What makes a cosmic filament? The dynamical origin and identity of filaments I. fundamentals in 2D

Cosmic filaments are the main transport channels of matter in the Megaparsec universe, and represent the most prominent structural feature in the matter and galaxy distribution. Here we describe and define the physical and dynamical nature of cosmic filaments. It is based on the realization that the complex spatial pattern and connectivity of the cosmic web are already visible in the primordial random Gaussian density field, in the spatial pattern of the primordial tidal and deformation eigenvalue field. The filaments and other structural features in the cosmic web emerging from this are multistream features and structural singularities in phase-space. The caustic skeleton formalism allows a fully analytical classification, identification, and treatment of the nonlinear cosmic web. The caustic conditions yield the mathematical specification of weblike structures in terms of the primordial deformation tensor eigenvalue and eigenvector fields, in which filaments are identified -- in 2D -- with the so-called cusp caustics. These are centered around points that are maximally stretched as a result of the tidal force field. The resulting mathematical conditions represent a complete characterization of filaments in terms of their formation history, dynamics, and orientation. We illustrate the workings of the formalism on the basis of a set of constrained $N$-body simulations of protofilament realizations. These realizations are analyzed in terms of spatial structure, density profiles, and multistream structure and compared to simpler density or potential field saddle point specifications. The presented formalism, and its 3D generalization, will facilitate the mining of the rich cosmological information contained in the observed weblike galaxy distribution, and be of key significance for the analysis of cosmological surveys such as SDSS, DESI, and Euclid.