On Kasner solution in Bianchi I f(T) cosmology
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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 \(T \ne 0\) 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 Belinskii–Khalatnikov–Lifshitz (BKL) scenario [3]. Therefore the Kasner set of solutions provides “building blocks” for the BKL picture.

If the equations of motion are of the second order, as in GR (that is, in Gauss–Bonnet gravity), the powerlaw solution for the scale factor is an asymptotic solution. In the highcurvature 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 lowcurvature 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].
This motivated studies of cosmological dynamics in f(T) gravity. Recently many papers on this topic have appeared, mostly concentrating on FRW cosmology (see, for example, [14, 15, 16, 17] and the references therein. Anisotropic cosmology is the next natural step in this direction. However, f(T) theory has its own problems connected with the lack of local Lorentz invariance [18, 19]. This leads to a situation when not all tetrads corresponding to the chosen metric form gives us the correct equations of motion [20, 21], so the separate problem of choosing a socalled “proper tetrad” [22] appears. Alternatively, proper nonzero spin connections must be associated with a given tetrad [23]. Careful investigation of this problem in connection with anisotropic metrics is still missing. In such a situation we can use the heuristic argument of [23] that if a tetrad (or spin connection) is chosen in a bad way, the resulting equations of motion should be (in some sense) pathological. The most common pathology is a requirement that the second derivative of f with respect to T vanishes, which evidently brings us back to TEGR. So, a reasonable current strategy may be in choosing the simplest tetrad associated with a Bianchi I metric, and in studying the corresponding equations of motion, if they do not show such pathologies.
This strategy has been implemented recently in several papers [24, 25, 26, 27]. The resulting equations of motion appear to be nonpathological. These papers have, however, some contradictory statements. In particular, the study of Ref. [24] concludes that a vacuum solution exists only for a particular \(f(T)=\sqrt{T}\) theory and it must be isotropic. On the contrary, Ref. [27] claims that a Kasner solution is still a solution for f(T) theory, though it is unstable. The fact that the Kasner solution remains as a solution can easily be checked by direct substitution to the equations of motion. Thus it seems that the second abovementioned alternative could be realized. However, the problem is that once the tetrad is fixed, the number of degrees of freedom in f(T) theory and in TEGR is the same. In f(R) gravity the instability of the Kasner solution (which is the exact solution in that theory) is due to the extra degrees of freedom, which are absent in f(T) theory in question. This contradiction needs a careful analysis, which is the goal of the present paper. We will see below that none of the two quadratic gravity alternatives regarding a Kasner solution can be true for f(T) cosmology where we meet a third, different situation.
We consider cosmological models in modified teleparallel gravity f(T), where (as in TEGR) the dynamical variables are tetrad fields \({\mathbf {e}_A(x^\mu )}\); here Greek indices are spacetime and capital Latin indices relate to the tangent spacetime. The metric tensor is given by \(g_{\mu \nu }=\eta _{\mathrm {AB}}\, e^A_\mu \, e^B_\nu ,\) where \(\eta _{\mathrm {AB}}=\mathrm {diag} (1,1,1,1)\).
It is easy to see that there is no de Sitter solution with \(H_a=H_b=H_c=H_{dS}=const\ne 0\) on this branch.
Since the constraint equation is an algebraic, but not a differential equation for T it is impossible to vary T, setting it to some nonzero value on the branch in question. So, all corrections to GR vanish on this branch of Bianchi I vacuum solutions. This situation has no analogs in both f(R) and Gauss–Bonnet gravity. Only if matter sources are taken into account, corrections to GR become nonvanishing (in this case T evidently is not zero). That is why the question of stability of a vacuum Kasner solution on the \(T=0\) branch is a meaningless question, because this branch contains no other vacuum solutions.
(2) The case of \(T^{N1}=\frac{1}{f_0(12 N)}=const\).
 (a)

if N is even, then \(f_0>0\),
 (b)

if N is odd, then \(f_0<0\).
To conclude we would like to underline on which points we agree and disagree with Ref. [27]. Since there are some subtleties, it is better to distinguish between the cases when the Kasner solution is an exact solution and an asymptotic solution.
We agree with [27] that the Kasner solution can be an exact solution in f(T) cosmology. However, it happens only on the \(T=0\) branch of vacuum solutions where the equations of motion coincide exactly with those for GR (any deviations from GR appears only when matter is taken into account). Therefore the question of the stability of the Kasner solution in this situation is meaningless—the Kasner solution is the general vacuum solution on the \(T=0\) branch.
On the contrary, the Kasner solution is an asymptotic solution on the second branch. In this case it is possible to ask for its stability, and we agree with [27] that it is unstable, and the cosmological evolution is generally directed from the Kasner to the de Sitter solution. However, in this situation the Kasner solution cannot be an exact solution of f(T) cosmology.
So, the statement that the Kasner solution satisfies the equations of motion in f(T) theory made in the Introduction of [27], and the statement that it is unstable (which is the main result of [27]) are both correct only when applied to appropriate branches of solutions. They are not correct for another branch. However, in [27] these statements have been formulated in general, without any reference to particular branches.
Our results can be understood in another way if we note that the vacuum equations of motion of the models \(f(T)=T+f_0T^N\) for the branch \(T=const\ne 0\) coincide with those of anisotropic Bianchi I models of GR with cosmological constant \(\Lambda \), where \(T>0\) corresponds to \(\Lambda <0\) (\(T<0\) corresponds to \(\Lambda >0\)). Therefore, a cosmological evolution is directed from a Kasner to a de Sitter solution in the case \(T<0\), and from one Kasner solution to the point of maximal expansion (see Fig. 7) and back to another Kasner solution in the case of positive T.
As for the statement of [24] that a vacuum solution can exist only for \(f(T)=\sqrt{T}\) theory, to our mind, it follows from an incorrect interpretation of the constraint equation (5). This equation may be considered as a differential equation which should get us such f(T) that the constraint is valid for all T. In this interpretation the authors of [24] are correct. However, this equation can be considered as an algebraic one which gives us some particular T for which the constraint equation has solutions. This situation is known in f(R) cosmology. The equation for a vacuum de Sitter solution \(2f(R)Rf'(R)=0\) when considered as a differential equation gives a particular \(f(R)=R^2\), for which a de Sitter solution with any R exists. On the contrary, the same equation being an algebraic equation gives for another f(R) some particular R for which a de Sitter solution exists. In f(R) theory this is restricted by de Sitter solutions only, because in a general situation there are other derivative terms. In f(T) the situation is general. If we search for any vacuum solution, there is no need for this solution to be valid for any T, so the constraint equation should be considered as an algebraic one.
We remind the reader that all results of the present paper have been obtained under the suggestion that the tetrad (1, a, b, c) is the proper tetrad for Bianchi I cosmology. Our results, showing that the Kasner solution is either the exact general solution or an asymptotic solution stable to the past, indicate that it is reasonable to search for an analog of BKL oscillations in anisotropic f(T) cosmology with spatial curvature. However, this needs to identify proper tetrads for other Bianchi metrics (or, in another formulation of the theory, correct nonzero spin connections for these cases), which is a separate interesting and currently unsolved problem of f(T) gravity.
Notes
Acknowledgements
The work was supported by RSF Grant No. 161210401 and by the Russian Government Program of Competitive Growth of Kazan Federal University.
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