We have constructed a series of Voronoi tessellations with different input parameters. Our realisations consist either only of structure elements or of structure elements and noise. We have chosen a number of points comparable with the situation in observational samples and simulations of large scale structure formation, i.e. our realisations correspond to models with all galaxies within structure elements and models with additional randomly distributed galaxies.
In the first two tessellations, only walls were constructed. Using the core-sampling method, we detected walls with a mean separation of 50 Mpc (see Table 1 (click here) and Fig. 2 (click here)) in these synthetic models. This corresponds exactly to the input parameter (the mean void size) of the Voronoi tesselation. The method provided an accurate reconstruction despite the additional noise which was added to the tessellation (Fig. 4 (click here), right). On the other hand, the scales of the spurious filaments in the tessellation (Fig. 5 (click here)) are sensitive to the addition of noise. The specific dependence found appears to be a property of the particular synthetic model generated.
The next four tessellations are all characterized by a mean cell size of 15 Mpc. The cells are surrounded only by filaments. Two tessellations were made with filaments of radius 0.2 Mpc, two more with 0.1 Mpc. Independent of the radii of filaments, we were able to determine the same mean distance of filaments (13.6 Mpc); however, this estimate was slightly reduced by adding randomly distributed points as background (cp. Fig. 8 (click here) , dashed lines). The detected spurious walls have distances of the order of the box size and very large errors so that they can be easily ruled out.
Our results demonstrate clearly that the core-sampling method allows one not only to determine the structure elements, but also to measure their characteristic mean separation or density almost independently of the influence of a substantial noise component. From the good agreement between the expected and recovered structure parameters, we conclude that the core-sampling method is a powerful tool for further investigations of observational surveys and the determination of structure elements such as filaments and walls of galaxies. Recently, the core-sampling method has been successfully applied both to numerical simulations (Doroshkevich et al. 1997) and the Las Campanas galaxy sample (Doroshkevich et al. 1996).
Acknowledgements
We would like to thank Rien Van de Weygaert for allowing us to use his code for generating Voronoi tessellations and Ron Kates for valuable discussion of the manuscript. This paper was supported in part by Danmark Grundforskningsfond through its support for the estableishment of the Theoretical Astrophysics Center. A.G.D. was partly supported by the INTAS grant 93-0068 and by the Center for Cosmoparticle Physics "Cosmion" in the framework of the project "Cosmoparticle Physics". S.G. wishes to express gratitude for the hospitality of the TAC Copenhagen.