The boundary of cosmic filaments

For decades, the boundary of cosmic filaments has been a subject of debate. In this work, we determine the physically motivated radii of filaments by constructing stacked galaxy number density profiles around the filament spines. We find that the slope of the profile changes with distance to the filament spine, reaching its minimum at approximately 1 Mpc at $z=0$ in both state-of-the-art hydrodynamical simulations and observational data. This can be taken as the average value of the filament radius. Furthermore, we note that the average filament radius rapidly decreases from $z=4$ to 1, and then slightly increases. Moreover, we find that the radius of the filament depends on the length of the filament, the distance from the connected clusters, and the masses of the clusters. These results suggest a two-phase formation scenario of cosmic filaments. The filaments experienced rapid contraction before $z=1$, but their density distribution has remained roughly stable since then. The subsequent mass transport along the filaments to the connected clusters is likely to have contributed to the formation of the clusters themselves.