Detecting Topology in a Nearly Flat Spherical Universe

Abstract

When the density parameter is close to unity, the universe has a large curvature radius independently of its being hyperbolic, flat, or spherical. Whatever the curvature, the universe may have either a simply connected or a multiply connected topology. In the flat case, the topology scale is arbitrary, and there is no a priori reason for this scale to be of the same order as the size of the observable universe. In the hyperbolic case any nontrivial topology would almost surely be on a length scale too large to detect. In the spherical case, by contrast, the topology could easily occur on a detectable scale. The present paper shows how, in the spherical case, the assumption of a nearly flat universe simplifies the algorithms for detecting a multiply connected topology, but also reduces the amount of topology that can be seen. This is of primary importance for the upcoming cosmic microwave background data analysis. This article shows that for spherical spaces one may restrict the search to diametrically opposite pairs of circles in the circles-in-the-sky method and still detect the cyclic factor in the standard factorization of the holonomy group. This vastly decreases the algorithm's run time. If the search is widened to include pairs of candidate circles whose centers are almost opposite and whose relative twist varies slightly, then the cyclic factor along with a cyclic subgroup of the general factor may also be detected. Unfortunately the full holonomy group is, in general, unobservable in a nearly flat spherical universe, and so a full 6-parameter search is unnecessary. Crystallographic methods could also potentially detect the cyclic factor and a cyclic subgroup of the general factor, but nothing else.

16 pages, 7 figures

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Cited by in corpus (25)

• Constraining the Topology of the Universe
• WMAP data and the curvature of space
• Interpretation of the Hubble diagram in a nonhomogeneous universe
• Cosmic microwave background anisotropies in multi-connected flat spaces
• Mapping the large-scale anisotropy in the WMAP data
• Simulating Cosmic Microwave Background maps in multi-connected spaces
• Generalized Chaplygin gas model, supernovae and cosmic topology
• A two-mass expanding exact space-time solution
• Cosmic microwave background constraints on multi-connected spherical spaces
• The circles-in-the-sky signature for three spherical universes
• Circles-in-the-sky searches and observable cosmic topology in a flat Universe
• A note on the large-angle anisotropies in the WMAP cut-sky maps
• What do very nearly flat detectable cosmic topologies look like?
• The spectral action and cosmic topology
• Topology beyond the horizon: how far can it be probed?
• Observational constraints on modified gravity models and the Poincaré dodecahedral topology
• Topological signatures in CMB temperature anisotropy maps
• Relativistic Effects of our Galaxy's Motion on Circles-in-the-sky
• Cosmic crystallography using short-lived objects - active galactic nuclei
• Detectability of Cosmic Topology in Generalized Chaplygin Gas Models
• Cosmological Parameters and Cosmic Topology
• Supernovae observations and cosmic topology
• Constraints on the cosmological density parameters and cosmic topology
• A constraint on any topological lensing hypothesis in the spherical case: it must be a root of the identity
• A topological interpretation of the color charge