 10−5 matching numerical results.The estimate uses the preferred δ = 1.36π and mixing angles, and agrees with the CP-phase calculation in table 5.Source: F Effect of the CP violating phase · p. 35, Table 5 · p. 35

Limitations

• Contributions proportional to sin δ require complex components of the zero-phase density matrix and therefore require the ATAO approximation to break down.The stated factorization and BBN conclusion are consequently limited by the assumptions underlying the approximation.Source: F Effect of the CP violating phase · p. 35