Cosmology · Baryons, gas and cosmic neutrinos

Neutrino mass and free streaming

Compare two mono-momentum collisionless test populations in an imposed expanding background, tracking their speeds, finite travel and angular phase mixing.

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

Relate a massive particle’s redshifting momentum to its local speed.
Calculate the meaning and units of a finite comoving travel integral.
Distinguish transport-induced phase mixing from particle loss and a full thermal-neutrino transfer prediction.

A massive relic slows as momentum redshifts

A decoupled massive particle can retain a redshifting momentum distribution while its speed changes. The local relation E² = p²c² + m²c⁴ gives v/c = pc/E.

For a fixed particle mass, decreasing momentum eventually makes the speed substantially smaller than c. A relic population contains a distribution of momenta, so different particles need not make that transition together.

Transport can change a density perturbation because particles arriving from different directions carry different spatial phases. Real neutrino structure calculations also include gravitational metric forcing and the momentum distribution.

Isolate the transport mechanism with two prescribed mono-momentum test populations. Their small density and absent perturbative metric forcing leave the background imposed by other matter; this is a conditional model rather than a thermal-neutrino fit.

Use the same expansion and reference momentum

Normalize a_ref = 1 and write a = q over 1 ≤ q ≤ 8. Supply H(q) = H_*q⁻³ᐟ², with arbitrary positive H_* in inverse proper seconds, representing a matter-only background.

Both independent populations have the same positive comoving momentum scale p_ref. Physical momentum is p = p_ref/q. Directions are isotropic and remain fixed along the unforced test trajectories.

Define η = mc/p_ref and choose η = 0.5 or 2. Their speeds obey v/c = 1/√[1+(ηq)²]. The larger η is slower at every displayed q.

These η values are dimensionless choices, not actual neutrino masses or eigenstate measurements. Neither a thermal distribution nor interaction history is supplied by assigning one reference momentum.

Travel depends on the whole supplied interval

Let χ be comoving distance accumulated from q = 1. With dt/dq = 1/(qH) and dχ/dt = v/a, its derivative is dχ/dq = v/(q²H).

Define X = H_*χ/c. Substitution gives X(q) = ∫₁ᑫdu/[√u√(1+(ηu)²)]. All displayed X values are dimensionless, while χ has comoving length units c/H_*.

At q = 8, the η = 0.5 population has X ≈ 1.883955 and the η = 2 population X ≈ 0.623866. The latter’s smaller speed makes its positive travel integrand smaller throughout the interval.

A light signal over the same imposed interval would have X_light = 2(√q−1), about 3.656854 at q = 8. The particle paths stay below that bound. None of these finite integrals is a particle horizon from the complete cosmic past or an instantaneous v/H scale.

Supplied mono-momentum test populations; X is travel from q = 1 and the mode coefficients use chosen κ = kc/H_* = 1
q	v/c, η = 0.5	v/c, η = 2	X, η = 0.5	X, η = 2	Mode ratio, η = 0.5	Mode ratio, η = 2
1	0.894427	0.447214	0	0	1	1
2	0.707107	0.242536	0.667494	0.274483	0.927379	0.987490
4	0.447214	0.124035	1.334988	0.478058	0.728341	0.962343
8	0.242536	0.0623783	1.883955	0.623866	0.504983	0.936383

Transport changes a contrast mode without removing particles

Supply a weak initial spatial density mode with amplitude δ_initial = 0.01 and the same isotropic directional distribution. Let its comoving wavenumber be k and choose κ = kc/H_* = 1.

Averaging unforced trajectories over direction gives δ(q)/δ_initial = j₀(kχ) = sin(κX)/(κX), with value one at X = 0. The coefficient is an angular phase average.

At q = 8 the coefficients are approximately 0.504983 and 0.936383 for the two η choices. The larger travel has mixed more of this supplied mode’s phase, while the heavier test population retains more of its initial contrast.

Comoving particle inventory is conserved and the proper mean number density dilutes as q⁻³. A reduced contrast coefficient does not mean particles were destroyed. The displayed phases remain below π; no globally monotonic suppression rule is inferred outside this finite range.

Two separate negligible-density collisionless test populations share a prescribed matter-only background a_ref =one, a =q and H(q) =H_*q^(−3/2), with positive H_* in inverse proper seconds. Display 1≤q≤8. Both have fixed isotropic directions and common positive comoving momentum p_ref; physical momentum is p_ref/q. Choose dimensionless masses η =mc/p_ref =0.5 or 2, not measured neutrino masses. The local dispersion E² =p²c²+m²c⁴ gives v/c =1/√[1+(ηq)²]. All panels use the same linear q axis from 1 to 8. Solid blue circles identify η =0.5, dashed orange squares η =two. Panel A has linear speed 0 to 1. Panel B has a separate linear dimensionless finite-travel scale X =H_*χ/c from zero to 4, with χ =zero at q =one. Proper time dt/dq =1/(qH) and comoving transport dχ/dt =v/a give X(q) =∫₁ᑫdu/[√u√(1+(ηu)²)]. It integrates the full supplied interval and is not instantaneous v/H or a horizon from the beginning of cosmic history. The dotted green diamond comparator in panel B only is light travel X_light =2(√q−1); particle paths remain below it. At q =1, 2, 4, 8, blue speeds are 0.894427, 0.707107, 0.447214, 0.242536; orange speeds 0.447214, 0.242536, 0.124035, 0.0623783. Blue X values are 0, 0.667494, 1.334988, 1.883955 and orange 0, 0.274483, 0.478058, 0.623866. At q =eight the light comparator is 3.656854. Panel C uses a separate linear angular density-mode ratio from zero to one. Supply δ_initial =0.01 and comoving κ =kc/H_* =one. Isotropic directional averaging gives δ/δ_initial =j₀(kχ) =sin(X)/X, with limit one at zero; coincident initial markers in panels B/C are exact. Blue mode ratios at q =1, 2, 4, 8 are 1, 0.927379, 0.728341, 0.504983, orange 1, 0.987490, 0.962343, 0.936383. All these test phases remain belowπ; no globally monotonic suppression is asserted beyond the finite display. Comoving particle inventory is conserved while proper mean density dilutes as q⁻³. A smaller contrast coefficient reflects angular phase mixing, not particle destruction or a change in total comoving count. This toy has no thermal momentum distribution, perturbative metric forcing or back-reaction. No observed free-streaming cutoff, real matter transfer, Jeans criterion, eigenstate mass, measured neutrino bound or calculated formation history follows. Original Ark teaching model.

A real prediction needs the coupled perturbation system

A thermal relic has a range of momenta and species properties. Its contribution to structure also depends on how its density and velocity perturbations interact gravitationally with other components.

The test populations omit that distribution and perturbative forcing. Their finite travel and sinc factors therefore do not supply an observed cutoff, a Jeans criterion, a real matter transfer function or a neutrino mass bound.

Different free-streaming length conventions can use instantaneous rates or integrated travel under stated conditions. Their definitions should be compared before their numerical values are treated as the same scale.

The useful mechanism is that mass and redshifting momentum control transport, and transport affects spatial contrast through phase mixing. The next lesson asks what homogeneous population moments can and cannot identify when mass and momentum scale are allowed to vary together.

Worked reasoning

Order the accumulated paths

At the same reference momentum and background, why does η = 2 travel less than η = 0.5 for every q > 1 in the displayed interval?

For each u, the factor √(1+(ηu)²) is larger for η = 2.
Its positive comoving-travel integrand is therefore smaller at every u from 1 to q.
Integrating over the same interval gives a smaller X, including 0.623866 versus 1.883955 at q = 8.

The mass-dependent speed orders the entire positive travel integral under the supplied common background.

Try the reasoning

Is X the instantaneous quantity v/H or a particle horizon from the beginning of cosmic history?
Show the answer and reasoning
Neither.
It is the normalized comoving path integrated only from the supplied q = 1 boundary.
Does a falling density-mode coefficient indicate that particles have been removed?
Show the answer and reasoning
No.
Directional phase averaging changes the contrast while the comoving particle count stays fixed.
Can the two mono-momentum curves directly provide a real thermal-neutrino mass constraint?
Show the answer and reasoning
No.
A thermal distribution, gravitational perturbations, data and a conditional inference model remain required.

What to carry forward

Mass and redshifting momentum jointly determine a collisionless particle’s speed.
Finite comoving transport is an integral over a specified interval, distinct from instantaneous scale conventions.
Angular phase mixing affects density contrast; full relic-neutrino inference needs the momentum distribution, gravity and observations.

References

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

Massive neutrinos and cosmologyJulien Lesgourgues; Sergio PastorTeaching resourcearxiv.orgRead source