Topological signatures in CMB temperature anisotropy maps

Abstract

We propose an alternative formalism to simulate CMB temperature maps in $\Lambda$CDM universes with nontrivial spatial topologies. This formalism avoids the need to explicitly compute the eigenmodes of the Laplacian operator in the spatial sections. Instead, the covariance matrix of the coefficients of the spherical harmonic decomposition of the temperature anisotropies is expressed in terms of the elements of the covering group of the space. We obtain a decomposition of the correlation matrix that isolates the topological contribution to the CMB temperature anisotropies out of the simply connected contribution. A further decomposition of the topological signature of the correlation matrix for an arbitrary topology allows us to compute it in terms of correlation matrices corresponding to simpler topologies, for which closed quadrature formulae might be derived. We also use this decomposition to show that CMB temperature maps of (not too large) multiply connected universes must show ``patterns of alignment'', and propose a method to look for these patterns, thus opening the door to the development of new methods for detecting the topology of our Universe even when the injectivity radius of space is slightly larger than the radius of the last scattering surface. We illustrate all these features with the simplest examples, those of flat homogeneous manifolds, i.e., tori, with special attention given to the cylinder, i.e., $T^1$ topology.

25 pages, 7 eps figures, revtex4, submitted to PRD

References in corpus (62)

• A high resolution foreground cleaned CMB map from WMAP
• The significance of the largest scale CMB fluctuations in WMAP
• The axis of evil
• Is the low-l microwave background cosmic?
• Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background
• Multipole Vectors--a new representation of the CMB sky and evidence for statistical anisotropy or non-Gaussianity at 2<=l<=8
• Circles in the Sky: Finding Topology with the Microwave Background Radiation
• Constraining the Topology of the Universe
• Measuring Statistical isotropy of the CMB anisotropy
• A Hint of Poincaré Dodecahedral Topology in the WMAP First Year Sky Map
• Statistical Isotropy of the Wilkinson Microwave Anisotropy Probe Data: A Bipolar Power Spectrum Analysis
• Can the lack of symmetry in the COBE/DMR maps constrain the topology of the universe?
• Cosmic microwave background anisotropies in multi-connected flat spaces
• CMB Anisotropy of the Poincare Dodecahedron
• Can COBE see the shape of the universe?
• Eigenmodes of 3-dimensional spherical spaces and their application to cosmology
• Constraints on the topology of the universe from the 2-yr COBE data
• CMB Anisotropy of Spherical Spaces
• Simulating Cosmic Microwave Background maps in multi-connected spaces
• The Multipole Vectors of WMAP, and their frames and invariants
• Polynomial Interpretation of Multipole Vectors
• Well-proportioned universes suppress CMB quadrupole
• Is The Universe Infinite Or Is It Just Really Big?
• Hyperbolic Universes with a Horned Topology and the CMB Anisotropy
• Flat Spots: topological signatures of an open universe in COBE sky maps
• Reconstructing the global topology of the universe from the cosmic microwave background
• Computing CMB Anisotropy in Compact Hyperbolic Spaces
• CMB Anisotropy in Compact Hyperbolic Universes I: Computing Correlation Functions
• The topology of the universe: the biggest manifold of them all
• CMB Anisotropy in Compact Hyperbolic Universes II: COBE Maps and Limits
• The Cosmic Microwave Background for a Nearly Flat Compact Hyperbolic Universe
• Detecting Topology in a Nearly Flat Spherical Universe
• Cosmic microwave background constraints on multi-connected spherical spaces
• Detectability of Cosmic Topology in Almost Flat Universes
• New Restrictions on Spatial Topology of the Universe from Microwave Background Temperature Fluctuations
• Computation of eigenmodes on a compact hyperbolic 3-space
• Are Small Hyperbolic Universes Observationally Detectable?
• How the Universe got its Spots
• Spikes in Cosmic Crystallography II: Topological Signature of Compact Flat Universes
• A Topological Signature in Cosmic Topology
• Spikes in Cosmic Crystallography
• The Fluctuations of the Cosmic Microwave Background for a Compact Hyperbolic Universe
• Can one hear the shape of the Universe?
• Detecting topology in a nearly flat hyperbolic universe
• Limits on the Detectability of Cosmic Topology in Hyperbolic Universes
• Laplacian eigenmodes for spherical spaces
• COBE and Global Topology: An Example of the Application of the Identified Circles Principle
• A small universe after all?
• What do very nearly flat detectable cosmic topologies look like?
• Statistical isotropy of the Cosmic Microwave Background
• A Counterexample to Claimed COBE Constraints on Compact Toroidal Universe Models
• Temperature correlations in a compact hyperbolic universe
• COBE Constraints on a Compact Toroidal Low-density Universe
• Detectability of Cosmic Topology in Flat Universes
• Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds
• How large is our universe?
• Are Compact Hyperbolic Models Observationally Ruled Out?
• Relativistic Effects of our Galaxy's Motion on Circles-in-the-sky
• Ringing the eigenmodes from compact manifolds
• Signature for the Shape of the Universe
• Geometric Gaussianity and Non-Gaussianity in the Cosmic Microwave Background
• Determining the shape of the Universe using discrete sources

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• Testing gaussianity, homogeneity and isotropy with the cosmic microwave background
• Constraining topology in harmonic space
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• Excess of lensing amplitude in the Planck CMB power spectrum
• Dark Energy Cosmology with the Alternative Cosmic Microwave Background Data
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