https://www.math.tamu.edu/~fulling/levin_topcosm.pdf

Topology and the cosmic microwave background

Establishing homogeneity of the universe in the shadow of dark energy

It has recently been suggested by Luminet et al. (2003) that the WMAP data are better matched by a geometry in which the topology is that of a Poincare dodecahedral model and the curvature is "slightly" spherical, rather than by an (effec- tively) infinite flat model. A general back-to-back matched circles analysis by Cornish et al. (2004) for angular radii in the range 25−90 ◦ , using a correlation statistic for signal detection, failed to support this. In this paper, a matched circles analysis specifi- cally designed to detect dodecahedral patterns of matched circles is performed over angular radii in the range 1−40 ◦ on the one- year WMAP data. Signal detection is attempted via a correlation statistic and an rms difference statistic. Extreme value distribu- tions of these statistics are calculated for one orientation of the 36 ◦ "screw motion" (Clifford translation) when matching circles, for the opposite screw motion, and for a zero (unphysical) rotation. The most correlated circles appear for circle radii of α = 11 ± 1 ◦ , for the left-handed screw motion, but not for the right-handed one, nor for the zero rotation. The favoured six dodecahedral face centres in galactic coordinates are (l II ,b II ) ≈ (252 ◦ , +65 ◦ ), (51 ◦ , +51 ◦ ), (144 ◦ , +38 ◦ ), (207 ◦ , +10 ◦ ), (271 ◦ , +3 ◦ ), (332 ◦ , +25 ◦ ) and their opposites. The six pairs of circles independently each favour a circle angular radius of 11 ± 1 ◦ . The temperature fluc- tuations along the matched circles are plotted and are clearly highly correlated. Whether or not these six circle pairs centred on dodecahedral faces match via a 36 ◦ rotation only due to unexpected statistical properties of the WMAP ILC map, or whether they match due to global geometry, it is clear that the WMAP ILC map has some unusual statistical properties which mimic a potentially interesting cosmological signal.

/pdf/a-hint-of-poincare-dodecahedral-topology-in-the-wmap-first-4i92nswpoe.pdf

A hint of Poincaré dodecahedral topology in the WMAP first year sky map

The 1997 Cleveland Workshop on Topology and Cosmology was a remarkable opportunity for mathematicians and cosmologist with overlapping interests in the topology of 3-space to discuss the state of knowledge in their fields This special issue reflects the wide range of talks given at the conference including both introductions/reviews of the relevant fields of mathematics and cosmology and new contributions to the scholarship in these fields In this introduction to the special issue, a brief history is given of the study of the topology of the universe Some attempt is also made to introduce and discuss each of the papers in the collection and to relate them to each other and to the wider conversation on the global topology of the universe

https://iopscience.iop.org/article/10.1088/0264-9381/15/9/002/pdf

Topology and cosmology

In this work we study to which extent the knowledge of spatial topology may place constraints on the parameters of the generalized Chaplygin gas (GCG) model for unification of dark energy and dark matter. By using both the Poincar\'e dodecahedral and binary octahedral spaces as the observable spatial topologies, we examine the current type Ia supernovae (SNe Ia) constraints on the GCG model parameters. We show that the knowledge of spatial topology does provide additional constraints on the ${A}_{s}$ parameter of the GCG model but does not lift the degeneracy of the $\ensuremath{\alpha}$ parameter.

Generalized Chaplygin gas model, supernovae, and cosmic topology

The spacelike hypersurface of the Universe at the present cosmological time is a three-dimensional manifold. A non-trivial global topology of this spacelike hypersurface would imply that the apparently observable universe (the sphere of particle horizon radius) could contain several images of the single, physical Universe. Recent three-dimensional techniques for constraining and/or detecting this topology are reviewed. Initial applications of these techniques using x-ray bright clusters of galaxies and quasars imply (weak) candidates for a non-trivial topology.

https://iopscience.iop.org/article/10.1088/0264-9381/15/9/012/pdf

Three-dimensional topology-independent methods to look for global topology

Compact, flat Friedmann-Lemaitre-Robertson-Walker (FLRW) models have recently regained interest as a good fit to the observed cosmic microwave background temperature fluctuations. However, it is generally thought that a globally, exactly-flat FLRW model is theoretically improbable. Here, in order to obtain a probability space on the set F of compact, comoving, 3-spatial sections of FLRW models, a physically motivated hypothesis is proposed, using the density parameter Omega as a derived rather than fundamental parameter. We assume that the processes that select the 3-manifold also select a global mass-energy and a Hubble parameter. The inferred range in Omega consists of a single real value for any 3-manifold. Thus, the obvious measure over F is the discrete measure. Hence, if the global mass-energy and Hubble parameter are a function of 3-manifold choice among compact FLRW models, then probability spaces parametrised by Omega do not, in general, give a zero probability of a flat model. Alternatively, parametrisation by the injectivity radius r_inj ("size") suggests the Lebesgue measure. In this case, the probability space over the injectivity radius implies that flat models occur almost surely (a.s.), in the sense of probability theory, and non-flat models a.s. do not occur.

A measure on the set of compact Friedmann-Lemaitre-Robertson-Walker models

The space-like hypersurface of the Universe at the present cosmological time is a three-dimensional manifold. A non-trivial global topology of this space-like hypersurface would imply that the apparently observable universe (the sphere of particle horizon radius) could contain several images of the single, physical Universe. Recent three-dimensional techniques for constraining and/or detecting this topology are reviewed. Initial applications of these techniques using X-ray bright clusters of galaxies and quasars imply (weak) candidates for a non-trivial topology.

Three-dimensional Topology-Independent Methods to Look for Global Topology

The approximate homogeneity of spatial sections of the Universe is well supported observationally, but the inhomogeneity of the spatial sections is even better supported. Here, we consider the implications of inhomogeneity in dust models for the connectedness of spatial sections at early times. We consider a non-global Lemaitre-Tolman-Bondi (LTB) model designed to match observations, a more general, heuristic model motivated by the former, and two specific, global LTB models. We propose that the generic class of solutions of the Einstein equations projected back in time from the spatial section at the present epoch includes subclasses in which the spatial section evolves (with increasing time) smoothly (i) from being disconnected to being connected, or (ii) from being simply connected to being multiply connected, where the coordinate system is comoving and synchronous. We show that (i) and (ii) each contain at least one exact solution. These subclasses exist because the Einstein equations allow non-simultaneous big bang times. The two types of topology evolution occur over time slices that include significantly post-quantum epochs if the bang time varies by much more than a Planck time. In this sense, it is possible for cosmic topology evolution to be "mostly" classical.

Vincent Blanloeil

Papers

Three-dimensional topology-independent methods to look for global topology

A measure on the set of compact Friedmann-Lemaitre-Robertson-Walker models

Three-dimensional Topology-Independent Methods to Look for Global Topology

On the topological implications of inhomogeneity