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Alexei Starobinsky investigated how energy stored in an oscillating inflaton can become particles after cosmic inflation. With Lev Kofman and Andrei Linde, he developed a treatment of preheating through parametric resonance in an expanding cosmological background.

Repeated oscillations can amplify a coupled scalar field much more rapidly than independent perturbative decays would suggest. The calculation follows both the growth of the produced particles and their backreaction on the oscillating field. That feedback, together with scattering, changes and eventually terminates the resonance.^[ref]

Andrei Linde develops models of cosmic inflation and the transition to the subsequent hot cosmic state. His work with Lev Kofman and Alexei Starobinsky explained how coherent inflaton oscillations can transfer energy through parametric resonance.

In an expanding background the amplification need not follow the simple periodic pattern of a fixed-frequency oscillator. Their study treated successive production events, backreaction and rescattering, establishing when efficient preheating occurs within the model. Producing particles and reaching thermal equilibrium remain distinct stages.^[ref]

For $u^{\prime \prime }+[1+k{ϵ}^2+ϵ\cos t]u=0$, set $t=2s$. The Mathieu equation has $a=4+4k{ϵ}^2$ and $q=-2ϵ$. Near the value $4$, the orthonormal even Fourier series basis $1,\cos 2s,\cos 4s,\cos 6s$ gives diagonal entries $0,4,16,36$ and adjacent couplings $\sqrt{2}q,q,q$. The odd basis $\sin 2s,\sin 4s,\sin 6s$ gives diagonal entries $4,16,36$ and adjacent couplings $q,q$. Higher modes require at least six coupling steps to affect the root near $4$, so these truncations determine its coefficients through $q^4$.

Substituting $a=4+c_2{q}^2+c_4{q}^4$ into the two characteristic polynomials and equating coefficients yieldsConverting back gives the displayed gap edges with remainders $O({ϵ}^4)$ in $k$. In a fixed bounded neighborhood of this parametric resonance, the true unstable gap is $k_{-}<k<k_{+}$. The leading slow-amplitude calculation cannot decide exact stability at $k=-1/12$ or $5/12$; the fourth-order shifts put those fixed values just outside the gap for sufficiently small positive $ϵ$. This does not provide a bound uniform in $ϵ$ for their transient amplification.

Lev Kofman co-developed models of energy transfer after cosmic inflation, including particle production amplified by an oscillating inflaton. The reheating analysis treats effects beyond a purely perturbative decay calculation, with parametric resonance and the response of the produced particles changing the subsequent evolution.^[ref]

The physical lesson is that ending accelerated expansion does not by itself establish a thermal universe. One must explain how energy moves into other degrees of freedom and how the resulting distribution evolves.

The Mathieu equation is the displayed linear differential equation with periodic coefficients. Floquet theory describes its bounded and exponentially growing solutions, while its periodic Mathieu characteristic values mark the edges of parametric resonance gaps. The forcing normalization matters: replacing $t$ by $t/2$ rescales both the constant and periodic coefficients.

Parametric resonance is growth caused by periodic variation of a system coefficient, rather than an additive forcing term. In the Mathieu equationthe stiffness varies near twice the natural angular frequency. Put $T=ϵt$ and $u_0=a(T)\cos t+b(T)\sin t$. The order-$ϵ$ correction has resonant coefficients $-2b_T-a$ in the cosine component and $2a_T+b$ in the sine component. Eliminating them givesThe slow eigenvalues are $\pm 1/2$, so an initial component in the growing direction is amplified as $e^{ϵt/2}$ at leading order. The exactly zero solution remains zero. Detuning competes with this coupling and produces an instability tongue, while Floquet theory gives the exact periodic-coefficient stability criterion. A weak-parameter growth approximation needs a stated time interval and can fail once neglected nonlinearities become important.^[ref]

Floquet theory suggests a sequence of instability tongues nearAt these limiting frequencies the unperturbed Floquet multipliers $e^{\pm 2\pi i\omega }$ coincide. Each multiplication by $\cos t$ changes a Fourier frequency by one, so $m$ forcing interactions are needed to couple $+m/2$ and $-m/2$. This explains both the sequence of parametric resonances and the successive powers of $ϵ$ in their widths.

There is no implication that the exact half-integers themselves lie in every finite-$ϵ$ growing interval. The tongue centers generally shift by $O({ϵ}^2)$, which exceeds the width when $m>2$. It is the narrow intervals near the shifted centers that contain exponentially growing solutions. Frequencies staying a fixed distance from every half-integer are stable for sufficiently small $ϵ$.

Away from the first parametric resonance, one forcing interaction changes a harmonic frequency by one and produces no resonant coupling of the two conjugate oscillations. Two interactions can couple frequency $+1$ to frequency $-1$, so the next parametric resonance is near $\omega =1$ and its effect first appears at order ${ϵ}^2$. This gives $p=2$ and the appropriate slow time $T={ϵ}^{2}t$.

Write $\omega =1+\kappa {ϵ}^2$ and $y_0=a(T)\cos t+b(T)\sin t$. The first correction satisfies $({\partial }_t^2+1)y_1=-\cos t{y}_0$, whose bounded particular solution isAt second order,The resonant part of $-\cos t{y}_1$ is $(5a/12)\cos t-(b/12)\sin t$. Applying the solvability condition in the method of multiple scales givesThe squared slow eigenvalue isThe second instability tongue of a weak Mathieu oscillator therefore has the leading intervalHere the two little-$o$ terms describe the respective exact edge locations; the strict inequality in $\kappa$ is the leading interior criterion. In particular $\omega =1$ is unstable, with physical growth rate ${ϵ}^2\sqrt{5}/24+o({ϵ}^2)$. The general leading solution for bounded $T$ again has $y_0=a(T)\cos t+b(T)\sin t$, where the matrix exponential of the displayed two-by-two amplitude system evolves arbitrary $a(0),b(0)$. Adding $ϵy_1$ identifies the first nonresonant harmonic correction. The leading expression alone has error $O(ϵ)$ on $t=O({ϵ}^{-2})$.

For comparison, at fixed $\omega \ne 1/2,1$, write $y_0=A(T)e^{i\omega t}+\overline{A}(T)e^{-i\omega t}$. The first forced harmonics have coefficients $A/[2(2\omega +1)]$ and $A/[2(1-2\omega )]$. Their return to the original harmonic at the next order givesIt is a phase correction, not amplitude growth. Consequently, away from the first two resonance neighborhoods, a general leading solution valid for bounded ${ϵ}^{2}t$ isNarrower higher-order instability intervals are invisible on this time scale. The detuned resonant calculation above is needed near $\omega =1$ because both the phase shift and the conjugate-mode coupling then matter.

The method of multiple scales applies when a small perturbation acts cumulatively over a long time, as in weakly perturbed harmonic oscillators or slowly modulated waves. A regular asymptotic expansion can contain secular terms, such as $ϵt\sin (\omega t)$, which cease to be small on the time scale of interest. Introduce independent fast and slow times, for example $t$ and $T=ϵt$, and allow the leading amplitude and phase to depend on $T$. The solvability condition in the method of multiple scales removes resonant forcing at each order and determines that slow evolution. For the Mathieu equation, these amplitude equations reveal parametric resonance and its instability intervals.

For a homogeneous canonical scalar field, ${\rho }_{\varphi }={\dot{\varphi }}^2/2+V$ and $P_{\varphi }={\dot{\varphi }}^2/2-V$. Potential-energy domination gives $P_{\varphi }\simeq -{\rho }_{\varphi }$, accelerated expansion and a decreasing comoving Hubble radius. Hubble friction can permit slow roll down a sufficiently flat potential. Enough inflation stretches an initially small causally connected region to encompass our observable universe, suppresses the relative spatial-curvature term and dilutes unwanted massive relics. quantum fluctuations supply possible seeds for cosmological structure; a successful model must also end inflation and create the hot post-inflationary universe. These statements do not imply that arbitrary initial conditions inevitably inflate.

Use the stated reduced-Planck units throughout the field calculation. Differentiating the Friedmann constraint and substituting the field equation givesDivision assumes the nontrivial expanding branch $H>0$; the vacuum endpoint follows continuously.

During rapid oscillations, require $m\gg H$ and $|\dot{A}|\ll mA$. To leading order, $\dot{\varphi }\simeq -mA\sin mt$, while the Friedmann constraint gives $H_m^2\simeq m^2{A}^2/6$. ThusAveraging over many oscillations on a timescale still short compared with $H^{-1}$ givesThe quadratic-inflaton Hubble oscillations obey, to leading order in $H_m/m$,Differentiating this expression reproduces the cosine term, with smaller corrections from the slowly varying amplitude. Instantaneously $H$ remains nonincreasing because its exact derivative is $-{\dot{\varphi }}^2/2$; an oscillatory correction does not mean oscillatory growth of the total Hubble parameter.

The mean equation integrates toThe shift $t_{\ast }$ is set by matching to the earlier evolution, not necessarily the Big Bang origin of the full solution. Averaged kinetic and potential energies are equal, so $⟨P_{\varphi }⟩\simeq 0$, $⟨{\rho }_{\varphi }⟩\propto a^{-3}$ and $A\propto a^{-3/2}$. The coherent quadratic oscillations behave as pressureless matter, not radiation, even though they are a scalar field rather than a gas of cold particles. radiation would instead give $a\propto t^{1/2}$. The averaged approximation is controlled only after oscillations are fast compared with expansion.

reheating requires interactions beyond the isolated field equations above. The oscillating inflaton can decay into other particles, whose scattering and further decays establish a thermal bath. The perturbative inflaton reheating equations areExpansion redshifts the components differently; transfer becomes efficient around $H\sim \Gamma$, and radiation domination follows when the transferred relativistic energy dominates. For suitable couplings, nonperturbative preheating by parametric resonance can precede thermalization, with backreaction and rescattering affecting its completion. Neither the decay rate nor thermalization is specified by $m$ and $V$ alone. Instantaneous reheating idealizes all this as immediate conversion of the end-of-inflation energy to radiation.^[ref]

For instantaneous-reheating horizon matching, specify the comparison: take the initial patch to be of order $H_i^{-1}$, compare the same-shaped inflationary history with approximately fixed $H_i/H_{\mathrm{end}}$, and assume instantaneous reheating followed by adiabatic expansion. Ignoring changes in relativistic degrees of freedom, $a_0/a_{\mathrm{reh}}\propto T_{\mathrm{reh}}$, while energy conservation and the Friedmann equation give $H_{\mathrm{end}}\propto T_{\mathrm{reh}}^2$. The present size of this inflated patch therefore scales asAt fixed present Hubble radius, a higher instantaneous reheating scale requires more e-folds in this standard comparison, approximately $\Delta N\simeq \ln (T_2/T_1)$, up to slow scale and entropy-degree corrections. Hotter reheating gives more subsequent expansion, but the associated initial Hubble patch is smaller by a stronger power of temperature. If instead one artificially holds the initial physical patch or $H_i$ fixed while varying the end energy independently, the post-inflationary geometric effect alone would give the opposite trend. temperature without an inflation-scale comparison convention does not fix that alternative problem; this is why the matching assumptions must be stated.^[ref]

Nonperturbative particle production after cosmic inflation, for example by parametric resonance caused by an oscillating inflaton. Large occupation numbers can grow before a thermal bath exists. Backreaction, rescattering and subsequent thermalization affect the outcome, so an effective perturbative decay rate need not describe every stage.^[ref]