Universality of the Network and Bubble Topology in Cosmological Gravitational Simulations

Abstract

Using percolation statistics we, for the first time, demonstrate the universal character of a network pattern in the real space, mass distributions resulting from nonlinear gravitational instability of initial Gaussian fluctuations. Percolation analysis of five stages of the nonlinear evolution of five power law models (P (k) ∝ k n with n = +3, + 1, 0, − 1, and −2 in an Ω = 1 universe) reveals that all models show a shift toward a network topology if seen with high enough resolution. However, quantitatively, the shift is significantly different in different models: the smaller the spectral index n the stronger the shift. On the contrary, the shift toward the bubble topology is characteristic only for the n ≤ −1 models.We find that the mean density of the percolating structures in the nonlinear density distributions generally is very different from the density threshold used to identify them and corresponds much better to a visual impression. We also find that the maximum of the number of structures (connected regions above or below a specified density threshold) in the evolved, nonlinear distributions is always smaller than in Gaussian fields with the same spectrum, and is determined by the effective slope at the cutoff frequency.

Tree Structure of a Percolating Universe

“…These results should be contrasted with that of Ref. [12], where |δ c /σ| 1 is obtained independently of n. However, the measurements in Ref. [12] use a smaller grid, N g = 64, and might be affected by effects (ii) mentioned above.…”

“…From our Gaussian samples, we find that combination of these two competing effects is inconsequential if /L < ∼ 0.01. [12,13], the asymmetry due to gravitational clustering between overdense (+) and underdense regions (−), δ + c > |δ − c |, increases with the level of non linearity and with −n. It is very small for n = 0.…”

“…[12], where |δ c /σ| 1 is obtained independently of n. However, the measurements in Ref. [12] use a smaller grid, N g = 64, and might be affected by effects (ii) mentioned above. The other difference is that the field is softened by a power-spectrum cut-off at small scales instead of our Gaussian smoothing.…”

Shapefinders: A New Shape Diagnostic for Large-Scale Structure

“…Percolation theory demonstrates that the number of clusters/voids in a simulation always peaks at a threshold just above/below percolation (Yess & Shandarin 1996). Thus, it makes sense to study shapes of individual clusters/voids at this threshold because the number of distinct isolated surfaces (boundaries of clusters/voids) is largest.…”

The cluster distribution as a test of dark matter models - IV. Topology and geometry

“…b * ) all the properties of the transition from the unpercolated distribution to a percolated one. To remedy this shortcoming, and to keep the analysis as comparable as possible to that of the preceding Section on the EPC, we adopt a more sophisticated approach, as described by Klypin & Shandarin (1993); see also Mo & Börner (1990), de Lapparent et al (1991, Yess & Shandarin (1996), Sathyaprakash, Sahni & Shandarin (1996) and Sahni, Sathyaprakash and Shandarin (1997). In this approach, we consider the application of percolation techniques to a cubic lattice on which, according to some density threshold criterion, cells are labelled as either 'filled' or 'empty'.…”