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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. G\"odel 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., Comment: 11 pages

Since the late 1950s a growing number of composers have engaged with scientific research and have tried to incorporate their understanding of various models and theories into their musical works. A...

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 Poincare Dodecahedral Space, the flat hypertorus or the hyperbolic Picard horn. We review the theoretical and observational status of the field., Comment: 9 pages, 3 figures. arXiv admin note: text overlap with arXiv:1310.1245. in Universe 2016, 2(1), 1

International audience; Cosmic Topology is the name given to the study of the overall shape of the universe, which involves both global topological features and more local geometrical properties such as curvature. Whether space is finite or infinite, simply-connected or multi-connected like a torus, smaller or greater than the portion of the universe that we can directly observe, are questions that refer to topology rather than curvature. A striking feature of some relativistic, multi-connected "small" universe models is to create multiples images of faraway cosmic sources. While the most recent cosmological data fit the simplest model of a zero-curvature, infinite space model, they are also consistent with compact topologies of the three homogeneous and isotropic geometries of constant curvature, such as, for instance, the spherical Poincaré Dodecahedral Space, the flat hypertorus or the hyperbolic Picard horn. After a "dark age" period, the field of Cosmic Topology has recently become one of the major concerns in cosmology, not only for theorists but also for observational astronomers, leaving open a number of unsolved issues.

What is the shape of the Universe? Is it finite or infinite ? Is space multi-connected to create ghost images of faraway cosmic sources? After a "dark age" period, the field of cosmic topology has now become one of the major concerns in astronomy and cosmology, not only from theorists but also from observational astronomers. Here I give a personal account of the spectacular progress in the field since the beginning of the 1990's, when I started to work in it., 14 pages. Contribution to a special issue of the journal "Gravitation and Cosmology" devoted to the 75th birthday of Prof. Andrey Grib. Version 2 (some sentences have been rephrased to avoid text overlap with arXiv:1303.5086, arXiv:1303.4226, arXiv:1104.0015 by other authors). arXiv admin note: text overlap with arXiv:0802.2236

This is an editorial note to accompany printing as a Golden Oldie in the Journal of General Relativity and Gravitation of the fundamental article by Georges Lema\^itre first published in French in 1927, in which the author provided the first explanation of the observations on the recession velocities of galaxies as a natural consequence of dynamical cosmological solutions of Einstein's field equations, and discovered the so-called Hubble law. We analyze in detail the scientific contents of this outstanding work, we describe how it remained unread or poorly appreciated until 1930, and we list and explain the differences between the 1927 and 1931 versions. Indeed the English translation published in 1931 in MNRAS was not perfectly faithful to the original text - it was updated. As it turned out very recently, the updates were done by Lema\^itre himself, but the discrepancies between the two texts caused a temporary stir among historians. Our new translation - given in the Appendix - follows the 1927 version exactly.

A widespread myth asserts that all small universe models suppress the CMB quadrupole. In actual fact, some models suppress the quadrupole while others elevate it, according to whether their low-order modes are weak or strong relative to their high-order modes. Elementary geometrical reasoning shows that a model’s largest dimension determines the rough value lmin at which the CMB power spectrum l(l + 1)Cl/2π effectively begins; for cosmologically relevant models, lmin ≤ 3. More surprisingly, elementary geometrical reasoning shows that further reduction of a model’s smaller dimensions – with its largest dimension held fixed – serves to elevate modes in the neighborhood of lmin relative to the high-l portion of the spectrum, rather than suppressing them as one might naively expect. Thus among the models whose largest dimension is comparable to or less than the horizon diameter, the low-order Cl tend to be relatively weak in well-proportioned spaces (spaces whose dimensions are approximately equal in all directions) but relatively strong in oddly-proportioned spaces (spaces that are significantly longer in some directions and shorter in others). We illustrate this principle in detail for the special cases of rectangular 3-tori and spherical spaces. We conclude that well-proportioned spaces make the best candidates for a topological explanation of the low CMB quadrupole observed by COBE and WMAP.