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

The ongoing Euclid mission is aimed at measuring spectroscopic redshifts for approximately two million galaxies using the, ^ line emission detected in near-infrared slitless spectroscopic data from the Euclid Deep Fields, leveraging both the red and blue grisms. These measurements will reach a flux limit of 5, , ^ 10^ -17 ̊m erg ̊m cm -2 ̊m s -1 in the redshift range 0. 4<z<1. 8, paving the way to numerous scientific investigations involving galaxy evolution, extending well beyond the mission's core objectives. The achieved luminosity depth will lead to a sufficiently high sampling, enabling the reconstruction of the large-scale galaxy environment. Here, we assess the quality of the reconstruction of the galaxy cosmic web environment with the expected spectroscopic dataset in Euclid Deep Fields. The analysis was carried out on the Flagship and galaxy mock catalogues. The quality of the reconstruction was first evaluated using simple geometrical and topological statistics measured on the cosmic web network; namely, the length of filaments, the area of walls, the volume of voids, and its connectivity and multiplicity. We then quantified how accurately gradients in galaxy properties can be recovered, with respect to the distance from filaments. As expected, the small-scale redshift-space distortions, such as Fingers of God (FoG) effects, have a strong impact on filament lengths and connectivity; however, they can be mitigated by compressing galaxy groups identified with an anisotropic group finder prior to a skeleton extraction. The cosmic web reconstruction is biased when relying solely on emitters. This limitation can be mitigated by applying stellar mass weighting during the cosmic web 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, although the well-established stellar mass transverse gradients towards filaments can still be observed to a lesser extent.