1 Introduction

In [1] Kantowski and Sachs, exact solutions of the field equations in General Relativity with a spatially homogeneous, anisotropic and irrotational spacetime were investigated. The spacetime admits a four-dimensional isometry group transitive on three-dimensional spacelike hypersurfaces. The isometries comprise a translation symmetry and a three-dimensional subgroup the orbits of which are two-dimensional surfaces of constant curvature. The exact solution determined in [1] is a non-vacuum solution for which the energy momentum tensor in the Einstein field equations describes a dust fluid source. An important characteristic of the Kantowski–Sachs spacetimes is that they can be seen as extended Schwarzschild manifolds [2].

The Kantowski–Sachs spacetime is related with two other line elements of cosmological interest. Indeed, from the line element of the locally rotationally symmetric mixmaster universe, that is, the Bianchi IX universe, after a Lie contraction the Kantowski–Sachs element is recovered. Furthermore, when the shear vanishes in the latter spacetime, that is, the spacetime is isotropic, the spacetime describes the closed Friedmann–Lemaître–Robertson–Walker (FLRW) space [3]. In [4] it was found that the Kantowski–Sachs universe becomes isotropic when a cosmological constant term is introduced into the field equations [4], similarly to the result found by Wald in [5]. Moreover, the introduction of a positive cosmological constant indicates the existence of an initial singularity for the spacetime which evolves to the de Sitter universe [6, 7]. In [8] it was found that the cosmic no-hair conjecture is widely valid for Kantowski–Sachs geometries which asymptotically approach the de Sitter universe

Additionally, the dynamics of Kantowski–Sachs model can describe the time-evolution for the physical variables in anisotropic and inhomogeneous models [9]. There exist a family of Szekeres solutions [10] which describe inhomogeneous Kantowski–Sachslike geometries without any isometry [11]. Kantowski–Sachs geometry can be used for the description of the universe in the very early stages and near to the cosmological singularity. Thus, quantum gravity in this specific anisotropic geometry has been widely studied [12,13,14,15]. There is a plethora of important results in the literature for the Kantowski–Sachs geometry with other kinds of matter sources, exact solution with a radiation fluid was found earlier in [16], while a stiff fluid source was considered in [17] and later in [18] in wh the scalar field was assumed to be that of a massless scalar field. See also [19,20,21,22,23,24,25] and references therein.

The cosmological constant in the context of General Relativity is a simple dynamical approach in order to reproduce expansion in the cosmological parameters. Nevertheless, the cosmological constant suffers from two major problems [26, 27]. Consequently, cosmologists have considered other ways to explain the early-time and late-time acceleration phases of the universe [28]. There are various approaches in the literature for the description of the observed phenomena. Modified theories of gravity form a large family of cosmological models in which the acceleration is attributed to geometrodynamical of freedom [29]. The latter follow from the new invariant functions which are introduced to modify the Einstein–Hilbert Action Integral. The modification of the gravitational Action Integral with the introduction of the squared Ricci scalar term is a simple modification which can provide a simple mechanism for the explanation of the inflationary epoch [30, 31]. This inflationary model belongs to a more general family of theories known as \(f\left( R\right) \)-theory of gravity [32,33,34,35].

Apart from the Ricci scalar there are other invariants which have been introduced to modify the gravitational theory [36,37,38,39,40]. The Ricci scalar is constructed by the Levi–Civita connection, which is the fundamental geometric object of General Relativity. However, a more general connection can be considered as the basis of the gravitational theory. The Weitzenböck connection [41] is a curvature-less connection which leads to the Teleparallel Equivalent of General Relativity (TEGR) [42]. In TEGR the gravitational Lagrangian is defined by the torsion scalar T [43]. On the other hand, a torsion-free connection which describes a flat geometry and nonmetricity scalar Q leads to the so-called Symmetric Teleparallel General Relativity (STGR) [44]. The Kantowski–Sachs geometry has also been investigated and in modified theories of gravity [45,46,47,48,49,50].

In this piece of study we are interested on the evolution of the cosmological parameters in a Kantowski–Sachs background geometry in scalar-torsion theory [51]. Scalar-torsion is the analogue of scalar-tensor theory [52] in teleparallelism [53], in which a non-minimally coupled scalar field is introduced into the gravitational Action Integral and it is coupled to the torsion scalar T. Scalar-torsion is a theory of special interest in cosmological studies because it provides a geometric mechanism for the explanation of the acceleration phases of the universe [54, 55], as a unification in the dark sector components of the universe [56]. While there seem to be many similarities between the scalar-torsion and the scalar-tensor theories, in terms of background dynamics [57, 58], the two theories are complete different [59,60,61]. Recently, in [62] with the use of the Noether symmetry analysis new analytic and exact solutions for the field equations for the classical and quantum level in scalar-torsion theory were derived for a Kantowski–Sachs geometry. In the following, we present a detailed analysis of the evolution of the dynamics. We study the phase-space of the cosmological parameters by determining the stationary points for the field equations and investigating their stability properties. This approach has been widely studied in various theories and it has provided important information for cosmological evolution [63,64,65,66,67,68]. Homogeneous and anisotropic spacetimes have not been widely studied in the literature. Such an analysis is important in order to understand the dynamical effects of scalar-torsion in the very early universe. The structure of the paper is as follows.

In Sect. 2 we present the basic properties and definition of teleparallelism, and we give the Action Integral of the gravitational theory that we discuss, the scalar-torsion theory. The Kantowski–Sachs geometry is considered in Sect. 3. We select a specific frame for the vierbein fields in which the limit of General Relativity is recovered. The field equations are determined. Moreover, we rewrite the field equations with the use of a new set of dimensional variables. Hence, the field equations are written in the equivalent form of an algebraic-differential system. The dynamical analysis is presented for two forms of the scalar field potential, the exponential potential and the power-law potential. The results are given in Sects. 4 and 5, respectively. Finally, in 6 we summarize our results and we draw our conclusions.

2 Teleparallel theory of gravity

Consider now the vierbein fields \({{\textbf{e}}_{\mu }(x^{\sigma })}\) with commutator relations \([e_{\mu },e_{\nu }]=c_{\nu \mu }^{\beta }e_{\beta }~\)  where \(c_{\left( \nu \mu \right) }^{\beta }=0.\) The vierbein fields form a basis for the tangent space at each point P with the relation \((e_{\mu },e_{\nu })={\textbf{e}}_{\mu }\cdot {\textbf{e}}_{\nu }=g_{\mu \nu }\). In the nonholonomic coordinates we consider the covariant derivative \(\nabla _{\mu }\) defined with the connection

$$\begin{aligned} \mathring{\Gamma }_{\nu \beta }^{\mu }=\{_{\nu \beta }^{\mu }\}+\mathring{\Gamma }_{\nu \beta }^{\mu }, \end{aligned}$$
(1)

where \(\{_{\nu \beta }^{\mu }\}\) is the symmetric Levi–Civita and

$$\begin{aligned} \mathring{\Gamma }_{\nu \beta }^{\mu }=\frac{1}{2}g^{\mu \sigma }(c_{\nu \sigma ,\beta }+c_{\sigma \beta ,\nu }-c_{\mu \beta ,\sigma }), \end{aligned}$$
(2)

where \(\mathring{\Gamma }_{\nu \beta }^{\mu }\) are antisymmetric in the two first indices, that is \(~\mathring{\Gamma }_{\mu \nu \beta }=-\mathring{\Gamma }_{\nu \mu \beta },~\mathring{\Gamma }_{\mu \nu \beta }=\eta _{\mu \sigma }\mathring{\Gamma }_{\nu \beta }^{\mu },\) and it is related to the commutator relations of the vierbein fields.

In teleparallelism the Weitzenböck connection is considered [41], which means that the connection defines the flat space, i.e., \({\textbf{e}}_{\mu }\cdot {\textbf{e}}_{\nu }=\eta _{\mu \nu }\), where now definition (2) is

$$\begin{aligned} \mathring{\Gamma }_{\nu \beta }^{\mu }=\frac{1}{2}\eta ^{\mu \sigma }(c_{\nu \sigma ,\beta }+c_{\sigma \beta ,\nu }-c_{\mu \beta ,\sigma }). \end{aligned}$$
(3)

We define the nonnull torsion tensor from the relation

$$\begin{aligned} T_{\mu \nu }^{\beta }=2\mathring{\Gamma }_{\left[ \nu \mu \right] }^{\beta } \end{aligned}$$
(4)

and the torsion scalar [43]

$$\begin{aligned} T=\frac{1}{2}({K^{\mu \nu }}_{\beta }+\delta _{\beta }^{\mu }{T^{\theta \nu }} _{\theta }-\delta _{\beta }^{\nu }{T^{\theta \mu }}_{\theta }){T^{\beta }}_{\mu \nu } \end{aligned}$$
(5)

in which \(K_{~~~\beta }^{\mu \nu }=-\frac{1}{2}({T^{\mu \nu }}_{\beta }-{T^{\nu \mu } }_{\beta }-{T_{\beta }}^{\mu \nu })\) is the contorsion tensor that equals the difference between the connections in the holonomic and the non-holonomic frame.

The fundamental Action Integral in teleparallelism is [42]

$$\begin{aligned} S_{T}=\frac{1}{16\pi G}\int d^{4}xeT~+S_{m}, \quad e=\det (e_{\mu }), \end{aligned}$$
(6)

where \(e=\det (e_{\mu })\) is the determinant of the vierbein fields and \(S_{m}\) is the Action Integral corresponding to the matter source. The matter source can be a dust fluid, radiation, Chaplygin gas, the cosmological constant, scalar field and many others. In the latter cases, the gravitational field equations which follow from the variation of (6) are equivalent with those of General Relativity.

2.1 Scalar-torsion theory

Inspired by the scalar-tensor theory which is defined by Mach’s principle, consider now a nonminimally coupled scalar field \(\phi \) which interact with the Lagrangian of TEGR. The resulting Machian teleparallel gravitational theory is defined by the Action Integral [55]

$$\begin{aligned} S=\frac{1}{16\pi G}\int d^{4}xe\left[ {\hat{F}}\left( \psi \right) T+\frac{1}{2}\psi _{;\mu }\psi ^{\mu }+{\hat{V}}\left( \psi \right) \right] \end{aligned}$$
(7)

and it is known as scalar-torsion theory. \({\hat{V}}\left( \psi \right) \) is the scalar field potential and \({\hat{F}}\left( \psi \right) \) is the coupling function of the scalar field with the torsion scalar.

Without loss of generality we can perform a change of variables \(\psi \rightarrow \phi \), such that we can write the Action Integral (7) in the following form

$$\begin{aligned} S=\frac{1}{16\pi G}\int d^{4}xe\left[ F\left( \phi \right) \left( T+\frac{\omega }{2}\phi _{;\mu }\phi ^{\mu }+V\left( \phi \right) \right) \right] , \end{aligned}$$
(8)

\({\hat{V}}\left( \psi \right) =F\left( \phi \right) V\left( \phi \right) \), and \(d\psi =\sqrt{\omega F\left( \phi \right) }d\phi \). The parameter \(\omega \) is a constant parameter similar to the Brans–Dicke parameter [52]. From (8) we conclude that scalar-torsion theory is a second-order theory of gravity.

Scalar-tensor theory is equivalent with a scalar field in Einstein-frame under the application of a conformal transformation. That property is not true for the scalar-torsion theory. Consequently, the scalar-tensor and scalar-torsion theories are not related through a conformal transformation. It is known that the additional degrees of freedom in \(f\left( R\right) \)-gravity can be attributed to a scalar field non-minimally coupled to the Ricciscalar, the equivalent higher-order theory in teleparallelism is the \(f\left( T,B\right) \)-gravity, where B is the boundary term and differentiates the torsion T and the Ricci scalar [69, 70].

3 Kantowski–Sachs cosmology

The Kantowski–Sachs geometry in the Misner-like variables is described by the line element

$$\begin{aligned}{} & {} ds^{2}=-N^{2}\left( t\right) dt^{2}\nonumber \\ {}{} & {} \quad +e^{2\alpha \left( t\right) }\left( e^{2\beta \left( t\right) }dx^{2}+e^{-\beta \left( t\right) }\left( dy^{2}+f^{2}\left( y\right) dz^{2}\right) \right) , \end{aligned}$$
(9)

where \(N\left( t\right) \) is the lapse function, \(\alpha \left( t\right) \) is the scale factor which describes the size of the three-dimensional hypersurface and \(\beta \left( t\right) \) is the anisotropic parameter.

The definition of the proper vierbein fields is essential in order that the limit of General Relativity be recovered. If we follow the definition given in [71, 72], we see that the limit of General Relativity does not exist.

We follow [62] and we assume the vierbein fields

$$\begin{aligned} e^{1}&=Ndt ,\\ e^{2}&=e^{a+\beta }\cos z\sin y~dx+e^{a-\frac{\beta }{2}}\\ {}&\quad \left( \cos y\cos z~dy-\sin y\sin z~dz\right) ,\\ e^{3}&=e^{a+\beta }\sin y\sin z~dx+e^{a-\frac{\beta }{2}}\\ {}&\quad \left( \cos y\sin z~dy-\sin y\cos z~dz\right) ,\\ e^{4}&=e^{a+\beta }\cos y~dx-e^{a-\frac{\beta }{2}}\sin y~dy. \end{aligned}$$

Hence, the torsion scalar for the line element (9) with the use of these vierbein fields is calculated

$$\begin{aligned} T=\frac{1}{N^{2}}\left( 6{\dot{\alpha }}^{2}-\frac{3}{2}{\dot{\beta }}^{2}\right) -2e^{-2\alpha +\beta }. \end{aligned}$$
(10)

Thus, with the use of (10) and by assuming that the scalar field inherits the symmetries of the background space the Action Integral (8) is

$$\begin{aligned} S= & {} \frac{1}{16\pi G}\int dt\left( F\left( \phi \right) e^{3\alpha }\left( \frac{1}{N}\left( 6{\dot{\alpha }}^{2}-\frac{3}{2}{\dot{\beta }}^{2}-\frac{\omega }{2}{\dot{\phi }}^{2}\right) \right. \right. \nonumber \\ {}{} & {} \left. \left. +N\left( V\left( \phi \right) -2e^{-2\alpha +\beta }\right) \right) \right) . \end{aligned}$$
(11)

By variation of the Action Integral (11) with respect to the dynamical variables \(N,~\alpha ,~\beta \) and \(\phi \) we derive the field equations. They are

$$\begin{aligned} 0= & {} \ddot{\alpha }+\frac{3}{2}{\dot{\alpha }}^{2}+\frac{3}{8}{\dot{\beta }}^{2} +\frac{1}{4}\left( \frac{\omega }{2}{\dot{\phi }}^{2}-V\left( \phi \right) \right) \nonumber \\ {}{} & {} +\frac{d}{dt}\left( \ln F\left( \phi \right) \right) \dot{\alpha }+\frac{1}{6}e^{-2\alpha +\beta }, \end{aligned}$$
(12)
$$\begin{aligned} 0= & {} \ddot{\beta }+3{\dot{\alpha }}{\dot{\beta }}+\frac{d}{dt}\left( \ln F\left( \phi \right) \right) {\dot{\beta }}-\frac{2}{3}e^{-2\alpha +\beta } , \end{aligned}$$
(13)
$$\begin{aligned} 0= & {} \omega \left( \ddot{\phi }+3{\dot{\alpha }}{\dot{\phi }}\right) {\dot{\phi }}+\dot{V}\nonumber \\ {}{} & {} +\frac{d}{dt}\left( \ln F\left( \phi \right) \right) \left( 6\dot{a}^{2}-\frac{3}{2}{\dot{\beta }}^{2}+V\left( \phi \right) -2e^{-2\alpha +\beta }\right) ~ \nonumber \\ \end{aligned}$$
(14)

and the constraint

$$\begin{aligned} 0=F\left( \phi \right) e^{3\alpha }\left( 6{\dot{\alpha }}^{2}-\frac{3}{2} {\dot{\beta }}^{2}-\frac{\omega }{2}{\dot{\phi }}^{2}-V\left( \phi \right) +2e^{-2\alpha +\beta }\right) , \nonumber \\ \end{aligned}$$
(15)

where without loss of generality we have selected \(N\left( t\right) =1\). Recall that in the Misner variables the Hubble function is \(H={\dot{a}}\) and the shear \(\sigma ^{2}\simeq {\dot{\beta }}^{2}\)

We define the following set of new variables

$$\begin{aligned} \Sigma= & {} \frac{{\dot{\beta }}}{2D}, \quad x=\frac{{\dot{\phi }}}{12D}, \quad \ y=\sqrt{\frac{V\left( \phi \right) }{6D^{2}}}, \\ \eta= & {} \frac{H}{D}, \quad D=\sqrt{H^{2}+\frac{1}{3}e^{-2\alpha +\beta },} \end{aligned}$$

where D is the normalization parameter. Hence, the Hubble function is \(H=e^{\frac{\beta }{2}-\alpha }\frac{\eta }{\sqrt{1-\eta ^{2}}}\), from which we infer that \(\eta ^{2}\le 1\). In the limit \(\eta ^{2}=1\), it follows that \(H^{2}+\frac{1}{3}e^{-2\alpha +\beta }\simeq H^{2}\), that is, the asymptotic solution has a spatially flat three-dimensional hypersurface. Hence, the Kantowski–Sachs geometry is reduced to a Bianchi I spacetime.

In the new variables \(\left\{ \Sigma ,x,y,\eta ,\lambda \right\} \) and for the new dependent variable \(d\tau =Ddt\), the field equations (12)–(14) are expressed as follow

$$\begin{aligned} \frac{d\Sigma }{d\tau }= & {} 1-\Sigma \left( 4\sqrt{3}x+\Sigma \right) -\frac{3}{2}\eta \Sigma \left( 1-\omega x^{2}+y^{2}-\Sigma ^{2}\right) \nonumber \\ {}{} & {} +\eta ^{2}\left( 4\sqrt{3}x\Sigma +\Sigma ^{2}-1\right) , \end{aligned}$$
(16)
$$\begin{aligned} \frac{dx}{d\tau }&=x^{2}\left( \frac{\omega }{2}x\eta +4\sqrt{3}\left( 2\eta ^{2}-1\right) \right) \nonumber \\ {}&\quad -\frac{2\sqrt{3}}{\omega }\left( \left( 2+\lambda \right) y^{2}+2\left( 2\eta ^{2}-\Sigma ^{2}\right) \right) \nonumber \\&\quad +x\left( 3\eta \Sigma ^{2}-3\eta \left( 1+y^{2}\right) -2\left( 1-\eta ^{2}\right) \right) , \end{aligned}$$
(17)
$$\begin{aligned} \frac{dy}{d\tau }= & {} \frac{1}{2}y\left( x\left( 3\omega x\eta +2\sqrt{3}\left( \lambda +4\eta ^{2}\right) \right. \right. \nonumber \\ {}{} & {} \quad \left. \left. -2\Sigma +\eta \left( 3\left( 1-y^{2}\right) +2\eta \Sigma +3\Sigma ^{2}\right) \right) \right) , \end{aligned}$$
(18)
$$\begin{aligned} \frac{d\eta }{d\tau }= & {} \frac{1}{2}\left( \eta ^{2}-1\right) \left( 1+3\left( \omega x^{2}-y^{2}+\Sigma ^{2}\right) \right. \nonumber \\ {}{} & {} \quad \left. +2\eta \left( 4\sqrt{3}x+\Sigma \right) \right) \end{aligned}$$
(19)

with the algebraic constraint

$$\begin{aligned} 1-\omega x^{2}-y^{2}-\Sigma ^{2}=0. \end{aligned}$$
(20)

The parameter \(\lambda \) is defined as \(\lambda =\left( \ln V\left( \phi \right) \right) _{,\phi }\). For the exponential potential, \(V\left( \phi \right) =V_{0}e^{\lambda _{0}\phi }\), it follows that \(\lambda =\lambda _{0} \), which means that \(\lambda \) is always a constant parameter. However for other potential functions \(\lambda \) varies in terms of time. In particular, we find that

$$\begin{aligned} \frac{d\lambda }{d\tau }=2\sqrt{3}x\lambda ^{2}\left( \Gamma \left( \lambda \right) -1\right) , \quad \Gamma \left( \lambda \left( \phi \right) \right) =\frac{V_{,\phi \phi }V}{\left( V_{,\phi }\right) ^{2}}. \end{aligned}$$
(21)

The cosmological parameters in the new variables are expressed as follow

$$\begin{aligned} w_{eff}\left( \Sigma ,x,y,\eta \right)= & {} \frac{1}{3\eta ^{2}}\left( 1+3\left( \omega x^{2}-y^{2}+\Sigma ^{2}\right) \right. \nonumber \\ {}{} & {} \left. +\eta \left( 8\sqrt{3}x-\eta \right) \right) , \end{aligned}$$
(22)

and

$$\begin{aligned} q\left( \Sigma ,x,y,\eta \right)= & {} \frac{4\eta ^{2}-3}{2\eta ^{4}}\left( 1+3\left( \omega x^{2}-y^{2}+\Sigma ^{2}\right) \right. \nonumber \\ {}{} & {} \left. +\eta \left( 8\sqrt{3} x-\eta \right) \right) \end{aligned}$$
(23)

in which \(w_{eff}\left( \Sigma ,x,y,\eta \right) \) is the equation of state parameter for the effective fluid and \(q\left( \Sigma ,x,y,\eta \right) \) is the deceleration parameter.

4 Dynamical analysis for the exponential potential

In the following we study the asymptotic behaviour of the dynamical parameters described by the field equations (16)–(19) in which for the scalar field potential we consider the exponential function, \(V\left( \phi \right) =V_{0}e^{\lambda \phi }\).

We determine all the stationary points of the phase-space and we study their stability properties. From the latter we reconstruct the cosmological history and we get important information for the initial value problem. Each stationary point describes an asymptotic solution for the field equations which corresponds to a specific epoch of the cosmological evolution.

Let \(\frac{d}{d\tau }{\textbf{X}}={\textbf{F}}\left( {\textbf{X}}\right) \) be the dynamical system (16)–(19), \({\textbf{X}}=\left( \Sigma ,x,y,\eta \right) ^{T}\). Consider the point P which satisfies the condition \({\textbf{F}}\left( {\textbf{X}}\left( P\right) \right) =0\). Then P is a stationary/critical point for the dynamical system. At point P, we can calculate the physical parameters \(w_{eff}\left( P\right) \) and \(q\left( P\right) \) in order to infer the physical properties of the asymptotic solution. For instance, when \(q\left( P\right) <0\), the asymptotic solution describes an accelerated universe, while, if \(\Sigma \left( P\right) =0\), the asymptotic solution describes an isotropic solution.

In order to determine the stability properties of the dynamical system near to the stationary point P, we perform a linearization of the dynamical system near to the point P and we calculate the eigenvalues of the linearized matrix. The point P is an attractor when all the eigenvalues have negative real parts. Point P is characterized as a source when all the eigenvalues have positive real parts, otherwise is a saddle point. The knowledge of the stability properties is necessary in order to infer the physical properties of the gravitational theory that we study, as also to get constraints for the initial conditions.

The dynamical variables of the field equations (16)–(19) satisfy the constraint equation (20). Hence, with the use of the constraint the dimension of the phase-space is reduced by one. We remark that parameter y is always positive. For \(\omega >0\) the constraint equation (20), the dynamical variables take values on a three-dimensional ellipsoid, which reduces to a sphere for \(\omega =1\), which means that variables \(\left( \Sigma ,x,y\right) \) are defined in the finite regime. On the other hand for \(\omega <0\) the dynamical variables are not constrainted which means that we should investigate the existence of stationary points at the infinity regime.

4.1 Stationary points at the finite regime

At the finite regime the stationary points P of the dynamical system which satisfy the algebraic constraint (20) are

$$\begin{aligned} P_{1}^{\pm }=\left( \pm \sqrt{1-\omega \left( x_{1}\right) ^{2}},x_{1},0,1\right) , \end{aligned}$$
(24)

where \(x_{1}\) is arbitrary and \(1-\omega \left( x_{1}\right) ^{2}\ge 0\) in order that the points be real and physically accepted. For \(\omega >0\), \(\left( x_{1}\right) ^{2}\le \frac{1}{\omega }\), while for \(\omega <0\) points \(P_{1}^{\pm }\) are always real. Points \(P_{1}^{\pm }\) describe Kasner-like anisotropic solutions of Bianchi I geometry with \(q\left( P_{1}^{\pm }\right) =2+4\sqrt{3}x_{1}\).

$$\begin{aligned} P_{2}^{\pm }=\left( \pm \sqrt{1-\omega \left( x_{2}\right) ^{2}},x_{2},0,-1\right) , \end{aligned}$$
(25)

in which \(x_{2}\) is arbitrary and constrained by the algebraic relation \(1-\omega \left( x_{2}\right) ^{2}\ge 0\). Similarly to above for \(\omega >0\), \(\left( x_{2}\right) ^{2}\le \frac{1}{\omega }\), while for \(\omega <0\) points\(~P_{2}^{\pm }\) are always real. The asymptotic solutions are those of Kasner-like anisotropic spacetimes with \(q\left( P_{2}^{\pm }\right) =2-4\sqrt{3}x_{2}.\)

$$\begin{aligned} P_{3}^{\pm }=\left( \mp \sqrt{\frac{\omega }{\omega +48}},\mp \frac{4\sqrt{3} }{\sqrt{\omega \left( \omega +48\right) }},0,\pm 2\sqrt{\frac{\omega }{\omega +48}}\right) \end{aligned}$$

\(P_{3}^{\pm }~\)are real points and the asymptotic solutions are physically accepted for \(\omega \left( \omega +48\right) >0\); that is, \(\omega >0\) or \(\omega <-48.\)However, from the definition of \(\eta ^{2}\le 1\) it follows \(0<\omega \le 16.\) Hence, the points exist only for positive \(\omega \). The asymptotic solutions at the points \(P_{3}^{\pm }\) describe anisotropic Kantowski–Sachs geometries with \(w_{eff}\left( P_{3}^{\pm }\right) =0\) and \(q\left( P_{3}^{\pm }\right) =\frac{13}{8}-\frac{18}{\omega }\). We conclude that the asymptotic solutions at points \(P_{3}^{\pm }~\)describe accelerated universes for \(0<\omega <\frac{144}{13}\).

Fig. 1
figure 1

Region plot in the space \(\left( \lambda ,\omega \right) \) the stationary points \(P_{3}^{\pm }\) are attractors and the Kantowski–Sachs solutions are stable

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4.1.1 Stability analysis

We now proceed with the stability analysis for the stationary points. We make use of the algebraic constraint (20) and we reduce by one the dimension of the phase-space, such that the stationary points are on the three dimensional manifold of the variables \(\left\{ \Sigma ,x,\eta \right\} \).

The eigenvalues of the linearized dynamical system near to the stationary points \(P_{1}^{\pm }\) are

$$\begin{aligned}{} & {} e_{1}\left( P_{1}^{\pm }\right) =0, \quad e_{2}\left( P_{1}^{\pm }\right) =6+2\sqrt{3}\left( 4+\lambda \right) x_{1},\\ {}{} & {} e_{3}\left( P_{1}^{\pm }\right) =2\left( 2+4\sqrt{3}x_{1}\pm \sqrt{1-\omega \left( x_{1}\right) ^{2}}\right) . \end{aligned}$$

Similarly, for the points \(P_{2}^{\pm }\) the eigenvalues are derived to be

$$\begin{aligned}{} & {} e_{1}\left( P_{2}^{\pm }\right) =0, \quad e_{2}\left( P_{2}^{\pm }\right) =-6+2\sqrt{3}\left( 4+\lambda \right) x_{2},\\ {}{} & {} e_{3}\left( P_{2}^{\pm }\right) =2\left( -2+4\sqrt{3}x_{1}\pm \sqrt{1-\omega \left( x_{2}\right) ^{2}}\right) . \end{aligned}$$

These two sets of points have at least one eigenvalue with zero real part, that is eigenvalue \(e_{1}\), while there are ranges of the free parameters for which the eigenvalues \(e_{2}\) and \(e_{3}\) can have negative real parts. The latter means that there may exist a stable submanifold in the dynamical system. However, the derivation of this submanifold is of mathematical interest and does not contribute in the physical discussion. Thus, the stability properties for these two set of points are investigated numerically.

For the points \(P_{3}^{\pm }\) the eigenvalues are determined numerically. In Fig. 1 we present the region plots in the space of the free variables \(\left( \lambda ,\omega \right) \) for which the points have all the eigenvalues with negative real parts.

In the series of Figs. 2 and 3 we present the evolution of the trajectories for the field equations in the three dimensional space \(\left( \Sigma ,x,\eta \right) \). In Fig. 2 the plots are for positive value \(\omega \), where we observe that points \(P_{3}^{\pm }\) can be attractors. On the other hand Fig. 3 is for \(\omega <0\) there are no attractors in the finite regime and the trajectories move to infinity. Finally, for the points \(P_{1}^{\pm }\) and \(P_{2}^{\pm }\) from the numerical simulations we can conclude that the stationary points do not describe stable solutions.

Fig. 2
figure 2

Phase-space portraits for the three-dimensional dynamical system in the space \(\left( \Sigma ,x,\eta \right) \) for the Kantowski–Sachs spacetime and for \(\omega =1\). In the first row the left figure is for \(\lambda =-\,2\) and the right figure is for \(\lambda =2\). In the second row the left figure is for \(\lambda =-\,5\) and the right figure is for \(\lambda =-\,2\). We observe that points \(P_{3}^{\pm }\) can be the attractors in the finite regime

Full size image
Fig. 3
figure 3

Phase-space portraits for the three-dimensional dynamical system in the space \(\left( \Sigma ,x,\eta \right) \) for the Kantowski–Sachs spacetime and for \(\omega =-\,1\). In the first row the left figure is for \(\lambda =-\,2\) and the right figure is for \(\lambda =2\). In the second row the left figure is for \(\lambda =-\,5\) and the right figure is for \(\lambda =-\,2\). We observe that the trajectories move to infinity

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4.2 Stationary points at the infinity

Consider now the case for which \(\omega <0\). In order to investigate the existence of stationary points at the infinity regime, we consider the new set of Poincare variables

$$\begin{aligned} \Sigma =\frac{\rho }{\sqrt{1-\rho ^{2}}}\sin \theta , \quad x=\frac{\rho }{\sqrt{1-\rho ^{2}}}\cos \theta , \quad \omega =-\Omega ^{2}\text {} \end{aligned}$$

with \(\rho \in \left[ 0,1\right] \) and \(\theta \in [0,2\pi )\).

Moreover, we define the new independent variable, \(dT=\sqrt{1-\rho ^{2}}d\tau \), such that the field equations are

$$\begin{aligned}&4\Omega ^{2}\frac{d\rho }{dT} \nonumber \\&\quad =-6\Omega ^{2}\eta \rho \sqrt{1-\rho ^{2}}\nonumber \\&\qquad \left( 2+\left( \Omega ^{2}-3+\left( 1+\Omega ^{2}\right) \cos \left( 2\theta \right) \right) \rho ^{2}\right) \nonumber \\&\qquad +\left( 1-\rho ^{2}\right) \left( 4\left( \sqrt{3\lambda }\cos \theta +\Omega ^{2}\sin \theta \right) \right. \nonumber \\&\qquad \left. +\left( \sqrt{3}\left( \Omega ^{2}\left( 3\lambda -4\right) -\left( 4+5\lambda \right) \right) \cos \theta \right) \rho ^{2}\right) +\nonumber \\&\qquad +\left( 1-\rho ^{2}\right) \rho ^{2}\left( \sqrt{3}\left( 4+\lambda \right) \left( 1+\Omega ^{2}\right) \right. \nonumber \\&\qquad \left. \cos \left( 3\theta \right) -8\Omega ^{2}\sin \theta \right) +\nonumber \\&\qquad +4\eta ^{2}\left( 1-\rho ^{2}\right) \left( \Omega ^{2}\sin \theta \left( 2\rho ^{2}-1\right) \right. \nonumber \\ {}&\qquad \left. +4\sqrt{3}\cos \theta \left( 1+\left( \Omega ^{2}-1\right) \rho ^{2}\right) \right) , \end{aligned}$$
(26)
$$\begin{aligned} \frac{\Omega ^{2}\rho }{\sin \theta }\frac{d\theta }{dT}&=\left( 1-\rho ^{2}\right) \left( \Omega ^{2}\cot \theta -\left( 4\sqrt{3}+\Omega ^{2} \cot \theta \right) \eta ^{2}\right) \nonumber \\&\qquad +\sqrt{3}\left( \left( \lambda -\left( 4+\lambda \right) \right. \right. \nonumber \\ {}&\quad \left. \left. \left( \Omega ^{2}\cos ^{2}\theta -\sin ^{2}\theta \right) \right) \rho ^{2} -\lambda \right) , \end{aligned}$$
(27)
$$\begin{aligned} \frac{d\eta }{dT}&=\left( 1-\eta ^{2}\right) \left( -\left( 4\sqrt{3}\cos \theta +\sin \theta \right) \eta \rho \right) \nonumber \\&\qquad +\frac{\left( 1-\eta ^{2}\right) }{\sqrt{1-\rho ^{2}}}\left( 1+\left( 3 \right. \right. \nonumber \\&\qquad \left. \left. \left( \Omega ^{2}\cos ^{2}\theta -\sin ^{2}\theta \right) -1\right) \rho ^{2}\right) . \end{aligned}$$
(28)

Infinity is reached when \(\rho \rightarrow 1\). However, from (28) we observe that stationary points exist only on the surfaces with \(\eta ^{2}=1\), or \(\eta =0\) and \(\left( 1+\left( 3\left( \Omega ^{2}\cos ^{2}\theta -\sin ^{2}\theta \right) -1\right) \rho ^{2}\right) =0\).

For \(\eta ^{2}=1\) and \(\rho \rightarrow 1\), the field equations become

$$\begin{aligned}{} & {} \frac{d\rho }{dT}=0 , \quad \frac{d\eta }{dT}=0 \end{aligned}$$
(29)
$$\begin{aligned}{} & {} \frac{d\theta }{dT}=-\frac{\sqrt{3}}{2\Omega ^{2}}\sin \theta \left( \Omega ^{2}-1+\left( 1+\Omega ^{2}\right) \cos 2\theta \right) . \end{aligned}$$
(30)

Consequently, stationary points exist for \(\theta _{1}=0~\) or\(~\theta _{2}=\frac{1}{2}\arccos \frac{1-\Omega ^{2}}{1+\Omega ^{2}}\). The point with \(\theta _{1}=0\) describes an isotropic spatially flat FLRW spacetime, while the asymptotic solution at \(\theta _{2}=\frac{1}{2}\arccos \frac{1-\Omega ^{2} }{1+\Omega ^{2}}\) is a Kasner-like solution in Bianchi I geometry.

One of the eigenvalues of the linearized system at the isotropic solutions has always positive real part, while one eigenvalue has always negative real part, consequently, the stationary point is a saddle point and the solution is unstable. On the other hand, for the Bianchi I point with \(\theta _{2}=\frac{1}{2}\arccos \frac{1-\Omega ^{2}}{1+\Omega ^{2}}\) one of the eigenvalues is found to be always positive, which means that the point describes always an unstable solution.

The surface with \(\eta =0\) describe static spacetimes. The stationary points are \(\theta _{3}=0~\)and\(~\theta _{4}=\frac{1}{2}\arccos \frac{1-\Omega ^{2} }{1+\Omega ^{2}}\) with similar physical properties as before. Indeed, for \(\theta =\theta _{3}\) the asymptotic solution is the static isotropic closed FLRW spacetime, while for \(\theta =\theta _{4}\) the asymptotic solution describes the static Kantowski–Sachs universe. The stability properties of these stationary points are investigated numerically, from where it follows that the static asymptotic solutions are always unstable.

In Figs. 4 and 5 we present three-dimensional phase-space portraits for the field equations in the Poincare variables \(\left( \rho ,\theta ,\eta \right) \) and different values of \(\lambda \). We observe that there is not any attractor for the dynamical system at the infinity, neither in the finite regime as we derived in the previous Section for \(\omega <0\). Hence, the trajectories start from the finite regime, reach infinity and vice-versa.

Fig. 4
figure 4

Phase-space portraits for the three-dimensional dynamical system in the space \(\left( \rho ,\theta ,\eta \right) \) for the Kantowski–Sachs spacetime and for \(\omega =-\,1,\) that is \(\Omega =1\). First row are plots for \(\lambda =1\) while in the second row are plots with \(\lambda =5.\) The plots of the left and right columns are the same but from different view point. It is clear that in there are not attractos for the dynamical system and the trajectories start from the finite regime, reach infinity and vice-versa

Full size image
Fig. 5
figure 5

Phase-space portraits for the three-dimensional dynamical system in the space \(\left( \rho ,\theta ,\eta \right) \) for the Kantowski–Sachs spacetime and for \(\omega =-\,1,\) that is \(\Omega =1\). First row are plots with \(\lambda =-\,1\) while in the second row are plots with \(\lambda =-\,5.\) The plots of the left and right columns are the same but from a different view point. It is clear that there are no attractors for the dynamical system and the trajectories start from the finite regime, reach infinity and vice-versa

Full size image
Fig. 6
figure 6

Phase-space portraits for the dynamical system for the power-law potential in the three-dimensional space \(\left( \Sigma ,x,\eta \right) .\) Plots are for \(\omega <0\) and \(\omega \nu >0\). It is clear that the de Sitter point\(~Q_{4}^{+}\) is an attractor at the finite regime. First row are plots for \(\omega =-\,1\) while in the second row are plots with \(\omega =-\,5.\) The plots of the left and right columns are the same but from a different view point

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5 Dynamical analysis for the power-law potential

Consider now the power-law potential \(V\left( \phi \right) =V_{0}\phi ^{\nu }\). Then the dynamical equation (21) is

$$\begin{aligned} \frac{d\lambda }{d\tau }=-\frac{2\sqrt{3}}{\nu }\lambda ^{2}x. \end{aligned}$$
(31)

We follow the same procedure as above and we determine the stationary points \(Q=\left( \Sigma \left( Q\right) ,x\left( Q\right) ,y\left( Q\right) ,\eta \left( Q\right) ,\lambda \right) \) for the dynamical system (16)–(19) and (31) at the finite and infinity regimes

5.1 Stationary points at the finite regime

At the finite regime, the physically acceptable stationary points are

$$\begin{aligned} Q_{1}^{\pm }=\left( \pm \sqrt{1-\omega \left( x_{1}\right) ^{2}},x_{1},0,1,0\right) , \end{aligned}$$

with the same existence conditions and physical properties as points \(P_{1}^{\pm }\).      

$$\begin{aligned} Q_{2}^{\pm }=\left( \pm \sqrt{1-\omega \left( x_{2}\right) ^{2}},x_{2},0,-1,0\right) , \end{aligned}$$

with the same existence conditions and physical properties as points \(P_{2}^{\pm }\).

$$\begin{aligned} Q_{3}^{\pm }=\left( \mp \sqrt{\frac{\omega }{\omega +48}},\mp \frac{4\sqrt{3} }{\sqrt{\omega \left( \omega +48\right) }},0,\pm 2\sqrt{\frac{\omega }{\omega +48}}\right) , \end{aligned}$$

with the same existence conditions and physical properties as points \(P_{3}^{\pm }\).

$$\begin{aligned} Q_{4}^{\pm }=\left( 0,0,1,\pm 1,-4\right) , \end{aligned}$$

describes spatially flat FLRW geometry with \(q\left( Q_{4}^{\pm }\right) =-1\), that is, the asymptotic solutions are de Sitter spacetimes

The main difference with the exponential potential above is the de Sitter solutions. However, because the dimension of the dynamical system is greater and a new dynamical variable exists the stability properties should be investigated. Briefly, we discuss the stability properties for the new points \(Q_{4}^{\pm }\).

5.1.1 Stability analysis

For points \(Q_{4}^{\pm }\) the eigenvalues of the linearized system are

$$\begin{aligned}{} & {} e_{1}\left( Q_{4}^{\pm }\right) =-3 , \quad e_{2}\left( Q_{4}^{\pm }\right) =-2 \\{} & {} e_{3}\left( Q_{4}^{\pm }\right) =\mp \frac{3}{2}+\frac{1}{2}\sqrt{\frac{3\left( 128+3\nu \omega \right) }{\nu \omega }} ,\\ {}{} & {} e_{4}\left( Q_{4} ^{\pm }\right) =\mp \frac{3}{2}-\frac{1}{2}\sqrt{\frac{3\left( 128+3\nu \omega \right) }{\nu \omega }} \end{aligned}$$

from which we conclude that the stationary point \(Q_{4}^{-}\) is always a saddle point, and the expanding de Sitter universe described by \(Q_{4}^{+}\) is an attractor for \(\omega \nu >0\).

In Fig. 6 we present the three-dimensional phase-portrait for the field equations in which we observe that the de Sitter point \(Q_{4}^{+}\) is an attractor. The plots are for \(\omega <0\) from which it is clear that the an attractor exists at the finite regime.

5.2 Stationary points at the infinity

For \(\omega <0\) the dynamical system can reach infinity. We consider the same Poincare variables as before. We determine the same stationary points as before where now \(\lambda =0\).

The stability properties are the same hence we omit the presentation of the analysis.

We conclude that for the power-law potential the de Sitter solution can be an attractor. In this case, the scalar-field potential acts as a cosmological constant, which means that in the scalar-torsion theory the introduction of the cosmological constant can solve the flatness and isotropic problem.

6 Conclusions

We investigated the phase-space for the gravitational field equations in scalar-torsion theory with a Kantowski–Sachs background spacetime. Specifically, we considered that the scalar field potential to be the exponential potential, or the power-law potential, while we assumed that there is not any other matter source. The scalar-torsion theory introduce a constant parameter \(\omega \) which indicate the coupling of the scalar field to the teleparallel Lagrangian, in a similar way to the Brans–Dicke parameter, in scalar-tensor theory and the Ricciscalar.

For the Kantowski–Sachs spacetime we wrote the field equations in the equivalent way of an algebraic-differential system, where we have used dimensionless variables similar to that of the Hubble normalization. For the latter dynamical system we derived the stationary points and we investigated their stability properties. This analysis was necessary in order to reconstruct the cosmological history. We remark that for \(\omega >0\), the phase-space is compact, however, when \(\omega <0\) the dynamical variables can reach the infinity regime.

For the exponential potential, and \(\omega >0\) we found that there exist a stationary point which can be an attractor, where the asymptotic solution describes a Kantowski–Sachs universe which can be accelerated. Additionally, there exist two set of points which describe unstable Kasner-like solutions. Recall that a Kasner-like solution can describe the evolution of the physical variables near to the cosmological singularity. Hence, for \(\omega >0\) and the exponential potential, the evolution of the Kantowski–Sachs universe is simple. It can start from a singular solution and the dynamics to end into an anisotropic Kantowski–Sachs space. On the other hand, for \(\omega <0\) and the exponential potential we found that there are not any attractors. Thus, the trajectories of the dynamical system start from the finite regime, reach infinity and vice-versa.

In the case of the power-law potential the behaviour of the dynamics is different. In this case, the exist a stationary point which describe an asymptotic isotropic, homogeneous and accelerated cosmological model, the de Sitter spacetime. The de Sitter point exists in the de Sitter regime and it is an attractor for \(\omega >0\) and \(\omega <0\). Finally, for \(\omega <0\) the analysis at the finite regime for the power-law potential is similar to the analysis for the exponential potential.

We conclude that the scalar-tensor theory for an appropriate potential function can solve the isotropization of the universe as also the zero valued spatial curvature. In a future work we plan to investigated further the evolution of anisotropies in scalar-tensor theory by studying the dynamics in Bianchi geometries.