基于持久Betti数的宇宙网拓扑
The Topology of the Cosmic Web in Terms of Persistent Betti Numbers
摘要：
我们介绍了一个多尺度的拓扑描述兆秒差距网状宇宙物质分布。贝蒂数和拓扑持久性提供了一种强有力的手段来描述宇宙网的丰富的连通性结构及其物质和星系的多尺度排列。Betti数和持久性图源自代数拓扑和Morse理论，是宇宙学上熟悉的拓扑亏格测度和相关几何Minkowski泛函的扩展和深化。除了描述的数学背景，本研究提出了计算贝蒂数和持久性图的密度场过滤的计算程序。该场可以从星系或模拟粒子的离散空间分布开始计算。本研究的主要重点是广泛而系统地探索不同的网状形态和不同层次的多尺度聚类在相应的计算贝蒂数和持久性图的印记。为此，我们使用Voronoi聚类模型作为模板的丰富多样的网状配置，和分形的Soneira-Peebles模型的范围内的多尺度配置。我们已经确定了清晰的印记集群节点，细丝，墙壁，并在持久性图中的空隙，随着嵌套层次结构的多尺度点分布。最后，我们概述了持久的拓扑结构的潜力，了解宇宙网络的连接结构，在大型模拟宇宙结构的形成，并在具有挑战性的背景下观察到的星系分布在大型星系调查。
英文摘要：
We introduce a multiscale topological description of the Megaparsec weblike cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multiscale arrangement of matter and galaxies. Emanating from algebraic topology and Morse theory, Betti numbers and persistence diagrams represent an extension and deepening of the cosmologically familiar topological genus measure, and the related geometric Minkowski functionals. In addition to a description of the mathematical background, this study presents the computational procedure for computing Betti numbers and persistence diagrams for density field filtrations. The field may be computed starting from a discrete spatial distribution of galaxies or simulation particles. The main emphasis of this study concerns an extensive and systematic exploration of the imprint of different weblike morphologies and different levels of multiscale clustering in the corresponding computed Betti numbers and persistence diagrams. To this end, we use Voronoi clustering models as templates for a rich variety of weblike configurations, and the fractal-like Soneira-Peebles models exemplify a range of multiscale configurations. We have identified the clear imprint of cluster nodes, filaments, walls, and voids in persistence diagrams, along with that of the nested hierarchy of structures in multiscale point distributions. We conclude by outlining the potential of persistent topology for understanding the connectivity structure of the cosmic web, in large simulations of cosmic structure formation and in the challenging context of the observed galaxy distribution in large galaxy surveys.
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