Constraining topology in harmonic space

Abstract

We consider several ways to test for topology directly in harmonic space by comparing the measured a_lm with the expected correlation matrices. Two tests are of a frequentist nature while we compute the Bayesian evidence as the third test. Using correlation matrices for cubic and slab-space tori, we study how these tests behave as a function of the minimal scale probed and as a function of the size of the universe. We also apply them to different first-year WMAP CMB maps and confirm that the universe is compatible with being infinitely big for the cases considered. We argue that there is an information theoretical limit (given by the Kullback-Leibler divergence) on the size of the topologies that can be detected.

19 pages, 16 figures. Updated to match PRD version

References in corpus (17)

• First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Foreground Emission
• First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Angular Power Spectrum
• A high resolution foreground cleaned CMB map from WMAP
• Cosmic Topology
• On Foreground Removal from the Wilkinson Microwave Anisotropy Probe Data by an Internal Linear Combination Method: Limitations and Implications
• Circles in the Sky: Finding Topology with the Microwave Background Radiation
• Constraining the Topology of the Universe
• Topology and the Cosmic Microwave Background
• Fast Convolution on the Sphere
• A Hint of Poincaré Dodecahedral Topology in the WMAP First Year Sky Map
• Cosmic microwave background anisotropies in multi-connected flat spaces
• Constraints on the topology of the universe from the 2-yr COBE data
• Simulating Cosmic Microwave Background maps in multi-connected spaces
• New Restrictions on Spatial Topology of the Universe from Microwave Background Temperature Fluctuations
• Non-Random Phases in Non-Trivial Topologies
• Topological signatures in CMB temperature anisotropy maps
• How large is our universe?

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• A test of the Poincare dodecahedral space topology hypothesis with the WMAP CMB data
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• Bayesian analysis of anisotropic cosmologies: Bianchi VII_h and WMAP
• Circles-in-the-sky searches and observable cosmic topology in a flat Universe
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• Cosmic Microwave Background anisotropies: the power spectrum and beyond