Cosmic topology - Part IVa - Classification of manifolds using machine learning: a case study with small toroidal universes

Q: What cosmic topology case study is used for training and testing the machine-learning classifiers?

The study focuses on Euclidean E1 topology, corresponding to a 3-torus, using harmonic-space realizations of the cosmic microwave background with a cubic fundamental domain.

Q: How accurate are the classifiers when the topology orientation is known?

When orientation is known and the topology scale is about half the last-scattering diameter, the classifiers achieve accuracy greater than 99%.

Q: Which algorithm performs best for distinguishing random rotations from covering-space realizations?

The extreme gradient boosting classifier performs best for random rotations, reaching about 88% accuracy.

Aug 4, 2026

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Non-trivial spatial topology of the Universe can leave measurable imprints in the cosmic microwave background. This work tests machine-learning methods for classifying harmonic-space realizations of the microwave background using the Euclidean E1 topology, i.e., a 3-torus with a cubic fundamental domain whose size is smaller than the last-scattering surface diameter. With known orientation, classification accuracy exceeds 99% when the topology scale is about half the last-scattering diameter. For random sky rotations, extreme gradient boosting achieves 88%, while random forests and 1D/2D CNN variants reach about 83%–87%. The approach can also classify non-rotated cases with slightly larger topology scales given sufficient training data, offering computationally cheaper likelihood-free inference at the cost of handling arbitrary orientations.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages36

Reading Time1 h 31 min

AccessFree to download

Introduction

The dataset

Properties of the E1 topology

Generating a ℓm realizations

Features of the a ℓm realization data

Machine learning algorithms

Random forests and extreme gradient boosting classifier trained on a ℓm data

1D convolutional neural networks trained on a ℓm data

2D convolutional neural networks trained on Cℓmℓ′m′ data

2D complex convolutional neural networks trained on Cℓmℓ′m′ data

Results

E1 realizations with L < LLSS

E1 realizations with L ≳ LLSS

Discussion and conclusions

Appendices

arXiv :2404 .01236v2 [ astro-ph .CO] 24 Sep 2024 IFT-UAM/CSIC-24-47 Cosmic topology. Part IVa . Classification of manifolds using machine learning: a case study with small toroidal universes (COMPACT Collaboration) Andrius Tamosiunas,a Fernando Cornet-Gomez,a Yashar Akrami,b,a,c Stefano Anselmi,d,e,f Javier Carrón Duque,b Craig J. Copi,a Johannes R. Eskilt,g,c Özenç Güngör,a Andrew H. Jaffe,c Arthur Kosowsky,h Mikel Martin Barandiaran,b James B. Mertens,a Deyan P. Mihaylov,a Thiago S. Pereira,i Samanta Saha,a Amirhossein Samandar,a Glenn D. Starkman,a Quinn Taylor,a and Valeri Vardanyanj aCERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106, USA bInstituto de Física Teórica (IFT) UAM-CSIC, C/ Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid, Spain cAstrophysics Group & Imperial Centre for Inference and Cosmology, Department of Physics, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2AZ, United Kingdom dINFN, Sezione di Padova, via Marzolo 8, I-35131 Padova, Italy eDipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova, via Marzolo 8, I-35131 Padova, Italy f Laboratoire Univers et Théories, Observatoire de Paris, Université PSL, Université Paris Cité, CNRS, F-92190 Meudon, France g Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway h Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA i Departamento de Física, Universidade Estadual de Londrina, Rod. Celso Garcia Cid, Km 380, 86057-970, Londrina, Paraná, Brazil jKavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo, Chiba 277-8583, Japan E-mail: [andrius.tamosiunas@case.edu](andrius.tamosiunas@case.edu), [fernando.cornetgomez@case.edu](fernando.cornetgomez@case.edu), [yashar.akrami@csic.es](yashar.akrami@csic.es), [stefano.anselmi@pd.infn.it](stefano.anselmi@pd.infn.it), [javier.carron@csic.es](javier.carron@csic.es), [craig.copi@case.edu](craig.copi@case.edu), [j.r.eskilt@astro.uio.no](j.r.eskilt@astro.uio.no), [ozenc.gungor@case.edu](ozenc.gungor@case.edu), [a.jaffe@imperial.ac.uk](a.jaffe@imperial.ac.uk), [kosowsky@pitt.edu](kosowsky@pitt.edu), [mikel.martin@uam.es](mikel.martin@uam.es), [james.mertens@case.edu](james.mertens@case.edu), [deyan.mihaylov@case.edu](deyan.mihaylov@case.edu), [tspereira@uel.br](tspereira@uel.br), [samanta.saha@case.edu](samanta.saha@case.edu), [amirhossein.samandar@case.edu](amirhossein.samandar@case.edu), [glenn.starkman@case.edu](glenn.starkman@case.edu), [qxt42@case.edu](qxt42@case.edu), [valeri.vardanyan@ipmu.jp](valeri.vardanyan@ipmu.jp) Abstract. Non-trivial spatial topology of the Universe may give rise to potentially measurable signatures in the cosmic microwave background. We explore different machine learning approaches to classify harmonic-space realizations of the microwave background in the test case of Euclidean E1 topology (the 3-torus) with a cubic fundamental domain of a size scale significantly smaller than the diameter of the last scattering surface. This is the first step toward developing a machine learning approach to classification of cosmic topology and likelihood-free inference of topological parameters. Different machine learning approaches are capable of classifying the harmonic-space realizations with accuracy greater than 99% if the topology scale is half of the diameter of the last-scattering surface and orientation of the topology is known. For distinguishing random rotations of these sky realizations from realizations of the covering space, the extreme gradient boosting classifier algorithm performs best with an accuracy of 88% . Slightly lower accuracies of 83% to 87% are obtained with the random forest classifier along with one-and two-dimensional convolutional neural networks. The techniques pres

FAQs

What cosmic topology case study is used for training and testing the machine-learning classifiers?

The study focuses on Euclidean E1 topology, corresponding to a 3-torus, using harmonic-space realizations of the cosmic microwave background with a cubic fundamental domain.

How accurate are the classifiers when the topology orientation is known?

When orientation is known and the topology scale is about half the last-scattering diameter, the classifiers achieve accuracy greater than 99%.

Which algorithm performs best for distinguishing random rotations from covering-space realizations?

The extreme gradient boosting classifier performs best for random rotations, reaching about 88% accuracy.

Aug 4, 2026

Upload by Eliana