Bouncing solutions in $f(T)$ gravity

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.


Introduction
Modified gravity can lead to some kinds of cosmological dynamics which are impossible in general relativity (GR), at least for usual matter content of the Universe, like a perfect fluid with positive energy density. One of well-known examples is the so-called non-standard singularity where the scale factor a, Hubble parameter H and the matter energy density ρ remain constant, whileḢ 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: 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 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

The equations of motion
In the present paper we consider the cosmological models in f (T ) gravity with matter. The action of this theory is where e = det(e A µ ) = √ −g is the determinant, which consists of the tetrad components e A µ , f (T ) is a general differentiable function of the torsion scalar T , S m is the matter action and K = 8πG. Here the units = c = 1 are used.
The line element of a non-flat homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker Universe is where a(t) is the scale factor, the parameter κ = 1 for ta closed Universe and κ = i for an open Universe. We write down the following diagonal tetrad (see [18], [31]), which relates to the metric (2): where for κ = 1 E 1 = − cos θdψ + sin ψ sin θ(cos ψdθ − sin ψ sin θdϕ), − sin ψ (sin ψ sin ϕ − cos ψ cos θ cos ϕ)dθ + (cos ψ sin ϕ + sin ψ cos θ cos ϕ) sin θdϕ , − sin ψ (sin ψ cos ϕ + cos ψ cos θ sin ϕ)dθ + (cos ψ cos ϕ − sin ψ cos θ sin ϕ) sin θdϕ (4) and for κ = i E 1 = cos θdψ + sinh ψ sin θ(− cosh ψdθ + i sinh ψ sin θdϕ), E 2 = − sin θ cos ϕdψ+ + sinh ψ (i sinh ψ sin ϕ − cosh ψ cos θ cos ϕ)dθ + (cosh ψ sin ϕ + i sinh ψ cos θ cos ϕ) sin θdϕ , In what follows we will use the more convenient curvature parameter k, such that k = 1 for a closed Universe and k = −1 for an open Universe. The torsion scalar for the chosen tetrad (3) is where H a ≡˙a a is the Hubble parameter, a dot denotes the derivative with respect to time. Then the time derivative of the torsion scalar has the forṁ The equations of motion can be found by varying the action (1) with respect to the chosen tetrad (3) (see, [31], [32]) where ρ is the matter energy density, p is its pressure, the equation of state is p = wρ where w is a constant. In the present paper we consider only usual matter, so the equation of state parameter is bounded within the region of thermodynamical stability of the matter w ∈ [−1; 1]. Here The continuity equation for matter iṡ For the case of f (T ) = T + f 0 T N the equations of motion (8), (9) have the following form: In the present paper we consider only integer N > 1. Equation (11) tells us now that a bounce cannot occur in a flat Universe with ordinary matter, since for the flat metric T = 0 at the point of bounce, and regularity of f T (0) indicates that left-hand side of Eq. (11) vanishes at the bounce, which is incompatible with positivity of ρ.

Bouncing solutions and a static Universe
Looking at Eq. (11) we can see two important points. First, there are curvature-containing terms, originating from the last term in the right-hand side of (11) which grow faster than 1/a 2 near the singularity a = 0. The fastest term is equal to (6k) N /a 2N , and this is the only correction term which does not vanish at a bounce. Thus, this term drives the bounce which we study in the present paper. Note that we need both f (T ) corrections and a spatial curvature k = 0 for this type of bounce. The change of power index from 2 for GR curvature term to 2N for the curvature f (T )-corrections term leads to the following: already in quadratic gravity the influence of the corrections of the curvature term may overcome the influence of matter near a singularity for w < 1/3 instead of w < −1/3 in GR, and starting from N = 3 this property covers the whole allowed range w ∈ [−1; 1]. This can induce bounces for a wider range of w than in GR; however, since there are also other terms in (11), this possibility is not satisfied automatically and needs further analysis. Second, the sign of the fastest growing curvature term is equal to k N , so it is the same for positive and negative curvature if N is even. This means that we can expect bouncing solutions in quadratic gravity even for k = −1. Motivated by this qualitative considerations, we start our analysis by analytical methods. In the next section we present results of numerical investigations.
We can expressḢ from the equation of motion (12) for the model f (T ) = T + f 0 T N : Taking into account (6) and (11) we finḋ From this equation we can finally exclude ρ using (11) The equation forȧ is obviouslyȧ = aH.
The dynamical system (15), (16) has one stationary point with the following coordinates: It follows from the expression for the coordinate a 0 that this point exists for We substitute the coordinates of the fixed point H = 0, a = a 0 to the constraint equation (11) and taking into account positivity of the matter energy density ρ 0 it is easy to get Uniting conditions (17), (18) we find finally that there are three different cases for which the static cosmological solution exists: 3 ; 1 . Note that the negative curvature case is realized only for N = 2 with w ∈ (1/3; 1]. Formally, similar solutions can exist for other even power indices, however, this requires an exotic matter with w > 1. The eigenvalues for the Jacobian matrix associated with the system (15), (16) in this critical point are From this expression we can see that the obtained critical point is either a center (for the cases 1 and 3) or a saddle (for the case 2). In the case of a center the static solution is stable.
However, due to properties of a center fixed point the nearby trajectories do not converge to the static solution, but rotate around it, realising oscillating solutions. The question of how close to the static solution should a trajectory be to be trapped into infinite oscillations cannot be solved in a local analysis and requires numerical investigations, which are described in the following section. Now we find the conditions of the existence of a bounce substituting H = 0 to (11), (15).
Applying the conditionsḢ > 0 and ρ 0 to (20), (21) we see that the bounces exist for where a). f 0 > 0 for w − 1 3 and b). f 0 can be positive as well as negative for w < − 1 3 .
where a). f 0 > 0 for even N and b). f 0 < 0 for odd N .
We can see that for the positive spatial curvature bounces exist in a larger range of w than the range where a static solution (stable or unstable) exists. In the case of a negative spatial curvature the conditions on the equation of state for bounce and static solutions to exist coincide for even N . As for the case of odd N , it always requires an exotic matter with w > 1 and is not considered in the present paper.
The case N = 2 and positive f 0 is shown in Fig. 1. The left panel corresponds to the case of w = 0 when stable stationary solution exists. In the vicinity of this solution trajectories move around it. Note, however, that this happens only for nearby trajectories, and the fate of trajectories outside of this basin is different. If a trajectory starts from a rather big scale factor, instead of bouncing due to the presence of an 1/a 4 term, it meets a non-standard singularity. In the non-standard singularityḢ diverges, while H and a are finite, so in the a(H) plot it corresponds to the sudden disappearance of the phase trajectory. This means that the possibility of a bounce is not realized for a contracting Universe that is big enough initially.
The right panel of Fig. 1 presents the situation for w = 1/3 when the stationary solution does not exist. Bouncing solutions disappeared as well. However, this does not change much the fate of a contracting large Universe, which still ends its evolution in a non-standard singularity.
Negative f 0 leads to instability of the stationary solution if it exists (Fig. 2, left panel, w = −0.5). A contracting Universe either experiences a bounce or falls into a standard singularity. The right panel represents the phase portrait without the stationary solution (w = 0). Since the conditions for a bounce are less restricted than the conditions for the stationary solution to exist, bounces are still possible. However, all trajectories with bounces begin and end in a non-standard singularity, and the fate for a big enough contracting Universe is unique -a standard singularity.
The two following plots show examples with an odd N . The stationary solution exists for N = 3, f 0 > 0, w > −1/3 (Fig. 3, right panel). A big contracting Universe ultimately falls into a standard singularity. An example of a phase portrait without stationary solutions is shown in the left panel.
Negative f 0 leads either to an unstable stationary solution (Fig. 4, left panel) or to dynamics without stationary solutions (Fig. 4, right panel). In the former case a big enough Universe can either go through a bounce or meet a non-standard singularity, in the latter case a non-standard singularity is the unique possibility.
The phase diagram for the case of N = 2 and a negative spatial curvature is presented in Fig. 5. It us easy to see from (11) that the maximum value of the scale factor at the point H = 0 is equal to a 2 max = 6f 0 . Larger values of the scale factor cannot correspond to its extremum. The allowed zone in the (a, H) plane is divided into three separate regions. Any big contracting Universe ends its evolution in a non-standard singularity, and we have now a simple criterion for a "big" Universe: a > a max .

Conclusions
In the present paper we have considered some peculiarities of a spatially curved isotropic Universe in f (T ) gravity. The function f (T ) has been chosen in the form f (T ) = T + f 0 T N ; the results obtained crucially depend on the parity of N and the sign of f 0 . One of the most interesting features of the dynamics is the existence of a stable static solution. For positive spatial curvature it exists for a rather wide interval of the equation of state parameter w, and for N > 3 it covers the whole thermodynamically possible interval of decelerating expansion w ∈ (1/3; 1]. Moreover, for even N and positive f 0 the stable static solution exists for a negatively curved Universe also; in this case the matter should be stiffer than an ultra-relativistic matter.
The conditions for possible bouncing solutions appear to be much wider than the condition w < −1/3 in GR. The stable static solution implies bounces, because nearby trajectories rotate around it, experiencing one bounce and one recollapse point per period. For a positive curvature bounces exist for a wider range of w than the static solutions. For negative spatial curvature with an even N the two conditions are the same. An odd N with negative spatial curvature needs an exotic matter with w > 1 for bounce.
Global analysis with numerical integration of the equations of motion and constructing phase portraits show, however, that the significance of bounce solutions is less than may be thought using the results of local analysis only. All phase portraits show a common feature: any trajectory going through a bounce driven by a correction term then experiences either recollapse or a non-standard singularity rather soon. For values of the coupling constant f 0 of the order of unity, studied in the present paper, the maximum value of the scale factor is of the same order of magnitude as the scale factor in the bounce point. On the other side, this means that a contracting Universe with large enough initial scale factor can go through a bounce only if w lies within the GR allowed interval w < −1/3. For stiffer matter bouncing trajectories cannot reach the low-curvature regime of a relatively big Universe. Further studies are needed to correctly quantify this qualitative result and determine how the maximum value of the scale factor after a bounce depends on f 0 . Currently, it seems highly unlikely that new bounce solutions can be important for the future of our "big" Universe if it starts at some time to contract.
For the negative curvature case the structure of the phase space is different and allows for a simple analytical expression of the maximal scale factor after bounce. The phase source is divided into three disconnected zones, one zone in the small scale factor range where trajectories move around static solution, and two symmetric zones in the large scale factor range where evolution of scale factor is monotonous. Since all bounces are located in the first zone, a Universe large enough to be located in the second zone cannot experience any bounce. The maximum scale factor of the first zone is a max = √ 6f 0 , giving an upper bound for the scale factors of bouncing trajectories.