Large $n$-point Functions in Resonant Inflation

TL;DR

The paper identifies a novel high-frequency resonant inflation regime in which information shifts from the power spectrum to large-$n$-point non-Gaussianities (with $3 \lesssim n \lesssim 9$). Using a decoupling-limit EFT for a canonical single-field model with an oscillatory potential, the authors derive a revised unitarity cutoff that exceeds the naive $4\pi f$ scale (by a logarithmic factor) and compute the signal-to-noise for all relevant $n$-point functions via an optimal estimator that sums in quadrature. They show that the total detectability scales with the number of observed modes $N_{\rm modes}$ and has a peak at $n_{\max} \approx (\omega/(4\pi f))^{2/3}$, enabling potentially observable resonant non-Gaussianities for $350 \lesssim \alpha \lesssim 1000$ with upcoming large-scale structure surveys. Observationally, this opens a window where higher-point correlators could provide the dominant signal while power-spectrum oscillations remain subdominant, though it requires navigating very fast log-periodic oscillations and plausible UV completions; future work may explore multi-field realizations and field-space boundaries to further constrain such signals.

Abstract

We investigate a qualitatively new regime of inflationary models with small and rapid oscillations in the potential-resonant non-Gaussianity. In contrast to the standard scenario, where most of the observable information is encoded in the power spectrum, in this regime the oscillatory signal predominantly appears in higher-order correlation functions with large $n$. This behavior emerges when the oscillation frequency $ω$ exceeds the naive cutoff of the theory, $4πf$. However, as noted by Hook and Rattazzi [2306.12489], the actual cutoff is somewhat higher -- though only logarithmically -- when the amplitude of the oscillations is small. We identify a phenomenologically relevant window in which $n$-point functions with $3 \lesssim n \lesssim 9$ are potentially observable. In this regime, the signal exhibits 350-1000 oscillations per decade in $k$.

Figures (2)

• Figure 1: Signal-to-noise $(S/N)^2_n$, Eq. \ref{['eq:SNR_final']}, over the number of modes $N_{\rm modes}$, as a function of $n$, evaluated at cutoff frequencies $\omega = \Lambda_{\rm cutoff}$ for different values of $\tilde{b}$. The cutoff is obtained numerically as explained in the text.
• Figure 2: Regions of detectability for the resonant non-Gaussianity correlators in the parameter space $(\tilde{b}$, $\omega)$. The blue region corresponds to detectable total signal-to-noise ratio in Eq. \ref{['total-SNR']} (for $N_{\rm modes} = 10^{9}$) and frequencies below the cutoff ($|\mathcal{M}_{n \to n}| < 1$ for all $n$'s). Black dashed lines labelled from $1$ to $10^{8}$ indicate the values of ${(S/N)}_{\rm tot}^2$. The horizontal dashed lines, labelled by the numbers $2, 3,\ldots 10$, denote which correlation function dominates the $(S/N)$ ratio. For instance, in region 3 we have $(S/N)^2_{ n = 3} > (S/N)^2_{n =2}$. In the green region, the frequency $\omega$ is above the cutoff.