阿尔法、贝蒂与百万秒差距宇宙：论宇宙网的拓扑结构
Alpha, Betti and the Megaparsec Universe: on the Topology of the Cosmic Web
摘要：
我们研究的Megaparsec宇宙网的拓扑结构的尺度相关的贝蒂数，形式化的宇宙质量分布的拓扑信息内容。虽然贝蒂数不能完全量化拓扑，但它们在亏格和欧拉特征方面扩展了传统拓扑宇宙学研究的信息。贝蒂数的信息量越丰富，计算它们的快速算法就越容易实现。对于连续的密度场，我们确定贝蒂数的尺度依赖性调用宇宙学熟悉的过滤定义的密度阈值的子级或超级集。然而，对于离散星系分布，分析是基于粒子的α形状。这些单纯复形构成Delaunay曲面细分的嵌套子集的有序序列，Delaunay曲面细分是由尺度参数$α$定义的过滤。由于它们同伦等价于距离场的子级集，它们是评估离散点分布拓扑结构的一个很好的工具。为了开发一个直观的理解的行为贝蒂数作为$α$的函数，以及它们之间的关系，在宇宙网络的形态模式，我们首先研究它们的背景下，简单的启发式Voronoi聚类模型。随后，我们解决的拓扑结构出现在标准的LCDM方案和宇宙学的方案与替代暗能量内容。贝蒂数的演化和尺度依赖性反映了宇宙网络的层次演化，并产生了一个有前途的宇宙学参数的测量。我们还讨论了期望贝蒂数作为密度阈值的函数的超水平集的高斯随机场。
英文摘要：
We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of Betti numbers goes along the availability of fast algorithms to compute them. For continuous density fields, we determine the scale-dependence of Betti numbers by invoking the cosmologically familiar filtration of sublevel or superlevel sets defined by density thresholds. For the discrete galaxy distribution, however, the analysis is based on the alpha shapes of the particles. These simplicial complexes constitute an ordered sequence of nested subsets of the Delaunay tessellation, a filtration defined by the scale parameter, $α$. As they are homotopy equivalent to the sublevel sets of the distance field, they are an excellent tool for assessing the topological structure of a discrete point distribution. In order to develop an intuitive understanding for the behavior of Betti numbers as a function of $α$, and their relation to the morphological patterns in the Cosmic Web, we first study them within the context of simple heuristic Voronoi clustering models. Subsequently, we address the topology of structures emerging in the standard LCDM scenario and in cosmological scenarios with alternative dark energy content. The evolution and scale-dependence of the Betti numbers is shown to reflect the hierarchical evolution of the Cosmic Web and yields a promising measure of cosmological parameters. We also discuss the expected Betti numbers as a function of the density threshold for superlevel sets of a Gaussian random field.
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