Project Team Geometrica

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## Section: New Results

### Applications

#### Study of the cosmic web

Participant : Monique Teillaud.

In collaboration with many coauthors: members of the OrbiCG Associate Team (Section  8.3.1.3 ), Herbert Edelsbrunner (IST Austria, Duke University, and Geomagic Inc.), and others

We introduce a new descriptor of the weblike pattern in the distribution of galaxies and matter: the scale dependent Betti numbers which formalize the topological information content of the cosmic mass distribution (see Figure 7 . While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic used in earlier analyses of cosmological models. The richer information content of Betti numbers goes along with the availability of fast algorithms to compute them. When measured as a function of scale they provide a “Betti signature” for a point distribution that is a sensitive yet robust discriminator of structure. The signature is highly effective in revealing differences in structure arising in different cosmological models, and is exploited towards distinguishing between different dark energy models and may likewise be used to trace primordial non-Gaussianities. In this study we demonstrate the potential of Betti numbers by studying their behaviour in simulations of cosmologies differing in the nature of their dark energy [48] , [41] . This work uses previous results obtained in Geometrica [49] , [50] .

Figure 7. Four $\alpha$-shapes of a Voronoi filament model realization. It concerns a sample of 200000 particles in a periodic box of 50 ${h}^{␣1}$Mpc size with 8 Voronoi cells. From top left to bottom right: $\alpha =0.5×{10}^{␣4},1.0×{10}^{␣4},2×{10}^{␣4}and4.0×{10}^{␣4}$.