Euclid preparation. Full-shape modelling of 2-point and 3-point correlation functions in real space

1. Problem Statement

The Euclid mission aims to map the large-scale structure (LSS) of the Universe using galaxy clustering and weak lensing. While the 2-point correlation function (2PCF) and its Fourier counterpart (power spectrum) are standard tools, higher-order statistics like the 3-point correlation function (3PCF) or bispectrum contain crucial non-Gaussian information regarding galaxy bias, nonlinear evolution, and primordial physics.

However, a full-shape analysis (extracting cosmological parameters from the entire shape of the correlation function, not just specific features like BAO) of the 3PCF in configuration space has been historically hindered by:

1. Computational Cost: Evaluating theoretical models for the 3PCF is significantly more expensive than for the 2PCF, especially when integrating over cosmological parameters in a likelihood framework.
2. Model Validity: The range of validity of perturbative models (specifically at small scales) for the 3PCF in real space had not been rigorously tested against synthetic data mimicking the Euclid spectroscopic sample.
3. Lack of Joint Analysis: Previous studies often treated 2PCF and 3PCF separately or used template fitting (fixed shapes) rather than full-shape parameter inference.

2. Methodology

Data Source

The authors utilized four redshift snapshots ($z = 0.9, 1.2, 1.5, 1.8$) from the Euclid Flagship I N-body simulation.

• Simulation: $3780 \, h^{-1}\text{Mpc}$ box with $2 \times 10^{12}$ dark matter particles.
• Galaxy Catalogs: Generated using a Halo Occupation Distribution (HOD) prescription (Model 3 of Pozzetti et al. 2016) to mimic Euclid's $H\alpha$-emitting galaxy sample ($f_{H\alpha} > 2 \times 10^{-16} \text{ erg cm}^{-2} \text{ s}^{-1}$).
• Idealization: The analysis assumes an ideal case without observational systematics (purity, completeness, angular footprint) to isolate theoretical model performance.

Theoretical Models

• 2PCF: Modeled using Effective Field Theory of Large Scale Structure (EFTofLSS) at next-to-leading order (NLO) (one-loop). This includes linear bias ($b_1$), quadratic bias ($b_2$), non-local bias ($b_{G2}, b_{\Gamma3}$), and counterterms ($c_0$).
• 3PCF: Modeled at tree-level (leading order) in perturbation theory.
• Infrared Resummation: Applied to both statistics to account for the smearing of Baryon Acoustic Oscillations (BAO) due to nonlinear evolution.
• Bias Relations: The study tests the impact of fixing bias parameters using theoretical relations (e.g., excursion-set predictions for $b_{G2}$ and co-evolution for $b_{\Gamma3}$) to reduce the parameter space.

Estimators and Emulation

• 3PCF Estimator: Utilized the Spherical Harmonics Decomposition (SHD) estimator (Slepian & Eisenstein 2015), which expands the angular dependence of triangles into Legendre polynomials, reducing computational complexity from $O(N_g^3)$ to $O(N_g^2)$.
• Emulator: A critical contribution was the development of a PyTorch-based emulator capable of generating fast, robust predictions for both 2PCF and 3PCF as a function of three cosmological parameters ($A_s, \omega_{cdm}, h$) and nuisance parameters. This bypassed the need for slow numerical integration during MCMC sampling.

Analysis Framework

• Likelihood: Joint Gaussian likelihood for 2PCF and 3PCF (neglecting cross-covariance for this validation step).
• Metrics:
  • Goodness of Fit (GoF): $\chi^2$ test.
  • Figure of Bias (FoB): Quantifies systematic shifts in recovered parameters relative to fiducial values.
  • Figure of Merit (FoM): Inverse square root of the determinant of the covariance matrix, measuring constraining power.
• Scale Cuts: Systematically varied the minimum separation ($r_{min}$) and the exclusion of nearly isosceles triangles (parameterized by $\eta_{min}$) to determine the validity range.

3. Key Contributions

1. First Joint Full-Shape Analysis in Real Space: This is the first study to perform a joint full-shape likelihood analysis of real-space 2PCF and 3PCF to constrain cosmological parameters ($A_s, \omega_{cdm}, h$) simultaneously, rather than relying on template fitting.
2. Development of a 3PCF Emulator: The authors created a novel emulator that enables efficient sampling of the joint likelihood function, overcoming the computational bottleneck that previously prevented full-shape 3PCF analysis.
3. Validation of Perturbative Models: Provided a rigorous validation of the perturbative bias expansion for 3PCF in real space against high-fidelity N-body simulations, establishing the scale cuts required for unbiased results.
4. Quantification of Bias Relations: Demonstrated how theoretical bias relations can reduce the dimensionality of the nuisance parameter space without introducing significant systematic errors, thereby improving cosmological constraints.

4. Key Results

Model Validity and Scale Cuts

• 2PCF: The NLO model provides a good fit down to scales of $20 \, h^{-1}\text{Mpc}$ across all redshifts.
• 3PCF: The tree-level model fails at small scales.
  • For $\eta_{min} = 1$ (including nearly isosceles triangles), the model is valid only for $r_{min} > 20 \, h^{-1}\text{Mpc}$ at $z=0.9$ and $r_{min} > 10 \, h^{-1}\text{Mpc}$ at higher redshifts.
  • Optimal Cut: The authors recommend a conservative scale cut of $r_{min} = 30 \, h^{-1}\text{Mpc}$ for the 3PCF when excluding nearly isosceles configurations ($\eta_{min} \ge 3$). A more aggressive cut ($r_{min} = 30 \, h^{-1}\text{Mpc}, \eta_{min} = 2$) is acceptable but shows slight tension with the fiducial $A_s$ at 68% confidence.
• Bias Relations: Using relations for $b_{G2}$ and $b_{\Gamma3}$ reduces the parameter space effectively. While it does not significantly alter the bias constraints themselves, it tightens constraints on the counterterm parameter $c_0$ and improves the precision of cosmological parameter estimation.

Cosmological Constraints

• Breaking Degeneracies: The joint analysis of 2PCF and 3PCF significantly breaks the degeneracy between the amplitude of scalar perturbations ($A_s$) and the linear bias ($b_1$), which is a major limitation in 2PCF-only analyses.
• Accuracy: The model recovers the fiducial values of $h$ and $\omega_{cdm}$ accurately across all redshifts.
• Tension: At the lowest redshift ($z=0.9$) and with aggressive scale cuts ($r_{min}=30, \eta_{min}=2$), a slight tension ($\sim 68\%$ CL) appears in the recovery of $A_s$, suggesting that small-scale systematics or the tree-level approximation for 3PCF may still be limiting.
• Constraining Power: The Figure of Merit (FoM) for the joint analysis increases significantly (by orders of magnitude on a log scale) compared to 2PCF alone, particularly at lower redshifts where galaxy density is higher.

5. Significance

This work represents a major step forward in preparing for the Euclid mission. By validating the theoretical models and developing the necessary computational tools (emulators), the authors have paved the way for:

• Configuration Space Analysis: Establishing that full-shape analysis in configuration space (real space) is feasible and competitive with Fourier-space (bispectrum) analyses.
• Future Redshift Space Studies: The methods and emulators developed here are directly applicable to the more complex redshift-space analysis (which includes redshift-space distortions), which is the primary goal for Euclid's spectroscopic sample.
• Robust Cosmology: Demonstrating that combining 2PCF and 3PCF provides superior constraints on cosmological parameters, specifically helping to disentangle galaxy bias from cosmological physics, which is essential for the precision cosmology goals of Stage IV surveys.

In summary, the paper successfully validates the perturbative framework for 3PCF in real space, defines the necessary scale cuts to avoid systematic biases, and proves that a joint full-shape analysis is a viable and powerful strategy for the upcoming Euclid data release.