Laplacian eigenmodes for spherical spaces

Abstract

The possibility that our space is multi - rather than singly - connected has gained a renewed interest after the discovery of the low power for the first multipoles of the CMB by WMAP. To test the possibility that our space is a multi-connected spherical space, it is necessary to know the eigenmodes of such spaces. Excepted for lens and prism space, and in some extent for dodecahedral space, this remains an open problem. Here we derive the eigenmodes of all spherical spaces. For dodecahedral space, the demonstration is much shorter, and the calculation method much simpler than before. We also apply to tetrahedric, octahedric and icosahedric spaces. This completes the knowledge of eigenmodes for spherical spaces, and opens the door to new observational tests of cosmic topology. The vector space V^k of the eigenfunctions of the Laplacian on the three-sphere S^3, corresponding to the same eigenvalue λ_k = -k (k+2), has dimension (k+1)^2. We show that the Wigner functions provide a basis for such space. Using the properties of the latter, we express the behavior of a general function of V^k under an arbitrary rotation G of SO(4). This offers the possibility to select those functions of V^k which remain invariant under G. Specifying G to be a generator of the holonomy group of a spherical space X, we give the expression of the vector space V_X^k of the eigenfunctions of X. We provide a method to calculate the eigenmodes up to arbitrary order. As an illustration, we give the first modes for the spherical spaces mentioned.

17 pages, no figure, to appear in CQG

References in corpus (5)

• Cosmic Topology
• Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background
• Eigenmodes of 3-dimensional spherical spaces and their application to cosmology
• Simulating Cosmic Microwave Background maps in multi-connected spaces
• A new basis for eigenmodes on the Sphere

Cited by in corpus (19)

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• Spacetime Curvature in terms of Scalar Field Propagators
• Scalar, Vector and Tensor Harmonics on the Three-Sphere
• Explicit vector spherical harmonics on the 3-sphere
• Topological signatures in CMB temperature anisotropy maps
• Melting holographic mesons by cooling a magnetized quark gluon plasma
• Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds
• On generalised D1-D5 near horizons and their spectra
• Cosmic Topology of Polyhedral Double-Action Manifolds
• Exact Polynomial Eigenmodes for Homogeneous Spherical 3-Manifolds
• The spectral geometry of the canonical Riemannian submersion of a compact Lie Group
• Cosmic topology. Part IIIa. Microwave background parity violation without parity-violating microphysics
• Elements of dodecahedral cosmology
• On momentum operators given by Killing vectors whose integral curves are geodesics
• Oscillating Multiple Giants
• Scalar, Vector and Tensor Harmonics on the Flat Compact Orientable Three-Manifolds
• Complexity of Magnetization and Magnetic Simplification
• Cosmic topology. Part IIIb. Eigenmodes and correlation matrices of spin-2 perturbations in orientable Euclidean manifolds
• Orbifold construction of the modes of the Poincare dodecahedral space