**Shape of the Universe**

From **The Status of Cosmic Topology after Planck Data**

In the last decade, the study of the overall shape of the universe, called Cosmic Topology, has become testable by astronomical observations, especially the data from the Cosmic Microwave Background (hereafter CMB) obtained by WMAP and Planck telescopes. Cosmic Topology involves both global topological features and more local geometrical properties such as curvature. It deals with questions such as whether space is finite or infinite, simply connected or multi connected, and smaller or greater than its observable counterpart. A striking feature of some relativistic, multi connected small universe models is to create multiples images of faraway cosmic sources. While the last CMB (Planck) data fit well the simplest model of a zero curvature, infinite space model, they remain consistent with more complex shapes such as the spherical Poincaré Dodecahedral Space, the flat hypertorus or the hyperbolic Picard horn.

From **The Status of Cosmic Topology after Planck Data**

The overall topology of the universe has become an important concern in astronomy and cosmology. Even if particularly simple and elegant models such as the PDS and the hypertorus are now claimed to be ruled out at a subhorizon scale, many more complex models of multi-connected space cannot be eliminated as such. In addition, even if the size of a multi-connected space is larger (but not too much) than that of the observable universe, we could still discover an imprint in the CMB, even while no pair of circles, much less ghost galaxy images, would remain. The topology of the universe could therefore provide information on what happens outside of the cosmological horizon [35].

Whatever the observational constraints, a lot of unsolved theoretical questions remain. The most fundamental one is the expected link between the present-day topology of space and its quantum origin, since classical general relativity does not allow for topological changes during the course of cosmic evolution. Theories of quantum gravity should allow us to address the problem of a quantum origin of space topology. For instance, in quantum cosmology, the question of the topology of the universe is completely natural. Quantum cosmologists seek to understand the quantum mechanism whereby our universe (as well as other ones in the framework of multiverse theories) came into being, endowed with a given geometrical and topological structure. We do not yet have a correct quantum theory of gravity, but there is no sign that such a theory would a priori demand that the universe have a trivial topology. Wheeler first suggested that the topology of space-time might fluctuate at a quantum level, leading to the notion of a space-time foam [36]. Additionally, some simplified solutions of the Wheeler-‐‑de Witt equations for quantum cosmology show that the sum over all topologies involved in the calculation of the wavefunction of the universe is dominated by spaces with small volumes and multi-connected topologies [37]. In the approach of brane worlds in string/M-theories, the extra-dimensions are often assumed to form a compact Calabi-Yau manifold; in such a case, it would be strange that only the ordinary, large dimensions of our 3-brane would not be compact like the extra ones. However, still at an early stage of development, string quantum cosmology can only provide heuristic indications on the way multi-connected spaces would be favored.

**Key People:**

**Jeffrey Weeks****Jean-Pierre Luminet****Neil Cornish**

**Key Sources of Research:**

**The Poincaré Dodecahedral Space and the Mystery of the Missing Fluctuations**

Jeffrey Weeks

http://www.ams.org/notices/200406/fea-weeks.pdf

**Dodecahedral space topology as anexplanation for weak wide-angletemperature correlations in thecosmic microwave background**

Jean-Pierre Luminet, Jeffrey R. Weeks, Alain Riazuelo ,Roland Lehoucq & Jean-Philippe Uzan

http://dpnc.unige.ch/users/meunier/DATA/DOCS/luminet-nat.pdf

**A cosmic hall of mirrors**

**The Shape and Topology of the Universe**

Jean-Pierre Luminet

2008

http://arxiv.org/pdf/0802.2236.pdf

Luminet, Jean-Pierre.

**The wraparound universe. **

CRC Press, 2008.

**Cosmic Topology : Twenty Years After**

Jean-Pierre Luminet

2013

http://arxiv.org/pdf/1310.1245.pdf

**The Shape of Space after WMAP data**

Jean-Pierre Luminet

http://www.sbfisica.org.br/bjp/files/v36_107.pdf

**The Status of Cosmic Topology after Planck Data**

Jean-Pierre Luminet

http://www.mdpi.com/2218-1997/2/1/1/htm

http://arxiv.org/pdf/1601.03884v2.pdf

**Geometry and Topology in Relativistic Cosmology**

Jean-Pierre Luminet

**Constraints on the Topology of the Universe: Extension to General Geometries**

Pascal M. Vaudrevange, Glenn D. Starkman, Neil J. Cornish, and David N. Spergel

http://arxiv.org/pdf/1206.2939.pdf

**Topology of compact space forms from Platonic solids. I.**

A. Cavicchioli∗, F. Spaggiari, A.I. Telloni

**Topology of compact space forms from Platonic solids. II**

A. Cavicchioli∗, F. Spaggiari, A.I. Telloni

**Ancient Map of Universe and Modern Science **

Jeoraj Jain

**Discovering the Total Contents of the Universe**

Jeoraj Jain

Poundstone, William.

**The recursive universe: cosmic complexity and the limits of scientific knowledge. **

Courier Corporation, 2013.