Optimising the sample selection for photometric galaxy surveys

1. Problem Statement

The next generation of cosmological surveys (Stage-IV, e.g., Euclid, LSST) aims to constrain cosmological parameters, particularly the Dark Energy (DE) equation of state ($w_0, w_a$), with high precision. A critical component of these surveys is the 3x2pt analysis, which combines three probes:

1. Weak Lensing (WL)
2. Galaxy Clustering (GC)
3. Galaxy-Galaxy Lensing (GGL)

To perform these analyses, galaxies must be grouped into tomographic redshift bins. The choice of binning strategy (e.g., equally spaced vs. equally populated) significantly impacts the constraining power of the survey.

• The Challenge: Optimizing the boundaries of these redshift bins for a combined 3x2pt analysis is an open problem. A configuration optimal for WL may not be optimal for GC. Furthermore, finding the global optimum via brute-force search is computationally intractable due to the high dimensionality of the parameter space and the cost of running Bayesian inference or Fisher matrix calculations at every step.
• The Goal: Develop an efficient, automated pipeline to find the optimal redshift bin edges for source (WL) and lens (GC/GGL) galaxies separately, maximizing the Figure of Merit (FoM) for DE parameters without incurring prohibitive computational costs.

2. Methodology

The authors propose an end-to-end pipeline that integrates photometric redshift estimation with an iterative optimization algorithm.

A. Data Generation and Photometric Redshifts (Photo-z)

• Simulation Source: The study utilizes the DESC Data Challenge 2 (DC2) catalogue, based on the CosmoDC2 simulation (Outer Rim simulation), covering $\sim440$ square degrees with galaxies up to $z=3$.
• Photo-z Estimation: Since the raw simulation data does not include photo-zs, the authors generated them using a Self-Organizing Map (SOM).
  • Input: Galaxy colors ($u-g, g-r, r-i, i-z, z-y$) derived from $ugrizy$ fluxes.
  • Process: The SOM reduces the dimensionality of the color space, clustering galaxies with similar "color phenotypes."
  • Assignment: Galaxies are assigned to voxels (neurons) based on color distance. The redshift distribution $n(z)$ of the training spectroscopic sample within a voxel is used to estimate the photo-z for galaxies assigned to that voxel.
  • Performance: The resulting catalogue achieved an outlier rate ($\mu_{0.15}$) of 0.013 and a normalized median absolute deviation ($\sigma_{NMAD}$) of 0.025, comparable to optimistic Stage-IV forecasts.

B. The 3x2pt Analysis Pipeline

• Framework: The analysis uses CosmoSIS to compute cosmological parameter constraints.
• Observables: The pipeline calculates angular power spectra for cosmic shear ($C_{\ell}^{\gamma\gamma}$), galaxy clustering ($C_{\ell}^{gg}$), and their cross-correlation ($C_{\ell}^{g\gamma}$) in the multipole range $\ell \in [10, 1500]$.
• Systematics Modeling: The pipeline includes realistic systematic effects:
  • Intrinsic Alignments (IA): Modeled using the TATT (Tidal Alignment and Tidal Torque) model.
  • Shear Calibration: Multiplicative bias ($m_i$).
  • Photo-z Errors: Shifts in the mean ($\Delta z_i$) and width ($\sigma_i$) of the redshift distributions.
  • Magnification Bias: Included in the clustering and GGL power spectra.
• Fisher Formalism: Constraints are estimated using the Fisher matrix ($F_{\alpha\beta}$), assuming Gaussian posteriors. The Figure of Merit (FoM) is defined as the square root of the determinant of the marginalized Fisher sub-matrix for the parameters of interest (e.g., $w_0, w_a$).

C. The Optimization Algorithm

Instead of a brute-force search, the authors employ an iterative, alternating optimization strategy:

1. Initialization: Start with a fiducial case of equally spaced redshift bins.
2. Iterative Perturbation:
   • Fix the lens bin configuration and perturb the source bin edges by adding Gaussian noise (standard deviation = half bin width).
   • Sample 200 random configurations per iteration.
   • Select the configuration yielding the highest FoM and fix it.
   • Repeat the process for the lens bins while fixing the optimized source bins.
3. Convergence: The process alternates between source and lens optimization. The authors found that the FoM converges rapidly, typically stabilizing after 4 iterations (2 for source, 2 for lens).

3. Key Contributions

1. Novel Optimization Framework: The paper presents a flexible, automated pipeline that optimizes tomographic bin edges for source and lens populations separately and iteratively. This avoids the computational intractability of simultaneous optimization while capturing the interplay between WL and GC.
2. Proof of Concept with Realistic Data: The method is validated using a Stage-IV-like photo-z catalogue generated from the DESC DC2 simulation via SOM, ensuring the input data reflects realistic survey characteristics and systematics.
3. Efficiency: The method achieves robust convergence with a feasible computational cost ($\sim1200$ CPU hours for the most expensive case), making it applicable to future survey planning.
4. Physical Insight: The study provides physical interpretations for the optimized configurations (e.g., wider bins at high redshift for sources, narrower bins at low-mid redshift for lenses).

4. Results

• Performance Gain: The optimized sample selection increases the FoM for the Dark Energy equation of state ($w_0, w_a$) by a factor of $\sim2$ (specifically $\sim2.45$ in some cases) compared to standard equally spaced bins.
• Comparison to Alternatives:
  • The improvement is significantly larger than the difference between "equally spaced" and "equally populated" bins (which showed only marginal gains).
  • The gain is comparable to increasing the survey area by a factor of $\sim4$ for non-optimized analyses.
• Parameter Constraints:
  • Targeted Optimization: Optimizing specifically for $w_0, w_a$ yields a $\sim25\%$ improvement in individual parameter constraints.
  • Global Optimization: Optimizing for the full $w_0wa\text{CDM}$ model (including $\Omega_m, \Omega_b, n_s, h, A_s$) yields a similar relative gain ($\sim2.25$), indicating that the optimization does not degrade constraints on other parameters.
  • Off-Target Parameters: Optimizing for $w_0, w_a$ does not negatively impact the constraints on the full set of cosmological parameters; in fact, constraints on off-target parameters also improve.
• Convergence Behavior: The algorithm converges quickly. For $N_{bins} > 5$, the gains stabilize after 4 iterations. The optimal configuration tends to create wider bins at high redshift for source galaxies (where WL signal is weaker and photo-z errors are larger) and narrower bins at low-to-mid redshift for lens galaxies (to maximize clustering signal).

5. Significance

• Maximizing Scientific Return: The study demonstrates that simply increasing survey area is not the only way to improve cosmological constraints; optimizing sample selection is a powerful lever. A factor of 2 improvement in FoM is a massive gain for Stage-IV surveys.
• Beyond Standard Paradigms: The results challenge the standard practice of using equally spaced or equally populated bins, showing that a customized, iterative approach yields superior science outcomes.
• Applicability: The pipeline is general enough to be applied to various survey configurations and science goals (e.g., dynamical DE vs. standard $\Lambda$CDM).
• Future Impact: The authors conclude that this optimization method should be integrated into the analysis pipelines of current and future photometric surveys (like Euclid and LSST) to ensure the most competitive cosmological constraints are derived from the data.