Chapter 9
Towards the “Shape” of Cosmological
Observables and the String Theory
Landscape with Topological Data
Analysis
Alex Cole and Gary Shiu
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 highdimensional 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.
A. Cole
GRAPPA and ITFA, Institute of Physics, University of Amsterdam, 1090 Amsterdam, WX,,
Netherlands
A. Cole · G. Shiu (B)
University of Wisconsin, Madison, WI 53706, USA
e-mail: shiu@physics.wisc.edu
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_9
219
Towards the “Shape” of Cosmological
Observables and the String Theory
Landscape with Topological Data
Analysis
Alex Cole and Gary Shiu
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 highdimensional 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.
A. Cole
GRAPPA and ITFA, Institute of Physics, University of Amsterdam, 1090 Amsterdam, WX,,
Netherlands
A. Cole · G. Shiu (B)
University of Wisconsin, Madison, WI 53706, USA
e-mail: shiu@physics.wisc.edu
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_9
219