Cosmic topology. Part IIb. Eigenmodes, correlation matrices, and detectability of non-orientable Euclidean manifolds

Journal of Cosmology and Astroparticle Physics, vol. 2026, pp. 004

Abstract

Abstract If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the amplitudes of Fourier modes of primordial scalar curvature perturbations. Topological boundary conditions replace the full set of Fourier modes with specific linear combinations of selected Fourier modes as the eigenmodes of the scalar Laplacian. In an earlier work we provided a comprehensive treatment of orientable Euclidean three-manifolds; but a thorough exploration of cosmic topology must include non-orientable three-manifolds as candidates for the geometry of space. In this paper we consider the non-orientable Euclidean topologies E 7 – E 10 , E 13 – E 15 , and E 17 , encompassing the full range of manifold parameters and observer positions, generalizing previous treatments. Under the assumption that the amplitudes of primordial scalar curvature eigenmodes are independent random variables, for each topology we obtain the correlation matrices of Fourier-mode amplitudes (of scalar fields linearly related to the scalar curvature) and the correlation matrices of spherical-harmonic coefficients of such fields sampled on a sphere, such as the temperature of the cosmic microwave background (CMB). We evaluate the detectability of these correlations given the cosmic variance of the CMB sky. As for orientable three-manifolds, we find that in manifolds where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB, we expect a correlation signal that is larger than cosmic variance noise in the CMB. The parameter space of the non-orientable Euclidean manifolds is quite rich, supporting, for example, complex dependencies of clone distances on those parameters. Our limited selection of manifold parameters — both the values of those we fix, and the choices of which to vary — are therefore exemplary of interesting behaviors (e.g., of how well the manifold can be distinguished from the covering space), but not necessarily representative. Future searches for topology will certainly require a much more thorough exploration of the parameter space to determine what values of the parameters predict statistical correlations that are convincingly attributable to topology.

Authors 15

1. Craig J. Copi

   Case Western Reserve University

   Affiliation as printed

   CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106,

2. Amirhossein Samandar

   Case Western Reserve University

   Affiliation as printed

   CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106,

3. Glenn D. Starkman

   Case Western Reserve University

   Affiliation as printed

   CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106,

4. Javier Carrón Duque

   Consejo Superior de Investigaciones Científicas · Instituto de Física Teórica · Universidad Autónoma de Madrid

   Affiliation as printed

   Instituto de Física Teórica (IFT) UAM-CSIC, C/ Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid,

5. Y. Akrami

   Consejo Superior de Investigaciones Científicas · Instituto de Física Teórica · Imperial College London · Case Western Reserve University · Universidad Autónoma de Madrid

   Affiliation as printed

   Astrophysics Group & Imperial Centre for Inference and Cosmology, Department of Physics, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2AZ,

   CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106,

   Instituto de Física Teórica (IFT) UAM-CSIC, C/ Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid,

6. S Anselmi

   Centre National de la Recherche Scientifique · University of Padua · Université Paris Cité · Université Paris Sciences et Lettres · Observatoire de Paris · Laboratoire Univers et Théories · Istituto Nazionale di Fisica Nucleare, Sezione di Padova

   Affiliation as printed

   Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova, via Marzolo 8, I-35131 Padova,

   INFN, Sezione di Padova, via Marzolo 8, I-35131 Padova,

   Laboratoire Univers et Théories, Observatoire de Paris, Université PSL, Université Paris Cité, CNRS, F-92190 Meudon,

   LUTH (UMR_8102) - Laboratoire Univers et Théories (5 place Jules Janssen 92195 Meudon cedex - France)

7. Arthur B. Kosowsky

   University of Pittsburgh

   Affiliation as printed

   Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260,

8. Fernando Cornet-Gomez

   University of Córdoba · Case Western Reserve University

   Affiliation as printed

   CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106,

   Departamento de Física, Universidad de Córdoba, Campus Universitario de Rabanales, Ctra. N-IV Km. 396, E-14071 Córdoba,

9. Johannes R. Eskilt

   University of Oslo

   Affiliation as printed

   Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo,

10. Mikel Martin Barandiaran

    Consejo Superior de Investigaciones Científicas · Instituto de Física Teórica · Universidad Autónoma de Madrid

    Affiliation as printed

    Departamento de Física Teórica, Universidad Autónoma de Madrid, 28049 Madrid,

    Instituto de Física Teórica (IFT) UAM-CSIC, C/ Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid,

11. Deyan P. Mihaylov

    Case Western Reserve University

    Affiliation as printed

    CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106,

12. Anna Negro

    Case Western Reserve University

    Affiliation as printed

    CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106,

13. Joline Noltmann Aachen Institute for Theoretical Particle Physics and Cosmology

    Consejo Superior de Investigaciones Científicas · Instituto de Física Teórica · Universidad Autónoma de Madrid · RWTH Aachen University

    Affiliation as printed

    Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen University, Templergraben 55, 52062 Aachen,

    Instituto de Física Teórica (IFT) UAM-CSIC, C/ Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid,

14. Thiago S. Pereira

    Universidade Estadual de Londrina

    Affiliation as printed

    Departamento de Física, Universidade Estadual de Londrina, Rod. Celso Garcia Cid, Km 380, 86057-970, Londrina, Paraná,

15. Andrius Tamosiunas

    Consejo Superior de Investigaciones Científicas · Instituto de Física Teórica · Case Western Reserve University · Universidad Autónoma de Madrid

    Affiliation as printed

    CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106,

    Instituto de Física Teórica (IFT) UAM-CSIC, C/ Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid,