Extending the Cosmological Collider: New Scaling Regimes and Constraints from BOSS

1. Problem Statement

The "Cosmological Collider" program aims to detect heavy particles (mass $m > 3H/2$) present during inflation by observing characteristic oscillatory signatures in primordial non-Gaussianity (PNG).

• The Challenge: In standard models (e.g., quasi-single-field inflation with de Sitter invariance), the signal from heavy fields in the squeezed limit of the bispectrum is doubly suppressed:
  1. Exponential Suppression: The amplitude is suppressed by a Boltzmann factor $\sim e^{-\pi \nu}$, where $\nu = \sqrt{m^2/H^2 - 9/4}$.
  2. Power-Law Suppression: The signal scales as $(k_1/k_2)^{3/2}$ in the squeezed limit ($k_1 \ll k_2$), causing it to vanish as $k_1 \to 0$.
• The Gap: These suppressions make the standard cosmological collider signal difficult to detect with current Large Scale Structure (LSS) surveys, which are often dominated by constraints on the equilateral bispectrum or local PNG. Furthermore, the theoretical parameter space has been largely restricted to scenarios where the scaling dimension $\Delta$ has a real part $\text{Re}(\Delta) = 3/2$.
• The Opportunity: Recent theoretical work suggests that breaking de Sitter isometries (e.g., via time-dependent mixing or Lorentz violation) allows for complex scaling dimensions $\Delta = \alpha + i\nu$ where the real part $\alpha$ can be significantly smaller than $3/2$ (even $\alpha \approx 0$ or negative). This regime could alleviate the power-law suppression and potentially the exponential suppression, opening a new window for detection.

2. Methodology

The authors employ a multi-stage approach combining theoretical modeling, Fisher forecast analysis, and data analysis of existing surveys.

A. Theoretical Modeling (Section 2)

• Two-Field Model: They analyze a model with two mixing scalar fields $\sigma_1, \sigma_2$ with a Lagrangian that breaks de Sitter isometries via a mixing term $\rho(\dot{\sigma}_1\sigma_2 - \dot{\sigma}_2\sigma_1)$.
• Scaling Dimensions: They solve the equations of motion to find that the scaling dimension $\Delta = \alpha + i\nu$ can be complex. Crucially, they demonstrate that for tachyonic masses ($m^2 < 0$) and specific mixing parameters, one can achieve regimes where $\alpha \approx 0$ (or even negative) while maintaining large frequencies $\nu$.
• Amplitude Calculation: Using numerical solutions for mode functions (matching WKB initial conditions to the Bunch-Davies vacuum), they calculate the amplitude of the squeezed bispectrum. They confirm that while the vacuum fluctuation amplitude is suppressed, direct couplings to the inflaton can overcome this, leading to observable signals with $\alpha < 3/2$.

B. Observational Signatures (Section 3)

• Scale-Dependent Bias: They derive how the primordial bispectrum in the squeezed limit modifies the galaxy power spectrum $P_g(k)$ via a scale-dependent bias.
• The Signal: The bias takes the form of a power law modulated by logarithmic oscillations:
  $$b_{NG}(k) \propto \frac{(k R_M)^\alpha}{k^2 T(k)} \left[ A_{\cos} \cos(\nu \ln(k R_M)) + A_{\sin} \sin(\nu \ln(k R_M)) \right]$$
  Here, $\alpha$ sets the envelope (power law) and $\nu$ sets the oscillation frequency.
• Fisher Forecasts: They compute Fisher matrices for current (BOSS) and future (DESI, SPHEREx, and a hypothetical billion-object survey) galaxy surveys.
  • Bandpowers: To handle high-frequency oscillations, they compute band-averaged power spectra rather than point estimates to avoid aliasing.
  • Parameterization: They use an amplitude parametrization ($A_{\cos}, A_{\sin}$) rather than phase/amplitude to handle the degeneracy at low frequencies and define a 95% exclusion limit on the total amplitude $A_{f_{NL}} = \sqrt{A_{\cos}^2 + A_{\sin}^2}$.

C. Data Analysis (Section 4)

• Dataset: They analyze the final release of the Baryon Oscillation Spectroscopic Survey (BOSS DR12), covering $\sim 1.2$ million galaxies in two redshift bins ($z \in [0.2, 0.5]$ and $[0.5, 0.75]$).
• Pipeline: They extend the standard PNG analysis pipeline (used in previous works like [24]) to include the oscillatory template.
• Inference: Using Markov Chain Monte Carlo (MCMC) with the Metropolis-Hastings algorithm, they marginalize over cosmological parameters (fixed to Planck 2018 best-fit), bias parameters, and nuisance parameters. They scan a grid of $\alpha \in [-1, 3]$ and $\nu \in [0.1, 50]$.
• Limit Setting: They calculate the 95% exclusion limit on $A_{f_{NL}}$ by finding the radius of a circle in the $(A_{\cos}, A_{\sin})$ plane that encloses 95% of the posterior probability, effectively marginalizing over the unknown phase.

3. Key Contributions

1. Extension of Parameter Space: The paper theoretically validates and phenomenologically explores a regime of the cosmological collider where the real part of the scaling dimension $\alpha$ is small ($\alpha \lesssim 1$) or negative, distinct from the standard $\alpha = 3/2$ heavy field scenario.
2. Oscillatory vs. Power-Law Sensitivity: They demonstrate that for $\alpha \gtrsim 0.5$ and high frequencies ($\nu \gtrsim 10$), the oscillatory nature of the signal breaks degeneracies with the smooth broadband galaxy power spectrum and nonlinear bias terms. This allows for significantly tighter constraints compared to non-oscillatory (pure power-law) PNG scenarios.
3. First Constraints: This work provides the first observational constraints on this extended class of oscillatory primordial non-Gaussianity using galaxy clustering data.
4. Methodological Robustness: They introduce robust statistical techniques for handling logarithmic oscillations in discrete survey data, including the use of bandpowers and phase-marginalized amplitude limits.

4. Results

• No Detection: The analysis of BOSS DR12 data finds no evidence for a non-zero oscillatory PNG signal. The posterior distributions for $A_{\cos}$ and $A_{\sin}$ are consistent with zero across the scanned parameter space.
• Constraints: They place 95% upper limits on the amplitude $A_{f_{NL}}$.
  • The tightest constraints are found for $\alpha = -1$ (scaling as $k^{-1}$), reaching $A_{f_{NL}} < \mathcal{O}(1)$.
  • Constraints degrade for larger $\alpha$ (up to $\alpha = 3$) and very small frequencies ($\nu \lesssim 1$).
  • For $\alpha \ge 0.5$ and $\nu \gtrsim 10$, the constraints on the oscillatory signal are 5 to 23 times tighter than the constraints on equivalent non-oscillatory power-law PNG.
• Survey Sensitivity:
  • BOSS: Limited by volume and resolution, but already shows sensitivity improvements for high $\nu$.
  • Future Surveys (DESI, SPHEREx): SPHEREx is projected to improve constraints by more than an order of magnitude for $\alpha \approx 0$. For $\alpha \approx 1$, future surveys will see massive gains at high frequencies due to the ability to resolve many oscillation periods.
• Degeneracy Behavior: At low frequencies ($\nu \lesssim 1$), the constraints weaken significantly due to the strong degeneracy between the sine and cosine amplitudes (phase degeneracy) and the signal peaking at the largest scales (low $k$) where cosmic variance is high.

5. Significance

• New Discovery Channel: The paper establishes that "extended" cosmological collider models (with broken de Sitter invariance) are not only theoretically viable but potentially more accessible to observation than standard heavy-field models. The oscillatory feature acts as a "smoking gun" that distinguishes primordial physics from late-time astrophysical effects.
• Optimization of Surveys: The results suggest that future galaxy surveys should prioritize high-frequency resolution and large volumes to maximize sensitivity to these signals, particularly for $\alpha \sim 1$.
• Theoretical Guidance: By constraining the $(\alpha, \nu)$ plane, the paper provides direct feedback to model builders regarding which inflationary scenarios (e.g., specific mixing strengths or tachyonic masses) are ruled out or favored.
• Foundation for Future Work: The methodology developed here (bandpowers, phase-marginalized limits) is directly applicable to upcoming data from DESI, Euclid, and SPHEREx, paving the way for a comprehensive search for ultra-high-energy physics in the early universe.

In summary, this paper successfully bridges the gap between a novel theoretical extension of the cosmological collider and observational reality, demonstrating that while no signal was found in BOSS data, the search strategy is highly promising and significantly more powerful than previous non-oscillatory approaches for specific parameter regimes.