Preheating: How Parametric Resonance Drains the Inflaton

What Preheating Is: The Physical Basis

When inflation ends, the field that drove it — the inflaton, φ — does not simply vanish. It rolls to the bottom of its potential and begins to oscillate coherently about the minimum, behaving like a gigantic homogeneous condensate of zero-momentum particles. In the simplest chaotic-inflation model V = ½m²φ², the field oscillates with frequency set by the inflaton mass, m ≈ 10¹³ GeV (about 10⁻⁶ times the Planck mass), and an initial amplitude Φ near the Planck scale.

The old picture, due to reheating, treated this as slow perturbative decay: each inflaton quantum independently decays, φ → χχ, at a rate Γ, and the universe heats when Γ ≈ H. Preheating, introduced by Kofman, Linde and Starobinsky in 1994, showed this misses the dominant physics. Because φ is a huge classical amplitude, it drives the coupled fields collectively and coherently. Quantum fluctuations of a field χ coupled via g²φ²χ² feel a periodically varying mass and are amplified by parametric resonance — the same instability that pumps a swing — producing particles exponentially fast and far from equilibrium.

The Mechanism: Mathieu's Equation and Floquet Growth

Consider a scalar χ coupled to the inflaton by an interaction ½g²φ²χ². Each Fourier mode χ_k obeys an oscillator equation whose frequency depends on the oscillating background φ(t) = Φ sin(mt):

• χ_k″ + (k²/a² + g²φ²(t)) χ_k = 0

Rescaling time as z = mt turns this into the Mathieu equation:

• χ_k″ + (A_k − 2q cos 2z) χ_k = 0, with q = g²Φ²/(4m²) and A_k = k²/(m²a²) + 2q.

By Floquet's theorem, solutions take the form e^(μ_k z) P(z) with P periodic. Wherever the Floquet exponent μ_k has a positive real part, that mode grows exponentially, and the occupation number balloons as n_k ∝ e^(2μ_k m t). On the Mathieu instability chart these unstable modes form resonance bands. Physically, χ quanta are created in bursts each time φ sweeps through zero and its effective mass changes non-adiabatically; the phases of successive bursts add stochastically, giving a random-walk that still grows exponentially on average.

Key Quantities: Broad Resonance and a Worked Example

Everything hinges on the resonance parameter q = g²Φ²/(4m²).

• Narrow resonance (q ≲ 1): only the first thin band near k ≈ m is unstable; μ is small (≲ q/2) and cosmic expansion easily redshifts modes out of the band, choking the growth.
• Broad resonance (q ≫ 1): many bands overlap, the typical Floquet exponent saturates at μ ≈ 0.1–0.25 independent of q, and momenta up to k* ≈ (g m Φ)^(1/2) ~ q^(1/4) m are amplified.

Worked numbers: take m = 10¹³ GeV, Φ ≈ 0.1 M_Pl ≈ 10¹⁸ GeV, and coupling g = 10⁻³. Then q = g²Φ²/(4m²) ≈ (10⁻⁶)(10⁴)/4 ≈ 2.5×10³ — deeply broad. With μ ≈ 0.2, the occupation number grows by e^(2·0.2) ≈ 1.5 per oscillation, so after only ~50–70 oscillations — a span of order 10⁻³⁵ s — n_k reaches ~10²⁰ and the χ energy density catches up to the inflaton's, near ρ ~ (10¹⁶ GeV)⁴. That is the whole point: preheating is essentially instantaneous on cosmological timescales.

How It's Studied: Floquet Charts and Lattice Simulations

Preheating leaves no direct fossil we can point a telescope at (yet), so it is probed theoretically and numerically. The first pass is linear Floquet analysis: solve the mode equation over one period and map μ_k across parameter space to locate the resonance bands and read off growth rates.

But exponential growth quickly makes the fields large, and three effects break the linear treatment: backreaction (produced χ quanta shift the effective inflaton mass and detune the resonance), rescattering (χ and φ quanta scatter, feeding φ fluctuations and fragmenting the condensate), and expansion (redshift sweeping modes through bands). Capturing these requires full nonlinear classical lattice simulations — codes such as LATTICEEASY (Felder & Tkachev), PSpectRe, DEFROST, GABE, and the modern CosmoLattice. These simulations show resonance shutting itself off once ρ_χ ≈ ρ_φ, followed by turbulent thermalization. Excitingly, the violent, inhomogeneous field motion can source a stochastic gravitational-wave background — a potential future observational window on the preheating epoch.

Preheating Versus Its Cousins

Preheating is one member of a family of post-inflationary energy-transfer mechanisms, and it is worth separating them:

• Perturbative reheating is the slow, thermal, one-quantum-at-a-time φ→χχ decay that completes the job after preheating exhausts itself. Preheating does not thermalize the universe; it drains the coherent inflaton into a non-thermal spectrum that later thermalizes.
• Tachyonic (spinodal) preheating occurs when the effective mass-squared goes negative, as in hybrid or hilltop models. Long-wavelength modes grow with no oscillation needed, and symmetry breaking can finish in a single field swing — even faster than parametric resonance.
• Instant preheating (Felder, Kofman, Linde) exploits a large coupling so that a single zero-crossing of φ creates very heavy χ quanta, which then decay before the next crossing.

The unifying theme is non-perturbative, non-adiabatic particle creation driven by a fast-changing background — in stark contrast to the adiabatic, Boltzmann-style decay of ordinary reheating.

Significance and Open Questions

Preheating reshaped how cosmologists think about the transition from inflation to the hot Big Bang. Because it is so efficient and so non-thermal, it opens doors that slow reheating cannot:

• Non-thermal relics: it can produce superheavy particles, dark matter, or even GUT-scale bosons whose out-of-equilibrium decays may drive baryogenesis.
• Symmetry restoration and defects: the huge fluctuation variance can restore broken symmetries and later re-form topological defects (cosmic strings, monopoles), tightening model constraints.
• Gravitational waves: the anisotropic stress of the fragmenting fields sources GWs whose peak frequency today depends on the energy scale — a possible smoking gun.

Open questions remain: precisely how and when the non-thermal spectrum thermalizes to fix the reheating temperature T_reh (bounded below by ~4 MeV for nucleosynthesis); how preheating operates in realistic Starobinsky R² and α-attractor models favored by Planck data; whether metric perturbations amplified during preheating can affect the CMB; and whether next-generation GW detectors could ever reach the ~MHz–GHz frequencies typical of this epoch. Preheating remains an active, model-dependent frontier.

Preheating regimes and their close cousin, perturbative reheating

Mechanism	Key parameter	Efficiency / timescale	Distinguishing feature
Narrow parametric resonance	q = g²Φ²/4m² ≲ 1	Slow; growth only in thin bands near k ≈ m, easily killed by expansion	Resonance in first instability band only; perturbative-like
Broad parametric resonance	q ≫ 1 (e.g. q ~ 10³–10⁵)	Explosive; n_k ~ e^(2μ_k m t) grows in tens of oscillations	Particle bursts each time φ crosses zero; stochastic; μ ~ 0.1–0.25
Tachyonic preheating	Negative m²_eff (spinodal)	Fastest; completes within ~1 field oscillation	Symmetry-breaking / hilltop; not oscillation-driven
Instant preheating	g very large, single crossing	One pass through φ=0 makes heavy quanta	Produced particles decay before next crossing
Perturbative reheating	Decay rate Γ vs H	Slow; sets T_reh when Γ ≈ H	Born-level φ→χχ decay; thermal, one quantum at a time