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The classification of the cosmic web into different environments is both atool to study in more detail the formation of halos and galaxies via the linkbetween their properties and the large-scale environment and as a class ofobjects whose statistics contain cosmological information. In this paper, wepresent an analytical framework to compute the probability of the differentenvironments in the cosmic web based on the T-web formalism that classifiesstructures in four different classes (voids, walls, filaments, knots) bystudying the eigenvalues of the tidal tensor (Hessian of the gravitationalpotential). This method relies on studying the eigenvalues of the tidal tensorwith respect to a given threshold and thus requires the knowledge of the JPDFof those eigenvalues. We perform a change of variables in terms of minimallycorrelated rotational invariants and we study their distribution in the linearregime of structure formation, and in the quasi-linear regime with the help ofa Gram-Charlier expansion and tree-order Eulerian perturbation theory. Thisexpansion allows us to predict the probability of the different environments inthe density field at a given smoothing scale as a function of the chosenthreshold and redshift. We check the validity of our predictions by comparingthose predictions to measurements made in the N-body Quijote simulations. Wenotably find that scaling the threshold value with the non-linear amplitude offluctuations allows us to capture almost entirely the redshift evolution of theprobability of the environments, even if we assume that the density field isGaussian (corresponding to the linear regime of structure formation). We alsoshow that adding mild non-Gaussian corrections in the form of third-ordercumulants of the field provides even more precise predictions for cosmic webabundances up to scales as small as ~5 Mpc/h and redshifts down to z~0.

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