Reheating and the Expansion History

Reheating connects the cold, accelerated inflationary background to the hot radiation bath. That connection is not a single instant in general: coherent field oscillations, perturbative decays, resonant particle production, rescattering, and thermalization can occur at distinct times. The resulting expansion history changes which field value produced a fixed observed wavenumber, so uncertainty after inflation propagates backward into inflationary predictions.

Required background. The Einstein Equation supplies Einstein Equations for the background constraint and energy conservation. Classical Field Modes and Canonical Variables supplies Flat-Spacetime Quantum Field Theory for decay rates and occupation modes.

Helpful background. Relativistic Matter and Compact Objects supplies stress–energy and equation-of-state language, while Relativistic Cosmology supplies the FLRW expansion history. Thermodynamics in an Expanding Universe develops the entropy and effective-degree-of-freedom conventions used below.

From an inflaton condensate to radiation

Let $t_{\rm end}$ mark $\epsilon_H=1$ and let $t_{\rm re}$ mark the first time after which a thermal radiation bath dominates and evolves adiabatically. These definitions separate the end of accelerated expansion from the end of reheating. Denote their energy densities by $\rho_{\rm end}$ and $\rho_{\rm re}$.

Near a quadratic minimum, a canonical inflaton oscillates rapidly compared with the Hubble time. Averaging over oscillations gives $\langle p_\phi\rangle\simeq0$, so the undamped condensate redshifts like matter. More generally, canonical oscillations in $V\propto\lvert\phi\rvert^p$ with $p\gt 0$ have the averaged equation of state

$$\overline w_\phi=\frac{p-2}{p+2}$$

when the oscillation period is short and backreaction is negligible. Over one cycle, the average of $d(\phi\dot\phi)/dt$ vanishes to this accuracy. The field equation then gives $\langle\dot\phi^2\rangle \simeq\langle\phi V_{,\phi}\rangle=p\langle V\rangle$, so $\langle p_\phi\rangle/\langle\rho_\phi\rangle=(p/2-1)/(p/2+1)$. Here the exponent $p$ is a fixed number and $p_\phi$ denotes pressure. Define $\Gamma_{\rm eff}$ as the condensate’s averaged fractional energy-loss rate. Perturbative energy transfer can then be represented schematically by

$$\dot\rho_\phi+3H(1+\overline w_\phi)\rho_\phi =-\Gamma_{\rm eff}\rho_\phi,$$ $$\dot\rho_r+4H\rho_r=+\Gamma_{\rm eff}\rho_\phi.$$

This description assumes a decay rate meaningful in a slowly changing background and does not prove that the daughters have thermalized. The temperature attained transiently during production can exceed the temperature at radiation domination; $T_{\rm re}$ below refers to the latter. For a quadratic condensate, perturbative decay, and parametric preheating, see Ellis, Vennin, and Wands 2025/2026, §23.2.2, printed pp. 5–6, eqs. (23.18)–(23.23) (Official PDF). The coarse-grained decay equations appear in Baumann 2012, §6.4, printed p. 35, eqs. (91)–(92) (Open PDF).

Preheating is a different regime. A periodically varying effective mass can produce quanta nonadiabatically and amplify selected bands. Linear Floquet analysis controls the early resonance, but occupation growth changes the background and induces rescattering. The nonlinear stage generally requires a specified interacting theory, initial quantum-to-classical prescription, lattice resolution, and convergence tests. A rapid transfer of energy is not the same as instantaneous thermal equilibrium.

The exact scale-matching identity

For a canonical scalar mode that crosses at $k=a_*H_*$, define

$$N_*\equiv\ln\left(\frac{a_{\rm end}}{a_*}\right), \qquad N_{\rm re}\equiv\ln\left(\frac{a_{\rm re}}{a_{\rm end}}\right).$$

Taking $a_0=1$, the scale-factor chain gives the exact bookkeeping identity

$$\frac{k}{a_0H_0} =e^{-N_*}e^{-N_{\rm re}} \frac{a_{\rm re}}{a_0}\frac{H_*}{H_0},$$

or

$$\boxed{ N_* =-\ln\left(\frac{k}{a_0H_0}\right) -N_{\rm re} +\ln\left(\frac{a_{\rm re}}{a_0}\right) +\ln\left(\frac{H_*}{H_0}\right) }.$$

Once a thermal bath evolves adiabatically,

$$\frac{a_{\rm re}}{a_0} =\frac{T_0}{T_{\rm re}} \left(\frac{g_{*s,0}}{g_{*s,\rm re}}\right)^{1/3}, \qquad \rho_{\rm re}=\frac{\pi^2}{30}g_{*,\rm re}T_{\rm re}^4.$$

Here $g_*$ and $g_{*s}$ count energy and entropy degrees of freedom in the thermalized reference bath. Entropy production after $t_{\rm re}$ would invalidate the first relation and must be inserted as an additional dilution factor.

Constant effective equation of state

Suppose the interval from $t_{\rm end}$ to $t_{\rm re}$ can be represented by a constant effective equation of state $w_{\rm re}\ne-1$. Energy conservation then gives

$$N_{\rm re} =\frac{1}{3(1+w_{\rm re})} \ln\left(\frac{\rho_{\rm end}}{\rho_{\rm re}}\right).$$

Holding $H_*$, $\rho_{\rm end}$, and the relativistic degrees of freedom fixed, eliminating $T_{\rm re}$ yields the reheating-dependent shift relative to a radiation-like history:

$$\boxed{ \Delta N_* =\frac{1-3w_{\rm re}}{12(1+w_{\rm re})} \ln\left(\frac{\rho_{\rm re}}{\rho_{\rm end}}\right) }.$$

The signs are instructive. Since $\rho_{\rm re}\lt \rho_{\rm end}$, a matter-like stage lowers $N_*$, a radiation-like stage leaves this particular correction zero, and a stiff stage raises $N_*$. The formula is background bookkeeping, not a claim that a violent nonthermal stage truly has one constant $w_{\rm re}$. For a time-dependent total equation of state, write $N=\ln(a/a_{\rm end})$ and define the e-fold average

$$\overline w_N=\frac{1}{N_{\rm re}} \int_0^{N_{\rm re}}w(N)\,dN.$$

Integrating $d\ln\rho=-3(1+w)dN$ then gives $\ln(\rho_{\rm end}/\rho_{\rm re})=3(1+\overline w_N)N_{\rm re}$. This e-fold average reproduces the background endpoint relation exactly; an unweighted average over cosmic time, or over the values of the component equations of state, generally does not. It still cannot reproduce the microphysics of particle production or entropy generation.

The model dependence of the $N_*$–$k$ relation and its effect on predicted spectra are reviewed in Ellis, Vennin, and Wands 2025/2026, §23.3.4 and §23.6, printed pp. 10–11 and 21–22, eqs. (23.54) and (23.83)–(23.84) (Official PDF). Those constraints are conditional on the inflationary potential, decay model, late-time likelihood, and priors; they are not a direct thermometer for an otherwise unspecified reheating era.

Results and limitations through 2026-08-19

The evidence through the 2026-08-19 cutoff supports energy conservation, entropy matching after thermalization, and controlled calculations within specified microphysical regimes. It does not establish one universal reheating mechanism or a model-independent duration. The nonlinear approach to equilibrium and the couplings of the inflaton remain open questions.

Process or background	Evidence and supported result	Scope, limitation, or missing test
Instantaneous reheating	A useful boundary prescription with $N_{\rm re}=0$ and $\rho_{\rm re}=\rho_{\rm end}$; it fixes the maximal prompt-matching case.	It is not a dynamical mechanism and does not demonstrate particle production or thermalization.
Perturbative oscillatory decay	Analytic decay-rate and Boltzmann equations are controlled for weak coupling, separated microscopic and Hubble timescales, and negligible resonance; quadratic oscillations give matter-like dilution.	The result depends on the interaction, daughter masses, plasma effects, and thermalization rate; $\Gamma_{\rm eff}$ alone is incomplete.
Preheating	Linear resonance analysis can establish early instability bands, while nonlinear lattice simulations can test backreaction and rescattering for a specified model.	Simulation results depend on fields, couplings, initial state, dynamic range, lattice spacing, and ultraviolet treatment; no universal $T_{\rm re}$ follows.
Prolonged matter-like epoch	The constant-$w$ mapping is analytically controlled and shifts $N_*$ downward for fixed endpoint densities.	Condensate fragmentation, decay products, or a nonquadratic potential can invalidate $w=0$.
Stiff epoch	The same conservation law controls $w\gt 1/3$ backgrounds and shifts $N_*$ upward; relic and gravitational-wave amplification can provide indirect tests.	The source of stiffness, its duration, stability, and consistency with BBN are model dependent.
Model-agnostic effective-$w$ history	It exposes how endpoint density and expansion alter $N_*$ without committing to one decay Lagrangian.	Many inequivalent microphysical histories are degenerate at the background level; it cannot predict nonthermal relics or fluctuation transfer.

Current observations enter indirectly: primordial-spectrum measurements constrain $N_*$ only after a potential and reheating parameterization are chosen, while BBN requires a sufficiently early radiation bath. No direct reheating relic or background has yet supplied a generally accepted, model-independent detection at this cutoff. A standard thermal history must reach approximately

$$T_{\rm re}\gt 4\ \mathrm{MeV}$$

to preserve successful BBN in the usual low-temperature-reheating analyses Domènech 2021, §6, printed p. 50, text following eq. (6.27) (Open PDF). This lower bound assumes the usual particle content and neutrino thermalization; exotic decay products or a modified expansion rate require a dedicated BBN calculation.

The interpretation could change after a new primordial-spectrum constraint materially changes the allowed $N_*$ range, a direct reheating relic or background is established, or a converged nonlinear simulation changes the effective equation-of-state range for a specified interaction and initial state.

A controlled sensitivity example

Take the quadratic potential only as a mapping diagnostic. Let a radiation-like reference history give $N_*=55$, and fix $\rho_{\rm re}/\rho_{\rm end}=10^{-12}$. The correction is

$$\Delta N_*=\frac{1-3w_{\rm re}}{12(1+w_{\rm re})}\ln(10^{-12}).$$

For $w_{\rm re}=0,1/3,1$, respectively,

$$\Delta N_*=(-2.302585,\ 0,\ +2.302585).$$

Using the leading quadratic relations $n_s=1-4/(2N_*+1)$ and $r=16/(2N_*+1)$ gives

$w_{\rm re}$	$N_*$	$n_s$	$r$
$0$	$52.6974$	$0.962404$	$0.150383$
$1/3$	$55.0000$	$0.963964$	$0.144144$
$1$	$57.3026$	$0.965399$	$0.138402$

The calculation isolates the expansion-history effect; it is not a fit to data, and the quadratic potential is not being promoted as a preferred model. Changing $w_{\rm re}$ while silently holding $N_*$ fixed would miss the very dependence the example is designed to expose.

Exercises

Two descriptions have the same endpoint ratio $\rho_{\rm re}/\rho_{\rm end}=10^{-12}$. Description A is a constant $w=0$ epoch. Description B transfers energy rapidly by preheating but remains nonthermal before settling into a radiation-like background. Which parts of the constant-$w$ calculation can be applied to each, and what additional information is required for B?

Solution

For A, energy conservation gives

$$N_{\rm re}=\frac13\ln(10^{12})=9.21034, \qquad \Delta N_*=-2.302585.$$

The entropy relation can then be applied at the declared thermalization endpoint once $g_{*,\rm re}$ and $g_{*s,\rm re}$ are specified.

For B, “rapid energy transfer” does not determine either the expansion or the thermalization endpoint. One must know $w(a)$, the residual condensate and daughter energy densities, when a common temperature becomes meaningful, the entropy produced during rescattering, and the relevant $g_*$ and $g_{*s}$. Only after integrating $d\ln\rho=-3[1+w(a)]d\ln a$ to that endpoint may one use the same exact scale-factor chain. Assigning B the value $w=1/3$ solely because particles were produced would be an uncontrolled formulation choice.

References

• Baumann, D., “TASI Lectures on Inflation,” arXiv:0907.5424v2 (2012), Open PDF.
• Domènech, G., “Scalar Induced Gravitational Waves Review,” Universe 7, 398 (2021), DOI, Open PDF.
• Ellis, J., V. Vennin, and D. Wands, “Inflation,” in F. Takahashi et al. (Particle Data Group), Review of Particle Physics, International Journal of Modern Physics A 41, 2630011 (2026), revised August 2025, Official PDF.