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

The ongoing Euclid mission aims to measure spectroscopic redshifts for approximately two million galaxies using the H $\alpha$ line emission detected in near-infrared slitless spectroscopic data from the Euclid Deep Fields (EDFs). These measurements will reach a flux limit of $5\times 10^{-17}\,{\rm erg}\,{\rm cm}^{-2}\,{\rm s}^{-1}$ in the redshift range $0.4<z<1.8$, opening the door to numerous investigations involving galaxy evolution, extending well beyond the mission's core objectives. The achieved H $\alpha$ luminosity depth will lead to a sufficiently high sampling, enabling the reconstruction of the large-scale galaxy environment. We assess the quality of the reconstruction of the galaxy cosmic web environment with the expected spectroscopic dataset in EDFs. The analysis is carried out on the Flagship and GAEA galaxy mock catalogues. The quality of the reconstruction is first evaluated using geometrical and topological statistics measured on the cosmic web, namely the length of filaments, the area of walls, the volume of voids, and its connectivity and multiplicity. We then quantify how accurately gradients in galaxy properties with distance from filaments can be recovered. As expected, the small-scale redshift-space distortions, have a strong impact on filament lengths and connectivity, but can be mitigated by compressing galaxy groups before skeleton extraction. The cosmic web reconstruction is biased when relying solely on H $\alpha$ emitters. This limitation can be mitigated by applying stellar mass weighting during the reconstruction. However, this approach introduces non-trivial biases that need to be accounted for when comparing to theoretical predictions. Redshift uncertainties pose the greatest challenge in recovering the expected dependence of galaxy properties, though the well-established stellar mass transverse gradients towards filaments can still be observed.