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Register a new userTopological signatures of interstellar magnetic fields - I. Betti numbers and persistence diagrams
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Added by
Luiz Felippe Santiago Rodrigues
Publication date
2017
fields
Physics
and research's language is
English
Authors
Irina Makarenko
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The interstellar medium (ISM) is a magnetised system in which transonic or supersonic turbulence is driven by supernova explosions. This leads to the production of intermittent, filamentary structures in the ISM gas density, whilst the associated dynamo action also produces intermittent magnetic fields. The traditional theory of random functions, restricted to second-order statistical moments (or power spectra), does not adequately describe such systems. We apply topological data analysis (TDA), sensitive to all statistical moments and independent of the assumption of Gaussian statistics, to the gas density fluctuations in a magnetohydrodynamic (MHD) simulation of the multi-phase ISM. This simulation admits dynamo action, so produces physically realistic magnetic fields. The topology of the gas distribution, with and without magnetic fields, is quantified in terms of Betti numbers and persistence diagrams. Like the more standard correlation analysis, TDA shows that the ISM gas density is sensitive to the presence of magnetic fields. However, TDA gives us important additional information that cannot be obtained from correlation functions. In particular, the Betti numbers per correlation cell are shown to be physically informative. Magnetic fields make the ISM more homogeneous, reducing the abundance of both isolated gas clouds and cavities, with a stronger effect on the cavities. Remarkably, the modification of the gas distribution by magnetic fields is captured by the Betti numbers even in regions more than 300 pc from the midplane, where the magnetic field is weaker and correlation analysis fails to detect any signatures of magnetic effects.
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The topology and geometry of random fields - in terms of the Euler characteristic and the Minkowski functionals - has received a lot of attention in the context of the Cosmic Microwave Background (CMB), as the detection of primordial non-Gaussianities would form a valuable clue on the physics of the early Universe. The virtue of both the Euler characteristic and the Minkowski functionals in general, lies in the fact that there exist closed form expressions for their expectation values for Gaussian random fields. However, the Euler characteristic and Minkowski functionals are summarizing characteristics of topology and geometry. Considerably more topological information is contained in the homology of the random field, as it completely describes the creation, merging and disappearance of topological features in superlevel set filtrations. In the present study we extend the topological analysis of the superlevel set filtrations of two-dimensional Gaussian random fields by analysing the statistical properties of the Betti numbers - counting the number of connected components and loops - and the persistence diagrams - describing the creation and mergers of homological features. Using the link between homology and the critical points of a function - as illustrated by the Morse-Smale complex - we derive a one-parameter fitting formula for the expectation value of the Betti numbers and forward this formalism to the persistent diagrams. We, moreover, numerically demonstrate the sensitivity of the Betti numbers and persistence diagrams to the presence of non-Gaussianities.
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; $beta_0$ is the number of connected regions, $beta_1$ is the number of circular holes, and $beta_2$ is the number of three-dimensional voids. 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. $beta_0$ dominates the high-threshold part of the genus curve measuring the abundance of high density regions (clusters). $beta_1$ dominates the genus near the median thresholds which measures the topology of negatively curved iso-density surfaces, and $beta_2$ 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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We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Mobius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams and graded persistence diagrams are integer-valued functions on the Cartesian plane. Whereas the persistence diagram takes non-negative values, the graded persistence diagram takes values of 0, 1, or -1. The sum of the graded persistence diagrams is the persistence diagram. We show that the positive and negative points in the k-th graded persistence diagram correspond to the local maxima and minima, respectively, of the k-th persistence landscape. We prove a stability theorem for graded persistence diagrams: the 1-Wasserstein distance between k-th graded persistence diagrams is bounded by twice the 1-Wasserstein distance between the corresponding persistence diagrams, and this bound is attained. In the other direction, the 1-Wasserstein distance is a lower bound for the sum of the 1-Wasserstein distances between the k-th graded persistence diagrams. In fact, the 1-Wasserstein distance for graded persistence diagrams is more discriminative than the 1-Wasserstein distance for the corresponding persistence diagrams.
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