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Progress in Information Geometry pp 219–244Cite as

Towards the “Shape” of Cosmological Observables and the String Theory Landscape with Topological Data Analysis

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Abstract

Persistent homology is a technique from Topological Data Analysis that computes the multiscale “shape” of a data set. We review the basic formalism of persistent homology as well as several applications to cosmology and string theory. We describe how persistent homology provides efficient and robust morphological summaries of cosmological observables including the Cosmic Microwave Background and Large-Scale Structure, and how statistical pipelines involving persistent homology can potentially constrain cosmological parameters. We also review the string theory landscape as an interesting data set displaying complex features in high-dimensional spaces, and thus an ideal setting for persistent homology. We describe the characterization of distributions of flux vacua using persistent homology.

Keywords

  • Topological Data Analysis
  • Persistent homology
  • Inflationary cosmology
  • Cosmic microwave background
  • Large scale structure
  • Non-gaussianity
  • String landscape

This research was supported in part by the DOE grant DE-SC0017647, the Heising-Simons Foundation, the Simons Foundation, National Science Foundation Grant No. NSF PHY-1748958, and the Kellett Award of the University of Wisconsin. GS gratefully acknowledges the hospitality of the Kavli Institute for Theoretical Physics during the final stage of this work. AC is grateful to have been supported by a Straka Fellowship at UW-Madison.

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Notes

  1. 1.

    One may also consider (persistent) homology defined over other coefficient fields. For simplicity, we restrict to \(\mathbb {Z}_2\). In fact, this is the most efficient choice if the underlying space does not have torsion, and the results computed for \(\mathbb {Z}_2\) can be translated to homology over other coefficient fields. See Sect. 4.3 of [103] for a discussion.

  2. 2.

    Strictly speaking, the 0-th homology group counts the independent components of the simplicial complex, i.e. disconnected clusters.

  3. 3.

    Strictly speaking, to use the simplcial machinery we have considered so far, M and f are suitably discretized.

  4. 4.

    As was also described in Sect. 9.2, the distinction between “point-like” and “field-like” observables is somewhat artificial, since collections of points can be used to define “field-like” observables and vice versa. In practice, however, transforming between these viewpoints may require nontrivial assumptions about the underlying physics, for examples regarding the bias of various tracers.

  5. 5.

    For a sense of scale, it is estimated that the number of flux vacua for a typical geometry is around \(10^{500}\) [8, 34] and can be as large as \(10^{272,000}\) [98]. The number of geometries in a particular of F-theory ensemble is bounded below by \(\frac{4}{3}\times 2.96\times 10^{755}\) [55].

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  1. GRAPPA and ITFA, Institute of Physics, University of Amsterdam, 1090, Amsterdam, WX, Netherlands

    Alex Cole

  2. University of Wisconsin, Madison, WI, 53706, USA

    Alex Cole & Gary Shiu

  3. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, 93106, USA

    Alex Cole & Gary Shiu

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Cole, A., Shiu, G. (2021). Towards the “Shape” of Cosmological Observables and the String Theory Landscape with Topological Data Analysis. In: Nielsen, F. (eds) Progress in Information Geometry. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-65459-7_9

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