advanced18 min studyLesson 854 in Physics

Thermal reaction rates versus Hubble expansion

Use rate comparisons without mistaking a crossing estimate for a complete kinetic solution.

Before you begin

On this page

The question for today

What you’ll understand

Identify the interaction rate relevant to a specified equilibrium condition.
Derive temperature scaling of a rate-to-expansion ratio.
Distinguish chemical freezeout, kinetic decoupling, and possible recoupling.

Equilibrium is a competition between relaxation and changing conditions. In an expanding background, collisions must reorganize distributions quickly enough to follow dilution, cooling, and shifting equilibrium abundances. The familiar comparison between an interaction rate and the Hubble rate is a useful first test, but the relevant interaction depends on the question. Elastic momentum exchange, number-changing reactions, flavor conversion, and radiative transitions can have different relaxation times. Throughout this lesson, rates are per-particle inverse times, rather than event counts per volume, and all temperature scalings hold only within their stated particle and expansion regimes.

Choose the relaxation process before comparing rates

For scattering on a dilute target population, multiply target number density by a thermally averaged cross section times relative velocity. The result has units of inverse time. A collision density instead includes the density of incoming particles as well; confusing it with a per-particle rate introduces an unwanted density factor. For identical incident particles, event-counting symmetry factors must likewise be separated from the rate experienced by one tagged particle.

Frequent elastic collisions can establish a common kinetic temperature while leaving a chemical potential undetermined. Number-changing reactions are needed to impose the corresponding chemical-equilibrium relation. Moreover, many tiny-angle collisions may transfer momentum inefficiently, requiring a transport-weighted rate rather than the total scattering rate. Asking whether a species is in equilibrium without naming the distribution feature or conserved quantity therefore hides an essential physical choice.

\Gamma=n_{\rm target}\langle\sigma v_{\rm rel}\rangle,\qquad t_{\rm int}=\Gamma^{-1},\qquad R=\frac{\Gamma}{H}

The appropriate collision or transport average depends on the proposed relaxation mechanism. A large

R

is a timescale requirement, not by itself proof that every chemical and kinetic mode has equilibrated.↔ Scroll wide equations horizontally.

The ratio can rise or fall as the universe cools

In a flat radiation-dominated universe with approximately fixed relativistic counts, the expansion rate scales as temperature squared. A relativistic target density scales as temperature cubed. At energies well below a heavy weak mediator mass, an effective four-fermion interaction gives a cross section proportional to temperature squared, so the weak collision rate scales as the fifth power. Its ratio to expansion falls rapidly during cooling, making decoupling plausible.

The same extrapolation fails above the mediator scale. A dimensionless-coupling interaction in an appropriate relativistic regime can instead have a cross section proportional to inverse temperature squared, giving a collision rate proportional to temperature. Its ratio to expansion then grows during cooling. Such a sector may first enter equilibrium rather than leave it. Mass thresholds, resonances, changes in entropy content, and a nonradiation expansion history can all alter these simple powers.

H\propto T^2,\qquad \Gamma\propto T^m\ \Longrightarrow\ R\propto T^{m-2},\qquad \Gamma_{\rm weak}\sim G_F^2T^5\ \Longrightarrow\ R_{\rm weak}\propto T^3

The last estimate uses the low-energy weak effective theory, relativistic targets, and constant radiation counts.

G_F

is the Fermi constant; the expression is a scaling estimate without its process-dependent coefficient.↔ Scroll wide equations horizontally.

A rate crossing is an estimate, not an instantaneous event

The criterion $R\sim1$ estimates when an interaction becomes ineffective over one expansion time. A species does not switch abruptly from a perfectly thermal gas to a perfectly collisionless gas at that moment. Its collision term continues to act over an interval. Following an equilibrium abundance that changes exponentially with inverse temperature can also require a faster relaxation rate than following the scale factor alone, so precision chemical freezeout requires the kinetic equation.

Integrated collision probabilities clarify the limitation. Even a modest local rate can accumulate over a long interval, while a large rate acting briefly may not erase an initial distortion. The initial state matters: a sector that never thermalized cannot inherit a thermal distribution merely by passing through a rate crossing. For observations, one usually propagates distributions or moments through the changing background and compares the resulting relic quantities, rather than treating the crossing temperature as the complete prediction.

Work it through

Compare two temperatures on either side of a toy crossing

Take a stipulated radiation era with fixed counts, $R(T)=(T/2\,\mathrm{MeV})^3$ , and $H(4\,\mathrm{MeV})=10\,\mathrm{s}^{-1}$ . Find the collision rates and both relevant timescales at $4\,\mathrm{MeV}$ and $1\,\mathrm{MeV}$ . Locate the approximate rate crossing; no precision neutrino temperature is being inferred.

At the higher temperature the ratio is $R=8$ . Therefore $\Gamma=80\,\mathrm{s}^{-1}$ , the interaction time is $0.0125\,\mathrm{s}$ , and the expansion time is $0.1\,\mathrm{s}$ . Several interactions fit within one expansion time.
At the lower temperature, use the stipulated radiation scaling to obtain $H=10(1/4)^2=0.625\,\mathrm{s}^{-1}$ . The ratio is $R=1/8$ , hence $\Gamma=0.078125\,\mathrm{s}^{-1}$ . Its inverse is $12.8\,\mathrm{s}$ , compared with the expansion time $1.6\,\mathrm{s}$ .
Set the ratio to unity to estimate the crossing at $2\,\mathrm{MeV}$ . The decreasing ratio indicates loss of contact during cooling. The calculation does not specify which distortion relaxes or integrate the residual collision term, so it cannot supply a sharp last-collision surface.

The result

The collision rates are $80\,\mathrm{s}^{-1}$ and $0.078125\,\mathrm{s}^{-1}$ , with interaction times shorter and longer than the respective expansion times. The approximate crossing lies at $2\,\mathrm{MeV}$ .

The living notebook

Explore the model

Interactive lab

Interactive lab · markov

Same stationary state, different relaxation speed

Reset and set both controls to $p=0.1$ , $q=0.1$ , then press Step four times. The exact probability of state 1 is $0.2952$ , with stationary value $0.5$ . Reset and set both controls to $p=0.4$ , $q=0.4$ , then Step four times: that probability is $0.4992$ , with the same stationary value. Each Step is one discrete transition here. This fixed two-state chain illustrates relaxation, not an expanding thermal plasma or a collision integral.

P(state 0)P(state 1)Stationary distribution

P(state 1) at step 0: 0%
Stationary P(state 1): 60%
Chain behavior: Monotone convergence

Transition p · 0 → 10.3Transition q · 1 → 00.2

t = 0 s

Exact finite-step probabilities for a two-state chain starting in state 0, not sampled trajectories. Playback advances two transitions per second. When p = q = 0 every distribution is stationary; when p = q = 1 the chain alternates forever. A single zero transition creates an absorbing state. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forKnowing an equilibrium state does not establish that finite-time dynamics has reached it.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Target density contributes

T^3

, giving

\Gamma\propto T

. Dividing by

H\propto T^2

yields

T^{-1}

within this specified regime.

Your turn

Try it yourself

For a different stipulated process, let $R(T)=3\,\mathrm{MeV}/T$ throughout a radiation era. Evaluate the ratio at $12\,\mathrm{MeV}$ and $1.5\,\mathrm{MeV}$ , find its crossing, and identify whether cooling improves or weakens the prospect of equilibration.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Track the inverse temperature dependence rather than importing the weak-interaction scaling.

Compare with the solution

The ratios are $1/4$ and $2$ . Equality occurs at $T=3\,\mathrm{MeV}$ . Cooling increases the number of interactions available per expansion time, so this process can begin to establish contact rather than decouple. Actual equilibration still requires a sufficiently long interval and interactions that relax the relevant distribution. An initial thermal abundance has not been assumed, and the crossing alone does not manufacture one.

Keep the idea

Take it with you

Specify the relaxation mode and a per-particle rate.
Cooling can cause decoupling or recoupling depending on the interaction regime.
A rate crossing guides a kinetic calculation but does not replace it.

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