Prolonged Reheating Epoch in Cosmology

• Prolonged reheating epoch is a phase in early universe cosmology where energy is gradually transferred from inflation to hot plasma, affecting the universe's thermal evolution.
• It is characterized by key parameters such as the number of e-folds, mean equation of state, and reheating temperature, which provide insights into the underlying microphysics.
• Observational constraints, notably from the CMB, refine inflationary model predictions by linking reheating dynamics to measurable cosmological parameters.

A prolonged reheating epoch refers to the situation in early-universe cosmology where the transfer of energy from the inflationary sector to the hot Big Bang plasma is not instantaneous but takes place over an extended duration with potentially nontrivial dynamics. In this regime, the energy is stored for a non-negligible period in non-thermal degrees of freedom—such as coherently oscillating fields, condensates, or moduli—and the transition to thermal radiation is governed by microphysical processes characterized by their own equations of state (EoS), decay rates, and coupling structure. This epoch acts as a crucial bridge between inflation and the standard thermal evolution of the universe, and its duration, dynamics, and microphysics imprint subtle but consequential effects on cosmological observables and constrain the parameter space of inflationary and particle physics models.

1. Parameterization and Kinematic Structure

A generic and widely used parameterization of reheating describes the evolution in terms of three central quantities: the duration (measured by the number of e-folds, $N_{\text{re}}$ or $\Delta N$), the mean equation of state $\overline{w}_{\mathrm{reh}}$ during reheating, and the reheating temperature $T_{\mathrm{reh}}$ (or equivalently, the energy density at the end of reheating, $\rho_{\mathrm{reh}}$). The evolution of the energy density $\rho$ during this interval follows

$$\rho(a) \propto a^{-3(1+\overline{w}_{\mathrm{reh}})},$$

where $a$ is the scale factor. The relation between quantities at the end of inflation and reheating is then

$$N_{\mathrm{reh}} = \frac{1}{3(1+\overline{w}_{\mathrm{reh}})} \ln\left(\frac{\rho_{\mathrm{end}}}{\rho_{\mathrm{reh}}}\right),$$

with $\rho_{\mathrm{end}}$ denoting the energy density at the end of inflation. The reheating temperature is set by

$$\rho_{\mathrm{reh}} = \frac{\pi^2}{30} g_{\mathrm{reh}} T_{\mathrm{reh}}^4,$$

where $g_{\mathrm{reh}}$ is the effective number of relativistic degrees of freedom.

A key kinematic parameter is the rescaled reheating parameter $R_{\mathrm{reh}}$, which controls the mapping between comoving scales that exited the Hubble radius during inflation and those observed in the Cosmic Microwave Background (CMB). The redshift at the end of inflation can be expressed as

$$1 + z_{\mathrm{end}} = \frac{1}{R_{\mathrm{rad}}}\left(\frac{\rho_{\mathrm{end}}}{\tilde{\rho}_{\gamma}}\right)^{1/4},$$

where $R_{\mathrm{rad}}$ is a dimensionless parameter and $\tilde{\rho}_{\gamma}$ is the present-day radiation energy density (appropriately rescaled) (Martin et al., 2014).

The mean equation of state

$$\overline{w}_{\mathrm{reh}} = \frac{1}{\Delta N} \int_{N_{\mathrm{end}}}^{N_{\mathrm{reh}}} \frac{P(n)}{\rho(n)}\, dn$$

is the time-averaged pressure-to-density ratio reflecting the microphysics of the dominant matter content. This EoS may range from approximately 0 (“matter-like”) for quadratic inflaton potentials to values above 1/3 for stiffer scenarios (e.g., $V \propto \phi^k$ with $k > 4$) (Cook et al., 2015, Garcia et al., 2023).

2. Observational Constraints from the CMB

CMB observations, particularly those from the Planck satellite, enable constraints on the kinematics of the reheating era by tightening the mapping between inflationary observables (e.g., the scalar spectral index $n_s$, amplitude $A_s$, and tensor-to-scalar ratio $r$) and the underlying inflationary model parameters.

The reheating history alters the precise correspondence between the scale that left the horizon during inflation and the present-day CMB pivot scale:

$$\ln\left(\frac{k}{a_0 H_0}\right) = -N_k - N_{\mathrm{reh}} - N_{\mathrm{RD}} + \cdots,$$

where $N_{\mathrm{RD}}$ is the number of e-folds during radiation domination, and $N_{\mathrm{reh}}$ is determined by reheating dynamics (Cook et al., 2015).

Planck 2013 and later datasets have been used to derive marginalized posterior distributions for $R_{\mathrm{reh}}$ across ensembles of inflationary models. Weighted model likelihoods demonstrate that current CMB data constrain the posterior-to-prior volume of the reheating parameter by on average 40%, with a width reduction factor $\sim 1.66$, indicating that the CMB anisotropy data are now sensitive enough to probe the details—even for prolonged or nonstandard reheating epochs (Martin et al., 2014).

When the mean equation of state is varied, the Bayesian evidence for given inflationary potentials demonstrates strong sensitivity to the reheating assumptions. For example, loop inflation sees a model evidence shift with $\ln(\text{Evidence ratio})$ from $-0.41$ (soft EoS, $\overline{w}_{\mathrm{reh}}=-0.3$) to $-3.27$ (stiffer, $\overline{w}_{\mathrm{reh}}=0.2$) (Martin et al., 2014).

3. Microphysics and Model Dependence

The microphysical realization of a prolonged reheating epoch is highly model-dependent. In single-field inflationary models with monomial or plateau-like potentials, the mean EoS is set by the shape near the minimum: $w = (k-2)/(k+2)$ for $V(\phi)\propto\phi^k$ (Garcia et al., 2023). For $k\geq4$, self-interactions can lead to fragmentation of the inflaton condensate, producing a rapid transition to $w\to1/3$ as the inflaton disperses into relativistic quanta.

Different classes of models yield distinct consequences:

• Small-field and certain supergravity brane inflation scenarios naturally prefer higher reheating temperatures ($T_{\mathrm{reh}}>400$ TeV up to $3\times10^6$ TeV) and sometimes prolonged, non-instantaneous reheating (Martin et al., 2014).
• Braneworld scenarios introduce further constraints: the requirement $$T_{\mathrm{reh}}<T_{\mathrm{reh}}^{\text{(crit)}} \sim 10^{15}$$ GeV sets a stringent upper bound on the 5D Planck mass scale $M_5$ and severely curtails the allowed parameter space for prolonged reheating (Bhattacharya et al., 2019).
• Kähler moduli inflation models feature an extended period of modulus domination, effectively increasing the length of reheating with $w_{\mathrm{re}}=0$ (matter-like), thereby shifting the predicted $n_s$ to lower (redder) values and imposing the need for "exotic" reheating with $w_{\mathrm{re}}>1/3$ if PLANCK-favored $n_s$ is to be matched (Bhattacharya et al., 2017).

4. Impact on Cosmological Observables and Model Differentiation

By connecting the duration and microphysics of reheating to observable inflationary signatures, one can break degeneracies in $(n_s, r)$ predictions among models that are otherwise indistinguishable at leading order. For instance:

• Quadratic ($k=2$) potentials typically yield $r\sim0.11-0.14$ for reasonable $w_{\mathrm{re}}$ and are disfavored by CMB bounds unless reheating is prolonged with $w_{\mathrm{re}}\ll1/3$ (Cook et al., 2015, Yadav et al., 2023).
• Plateau-style (Starobinsky/Higgs-type) potentials naturally accommodate larger $N_{\mathrm{reh}}$ and smaller $r$, fitting within observationally allowed regions.

Prolonged reheating, characterized by large $N_{\mathrm{reh}}$ and low $w_{\mathrm{re}}$, decreases $T_{\mathrm{reh}}$ and delays the onset of the radiation-dominated epoch. This affects the matching between horizon-exit and CMB scales, alters slow-roll parameter extraction, and narrows allowed $n_s$ values. Observational priors (e.g., $T_{\mathrm{reh}} \gtrsim 10$ MeV for successful BBN) further limit viable reheating durations (Cook et al., 2015, Saha, 2021).

Bayesian model evidence calculations reveal that models requiring extensive (prolonged) reheating can be disfavored if the data "prefer" rapid or radiation-like transitions. Thus, a full assessment of model viability mandates explicit inclusion of reheating history (Martin et al., 2014).

5. Phenomenological and Particle Physics Implications

The nature of extended reheating phases can have profound downstream implications:

• Baryogenesis/leptogenesis is sensitive to the expansion rate during reheating. Prolonged reheating delays Yukawa equilibration of charged leptons, potentially modifying the flavor regime of leptogenesis, e.g., shifting from flavored to unflavored scenarios (Datta et al., 2022).
• Relic abundances of dark matter and unwanted relics are shaped by $T_{\mathrm{reh}}$ and the expansion/entropy injection profile. Overly prolonged reheating risks overproducing relics or impeding their proper dilution.
• The detailed microphysics—such as fragmentation, cannibalization, or extra thermal sources like a Higgs condensate—can further extend or modify the duration and impact of reheating, resulting in observable consequences for the relic density, isocurvature fluctuation spectrum, or their detectability in laboratory and astrophysical settings (Passaglia et al., 2021, Bernal et al., 10 Jun 2025, Bhattacharya et al., 17 Sep 2025).

6. Extended Reheating in Non-Standard Theories and Future Probes

Reheating dynamics can be further complicated in modified gravity or braneworld scenarios. For example, in Starobinsky $R^2$ inflation with additional scalar fields, gravitational corrections alter the expansion rate, modulating both the reheating temperature and the duration of the phase, and affect the efficiency of preheating by modifying resonance structures (Bruck et al., 2016). In Einstein-Gauss-Bonnet gravity, the coupling function’s higher derivatives act as a dynamical correction to the effective potential, allowing for a broader range of viable EoS during reheating and directly linking model parameters to observational constraints and reheating viability (Venikoudis et al., 2022).

Probes of the reheating epoch include not only the CMB but also gravitational wave backgrounds (e.g., those sourced by primordial black hole formation during a matter-dominated, prolonged reheating phase), dark matter relic abundance measurements, collider experiments (sensitive to relic–SM couplings set by freeze-out during reheating), and potentially primordial cosmic complexity diagnostics embedded in the quantum properties of initial fluctuations (Saha et al., 2022, Padilla et al., 29 May 2024, Bhattacharya et al., 17 Sep 2025).

7. Theoretical and Observational Importance

As contemporary and future cosmological observations reach higher precision, the reheating epoch can no longer be treated as an uncertainty to be marginalized over. Its duration, effective EoS, and thermal history are not only testable but also play a decisive role in model selection, theoretical consistency, and the correct interpretation of cosmological data. Analyses now routinely include constraints on $$(N_{\mathrm{reh}}, w_{\mathrm{reh}}, T_{\mathrm{reh}})$$, with implications ranging from gravitational wave detection prospects to the solution of the matter-antimatter asymmetry problem. Advances in CMB, gravitational wave observatories, and laboratory experiments are expected to further tighten these constraints and illuminate the detailed physics of the prolonged reheating epoch (Martin et al., 2014, Cook et al., 2015, Ye et al., 27 Jul 2025).