On Kasner solution in Bianchi I f(T) cosmology

Abstract

Recently the cosmological dynamics of an anisotropic Universe in f (T ) gravity became an area of intense investigations. Some earlier papers devoted to this issue contain contradictory claims about the nature and propertied of vacuum solutions in this theory. The goal of the present paper is to clarify this situation. We compare properties of f (T ) and f (R) vacuum solutions and outline differences between them. The Kasner solution appears to be an exact solution for the T = 0 branch, and an asymptotic solution for the T = 0 branch. It is shown that the Kasner solution is a past attractor if T < 0, being a past and future attractor for the T > 0 branch.The Kasner solution, being one of the first known exact solutions in relativistic cosmology [1] continues to be one of the most important exact solutions in general relativity (GR) or its modifications. One of the reasons is that despite this being a vacuum solution, it is a good approximation near a cosmological singularity for almost all matter sources (except for a stiff fluid) in a flat anisotropic Universe. Moreover, a general cosmological singularity is believed to be constructed as an infinite series of consecutive epochs, each of them being a particular Kasner solution with a good accuracy (though a mathematical proof of this scenario is still absent in full details; see, for example, [2])-the famous BelinskiiKhalatnikov-Lifshitz (BKL) scenario [3]. Therefore the Kasner set of solutions provides "building blocks" for the BKL picture.If we assume that GR needs some modifications at UV scale, it is natural to expect that such modifications should change the behavior near a cosmological singularity significantly. That is why the fate of the Kasner solution in modified gravity theories is an area of intense investigations. A lot of efforts have been devoted to Kasner solutions and their modifications in quadratic gravity. We remind the reader that the Kasner solution is a solution for an anisotropically expanda e-mail: masha-sk@mail.ru b e-mail: atopor@rambler.ru ing Universe with scale factors changing as powers of time. These power exponents are subject of two conditions giving us their sum as well as the sum of their squares (both sums are equal to unity). In quadratic gravity, two different situations were identified:• If the equations of motion are of the second order, as in GR (that is, in Gauss-Bonnet gravity), the power-law solution for the scale factor is an asymptotic solution.In the high-curvature regime, these two conditions for power exponents are different from those in the GR Kasner solution [4][5][6], while the GR Kasner solution is an asymptotic solution in the low-curvature regime.• In fourth order gravity (like R + R 2 or a general quadratic gravity) the Kasner solution (with the same conditions for the exponents) is an exact vacuum solution. However, since the phase space has two additional dimensions in comparison with GR, a Kasner solution in quadratic gravity may be in some situations unstable [7,8].Recently a new class of mod...

Covariant formulation of scalar-torsion gravity

“…makes the connection condition (31) satisfied, which vindicates the respective studies in f (T ) gravity [54][55][56][57]. But for more general anisotropic models one has to check the compatibility of the spin connection.…”

Teleparallel Gravity: From Theory to Cosmology

“…In the absence of matter fluids, the Kasner cosmology is a solution to the field equations if f (0) = 0 provided that |f T (0)|, |f T T (0)| < ∞ 25 . This allows for an extensive class of functions which can host the Kasner cosmology, including (i) power-law f (T ) ∝ T n for n > 0, and (ii) exponential f (T ) = −T − T e −pT , p is some constant [861,984]. Alternatively, a Kasner metric is obtained through the modification of Kasner's relations [861].…”

“…In Refs. [984,863], the background cosmological stability of a Kasner universe in the context of f (T ) gravity was investigated in the case of vacuum. In Ref.…”

Dynamical properties of Bianchi-I spacetimes in f(R) gravity

“…Goedel and Einstein universes are natural candidates, since there already exist some existence results in the literature [49]. We could also consider different geometric situations as, for instance, the case of Bianchi-IX metrics [50] or the presence of torsion [51,52]. Some of these points are now under investigation.…”