I enjoyed reading this paper on the shapes of cosmic structures. The authors take an Nbody simulation with 150 [tex]h^{1}[/tex]Mpc on a side (which they consider large, but I would rather call the size moderate, as it's not much larger than the observed homogeneity scale), and produce a density field from the point distribution using a Delaunay tessellation. This continuous field is then analysed on different scales to put together a multiscale picture (in a manner I didn't quite understand).
In addition to providing nice visualisations, the method provides (among other statistical quantities) an inventory of cosmic structures in terms of clusters, filaments, walls and voids, with percentages of mass and volume. It's clear that the precise numbers depend on the adopted definitions, but at least one can get some idea. The authors emphasise that these features overlap in terms of density (which highlights the role of shear in structure formation).
An analysis of the density profiles of filaments in the direction orthogonal to the filament was an interesting application as well.
[1007.0742] Multiscale Phenomenology of the Cosmic Web
Authors:  Miguel A. AragonCalvo (1), Rien van de Weygaert (2), Bernard J.T. Jones (2) ((1) Dept. Physics & Astronomy, the Johns Hopkins University, Baltimore, U.S.A., (2) Kapteyn Astronomical Institute, University of Groningen, the Netherlands) 
Abstract:  We analyze the structure and connectivity of the distinct morphologies that define the Cosmic Web. With the help of our Multiscale Morphology Filter (MMF), we dissect the matter distribution of a cosmological $\Lambda$CDM Nbody computer simulation into cluster, filaments and walls. The MMF is ideally suited to adress both the anisotropic morphological character of filaments and sheets, as well as the multiscale nature of the hierarchically evolved cosmic matter distribution. The results of our study may be summarized as follows: i). While all morphologies occupy a roughly well defined range in density, this alone is not sufficient to differentiate between them given their overlap. Environment defined only in terms of density fails to incorporate the intrinsic dynamics of each morphology. This plays an important role in both linear and non linear interactions between haloes. ii). Most of the mass in the Universe is concentrated in filaments, narrowly followed by clusters. In terms of volume, clusters only represent a minute fraction, and filaments not more than 9%. Walls are relatively inconspicous in terms of mass and volume. iii). On average, massive clusters are connected to more filaments than low mass clusters. Clusters with $M \sim 10^{14}$ M$_{\odot}$ h$^{1}$ have on average two connecting filaments, while clusters with $M \geq 10^{15}$ M$_{\odot}$ h$^{1}$ have on average five connecting filaments. iv). Density profiles indicate that the typical width of filaments is 2$\Mpch$. Walls have less well defined boundaries with widths between 58 Mpc h$^{1}$. In their interior, filaments have a powerlaw density profile with slope ${\gamma}\approx 1$, corresponding to an isothermal density profile. 
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 Joined: March 02 2005
 Affiliation: University of Helsinki