---
title: Maximum Reheating Temperature
url: https://www.emergentmind.com/topics/maximum-reheating-temperature
type: topic
---

# Maximum Reheating Temperature

The maximum reheating temperature is a fundamental concept in cosmological model building, quantifying the highest temperature attained by the Universe during the transition from the inflationary epoch to the radiation-dominated phase. This parameter is of crucial interest for theories of baryogenesis, dark-matter genesis, and the thermal history of the early Universe. Its precise definition, calculation, and model dependence are central in constraining inflationary potentials and new physics beyond the Standard Model.

## 1. Definition and Distinction: $T_{\rm max}$ vs $T_{\rm RH}$

Following the end of inflation (at scale factor $a_{\rm end}$), the inflaton field $\Phi$ (or $\phi$) enters a phase of damped oscillations and decays into a bath of relativistic particles, whose energy density $\rho_R$ increases from zero. During this stage:

- **Maximum reheating temperature ($T_{\rm max}$):** The peak temperature reached by the thermal bath, defined at $a=a_{\rm max}$, where $T(a)$ (the instantaneous temperature) is maximized: $T_{\rm max} \equiv T(a_{\rm max})$.
- **Reheating temperature ($T_{\rm RH}$):** The temperature at $a=a_{\rm reh}$ where the radiation energy density overtakes the inflaton and the Universe becomes radiation dominated, i.e., $\rho_R(a_{\rm reh}) = \rho_\Phi(a_{\rm reh})$. Hence, $T_{\rm RH} \equiv T(a_{\rm reh})$ [2004.08404], [1709.00251].

These two scales can differ by several orders of magnitude depending on the inflaton potential and decay processes.

## 2. Dynamics of Energy Density and Temperature Evolution

The Boltzmann–Friedmann system governs the evolution of the inflaton and radiation energy densities:
\[
\begin{aligned}
\dot\rho_\Phi + 3H(1+w)\rho_\Phi &= -\Gamma_\Phi \rho_\Phi, \\
\dot\rho_R + 4H\rho_R &= \Gamma_\Phi \rho_\Phi, \\
H^2 &\simeq \frac{\rho_\Phi}{3 M_P^2} \quad (\rho_\Phi \gg \rho_R),
\end{aligned}
\]
where $w = (k-2)/(k+2)$ for inflaton potentials $V(\Phi) = \mu^{4-k}\Phi^k$. $\Gamma_\Phi$ is the decay width, which may be constant or temperature/time-dependent depending on the couplings [2004.08404].

- At early times, $\Gamma_\Phi$ can be neglected, resulting in $\rho_\Phi(a) = \rho_{\rm end} (a/a_{\rm end})^{-6k/(k+2)}$.
- The radiation grows as $\rho_R(a) \propto a^{(6-6k)/(k+2)}$ for Yukawa-type decay.
- The temperature evolution, assuming instantaneous thermalization, is $T(a) \propto a^{-(3k-3)/(2k+4)}$ for $a \gg a_{\rm end}$.
    - For $k=2$ (quadratic, matter-like): $T(a) \propto a^{-3/8}$.
    - For $k=4$: $T(a) \propto a^{-3/4}$ [2004.08404].

## 3. Analytic Expressions for $T_{\rm max}$ and $T_{\rm RH}$

Extremizing $T(a)$ yields the scale factor where the maximum is reached, and the corresponding $T_{\rm max}$:
\[
T_{\rm max}^4 = \frac{15\, y^2}{16\pi^3 g_*} \sqrt{3k(k-1)}\,\mu^{4/k} M_P^{4/k} \rho_{\rm end}^{(k-1)/k} \left(\frac{3k-3}{2k+4}\right)^{3(k-1)/(7-k)},
\]
where $y$ is the inflaton coupling to decay products and $g_*$ is the effective number of relativistic degrees of freedom.

The reheating temperature is given by
\[
T_{\rm RH}^4 = \frac{15\, y^{2k}}{2^{4k-1} \pi^{2+k} g_*} [3k(k-1)]^{k/2} \mu^4,
\]
which depends more sensitively on $y$ and $k$ [2004.08404].

**For $k=2$ (standard matter-dominated period):**
- $T_{\rm RH} \propto y\mu$,
- $T_{\rm max} \propto y^{1/2}$,
- $T_{\rm max}/T_{\rm RH} \propto y^{-1/2}$.

**For $k>2$:**
- $\rho_\Phi$ redshifts faster,
- $\Gamma_\Phi \propto m_\Phi(t)$ decreases,
- Reheating is delayed,
- $T_{\rm RH} \propto y^k$; $T_{\rm max} \sim O(10^{12}\,{\rm GeV}) (y/10^{-5})^{3/8}$ for $k=4$ [2004.08404].

## 4. Bounds, Model Dependence, and Key Results

### 4.1 General Maximum Value

- For $\mu \sim M_P$ and $y \sim 10^{-5}$, $T_{\rm max} \simeq 2 \times 10^{12}$ GeV,
- $T_{\rm max}$ is essentially independent of $k$ for typical couplings,
- $T_{\rm RH}$ depends strongly on $k$, dropping from $\sim 10^{10}$ GeV ($k=2$) to $\sim 10^{4}$ GeV ($k=4$) for $y=10^{-5}$ [2004.08404],
- $T_{\rm max}$ is set early, i.e., "well before" radiation domination and decoupled from $\Gamma_\Phi$ and details of the decay process once $y$ is fixed.

### 4.2 Physical and Phenomenological Constraints

- Big Bang Nucleosynthesis (BBN) imposes $T_{\rm RH} \gtrsim \mathcal{O}(1\,{\rm MeV})$,
- For $y \gtrsim 10^{-5}$, perturbative reheating breaks down,
- Dark matter production rates $\propto T^n\ (n\geq 1)$ can be enhanced by factors $\sim T_{\rm max}/T_{\rm RH}$ if $T_{\rm max} \gg T_{\rm RH}$, impacting freeze-in and related mechanisms [2004.08404].

### 4.3 Open Questions and Limitations

- Thermal masses: the effect of temperature-dependent masses on $\Gamma_\Phi$ can alter $T_{\rm max}$,
- Instantaneous thermalization: the assumption may fail, necessitating kinetic or Boltzmann analyses,
- Nonperturbative preheating and nontrivial potential shapes can change the maximum temperature,
- UV completions and the possible suppression or enhancement of $T_{\rm max}$ are model-dependent and require further studies [2004.08404].

## 5. Broader Implications and Model-Agnostic Summary

- $T_{\rm max}$ is an upper bound for the temperature attained by the early Universe following inflation but prior to full radiation domination.
- Baryogenesis or new particle production that depend on temperatures beyond $T_{\rm RH}$ but below $T_{\rm max}$ remain viable in this thermal window.
- $T_{\rm max}$, being largely independent of specific microphysical details (for fixed $y$ and $\mu$), provides a robust target for assessing the viability of high-scale, temperature-dependent early-Universe phenomena.

| Parameter          | $k=2$ (Quadratic)        | $k=4$ (Quartic)             | $k=3$                   |
|--------------------|--------------------------|-----------------------------|-------------------------|
| $T_{\rm max}$      | $2 \times 10^{12}$ GeV   | $2 \times 10^{12}$ GeV      | $2 \times 10^{12}$ GeV  |
| $T_{\rm RH}$       | $10^{10}$ GeV            | $10^{4}$ GeV                | ~intermediate           |
| $T_{\rm max}/T_{\rm RH}$ | $y^{-1/2}$              | strong $k$ dependence        |  —                      |
| $y$ (limit)        | $<10^{-5}$ (perturbative)| $<10^{-5}$                  | $<10^{-5}$              |

All temperature scalings here assume $\mu \sim M_P$, $y \sim 10^{-5}$, and $g_* \sim 100$ [2004.08404].

## 6. Summary of Key Analytical Results

\[
\boxed{
\begin{aligned}
& T_{\rm max}^4 = \frac{15\, y^2}{16\pi^3 g_*} \sqrt{3k(k-1)}\,\mu^{4/k} M_P^{4/k} \rho_{\rm end}^{(k-1)/k} \left(\frac{3k-3}{2k+4}\right)^{3(k-1)/(7-k)} \\
& T_{\rm RH}^4 = \frac{15\, y^{2k}}{2^{4k-1} \pi^{2+k} g_*} [3k(k-1)]^{k/2} \mu^4
\end{aligned}
}
\]
with $T_{\rm max}$ only mildly sensitive to $k$ and $T_{\rm RH}$ highly sensitive to both $y$ and $k$ [2004.08404].

The maximum reheating temperature is thus a pivotal scale for post-inflationary cosmology, controlling early-Universe thermal processes, setting benchmarks for new physics, and constraining model space via cosmological observables and particle physics requirements. Its rigorous, model-dependent computation remains an active area of research, with outstanding questions in the validity of instantaneous thermalization, the effects of non-perturbative phenomena, and the precise role of thermal masses.

Source: https://www.emergentmind.com/topics/maximum-reheating-temperature