Maximum Reheating Temperature

• Maximum reheating temperature is defined as the peak temperature attained during inflaton decay, marking the transition before full radiation domination.
• It is calculated by extremizing the temperature evolution function, thereby constraining inflationary potentials and setting benchmarks for baryogenesis and dark matter production.
• This parameter, largely independent of microphysical details for fixed couplings, serves as a robust target for probing early-universe thermal dynamics and new physics.

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, $T_{\rm RH} \equiv T(a_{\rm reh})$ (Garcia et al., 2020, Maity, 2017).

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 (Garcia et al., 2020).

• 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 $k=4$: $T(a) \propto a^{-3/4}$ (Garcia et al., 2020).

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 $k$ (Garcia et al., 2020).

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 $k=4$ (Garcia et al., 2020).

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 $y=10^{-5}$ (Garcia et al., 2020),
• $T_{\rm max}$ 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 (Garcia et al., 2020).

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 (Garcia et al., 2020).

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 $g_* \sim 100$ (Garcia et al., 2020).

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 $k$ (Garcia et al., 2020).

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.