# Institute of Theoretical Physics

# Frank Steiner's Group

Ulm University | and | Observatoire de Lyon Centre de Recherche Astrophysique de Lyon Ecole Normale Supérieure de Lyon, Université Lyon 1, CNRS 9, avenue Charles André F-69230 Saint-Genis Laval France |

# Cosmic Microwave Background (CMB) Radiation

The cosmic microwave background (CMB) radiation is the earliest electromagnetic radiation after the Big Bang which can be observed e.g. by the NASA-satellite WMAP and the ESA-satellite Planck. The tiny anisotropies in the CMB radiation are believed to be generated by quantum fluctuations in the early Universe and represent the seeds for the large scale structure formation that is the cause for the formation of galaxy clusters and galaxies. An introduction into the complex physics can be found here.

Our focus is on cosmic topology. It addresses the question whether a non-trivial topological structure of the Universe is betrayed by cosmological observations. The best chances for such a detection are provided by anisotropies found in the CMB radiation. A non-trivial topology of the Universe can lead to a finite volume which in turn implies a suppression of anisotropies on the largest scales. Such a suppression is indeed observed in the CMB radiation.

In the case of a spatially flat Universe the simplest non-trivial example is that of a 3-torus which can be interpreted as a 3-dimensional box where three pairs of opposing sides of the box are identified. In this way a flat space of finite volume without a boundary is constructed. Below a CMB simulation for such a 3-torus topology is shown.

A cosmic microwave background simulation is shown for a 3-torus topology using more than 5.5 million eigenfunctions. The anisotropies of the CMB are encoded as colours whereby red means hotter than the average temperature and blue cooler temperatures. The average temperature is approximately 2.7 Kelvin.

An example for a non-trivial topology in a space with constant positive curvature is provided by the Poincaré dodecahedron. A cosmic microwave background simulation for this topology is displayed here.

# Inhomogeneous Manifolds

A non-trivial topology is specified by a group Γ of transformations which determines how the spatial points are connected. A cosmological observer constructs his fundamental domain in such a way that the domain does not contain points that can be transformed by the group Γ closer to the observer. This natural construction leads to a fundamental domain which is also called Dirichlet domain or Voronoi cell. The interesting point is that there are two classes of manifolds. On one hand those for which all observers obtain independent of their position the same Voronoi cell. Such spaces are called homogeneous. On the other hand those for which the shape of the Voronoi cell can depend on the position of the observer. These are inhomogeneous manifolds. The crucial point for cosmic topology is that the statistical properties of the CMB anisotropies vary in the latter case with the position of the observer. The comparison of such models with the measured CMB anisotropies is then much more complex.

Such an example in flat space is provided by the so-called half-turn space where, in contrast to the above mentioned 3-torus, one pair of sides is rotated by 180° before the sides are identified. The animation below displays the Voronoi cell of the half-turn space whereby the observer is moved along a curve parameterised by Δ. This shows the varability of the fundamental domain.