Cosmological Collider Signals at Strong Mixing

Technical Summary: Cosmological Collider Signals at Strong Mixing

Problem Statement
Cosmological collider physics aims to probe high-energy degrees of freedom during inflation through nonanalytic momentum dependencies in correlation functions. While previous studies established that massive fields coupled to the curvature perturbation via a turning trajectory generate characteristic signals, these analyses often relied on perturbative expansions in the turn rate. This approach breaks down in regimes of rapid turning (strong mixing), where the quadratic mixing between the canonically normalized curvature fluctuation ($q$) and a massive isocurvature field ($\sigma$) cannot be truncated. The challenge lies in treating this derivative mixing nonperturbatively to accurately determine the amplitude and phase of the resulting cosmological collider signals, specifically the squeezed bispectrum.

Methodology
The authors build upon the exact analytic solutions for the coupled linear curvature-isocurvature system derived by Huenupi et al. [16], which treat the dimensionless mixing parameter $\lambda$ and the isocurvature mass $\mu$ nonperturbatively in quasi-de Sitter space. The methodology proceeds as follows:

1. Exact Linear Basis: The study utilizes the exact Bunch-Davies mode solutions for the two-field system. These solutions are expressed in terms of hypergeometric functions and Tricomi functions, allowing for the construction of exact Wightman functions and mixed propagators without resorting to perturbative interaction insertions.
2. Mixed External Leg Construction: The authors construct the mixed external leg propagator, $K_{\sigma q}$, which connects the isocurvature field to the curvature mode. This is achieved by summing over the exact mode functions, effectively resumming the derivative mixing.
3. Mellin-Barnes Representation: To isolate the late-time behavior relevant for cosmological collider signals, a Mellin-Barnes representation is applied directly to the exact mixed external leg. This technique separates analytic terms from the nonanalytic late-time branches ($z^{3/2 \pm i\rho}$), where $\rho$ is related to the heavy mass.
4. Bispectrum Calculation: The tree-level bispectrum is computed for a cubic isocurvature self-interaction ($g\sigma^3$). The calculation reduces to a single time integral involving the product of three mixed external legs.
5. Squeezed Limit Analysis: The squeezed limit ($k_1, k_2 \gg k_3$) is evaluated analytically. The nonanalytic part of the signal is isolated, yielding a power-law envelope modulated by logarithmic oscillations.
6. Large-Mass Expansion: For fixed mixing $\lambda$ and large mass ($\rho \to \infty$), the authors perform a JWKB analysis of the "hard" integral (the integral over the hard momenta). This involves matching the JWKB waves to the endpoint Bessel layer at $t=0$ to derive a controlled asymptotic expansion.

Key Contributions and Results

• Exact Mixed Propagator: The paper derives a closed-form expression for the mixed external leg $K_{\sigma q}$ (Eq. 1.1) that treats the derivative mixing exactly. This formula resums the mixing effects, providing a nonperturbative description of the transfer from the isocurvature field to the curvature perturbation.
• Nonanalytic Squeezed Bispectrum: For the cubic interaction, the squeezed bispectrum is shown to take the form:
  $$B_{q, \text{nonan}}^{(\sigma^3)} \propto \frac{1}{c^{3/2}} \sin(\rho \ln c + \theta_{\lambda, \rho})$$
  where $c = k_{\text{soft}}/k_{\text{hard}}$. The frequency of the logarithmic oscillation ($\rho$) encodes the heavy mass, while the amplitude and phase ($\theta_{\lambda, \rho}$) retain the full, nonperturbative dependence on the mixing strength $\lambda$.
• Analytic Hard Integral: The authors provide an exact series representation for the hard integral $J_{\lambda, \rho}$ (Eq. 4.17) using hypergeometric functions. This series is absolutely convergent and allows for precise numerical evaluation.
• Large-Mass JWKB Expansion: In the limit of large mass at fixed mixing, the authors derive the asymptotic expansion of the squeezed correlator. The leading algebraic contribution arises from the positive-cycle endpoint rather than a finite complex saddle. The resulting phase and amplitude (Eqs. 5.3–5.4) reveal that the Boltzmann suppression ($e^{-\pi\rho}$) may be lifted when the mixing-to-mass ratio $\lambda/\rho$ becomes of order unity, though this regime lies beyond the strict validity of the current approximation.
• Numerical Validation: The analytic series and JWKB approximations are compared against independent finite-cutoff numerical integrations. The results show excellent agreement across representative masses ($\rho = 1, 3, 5$) and mixing strengths, validating the analytic methods.

Significance
The paper claims significance in extending the toolkit for cosmological collider physics beyond perturbative treatments of turning trajectories. By utilizing exact linear solutions, the authors demonstrate that the full mixing dependence of the signal's amplitude and phase can be captured without truncation. This allows for a more robust extraction of inflationary parameters (mass and mixing) from future observational data, particularly in scenarios where the turn rate is large. The work also provides a concrete framework for handling derivative mixing in higher-point functions, moving beyond the power spectrum to non-Gaussianity.

The authors explicitly note that the research idea was suggested by ARC (a research assistant), the calculations were performed by GPT, and the results were checked by the authors, highlighting a novel workflow in theoretical physics research. The paper concludes by acknowledging concurrent works [21, 22] addressing related topics.