Euclid preparation. 3D reconstruction of the cosmic web with simulated Euclid Deep spectroscopic samples

1. Problem Statement

The paper addresses the challenge of reconstructing the three-dimensional (3D) large-scale structure of the Universe (the "cosmic web") using the upcoming spectroscopic data from the Euclid Deep Fields (EDF). While previous surveys have mapped the cosmic web at low redshifts ($z \lesssim 0.3$), the redshift range $0.4 < z < 1.8$—corresponding to the peak epoch of star formation ("cosmic noon")—remains largely uncharted due to the difficulty of obtaining high-completeness spectroscopic redshifts.

The specific challenges identified for the Euclid mission include:

• Selection Bias: The Euclid spectroscopic sample will be limited by H$\alpha$ flux rather than stellar mass. This selection function may miss quiescent massive galaxies and bias the sampling of the cosmic web.
• Redshift-Space Distortions (RSD): Small-scale random motions of galaxies within virialized halos create the "Fingers-of-God" (FoG) effect, artificially elongating structures along the line of sight and distorting the filamentary network.
• Observational Limitations: The data will suffer from redshift measurement uncertainties ($\sigma_z \approx 0.001(1+z)$) and incompleteness (estimated at 60% due to reliability cuts).

The authors aim to assess the quality of cosmic web reconstruction under these conditions and provide guidelines for future analysis.

2. Methodology

The study utilizes mock galaxy catalogues to simulate the Euclid Deep Survey environment and test reconstruction algorithms.

• Simulations: Two distinct mock catalogues were used:

  • Euclid Flagship: An N-body simulation ($4 \times 10^{12}$ particles) populated with galaxies using Halo Occupation Distribution (HOD) and abundance matching.
  • GAEA: A semi-analytic model run on the Millennium simulation, featuring different prescriptions for AGN feedback.
  • Both models were calibrated to match observational data (stellar mass functions, H$\alpha$ luminosity functions, and the star-forming main sequence).

• Mocking the Survey: The authors applied specific constraints to the mocks to mimic the EDF:

  • Flux Limit: $H\alpha$ flux limit of $6 \times 10^{-17}$ erg cm$^{-2}$ s$^{-1}$.
  • Completeness: Randomly removed 40% of galaxies to simulate a 60% completeness rate.
  • Redshift Errors: Added Gaussian noise with $\sigma_z = 0.001(1+z)$.
  • Comparison Samples: Created stellar-mass-limited ($M^*$) counterparts to isolate the effects of the flux-limited selection.

• Reconstruction Algorithm:

  • Group Finding: An anisotropic Friends-of-Friends (FoF) algorithm was used to identify galaxy groups and compress them radially to mitigate the FoG effect.
  • Skeleton Extraction: The DisPerSE (Discrete Persistent Structures Extractor) algorithm was used to identify the cosmic web skeleton (filaments, walls, voids, nodes) based on Delaunay Tessellation Field Estimator (DTFE).
  • Weighting: Reconstructions were performed both with and without stellar mass weighting of the tessellation to test its efficacy in correcting for selection biases.
  • Persistence: A topological persistence threshold of $N_\sigma = 5$ was used to filter out noise-induced spurious features.

• Metrics: The quality of reconstruction was evaluated using:

  • Geometrical: Filament lengths, wall areas, void volumes.
  • Topological: Connectivity (number of filaments connected to a node) and Multiplicity (connectivity minus bifurcation points).
  • Physical: Transverse gradients of stellar mass relative to filaments.

3. Key Contributions

• Quantification of Systematic Biases: The paper provides the first comprehensive assessment of how H$\alpha$ flux selection, redshift errors, and incompleteness specifically degrade 3D cosmic web reconstruction in the $z \sim 1$ regime.
• Optimization of FoG Correction: It demonstrates that while FoG correction is essential, its success depends heavily on the galaxy selection function. The authors show that standard correction methods work well for mass-limited samples but struggle with flux-limited samples unless stellar mass weighting is applied.
• Guidelines for Euclid: The study establishes a set of practical guidelines for the Euclid Consortium, recommending specific processing steps (e.g., when to use stellar mass weighting) for different scientific goals.
• Robustness of Multiplicity: A novel finding is that multiplicity (local connectivity) is a more robust topological statistic than global connectivity when dealing with noisy, flux-limited data.

4. Results

Geometrical Properties

• FoG Impact: The FoG effect significantly elongates filaments, shifting their length distribution to larger values.
• Correction Efficacy: Correcting for FoG by compressing galaxy groups restores the filament length distribution to match the reference (true) sample.
• Role of Weighting: For H$\alpha$-limited samples, stellar mass weighting of the Delaunay tessellation is crucial. Without it, the correction for FoG is incomplete, and the reconstructed geometry remains biased. For mass-limited samples, weighting is less critical.
• Redshift Errors: Redshift uncertainties hinder the ability to correct for FoG, particularly for flux-limited samples.

Topological Properties (Connectivity & Multiplicity)

• Connectivity: The FoG effect strongly alters the distribution of connectivity. While correction improves the results, stellar mass weighting for H$\alpha$ samples artificially inflates connectivity values, making direct comparison with theoretical predictions difficult.
• Multiplicity: Unlike connectivity, multiplicity is highly robust. It recovers the expected trend (increasing multiplicity with stellar mass) even without tessellation weighting and in the presence of redshift errors and incompleteness.
• Recommendation: The authors recommend using multiplicity over connectivity for H$\alpha$-limited surveys to avoid biases introduced by weighting.

Stellar Mass Gradients

• Trend Recovery: The expected trend—that more massive galaxies reside closer to filaments—is recovered in the reference sample.
• Degradation: Redshift errors and 60% sampling significantly weaken this signal.
• Weighting Effect: While stellar mass weighting helps recover the gradient signal in noisy data, it introduces a "by-construction" bias (massive galaxies are forced to follow the skeleton more closely). However, the authors confirm the gradient is an intrinsic physical feature, not an artifact of the weighting.

5. Significance

This paper is critical for the success of the Euclid mission's "by-product" science regarding galaxy evolution and large-scale structure.

• Scientific Impact: It validates that despite the challenges of flux-limited spectroscopy and redshift errors, the Euclid Deep Fields will allow for the first detailed 3D mapping of the cosmic web at $z \sim 1-1.8$.
• Methodological Impact: It provides a roadmap for handling RSD and selection biases, ensuring that future measurements of galaxy co-evolution (e.g., star formation quenching, morphology, angular momentum alignment) relative to the cosmic web are not systematically biased.
• Synergy: The results highlight the complementarity of Euclid with other surveys (DESI, Roman, MOONS, PFS) to cover different redshift regimes and selection functions, enabling a complete picture of cosmic structure formation.

In conclusion, while the reconstruction of the cosmic web with Euclid's H$\alpha$ data faces significant hurdles from selection effects and redshift uncertainties, the application of FoG compression and stellar mass weighting (specifically for geometrical measures and connectivity) allows for a reliable recovery of the large-scale structure, opening a new window into the co-evolution of galaxies and their environment during the universe's peak star-forming era.