\textit{Euclid} preparation. Baryon acoustic oscillations extraction techniques: comparison and optimisation

Technical Summary: Euclid Preparation. Baryon Acoustic Oscillations Extraction Techniques: Comparison and Optimisation

Problem Statement
The Euclid mission aims to map approximately 30 million H$\alpha$ emission-line galaxies to constrain dark energy and cosmic expansion via Baryon Acoustic Oscillations (BAO). While BAO reconstruction techniques based on the Zeldovich approximation (ZA) have been shown to sharpen the acoustic peak and improve distance measurements (e.g., in SDSS and DESI), their application to Euclid's specific data characteristics requires rigorous validation. Key challenges include optimizing analysis choices (smoothing scales, grid resolution), comparing reconstruction implementations (RecSym vs. RecIso), ensuring robustness against fiducial cosmology mismatches, and managing computational costs associated with covariance estimation and parameter inference for large-scale surveys.

Methodology
The authors present the first end-to-end validation of the Euclid BAO analysis pipeline using eight mock catalogues per redshift snapshot extracted from the Flagship 1 (FS1) full N-body simulation. The pipeline integrates four main components:

1. Reconstruction Algorithms: The study implements two standard ZA-based reconstruction schemes: RecSym (shifting random catalogues with the same redshift-space distortion factor as the data) and RecIso (shifting random catalogues using real-space displacements, neglecting line-of-sight velocity contributions). These are executed using the MultiGridReconstruction algorithm, which solves the displacement field via a multigrid relaxation scheme in configuration space.
2. Clustering Measurement: The pipeline measures the anisotropic two-point correlation function (2PCF) and its multipoles ($\ell=0, 2, 4$) using the Landy–Szalay estimator.
3. Covariance Estimation: To address the computational cost of generating thousands of mocks, the authors employ a semi-analytical covariance estimator combining BeXiCov (which fits a physical template to the 2PCF multipoles to model the clustering signal) and WinCov (which convolves the periodic Gaussian covariance with the survey window function). This approach is calibrated on only eight mock realisations.
4. Parameter Inference: A novel Bora.jl emulator, a neural network trained on 10,000 theoretical realisations, approximates the physical 2PCF template. This emulator is paired with a No-U-Turn Sampler (NUTS), a Hamiltonian Monte Carlo (HMC) method, to infer cosmological parameters ($\Omega_m, H_0r_s$) and BAO scaling parameters ($\alpha_\parallel, \alpha_\perp$).

Key Contributions

• Computational Efficiency: The integration of the Bora.jl emulator with the NUTS sampler achieves a speed-up of over 500-fold compared to standard Markov Chain Monte Carlo (MCMC) methods while maintaining accuracy.
• Robust Covariance Estimation: The BeXiCov+WinCov framework enables robust error estimation using only eight mock realisations. It reduces the bias in BAO uncertainty estimates from $\sim20\%$ (when using non-iterated theoretical covariances) to below 5%, even when the fiducial cosmology differs significantly from the true cosmology.
• Reconstruction Scheme Comparison: A systematic comparison of RecSym and RecIso across four redshift bins ($0.9 \le z \le 1.8$) demonstrates that both schemes yield unbiased BAO measurements. However, RecSym is found to be less sensitive to variations in the smoothing scale and fiducial cosmology, simplifying the pipeline for testing phases.
• Optimal Smoothing Scale: The study validates that the BAO scale recovery is weakly sensitive to the smoothing scale $R_s$ within the range $[R_{opt}, R_{eq}]$. A baseline of $R_s = 15 \, h^{-1}\text{Mpc}$ is adopted to balance computational efficiency (allowing coarser grids) and theoretical consistency with the damping model.

Results

• Performance Gains: Reconstruction enhances the Figure of Merit (FoM) for the parameters $\{\Omega_m, H_0r_s\}$ by a factor of $\sim3$ in individual redshift bins compared to pre-reconstruction analyses, effectively tripling the survey volume.
• Cosmological Constraints: Combining the four redshift bins, the pipeline forecasts a precision of $\sim10\%$ on $\Omega_m$ and $\sim3\%$ on $H_0r_s$ for Euclid Data Release 1 (DR1).
• Robustness: The pipeline exhibits no statistically significant bias in recovered BAO scales when the fiducial cosmology is varied by up to $5\sigma$ from the true value. The Figure of Bias (FoB) remains below the acceptance threshold for all tested configurations.
• Consistency: Results from RecSym and RecIso are consistent within uncertainties, though RecSym is recommended for testing due to its reduced sensitivity to covariance variations and the lack of need for covariance recalibration at every smoothing scale.

Significance
This work establishes the accuracy, robustness, and scalability of the Euclid BAO pipeline for the upcoming Data Release 1. By validating the full chain from reconstruction to cosmological inference on realistic mock data, the authors confirm that the pipeline is scientifically ready to deliver unbiased constraints on dark energy. The methodological advances—specifically the emulator-based inference and semi-analytical covariance—provide a scalable foundation for future high-precision cosmological analyses, ensuring that Euclid's first cosmological results will be derived from a fully validated and computationally efficient framework. The findings align with legacy SDSS benchmarks, confirming the maturity of BAO reconstruction techniques for Stage-IV spectroscopic surveys.