Bouncing solutions in f ( T ) gravity
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Regular Article - Theoretical Physics
Bouncing solutions in f (T ) gravity Maria A. Skugoreva1,a , Alexey V. Toporensky1,2,b 1 2
Kazan Federal University, Kremlevskaya 18, Kazan 420008, Russia Sternberg Astronomical Institute, Lomonosov Moscow State University, Moscow 119991, Russia
Received: 1 October 2020 / Accepted: 31 October 2020 © The Author(s) 2020
Abstract We consider certain aspects of cosmological dynamics of a spatially curved Universe in f (T ) gravity. Local analysis allows us to find conditions for bounces and for static solutions; these conditions appear to be in general less restrictive than in general relativity. We also provide a global analysis of the corresponding cosmological dynamics in the cases when bounces and static configurations exist, by constructing phase diagrams. These diagrams indicate that the fate of a big contracting Universe is not altered significantly when bounces become possible, since they appear to be inaccessible by a sufficiently big Universe.
1 Introduction Modified gravity can lead to some kinds of cosmological dynamics which are impossible in general relativity (GR), at least for the usual matter content of the Universe, like a perfect fluid with positive energy density. One of wellknown examples is the so-called non-standard singularity where the scale factor a, the Hubble parameter H and the matter energy density ρ remain constant, while H˙ diverges [1]. Evolution of the Universe cannot be prolonged through this point. A non-zero spatial curvature gives even more diversity in possible dynamical regimes. We should remind the reader that in GR the influence of the spatial curvature upon the cosmological dynamics of an isotopic Universe filled with a perfect fluid is quite easy to explain. The Friedmann equation contains only three terms: k 3 M Pl 2 H 2 + 2 = ρ, 8π a where k = 0, 1, −1 for zero, positive and negative spatial curvature, respectively. Qualitative features of the dynamics are determined completely if we know which term (with a e-mail:
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the curvature or the matter energy density) dominates at the particular epoch. Since the energy density of a perfect fluid with the equation of state parameter w falls as a 2/[3(w+1)] , the matter energy density dominates at small a for w > −1/3 and for large a in the opposite case of w < −1/3. For positive spatial curvature this leads to an ultimate bounce in the case of w < −1/3 and an ultimate recollapse for w > −1/3. A particular value of the energy density of matter makes possible a static solution in the w < −1/3 case which is known to be unstable. On the contrary, negative spatial curvature does not change the general evolution of the Universe from Big Bang to an eternal expansion, and the only difference between the two cases, w > −1/3 and w < −1/3, is the time location of the curvature dominated Milne asymptotic a ∝ t, which occurs near Big Bang for w < −1/3 and at late times for w > −1/3. In modified gravity the e
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