This article investigates the signature of the seventeen multiconnected flat spaces in cosmic microwave background (CMB) maps. For each such space it recalls a fundamental domain and a set of generating matrices, and then goes on to find an orthonormal basis for the set of eigenmodes of the Laplace operator on that space. The basis eigenmodes are expressed as linear combinations of eigenmodes of the simply connected Euclidean space. A preceding work, which provides a general method for implementing multiconnected topologies in standard CMB codes, is then applied to simulate CMB maps and angular power spectra for each space. Unlike in the 3-torus, the results in most multiconnected flat spaces depend on the location of the observer. This effect is discussed in detail. In particular, it is shown that the correlated circles on a CMB map are generically not back to back, so that negative search of back-to-back circles in the Wilkinson Microwave Anisotropy Probe data does not exclude a vast majority of flat or nearly flat topologies.

This article investigates the computation of the eigenmodes of the Laplacian operator in multi-connected three-dimensional spherical spaces. General mathematical results and analytical solutions for lens and prism spaces are presented. Three complementary numerical methods are developed and compared with our analytic results and previous investigations. The cosmological applications of these results are discussed, focusing on the cosmic microwave background (CMB) anisotropies. In particular, whereas in the Euclidean case too small universes are excluded by present CMB data, in the spherical case there will always exist candidate topologies even if the total energy density parameter of the universe is very close to unity.

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. We review the theoretical and observational status of the field.

A widespread myth asserts that all small universe models suppress the cosmic microwave background (CMB) quadrupole. In actual fact, some models suppre

I give a historical survey of the discussions about the existence of closed timelike curves in general relativistic models of the universe, opening the physical possibility of time travel in the past, as first recognized by K. Gödel in his rotating universe model of 1949. I emphasize that journeying into the past is intimately linked to spacetime models devoid of timelike singularities. Since such singularities arise as an inevitable consequence of the equations of general relativity given physically reasonable assumptions, time travel in the past becomes possible only when one or another of these assumptions is violated. It is the case with wormhole-type solutions. S. Hawking and other authors have tried to “save” the paradoxical consequences of time travel in the past by advocating physical mechanisms of chronological protection; however, such mechanisms remain presently unknown, even when quantum fluctuations near horizons are taken into account. I close the survey by a brief and pedestrian discussion of Causal Dynamical Triangulations, an approach to quantum gravity in which causality plays a seminal role.