Narrow Results

We reexamine the possibility of the detection of the cosmic topology in nearly flat hyperbolic Friedmann-Lemaître-Robertson-Walker (FLRW) universes by using patterns repetition. We update and extend our recent results in two important ways: by employing recent observational constraints on the cosmological density parameters as well as the recent mathematical results concerning small hyperbolic 3-manifolds. This produces new bounds with consequences for the detectability of the cosmic topology. In addition to obtaining new bounds, we also give a concrete example of the sensitive dependence of detectability of cosmic topology on the uncertainties in the observational values of the density parameters.

We address three relevant questions related to the search for the size and shape of our Universe: (i) How do the actual observation of multiple images of certain cosmic objects, e.g. galaxy clusters, constrain the possible models for the shape of our Universe?, (ii) What kind of predictions can be done once a pair of cosmic objects have been identified to be topological images related by a translation?, and (iii) Is it possible to determine if two regions of space are topologically identified, even when distortions on the distributions of cosmic sources due to observational limitations are not negligible? We give examples answering the first two questions using the suggestion of Roukema and Edge that the clusters RXJ 1347.5-1145 and CL 09104+4109 might be topological images of the Coma cluster. For the third question, we suggest a method based on the analysis of PSH's noise correlations which seems to give a positive answer.

Recent high precision data by WMAP and SDSS have provided strong evidence to suggest that the universe is nearly flat. They are also making it possible to probe the topology of the universe. Motivated by these results, we have recently studied the consequences of taking the inflationary limit, i.e. |Ω₀ - 1| ≪ 0. We have shown that in this limit a generic detectable spherical or hyperbolic topology is locally indistinguishable from either or , irrespective of its global shape. Here we briefly present these results and further discuss their observational implications.

A nontrivial topology of the spatial section of the universe is an observable which can be probed for all homogeneous and isotropic universes, without any assumption on the cosmological density parameters. We discuss how one can use this observable to set constraints on the density parameters of the universe by using a specific spatial topology along with type Ia supernovae and X-ray gas mass fraction data sets.

This is an overview of new and ongoing research developments aimed at constructing cosmological models based on noncommutative geometry, via the spectral action functional, thought of as a modified gravity action, which includes the coupling with matter when computed on an almost commutative geometry. This survey is mostly based on recent results obtained in collaboration with Elena Pierpaoli and Kevin Teh. We describe various aspects of cosmological models of the very early universe, developed by the author and Pierpaoli, based on the asymptotic expansion of the spectral action functional and on renormalization group analysis of the associated particle physics model (an extension of the standard model with right-handed neutrinos and Majorana mass terms previously developed in collaboration with Chamseddine and Connes). We also describe nonperturbative results, more recently obtained by Pierpaoli, Teh, and the author, which extend to the more modern universe, which show that, for different candidate cosmic topologies, the form of the slow-roll inflation potentials obtained from the nonperturbative calculation of the spectral action are strongly coupled to the underlying geometry and topology. We discuss some ongoing directions of research and open questions in this new field of "noncommutative cosmology". The paper is based on the talk given by the author at the conference "Geometry and Quantum Field Theory" at the MPI, in honor of Alan Carey.

The immediate observational consequence of a non-trivial spatial topology of the Universe is that an observer could potentially detect multiple images of radiating sources. In particular, a non-trivial topology will generate pairs of correlated circles of temperature fluctuations in the anisotropies maps of the cosmic microwave background (CMB), the so-called circles-in-the-sky. In this way, a detectable non-trivial spatial topology may be seen as an observable attribute, which can be probed through the circles-in-the-sky for all locally homogeneous and isotropic universes with no assumptions on the cosmological dark energy (DE) equation of state (EoS) parameters. We show that the knowledge of the spatial topology through the circles-in-the-sky offers an effective way of reducing the degeneracies in the DE EoS parameters. We concretely illustrate the topological role by assuming a Poincaré dodecahedral space topology and reanalyzing the constraints on the parameters of a specific EoS which arise from the supernovae type Ia, baryon acoustic oscillations and the CMB plus the statistical topological contribution.