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Astron. Astrophys. 363, L29-L32 (2000)

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3. Analysis of a cosmological N-body simulation

Cosmological N-body simulations provide an ideal template for illustrating the virtues of our method. They tend to contain a large variety of objects, with diverse morphologies, a large reach of densities, spanning over a vast range of scales. They display low density regions, sparsely filled with particles, as well as highly dense and compact clumps, represented by a large number of particles. Moderate density regions typically include strongly anisotropic structures such as filaments and walls.

Each of these features have their own individual characteristics, and often these may only be sufficiently highlighted by some specifically designed analysis tool. Conventional methods are usually only tuned for uncovering one or a few aspects of the full array of properties. Instead of artificial tailor-made methods, which may be insensitive to unsuspected but intrinsically important structural elements, our Delaunay method is uniquely defined and fully self-adaptive. Its outstanding performance is clearly illustrated by Fig. 2. Here we have analyzed an N-body simulation of structure formation in a standard CDM scenario ([FORMULA] = 1, [FORMULA] = 50 km/s/Mpc). It shows the resulting distribution of [FORMULA] particles in a cubic simulation volume of 100[FORMULA] Mpc, at a cosmic epoch at which [FORMULA]. The figure depicts a [FORMULA] slice through the center of the box. The lefthand column shows the particle distribution in a sequence of frames at increasingly fine resolution. Specifically we zoomed in on the richest cluster in the region. The righthand column shows the corresponding density field reconstruction on the basis of the grid-based Triangular-Shaped Clouds (TSC) method, here evaluated on a [FORMULA] grid. For the TSC method, one of the most frequently applied algorithms, we refer to the description in Hockney & Eastwood 1981. A comparison with other, more elaborate methods which have been developed to deal with the various aspects that we mentioned, of which Adaptive Grid methods and SPH based methods have already acquired some standing, will be presented in Schaap & van de Weygaert 2000.

[FIGURE] Fig. 2. A 9-frame mosaic comparing the performance of the Delaunay density estimating technique with a conventional grid-based TSC method in analyzing a cosmological N-body simulation. Left column: the particle distribution in a [FORMULA] wide central slice through the simulation box. Central column: the corresponding Delaunay density field reconstruction. Right column: the TSC rendered density field reconstruction. The colour scale of the density fields is logarithmic, ranging from [FORMULA].

A comparison of the lefthand and righthand columns with the central column, i.e. the Delaunay estimated density fields, reveals the striking improvement rendered by our new procedure. Going down from the top to the bottom in the central column, we observe seemingly comparable levels of resolved detail. The self-adaptive skills of the Delaunay reconstruction evidently succeed in outlining the full hierarchy of structure present in the particle distribution, at every spatial scale represented in the simulation. The contrast with the achievements of the fixed grid TSC method in the righthand column is striking, in particular when focus tunes in on the finer structures. The central cluster appears to be a mere featureless blob! In addition, low density regions are rendered as slowly varying regions at moderately low values. This realistic conduct should be set off against the erratic behaviour of the TSC reconstructions, plagued by annoying shot-noise effects.

Fig. 2 also bears witness to another virtue of the Delaunay technique. It evidently succeeds in reproducing sharp, edgy and clumpy filamentary and wall-like features. Automatically it resolves the fine details of their anisotropic geometry, seemlessly coupling sharp contrasts along one or two compact directions with the mildy varying density values along the extended direction(s). Moreover, it also manages to deal succesfully with the substructures residing within these structures. The well-known poor operation of e.g. the TSC method is clearly borne out by the central righthand frame. Its fixed and inflexible "filtering" characteristics tend to blur the finer aspects of such anisotropic structures. Such methods are therefore unsuited for an objective and unbiased scrutiny of the foamlike geometry which so pre-eminently figures in both the observed galaxy distribution as well as in the matter distribution in most viable models of structure formation.

Not only qualitatively, but also quantitatively our method turns out to compare favourably with respect to conventional methods. We are in the process of carefully scrutinizing our method by means of an array of quantitative tests. A full discussion will be presented in Schaap & van de Weygaert 2000. Here we mention the fact that the method recovers the density distribution function over many orders of magnitude. The grid-based methods, on the other hand, only managed to approach the appropriate distribution in an asymptotic fashion and yielded reliable estimates of the distribution function over a mere restricted range of density values. Very importantly, on the basis of the continuous density field reconstruction of our Delaunay method, we obtained an estimate of the density autocorrelation function that closely adheres to the (discrete) two-point correlation function directly determined from the point distribution. Further assessments on the basis of well-known measures like the Kullback-Leibler divergence (Kullbach & Leibler 1951), an objective statistic for quantifying the difference between two continuous fields, will also be presented in Schaap & van de Weygaert (2000). Finally, we may also note that in addition to its statistical accomplishments, we should also consider the computational requirements of the various methods. Given a particle distribution, the basic action of computing the corresponding Delaunay tessellation, itself an [FORMULA] routine (van de Weygaert 1991), the subsequent interpolation steps, at any desired resolution, are considerably less CPU intensive than the TSC method (both also [FORMULA]). In the case of Fig. 2 the Delaunay method is about a factor of 10 faster. In the present implementation, the bottleneck is Delaunay's substantial memory requirement ([FORMULA] the TSC operation), but a more efficient algorithm will be available in short order. These issues will be treated extensively in our upcoming publication.

The preceding is ample testimony of the promise of tessellation methods for the aim of continuous field reconstruction. The presented method, following up on earlier work by Bernardeau & van de Weygaert (1996), may be seen as a first step towards yet more advanced tessellation methods. One suggested improvement will be a second-order method rendering a continuously differentiable field reconstruction, which would dispose of the rather conspicuous triangular patches that form an inherent property of the linear procedure with discontinous gradients. In particular, we may refer to similar attempts to deal with related problems, along the lines of natural neighbour interpolation (Sibson 1981), such as implemented in the field of geophysics (Sambridge et al. 1995; Braun & Sambridge 1995) and in engineering mechanics (Sukumar 1998). As multidimensional discrete data sets are a major source of astrophysical information, we wish to promote such tessellation methods as a natural instrument for astronomical data analysis.

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000