Parametric Resonance in Coupled Scalar Fields

• Parametric Resonance Condition is defined as the periodic modulation of system parameters that leads to exponential field amplification when resonance criteria are met.
• The analytic framework divides the evolution into rolling and zero-crossing phases, quantifying energy boosts and the resonance termination through backreaction.
• This mechanism underpins preheating in cosmology, enhances inhomogeneities, and affects inflation stability, making it critical for early Universe models.

Parametric resonance is a dynamical instability that occurs in systems whose parameters—such as effective mass, coupling, or frequency—are modulated periodically in time. Parametric resonance manifests as exponential amplification of certain modes or fields, provided well-defined resonance conditions are met. Its mechanisms and implications are especially important in cosmological preheating, particle production, and coupled-field dynamics in the early Universe.

1. Analytic Structure and Regimes of Parametric Resonance

The paradigm is a two-scalar field model coupled via a quartic interaction: $$\mathcal{L} = -\frac{1}{2} m^2 \phi^2 - \frac{1}{2} M^2 \chi^2 - \lambda \phi^2 \chi^2 - \frac{1}{2} (\partial \phi)^2 - \frac{1}{2} (\partial \chi)^2$$ with a mass hierarchy $\lambda|\phi|^2 \gg m^2 \gg \lambda \chi^2$ and $m^2 \gg M^2$. The background field $\phi$ oscillates as a harmonic oscillator, $\phi(t) = |\phi| \sin(mt)$, dominating the effective mass of $\chi$: $$m_\text{eff}^2 = M^2 + 2\lambda \phi^2 \approx 2\lambda \phi^2$$ This time-dependence converts the $\chi$ equation into a Mathieu-type form, inherently susceptible to parametric resonance.

The analytic framework distinguishes two distinct phases for $\chi$:

• Rolling Stage: For $|\phi|$ away from zero, $\chi$ behaves as a slow-varying harmonic oscillator with adiabatic evolution. Amplitude and phase corrections scale as

$$\frac{d|\chi|}{dt} \propto -\frac{\lambda\phi \dot{\phi}}{m_\text{eff}^2}|\chi|, \quad \varphi_\chi = \frac{\pi^2 \lambda \phi \dot{\phi}}{m_\text{eff}^3}$$

• Zero-Crossing Stage: When $|\phi|$ approaches zero, $m_\text{eff}$ varies rapidly and $\chi$ evolves according to

$$\chi'' + \tau^2 \chi = 0, \quad \tau \equiv t(2\lambda |\phi|^2 m^2)^{1/4}$$

The solution, in terms of confluent hypergeometric functions, enables calculation of the energy "boost" parameter $\eta$.

The average "boost" for $\chi$ per half-cycle of $\phi$ is quantified by $\langle \eta \rangle \approx 0.346$, corresponding to a near doubling of $\chi$'s amplitude each cycle. The maximum boost per crossing is $\eta_m = \ln(1+\sqrt{2}) \approx 0.881$.

2. Exponential Amplification and Preheating Dynamics

The critical consequence of parametric resonance is the exponential amplification of $\chi$: $$|\chi| \rightarrow e^\eta |\chi|, \quad \langle \eta \rangle \approx 0.346$$ on average for each half oscillation of $\phi$, so that the energy increases by a factor $\approx 2$ per cycle. This is the essence of preheating—a stage following inflation during which energy is explosively transferred from the inflaton $\phi$ to other fields via parametric resonance.

Distinct Preheating Stages

• Large $\delta'$ (Randomized Phase) Stage: When the rotation angle $\delta$ shifts significantly between cycles, the boost is statistically distributed, and the net amplification is described by eigenvalues of the transfer matrix

$$T = R(\delta) \cdot \text{diag}(e^{\eta_m}, e^{-\eta_m})$$

The mean exponential rate is again $\langle \eta_l \rangle \approx 0.346$.

• Small $\delta'$ (Fixed Phase) Stage: When $\delta$ changes only slowly, the system converges toward a fixed-point phase and amplification occurs in a stepwise, platformed fashion. However, the mean growth rate remains the same.

3. Backreaction and Termination of Resonance

As $\chi$ grows, backreaction effects—specifically the correction to $\phi$'s effective potential by growing $\lambda \chi^2$—become important. The resonance terminates when the induced mass correction is no longer negligible: $$V_\text{eff}(\phi) \simeq \frac{1}{2} m^2 \phi^2 + \lambda \phi^2 \chi^2$$ Near zero-crossing, the feedback accelerates $\phi$'s oscillation, creating positive feedback that halts further resonance. For quadratic slow-roll inflation (inflaton mass $m\sim10^{-6}$), termination typically occurs after $\sim5$ e-folds of resonant amplification. Perturbative methods allow analytical estimates for the shift in $\phi$'s period and amplitude and the backreaction-shaped evolution of resonance.

4. Amplification of Inhomogeneities

Parametric resonance introduces sensitivity to initial inhomogeneities. Small spatial variations in $\phi$ result in differences in the phase variable $\beta$ for $\chi$, with the boost $\eta(\beta)$ being phase dependent. Consequently, energy density perturbations in $\chi$ are generated, which are converted to curvature perturbations such that: $$\left(\frac{\delta \rho}{\rho}\right)_k \sim \frac{6 \Delta w H}{m} \frac{\Delta\langle\eta_\text{acute}\rangle}{\langle\eta\rangle} \frac{\delta \phi_k}{\phi_k}$$ where $\Delta w$ is the change in equation-of-state. These amplified inhomogeneities can strongly constrain model parameters. For quadratic slow-roll inflation, the observed constraint ($\delta \rho/\rho \lesssim 10^{-5}$) implies $\lambda < 10^{-14}$. This restricts the utility of weakly coupled models for efficient preheating without exceeding cosmological bounds.

5. Suppression of Resonant Transfer in Locked Inflation

"Locked inflation" refers to scenarios with a multifield fast-rolling $\phi$ locking another field $\chi$ in a false minimum to drive inflation. In single-component $\phi$, zero-crossings trigger strong parametric resonance and can rapidly transfer energy, thus prematurely terminating inflation.

However, if $\phi$ is a multiplet (e.g., $\phi_1$, $\phi_2$), the effective mass for $\chi$ never reaches zero at any one instant: $$m_\text{eff}^2 = M^2 + 2\lambda(\phi_1^2 + \phi_2^2) = M^2 + 2\lambda|\phi|^2(1+\sin2\varphi \sin2mt)$$ Since at least one component is always nonzero, the resonance condition $|\dot{m}_\text{eff}|/m_\text{eff}^2 \ll 1$ holds at all times and the "energy kick" to $\chi$ never occurs. Therefore, exponential parametric energy transfer is suppressed, resolving the resonance-induced instability of locked inflation for multiplet scenarios.

6. Analytical Control and Physical Implications

The analytic framework deployed—splitting the evolution into rolling and zero-crossing regimes, and employing statistical averaging over phase variables—enables explicit calculation of resonance rates, parameter constraints, and termination conditions. The distinctions between preheating stages and the treatment of backreaction are essential for predicting the timing of reheating completion and ensuing radiation-dominated evolution.

The amplification of inhomogeneities via phase-dependent resonance parameters demonstrates sensitivity to initial field fluctuations, crucial for model-building in cosmology. Furthermore, the suppression of resonance in multiplet $\phi$ fields provides a concrete mechanism for stabilizing inflationary models against premature termination.

Summary Table: Key Features of the Parametric Resonance Condition in $\lambda\phi^2\chi^2$ Models

Phenomenon	Analytical Expression	Physical Consequence
Resonant boost per half-cycle	$\langle \eta \rangle \approx 0.346$	$\chi$ amplitude doubles per cycle
Resonance termination	$\lambda \chi^2 \sim m^2$ (backreaction)	Resonance halts after $\sim$5 e-folds
Inhomogeneity amplification	$(\delta \rho/\rho)_k \propto \delta \phi_k$	Stringent bounds on coupling $\lambda$
Multiplet $\phi$ suppression	$m_\text{eff}^2$ never vanishes	Parametric resonance inhibited

These results provide a quantitative and physically transparent criterion for the parametric resonance condition in coupled scalar systems, with direct relevance for preheating, the evolution of inhomogeneities, and inflationary model engineering (Wang, 2011).