Large $n$-point Functions in Resonant Inflation

Technical Summary: Large n-point Functions in Resonant Inflation

Problem Statement
Standard analyses of inflationary models typically assume that the majority of observable information is encoded in low-order correlation functions, specifically the power spectrum ($n=2$) and the bispectrum ($n=3$). In resonant inflation models, characterized by a potential with small, rapid oscillations ($V(\phi) = V_{sr}(\phi) + \Lambda^4 \cos(\phi/f)$), previous effective field theory (EFT) studies suggested that the oscillatory signal in the power spectrum dominates over higher-order $n$-point functions. This conclusion relied on the assumption that the EFT breaks down at a naive ultraviolet (UV) cutoff scale of $4\pi f$. However, recent work by Hook and Rattazzi [1] demonstrated that for canonical scalar fields with oscillatory potentials, the perturbative unitarity cutoff is actually higher than $4\pi f$ by a logarithmic factor when the oscillation amplitude is small. This discrepancy opens a phenomenological window where the oscillatory signal in higher-order $n$-point functions ($n \gg 2$) might become observable and potentially dominate the signal in the power spectrum, a regime this paper investigates.

Methodology
The authors analyze a single-field resonant inflation model where a canonical scalar field $\phi$ drives inflation with an oscillatory potential. They focus on the "decoupling limit" where the slow-roll parameter $\epsilon$ is negligible, allowing the background metric to be treated as de Sitter. The dynamics are described by the Goldstone boson action $\pi$ in the EFT of inflation.

The methodology proceeds in three main steps:

1. Unitarity Cutoff Calculation: The authors revisit the derivation of the perturbative unitarity bound for $n \to n$ scattering amplitudes. By calculating the scattering amplitude $M_{n \to n}$ in flat space and imposing the condition $|M_{n \to n}| < 1$, they determine the maximum energy scale $\Lambda_{\text{cutoff}}$ allowed before the theory becomes strongly coupled. They find that $\Lambda_{\text{cutoff}}$ exceeds the naive $4\pi f$ scale logarithmically, depending on the oscillation amplitude $\tilde{b}$.
2. Signal-to-Noise (S/N) Ratio Derivation: The authors compute the connected $n$-point correlation functions of the curvature perturbation $\zeta$ to first order in the oscillation amplitude $\tilde{b}$. Using the in-in formalism and a saddle-point approximation (valid for large oscillation frequency $\omega$), they derive the explicit form of the $n$-point functions. They then construct an optimal estimator for the parameter $\tilde{b}$ and calculate the signal-to-noise ratio $(S/N)_n$ for each $n$-point function. This involves integrating over momentum space with an exponential window function to model experimental cutoffs ($k_{\text{max}}$).
3. Parameter Space Scan: The authors numerically evaluate the $(S/N)$ ratio across the parameter space defined by the oscillation amplitude $\tilde{b}$ and frequency $\omega$. They identify the region where the theory remains perturbatively unitary ($\omega < \Lambda_{\text{cutoff}}$) and determine which $n$-point functions provide the dominant signal.

Key Contributions and Results

• Revised Cutoff and Observable Window: The paper confirms that the unitarity cutoff $\Lambda_{\text{cutoff}}$ is significantly higher than $4\pi f$ for small amplitudes. This creates a regime where the oscillation frequency $\omega$ can exceed $4\pi f$ without violating unitarity, provided the amplitude $\tilde{b}$ is exponentially small.
• Dominance of Large $n$-point Functions: In this high-frequency regime ($\omega > 4\pi f$), the signal-to-noise ratio for $n$-point functions initially grows with $n$ before decaying exponentially. The authors identify a specific "sweet spot" where the signal is maximized.
• Detectable Range: The analysis reveals a phenomenologically relevant window where $n$-point functions with $3 \lesssim n \lesssim 9$ are potentially observable. In this regime, the oscillatory signal in these higher-order correlators dominates over the signal in the power spectrum.
• Oscillation Frequency: The detectable regime corresponds to oscillation frequencies of approximately $350 \lesssim \alpha \lesssim 1000$ (where $\alpha = \omega/H_0$), implying roughly 350–1000 oscillations per decade in wavenumber $k$.
• Scaling Relations: The authors derive an analytic expression for the maximum $(S/N)$ ratio, showing that it scales with the number of modes $N_{\text{modes}}$ and depends exponentially on the maximum contributing $n$ ($n_{\text{max}}$). They find that $n_{\text{max}} \simeq (\omega/4\pi f)^{2/3}$, which coincides with the value maximizing the scattering amplitude.
• Constraints: The paper notes that current observational bounds on the power spectrum oscillations ($\alpha \lesssim 100$) are below the threshold for this new regime. However, upcoming large-scale structure surveys with higher resolution ($k_{\text{max}}$) could potentially reach $\alpha \sim 360$, entering the region where the bispectrum ($n=3$) and higher-order functions begin to dominate.

Significance
The paper claims to identify a qualitatively new regime of resonant non-Gaussianity. Its primary significance lies in challenging the standard paradigm that the power spectrum is the primary carrier of information in resonant inflation. By leveraging the revised unitarity bound, the authors demonstrate that for specific parameter choices, the most observable signature of the model may not be the power spectrum oscillations, but rather the large $n$-point correlation functions. This suggests that future analyses of cosmological data should consider higher-order statistics to fully probe the parameter space of resonant inflation models, particularly in the high-frequency, small-amplitude limit. The authors also highlight that while the theory remains perturbatively unitary in this regime, it may still be cosmologically Gaussian, decoupling the concepts of strong coupling and large non-Gaussianity in this specific context.