Scalar Field Source Teleparallel Robertson-Walker F(T)-Gravity Solutions

1. Introduction

The teleparallel theories of gravity are an important and promising class of alternative theories where all quantities and symmetries are defined in terms of the coframe and the spin-connection [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. The most appropriate teleparallel geometry definition is as affine symmetry. A such frame-based symmetry on the frame bundle is defined by a coframe/spin-connection pair and a field $\mathbf{X}$ satisfying some fundamental Lie derivative relations [7,8,10,11]. These such relationship definitions are considered as the frame-dependent analogue of the symmetry definition as presented in refs. [3,4]. For a pure teleparallel geometry, a coframe/spin-connection pair will have in addition to satisfy the null Riemann curvature criteria relation $R_{b\mu \nu }^a=0$, which leads to the spin-connection solution ${\omega }_{b\mu }^a={\Lambda }_c^a{\partial }_{\mu }{\Lambda }_b^c$ in terms of a typical Lorentz transformation ${\Lambda }_b^a$ [11].

By using the Cartan-Karlhede (CK) algorithm, we can construct some invariant coframe/spin-connection pairs satisfying the affine frame symmetries. However, there are cosmological teleparallel spacetime geometries classes invariant under the full $G_6$ Lie algebra of affine symmetries [7,8,9]. In addition, the proper coframe and the pure symmetric field equations (FE) were ever obtained and described by a teleparallel Robertson-Walker (TRW) geometry, the Robertson-Walker (RW) metric $g_{\mu \nu }$ and a parameter $k=(-1,0,1)$[7,8,9]. The parameter k is defined as the constant spatial curvature in the RW pseudo-Riemannian metric, but $R_{b\mu \nu }^a=0$ in teleparallel spacetimes. The parameter k is usually considered as a 3-dimensional space curvature in the metric-based approach and a part of the torsion scalar in 4-dimensional teleparallel spacetimes. In TRW geometries, an appropriate coframe/spin-connection pair leads to the trivial antisymmetric FEs parts. There are some TRW geometries where the coframe/spin-connection pair admits the full $G_6$ Lie algebra defined by the 6 Killing Vectors (KVs). In the literature, there are papers for $k=\pm 1$ cases some investigated geometries not yielding a $G_6$ Lie algebra. There are also some solutions involving the inappropriate coframes and/or spin-connections using. Recently, new coframe/spin-connection pairs satisfying a $G_6$ symmetry group have been constructed [3,4,19,20]. The most recent achievements on that leading to new teleparallel solutions were in refs [7,8,9,11].

A lot of works in the literature have been made on $k=0$ TRW cosmological model cases (refs. [5,6] and references within). In particular, specific forms for $F\left(T\right)$ have been investigated by using specific ansatz, and reconstruction methods have been explored extensively (where the function $F\left(T\right)$ is reconstructed from assumptions on the models). Dynamical systems methods, especially fixed point and stability analysis, in flat TRW models have been used with the stability conditions study of the standard de Sitter fixed point [5,21,22,23,24,25]. Then $k\ne 0$ solutions have been recently studied in bounce and inflation models [26,27] (i.e. the analysis is only applicable for $k=1$ case). The perturbations have also been studied in non-flat cosmology [28]. In a recent paper, we found as exact $k=0$ solutions of the FEs a combination of two power-law terms with the cosmological constant for a linear perfect fluid [8]. For $k=\pm 1$ cases, the differential equation has been linearized and a rigorous stability test has been performed for determining specific conditions on possible non-flat cosmological solutions. This test has been made with the $k=0$ exact solution (dominating term) and a linear correction term by using a power-law ansatz, perfect fluid Equation of State (EoS) and cosmological parameters for stable TRW solution and model requirements [8]. After the development of $F\left(T\right)$ TRW geometries and solutions for linear perfect fluids, we can go further by using the same approach. There are new papers on cosmological teleparallel $F(T,B)$ solutions for linear and non-linear perfect fluids and also for some scalar field sources [9]. We can also add recent papers on Kantowski-Sachs (KS) (pure time-dependent spacetimes) teleparallel $F\left(T\right)$ solutions for perfect fluids and scalar fields [14,15,16]. Therefore, we have already suggested as a achievable research works in ref. [8] to study the scalar field TRW $F\left(T\right)$ solutions and this deserves to be done as primary aim of the current paper. But there are also physical motivations for this new paper.

The most important physical motivations are concerning the possible teleparallel dark energy (DE) models. The first motivation is the quintessence DE physical process described by the linear perfect fluid EoS $P_{\varphi }={\alpha }_Q{\rho }_{\varphi }$ where $-1<{\alpha }_Q<-{\displaystyle \frac{1}{3}}$[29,30,31,32,33,34,35,36,37,38]. This perfect fluid described DE model assumes a fundamental scalar field inducing and explaining this physical process and constitutes the first possible form of DE satisfying the $P+\rho \ge 0$ energy condition [16]. There are also some teleparallel extension theories based on scalar field as a boundary variable or scalar-torsion theories to name only a few [39,40]. As a lower limit, there is the cosmological constant defined by a ${\alpha }_Q=-1$ perfect fluid as another form of DE and constitutes a some boundary of quintessence process in terms of EoS. The other main DE form is the phantom energy (or negative energy) where ${\alpha }_Q<-1$ and the energy condition will usually be violated ($P+\rho \ngeqq 0$) [41,42,43,44,45,46,47,48]. A phantom energy-based cosmological model is described as an strong accelerated expanding universe leading after a finite-time to the Big Rip physical process (great breakdown) and constitutes a extreme cosmological scenario. It is often occurring where we assume a non-linear perfect fluid teleparallel cosmological model [14,16]. After type of DE determination and study, there are interests on the combination (or mix) of the quintessence and phantom DE models with the cosmological constant as an intermediate limit: the quintom physical process models [49,50,51,52,53,54]. This is often described by some two-scalar field models: one field for quintessence and the second for phantom, or one for unified process and a second for the coupling. There is even a study on quintom oscillating model between quintessence and phantom energy where the cosmological constant state is the mid-point oscillating amplitude position [51]. However, there is only one relevant paper concerning teleparallel quintom models [54]. All these DE physical process works are well justifying a study on possible scalar field TRW $F\left(T\right)$ solution classes.

For this paper, we will first summarize in Section 2 the teleparallel $F\left(T\right)$ gravity FEs, the TRW coframe/spin-connection pair used and the scalar field source parameter equations. In Section 3, we will solve the FEs for flat cosmological cases ($k=0$) and allowing some easy-to-compute $F\left(T\right)$ solutions. We will enchain in Section 4 and Section 5 with $k=-1$ and $+1$ solutions, but the FEs will only allow analytical $F\left(T\right)$ solutions for some specific subcases. We will conclude this new development in Section 6 on possible impacts in favor of future DE models like quintessence, phantom and quintom physical processes.

2. Summary of Teleparallel Gravity and Field Equations

2.1. Teleparallel $F\left(T\right)$-Gravity Theory

The teleparallel $F\left(T\right)$-type gravity action integral with any gravitational source is [1,5,12,13,14,15,18]:

$$S_{F\left(T\right)}=\int d^{4}x\left[{\displaystyle \frac{h}{2\kappa }}F\left(T\right)+{\mathcal{L}}_{Source}\right],$$

(1)

where h is the coframe determinant, $\kappa$ is the coupling constant and ${\mathcal{L}}_{Source}$ is the gravitational source term. We will apply the least-action principle on the eqn (1) to find the symmetric and antisymmetric parts of FEs as [12,13,14,15]:

$$\begin{array}{ccc}\hfill \kappa {\Theta }_{\left(ab\right)}& =& F_T\left(T\right){\stackrel{\circ }{G}}_{ab}+F_{TT}\left(T\right)S_{\left(ab\right)}^{\mu }{\partial }_{\mu }T+{\displaystyle \frac{g_{ab}}{2}}\left[F\left(T\right)-T{F}_T\left(T\right)\right],\hfill \end{array}$$

(2a)

$$\begin{array}{ccc}\hfill 0& =& F_{TT}\left(T\right)S_{\left[ab\right]}^{\mu }{\partial }_{\mu }T,\hfill \end{array}$$

(2b)

with ${\stackrel{\circ }{G}}_{ab}$ the Einstein tensor, ${\Theta }_{\left(ab\right)}$ the energy-momentum, T the torsion scalar, $g_{ab}$ the gauge metric, $S_{ab}^{\mu }$ the superpotential (torsion dependent) and $\kappa$ the coupling constant. The canonical energy-momentum and its GR conservation law are obtained from ${\mathcal{L}}_{Matter}$ term of eqn (1) as [5,18]:

$$\begin{array}{cc}\hfill {\Theta }_a^{\mu }=& {\displaystyle \frac{1}{h}}{\displaystyle \frac{\delta {\mathcal{L}}_{Source}}{\delta h_{\mu }^a}},\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill \Rightarrow & {\stackrel{\circ }{\nabla }}_{\nu }\left({\Theta }^{\mu \nu }\right)=0,\hfill \end{array}$$

(3b)

where ${\stackrel{\circ }{\nabla }}_{\nu }$ the covariant derivative and ${\Theta }^{\mu \nu }$ the conserved energy-momentum tensor. The eqn (3a) antisymmetric and symmetric parts are [12]:

$${\Theta }_{\left[ab\right]}=0,{\Theta }_{\left(ab\right)}=T_{ab},$$

(4)

where $T_{ab}$ is the symmetric part of ${\Theta }^{\mu \nu }$. The eqn (3b) also imposes the symmetry of ${\Theta }^{\mu \nu }$ and then eqns (4) condition. Eqn (4) is valid only when the matter field interacts with the metric $g_{\mu \nu }$ defined from the coframe $h_{\mu }^a$ and the gauge $g_{ab}$, and is not directly coupled to the $F\left(T\right)$ gravity. This consideration is only valid for the null hypermomentum case (i.e. ${\mathfrak{T}}^{\mu \nu }=0$) as discussed in refs. [13,14,15,55]. This last condition on hypermomentum is defined from eqns (2a) and (2b) as [55]:

$$\begin{array}{c}\hfill {\mathfrak{T}}_{ab}=\kappa {\Theta }_{ab}-F_T\left(T\right){\stackrel{\circ }{G}}_{ab}-F_{TT}\left(T\right)S_{ab}^{\mu }{\partial }_{\mu }T-{\displaystyle \frac{g_{ab}}{2}}\left[F\left(T\right)-T{F}_T\left(T\right)\right]=0.\end{array}$$

(5)

There are more general teleparallel ${\mathfrak{T}}^{\mu \nu }$ definition and ${\mathfrak{T}}^{\mu \nu }\ne 0$ conservation law, but this is not really concerning the teleparallel $F\left(T\right)$-gravity situation [55,56,57,58].

2.2. Teleparallel Robertson-Walker Spacetime Geometry

In teleparallel gravity, any frame-based geometry on a frame bundle defined by a coframe/spin-connection pair and a field $\mathbf{X}$ must satisfy the fundamental Lie Derivative-based equations [7,8,10,11]:

$${\mathcal{L}}_{\mathbf{X}}{\mathbf{h}}_a={\lambda }_a^b{\mathbf{h}}_{b}and{\mathcal{L}}_{\mathbf{X}}{\omega }_{bc}^a=0,$$

(6)

where ${\omega }_{bc}^a$ is the spin-connection in terms of the differential coframe ${\mathbf{h}}_a$ and ${\lambda }_a^b$ is the linear isotropy group component. For TRW spacetime geometries on an orthonormal frame, the coframe $h_{\mu }^a$ solution will be [7,8] :

$$h_{\mu }^a=Diag\left[1,a\left(t\right){\left(1-k{r}^2\right)}^{-1/2},a\left(t\right)r,a\left(t\right)r\sin \theta \right].$$

(7)

The solution for spin-connection components will also be [7,8] :

$$\begin{array}{cccc}\hfill {\omega }_{122}=& {\omega }_{133}={\omega }_{144}=W_1\left(t\right),\hfill & \hfill {\omega }_{234}=& -{\omega }_{243}={\omega }_{342}=W_2\left(t\right),\hfill \\ \hfill {\omega }_{233}=& {\omega }_{244}=-{\displaystyle \frac{\sqrt{1-k{r}^2}}{a\left(t\right)r}},\hfill & \hfill {\omega }_{344}=& {\displaystyle \frac{cos\left(\theta \right)}{a\left(t\right)r\sin \left(\theta \right)}},\hfill \end{array}$$

(8)

where $W_1$ and $W_2$ are depending on k-parameter and defined by:

• $k=0$: $W_1=W_2=0$,
• $k=+1$: $W_1=0$ and $W_2\left(t\right)=\pm {\displaystyle \frac{\sqrt{k}}{a\left(t\right)}}$,
• $k=-1$: $W_1\left(t\right)=\pm {\displaystyle \frac{\sqrt{-k}}{a\left(t\right)}}$ and $W_2=0$.

For any $W_1$ and $W_2$, we will obtain the same symmetric FEs set to solve for each subcases depending on k-parameter. These previous coframe and spin-connection expressions were found by solving the eqns (6) and imposing the null Riemann curvature condition (i.e., $R_{b\mu \nu }^a=0$ as stated in ref. [11]). These solutions were also used in $F\left(T\right)$ TRW spacetime recent works [7,8]. The FEs to be solved in the current paper are defined for each k-parameter cases and will lead to different new teleparallel $F\left(T\right)$ solution classes. These FEs are also purely symmetric and valid on proper frames as clearly showed in refs. [7,8].

2.3. Scalar Field Source Conservation Laws Solutions

The scalar field source Lagrangian density ${\mathcal{L}}_{Source}$ term is defined as [5,8,9,16,40]:

$$\begin{array}{c}\hfill {\mathcal{L}}_{Source}={\displaystyle \frac{h}{2}}\stackrel{\circ }{\nabla }\nu \varphi \stackrel{\circ }{\nabla }\nu \varphi -hV\left(\varphi \right).\end{array}$$

(9)

The eqn (9) for a cosmological-like spacetime and a time-dependent $\varphi =\varphi \left(t\right)$ scalar field source is [9,16]:

$$\begin{array}{c}\hfill {\mathcal{L}}_{Source}=h{\displaystyle \frac{{\dot{\varphi }}^2}{2}}-hV\left(\varphi \right).\end{array}$$

(10)

where $\dot{\varphi }={\partial }_t\varphi$. By applying the least-action principle to eqn (10), the energy-momentum tensor $T_{ab}$ is [13,14,15,21,59]:

$$\begin{array}{c}\hfill T_{ab}=\left(P_{\varphi }+{\rho }_{\varphi }\right)u_a{u}_b+g_{ab}{P}_{\varphi },\end{array}$$

(11)

where $u_a=(-1,0,0,0)$, the pressure $P_{\varphi }$ and density ${\rho }_{\varphi }$ are [8,16]:

$$\begin{array}{c}\hfill P_{\varphi }={\displaystyle \frac{{\dot{\varphi }}^2}{2}}-V\left(\varphi \right)\mathrm{and}{\rho }_{\varphi }={\displaystyle \frac{{\dot{\varphi }}^2}{2}}+V\left(\varphi \right),\end{array}$$

(12)

with $\dot{\varphi }={\varphi }_t$ and $V=V\left(\varphi \right)$. The TRW spacetimes scalar field conservation law is [8,9]:

$$\begin{array}{c}\hfill 3H\dot{\varphi }+\ddot{\varphi }+{\displaystyle \frac{dV}{d\varphi }}=0,\end{array}$$

(13)

The eqn (13) is valid for any $\varphi \left(t\right)$ scalar field expression. For coming steps, we will solve the FEs by solving and satisfying the eqn (13).

To complete the discussion on physical implications, from eqns (12) and making the parallel between scalar field $\varphi$ and the DE linear perfect fluids EoS equivalent $P_{\varphi }={\alpha }_Q{\rho }_{\varphi }$, we need to define the quintessence coefficient index (or DE index) ${\alpha }_Q$ as [16,29,30,31,36]:

$$\begin{array}{cc}\hfill {\alpha }_Q=& {\displaystyle \frac{P_{\varphi }}{{\rho }_{\varphi }}}={\displaystyle \frac{{\dot{\varphi }}^2-2V\left(\varphi \right)}{{\dot{\varphi }}^2+2V\left(\varphi \right)}},\hfill \end{array}$$

(14)

where the usual types of DE are defined:

• Quintessence $-\mathbf{1}<{\alpha }_Q<-{\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}$: It describes a controlled accelerating universe expansion where energy conditions are always satisfied, i.e. $P_{\varphi }+{\rho }_{\varphi }>0$ [29,30,31,36].
• Phantom energy ${\alpha }_Q<-\mathbf{1}$: It usually describes an uncontrolled accelerating universe expansion going to the Big Rip event [42,43,44]. The energy condition is violated, i.e. $P_{\varphi }+{\rho }_{\varphi }\ngeqq 0$.
• Cosmological constant ${\alpha }_Q=-\mathbf{1}$: This is an intermediate limit between the two previous and main types of DE where $P_{\varphi }+{\rho }_{\varphi }=0$. A constant scalar field source $\varphi ={\varphi }_0$ will directly lead to this case according to eqn (14).
• Quintom models: Mixture of previous DE types usually described by some double scalar-field models [49,50,51,52,53].

By using eqn (14), we can find the perfect fluid equivalent for any new teleparallel $F\left(T\right)$ solutions, potential $V\left(\varphi \right)$ and ansatz. The eqn (14) is useful for making new teleparallel $F\left(T\right)$ solutions classification in terms of DE quintessence, phantom and quintom processes in cosmological model studies within the teleparallel framework.

3. $k=0$ Cosmological Solutions

The FEs and torsion scalar expressions for $\rho ={\rho }_{\varphi }$ and $P=P_{\varphi }$ defined by eqns (12) are [8]:

$$\begin{array}{cc}\hfill \kappa \left({\displaystyle \frac{{\dot{\varphi }}^2}{2}}+V\left(\varphi \right)\right)=& -{\displaystyle \frac{F\left(T\right)}{2}}+6H^2{F}_T,\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill \kappa ({\dot{\varphi }}^2-V\left(\varphi \right))=& {\displaystyle \frac{F\left(T\right)}{2}}-3\left(\dot{H}+2H^2\right)F_T-3H{\dot{F}}_T,\hfill \end{array}$$

(15b)

$$\begin{array}{cc}\hfill T=& 6H^2,\hfill \end{array}$$

(15c)

where $H={\displaystyle \frac{\dot{a}}{a}}$ is the Hubble parameter. By merging eqns (15a) and (15b) and then using eqn (15c), we find as unified FE:

$$\begin{array}{cc}\hfill -{\displaystyle \frac{\sqrt{6}\kappa {\dot{\varphi }}^2}{2}}=& {\partial }_t\left(\sqrt{T}{F}_T\right).\hfill \end{array}$$

(16)

For any $\varphi =\varphi \left(t\right(T\left)\right)$ scalar field definition and any potential $V\left(\varphi \right)$ expression, we will use the $a\left(t\right)=a_0{t}^n$ and eqn (15c) to find that $t\left(T\right)={\displaystyle \frac{\sqrt{6}n}{\sqrt{T}}}$. For any scalar field $\varphi =\varphi \left(t\right(T\left)\right)$ and potential $V\left(\varphi \right)$ expressions, the general $F\left(T\right)$ solution formula will be:

$$\begin{array}{c}\hfill F\left(T\right)=-{\Lambda }_0+B\sqrt{T}-{\displaystyle \frac{\sqrt{6}\kappa }{2}}\left[\int d{T}^{\prime }{T}^{\prime -1/2}\left[{\int }_{t\left(T^{\prime }\right)}d{t}^{\prime }{\dot{\varphi }}^2\left(t^{\prime }\right)\right]\right].\end{array}$$

(17)

The first two terms of eqn (17) are identical to those from the linear perfect fluid $k=0$ TRW $F\left(T\right)$ solution found in ref [8]. The last term of eqn (17) is clearly the scalar field source contribution.

We can compute the $F\left(T\right)$ solutions by using eqn (17) and the $a\left(t\right)=a_0{t}^n$ ansatz yielding to $t\left(T\right)={\displaystyle \frac{\sqrt{6}n}{\sqrt{T}}}$ for the scalar field cases:

• Power-law general: For $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}{\mathbf{t}}^{\mathbf{p}}$ and $p\ne {\displaystyle \frac{1}{2}}$, we find:

  $$\begin{array}{c}\hfill F\left(T\right)=-{\Lambda }_0+B\sqrt{T}+{\displaystyle \frac{\sqrt{6}\kappa p_0^2{p}^2{\left(6n^2\right)}^{p-\frac{1}{2}}}{2(2p-1)(p-1)}}{T}^{1-p}.\end{array}$$

  (18)
• Power-law special: For $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}{\mathbf{t}}^{{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}}$, we find:

  $$\begin{array}{c}\hfill F\left(T\right)=-{\Lambda }_0+B\sqrt{T}+{\displaystyle \frac{\sqrt{6}\kappa p_0^2}{8}}\sqrt{T}ln\left(T\right).\end{array}$$

  (19)
• Logarithmic: For a field defined as $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}ln\left(\mathbf{t}\right)$, we find:

  $$\begin{array}{c}\hfill F\left(T\right)=-{\Lambda }_0+B\sqrt{T}+{\displaystyle \frac{\kappa p_0^2}{2n}}T.\end{array}$$

  (20)
• Exponential: For $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}exp\left(\mathbf{p}\mathbf{t}\right)$, we find:

  $$\begin{array}{c}\hfill F\left(T\right)=-{\Lambda }_0+B\sqrt{T}-{\displaystyle \frac{\sqrt{6}\kappa }{2}}{p}_0^{2}p\left[2\sqrt{6}np{Ei}_1\left(-{\displaystyle \frac{2p\sqrt{6}n}{\sqrt{T}}}\right)+\sqrt{T}exp\left({\displaystyle \frac{2p\sqrt{6}n}{\sqrt{T}}}\right)\right],\end{array}$$

  (21)

  where ${Ei}_1\left(x\right)$ is an exponential integral function.

There are several other possible $\varphi \left(t\right)$ source terms which may lead to new teleparallel $F\left(T\right)$ solutions using eqn (17) general formula.

4. $k=-1$ Cosmological Solutions

The FEs and torsion scalar for $\rho ={\rho }_{\varphi }$ and $P=P_{\varphi }$ defined by eqns (12) are [8]:

$$\begin{array}{cc}\hfill \kappa \left({\displaystyle \frac{{\dot{\varphi }}^2}{2}}+V\left(\varphi \right)\right)=& -{\displaystyle \frac{F\left(T\right)}{2}}+6H\left(H+{\displaystyle \frac{\delta \sqrt{-k}}{a}}\right)F_T,\hfill \end{array}$$

(22a)

$$\begin{array}{cc}\hfill \kappa ({\dot{\varphi }}^2-V\left(\varphi \right))=& {\displaystyle \frac{F\left(T\right)}{2}}-3\left(\dot{H}+2H^2+2H{\displaystyle \frac{\delta \sqrt{-k}}{a}}-{\displaystyle \frac{k}{a^2}}\right)F_T-3\left(H+{\displaystyle \frac{\delta \sqrt{-k}}{a}}\right){\dot{F}}_T,\hfill \end{array}$$

(22b)

$$\begin{array}{cc}\hfill T=& 6{\left(H+{\displaystyle \frac{\delta \sqrt{-k}}{a}}\right)}^2.\hfill \end{array}$$

(22c)

By merging eqns (22a) and (22b), the unified FE will be expressed as:

$$\begin{array}{cc}\hfill -{\displaystyle \frac{\kappa {\dot{\varphi }}^2}{2}}=& \left(\dot{H}-{\displaystyle \frac{k}{a^2}}\right)F_T+\left(H+{\displaystyle \frac{\delta \sqrt{-k}}{a}}\right){\dot{F}}_T.\hfill \end{array}$$

(23)

The general $F\left(T\right)$ solution from eqn (23) to be computed will be expressed as:

$$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+\int dT\left[\right(C_1-{\displaystyle \frac{\kappa }{2}}{\int }_{t\left(T\right)}d{t}^{\prime }exp\left({\int }_{t^{\prime }}d{t}^{\prime \prime }\left({\displaystyle \frac{a^2\dot{H}-k}{a\left(aH+\delta \sqrt{-k}\right)}}\right)\right)\hfill \\ \hfill & \times {\displaystyle \frac{a{\dot{\varphi }}^2\left(t^{\prime }\right)}{\left(aH+\delta \sqrt{-k}\right)}})exp\left(-{\int }_{t\left(T\right)}d{t}^{\prime }\left({\displaystyle \frac{a^2\dot{H}-k}{a\left(aH+\delta \sqrt{-k}\right)}}\right)\right)].\hfill \end{array}$$

(24)

By applying a power-law ansatz $a\left(t\right)=a_0{t}^n$ to eqn (24), the $F\left(T\right)$ solution will simplify as:

$$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+\int dT[\left(C_1-\kappa {\int }_{t\left(T\right)}d{t}^{\prime }{\dot{\varphi }}^2\left(t^{\prime }\right)exp\left(-{\displaystyle \frac{\delta \sqrt{-k}}{a_0(n-1)}}{t}^{\prime 1-n}\right)\right)\hfill \\ \hfill & \times {\displaystyle \frac{exp\left(\frac{\delta \sqrt{-k}}{a_0(n-1)}{t}^{1-n}\left(T\right)\right)}{2n{t}^{-1}\left(T\right)+2\frac{\delta \sqrt{-k}}{a_0}{t}^{-n}\left(T\right)}}].\hfill \end{array}$$

(25)

From eqn (22c), the characteristic equation leading to $t\left(T\right)$ expressions is:

$$\begin{array}{c}\hfill 0={\delta }_1\sqrt{{\displaystyle \frac{T}{6}}}-n{t}^{-1}-{\displaystyle \frac{\delta \sqrt{-k}}{a_0}}{t}^{-n}.\end{array}$$

(26)

There are specific values of n leading to eqn (26)-based analytical teleparallel $F\left(T\right)$ solutions:

• $\mathbf{n}={\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$: Eqn (26) becomes:

  $$\begin{array}{cc}\hfill 0=& t^{-1}+{\displaystyle \frac{2\delta \sqrt{-k}}{a_0}}{t}^{-{\displaystyle \frac{1}{2}}}-{\delta }_1\sqrt{{\displaystyle \frac{2T}{3}}},\hfill \\ \hfill \Rightarrow & t^{-{\displaystyle \frac{1}{2}}}\left(T\right)=-{\displaystyle \frac{\delta \sqrt{-k}}{a_0}}+{\delta }_2\sqrt{-{\displaystyle \frac{k}{a_0^2}}+{\delta }_1\sqrt{{\displaystyle \frac{2T}{3}}}},\hfill \end{array}$$

  (27)

  where ${\delta }_2=\pm 1$. Then eqn (25) for $F\left(T\right)$ solution is:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+\int dT[\left(C_1-\kappa {\delta }_1\sqrt{{\displaystyle \frac{3}{2}}}{\int }_{t\left(T\right)}d{t}^{\prime }{\dot{\varphi }}^2\left(t^{\prime }\right)exp\left({\displaystyle \frac{2\delta \sqrt{-k}}{a_0}}{t}^{\prime {\displaystyle \frac{1}{2}}}\right)\right)\hfill \\ \hfill & \times T^{-1/2}exp\left(2{\left[1-{\delta }_2\sqrt{1-{\displaystyle \frac{a_0^2{\delta }_1}{k}}\sqrt{{\displaystyle \frac{2T}{3}}}}\right]}^{-1}\right)].\hfill \end{array}$$

  (28)
  Eqn (28) by setting ${\displaystyle \frac{\delta \sqrt{-k}}{a_0}}=1$ and a power-law scalar field $\varphi \left(t\right)=p_0{t}^p$ yields to new analytical $F\left(T\right)$ solutions for the subcases:

  (a)

  $\mathbf{p}={\displaystyle \frac{\mathbf{3}}{\mathbf{4}}}$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+C_{1}exp\left(2{\left[1-{\delta }_2\sqrt{1+{\delta }_1\sqrt{{\displaystyle \frac{2T}{3}}}}\right]}^{-1}\right){\left[1-{\delta }_2\sqrt{1+{\delta }_1\sqrt{{\displaystyle \frac{2T}{3}}}}\right]}^2\hfill \\ \hfill & -\kappa {\delta }_1{p}_0^2{\displaystyle \frac{9}{8}}\sqrt{{\displaystyle \frac{3}{2}}}{T}^{1/2}.\hfill \end{array}$$

  (29)

  (b)

  $\mathbf{p}=\mathbf{1}$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+C_{1}exp\left(2{\left[1-{\delta }_2\sqrt{1+{\delta }_1\sqrt{{\displaystyle \frac{2T}{3}}}}\right]}^{-1}\right){\left[1-{\delta }_2\sqrt{1+{\delta }_1\sqrt{{\displaystyle \frac{2T}{3}}}}\right]}^2+\kappa p_0^2\hfill \\ \hfill & \times ({\delta }_1\sqrt{{\displaystyle \frac{3}{2}}}{T}^{1/2}-2{\delta }_2\sqrt{9+3{\delta }_1\sqrt{6}\sqrt{T}}+3ln\left({\displaystyle \frac{1+{\delta }_2\sqrt{1+{\delta }_1\sqrt{\frac{2T}{3}}}}{1-{\delta }_2\sqrt{1+{\delta }_1\sqrt{\frac{2T}{3}}}}}\right)\hfill \\ \hfill & -{\displaystyle \frac{3}{2}}ln\left(-2T\right)).\hfill \end{array}$$

  (30)

  There are several other possible $F\left(T\right)$ solutions by setting other values of ${\displaystyle \frac{\delta \sqrt{-k}}{a_0}}$, p and/or other scalar field $\varphi \left(t\right)$ expressions inside eqn (28) general expression. However, we can expect that such cases will yield to more bigger expression of $F\left(T\right)$.
• $\mathbf{n}=\mathbf{1}$: Eqn (26) becomes:

  $$\begin{array}{cc}\hfill 0=& {\delta }_1\sqrt{{\displaystyle \frac{T}{6}}}-\left(1+{\displaystyle \frac{\delta \sqrt{-k}}{a_0}}\right)t^{-1},\hfill \\ \hfill \Rightarrow & t\left(T\right)={\delta }_1\left(1+{\displaystyle \frac{\delta \sqrt{-k}}{a_0}}\right){\displaystyle \frac{\sqrt{6}}{\sqrt{T}}},\hfill \end{array}$$

  (31)
  Eqn (25) for $F\left(T\right)$ solution becomes:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+B{T}^{{\displaystyle \frac{\delta \sqrt{-k}}{2a_0}}+{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & -{\displaystyle \frac{{\delta }_1\sqrt{6}\kappa }{2{\left({\delta }_1\sqrt{6}\left(1+\frac{\delta \sqrt{-k}}{a_0}\right)\right)}^{\frac{\delta \sqrt{-k}}{a_0}}}}\int dT{T}^{{\displaystyle \frac{\delta \sqrt{-k}}{2a_0}}-{\displaystyle \frac{1}{2}}}\left[{\int }_{t\left(T\right)}d{t}^{\prime }{\dot{\varphi }}^2\left(t^{\prime }\right)t^{\prime {\displaystyle \frac{\delta \sqrt{-k}}{a_0}}}\right].\hfill \end{array}$$

  (32)
  Eqn (32) yields to a new analytical $F\left(T\right)$ solutions for the cases:

  (a)

  General Power-law $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}{\mathbf{t}}^{\mathbf{p}}$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+B{T}^{{\displaystyle \frac{\delta \sqrt{-k}}{2a_0}}+{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{{\delta }_1\sqrt{6}\kappa p_0^2{p}^2{\left[{\delta }_1\left(1+\frac{\delta \sqrt{-k}}{a_0}\right)\sqrt{6}\right]}^{2p-1}}{2\left(2p-1+\frac{\delta \sqrt{-k}}{a_0}\right)(p-1)}}{T}^{1-p}.\hfill \end{array}$$

  (33)

  (b)

  Special Power-law $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}{t}^{{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}-{\displaystyle \frac{\delta \sqrt{-\mathbf{k}}}{\mathbf{2}{\mathbf{a}}_{\mathbf{0}}}}}$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+B{T}^{{\displaystyle \frac{\delta \sqrt{-k}}{2a_0}}+{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{\kappa p_0^2{\left(1-\frac{\delta \sqrt{-k}}{a_0}\right)}^2{T}^{\frac{\delta \sqrt{-k}}{2a_0}+\frac{1}{2}}ln\left(T\right)}{8{\left({\delta }_1\sqrt{6}\right)}^{\frac{\delta \sqrt{-k}}{a_0}-1}{\left(1+\frac{\delta \sqrt{-k}}{a_0}\right)}^{\frac{\delta \sqrt{-k}}{a_0}+1}}}.\hfill \end{array}$$

  (34)

  (c)

  Logarithmic $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}ln\left(\mathbf{pt}\right)$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+B{T}^{{\displaystyle \frac{\delta \sqrt{-k}}{2a_0}}+{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{\kappa p_0^2}{2\left(1+\frac{k}{a_0^2}\right)}}T.\hfill \end{array}$$

  (35)

  (d)

  Exponential $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}exp\left(\mathbf{pt}\right)$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+B{T}^{{\displaystyle \frac{\delta \sqrt{-k}}{2a_0}}+{\displaystyle \frac{1}{2}}}-{\displaystyle \frac{{\delta }_1\sqrt{6}\kappa p_0^2{p}^2}{2{\left({\delta }_1\sqrt{6}\left(1+\frac{\delta \sqrt{-k}}{a_0}\right)\right)}^{\frac{\delta \sqrt{-k}}{a_0}}}}\hfill \\ \hfill & \times \int dT{T}^{{\displaystyle \frac{\delta \sqrt{-k}}{2a_0}}-{\displaystyle \frac{1}{2}}}\left[{\int }_{t\left(T\right)}d{t}^{\prime }exp\left(2p{t}^{\prime }\right)t^{\prime {\displaystyle \frac{\delta \sqrt{-k}}{a_0}}}\right].\hfill \end{array}$$

  (36)

  There is no general solution for eqn (36). However, there are specific solutions:

  • ${\displaystyle \frac{\delta \sqrt{-k}}{a_0}}=\mathbf{1}$:

    $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+BT\hfill \\ \hfill & -{\displaystyle \frac{\kappa p_0^2}{16}}\left[96{Ei}_1\left(-{\displaystyle \frac{4p{\delta }_1\sqrt{6}}{\sqrt{T}}}\right)p^2-e^{{\displaystyle \frac{4p{\delta }_1\sqrt{6}}{\sqrt{T}}}}\left(T-4\sqrt{T}p{\delta }_1\sqrt{6}\right)\right].\hfill \end{array}$$

    (37)
  • ${\displaystyle \frac{\delta \sqrt{-k}}{a_0}}=\mathbf{2}$:

    $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+B{T}^{{\displaystyle \frac{3}{2}}}-{\displaystyle \frac{{\delta }_1\kappa p_0^2}{18p}}[108{\delta }_1{p}^3{Ei}_1\left(-{\displaystyle \frac{6p{\delta }_1\sqrt{6}}{\sqrt{T}}}\right)\hfill \\ \hfill & +e^{{\displaystyle \frac{6p{\delta }_1\sqrt{6}}{\sqrt{T}}}}\left({\displaystyle \frac{T^{\frac{3}{2}}\sqrt{6}}{36}}+3\sqrt{6}{p}^2\sqrt{T}-{\delta }_{1}pT\right)].\hfill \end{array}$$

    (38)

  There are several possible new $F\left(T\right)$ solutions for other values of ${\displaystyle \frac{\delta \sqrt{-k}}{a_0}}$ and using eqn (36).

  There are additional possible new $F\left(T\right)$ solutions from eqn (32) integral with other types of scalar field sources.
• $\mathbf{n}=\mathbf{2}$: Eqn (26) becomes:

  $$\begin{array}{cc}\hfill 0=& t^{-2}+{\displaystyle \frac{2\delta a_0}{\sqrt{-k}}}{t}^{-1}-{\delta }_1\delta a_0\sqrt{-{\displaystyle \frac{T}{6k}}},\hfill \\ \hfill \Rightarrow & t^{-1}\left(T\right)=-{\displaystyle \frac{\delta a_0}{\sqrt{-k}}}+{\delta }_2\sqrt{{\delta }_1\delta a_0\sqrt{-{\displaystyle \frac{T}{6k}}}-{\displaystyle \frac{a_0^2}{k}}},\hfill \end{array}$$

  (39)

  where ${\delta }_2=\pm 1$. Eqn (25) becomes:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+\int dT[\left(C_1-{\displaystyle \frac{{\delta }_1\kappa \sqrt{6}}{2}}{\int }_{t\left(T\right)}d{t}^{\prime }{\dot{\varphi }}^2\left(t^{\prime }\right)exp\left(-{\displaystyle \frac{\delta \sqrt{-k}}{a_0}}{t}^{\prime -1}\right)\right)\hfill \\ \hfill & \times {\displaystyle \frac{exp\left[-1+{\delta }_2\sqrt{1+\frac{{\delta }_1\delta }{a_0}\sqrt{\frac{-k}{6}}\sqrt{T}}\right]}{\sqrt{T}}}].\hfill \end{array}$$

  (40)
  Eqn (40) by using a power-law scalar field $\varphi \left(t\right)=p_0{t}^p$ and setting ${\displaystyle \frac{\delta \sqrt{-k}}{a_0}}=1$ yields to a new analytical $F\left(T\right)$ solutions for some $p<0$ subcases:

  (a)

  $\mathbf{p}=-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+C_1\left[-1+{\delta }_2\sqrt{1+{\delta }_1\sqrt{{\displaystyle \frac{T}{6}}}}\right]exp\left[-1+{\delta }_2\sqrt{1+{\delta }_1\sqrt{{\displaystyle \frac{T}{6}}}}\right]\hfill \\ \hfill & -{\delta }_2\kappa p_0^2{\left(1+{\delta }_1\sqrt{{\displaystyle \frac{T}{6}}}\right)}^{3/2}.\hfill \end{array}$$

  (41)

  (b)

  $\mathbf{p}=-\mathbf{1}$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+C_1\left[-1+{\delta }_2\sqrt{1+{\delta }_1\sqrt{{\displaystyle \frac{T}{6}}}}\right]exp\left[-1+{\delta }_2\sqrt{1+{\delta }_1\sqrt{{\displaystyle \frac{T}{6}}}}\right]\hfill \\ \hfill & -\kappa p_0^2\left(2\sqrt{6}{\delta }_1{T}^{1/2}+{\displaystyle \frac{T}{2}}\right).\hfill \end{array}$$

  (42)

  There are several other possible $F\left(T\right)$ solutions by setting other values of ${\displaystyle \frac{\delta \sqrt{-k}}{a_0}}$, p and/or other scalar field $\varphi \left(t\right)$ expressions. However, we can expect that such cases will yield to more bigger expression of $F\left(T\right)$.
• $\mathbf{n}\gg \mathbf{1}$: Eqn (26) becomes:

  $$\begin{array}{cc}\hfill 0=& {\delta }_1\sqrt{{\displaystyle \frac{T}{6}}}-n{t}^{-1},\hfill \\ \hfill \Rightarrow & t\left(T\right)=n{\delta }_1\sqrt{{\displaystyle \frac{6}{T}}}.\hfill \end{array}$$

  (43)
  Eqn (25) becomes for $n\to \infty$:

  $$\begin{array}{cc}\hfill F\left(T\right)\approx & -{\Lambda }_0+B\sqrt{T}-{\displaystyle \frac{\kappa {\delta }_1\sqrt{6}}{2}}\int dT{T}^{-1/2}\left[{\int }_{t\left(T\right)}d{t}^{\prime }{\dot{\varphi }}^2\left(t^{\prime }\right)\right],\hfill \end{array}$$

  (44)

  where $exp\left(-{\displaystyle \frac{\delta \sqrt{-k}}{a_{0}n}}{t}^{\prime -n}\right)\to 1$ under the very large n limit. Eqn (44) yields to new analytical $F\left(T\right)$ solutions for the cases:

  (a)

  General Power-law $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}{\mathbf{t}}^{\mathbf{p}}$:

  $$\begin{array}{cc}\hfill F\left(T\right)\approx & -{\Lambda }_0+B\sqrt{T}-{\displaystyle \frac{\kappa {\left(n{\delta }_1\sqrt{6}\right)}^{2p}{p}_0^2{p}^2}{2n(2p-1)(1-p)}}{T}^{1-p}.\hfill \end{array}$$

  (45)

  (b)

  Special Power-law $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}{\mathbf{t}}^{\mathbf{1}/\mathbf{2}}$:

  $$\begin{array}{cc}\hfill F\left(T\right)\approx & -{\Lambda }_0+B\sqrt{T}+{\displaystyle \frac{\kappa {\delta }_1\sqrt{6}{p}_0^2}{4}}\sqrt{T}ln\left(T\right).\hfill \end{array}$$

  (46)

  (c)

  Logarithmic $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}ln\left(\mathbf{pt}\right)$:

  $$\begin{array}{cc}\hfill F\left(T\right)\approx & -{\Lambda }_0+B\sqrt{T}+{\displaystyle \frac{\kappa p_0^2}{2n}}T.\hfill \end{array}$$

  (47)

  (d)

  Exponential $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}exp\left(\mathbf{pt}\right)$:

  $$\begin{array}{cc}\hfill F\left(T\right)\approx & -{\Lambda }_0+B\sqrt{T}\hfill \\ \hfill & -{\displaystyle \frac{\kappa {\delta }_1\sqrt{6}{p}_0^{2}p}{2}}\left[2\sqrt{6}np{\delta }_1{Ei}_1\left(-{\displaystyle \frac{2pn{\delta }_1\sqrt{6}}{\sqrt{T}}}\right)+e^{{\displaystyle \frac{2pn{\delta }_1\sqrt{6}}{\sqrt{T}}}}\sqrt{T}\right].\hfill \end{array}$$

  (48)

  There are several possible new $F\left(T\right)$ solutions from eqn (44) integral with other types of scalar field sources. Therefore eqns (45)–(48) for large n yield to very similar solutions to eqns (18)–(21) ($\mathbf{k}=\mathbf{0}$ case solutions).

5. $k=+1$ Cosmological Solutions

The FEs and torsion scalar for $\rho ={\rho }_{\varphi }$ and $P=P_{\varphi }$ defined by eqns (12) are [8]:

$$\begin{array}{cc}\hfill \kappa \left({\displaystyle \frac{{\dot{\varphi }}^2}{2}}+V\left(\varphi \right)\right)=& -{\displaystyle \frac{F\left(T\right)}{2}}+6H^2{F}_T,\hfill \end{array}$$

(49a)

$$\begin{array}{cc}\hfill \kappa ({\dot{\varphi }}^2-V\left(\varphi \right))=& {\displaystyle \frac{F\left(T\right)}{2}}-3\left(\dot{H}+2H^2-{\displaystyle \frac{k}{a^2}}\right)F_T-3H{\dot{F}}_T,\hfill \end{array}$$

(49b)

$$\begin{array}{cc}\hfill T=& 6\left[H^2-{\displaystyle \frac{k}{a^2}}\right].\hfill \end{array}$$

(49c)

By merging eqns (49a) and (49b), the unified FE will be expressed as:

$$\begin{array}{cc}\hfill -{\displaystyle \frac{\kappa {\dot{\varphi }}^2}{2}}=& \left(\dot{H}-{\displaystyle \frac{k}{a^2}}\right)F_T+H{\dot{F}}_T,\hfill \end{array}$$

(50)

The general $F\left(T\right)$ solution from eqn (50) to be computed will be expressed as:

$$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+\int {\displaystyle \frac{dT}{H\left(t\right(T\left)\right)}}\left[C_1-{\displaystyle \frac{\kappa }{2}}{\int }_{t\left(T\right)}d{t}^{\prime }exp\left(-k{\int }_{t^{\prime }}{\displaystyle \frac{d{t}^{\prime \prime }}{H{a}^2}}\right){\dot{\varphi }}^2\left(t^{\prime }\right)\right]exp\left(k{\int }_{t\left(T\right)}{\displaystyle \frac{d{t}^{\prime }}{H{a}^2}}\right).\hfill \end{array}$$

(51)

Eqn (51) becomes by applying the power-law ansatz $a\left(t\right)=a_0{t}^n$ to eqn (51):

$$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+{\displaystyle \frac{1}{n}}\int dTt\left(T\right)\left[C_1-{\displaystyle \frac{\kappa }{2}}{\int }_{t\left(T\right)}d{t}^{\prime }{\dot{\varphi }}^2\left(t^{\prime }\right)exp\left(-{\displaystyle \frac{k{t}^{\prime 2(1-n)}}{2n(1-n)a_0^2}}\right)\right]\hfill \\ \hfill & \times exp\left({\displaystyle \frac{k{t}^{2(1-n)}\left(T\right)}{2n(1-n)a_0^2}}\right).\hfill \end{array}$$

(52)

From eqn (49c), the characteristic equation leading to $t\left(T\right)$ expressions is:

$$\begin{array}{cc}\hfill 0=& {\displaystyle \frac{T}{6}}-{\displaystyle \frac{n^2}{t^2}}+{\displaystyle \frac{k}{a_0^2}}{t}^{-2n}.\hfill \end{array}$$

(53)

There are specific values of n leading to eqn (53)-based analytical teleparallel $F\left(T\right)$ solutions:

• $\mathbf{n}={\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$: Eqn (53) becomes:

  $$\begin{array}{cc}\hfill 0=& t^{-2}-{\displaystyle \frac{4k}{a_0^2}}{t}^{-1}-{\displaystyle \frac{2T}{3}},\hfill \\ \hfill \Rightarrow & t^{-1}\left(T\right)={\displaystyle \frac{2k}{a_0^2}}+2{\delta }_2\sqrt{{\displaystyle \frac{k^2}{a_0^4}}+{\displaystyle \frac{T}{6}}},.\hfill \end{array}$$

  (54)

  where ${\delta }_2=\pm 1$. Eqn (52) becomes:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+\int dT\left[C_1-{\displaystyle \frac{\kappa a_0^2}{2k}}{\int }_{t\left(T\right)}d{t}^{\prime }{\dot{\varphi }}^2\left(t^{\prime }\right)exp\left(-{\displaystyle \frac{2k{t}^{\prime }}{a_0^2}}\right)\right]{\displaystyle \frac{exp\left(4{\left(1+{\delta }_2\sqrt{1+\frac{a_0^4}{6k^2}T}\right)}^{-1}\right)}{\left(1+{\delta }_2\sqrt{1+\frac{a_0^4}{6k^2}T}\right)}}.\hfill \end{array}$$

  (55)
  Eqn (55) by setting ${\displaystyle \frac{k}{a_0^2}}=1$ yields to new analytical $F\left(T\right)$ solutions for the scalar field:

  (a)

  Linear Power-law $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}\mathbf{t}$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+C_1[\left(1+{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}\right)exp\left(4{\left(1+{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}\right)}^{-1}\right)\hfill \\ \hfill & +{Ei}_1\left(-4{\left(1+{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}\right)}^{-1}\right)]\hfill \\ \hfill & +{\displaystyle \frac{3\kappa p^2{p}_0^2}{2}}\left[2{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}-ln\left(T\right)+ln\left({\displaystyle \frac{{\delta }_2\sqrt{1+\frac{T}{6}}-1}{{\delta }_2\sqrt{1+\frac{T}{6}}+1}}\right)\right].\hfill \end{array}$$

  (56)

  (b)

  Quadratic Power-law $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}{\mathbf{t}}^{\mathbf{2}}$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+C_1[\left(1+{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}\right)exp\left(4{\left(1+{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}\right)}^{-1}\right)\hfill \\ \hfill & +3{Ei}_1\left(-4{\left(1+{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}\right)}^{-1}\right)]\hfill \\ \hfill & +{\displaystyle \frac{9\kappa p^2{p}_0^2}{4T}}\left[{\displaystyle \frac{12}{T}}-1+4\left(1-{\displaystyle \frac{3}{T}}\right){\delta }_2{\left(1+{\displaystyle \frac{T}{6}}\right)}^{3/2}\right].\hfill \end{array}$$

  (57)

  (c)

  Exponential $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}exp\left(\mathbf{pt}\right)$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+C_1[\left(1+{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}\right)exp\left(4{\left(1+{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}\right)}^{-1}\right)\hfill \\ \hfill & +3{Ei}_1\left(-4{\left(1+{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}\right)}^{-1}\right)]\hfill \\ \hfill & -{\displaystyle \frac{3\kappa p^2{p}_0^2}{(p-1)}}[\left(1+{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}\right)exp\left(4p{\left(1+{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}\right)}^{-1}\right)\hfill \\ \hfill & +(4p-1){Ei}_1\left(-4p{\left(1+{\delta }_2\sqrt{1+{\displaystyle \frac{T}{6}}}\right)}^{-1}\right)].\hfill \end{array}$$

  (58)

  There are several other possible $F\left(T\right)$ solutions by setting other values of ${\displaystyle \frac{k}{a_0}}$, p and/or other scalar field $\varphi \left(t\right)$ expressions. However, we can expect that such cases will yield to more bigger expression of $F\left(T\right)$.
• $\mathbf{n}=\mathbf{1}$: Eqn (53) becomes:

  $$\begin{array}{cc}\hfill 0=& T-6\left(1-{\displaystyle \frac{k}{a_0^2}}\right)t^{-2},\hfill \\ \hfill \Rightarrow & t^2\left(T\right)={\displaystyle \frac{6\left(1-\frac{k}{a_0^2}\right)}{T}}.\hfill \end{array}$$

  (59)
  Eqn (52) becomes:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+B{T}^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{k}{2a_0^2}}}+{\displaystyle \frac{\kappa }{2}}{\left(6\left(1-{\displaystyle \frac{k}{a_0^2}}\right)\right)}^{{\displaystyle \frac{k}{2a_0^2}}+{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \times \int dT\left[{\int }_{t\left(T\right)}d{t}^{\prime }{\dot{\varphi }}^2\left(t^{\prime }\right)t^{\prime -{\displaystyle \frac{k}{a_0^2}}}\right]T^{-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{k}{2a_0^2}}}.\hfill \end{array}$$

  (60)
  Eqn (60) yields to a new analytical $F\left(T\right)$ solutions for the cases:

  (a)

  General Power-law $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}{\mathbf{t}}^{\mathbf{p}}$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+B{T}^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{k}{2a_0^2}}}+{\displaystyle \frac{\kappa p_0^2{p}^2}{2\left(2p-1-\frac{k}{a_0^2}\right)(1-p)}}{\left(6\left(1-{\displaystyle \frac{k}{a_0^2}}\right)\right)}^p{T}^{1-p}.\hfill \end{array}$$

  (61)

  (b)

  Special Power-law $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}{\mathbf{t}}^{{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}+{\displaystyle \frac{\mathbf{k}}{\mathbf{2}{\mathbf{a}}_{\mathbf{0}}^{\mathbf{2}}}}}$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+B{T}^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{k}{2a_0^2}}}\hfill \\ \hfill & +{\displaystyle \frac{\kappa p_0^2}{8}}{\displaystyle \frac{{\left(1+\frac{k}{a_0^2}\right)}^2}{{\left(1-\frac{k}{a_0^2}\right)}^2}}{\left(6\left(1-{\displaystyle \frac{k}{a_0^2}}\right)\right)}^{{\displaystyle \frac{k}{2a_0^2}}+{\displaystyle \frac{1}{2}}}\left[2-\left(1-{\displaystyle \frac{k}{a_0^2}}\right)ln\left(T\right)\right]T^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{k}{2a_0^2}}}.\hfill \end{array}$$

  (62)

  (c)

  Logarithmic $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}ln\left(\mathbf{pt}\right)$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+B{T}^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{k}{2a_0^2}}}-{\displaystyle \frac{\kappa p_0^2}{2\left(1+\frac{k}{a_0^2}\right)}}T.\hfill \end{array}$$

  (63)

  (d)

  Exponential $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}exp\left(\mathbf{pt}\right)$:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+B{T}^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{k}{2a_0^2}}}+{\displaystyle \frac{\kappa p_0^2{p}^2}{2}}{\left(6\left(1-{\displaystyle \frac{k}{a_0^2}}\right)\right)}^{{\displaystyle \frac{k}{2a_0^2}}+{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \times \int dT\left[{\int }_{t\left(T\right)}d{t}^{\prime }exp\left(2p{t}^{\prime }\right)t^{\prime -{\displaystyle \frac{k}{a_0^2}}}\right]T^{-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{k}{2a_0^2}}}.\hfill \end{array}$$

  (64)

  There is no general solution, but for specific values of ${\displaystyle \frac{k}{a_0^2}}>0$:

  • ${\displaystyle \frac{\mathbf{k}}{{\mathbf{a}}_{\mathbf{0}}^{\mathbf{2}}}}\to \mathbf{1}$:

    $$\begin{array}{cc}\hfill F\left(T\right)\approx & -{\Lambda }_0-3\kappa p_0^2{p}^2\left(\gamma +{\displaystyle \frac{3}{2}}ln\left(24\right)\right)ϵln\left(T\right),\hfill \end{array}$$

    (65)

    where $F\left(T\right)\to -{\Lambda }_0$ under $ϵ\to 0$ limit.
  • ${\displaystyle \frac{\mathbf{k}}{{\mathbf{a}}_{\mathbf{0}}^{\mathbf{2}}}}=\mathbf{2}$:

    $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+B{T}^{-{\displaystyle \frac{1}{2}}}-6\kappa p_0^2{p}^2[2\sqrt{6}p{(-T)}^{-1/2}{Ei}_1\left(-2p\sqrt{6}{(-T)}^{-1/2}\right)\hfill \\ \hfill & -{\mathrm{Ei}}_1\left(-2p\sqrt{6}{(-T)}^{-1/2}\right)+e^{2p\sqrt{6}{(-T)}^{-1/2}}-1].\hfill \end{array}$$

    (66)

  There are several possible new $F\left(T\right)$ solutions from eqn (60) integral with other scalar field sources.

• $\mathbf{n}=\mathbf{2}$: Eqn (53) becomes:

  $$\begin{array}{cc}\hfill 0=& {\displaystyle \frac{a_0^{2}T}{6k}}-{\displaystyle \frac{4a_0^2}{k}}{t}^{-2}+t^{-4},\hfill \\ \hfill \Rightarrow & t^{-2}\left(T\right)={\displaystyle \frac{2a_0^2}{k}}\left[1+{\delta }_2\sqrt{1-{\displaystyle \frac{k}{24a_0^2}}T}\right],\hfill \end{array}$$

  (67)

  where ${\delta }_2=\pm 1$. Eqn (52) becomes:

  $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+\int dT\left[C_1-{\displaystyle \frac{\kappa }{4}}{\int }_{t\left(T\right)}d{t}^{\prime }{\dot{\varphi }}^2\left(t^{\prime }\right)exp\left({\displaystyle \frac{k{t}^{\prime -2}}{4a_0^2}}\right)\right]{\left[{\displaystyle \frac{2a_0^2}{k}}\left[1+{\delta }_2\sqrt{1-{\displaystyle \frac{k}{24a_0^2}}T}\right]\right]}^{-1/2}\hfill \\ \hfill & \times exp\left(-{\displaystyle \frac{1}{2}}\left[1+{\delta }_2\sqrt{1-{\displaystyle \frac{k}{24a_0^2}}T}\right]\right).\hfill \end{array}$$

  (68)
  Eqn (68) for the power-law scalar field $\varphi \left(t\right)=p_0{t}^p$ and ${\displaystyle \frac{k}{a_0^2}}=1$ yields to new analytical $F\left(T\right)$ solutions in subcases:
  • $\mathbf{p}=-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$:

    $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+\sqrt{1+{\delta }_2\sqrt{1-{\displaystyle \frac{T}{24}}}}\hfill \\ \hfill & \times \left[C_{1}exp\left(-{\displaystyle \frac{1}{2}}\left[1+{\delta }_2\sqrt{1-{\displaystyle \frac{T}{24}}}\right]\right)+{\displaystyle \frac{4\kappa p_0^2}{\sqrt{2}}}\left({\delta }_2\sqrt{1-{\displaystyle \frac{T}{24}}}-2\right)\right].\hfill \end{array}$$

    (69)
  • $\mathbf{p}=-{\displaystyle \frac{\mathbf{3}}{\mathbf{2}}}$:

    $$\begin{array}{cc}\hfill F\left(T\right)=& -{\Lambda }_0+\sqrt{1+{\delta }_2\sqrt{1-{\displaystyle \frac{T}{24}}}}[C_{1}exp\left(-{\displaystyle \frac{1}{2}}\left[1+{\delta }_2\sqrt{1-{\displaystyle \frac{T}{24}}}\right]\right)+{\displaystyle \frac{144\kappa p_0^2{p}^2}{\sqrt{2}}}\hfill \\ \hfill & \times \left[\left({\delta }_2\sqrt{1-{\displaystyle \frac{T}{24}}}-2\right)+{\displaystyle \frac{1}{10}}\left(1+{\delta }_2\sqrt{1-{\displaystyle \frac{T}{24}}}\right)\left(3{\delta }_2\sqrt{1-{\displaystyle \frac{T}{24}}}-2\right)\right]].\hfill \end{array}$$

    (70)
  There are several other possible $F\left(T\right)$ solutions by setting other values of ${\displaystyle \frac{k}{a_0}}$, p and/or other scalar field expressions. However, we can expect that such cases will yield to more bigger expression of $F\left(T\right)$.
• $\mathbf{n}\gg \mathbf{1}$: Eqn (53) becomes for large n:

  $$\begin{array}{cc}\hfill 0=& {\displaystyle \frac{T}{6}}-{\displaystyle \frac{n^2}{t^2}},\hfill \\ \hfill \Rightarrow & t\left(T\right)={\delta }_2{\displaystyle \frac{\sqrt{6}n}{\sqrt{T}}},\hfill \end{array}$$

  (71)

  where ${\delta }_2=\pm 1$. Eqn (52) becomes for $n\to \infty$:

  $$\begin{array}{cc}\hfill F\left(T\right)\approx & -{\Lambda }_0+B\sqrt{T}-{\displaystyle \frac{\kappa {\delta }_2\sqrt{6}}{2}}\int dT{T}^{-1/2}\left[{\int }_{t\left(T\right)}d{t}^{\prime }{\dot{\varphi }}^2\left(t^{\prime }\right)\right],\hfill \end{array}$$

  (72)

  where $exp\left({\displaystyle \frac{k{t}^{\prime -2n}}{2n^2{a}_0^2}}\right)\to 1$ under the very large n limit. Eqn (68) yields to a new analytical $F\left(T\right)$ solutions for the cases:

  (a)

  General Power-law $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}{\mathbf{t}}^{\mathbf{p}}$:

  $$\begin{array}{cc}\hfill F\left(T\right)\approx & -{\Lambda }_0+B\sqrt{T}+{\displaystyle \frac{\kappa {\delta }_2\sqrt{6}{p}_0^2{p}^2{\left({\delta }_2\sqrt{6}n\right)}^{2p-1}}{2(2p-1)(p-1)}}{T}^{1-p}.\hfill \end{array}$$

  (73)

  (b)

  Special Power-law $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}{\mathbf{t}}^{\mathbf{1}/\mathbf{2}}$:

  $$\begin{array}{cc}\hfill F\left(T\right)\approx & -{\Lambda }_0+B\sqrt{T}+{\displaystyle \frac{\kappa {\delta }_2\sqrt{6}{p}_0^2}{8}}\sqrt{T}ln\left(T\right).\hfill \end{array}$$

  (74)

  (c)

  Logarithmic $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}ln\left(\mathbf{pt}\right)$:

  $$\begin{array}{cc}\hfill F\left(T\right)\approx & -{\Lambda }_0+B\sqrt{T}+{\displaystyle \frac{\kappa p_0^2}{2n}}T.\hfill \end{array}$$

  (75)

  (d)

  Exponential $\varphi \left(t\right)={\mathbf{p}}_{\mathbf{0}}exp\left(\mathbf{pt}\right)$:

  $$\begin{array}{cc}\hfill F\left(T\right)\approx & -{\Lambda }_0+B\sqrt{T}-{\displaystyle \frac{\kappa p_0^{2}p{\delta }_2\sqrt{6}}{2}}\left[2\sqrt{6}np{\delta }_2{Ei}_1\left(-{\displaystyle \frac{2pn\delta \sqrt{6}}{\sqrt{T}}}\right)+e^{{\displaystyle \frac{2pn\delta \sqrt{6}}{\sqrt{T}}}}\sqrt{T}\right].\hfill \end{array}$$

  (76)

  There are several possible new $F\left(T\right)$ solutions from eqn (72) integral with other types of scalar field sources. Therefore eqns (73)–(76) for large n yield to almost identical solutions comparing to eqns (18)–(21) ($\mathbf{k}=\mathbf{0}$case solutions) and eqns (45)–(48) ($\mathbf{k}=-\mathbf{1}$case solutions). We can conclude that the $\mathbf{k}=\pm \mathbf{1}$ teleparallel $F\left(T\right)$ solutions for $\mathbf{n}\to \infty$ will lead to flat cosmological ($\mathbf{k}=\mathbf{0}$) solutions as a general limit.

All the previous teleparallel $F\left(T\right)$ solutions found in Section 3 to Section 5 are new and go further than some recent research papers in the literature. Therefore there are several other possible subcases which might be leading to additional new $F\left(T\right)$ solutions and possibly going to the same direction.

6. Concluding Remarks

The present works first allowed to solve the TRW $F\left(T\right)$ FEs obtained in refs [7,8], this for a scalar field source $\varphi$ expressed in terms of density ${\rho }_{\varphi }$ and pressure $P_{\varphi }$. For flat cosmological spacetimes ($k=0$ case), we obtain by using the general eqn (17) with different scalar fields $\varphi \left(t\right)$ purely analytical and simple teleparallel $F\left(T\right)$ solutions as expressed by eqns (18)–(21). This was all possible by using the power ansatz for the scale factor $a\left(t\right)$ and then the relation $t\left(T\right)$ obtained with eqn () to express everything in terms of scalar torsion T. One could still use eqn (17) to obtain other $F\left(T\right)$ solutions from other scalar fields $\varphi \left(t\right)$. This type of result is a logical continuation of the literature, because there are many $k=0$ solutions for other types of sources not only in teleparallel $F\left(T\right)$ gravity, but also in its extensions. The most important thing here is obtaining the all-purpose eqn (17) to generate all the teleparallel $F\left(T\right)$ solutions needed to study and compare the different $k=0$ cosmological solutions. This flexibility will allow the study of cosmological phenomena with DE for any values of n and potential $V\left(\varphi \right)$ involved in the models. These are good advantages favoring our new $F\left(T\right)$ solutions.

We have also found general and multipurpose formulas for non-flat cosmological spacetimes ($k=-1$ and $+1$ cases) from the $k=\pm 1$ TRW $F\left(T\right)$ FEs and the power ansatz for $a\left(t\right)$, namely eqns (24) and (51) respectively. However, the characteristic eqns (26) and (53) from the scalar torsion eqns (22c) and (49c) ultimately allow only a limited number of purely analytic teleparallel $F\left(T\right)$ solutions defined for very specific values of n. Simple analytical solutions are obtained for $n=1$ and in the limit of very large n, which allows in the latter case to treat models involving the very rapid acceleration of the expansion of the universe. This type of situation will be very useful for the study of phantom energy models and more precisely the physical mechanisms leading to the Big Rip. The large n limit solutions for any $k=0$ and $\pm 1$ cases will lead to the same types of $F\left(T\right)$ solutions by comparing eqns (18)–(21), (45)–(48) and (73)–(76) for each scalar field source subcases. For lower values of n, the latter will be especially useful for the study of the DE quintessence for non-flat cosmological systems. Note that the cases of early ($n={\displaystyle \frac{1}{2}}$) and late ($n=2$) expansion lead to very different teleparallel $F\left(T\right)$ solutions: this represents an important benefit for studies on the different forms of DE quintessence. However, the expressions for $F\left(T\right)$ often involve various special functions, a sign that these are mathematically more sophisticated cases. The solutions $n=1$ here constitute an intermediate case between the early and late expansions of universe expansion also involving the DE quintessence process to thus better bridge the gap between the solutions $n={\displaystyle \frac{1}{2}}$ and 2. In principle, other values of n could have been used, but this would only have complicated the present approach without necessarily providing better results and conclusions.

The results of present work with those obtained in ref. [8] for linear perfect fluids will together allow to fully develop the TRW $F\left(T\right)$-type cosmological models involving the different forms of DE. There have been attempts at a similar development to this one in recent papers. These studies often focused on teleparallel extensions [9,23,24,25,26,27]. There have been very recent similar studies involving KS type spacetimes having however fewer symmetries and possible simplifications at the origin (4 KVs instead of 6) [14,15,16]. The present paper on TRW $F\left(T\right)$ solutions complements the various studies providing the essential ingredients of the perfect fluids and scalar field $F\left(T\right)$ solutions classes. We must keep in mind that the ultimate aims are the complete studies of teleparallel cosmological models involving the various DE forms. These are therefore the next steps in the development of Teleparallel Gravity and we must now go all the way.