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BETTI NUMBERS OF GAUSSIAN FIELDS

Artykuł
Czasopismo : Journal of the Korean Astronomical Society   Tom: 46, Zeszyt: 3, Strony: 125-131
Changbom Park [1] , Pratyush Pranav [2] , Pravabati Chingangbam [3] , Rien van de Weygaert [2] , Bernard Jones [2] , Gert Vegter [4] , Inkang Kim [5] , Johan Hidding [2] , Wojciech Hellwing [6] , [7]
  • [1]
    School of Physics, Korea Institute for Advanced Study, Seou l 130-722, Korea
  • [2]
    Kapteyn Astron. Inst., Univ. of Groningen, PO Box 800, 9700 A V Groningen, The Netherlands
  • [3]
    Indian Institute of Astrophysics, Koramangala II Block, Ba ngalore 560 034, India
  • [4]
    Johann Bernoulli Inst. for Mathematics and Computer Scienc e, Univ. of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands
  • [5]
    School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Korea
  • [6]
    Institute of Computational Cosmology, Department of Physi cs, Durham University, South Road, Durham DH1 3LE, United Kingdom
  • [7]
2013 angielski
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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; β₀ is the number of connected regions, β₁ is the number of circular holes (i.e., complement of solid tori), β₂ and 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. β₀ 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 though 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.
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