A theoretical view of the T-web statistical description of the cosmic web

The classification of the cosmic web into different environments is both a tool to study in more detail the formation of halos and galaxies via the link between their properties and the large-scale environment and as a class of objects whose statistics contain cosmological information. In this paper, we present an analytical framework to compute the probability of the different environments in the cosmic web based on the T-web formalism that classifies structures in four different classes (voids, walls, filaments, knots) by studying the eigenvalues of the tidal tensor (Hessian of the gravitational potential). This method relies on studying the eigenvalues of the tidal tensor with respect to a given threshold and thus requires the knowledge of the JPDF of those eigenvalues. We perform a change of variables in terms of minimally correlated rotational invariants and we study their distribution in the linear regime of structure formation, and in the quasi-linear regime with the help of a Gram-Charlier expansion and tree-order Eulerian perturbation theory. This expansion allows us to predict the probability of the different environments in the density field at a given smoothing scale as a function of the chosen threshold and redshift. We check the validity of our predictions by comparing those predictions to measurements made in the N-body Quijote simulations. We notably find that scaling the threshold value with the non-linear amplitude of fluctuations allows us to capture almost entirely the redshift evolution of the probability of the environments, even if we assume that the density field is Gaussian (corresponding to the linear regime of structure formation). We also show that adding mild non-Gaussian corrections in the form of third-order cumulants of the field provides even more precise predictions for cosmic web abundances up to scales as small as ~5 Mpc/h and redshifts down to z~0.