Reheating and the Post-Inflationary Universe
How the cold, empty universe left by inflation is repopulated with a hot thermal plasma: perturbative decay, parametric resonance, and connections to axion cosmology
Overview
At the end of slow-roll inflation, the inflaton oscillates around the minimum of its potential and decays into Standard Model particles, reheating the universe and initiating the hot Big Bang. The reheating temperature \(T_{\text{rh}}\) is a crucial parameter: it must be high enough for baryogenesis and BBN, yet its value depends on the inflaton's couplings to other fields. We examine perturbative reheating, the explosive non-perturbative process of preheating via parametric resonance, and the implications for axion cosmology.
1. The End of Inflation
Inflation ends when the slow-roll parameter \(\epsilon\) reaches unity. The inflaton then oscillates coherently around the minimum of \(V(\phi)\). For a quadratic minimum \(V \approx \frac{1}{2}m_\phi^2\phi^2\), the oscillating field behaves as pressureless matter:
$$\langle p_\phi \rangle = \langle \tfrac{1}{2}\dot{\phi}^2 - V \rangle = 0 \quad \Longrightarrow \quad \rho_\phi \propto a^{-3}$$
The universe is effectively matter-dominated during the oscillation phase, with the energy stored in a coherent condensate of zero-momentum inflaton particles.
2. Perturbative Reheating
If the inflaton has a small coupling to lighter fields (e.g., \(g\phi\bar\psi\psi\)or \(\sigma\phi\chi^2\)), it decays perturbatively with rate \(\Gamma_\phi\). The inflaton energy density evolves as:
$$\dot{\rho}_\phi + 3H\rho_\phi = -\Gamma_\phi\,\rho_\phi$$
while the radiation energy density produced by the decay satisfies:
$$\dot{\rho}_R + 4H\rho_R = +\Gamma_\phi\,\rho_\phi$$
Reheating completes when \(\Gamma_\phi \sim H\). At this point, the energy density is \(\rho \sim \Gamma_\phi^2 M_{\text{Pl}}^2\), and the reheating temperature is:
Reheating Temperature
$$\boxed{T_{\text{rh}} \approx 0.2\left(\frac{100}{g_*}\right)^{1/4}\sqrt{\Gamma_\phi\, M_{\text{Pl}}}}$$
The prefactor arises from \(\rho_R = (\pi^2/30)g_* T^4\) and \(H^2 = \rho/(3M_{\text{Pl}}^2)\). For gravitational-strength couplings (\(\Gamma_\phi \sim m_\phi^3/M_{\text{Pl}}^2\)),\(T_{\text{rh}}\) can be as low as \(\sim 1\) MeV.
BBN requires \(T_{\text{rh}} \gtrsim 4\) MeV to produce the observed light element abundances. Baryogenesis mechanisms typically require much higher temperatures: thermal leptogenesis needs \(T_{\text{rh}} \gtrsim 10^9\) GeV.
3. Preheating: Parametric Resonance
Before perturbative decay completes, the coherent oscillations of \(\phi\) can drive explosive, non-perturbative particle production. Consider a coupling \(g^2\phi^2\chi^2\)to a scalar \(\chi\). The mode equation for \(\chi_k\) in the oscillating background \(\phi(t) = \Phi(t)\sin(m_\phi t)\) is:
Mathieu Equation
$$\ddot{\chi}_k + \left(\frac{k^2}{a^2} + m_\chi^2 + g^2\Phi^2\sin^2(m_\phi t)\right)\chi_k = 0$$
Rescaling to dimensionless variables \(z = m_\phi t\), this takes the standard Mathieu form:
$$\chi_k'' + (A_k - 2q\cos 2z)\,\chi_k = 0$$
with \(q = g^2\Phi^2/(4m_\phi^2)\). For \(q \gg 1\) (broad resonance), modes in instability bands grow exponentially: \(\chi_k \propto e^{\mu_k z}\)with Floquet exponent \(\mu_k\). The number density of \(\chi\)particles grows as:
$$n_\chi \propto e^{2\mu_k m_\phi t}$$
This explosive particle production occurs on a timescale much shorter than the perturbative decay time, and can transfer a significant fraction of the inflaton energy into \(\chi\)particles within a few oscillation periods.
4. Thermalisation
The particles produced by preheating are highly non-thermal, occupying specific momentum bands. Subsequent scattering processes redistribute the energy and eventually produce a thermal distribution. The thermalisation timescale depends on the coupling strengths but is typically much shorter than the Hubble time:
$$t_{\text{therm}} \sim \frac{1}{\alpha^2 T} \ll H^{-1}$$
where \(\alpha\) is the relevant gauge coupling. After thermalisation, the universe enters the standard radiation-dominated epoch with temperature \(T_{\text{rh}}\).
5. Implications for Axion Cosmology
The reheating temperature has profound implications for axion dark matter. The QCD axion acquires its mass at the QCD phase transition (\(T_{\text{QCD}} \sim 150\) MeV). If the Peccei-Quinn symmetry is broken after inflation (i.e., \(T_{\text{rh}} > f_a\)where \(f_a \sim 10^{10}\text{--}10^{12}\) GeV is the axion decay constant), then the axion field takes different random values in different Hubble patches, and topological defects (axion strings and domain walls) form.
Conversely, if \(T_{\text{rh}} < f_a\), the PQ symmetry was broken during inflation and the axion field is homogeneous across the observable universe. In this “pre-inflationary” scenario, the relic axion abundance depends on the initial misalignment angle \(\theta_i\):
Axion Relic Density (Misalignment)
$$\Omega_a h^2 \approx 0.12\left(\frac{f_a}{10^{12}\;\text{GeV}}\right)^{7/6}\theta_i^2$$
For \(\theta_i \sim \mathcal{O}(1)\), the axion can constitute all of dark matter if \(f_a \sim 10^{12}\) GeV, corresponding to an axion mass\(m_a \sim 10\;\mu\text{eV}\).
The interplay between \(T_{\text{rh}}\) and \(f_a\) thus determines whether axion dark matter is produced via the misalignment mechanism alone, or whether string and domain wall decay also contribute — potentially changing the predicted axion mass window by an order of magnitude.
6. The Gravitino Problem
In supersymmetric theories, gravitinos are produced thermally after reheating with an abundance proportional to \(T_{\text{rh}}\):
$$\frac{n_{3/2}}{s} \sim 10^{-12}\left(\frac{T_{\text{rh}}}{10^{10}\;\text{GeV}}\right)$$
If the gravitino is unstable with mass \(m_{3/2} \sim 100\) GeV – 1 TeV, its late decays (\(\tau \sim M_{\text{Pl}}^2/m_{3/2}^3 \sim 10^8\) s) inject energetic particles that destroy the light elements produced during BBN. Avoiding this “gravitino problem” typically requires:
Gravitino Bound on Reheating
$$T_{\text{rh}} \lesssim 10^{6}\text{--}10^{9}\;\text{GeV}$$
The exact bound depends on the gravitino mass and the dominant decay channels. This creates tension with thermal leptogenesis, which requires \(T_{\text{rh}} \gtrsim 10^9\) GeV.