(The) Wiggles going non-linear

Technical Summary: The Wiggles Going Non-linear

Problem Statement
Current cosmological observations are consistent with simple slow-roll inflation, predicting a nearly scale-invariant primordial power spectrum (PPS). However, many well-motivated inflationary scenarios predict "wiggles"—localized features, oscillations, or scale-dependent structures—in the PPS. While Cosmic Microwave Background (CMB) observations constrain these features, they are limited by cosmic variance on large scales and photon diffusion on small scales. Large-scale structure (LSS) surveys (e.g., DESI, Euclid, LSST) offer a complementary probe with access to a vast number of Fourier modes.

The primary challenge in utilizing LSS data to detect primordial wiggles is the non-linear evolution of structure. On small scales ($k \gtrsim 0.1 \, h/\text{Mpc}$), gravitational evolution couples Fourier modes, smearing initially coherent oscillatory structures and damping the features in the late-time matter power spectrum. Standard linear theory fails in this regime. While N-body simulations provide accurate predictions, they are computationally prohibitive for the extensive parameter sweeps required in cosmological inference. Conversely, existing semi-analytic models often assume smooth power-law PPS and do not account for the specific damping of oscillatory features.

Methodology
The authors employ a multi-step strategy to model and validate the non-linear damping of primordial oscillations:

1. Theoretical Framework:

   • They define the PPS using phenomenological oscillatory templates: linear oscillations ($\cos[\omega k/k_* + \phi]$) and logarithmic oscillations ($\cos[\omega \ln(k/k_*) + \phi]$).
   • To mitigate simulation noise and numerical systematics, they utilize the relative matter power spectrum, $\delta(k) = [P_m(k, \text{feature}) - P_m(k, \text{ref})] / P_m(k, \text{ref})$. In linear theory, this quantity isolates the oscillatory feature directly.
   • They adopt a semi-analytic damping model (following Ballardini et al.) where the non-linear relative spectrum is modeled as the linear oscillation multiplied by a Gaussian damping envelope:
     $$\delta_{fit}(k) = A \cos[\omega f(k) + \phi] \exp\left[-\frac{k^2 \Sigma^2(\omega, z)}{2}\right]$$
     Here, $\Sigma(\omega, z)$ is an effective damping function dependent on frequency and redshift.

2. N-Body Simulations:

   • High-resolution, dark-matter-only simulations were performed using $1024^3$ particles in a $1024 \, h^{-1}\text{Mpc}$ box.
   • Matched-pair strategy: For each feature configuration, a "feature" simulation and a "reference" (featureless) simulation were generated using the same random seed to cancel sample variance.
   • Simulations covered wavenumbers $0.05 \le k \le 0.6 \, h/\text{Mpc}$ and redshifts $z \in \{0, 1, 2, 3, 4, 5\}$.
   • A modulation amplitude of $A=0.03$ was fixed, consistent with current CMB constraints.

3. Calibration and Validation:

   • The damping function $\Sigma(\omega, z)$ was calibrated by fitting the semi-analytic template to the simulated relative power spectra.
   • Diagnostics: The authors assessed the model's validity using residual diagnostics (maximum absolute residual and global RMS residual) against a target accuracy of $\epsilon_0 = 5 \times 10^{-3}$ (sub-percent level).
   • Uncertainty Quantification: Simulation variance was estimated using four additional matched pairs with independent seeds, smoothed via polynomial fitting to avoid noise-induced weighting artifacts in the likelihood.

4. Emulation:

   • Since likelihood analyses require continuous evaluation of $\Sigma(\omega, z)$, the authors used Gaussian Process Regression (GPR) to interpolate the discrete calibration grid.
   • The GPR emulator incorporates the statistical uncertainty of the calibration fits as training noise, providing a continuous, uncertainty-aware prediction of the damping function.

5. Information Recovery Test:

   • A Fisher-like analysis was performed using a Euclid-like spectroscopic setup to compare three scenarios:
     • Undamped Full Range: Idealized case with no damping.
     • Damped Full Range: Uses the GPR-calibrated damping model over the full $k$-range.
     • Linear Only: Restricts analysis to scales where damping is negligible (conservative cutoff).

Key Results

• Domain of Validity: The one-parameter Gaussian damping model is validated for:
  • Logarithmic oscillations: $0.8 \le \log_{10}\omega \le 2.0$.
  • Linear oscillations: $0.4 \le \log_{10}\omega \le 1.2$.
  • For logarithmic oscillations with $\log_{10}\omega < 0.8$, the residuals exceed the accuracy threshold at late times, indicating the single-parameter model is insufficient.
• Calibration Consistency: The calibrated $\Sigma$ values agree with previous work (Ballardini et al.) at the few-percent level where frequencies overlap.
• Emulator Performance: The GPR emulator successfully interpolates the damping function across the validated frequency-redshift domain, with predictive uncertainties comparable to the statistical uncertainty of the training data.
• Information Recovery: In the Euclid-like comparison for logarithmic wiggles:
  • Modeling the non-linear damping recovers a substantial fraction of the information lost by conservative linear-only cuts.
  • Amplitude ($A$): Uncertainty reduced by $\sim 40\%$ compared to the linear-only case.
  • Frequency ($\log_{10}\omega$): Uncertainty reduced by $\sim 65\%$ (nearly a factor of three improvement).
  • Phase ($\phi$): Uncertainty improved by $\sim 10\%$.
  • The damped full-range constraints remain within a factor of order unity of the idealized undamped case.

Significance and Claims
The paper claims to establish a practical, simulation-calibrated route for incorporating primordial wiggles into non-linear LSS likelihood analyses. By demonstrating that a simple, physically motivated damping model can predict the matter power spectrum with sub-percent accuracy (for sufficiently high frequencies), the authors argue that it is feasible to search for inflationary wiggles in upcoming LSS data without discarding the non-linear scales where the most information resides.

The work emphasizes the importance of identifying the domain of validity for such models. The authors caution that while the approach works well for high-frequency modulations, low-frequency logarithmic features require more flexible prescriptions. The study concludes that with the calibrated damping emulator, next-generation surveys can significantly improve constraints on primordial features compared to analyses restricted to the linear regime.