Reheating

The problem inflation leaves behind

At the end of slow-roll inflation, the universe is cold and almost empty. Whatever particles were present before inflation have been diluted by a factor of e^180 ≈ 10⁷⁸, leaving essentially zero density of anything except the inflaton field itself. The inflaton sits in a single coherent classical state holding 100% of the universe's energy density — perhaps ρ_φ ~ M_Pl² H_inf² ~ 10⁶⁴ GeV/m³ for high-scale models.

The hot Big Bang needs the opposite. Nucleosynthesis at t ~ 1 s requires a thermal plasma at T ~ 1 MeV, dominated by relativistic photons and neutrinos. The CMB at recombination requires a blackbody radiation spectrum with one part in 10⁵ anisotropies tracing acoustic oscillations of a baryon-photon fluid. Galaxy formation requires cold dark matter. None of this can happen if the universe is cold and inflaton-dominated.

Reheating is the bridge. The inflaton must convert its potential energy into a thermal bath of Standard Model particles, fast enough that nucleosynthesis happens on schedule, but not so fast that it overproduces dangerous relics (gravitinos, monopoles) or disrupts the inflationary smoothness.

Oscillation: the inflaton bounces in its well

Picture the inflaton's potential V(φ) as a smooth bowl with a flat plateau on one side. During inflation, φ rolls slowly along the plateau. Eventually it reaches the steep wall and slides down toward the minimum at φ = φ_min. The slow-roll approximation breaks; ε grows past 1; the kinetic energy of φ becomes comparable to its potential. Inflation ends.

What happens next depends on the shape of the potential near its minimum. For the canonical quadratic case V = ½ m_φ² φ², φ oscillates harmonically:

φ(t) ≈ Φ(t) · sin(m_φ t)

where Φ(t) is a slowly decreasing envelope. The energy density of the oscillating condensate scales as ρ_φ ∝ a⁻³ — matter-like, because a harmonically oscillating massive field has equation of state w = 0 averaged over a period. For a quartic potential V = (λ/4)φ⁴, ρ_φ ∝ a⁻⁴ — radiation-like, with w = 1/3. The universe behaves matter-dominated or radiation-dominated depending on which power of φ dominates V near its minimum.

During this oscillating phase, the universe is filled with a coherent condensate of zero-momentum 'inflaton particles' of energy m_φ each. The total particle number density is ρ_φ / m_φ ~ 10⁵¹ m⁻³ for m_φ ~ 10¹³ GeV at the end of inflation. Each oscillation is one shot at producing real particles.

Perturbative decay: the original picture

The simplest reheating model treats each inflaton 'particle' as decaying independently into Standard Model particles via its couplings:

φ → χ χ    (bosonic)
φ → ψ̄ ψ    (fermionic)

The perturbative decay rates are

Γ_φ→χχ  =  g⁴ Φ² / (8π m_φ)    (boson final state)
Γ_φ→ψ̄ψ  =  y² m_φ / (8π)        (fermion final state)

Particles are produced steadily as the inflaton condensate decays. The Boltzmann equation for the inflaton energy density is

dρ_φ / dt + 3 H ρ_φ = −Γ_φ ρ_φ

Roughly, the inflaton decays efficiently once H drops below Γ_φ. That happens at the time when

H_decay ~ Γ_φ ↔ ρ_φ ~ M_Pl² Γ_φ²

By energy conservation, that energy thermalises at temperature

T_RH = (90 / π² g_*)^(1/4) · √(M_Pl Γ_φ)

For a typical g_* ~ 200 (full Standard Model plus additions), this gives T_RH ≈ 0.5 √(M_Pl Γ_φ). With Γ_φ ~ 10⁻⁵ m_φ and m_φ ~ 10¹³ GeV, T_RH ~ 10¹⁰ GeV.

Preheating: the explosive first burst

The perturbative picture missed something important. In 1994, Linde, Kofman and Starobinsky pointed out that the oscillating inflaton is not a random soup of decaying particles — it is a coherent classical background. Other fields coupled to φ feel a periodic mass term:

m_χ²(t) = m_χ,0² + g² φ²(t) = m_χ,0² + g²Φ² sin²(m_φ t)

The mode equation for χ_k becomes a Mathieu equation with periodic coefficient. Mathieu equations have notorious 'parametric resonance' instability bands: certain wavenumbers k experience exponentially growing solutions, with occupation numbers n_k growing as exp(2μ_k m_φ t) where μ_k is the Floquet exponent of the resonance band. For wavenumbers in the resonance band, n_k can grow by factors of 10²⁰ in a few oscillations.

The physical picture: each oscillation peak of φ acts as a brief, very deep potential well for χ. Quanta in resonance bands get amplified coherently, like a child pumping a swing. The result is a fast, non-thermal, non-equilibrium production of χ particles with a very specific spectrum peaked at a few discrete wavenumbers.

Preheating completes in ~10⁻³⁰ s — milliseconds-fast on inflation timescales. After it ends, the universe contains a non-thermal mixture of inflaton remnants and amplified χ modes, far from equilibrium. Subsequent rescattering and perturbative decay drive the system toward a thermal distribution at T_RH.

A worked numerical example

Quantity	Value	Notes
Inflation scale H_inf	~10¹³ GeV	Bounded by tensor-to-scalar ratio r < 0.04
Inflaton mass m_φ	~10¹³ GeV	Set by potential curvature at minimum
Energy density at end of inflation	~10⁶⁴ GeV/m³	ρ ~ M_Pl² H_inf²
Yukawa-like coupling y	~10⁻⁵	Phenomenological parameter
Decay rate Γ_φ	~10⁷ GeV ≈ 10³¹ s⁻¹	Γ = y² m_φ / 8π
Reheat temperature T_RH	~10⁹ GeV	From energy conservation
Time of reheating	~10⁻³⁰ s after BB	t ~ 1/Γ_φ when H ~ Γ_φ
g_* at T_RH	~200	Full SM + extensions
e-folds during reheating	~few	Universe expands by O(10)

The exact numbers shift by orders of magnitude depending on whether the inflation scale is high or low and what couplings φ has to the rest of physics. The point is that T_RH is highly tunable, which is both a feature (allows model-building flexibility) and a bug (hard to falsify).

Inflation vs reheating: the contrast

Property	Inflation	Reheating
Driver	Slowly rolling inflaton, ε ≪ 1	Oscillating inflaton, ε ~ 1
Equation of state	w ≈ −1	w = 0 (quadratic) or 1/3 (quartic)
Expansion	Exponential, a(t) ∝ exp(Ht)	Matter- or radiation-like, a(t) ∝ t^(2/3) or t^(1/2)
Temperature	Diluted to ~0	Rises to T_RH and thermalises
Duration (e-folds)	~50–60	O(1–10)
Particle content	Inflaton condensate only	All Standard Model + relics
Energy in modes	Quantum vacuum + zero mode	Thermal distribution post-reheat

Constraints on the reheat temperature

• Big Bang nucleosynthesis (lower). Light element abundances (D/H, ⁴He, ⁷Li) form at T ~ 0.1–1 MeV. The universe must be radiation-dominated, in thermal equilibrium, by then. This forces T_RH > 4 MeV. Tighter constraints from CMB-era plasma physics push this slightly higher.
• Gravitino overproduction (upper, in SUSY). If the early universe is supersymmetric, gravitinos with mass ~100 GeV are produced thermally with abundance ~T_RH / M_Pl per Hubble volume. They are nearly stable but decay in ~10⁴ s — long after BBN. Their decay products would disrupt nucleosynthesis predictions. Avoiding the problem requires T_RH < 10⁶–10⁹ GeV depending on gravitino mass.
• Monopole avoidance. If reheating exceeds the GUT scale (~10¹⁵–10¹⁶ GeV), thermal production recreates the monopole disaster that inflation was supposed to dilute away. T_RH < 10¹⁵ GeV is therefore essentially required.
• Baryogenesis viability. Various baryogenesis scenarios require minimum reheat temperatures. Thermal leptogenesis via the seesaw mechanism with right-handed neutrinos of mass M_R needs T_RH > M_R ~ 10⁹ GeV.
• Scalar tilt n_s. The Planck-measured tilt n_s ≈ 0.9649 ± 0.0042 depends on the number of e-folds N_* between the end of inflation and the moment our observable Hubble volume left the horizon. N_* depends on the reheating history. Self-consistent inflation models constrain the joint (n_s, T_RH) plane.

History: 1982 to today

The original inflation proposal (Guth 1981) had no working reheating mechanism — its 'old inflation' relied on bubble nucleation that left a fractal pattern of bubble walls with no homogeneous radiation. This was a serious problem, and Guth himself acknowledged it. The 'new inflation' of Linde (1982) and Albrecht and Steinhardt (1982) replaced the first-order transition with a slow-roll scenario, and Albrecht, Steinhardt, Turner and Wilczek (1982) gave the first reheating calculation: perturbative decay of the inflaton with a calculable T_RH. This is the simple picture taught in most courses today.

The big revolution was 1994. Linde, Kofman and Starobinsky (Phys. Rev. Lett. 73, 3195) showed that the perturbative picture was incomplete — the coherent oscillation produces explosive non-perturbative particle production via parametric resonance long before the perturbative decay kicks in. They called this 'preheating'. The result re-opened essentially every calculation involving the universe's earliest moments: gravitational wave production, baryogenesis, relic abundance estimates, monopole and defect re-production. The full reheating problem became a major numerical-cosmology endeavor, with lattice simulations like LATTICEEASY, CosmoLattice, and GABE.

Modern reheating theory is rich: tachyonic preheating, instant preheating, fermionic preheating, gravitational reheating, geometric reheating. Each has different signatures in primordial gravitational waves or CMB μ-distortions. Constraining the reheating history through observations is an active frontier — the LiteBIRD and CMB-S4 surveys aim to pin n_s precisely enough that combined with inflation models they constrain T_RH.

Common pitfalls

• Treating 'reheating' as the start of the hot Big Bang. The hot Big Bang in standard cosmology textbooks really starts at T_RH. What happened before is inflation. The 'Big Bang singularity' at t = 0 is replaced in inflationary cosmology by an inflationary phase whose origins remain unclear.
• Ignoring preheating. The simple perturbative T_RH formula misses the dominant production mechanism in many models. Preheating completes in 10⁻³⁰ s; perturbative decay can take much longer. The full thermal history requires lattice simulations.
• Forgetting the gravitino bound in SUSY. Any supersymmetric inflation model must have T_RH ≲ 10⁹ GeV. This is a hard constraint that rules out many otherwise-plausible scenarios.
• Conflating T_RH with the inflation scale. Inflation can be high-scale (H ~ 10¹³ GeV) with low-scale reheating (T_RH ~ 10⁴ GeV) if the inflaton couples weakly to matter. The two scales are independent.
• Assuming reheating is observed. No direct observational signature of reheating has been detected. All current constraints are indirect, through BBN, CMB, or gravitino-style bounds. A direct GW signal from preheating is a major target of future detectors.