Cosmology · The thermal history of the early Universe

Interaction rates compared with expansion

Calculate two supplied rate-to-expansion ratios that evolve in opposite ways during cooling, and interpret the limits of a timescale crossing.

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

Compare a relevant interaction or relaxation rate with H using consistent units.
Calculate rate ratios and their associated timescales in a supplied radiation-background model.
Explain why a ratio crossing alone does not prove equilibrium or an instantaneous freeze-out.

Ask what an interaction must accomplish

A collision can change a particle’s direction without changing its number. Repeated small momentum transfers may relax a distribution at a different rate from the raw collision count. Number-changing reactions answer another question again.

Before comparing with expansion, identify the process that would restore the property of interest: momentum sharing, chemical abundance, a distribution shape or another quantity. Its relevant rate Γ has units of inverse time, just as H does.

The dimensionless ratio Γ/H compares a process timescale Γ⁻¹ with the local expansion timescale H⁻¹. A large ratio can permit rapid adjustment relative to expansion; a small ratio can indicate an inability to keep up under the specified mechanism.

H⁻¹ is not automatically the exact cosmic age. Nor does a large ratio alone prove that the population began in equilibrium or remained there long enough. Initial conditions, detailed balance and the full evolution still matter.

Cooling can move different processes in opposite directions

Let y = T/T_star, where T_star is a supplied reference temperature whose physical value is not assigned. Over 0.5 ≤ y ≤ 2, choose a radiation-background scaling H = H_star y² with H_star = 1 s⁻¹ and constant effective radiation degrees of freedom.

Choose hypothetical relevant rates Γ_A = H_star y⁵ and Γ_B = H_star y. These are teaching laws for two separate processes, not measured neutrino or other particle reaction rates.

Dividing by the shared H gives Γ_A/H = y³ and Γ_B/H = 1/y. As temperature falls from 2T_star to T_star/2, the A ratio decreases from 8 to 1/8, while the B ratio increases from 1/2 to 2.

Both ratios equal 1 at y = 1, but they cross in opposite directions during cooling. A statement that cooling always makes every interaction less effective relative to expansion is therefore too broad even within these simple supplied laws.

Read the ratio as a timescale comparison

At y = 1/2, H = 0.25 s⁻¹ and H⁻¹ = 4 s. The A rate is 0.03125 s⁻¹, so Γ_A⁻¹ = 32 s. Only one eighth of that process-A timescale fits within the local expansion timescale.

At the same temperature, Γ_B = 0.5 s⁻¹ and Γ_B⁻¹ = 2 s. Two process-B timescales fit within H⁻¹. The ratios follow from the units and supplied powers, without needing an assigned numerical T_star.

A rate interpreted as a relaxation coefficient need not count discrete events one for one. Describing Γ/H as the number of relevant timescales per H⁻¹ avoids silently substituting a raw collision count for a momentum or chemical relaxation rate.

The inverse timescale ratio is Γ⁻¹/H⁻¹ = H/Γ. A large Γ/H corresponds to a small process-to-expansion time ratio. Keeping the two ratios distinct prevents reversing the physical interpretation.

Two synthetic positive relevant rates in one supplied H = y² s⁻¹ background; decreasing y represents cooling
y = T/T_star	H (s⁻¹)	Γ_A (s⁻¹)	Γ_B (s⁻¹)	Γ_A/H	Γ_B/H
2	4	32	2	8	1/2
1	1	1	1	1	1
1/2	0.25	0.03125	0.5	1/8	2

The synthetic radiation-background model has temperature ratio y=T/T_star from 0.5 to 2, constant effective degrees of freedom, H=H_star y² and supplied H_star=1 inverse second. Two chosen relevant processes have Γ_A=H_star y⁵ and Γ_B=H_star y, giving dimensionless Γ_A/H=y³ and Γ_B/H=1/y. The horizontal axis is linear y; a leftward arrow explicitly identifies cooling from y=2 toward y=1/2. The vertical axis is logarithmic base two but labels the actual positive ratios 1/8,1/4,1/2,1,2,4,8. Solid blue circles identify A and dashed orange squares B. At y=2 their ratios are 8 and 1/2, at y=1 both are 1, and at y=1/2 they are 1/8 and 2. Thus cooling makes A less effective relative to expansion while B becomes more effective within these supplied laws. At y=1/2, H=0.25 s^(−1), Γ_A=0.03125 s^(−1), Γ_B=0.5 s^(−1), and their inverse times are 4,32,2 seconds. The comparison Γ/H equals the number of relevant process timescales per instantaneous expansion timescale, not necessarily a discrete collision count. H^(−1) is not asserted to be the exact age. Equality alone neither proves equilibrium, determines an actual freeze-out abundance nor supplies a discontinuous on/off switch; initial state, duration, target property, detailed balance and the relevant kinetic equation matter. No actual particle cross-section, decoupling temperature, observed rate, physical T_star or fitted uncertainty is assigned. Original Ark teaching model.

A diagnostic boundary does not replace evolution

Setting Γ = H often identifies a useful approximate stage to examine. It does not insert a physical on/off switch into a kinetic equation. With smooth rates, distributions and abundances can depart from equilibrium over a range of times.

A relaxation process also needs a target state and a mechanism that drives deviations toward it. A large total collision rate is insufficient if those collisions conserve the quantity whose chemical equilibrium is being tested.

A species initially far from equilibrium might not fully relax during a brief high-ratio interval. Conversely a distribution can retain a thermal form through later free propagation in special circumstances, as the preceding photon example demonstrated.

Calculating a freeze-out abundance or kinetic temperature therefore requires the relevant evolution equation, initial conditions and bath history. The simple equality is an organizing estimate whose accuracy must be checked for the particular problem.

Rate laws and expansion laws have domains

The assumed H proportional to T² scaling requires the stated radiation-background and effective-degree conditions. Entropy transfer, a different dominant component or additional energy density can alter the relation between temperature and expansion.

The interaction powers also depend on the process and regime. Particle masses, thresholds, mediator properties and changes in the distribution can prevent one simple power law from applying across a broad temperature range.

The table supplies no actual decoupling temperature, particle cross-section or measured relic population. Its opposing trends show why the assumptions must be stated before a rate comparison is turned into a history.

A useful analysis names the rate, units, bath, expansion law, temperature range and initial state, then solves enough evolution to test the estimate. Those steps connect a timescale argument to a defensible physical conclusion.

Worked reasoning

Compare process A at half the reference temperature

At y = 1/2, how many process-A relaxation timescales fit in H⁻¹?

H = (1/2)² s⁻¹ = 0.25 s⁻¹, so H⁻¹ = 4 s.
Γ_A = (1/2)⁵ s⁻¹ = 0.03125 s⁻¹, so Γ_A⁻¹ = 32 s.
Their ratio is Γ_A/H = 4/32 = 1/8.

One eighth of a process-A relaxation timescale fits in the local expansion timescale.

Try the reasoning

During cooling from y = 2 to y = 1/2, does Γ_B/H decrease?
Show the answer and reasoning
No; it rises from 1/2 to 2.
The supplied B ratio is 1/y.
Is the total collision rate automatically the rate that equilibrates particle number?
Show the answer and reasoning
No.
Number equilibrium requires the relevant number-changing reactions; elastic scattering can conserve number.
Does Γ/H > 1 by itself prove that a population was in equilibrium?
Show the answer and reasoning
No.
Initial conditions, the duration of the regime, detailed balance and the relevant relaxation mechanism also matter.

What to carry forward

Use the rate relevant to the physical property being tested.
Different rate laws can move in opposite directions relative to expansion during cooling.
A rate crossing is a timescale diagnostic whose physical interpretation requires evolution and conditions.

References

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