Eigenmodes of 3-dimensional spherical spaces and their application to cosmology

Abstract

This article investigates the computation of the eigenmodes of the Laplacian operator in multi-connected three-dimensional spherical spaces. General mathematical results and analytical solutions for lens and prism spaces are presented. Three complementary numerical methods are developed and compared with our analytic results and previous investigations. The cosmological applications of these results are discussed, focusing on the cosmic microwave background (CMB) anisotropies. In particular, whereas in the Euclidean case too small universes are excluded by present CMB data, in the spherical case there will always exist candidate topologies even if the total energy density parameter of the universe is very close to unity.

28 pages, 24 figures

References in corpus (2)

• Topology and the Cosmic Microwave Background
• Topological Lensing in Spherical Spaces

Cited by in corpus (48)

• First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters
• Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background
• Theory of cosmological perturbations in an anisotropic universe
• Predictions from an anisotropic inflationary era
• Planck 2013 results. XXVI. Background geometry and topology of the Universe
• Parametric amplification of metric fluctuations through a bouncing phase
• WMAP data and the curvature of space
• A Hint of Poincaré Dodecahedral Topology in the WMAP First Year Sky Map
• The optimal phase of the generalised Poincare dodecahedral space hypothesis implied by the spatial cross-correlation function of the WMAP sky maps
• Interpretation of the Hubble diagram in a nonhomogeneous universe
• Cosmic microwave background anisotropies in multi-connected flat spaces
• CMB Anisotropy of the Poincare Dodecahedron
• Simulating Cosmic Microwave Background maps in multi-connected spaces
• A two-mass expanding exact space-time solution
• Cosmic microwave background constraints on multi-connected spherical spaces
• Imprints of spherical non-trivial topologies on the CMB
• Power spectrum for perturbations in an inflationary model for a closed universe
• Laplacian eigenmodes for spherical spaces
• The residual gravity acceleration effect in the Poincare dodecahedral space
• Arrow of Time in String Theory
• Observational signatures of a non-singular bouncing cosmology
• The spectral action and cosmic topology
• Poincare dodecahedral space parameter estimates
• Tensor perturbations during inflation in a spatially closed Universe
• Constraints on the Detectability of Cosmic Topology from Observational Uncertainties
• Predicting the CMB power spectrum for binary polyhedral spaces
• Topological signatures in CMB temperature anisotropy maps
• Bifurcation of hyperbolic planforms
• Spherical Universe topology and the Casimir effect
• Stringy Instability of Topologically Non-Trivial Ads Black Holes and of desitter S-Brane Spacetimes
• The Most Probable Size of the Universe
• Casimir Effect in $E^3$ closed spaces
• Invariant operator due to F. Klein quantizes H. Poincare's dodecahedral 3-manifold
• Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds
• Platonic topology and CMB fluctuations: Homotopy, anisotropy, and multipole selection rules
• Exact Polynomial Eigenmodes for Homogeneous Spherical 3-Manifolds
• CMB radiation in an inhomogeneous spherical space
• Bounded Scalar Perturbations in Bouncing Brane World Cosmologies
• Cosmic topology. Part IIIa. Microwave background parity violation without parity-violating microphysics
• Cosmic Topology : Twenty Years After
• How well-proportioned are lens and prism spaces?
• Transitioning from a bounce to $R^2$ inflation
• Waves on accelerating dodecahedral universes
• Cosmological principle and honeycombs
• Wave computation on the Poincaré dodecahedral space
• Cosmic topology. Part IIIb. Eigenmodes and correlation matrices of spin-2 perturbations in orientable Euclidean manifolds
• Scalar Perturbations in Nonsingular Universes from Interacting Vacuum
• Pattern formation for the Swift-Hohenberg equation on the hyperbolic plane