Asymptotic cosmological regimes in scalar–torsion gravity with a perfect fluid
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Abstract
We consider the cosmological dynamics of a nonminimally coupled scalar field in scalar–torsion gravity in the presence of hydrodynamical matter. The potential of the scalar field have been chosen as power law with negative index, this type of potentials is usually used in quintessence scenarios. We identify several asymptotic regimes, including de Sitter, kinetic dominance, kinetic tracker, and tracker solutions and study the conditions for their existence and stability. We show that for each combination of coupling constant and potential power index one of the regimes studied in the present paper is stable to the future.
Keywords
Scalar Field Effective Potential Einstein Frame Asymptotic Regime Torsion Theory1 Introduction
Recently a new class of theories modifying General Relativity (GR) has started to attract great attention mainly in order to explain observation data indicating the accelerated expansion of the late Universe [1, 2, 3, 4, 5]. It is based on the rather old theory attributed to Einstein himself [6], however, forgotten for decades since it is in fact not a separate theory from the viewpoint of equation of motion, but rather a reformulation of GR. Instead of torsionfree LeviCivita connections it uses curvature free Weitzenböck connections [7]. It is well known that zerocurvature connections allow for the existence of a pathindependent definition of vector parallel transport, so this theory has got the name of the Teleparallel Equivalent of General Relativity (TEGR) [8, 9, 10, 11, 12]. What, however, has become clear rather recently is that the wellknown modifications of GR, such as a theory of a scalar field nonminimally coupled to gravity, when constructed on the base of TEGR, lead to different equations of motion (for the reason of this see [13]). Such a theory has been intensively investigated in many papers during several recent years.
Generalizations of TEGR are usually constructed in the same way as modifications of GR. It is possible to generalize the Lagrangian replacing in it the torsion scalar T with a function f(T) [14, 15, 16]. Different types of cosmological scenarios appear in scalar–torsion gravity, namely, in the class of models with nonminimal coupling between the torsion scalar and the scalar field \(\phi \) of the form \(\xi T F(\phi )\), where \(\xi \) is a coupling constant, \(F(\phi )\) is some function of a scalar field [17, 18, 19, 20, 21, 22, 23, 24, 25]. Recently some other modifications of TEGR including a nonminimal derivative coupling to torsion [26] and analogs of the Gauss–Bonnet invariant have appeared [27, 28, 29, 30]. There are also modifications of TEGR with no direct analogs of GR modification [13].
In our recent paper [31] we applied dynamical system methods to scalar–torsion theory in order to find some cosmological asymptotic regimes and describe the corresponding phase portraits, focusing mainly on growing scalar field potentials. The main qualitative result of that paper is that scalar–torsion coupling leads to much less variety of the possible dynamical regimes than scalar–curvature coupling [32, 33]. In the present paper we give an heuristic explanation of this feature, as well as show that more possibilities appears for decreasing scalar fields potentials, usually used in quintessence models. As such models have been created for a description of the late Universe, when the usual matter is important as well, we add hydrodynamical matter to the scalar field. We will use units with \(\hbar =c=1\).
The paper is organized as follows: in Sect. 2 we present the equation of motion in initial and expansion normalized variables, in Sect. 3 the analysis of stationary points and corresponding asymptotic solutions is given, and Sect. 4 gives a brief summary of the results obtained.
2 Main equations
2.1 The effective potential
In the standard scalar–curvature theory of a nonminimal coupling the conformal transformation to an Einstein frame is often used. In the Einstein frame the theory is equivalent to GR with a scalar field as a source, so the evolution of the scalar field in an expanding Universe is, as usual, directed to the minimum of the potential, which is not true in the initial nonminimal formulation, called the Jordan frame. This conformal transformation is usually divided into two steps: a redefinition of the potential via \(V_{\mathrm{eff}}(\phi )=V(\phi )/U^2(\phi )\), where \(U(\phi )=1+K\xi B(\phi )\) in our notations, and a redefinition of the scalar field in order to get the canonical kinetic term. The combination denoted here \(V_{\mathrm{eff}}(\phi )\) is rather interesting by itself. It is a conformal invariant, so it is not changed by any conformal transformation [34]. Moreover, it can give important information as regards the dynamics of the system without the second step to the Einstein frame (which is usually much more technically complicated than the very easily calculated first step). Namely, de Sitter solutions corresponds to the minima of the effective potential, their stability is determined by the sign of the second derivative of the effective potential in a standard way [33, 34].
2.2 Dimensionless variables
3 Stationary points and corresponding regimes
3.1 Stationary points analysis
 1.\(x=0\), \(y=1\), \(z=0\), \(m=0\), \(A={\frac{c}{K \xi }}\). We calculate the eigenvalues for the Jacobian matrix associated with the system (27) to findso this point is stable for nonphantom matter. For this point the quantity \(Y=\frac{\dot{H}}{H^2}=0\) is found using (26) and, therefore, \(\dot{H}=0\). Then we find the time dependence of the scale factor$$\begin{aligned} \lambda _1= & {} 3(1+\omega ),\nonumber \\ \lambda _{2,3}= & {} \frac{3}{2}\pm \frac{1}{2}\sqrt{924\xi (c+2)}, \end{aligned}$$(28)As the coordinate of this fixed point is \(A=\frac{N\phi ^N}{1+K\xi \phi ^N}\rightarrow {\frac{n}{K \xi }}\ne 0\), the scalar field approaches a constant$$\begin{aligned} a(t)=a_0 e^{H_0(tt_0)}. \end{aligned}$$(29)This is a de Sitter solution. The constants \(\phi _0\) and \(H_0\) are found with the substitution of this solution to the initial system of Eqs. (6)–(8):$$\begin{aligned} \phi =\phi _0. \end{aligned}$$(30)Clearly, this regime exists only for \(0>n>2\). Since it is stable it can be used for describing the latetime acceleration of our Universe. Definitely, the unnaturally low value of \(H_0\) (in natural units) should transform to a very low value of some of the parameters of the theory. Note, however, that in the theory under investigation we have more possibilities to get a very low \(H_0\)—this may happen due to either a very low \(V_0\), a very low \(\xi \) or a potential power index n being very close to \(2\). For the last possibility, if we denote \(n+2=\epsilon<< 1\), we get \(H_0 \sim \epsilon \sqrt{V_0 \xi }\), so small corrections to the \(\phi ^{2}\) potential would do the job.$$\begin{aligned} \phi _0=\pm \sqrt{\frac{n}{K\xi (2+n)}},\quad H_0=\pm \sqrt{\frac{nV_0{\phi _0}^{n2}}{6\xi }}. \end{aligned}$$
 2.\(x=1\), \(y=0\), \(z=0\), \(m=\sqrt{6\xi }\), \(A=\frac{2}{K\xi }\). The corresponding eigenvalues areUsing (26) the quantity Y at this stationary point is obtained to be \(Y_{\mathrm{stat}}=32\sqrt{6\xi }\) and now we can restore the time dependence of the Hubble parameter, \(H(t)=\frac{1}{Y_{\mathrm{stat}}(tt_0)}\), and the scale factor$$\begin{aligned} \lambda _1= & {} 2\sqrt{6\xi },\nonumber \\ \lambda _2= & {} 3(1\omega )+2\sqrt{6\xi },\nonumber \\ \lambda _3= & {} (c+2)\sqrt{6\xi }+6. \end{aligned}$$(31)The corresponding behavior of the scalar field is obtained applying the definition of the variable \(m=\frac{\dot{\phi }}{H\phi }\), then \(\frac{\dot{\phi }}{\phi }=m_{\mathrm{stat}}\frac{\dot{a}}{a}\), where \(m_{\mathrm{stat}}\) is the coordinate of a stationary point,$$\begin{aligned} a(t)=a_0{tt_0}^{\frac{1}{Y_{\mathrm{stat}}}}=a_0{tt_0}^{\frac{1}{3+2\sqrt{6\xi }}}. \end{aligned}$$(32)$$\begin{aligned} \phi (t)=\phi _0{tt_0}^{\frac{m_{\mathrm{stat}}}{Y_{\mathrm{stat}}}}=\phi _0{tt_0}^{\frac{\sqrt{6\xi }}{3+2\sqrt{6\xi }}}. \end{aligned}$$(33)
 3.\(x=1\), \(y=0\), \(z=0\), \(m=\sqrt{6\xi }\), \(A=\frac{2}{K\xi }\). The eigenvalues for this fixed point areWe get an asymptotic behavior a(t) and \(\phi (t)\) calculating the quantity \(Y_{\mathrm{stat}}=3+2\sqrt{6\xi }\) and using \(m=m_{\mathrm{stat}}=\sqrt{6\xi }\) at this point,$$\begin{aligned} \lambda _1= & {} 2\sqrt{6\xi },\nonumber \\ \lambda _2= & {} 3(1\omega )2\sqrt{6\xi },\nonumber \\ \lambda _3= & {} 6\sqrt{6\xi }(c+2). \end{aligned}$$(34)For \(\xi =\frac{3}{8}\) instead of this solution (35) we have the exponential one (because in this case \(Y=0\), \(H=H_0\), \(\frac{\dot{\phi }}{\phi }=m_{\mathrm{stat}}H_0\), and \(m_{\mathrm{stat}}=\frac{3}{2}\)),$$\begin{aligned} a(t)= & {} a_0{tt_0}^{\frac{1}{32\sqrt{6\xi }}},\nonumber \\ \phi (t)= & {} \phi _0{tt_0}^{\frac{\sqrt{6\xi }}{2\sqrt{6\xi }3}}. \end{aligned}$$(35)The points 2 and 3 represent a situation when the kinetic term of the scalar field dominates.$$\begin{aligned} a(t)= & {} a_0e^{H_0(tt_0)},\nonumber \\ \phi (t)= & {} \phi _0 e^{\frac{3}{2} H_0(tt_0)}. \end{aligned}$$(36)
 4.\(x=\frac{8\xi }{3(\omega 1)^2}\), \(y=0\), \(z=\frac{3+3\omega ^26\omega 8\xi }{3(\omega 1)^2}\), \(m=\frac{4\xi }{\omega 1}\), \(A=\frac{2}{K\xi }\). We find the eigenvalues for this point,As the quantity Y at this stationary point is \(Y_{\mathrm{stat}}=\frac{33\omega ^28\xi }{2(\omega 1)}\), we have$$\begin{aligned} \lambda _1= & {} \frac{3\omega ^23+4 c\xi }{\omega 1},\nonumber \\ \lambda _2= & {} \frac{3{(\omega 1)}^28\xi }{2(\omega 1)},\nonumber \\ \lambda _3= & {} \frac{8\xi }{1\omega }. \end{aligned}$$(37)The corresponding behavior \(\rho (t)\) is found using \(\dot{\rho }+3H\rho (1 +\omega )=0\), then \(\frac{\dot{\rho }}{\rho }=3(1+\omega )\frac{\dot{a}}{a}\) and$$\begin{aligned} a(t)=a_0{tt_0}^{\frac{2(\omega 1)}{3\omega ^23+8\xi }},\quad \phi (t)=\phi _0{tt_0}^{\frac{8\xi }{3\omega ^23+8\xi }}.\nonumber \\ \end{aligned}$$(38)This is a kinetic tracker regime when the potential term is negligible, and the ratio between the kinetic energy of the scalar field and the matter density remains constant. When \(\xi =\frac{3}{8}(1\omega ^2)\), the solution (38), (39) does not exist and the exponential one appears (\(m_{\mathrm{stat}}=\frac{3}{2}(1+\omega )\)),$$\begin{aligned} \rho (t)=\rho _0{tt_0}^{\frac{3(1+\omega )}{Y_{\mathrm{stat}}}}=\rho _0{tt_0}^{\frac{6(1\omega ^2)}{3\omega ^23+8\xi }}. \end{aligned}$$(39)where \({H_0}^2=\frac{4\rho _0}{9\phi _0^2\omega (1+\omega )}\) is found on substituting the exponential solution (40) into the initial system of equations (6)–(8).$$\begin{aligned} a(t)= & {} a_0 e^{H_0(tt_0)},\nonumber \\ \phi (t)= & {} \phi _0 e^{\frac{3}{2}(1+\omega )H_0(tt_0)},\nonumber \\ \rho (t)= & {} \rho _0 e^{3(1+\omega )H_0(tt_0)}, \end{aligned}$$(40)
 5.\(x=\frac{\xi {(c+2)}^2}{6}\), \(y=1\frac{\xi {(c+2)}^2}{6}\), \(z=0\), \(m=\xi (c+2)\), \(A=\frac{2}{K\xi }\). The eigenvalues are calculated,Similar to the previous points, using \(Y_{\mathrm{stat}}=\frac{1}{2}\xi (4c^2)\) and \(m_{\mathrm{stat}}=\xi (c+2)\), we find a(t), \(\phi (t)\) (\(c=n\)),$$\begin{aligned} \lambda _1= & {} 2\xi (c+2),\nonumber \\ \lambda _2= & {} 3+\frac{\xi {(c+2)}^2}{2},\nonumber \\ \lambda _3= & {} 3(1+\omega )+c\xi (c+2). \end{aligned}$$(41)$$\begin{aligned} a(t) = a_0{tt_0}^{\frac{2}{\xi (n^24)}},\quad \phi (t) = \phi _0{tt_0}^{{\frac{2}{2n}}}. \end{aligned}$$(42)
 6.\(x=\frac{3{(1+\omega )}^2}{2\xi c^2}\), \(y={\frac{33\omega ^24 c\xi }{2\xi c^2}}\), \(z=\frac{c^2\xi 33\omega +2 c\xi }{\xi c^2}\), \(m=\frac{3(1+\omega )}{c}\), \(A=\frac{2}{K\xi }\). We obtain the eigenvalues for this point,where \(f_1(\xi ,c)=27 c^2\xi 12\xi 60 c\xi +72\), \(f_2(\xi ,c)=24\xi 72+96 c\xi +6 c^2\xi \), and \(f_3(\xi ,c)=21 c^2\xi 72+156 c\xi 32\xi ^2 c^312\xi 64 c^2\xi ^2\). The quantities \(Y_{\mathrm{stat}}=\frac{3}{2 c}(1+w)(2c))\) and \(m_{\mathrm{stat}}=\frac{3(1+\omega )}{c}\) give us the behavior a(t), \(\phi (t)\), and \(\rho (t)\) (\(c=n\)),$$\begin{aligned} \lambda _1= & {} \frac{6(1+\omega )}{c},\nonumber \\ \lambda _{2,3}= & {} \frac{1}{4\xi c}\Bigg ( 3\xi (2c+\omega (c+2))\nonumber \\&\pm \sqrt{3\xi (72\omega ^3+f_1(\xi ,c)\omega ^2+f_2(\xi ,c)\omega +f_3(\xi ,c))}\Bigg ),\nonumber \\ \end{aligned}$$(43)This is a tracker when the potential and kinetic terms of the scalar field as well as the density of the matter are constant with respect to each other.$$\begin{aligned} a(t)= & {} a_0{tt_0}^{\frac{2 n}{3(1+\omega )(n2)}},\nonumber \\ \phi (t)= & {} \phi _0{tt_0}^{\frac{2}{2n}},\nonumber \\ \rho (t)= & {} \rho _0{tt_0}^{\frac{2 n}{2n}}. \end{aligned}$$(44)
3.2 Conditions of existence and stability of obtained solutions
With the substitution of the powerlaw solution \(a(t)=a_0{(tt_0)}^\alpha \), \(\phi (t)=\phi _0{(tt_0)}^{\alpha \beta }\), and \(\rho (t)=\rho _0(t\,t_0)^{3(1+\omega )\alpha }\) to the initial system of equations (6)–(8), where \(B(\phi )=\phi ^2\), \(V(\phi )=V_0\phi ^n\), we find the conditions of the existence and stability of the six solutions obtained from the previous subsection. We consider only the cases with \(n<0\), \(\xi >0\), and only positive values of \(\phi \). It is worth to recall that in general a fixed point does not necessary correspond to some solution of the initial system. First of all, all regimes found above (except the de Sitter solution, which is an exact, and not only an asymptotic solution) exist in the \(\phi \rightarrow \infty \) limit where we can neglect the Einstein term in comparison with the term originating from a nonminimal coupling in the denominator of the expansion in normalized variables we use. When the field is small (which is realized near a cosmological singularity for decreasing powerlaw potentials) the Universe expands according to GR, since the correction terms are less important. Such regimes for quintessence potentials are well known and are not included in our analysis here. As for the large \(\phi \) regimes, studied in the present paper, we require that the omitted terms in the equations of motion be negligible for large \(\phi \). Otherwise the regime cannot be realized as an asymptotic solution. For example, suppose we have a vacuum asymptotic solution. If the influence of matter grows with growing \(\phi \), this means that the asymptotic solution corresponding to this particular expansion dynamics is absent if we add any amount of matter. This does not mean that such a regime has no physical meaning at all—the Universe may follow it as transient one, if the amount of matter is small enough. However, such a situation does not belong to the asymptotic regimes, and we will not list it in the following.
Conditions of existence and stability of solutions in vacuum
No. of point  Conditions of existence  Type of stability 

1.  \(2<n<0\)  \(0<\xi \leqslant \frac{3}{8(n+2)}\) 
Stable node  
\(\xi >\frac{3}{8(n+2)}\)  
Stable focus  
2.  \(n<2\), \(\xi >\frac{6}{{(n+2)}^2}\), \(t\rightarrow \infty \)  Stable node 
3.  \(2<n<0\), 1. \(0<\xi <\frac{3}{8}\), \(t\rightarrow t_0\)  Unstable 
2. \(\frac{3}{8}<\xi <\frac{6}{{(n+2)}^2}\), \(t\rightarrow \infty \)  node  
\(n\leqslant 2\), 1. \(0<\xi <\frac{3}{8}\), \(t\rightarrow t_0\)  
2. \(\xi >\frac{3}{8}\), \(t\rightarrow \infty \)  
\(n<0\), \(\xi =\frac{3}{8}\), \(t\rightarrow \infty \)  
5.  \(2<n<0\), \(0<\xi <\frac{6}{{(n+2)}^2}\), \(t\rightarrow \infty \)  Saddle 
\(n<2\), \(0<\xi <\frac{6}{{(n+2)}^2}\), \(t\rightarrow \infty \)  Stable node 
Conditions of existence and stability of solutions for \(\rho \ne 0\)
No. of point  Conditions of existence  Type of stability 

1.  \(2<n<0, \omega \in (1;1]\)  \(0<\xi \leqslant \frac{3}{8(n+2)}\) 
Stable node  
\(\xi >\frac{3}{8(n+2)}\)  
Stable focus  
3.  \(n<0\), 1. \(\omega \in [1;0)\),  Unstable 
\(0<\xi <\frac{3{(1\omega )}^2}{8}\),  node  
\(t\rightarrow t_0\)  
2. \(\omega \in [0;1)\),  
\(\frac{3}{8}<\xi <\frac{6}{{(n+2)}^2}\),  
\(t\rightarrow \infty \)  
\(n<0\), \(\xi =\frac{3}{8}\), \(\omega \in [1;0)\),  
\(t\rightarrow \infty \)  
4.  \(n<0\), \(0<\xi <\frac{3(1\omega ^2)}{8}\),  Unstable 
\(\omega \in (1;1)\), \(t\rightarrow t_0\)  node  
\(n<0\), \(\xi =\frac{3}{8}(1\omega ^2)\),  Saddle  
\(\omega \in (1;0)\), \(t\rightarrow \infty \)  
5.  \(n<2\), \(0<\xi <\frac{3(1+\omega )}{n(n+2)}\),  Stable node 
\(\omega \in (1;1]\), \(t\rightarrow \infty \)  
6.  \(n<2\), \(\xi >\frac{3(1+\omega )}{n(n+2)}\),  \(\frac{3(1+\omega )}{n(n+2)}<\xi \leqslant \xi _0\) 
\(\omega \in (1;1]\), \(t\rightarrow \infty \)  Stable node  
\(\xi >\xi _0\)  
Stable focus 

For \(n>2\) a de Sitter solution exists and is stable.

For \(n<2\) and \(\xi < 6/{(n+2)}^2\) the regime of point 5 is stable.

For \(n<2\) and \(\xi >6/{(n+2)}^2\) the kinetic dominating regime of point 2 is stable.
In Table 2
\(\xi _0=\frac{3}{64n^2(n+2)}(f_4(\xi ,n)+\sqrt{f_5(\xi ,n)f_6(\xi ,n)})\), where \(f_4(\xi ,n)=52n7n^2+4+\omega ^2(9n^2+20n+4)+2\omega (416nn^2)\), \(f_5(\xi ,n)=\omega ^2(4+36n+81n^2)+\omega (126n^264n+8)+4100n+49n^2\), \(f_6(\xi ,n)={(2n+\omega (n+2))}^2\).
For \(\xi =\xi _0\) the square root equals zero in the eigenvalues \(\lambda _{2,3}\) of point 6 [see (43)].

For \(n>2\), as for the vacuum case, the future stable point is a de Sitter solution.

For \(n<2\) and \(\xi <3(1+\omega )/(n^2+2n)\) the regime of point 5 is stable, so the scalar field dominates in late time.

For \(n<2\) and \(\xi >3(1+\omega )/(n^2+2n)\) the tracker (point 6) is stable to the future.
4 Conclusions
In the present paper we considered the evolution of the Universe in a teleparallel version of a nonminimally coupled scalar field theory, focusing mostly on quintessence scalar field potentials. In our previous paper [31] we already pointed out the radical differences in the dynamics of “classical” and teleparallel cases of this theory. Here we provided a qualitative explanation of this difference using the notion of an effective scalar field potential. It appears as a first step from a Jordan to an Einstein frame (to be followed by a redefinition of the scalar field in order to transform the kinetic term to the canonical form) in the standard theory, though it already gives an intuitively clear picture of possible dynamical regimes without this second step. It is interesting that in the teleparallel version, where the Einstein frame does not exist at all [35], the effective potential nevertheless can be introduced in the same manner. The difference in the forms of this effective potential (being \(V(\phi )/U^2(\phi )\) for the classical case and \(V(\phi )U(\phi )\) for the teleparallel case, where \(V(\phi )\) is the scalar field potential and \(U(\phi )\) is the coupling function which includes the Einstein–Hilbert term) results in very different dynamics for the same scalar field potentials.
For the standard scalar–curvature coupling, the presence of \(U(\phi )\) [which is usually taken in the form \(U(\phi )=1+8\pi G\xi \phi ^2\)] in the denominator leads to a runaway solution for growing \(V(\phi )\), corresponding to \(V_{\mathrm{eff}}(\phi )\) decreasing for large \(\phi \). This may happen if \(U^2(\phi )\) grows more rapidly than V with \(\phi \) [33]. In the boundary case of \(U^2(\phi ) \sim V(\phi )\) for large \(\phi \) the effective potential is asymptotically flat, which is the reason for the existence of viable Higgs inflation models [37].
None of these features can exist for the effective potential in the form of \(U(\phi )V(\phi )\) as in scalar–torsion theory. For the usual quadratic coupling (we study only this form of the coupling in the present paper) \(V_{\mathrm{eff}}(\phi )\) is growing for an arbitrarily growing \(V(\phi )\), so the dynamics inevitably pushes the scalar field toward zero, as we have seen in our previous paper. On the contrary, for the decreasing potentials studied in the present paper the effective potential can have a minimum, resulting in a de Sitter solution. For a quadratic coupling this happens if \(V(\phi )\) scales less steep than \(\phi ^{2}\). This condition for a de Sitter solution to exist has been obtained earlier in [31] and in a more general context in [36], and we now have a very easy way to explain it.
Steeper potentials result in unlimited growing of the scalar field, similar to the quintessence scenario in GR. We have identified a tracker solution which is always an attractor in its range of existence. It is shown that when the coupling constant \(\xi \) is low enough for the tracker to exist (see the corresponding condition in Table 2), a vacuum regime is stable, so the scalar field dominates at late time. We also list several unstable asymptotic regimes. A full description of the scalar field dynamics requires matching solutions studied in the present paper with GR solutions valid for small \(\phi \). We leave this for future work.
Notes
Acknowledgments
The work was supported by RSF Grant No. 161210401 and by the Russian Government Program of Competitive Growth of Kazan Federal University. Authors are grateful to Emmanuel Saridakis for discussions.
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