Tree Structure of a Percolating Universe

Abstract

We present a numerical study of topological descriptors of initially Gaussian and scale-free density perturbations evolving via gravitational instability in an expanding universe. We carefully evaluate and avoid numerical contamination in making accurate measurements on simulated fields on a grid in a finite box. Independent of extent of non linearity, the measured Euler number of the excursion set at the percolation threshold, δc, is positive and nearly equal to the number of isolated components, suggesting that these structures are trees. Our study of critical point counts reconciles the clumpy appearance of the density field at δc with measured filamentary local curvature. In the Gaussian limit, we measure |δc| > σ in contrast to widely held belief that |δc| ∼ σ, where σ 2 is the variance of the density field.The large scale structure of the universe can be characterized topologically at various scales by studying the excursion sets of the density contrast δ(x); i.e., the setsAn interesting choice of δ th in either cases is the percolation threshold, δ c , for which the excursion set includes an infinite, connected structure [1].Accurate determination of δ c is a challenge both from the theoretical point of view -no analytic calculation of δ c exist so far in the three-dimensional case; the experimental point of view -numerous spurious effects, such as discreteness, finiteness of the sampled volume, etc., can contaminate the measurements, and the results depend on the method employed [2][3][4]. Here, we proceed as in Ref.[4] and consider site percolation on a grid, i.e., neighboring sites above/below the density threshold are connected through faces.We analyze the topology of E ± δth in terms of the three dimensional local curvature of δ(x) given by the Hessian matrix, ∂ 2 δ/∂x i ∂x j . The number of negative eigenvalues, I, of the Hessian matrix at any point classifies the local density structure into one of four distinct classes: region with I = 3 is in a "clump"(c); I = 2 in a "filament" (f); I = 1 in a "pancake" (p) and I = 0 in a "void"(v). In particular, this classification applies to critical points of the field, where the gradient ∂δ/∂x i = 0. Connectivity of the excursion set E ± δth is dictated solely by the number, C I , of critical points of each class I within E ± δth . For clarity, we replace the numeral index I of C I by the alphabetic label I = 0, 1, 2, 3 → v, p, f, c.At a simple qualitative level, the role of critical points in outlining the connectivity of excursion sets can be understood quite intuitively. Connectivity at percolation is along special field lines ("ridges" and "river beds") threading the critical points. Filament saddle points (fsaddles) are thus crucial for percolation in overdense regions (E + δth ). Indeed, a large fraction of them lie along ridges connecting two local maxima. When δ th > δ at a saddle point, the point drops out of the excursion set, thus disconnecting the corresponding clumps. Analogous reasoning can be applied to percolation in underdense regions where ...

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