The density distribution arising at the nonlinear stage of gravitational instability is similar to intermittency phenomena in acoustic turbulence. Initially small-amplitude density fluctuations of Gaussian type transform into thin dense pancakes, filaments, and compact clumps of matter. It is perhaps surprising that the motion of self-gravitating matter in the expanding universe is like that of noninteracting matter moving by inertia. A similar process is the distribution of light reflected or refracted from rippled water. The similarity of gravitational instability to acoustic turbulence is highlighted by the fact that late nonlinear stages of density perturbation growth can be described by the Burgers equation, which is well known in the theory of turbulence. The phenomena discussed in this article are closely related to the problem of the formation of large-scale structure of the universe, which is also discussed.

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The cosmic web is one of the most striking features of the distribution of galaxies and dark matter on the largest scales in the Universe. It is composed of dense regions packed full of galaxies, long filamentary bridges, flattened sheets and vast low density voids. The study of the cosmic web has focused primarily on the identification of such features, and on understanding the environmental effects on galaxy formation and halo assembly. As such, a variety of different methods have been devised to classify the cosmic web -depending on the data at hand, be it numerical simulations, large sky surveys or other. In this paper we bring twelve of these methods together and apply them to the same data set in order to understand how they compare. In general these cosmic web classifiers have been designed with different cosmological goals in mind, and to study different questions. Therefore one would not a priori expect agreement between different techniques however, many of these methods do converge on the identification of specific features. In this paper we study the agreements and disparities of the different methods. For example, each method finds that knots inhabit higher density regions than filaments, etc. and that voids have the lowest densities. For a given web environment, we find substantial overlap in the density range assigned by each web classification scheme. We also compare classifications on a halo-by-halo basis; for example, we find that 9 of 12 methods classify around a third of group-mass haloes (i.e. M halo ∼ 10 13.5 h −1 M ⊙ ) as being in filaments. Lastly, so that any future cosmic web classification scheme can be compared to the 12 methods used here, we have made all the data used in this paper public.

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