Cosmology · Inflation and primordial perturbations

Reheating and uncertain links to thermal history

Follow energy transfer between an oscillation-averaged scalar and a radiation-like component, distinguishing radiation domination from thermal equilibrium.

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

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After reading, you can

Describe why an early accelerated phase needs a connection to the later thermal history.
Read a supplied two-component energy-transfer calculation with expansion included.
Separate energy transfer, radiation-like pressure and kinetic or chemical equilibration.

Ending acceleration and making a thermal population are separate tasks

An early accelerating scalar-field phase does not by itself specify the hot population used in later thermal-history calculations. A larger proposal must explain how the accelerating phase ends and how energy becomes available to other populations.

Reheating is the name for this connection and the associated physical processes. Depending on the model, field oscillations, particle production, scattering and redistribution can all matter. One short energy-transfer equation cannot determine every stage.

This lesson supplies a finite example after a scalar has entered rapid oscillations around a quadratic minimum. Its oscillation-averaged equation of state is chosen to be zero. That assumption differs from the negative pressure of the preceding accelerating example.

The new example is a separate construction with its own initial state and couplings. It does not supply the dynamical exit missing from the previous exponential potential, and its chosen decay rate is not inferred from an observed particle or temperature.

Expansion and transfer both change the densities

Let ε_0 be the initial scalar energy density in joules per cubic metre. In flat general relativity with only the two stated components, define H_i² = 8πGε_0/(3c²), τ = H_i(t − t_i), p = ε_φ/ε_0 and r = ε_r/ε_0. Here p is a dimensionless scalar density variable, not a pressure symbol.

Set a_i = 1 and h = H/H_i = √(p + r). Supply Γ/H_i = γ = 0.5. The averaged equations are dp/dτ = −(3h + γ)p, dr/dτ = γp − 4hr and da/dτ = ah, with initial (p,r,a) = (1,0,1).

The scalar loses energy through both expansion dilution and decay. The radiation-like component receives γp and dilutes with the fourth power appropriate to its imposed pressure ε_r/3. The transfer terms cancel when the two density equations are added.

To make oscillation averaging meaningful in this construction, supply a microscopic oscillation angular frequency 100H_i, faster than the initial expansion and Γ; H subsequently decreases. Detailed oscillation phases, resonance, thermal corrections and particle couplings are outside this two-fluid approximation.

Radiation becomes dominant over an interval

Numerical integration of these supplied equations gives the table. At τ = 0 there is no radiation energy in this account. By τ = 1, the radiation-like component contains about one third of the remaining total energy density.

The two densities become equal at τ ≈ 1.77924, with p = r ≈ 0.0324556 and a ≈ 2.33051. This equality is a useful landmark in the stated calculation, rather than a universal definition of completed reheating.

At τ = 8, the radiation fraction r/(p + r) is about 0.960881. Both component densities are nevertheless far below ε_0 because the expansion and transfer occur together. A growing fraction does not require a growing physical density.

The time coordinate is dimensionless and tied to the arbitrary H_i. No number of seconds or temperature is assigned. The numerical values describe the chosen initial state and γ, not an observed reheating record.

Supplied averaged two-fluid example with γ = 0.5; p and r are energy densities divided by the same initial ε_0
τ	Scalar p	Radiation r	Radiation fraction	a/a_i	H/H_i
0	1	0	0	1	1
0.5	0.255754	0.0612901	0.193317	1.449444	0.563067
1	0.0993217	0.0494750	0.332501	1.827832	0.385742
2	0.0247441	0.0290001	0.539595	2.458920	0.231828
4	0.00331743	0.0123393	0.788115	3.442468	0.125127
8	0.000157003	0.00385646	0.960881	4.886198	0.0633519

An exact identity checks the numerical solution

Multiply the scalar equation by a³ and include the derivative of a³. The expansion dilution cancels, leaving d(a³p)/dτ = −γa³p. With the chosen initial value, a³p = exp(−γτ) exactly within this averaged model.

For example, at τ = 2 the value is exp(−1) ≈ 0.367879. Multiplying the table’s more precise scalar density by a³ reproduces this value. It provides a check on the calculation independently of the plotted curve.

Adding the two density equations instead gives d(p + r)/dτ = −3hp − 4hr. The scalar-to-radiation terms cancel locally, but the total physical density still falls. Even a³(p + r) is not constant when radiation pressure contributes to the expansion account.

This is the local continuity relation for the supplied components in an expanding geometry. It does not describe a box with moving material walls or require an external cosmic reservoir receiving a conserved global energy balance.

A separate supplied flat general-relativistic model has only a rapidly oscillating scalar with mean equation of state zero and a radiation-like component with imposed pressure ε_r/3. Let ε_0 be positive initial scalar energy density in joules per cubic metre, H_i² = 8πGε_0/(3c²), τ = H_i(t−t_i), p = ε_φ/ε_0, r = ε_r/ε_0, a_i = 1 and h = H/H_i = √(p+r). The symbol p is a density variable here, not pressure. Supply γ = Γ/H_i = 0.5 and microscopic oscillation angular frequency 100H_i. The averaged equations are p′ = −(3h+γ)p, r′ = γp−4hr, a′ = ah, initially (1, 0, 1). Panel A uses the full finite τ range 0 to 8 and a linear radiation fraction r/(p+r) axis from zero to one. Its solid green circle samples at τ = 0, 0.5, 1, 2, 4, 8 give fractions 0, 0.193317, 0.332501, 0.539595, 0.788115, 0.960881. Panel B separately magnifies only τ from 1 to 8; its linear density axis uses the common initial ε_0, from 0 to 0.1, rather than total-normalised shares. Solid blue circles and dashed orange squares show p and r. At τ = 1, 2, 4, 8 the p values are approximately 0.0993217, 0.0247441, 0.00331743, 0.000157003, while r values are 0.0494750, 0.0290001, 0.0123393, 0.00385646. The initial p = 1, r = 0 and earlier interval lie outside panel B, not outside the model. Diamonds mark p = r ≈ 0.0324556 at τ ≈ 1.77924, a ≈ 2.33051; panel A then has fraction 1/2. Both component densities decline in panel B while the radiation fraction grows. The exact scalar identity a³p = exp(−0.5τ) gives exp(−1) ≈ 0.367879 at τ = 2. Adding the equations cancels the transfer but leaves (p+r)′ = −3hp−4hr. This is a local continuity account; it does not require a box, material walls or external energy reservoir. Radiation domination is not thermal equilibrium, and no distribution function, kinetic or chemical equilibration, physical temperature, measured decay rate or absolute time is computed. The averaging excludes resonance and detailed field phases. This separate construction does not supply the exit missing from the preceding scalar potential. Original Ark teaching model.

Pressure and equilibrium impose different conditions

A component with pressure near one third of its energy density can be radiation-like before its particles form a thermal distribution. The energy-density equations used here impose that equation of state and do not solve a distribution function.

Kinetic equilibration concerns redistribution of momentum and energy among particles. Chemical equilibration also concerns number-changing processes and the relevant abundances. Those conditions need interaction information beyond the single scalar decay term.

Consequently, r > p does not prove either form of equilibrium. Turning a radiation density into a temperature additionally requires an appropriate thermal state and an account of which relativistic populations contribute. Neither is calculated in this table.

A complete reheating study therefore connects field dynamics, transfer and equilibration before using a thermal-history temperature. Keeping these steps separate makes the uncertainties in that connection explicit and prevents a simple fluid landmark from carrying more meaning than its equations support.

Worked reasoning

Derive a check on scalar dilution and decay

Why is a³p equal to exp(−γτ), rather than a constant?

Differentiate a³p and use da/dτ = ah, giving d(a³p)/dτ = a³ dp/dτ + 3ha³p.
Insert dp/dτ = −(3h + γ)p. The 3h terms cancel, leaving −γa³p.
Integrate from a³p = 1 at τ = 0 to obtain a³p = exp(−γτ); at τ = 2 with γ = 0.5 it is exp(−1).

Expansion dilution cancels in the comoving scalar account, while decay still removes scalar energy at the supplied rate.

Try the reasoning

Must ε_r increase whenever the radiation fraction increases?
Show the answer and reasoning
No.
A component’s fraction can grow while both physical densities decline through expansion.
Does radiation domination in this model prove thermal equilibrium?
Show the answer and reasoning
No.
The radiation-like pressure was imposed; momentum redistribution and number-changing interactions were not solved.
Can a physical reheating temperature be read directly from these dimensionless densities?
Show the answer and reasoning
No.
No physical reference energy density, thermal distribution or contributing particle count has been supplied for that conversion.

What to carry forward

Reheating connects an early field-dominated phase to later populations through model-dependent dynamics.
Local transfer terms cancel while expansion still changes the density account.
Radiation-like pressure and radiation domination do not by themselves establish thermalization.

References

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