Euclid: Optimising tomographic redshift binning for 3$\times$2pt power spectrum constraints on dark energy

Technical Summary: Optimising Tomographic Redshift Binning for Euclid DR1 3×2pt Power Spectrum Constraints

Problem Statement
The tomographic analysis of the 3×2pt signal (combining cosmic shear, angular galaxy clustering, and galaxy-galaxy lensing) requires dividing the observed galaxy sample into redshift bins. While increasing the number of bins generally captures more cosmological information, it also increases computational costs and Poisson shot noise. Previous studies investigating the "optimum" binning strategy for Stage IV surveys like Euclid have yielded inconsistent conclusions regarding the ideal number of bins and the optimal method for defining bin boundaries (e.g., equipopulated, equal redshift width, or equal comoving distance). Furthermore, existing literature often relies on theoretical signal exploration rather than simulating realistic survey observations with associated systematics. This paper aims to resolve these inconsistencies by presenting a simulation-based method to determine the optimum tomographic binning strategy specifically for the Euclid first data release (DR1), focusing on constraints for the dark energy equation of state parameters $(w_0, w_a)$.

Methodology
The authors developed a simulation pipeline (SWEPT) to generate mock Stage IV-like galaxy catalogues that mimic a realistic Euclid DR1 observation. The workflow involves:

1. Galaxy Population Generation: Sampling a galaxy population from a predicted Euclid-like redshift distribution $p(z)$ and spatial number density.
2. Theoretical Signal Calculation: Computing theoretical 3×2pt power spectra ($C^{\gamma\gamma}_{ij}$, $C^{\delta_g\delta_g}_{ij}$, $C^{\delta_g\gamma}_{ij}$) using CosmoSIS, CAMB, and halofit under a fiducial flat $w_0w_a$CDM cosmology ($\Omega_m=0.3, H_0=70, w_0=-1, w_a=0$).
3. Map Generation: Creating 2D HEALPix maps of the correlated cosmic shear and galaxy density fields on the sky using FLASK, approximating the full 3D cosmological information.
4. Mock Catalogue Construction: Poisson-sampling the density fields to place galaxies and assigning weak lensing observables (convergence and shear) from the field maps at the galaxy positions.
5. Noise Injection: Introducing realistic observational uncertainties, including Gaussian-distributed photometric redshift (photo-z) errors ($\sigma_z = \sigma_{phot}(1+z)$ with $\sigma_{phot}=0.05$) and Gaussian shape noise ($\sigma(\epsilon_{int})=0.3$). A subset of simulations also included 5% catastrophic photo-z outliers.
6. Analysis: Measuring Pseudo-$C_\ell$ power spectra for various tomographic configurations (varying bin counts and boundary definitions) using NaMaster. Constraints on $(w_0, w_a)$ were derived via a grid-based Gaussian likelihood analysis using a numerical covariance matrix derived from 3,000 realisations, corrected via the Hartlap factor.

Key Results
The study evaluated three binning strategies: equipopulated bins, bins equally spaced in redshift, and bins equally spaced in fiducial comoving distance.

• Optimal Binning Strategy:

  • Full 3×2pt and Angular Clustering: For both the full 3×2pt signal and the angular clustering component alone, equipopulated bins yield the tightest constraints on $(w_0, w_a)$. This holds true for both noise-free and realistic noisy simulations.
  • Cosmic Shear Only: The optimal strategy depends on the noise model. In noise-free simulations, equal redshift width bins provide the best constraints. However, in realistic simulations including photo-z and shape noise, bins equally spaced in fiducial comoving distance yield the best constraints, though the difference between binning choices is minimal (<5%).
  • Dominance of Clustering: The preference for equipopulated bins in the full 3×2pt analysis is driven primarily by the angular clustering component, which dominates the constraining power of the joint signal.

• Number of Bins and Information Saturation:

  • Information gain saturates beyond six to eight bins for all signal components (shear, clustering, and full 3×2pt).
  • Increasing the number of bins from 7 to 10 yields only marginal gains (~1–2% improvement).
  • Extending the analysis to 13 bins (a configuration proposed in other literature) results in only a ~5% further improvement in precision compared to 7 bins. The authors note that marginal gains from additional bins are likely limited by the increasing demand for accuracy in covariance matrix estimation and the presence of other systematics in real data.

• Impact of Catastrophic Photo-z Errors:

  • Introducing 5% catastrophic photo-z outliers without mitigation leads to severe biases.
  • For analyses with 5 or 10 bins, the resulting constraints on $(w_0, w_a)$ are inconsistent with the fiducial $\Lambda$CDM cosmology at $\gtrsim 3\sigma$, favoring a time-evolving dark energy model ($w_0 > -1, w_a < 0$).
  • The posterior distributions exhibit sharp, non-elliptical features, indicating that unmitigated catastrophic errors induce complex, non-trivial biases in the power spectra.

Significance and Claims
The paper claims to provide a robust, simulation-based framework that mimics a real survey observation, offering a more direct assessment of tomographic binning strategies than previous theoretical studies. The primary conclusions are:

1. Equipopulated bins are the recommended strategy for Euclid DR1 analyses involving the full 3×2pt signal or angular clustering, as they provide the most robust constraints.
2. The cosmic shear component is relatively insensitive to the specific binning methodology (differences are <5% in noisy scenarios), suggesting that the choice of binning for shear alone is less critical than for clustering.
3. There is diminishing return in increasing the number of bins beyond seven or eight; the marginal statistical gains are likely outweighed by the computational costs and the increased difficulty in modelling systematics and covariance matrices accurately.
4. Catastrophic photo-z errors pose a critical threat to cosmological inference. If unmitigated, they can induce biases that mimic a deviation from $\Lambda$CDM at high significance, necessitating rigorous modelling or mitigation strategies in the analysis pipeline.

The authors emphasize that while their results suggest equipopulated bins are optimal for the specific survey characteristics of Euclid DR1, the relative insensitivity of cosmic shear to binning choices and the saturation of information gain suggest that the exact binning strategy may be less critical than the accurate modelling of systematics and the covariance matrix.