Cosmology · Baryons, gas and cosmic neutrinos

Neutrino decoupling and entropy transfer

Use an ideal decoupled momentum distribution and a separately conserved electromagnetic entropy account to distinguish temperature, particle-number and relativistic energy ratios.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Explain how a collisionless relic can retain an initial momentum-distribution shape.
Derive ideal temperature ratios from a stated entropy convention.
Distinguish one-pair number and energy ratios from a precision species or effective-density inference.

Interactions and expansion compete over a history

Neutrinos can exchange energy and momentum with an early plasma through interactions. As conditions change, comparison of collision processes with expansion helps describe departures from the coupled state.

Actual decoupling is a kinetic process with momentum and flavour dependence. For a transparent teaching calculation, impose complete instantaneous decoupling before an idealized electron–positron entropy transfer.

Initially supply a zero-chemical-potential relativistic Fermi–Dirac momentum occupation f_ν(p) = 1/[exp(pc/(k_BT_ν))+1]. After collisionless decoupling, physical momentum redshifts as p proportional to 1/a.

The initial shape is then preserved with aT_ν constant. T_ν labels the momentum scale; retaining that shape does not establish continued collisions or a nonrelativistic kinetic-temperature law.

Entropy conservation fixes a comoving temperature ratio

Let a_D and T_D describe the common initial normalization. Photons remain thermal, while the decoupled neutrino population receives no subsequent entropy in this ideal account.

Define electromagnetic entropy density s_EM = (2π²/45)k_B[k_BT_γ/(ℏc)]³g_s,EM. The initial relativistic photon-plus-electron–positron count is g_s,EM = 2+(7/8)×4 = 11/2.

Impose conservation of a³s_EM while the electromagnetic sector ends with the two photon polarization states, g_s,EM = 2. This gives aT_γ/(a_DT_D) = [(11/2)/g_s,EM]¹ᐟ³, while aT_ν/(a_DT_D) = 1.

The growing comoving photon temperature describes heating relative to free 1/a redshifting. It does not assert an absolute temperature rise or an instantaneous heating jump at fixed a; no numerical scale factor or time is assigned.

Fermi statistics and powers of temperature give different ratios

Consider one populated neutrino-plus-antineutrino pair with two states in total, compared with two-state photons. At equal momentum-temperature scale, the number-density ratio is 3/4 and the relativistic energy-density ratio is 7/8.

At unequal scales, n_ν/n_γ = (3/4)(T_ν/T_γ)³ and ε_ν/ε_γ = (7/8)(T_ν/T_γ)⁴. The displayed energy ratio uses the relativistic limit; massive-particle energy later needs a momentum integral.

After the ideal transfer is complete, T_ν/T_γ = (4/11)¹ᐟ³ ≈ 0.713766. The one-pair number ratio is 3/11 ≈ 0.272727, while its relativistic energy ratio is about 0.227107.

These three ratios differ because they weight the distribution differently. They are not alternative estimates of the same quantity, and none is a numerical observational determination of the present relic population.

Abstract electromagnetic entropy states in the supplied ideal limit; intermediate rows are not assigned times or measured epochs
g_s,EM	aT_γ/(a_DT_D)	aT_ν/(a_DT_D)	T_ν/T_γ	One-pair n_ν/n_γ	One-pair ε_ν/ε_γ
11/2	1	1	1	3/4	0.875
4	1.111990	1	0.899289	6/11	0.572275
3	1.223903	1	0.817058	9/22	0.389959
2	1.401020	1	0.713766	3/11	0.227107

A fixed relic number and a changing photon number can coexist

The ideal collisionless neutrino comoving number remains constant. Photon comoving number instead scales as (aT_γ)³ and increases by 11/4 between the supplied endpoints.

The electromagnetic entropy remains constant despite that photon-number change. Entropy and particle number are different accounts, and photons can be produced or absorbed while thermal conditions redistribute energy.

The intermediate values g_s,EM = 4 and 3 are abstract effective entropy parameters. They do not solve the electron equation of state, an annihilation trajectory or a decoupling temperature.

Before the transfer ends, the changing temperature ratio and electron–positron contribution must be included explicitly. A late relativistic-density normalization containing the final 4/11 factor cannot simply replace that evolving account.

Impose complete instantaneous neutrino decoupling before an ideal electron–positron entropy transfer. Initially use a relativistic zero-chemical-potential Fermi–Dirac momentum shape with two populated neutrino-plus-antineutrino states, compared with two photon states. After collisionless redshifting, aT_ν stays constant as a momentum-distribution scale; this does not prove continuing collisions or a nonrelativistic kinetic-temperature law. The photon sector stays thermal and its comoving electromagnetic entropy a³s_EM is conserved, with s_EM =(2π²/45)k_B[k_BT_γ/(ℏc)]³g_s, EM. Initially g_s, EM =11/2; completed transfer leaves g_s, EM =two. Both panels use an ascending abstract parameter g_s, EM from 2 at left to 5.5 at right. The completed-transfer endpoint is therefore left and the initial endpoint right. No clock, redshift, physical scale factor, electron equation of state or real heating trajectory is assigned to intermediate g_s =3, 4. Panel A has a separate linear comoving temperature ordinate from 1 to 1.5. Solid blue circles show aT_γ/(a_DT_D) =[(11/2)/g_s]^(1/3); dashed orange squares show aT_ν/(a_DT_D) =one. At g_s =5.5, 4, 3, 2, photon values are 1, 1.111990, 1.223903, 1.401020 and neutrino values stay 1. Coincident initial markers at g_s =5.5 are exact, not missing points. Panel B uses a zero-based linear ratio scale from zero to one. Solid green circles show temperature T_ν/T_γ =[g_s/(11/2)]^(1/3), dashed violet squares one-pair number ratio(3/4)(T_ν/T_γ)³, and dotted orange diamonds one-pair relativistic energy ratio(7/8)(T_ν/T_γ)⁴. At g_s =5.5, 4, 3, 2, temperature ratios are 1, 0.899289, 0.817058, 0.713766; number ratios 3/4, 6/11, 9/22, 3/11; relativistic energy ratios 0.875, 0.572275, 0.389959, 0.227107. These observables differ by statistics factors and powers, not alternative estimates of one quantity. The 3/4 and 7/8 factors compare equal two-state Fermi and Bose integrals at equal scale. Photon comoving number rises by 11/4 across the supplied endpoints, while neutrino comoving number and electromagnetic comoving entropy each remain fixed. Comoving photon heating relative to 1/a redshifting need not mean absolute physical temperature rises. Energy ratios here are relativistic only; a massive species later requires a full momentum integral. No current relic temperature, actual N_eff, precision correction or measured species count is given. Real decoupling needs finite collisions, flavour and plasma/QED history beyond this ideal entropy account. Original Ark teaching model.

Precision relic calculations require the collision history

Finite interaction energy exchange, flavour oscillations and finite-temperature electromagnetic effects alter the precision relic spectrum from the instantaneous complete-decoupling limit.

Computing those corrections requires kinetic evolution and an explicitly defined plasma model. The entropy table supplies no collision solver, numerical precision correction or empirical present-day temperature.

An effective relativistic energy-density parameter summarizes energy under a declared convention. Different temperatures, occupation shapes or additional relativistic populations can prevent it from being a simple integer species count.

The next lessons separate the transport of massive collisionless particles from a homogeneous population-moment inference. The ideal ratios here provide a clearly defined starting point whose assumptions remain visible.

Worked reasoning

Recover the final number ratio

Why is the final ideal one-pair number ratio 3/11 rather than equal to T_ν/T_γ?

The two-state zero-chemical-potential Fermi–Dirac pair has number factor 3/4 relative to two-state photons at equal temperature scale.
Number density scales with the cube of that momentum-temperature scale.
At the endpoint, (T_ν/T_γ)³ = 4/11, giving (3/4)(4/11) = 3/11.

The number ratio includes both the statistics factor and a cubic temperature factor.

Try the reasoning

Does a retained Fermi–Dirac momentum shape after decoupling prove continued collisional equilibrium?
Show the answer and reasoning
No.
Collisionless redshifting preserves the initial shape when p and the momentum scale T_ν redshift together.
Is the table row g_s,EM = 3 a measured epoch or a solved annihilation time?
Show the answer and reasoning
No.
It is an abstract effective entropy state with no physical time or electron equation-of-state history assigned.
Does aT_γ rising by a factor about 1.401 prove that physical T_γ increased by that factor?
Show the answer and reasoning
No.
Physical temperature also divides by the changing scale factor; the table concerns a comoving normalization.

What to carry forward

A decoupled relic can preserve an initial momentum shape without ongoing thermal collisions.
A stated entropy account produces distinct ideal temperature, number and relativistic-energy ratios.
Precision spectra and effective population constraints require a fuller kinetic and statistical model.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.