New Bianchi type I cosmological solutions in Eddington-inspired-Born-Infeld theory
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New Bianchi type I cosmological solutions in Eddington-inspired-Born-Infeld theory Calvin Tadmon1,2 · Guichard Djiodjo-Seugmo1 Received: 16 December 2019 / Accepted: 24 September 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract We consider spherically symmetric metric tensor g, auxiliary tensor q and scalar field ψ in Eddington-inspired-Born-Infeld theory. We first recall the system of nonlinear partial differential equations derived in a previous work by Tadmon (Gen Relativ Gravit 51:15. https:// doi.org/10.1007/s10714-018-2495-9, 2019). Then assuming that the space time metric g, the auxiliary metric q and the scalar field depend only on the time coordinate, we deduce a system of nonlinear ordinary differential equations that we solve under specific assumptions to derive, by using elliptic integrals, new conformal Bianchi type I cosmological solutions together with a scalar field. Keywords Eddington-inspired-Born-Infeld theory · Scalar field · Bianchi type I cosmological solutions Mathematics Subject Classification 83D05 · 83F05
1 Introduction We can testify the veracity of the theory of General Relativity (GR) in the explanation of certain astrophysical and cosmological phenomena [1–3,31,36]. But the problem of dark matter and that of dark energy described in [29], which preoccupy many researchers today, are not yet satisfactorily elucidated by GR. This is the reason why several alternative theories (see [27,30,36,37]) to GR have emerged. Among all these different theories, the one that interests us here is the Eddington-inspired-Born-Infeld (EiBI) theory of bi-gravitation developed in [4–25,32–35]. We aim at finding new exact solutions in EiBI theory because a theory, after construction, merits to be consolidated by computing some exact solutions. The originality of this work, compared to that of Tadmon [28], stems from the type of exact solutions we are looking for. Here the looked for solutions are in the form of homogeneous
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Calvin Tadmon [email protected]
1
Departement of Mathematics and Computer Science, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon
2
The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, Trieste 34151, Italy
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C. Tadmon and G. Djiodjo-Seugmo
metrics, together with a scalar field. It is worth recalling that, assuming Friedmann-LemaîtreRobertson-Walker (FLRW) form for the metrics, in section 5 of [26], J. B. Jimenez and his collaborators provided a very good review of cosmological applications and exhibited, with many illustrative figures, the relevant phenomenology derived from EiBI theory of gravity. We use Schwarzschild coordinates (t, r , θ, ϕ) (see [22]), and consider in the EiBI gravity, a four-dimensional manifold V4 endowed with a space-time metric g and auxiliary metric q and a scalar field, all depending on the time coordinate t and radial coordinate r . It is assumed that the scalar field is coupled with the space-time metric g and not with the aux
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