Nearly a century after the discovery, the physical nature of dark matter remains one of the most pressing open questions in physics. Over the last several decades, an extensive research program has been brought forward, with the aim to determine the cosmological origin, fundamental constituents, and interaction mechanisms of dark matter. So far all Earth-based laboratory experiments had failed to catch and detect the elusive dark matter particles. Thus essential to keep in mind that the only direct, empirical measurements of dark matter properties to date come from astrophysical and cosmological observations. In the coming decade many grand-design observational campaigns such as LSST, Euclid, DESI, 4HS, SKA, will come on-line and assume operation. These will provide us with a deluge of new data of unprecedented scale and precision. This makes the goal of searching for astrophysical observables optimal for constraining the nature and physics of dark matter urgent and pressing. This project addresses the heart of this urgency by proposing an ambitious program for a novel systematic study of the theoretical connection between microscopic dark matter physics and macroscopic properties of haloes galaxies. We will search for observables offering the potential to discriminate among three main particle dark matter models: Cold Dark Matter (CDM), Warm Dark Matter (WDM), and Self-Interacting Dark matter (SIDM) There is a large number of potential halo and galaxy observables that can be used to infer and constrain the particle dark matter models. These include satellite luminosity functions, strong lens flux ratio anomalies, galaxy-galaxy strong lensing, red giant branch populations, gamma annihilation, and X-ray decay fluxes, and more. The observed variability of these observables to some extent is a result of the extra-cosmic variance induced by mixing together objects from different cosmic environments. In the COLAB project, we propose to search for optimal observational dark matter signature in dark matter halo and galaxy populations segregated by the large-scale Cosmic-Web environment in which they reside. We put forward two research hypotheses: (i) analysis of galaxy and dark matter halo populations split by the Cosmic Web environment they live in will significantly reduce the cosmic and sample variance impact. Thanks to this we will find and mark observables best suited for constraining different dark matter candidates; (ii) The separation of the galaxy population into more busy and more quiescent cosmic regions can help to reduce degeneracies due to non-linear baryonic processes. We will employ various Cosmic Web identification algorithms based on either eigenvalues of the Hessian matrix (such as NEXUS+, or T-Web, and V-Web) or on graph theory (such as Minimum Spanning Tree) on the new high-resolution simulations of the three dark matter variants. We will use the state-of-the-art both N-body and hydrodynamical schemes to generate artificial galaxy catalogs. This with a help of semi-analytical galaxy formation models and various hydrodynamical subgrid recipes will allow for modeling of mock observables such as galaxy and satellite luminosity functions, strong lens flux ratio anomalies, galaxy-galaxy strong lensing, red giant branch populations, gamma annihilation, and X-ray decay fluxes. Lastly, our galaxy catalogs, mock observables together with their Cosmic-Web environmental flags, will be used to forecast realistic observational detection and dark matter model discriminatory power of various observables and samples. With the final quest of finding the cleanest and most optimal pairs of environmental samples and observables for yielding the astrophysical dark matter nature detection.
