9/2/2023 0 Comments Positively curved space![]() No noticeable difference was found when compared to the no-bounce version of. In, we have studied the primordial tensor power spectrum with quantum fluctuations originating prior to the bounce. Quite impressively, the usual claims about the need of a quantum (or modified) theory of gravity to escape the unavoidable singularity are simply contradicted using only the hypotheses of the standard cosmological scenario (if space is positively curved). A further numerical investigation was carried out in. The singular origin of our universe is naturally regularized thanks to the positive curvature combined with inflation. ![]() The possible implementations of this scenario were studied in. It was concluded that a smaller-than-usual power is to be expected at very large scales.Īlthough straightforward to show, it is often forgotten that a de Sitter space-time with closed spatial sections evades the big bang singularity and instead, naturally leads to a bounce. The study of scalar and tensor perturbations was carried out in the case of a closed universe in. ![]() These initial fluctuations can be evolved using the theory of gauge-invariant linear perturbations and the primordial power spectrum can then be unambiguously calculated. In the inflationary model, the temperature anisotropies observed in the CMB are explained by quantum fluctuations of the inflaton scalar field. Strong constraints on non-Gaussianities in the CMB favor a single field scenario. While not fully confirmed, the theory of inflation is supported by strong observational evidences, the most obvious being the slightly red-tilted spectrum of primordial fluctuations. It is believed in the standard cosmological paradigm that at its earliest stage the universe grew exponentially fast in a quasi-de Sitter phase. Not to mention that some – speculative but reasonable – arguments from quantum gravity also favor this possibility (a finite space is a natural infrared regulator). Especially when taking into account, as we shall see, that this might cure naturally the initial singularity problem. Nonetheless, even ignoring the claims of, the possibility that the universe is positively curved is worth being phenomenologically considered as a possible situation. It is also dependent upon specific statistical priors that can be questioned. Although quite convincing when using CMB data alone, this result is in tension with other measurements, such as the baryonic acoustic oscillations. The latest data released by the Planck collaboration might favor a positively curved universe, described by a curvature density \(\Omega _K=-0.044\), with a high confidence level. It is still unanswered but it has attracted a lot of attention recently, based on the refined observations of the cosmic microwave background (CMB). All rights reserved.The question of the shape of the spatial sections of the Universe is an old one. Copyright © 2023, Columbia University Press. The Columbia Electronic Encyclopedia, 6th ed. One interesting feature of a universe described by Riemann's geometry is that it is finite but unbounded straight lines ultimately form closed curves, so that a ray of light could eventually return to its source. Similarly, in three dimensions the spaces corresponding to these three types of geometry also have zero, positive, or negative curvature, respectively.Īs to which of these systems is a valid description of our own three-dimensional space (or four-dimensional space-time), the choice can be made only on the basis of measurements made over very large, cosmological distances of a billion light-years or more the differences between a Euclidean universe of zero curvature and a non-Euclidean universe of very small positive or negative curvature are too small to be detected from ordinary measurements. What distinguishes the plane of Euclidean geometry from the surface of a sphere or a saddle surface is the curvature of each (see differential geometry) the plane has zero curvature, the surface of a sphere and other surfaces described by Riemann's geometry have positive curvature, and the saddle surface and other surfaces described by Lobachevsky's geometry have negative curvature.
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