Betti Numbers of Gaussian Fields
Betti Numbers of Gaussian Fields
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DOI：
10.5303/jkas.2013.46.3.125
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发表时间：
2013-06
影响因子：
1
通讯作者：
Changbom Park;Pratyush Pranav;P. Chingangbam;R. Weygaert;B. Jones;G. Vegter;Inkang Kim;J. Hidding;W. Hellwing
中科院分区：
文献类型：
--
作者：
Changbom Park;Pratyush Pranav;P. Chingangbam;R. Weygaert;B. Jones;G. Vegter;Inkang Kim;J. Hidding;W. Hellwing
给出了三维和二维光滑高斯随机场的游程集的亏格与Betti数之间的关系，并数值研究了Betti数作为阈值能级的函数。Betti数是图形的拓扑不变量，可以用来区分拓扑空间。在三维场的偏移集的情况下，有三个可能不为零的Betti数；β？是连通区域的数量，β1是圆形孔洞的数量(即，实心圆环的补足)，β₂是三维空洞的数量(即，三维偏移区域的补足)。它们的符号交替之和是游程区域曲面的亏格。结果表明，在一个特定的阈值范围内，每个Betti数对该属具有显性贡献。β？主宰属曲线的高门槛部分，衡量高密度区域(簇)的丰度。β₁在测量负弯曲等密度表面拓扑的中值阈值附近占主导地位，而β₂对应于测量空洞丰度的低阈值部分。我们平均了许多高斯场实现上的Betti数曲线(作为阈值水平的函数的Betti数)，发现Betti数曲线的幅度和形状都取决于功率谱n的斜率，随着n的减小，它们的形状变得更宽，其幅度的下降没有亏格那么陡峭。这一行为与以下事实形成对比：无论功率谱如何，亏格曲线的形状对于所有高斯场都是固定的。即使对于每个给定的功率谱都应该计算GaussBetti数曲线，我们也建议使用Betti数来更好地描述宇宙中大尺度结构的拓扑。
We present the relation between the genus in cosmology and the Betti numbers for excursion sets of three-and two-dimensional smooth Gaussian random fields, and numerically investigate the Betti numbers as a function of threshold level. Betti numbers are topological invariants of figures that can be used to distinguish topological spaces. In the case of the excursion sets of a three-dimensional field there are three possibly non-zero Betti numbers; β? is the number of connected regions, β 1 is the number of circular holes (i. e., complement of solid tori) and β₂ is the number of three-dimensional voids (i.e., complement of three-dimensional excursion regions). Their sum with alternating signs is the genus of the surface of excursion regions. It is found that each Betti number has a dominant contribution to the genus in a specific threshold range. β? dominates the high-threshold part of the genus curve measuring the abundance of high density regions (clusters). β₁ dominates the genus near the median thresholds which measures the topology of negatively curved iso-density surfaces, and β₂ corresponds to the low-threshold part measuring the void abundance. We average the Betti number curves (the Betti numbers as a function of the threshold level) over many realizations of Gaussian fields and find that both the amplitude and shape of the Betti number curves depend on the slope of the power spectrum n in such a way that their shape becomes broader and their amplitude drops less steeply than the genus as n decreases. This behaviour contrasts with the fact that the shape of the genus curve is fixed for all Gaussian fields regardless of the power spectrum. Even through the Gaussian Betti number curves should be calculated for each given power spectrum, we propose to use the Betti numbers for better specification of the topology of large scale structures in the universe.