Chaplygin and Polytropic Gases Teleparallel Robertson-Walker F(T) Gravity Solutions

1. Introduction

The teleparallel $F\left(T\right)$ gravity is a frame-based and alternative theory to general relativity (GR) fundamentally defined in terms of the coframe ${\mathbf{h}}^a$ and the spin-connection ${\omega }_{bc}^a$ [1,2,3,4,5,6,7]. These two last quantities define the torsion tensor $T_{bc}^a$ and torsion scalar T. We remind that GR is defined by the metric $g_{\mu \nu }$ and the spacetime curvatures $R_{b\mu \nu }^a$, $R_{\mu \nu }$ and R. Under some considerations, we can determine the symmetries for any independent coframe/spin-connection pairs, and then spacetime curvature and torsion are defined as geometric objects [4,5,6,7,8,9]. Any geometry described by a such pair whose curvature and non-metricity are both zero ($R_{b\mu \nu }^a=0$ and $Q_{a\mu \nu }=0$ conditions) is a teleparallel gauge-invariant geometry (valid for any $g_{ab}$). The fundamental pairs must satisfy two Lie derivative-based relations and we use the Cartan–Karlhede algorithm to solve these two fundamental equations for any teleparallel geometry. For a pure teleparallel $F\left(T\right)$ gravity spin-connection solution, we also solve the null Riemann curvature condition leading to a Lorentz transformation-based definition of the spin-connection ${\omega }_{b\mu }^a$. There is a direct equivalent to GR in teleparallel gravity: the teleparallel equivalent to GR (TEGR) generalizing to teleparallel $F\left(T\right)$-type gravity [7,9,10,11,12]. All the previous considerations are also adapted for the new general relativity (NGR) (refs. [13,14,15] and refs. therein), the symmetric teleparallel $F\left(Q\right)$-type gravity (refs. [16,17,18,19] and refs. therein) and some extended theories like $F(T,Q)$-type, $F(R,Q)$-type, $F(R,T)$-type, and several other ones (refs. [20,21,22,23,24,25,26,27,28,29] and refs. therein). In the current paper we will restrict our study to the teleparallel $F\left(T\right)$ gravity framework.

There are a large number of research papers on spherically symmetric spacetimes and solutions in teleparallel $F\left(T\right)$ gravity using a large number of approaches, energy-momentum sources and made for various purposes [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56]. But there are a special class of teleparallel spacetime which the field equations (FEs) are purely symmetric: the teleparallel Robertson-Walker (TRW) spacetime [55,56,57]. This type of spacetime is defined in terms of the k-parameter where $k=0$ is a flat cosmological spacetime, $k=\pm 1$ cases are respectively positive and negative space cosmological curvature [58,59,60,61]. We have proven that a such teleparallel geometry is described by a $G_6$ Lie algebra group, where the 4th to 6th Killing Vectors (KVs) are characteristic of this spacetime [56,57]. The main apparent consequence of additional KVs is the trivial antisymmetric parts of FEs. The same spacetime geometry structure also exists for some extensions such as teleparallel $F(T,B)$ gravity [62,63,64,65,66]. We have found teleparallel $F\left(T\right)$ and $F(T,B)$ solutions for perfect fluid (PF) and scalar field (SF) sources. In the last case, the teleparallel $F\left(T\right)$ and $F(T,B)$ solutions are scalar potential independent and only SF dependent [55,62]. But there are additional possible sources of energy-momentum which can lead to new further teleparallel solutions in $F\left(T\right)$-type, but also for extensions.

The Dark Energy (DE) states are usually studied by using the PF equivalent equation of state (EoS) $P_{\varphi }={\alpha }_Q{\rho }_{\varphi }$ where ${\alpha }_Q$ is the DE index (or quintessence index in some refs.) [67,68,69]. The possible DE forms in terms of ${\alpha }_Q$ are:

1.

Quintessence $-\mathbf{1}<{\alpha }_Q<-{\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}$: This describes a controlled accelerating universe expansion where energy conditions are always satisfied, i.e., $P_{\varphi }+{\rho }_{\varphi }>0$ [70,71,72,73,74,75,76,77,78,79,80,81,82]. This usual DE form has been significantly studied in the literature in recent decades for the fascination it provokes and the realism of the models.

2.

Phantom energy ${\alpha }_Q<-\mathbf{1}$: This describes an uncontrolled universe expansion accelerating toward a Big Rip event (or singularity) [83,84,85,86,87,88,89,90,91]. The energy condition is violated, i.e., $P_{\varphi }+{\rho }_{\varphi }\ngeqq 0$. But this DE form is fascinating because we can find new teleparallel solutions and physical models.

3.

Cosmological constant ${\alpha }_Q=-\mathbf{1}$: This is an intermediate limit between the quintessence and phantom DE states, where $P_{\varphi }+{\rho }_{\varphi }=0$. A constant SF source $\varphi ={\varphi }_0$ added by a positive scalar potential $V\left({\varphi }_0\right)>0$ will directly lead to this primary DE state. Note that a negative scalar potential (i.e., $V\left({\varphi }_0\right)\le 0$) will not lead to a positive cosmological constant and/or a DE solution.

4.

Quintom models: This is a mixture of previous DE types, usually described by double SF models [92,93,94,95,96,97]. This type of model is more complex to study and solve in general. Several types of models are in principle possible and these physical processes need further studies in the future.

There are in the recent literature in teleparallel gravity and its extension a number of paper on possible solution in static radial-dependent, time-dependent and cosmological teleparallel $F\left(T\right)$ solutions [48,49,50,51,52,55,56,62]. The most of those are suitable for DE models, in particular for quintessence DE state. We use cosmological PF as SF approaches, and we find that the solution classes are similar between the both cases.

The quartessence model is an unified models of DE and Dark Matter (DM) arising from a Chaplygin cosmological gas. The DE and DM are two states of a single and simple quartessence dark cosmological fluid [98,99,100,101,102]. This non-linear cosmological fluid model can be considered as an alternative and unified explanation of DE and DM evolving in the universe influencing the cosmological processes. For polytropic gases, there are additional possible physical models arising from this class of EoS-based cosmological solutions. [103,104,105,106,107,108,109,110]. But the Chaplygin, polytropic gas and any superposition of those models in general can explain not only DE and DM mixed or separate models, but also any non-linear cosmological gas and fluid system at the limit [111]. Most of the previous works have been done in the GR (or $F\left(R\right)$-type) framework, but these last cases may also arise in teleparallel theories of gravity. Under this last consideration, the current investigation concerning the Chaplygin and polytropic fluids deserves to be achieved in teleparallel framework and will constitute the main aim of the paper, in particular for $F\left(T\right)$-type case. Ultimately, we want to build some pure teleparallel quartessence and polytropic-based cosmology, allowing to study more realistic universe models.

Ultimately we want to study in detail the quartessence suitable teleparallel cosmological solutions and its physical impacts. Therefore we need at the current stage to find the possible Chaplygin and polytropic cosmological gas teleparallel $F\left(T\right)$ gravity solutions in a Robertson-Walker spacetime (TRW). We had found the TRW geometry and we had solved the TRW FEs and conservation laws (CL) for PF and SF solutions in teleparallel $F\left(T\right)$ and $F(T,B)$ gravities [55,56,57,62]. But we can do further and aim to solve for polytropic and Chaplygin cosmological gases teleparallel $F\left(T\right)$ solutions as the next step of development. For satisfying this main aim, we will use the same TRW geometry, FEs and CLs to develop the teleparallel Chaplygin $F\left(T\right)$ solutions in Section 3, the teleparallel polytropic $F\left(T\right)$ solutions in Section 4. We will then compare and highlight the similarities and differences by using graphs in Section 5 before concluding the current paper in Section 6.

2. Summary of Teleparallel Gravity and Field Equations

2.1. Teleparallel $F\left(T\right)$-Gravity Theory Field Equations and Torsional Quantities

The teleparallel $F\left(T\right)$-type gravity action integral with any gravitational source is [2,3,5,7,47,48,49,50,51,52,55]:

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

(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 Equation (1) to find the symmetric and antisymmetric parts of FEs as [47,48,49,50,51,52,55]:

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

(2)

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

(3)

with ${\stackrel{\circ }{G}}_{ab}$ the Einstein tensor, ${\mathrm{\Theta }}_{\left(ab\right)}$ the energy-momentum, $g_{ab}$ the gauge metric and $\kappa$ the coupling constant. The torsion tensor $T_{\mu \nu }^a$, the torsion scalar T and the super-potential $S_a^{\mu \nu }$ are defined as [5]:

$$\begin{array}{cc}\hfill T_{\mu \nu }^a=& {𝜕}_{\mu }{h}_{\nu }^a-{𝜕}_{\nu }{h}_{\mu }^a+{\omega }_{b\mu }^a{h}_{\nu }^b-{\omega }_{b\nu }^a{h}_{\mu }^b,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill S_a^{\mu \nu }=& {\displaystyle \frac{1}{2}}\left(T_a^{\mu \nu }+T_a^{\nu \mu }-T_a^{\mu \nu }\right)-h_a^{\nu }{T}_{\lambda }^{\lambda \mu }+h_a^{\mu }{T}_{\lambda }^{\lambda \nu },\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill T=& {\displaystyle \frac{1}{2}}{T}_{\mu \nu }^a{S}_a^{\mu \nu }.\hfill \end{array}$$

(6)

Equation (4) can be expressed in terms of the three irreducible parts of torsion tensor as:

$$\begin{array}{c}\hfill T_{abc}={\displaystyle \frac{2}{3}}\left(t_{abc}-t_{acb}\right)-{\displaystyle \frac{1}{3}}\left(g_{ab}{V}_c-g_{ac}{V}_b\right)+{ϵ}_{abcd}{A}^d\end{array}$$

(7)

where,

$$\begin{array}{cc}\hfill V_a=& T_{ba}^b,A^a={\displaystyle \frac{1}{6}}{ϵ}^{abcd}{T}_{bcd},t_{abc}={\displaystyle \frac{1}{2}}\left(T_{abc}+T_{bac}\right)-{\displaystyle \frac{1}{6}}\left(g_{ca}{V}_b+g_{cb}{V}_a\right)+{\displaystyle \frac{1}{3}}{V}_c.\hfill \end{array}$$

(8)

We usually solve in teleparallel $F\left(T\right)$ gravity the Equations (2)–(3). Therefore in refs. [55,56,57], we showed that Equation (3) is trivially satisfied despite a non-zero spin-connection, because the teleparallel geometry is purely symmetric. Only the Equations (2) is non-trivial and will be explicitly solved in detail.

2.2. Teleparallel Robertson-Walker Spacetime Geometry

Any frame-based geometry in teleparallel gravity on a frame bundle is defined by a coframe/spin-connection pair and a field $\mathbf{X}$. The geometry must satisfy the fundamental Lie Derivative-based equations [5,6,55,56,57]:

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

(9)

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. In addition for a pure teleparallel $F\left(T\right)$-type gravity, we must also satisfy the null Riemann curvature condition $R_{abc}^a=0$. For TRW spacetime geometries on an orthonormal frame, the coframe/spin-connection pair $h_{\mu }^a$ and ${\omega }_{abc}$ solutions are [55,56,57] :

$$\begin{array}{cc}\hfill h_{\mu }^a=& Diag\left[1,a\left(t\right){\left(1-k{r}^2\right)}^{-1/2},a\left(t\right)r,a\left(t\right)rsin\theta \right],\hfill \\ \hfill {\omega }_{122}=& {\omega }_{133}={\omega }_{144}=W_1\left(t\right),{\omega }_{234}=-{\omega }_{243}={\omega }_{342}=W_2\left(t\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\omega }_{233}=& {\omega }_{244}=-{\displaystyle \frac{\sqrt{1-k{r}^2}}{a\left(t\right)r}},{\omega }_{344}={\displaystyle \frac{cot\left(\theta \right)}{a\left(t\right)r}},\hfill \end{array}$$

(11)

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

1.

$k=0$: $W_1=W_2=0$,

2.

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

3.

$k=-1$: $W_1\left(t\right)=\pm {\displaystyle \frac{\sqrt{-k}}{a\left(t\right)}}$ 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. The previous coframe/ spin-connection pair was found by solving the Equations (9) and $R_{b\mu \nu }^a=0$ condition as defined in ref. [5]. These solutions were also used in $F\left(T\right)$ TRW spacetime recent works [55,56,57]. The TRW spacetime structure is typically explanable by a $G_6$ Lie algebra group. The FEs to be solved in the current paper are defined for each k-parameter cases and will lead to additional new teleparallel $F\left(T\right)$ solution classes. The FEs defined by Equations (2)–(3) are still purely symmetric and valid on proper frames as showed in refs. [55,56,57]. The Equations (3) are trivially satisfied and we will solve the Equations (2) for each k-parameter case.

2.3. Conservation Laws and Field Equations of Cosmological Perfect Fluids

The canonical energy-momentum and its GR CLs are obtained from ${\mathcal{L}}_{Source}$ term of Equation (1) as [3,7]:

$$\begin{array}{cc}\hfill {\mathrm{\Theta }}_a^{\mu }=& {\displaystyle \frac{1}{h}}{\displaystyle \frac{\delta {\mathcal{L}}_{Source}}{\delta h_{\mu }^a}},\Rightarrow {\stackrel{\circ }{\nabla }}_{\nu }\left({\mathrm{\Theta }}^{\mu \nu }\right)=0,\hfill \end{array}$$

(12)

where ${\stackrel{\circ }{\nabla }}_{\nu }$ the covariant derivative and ${\mathrm{\Theta }}^{\mu \nu }$ the conserved energy-momentum tensor. The antisymmetric and symmetric parts of ${\mathrm{\Theta }}_{ab}$ are [47,48,49,50,51,52,55]:

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

(13)

where $T_{ab}$ is the symmetric part of ${\mathrm{\Theta }}^{\mu \nu }$. The Equation (12) also imposes the symmetry of ${\mathrm{\Theta }}^{\mu \nu }$ and then Equations (13) condition. Equation (13) 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. [48,49,50,51,52,54,55]. This last condition on hypermomentum is defined from Equations (2)–(3) as [54]:

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

(14)

There are more general teleparallel ${\mathfrak{T}}^{\mu \nu }$ definitions and ${\mathfrak{T}}^{\mu \nu }\ne 0$ CLs, but this does not really concern the teleparallel $F\left(T\right)$-gravity situation [54,112,113,114].

For a TRW spacetime geometry defined by Equations (10)–(11), the Equation (12) for a $P=P\left(\rho \right)$ fluid is [55,56,57]:

$$\begin{array}{c}\hfill \dot{\rho }+3H\left(P\left(\rho \right)+\rho \right)=0,\end{array}$$

(15)

where $H={\displaystyle \frac{\dot{a}}{a}}$ is the Hubble parameter. The general FEs system for TRW cosmological spacetimes are [55,56,57]:

1.

$\mathbf{k}=\mathbf{0}$ flat or non-curved:

$$\begin{array}{cc}\hfill \kappa \rho =& -{\displaystyle \frac{F}{2}}+6H^2{F}_T,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \kappa (\rho +3P(\rho \left)\right)=& F-6\left(\dot{H}+2H^2\right)F_T-6H{F}_{TT}\dot{T},\hfill \end{array}$$

(17)

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

(18)

The Equation (18) yields to $H=\sqrt{{\displaystyle \frac{T}{6}}}$, and then Equations (16)–(17) become:

$$\begin{array}{cc}\hfill \kappa \rho =& -{\displaystyle \frac{F}{2}}+T{F}_T,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \kappa (\rho +3P(\rho \left)\right)=& F-\left(6\dot{H}+2T\right)F_T-12\dot{H}T{F}_{TT},\hfill \end{array}$$

(20)

The pure vacuum solution ($\rho =0$ and $P=0$) to Equations (19)–(20) is $F\left(T\right)=F_0\sqrt{T}$. However for any $P=P\left(\rho \right)$, we can set $a\left(t\right)=a_0{t}^n$ as cosmological scale and $F\left(T\right)=\sqrt{T}G\left(T\right)$ as solution ansatz, and we find the unified FE by merging Equations (19)–(20):

$$\begin{array}{cc}\hfill {\displaystyle \frac{3}{2}}\left(n-1+{\displaystyle \frac{nP\left(\rho \right)}{\rho }}\right)G_T=& T{G}_{TT}.\hfill \end{array}$$

(21)

Equation (21) is the general $k=0$ unified FE to solve for any EoS and CL. This is an easy-to-solve and the solution will be some easy-to-compute integral equation.

2.

$\mathbf{k}=-\mathbf{1}$ negative curved:

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

(22)

$$\begin{array}{cc}\hfill \kappa (\rho +3P(\rho \left)\right)=& F-6\left(\dot{H}+H^2+{\left(H+{\displaystyle \frac{\delta \sqrt{-k}}{a}}\right)}^2\right)F_T-6\left(H+{\displaystyle \frac{\delta \sqrt{-k}}{a}}\right)F_{TT}\dot{T},\hfill \end{array}$$

(23)

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

(24)

From Equation (24) and using $a\left(t\right)=a_0{t}^n$ ansatz, we find a characteristic equation yielding to $t\left(T\right)$ solutions:

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

(25)

The possible solutions of Equation (25) are:

(a)

$\mathbf{n}={\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$ (slow expansion):

$$\begin{array}{cc}\hfill & t^{-1}\left(T\right)={\left[-{\displaystyle \frac{\delta \sqrt{-k}}{a_0}}\pm \sqrt{-{\displaystyle \frac{k}{a_0^2}}+{\delta }_1\sqrt{{\displaystyle \frac{2T}{3}}}}\right]}^2.\hfill \end{array}$$

(26)

(b)

$\mathbf{n}=\mathbf{1}$ (linear expansion):

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

(27)

(c)

$\mathbf{n}=\mathbf{2}$ (fast expansion):

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

(28)

(d)

$\mathbf{n}\to \infty$ (very fast expansion limit):

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

(29)

Then Equations (22)–(23) become by substituting Equation (25) and the ansatz:

$$\begin{array}{cc}\hfill \kappa \rho =& -{\displaystyle \frac{F}{2}}+\sqrt{6}n{\delta }_1{t}^{-1}\left(T\right)\sqrt{T}{F}_T,\hfill \\ \hfill \kappa \rho \left(1+{\displaystyle \frac{3P\left(\rho \right)}{\rho }}\right)=& F-\left(6n(n-1)t^{-2}\left(T\right)+T\right)F_T\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & +12n{t}^{-1}\left(T\right)\left((1-n)t^{-1}\left(T\right)+{\delta }_1\sqrt{{\displaystyle \frac{T}{6}}}\right)T{F}_{TT},\hfill \end{array}$$

(31)

The unified FE from Equations (30)–(31) is:

$$\begin{array}{cc}\hfill & \left[-{\displaystyle \frac{F}{2}}+\sqrt{6}n{\delta }_1{t}^{-1}\left(T\right)\sqrt{T}{F}_T\right]\left(1+{\displaystyle \frac{3P\left(\rho \right)}{\rho }}\right)\hfill \\ \hfill & =F-\left(6n(n-1)t^{-2}\left(T\right)+T\right)F_T+12n{t}^{-1}\left(T\right)\left((1-n)t^{-1}\left(T\right)+{\delta }_1\sqrt{{\displaystyle \frac{T}{6}}}\right)T{F}_{TT},\hfill \end{array}$$

(32)

3.

$\mathbf{k}=+\mathbf{1}$ positive curved:

$$\begin{array}{cc}\hfill \kappa \rho =& -{\displaystyle \frac{F}{2}}+6H^2{F}_T,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill \kappa (\rho +3P(\rho \left)\right)=& F-6\left(\dot{H}+2H^2-{\displaystyle \frac{k}{a^2}}\right)F_T-6H{F}_{TT}\dot{T},\hfill \end{array}$$

(34)

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

(35)

From Equation (35) and using $a\left(t\right)=a_0{t}^n$ ansatz, we find the characteristic equation for $t^{-1}\left(T\right)$:

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

(36)

The possible solutions of Equation (36) are:

(a)

$\mathbf{n}={\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$ (slow expansion):

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

(37)

(b)

$\mathbf{n}=\mathbf{1}$ (linear expansion):

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

(38)

(c)

$\mathbf{n}=\mathbf{2}$ (fast expansion):

$$\begin{array}{cc}\hfill & t^{-2}\left(T\right)={\displaystyle \frac{2a_0^2}{k}}\pm \sqrt{{\displaystyle \frac{4a_0^4}{k^2}}-{\displaystyle \frac{a_0^{2}T}{6k}}}.\hfill \end{array}$$

(39)

(d)

$\mathbf{n}\to \infty$ (very fast expansion limit):

$$\begin{array}{c}\hfill t^{-2}\left(T\right)\approx {\displaystyle \frac{T}{6n^2}}\to 0.\end{array}$$

(40)

Then Equations (33)–(34) become by substituting Equation (36) and the ansatz:

$$\begin{array}{cc}\hfill \kappa \rho =& -{\displaystyle \frac{F}{2}}+6n^2{t}^{-2}\left(T\right)F_T,\hfill \\ \hfill \kappa \rho \left(1+{\displaystyle \frac{3P\left(\rho \right)}{\rho }}\right)=& F-\left(6n(n-1)t^{-2}\left(T\right)+T\right)F_T\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & +12n^2{t}^{-2}\left(T\right)\left(T-6n(n-1)t^{-2}\left(T\right)\right)F_{TT},\hfill \end{array}$$

(42)

The unified FE from Equations (41)–(42) is:

$$\begin{array}{cc}\hfill & \left[-{\displaystyle \frac{F}{2}}+6n^2{t}^{-2}\left(T\right)F_T\right]\left(1+{\displaystyle \frac{3P\left(\rho \right)}{\rho }}\right)\hfill \\ \hfill & =F-\left(6n(n-1)t^{-2}\left(T\right)+T\right)F_T+12n^2{t}^{-2}\left(T\right)\left(T-6n(n-1)t^{-2}\left(T\right)\right)F_{TT},\hfill \end{array}$$

(43)

2.4. Energy Conditions and Thermodynamic Laws in Teleparallel Gravity

Regardless of the definition of EoS or any pressure-density relationship, there are energy conditions to satisfy for any physical system based on a PF [115]:

• Weak Energy Condition (WEC): $\rho \ge 0$, $P_r+\rho \ge 0$ and $P_t+\rho \ge 0$.
• Strong Energy Condition (SEC): $P_r+2P_t+\rho \ge 0$, $P_r+\rho \ge 0$ and $P_t+\rho \ge 0$.
• Null Energy Condition (NEC): $P_r+\rho \ge 0$ and $P_t+\rho \ge 0$.
• Dominant Energy Condition (DEC): $\rho \ge |P_r|$ and $\rho \ge |P_t|$.

From this consideration and by setting $P_r=P_t$ for a uniform pressure cosmological fluid, we can summarize the energy conditions (ECs):

$$\begin{array}{cc}\hfill & \rho +P\ge 0,\rho +3P\ge 0,\rho \ge \left|P\right|,\rho \ge 0.\hfill \end{array}$$

(44)

We will apply in Section 3 and Section 4 the Equations (44) for any CL solutions. By this way, we will verify the physical consistency of all CL solutions found in the current paper.

3. Pure Chaplygin Gas Teleparallel Field Equation Solutions

3.1. Conservation Law Solutions and Energy Conditions

For a pure Chaplygin gas of EoS $P={\displaystyle \frac{P_0}{\rho }}$ (see refs. [98,99,100,101,102,111]) and by using a power-law ansatz $a\left(t\right)=a_0{t}^n$, the Equation (15) becomes:

$$\begin{array}{cc}\hfill & \dot{\rho }+{\displaystyle \frac{3n}{t}}\left({\displaystyle \frac{P_0}{\rho }}+\rho \right)=0,\hfill \\ \hfill & \Rightarrow \rho \left(t\right)={\left[\left(P_0+{\rho }_0^2\right)t^{-6n}-P_0\right]}^{1/2}.\hfill \end{array}$$

(45)

We will apply the ECs as stated by Equations (44) on Equation (45) and find that:

$$\begin{array}{cc}\hfill & \left(P_0+{\rho }_0^2\right)t^{-6n}\ge 0,\left(P_0+{\rho }_0^2\right)t^{-6n}\ge -2P_0,\left(P_0+{\rho }_0^2\right)t^{-6n}\ge P_0+\left|P_0\right|,\hfill \\ \hfill & \left(P_0+{\rho }_0^2\right)t^{-6n}\ge P_0.\hfill \end{array}$$

(46)

The dominating condition from Equations (46) is:

$$\begin{array}{cc}\hfill & \left(P_0+{\rho }_0^2\right)t^{-6n}\ge 2\left|P_0\right|.\hfill \end{array}$$

(47)

3.2. $k=0$ Cosmological Solutions

Using Equation (45) for CL and EoS and the torsion scalar defined by Equation (18), the unified FEs described by Equation (21) and the solution are:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{3}{2}}\left[n-1+{\displaystyle \frac{n{P}_0}{\left[\left(P_0+{\rho }_0^2\right)T^{3n}-P_0\right]}}\right]G_T=T{G}_{TT},\hfill \\ \hfill & \Rightarrow G_T\left(T\right)=G_T\left(0\right)T^{{\displaystyle \frac{3}{2}}(n-1)}\sqrt{\beta T^{3n}-P_0},\hfill \end{array}$$

(48)

where $\beta ={\displaystyle \frac{P_0+{\rho }_0^2}{{\left(6n^2\right)}^{3n}}}$. By integration and substitution into $F\left(T\right)$ ansatz, we find as final solution:

$$\begin{array}{c}\hfill F\left(T\right)=G\left(0\right)\sqrt{T}-2G_T\left(0\right)\sqrt{-P_0} {}_{2}F_1\left(-{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{6n}};1-{\displaystyle \frac{1}{6n}};{\displaystyle \frac{\beta }{P_0}}{T}^{3n}\right).\end{array}$$

(49)

As in refs [55,56], the $k=0$ flat cosmological case yields to an easy to compute analytical teleparallel $F\left(T\right)$ solution for any value of n.

3.3. $k=-1$ Cosmological Solutions

The Equation (32) with $P\left(\rho \right)={\displaystyle \frac{P_0}{\rho }}$ is by substituting Equation (45) solution:

$$\begin{array}{cc}\hfill & \left[-{\displaystyle \frac{F}{2}}+\sqrt{6}n{\delta }_1{t}^{-1}\left(T\right)\sqrt{T}{F}_T\right]\left({\displaystyle \frac{\left(P_0+{\rho }_0^2\right)t^{-6n}\left(T\right)+2P_0}{\left(P_0+{\rho }_0^2\right)t^{-6n}\left(T\right)-P_0}}\right)\hfill \\ \hfill & =F-\left(6n(n-1)t^{-2}\left(T\right)+T\right)F_T+12n{t}^{-1}\left(T\right)\left((1-n)t^{-1}\left(T\right)+{\delta }_1\sqrt{{\displaystyle \frac{T}{6}}}\right)T{F}_{TT}.\hfill \end{array}$$

(50)

The possible solution to Equation (50) are by using a power-law ansatz $F\left(T\right)=F_0{T}^r$:

1.

$\mathbf{n}={\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$: By using the approximation ${\displaystyle \frac{a_0^2{\delta }_1}{k}}\sqrt{{\displaystyle \frac{2T}{3}}}\ll 1$ and setting the + root, Equation (50) simplifies as:

$$\begin{array}{cc}\hfill & 0\approx C_1{T}^{2}F+\left(2C_2{C}_1{T}^{7/2}-2C_2\sqrt{T}+C_2^{2}T-1\right)F_T+2C_2\left(C_{2}T-\sqrt{T}\right)T{F}_{TT},\hfill \end{array}$$

(51)

where $C_1=\left(1+{\displaystyle \frac{{\rho }_0^2}{P_0}}\right){\displaystyle \frac{a_0^6}{216k^3}}$ and $C_2={\displaystyle \frac{{\delta }_1{a}_0^2}{2\sqrt{6}k}}$. The solution of Equation (51) is:

$$\begin{array}{cc}\hfill & F\left(T\right)\approx F_{+}{T}^{r_{+}}+F_{-}{T}^{r_{-}}\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill r_{\pm }=& {\displaystyle \frac{1}{4C_2(C_2-1)}}[-\left(2C_1{C}_2-10C_2+7C_2^2-1\right)\pm [{\left(2C_1{C}_2-10C_2+7C_2^2-1\right)}^2\hfill \\ \hfill & -8C_2(C_2-1)\left(C_1(1-C_2)-{\displaystyle \frac{25}{2}}{C}_2+6C_2^2-3\right){]}^{1/2}],\hfill \end{array}$$

(53)

where $C_2\ne \left\{0,1\right\}$.

2.

$\mathbf{n}=\mathbf{1}$: For $C_3\ll 1$, Equation (50) can be approximated as:

$$\begin{array}{cc}\hfill & 0\approx C_3{T}^{2}F-{\displaystyle \frac{2}{C_4}}\left(1+C_3{T}^3-{\displaystyle \frac{C_4}{2}}\right)F_T-{\displaystyle \frac{2}{C_4}}T{F}_{TT},\hfill \end{array}$$

(54)

where $C_3={\displaystyle \frac{\left(1+\frac{{\rho }_0^2}{P_0}\right)}{216{\left(\frac{\delta \sqrt{-k}}{a_0}+1\right)}^6}}$ and $C_4={\displaystyle \frac{\delta \sqrt{-k}}{a_0}}+1$. The solution of Equation (54) is described by Equation (52) with the $r_{\pm }$ roots:

$$\begin{array}{cc}\hfill r_{\pm }=& -{\displaystyle \frac{1}{2}}\left(6+C_3-{\displaystyle \frac{C_4}{2}}\right)\pm {\left[{\displaystyle \frac{1}{4}}{\left(6+C_3-{\displaystyle \frac{C_4}{2}}\right)}^2-\left(9-{\displaystyle \frac{3C_4}{2}}-{\displaystyle \frac{C_3{C}_4}{2}}\right)\right]}^{1/2}.\hfill \end{array}$$

(55)

3.

$\mathbf{n}=\mathbf{2}$: By using the approximation ${\displaystyle \frac{\delta {\delta }_1\sqrt{-k}}{a_0\sqrt{6}}}\sqrt{T}\ll 1$ and setting the + root, Equation (50) simplifies as:

$$\begin{array}{cc}\hfill & 0\approx C_5{T}^{5}F+\left({\displaystyle \frac{7}{2}}+2C_5{T}^6\right)F_T+3T{F}_{TT},\hfill \end{array}$$

(56)

where $C_5\approx 5,233\times {10}^{-9}\left(1+{\displaystyle \frac{{\rho }_0^2}{P_0}}\right)\ll 1$. The solution of Equation (54) is still the Equation (52) with $r_{\pm }$ roots:

$$\begin{array}{cc}\hfill r_{\pm }=& {\displaystyle \frac{1}{12}}\left[-\left(1+4C_5\right)\pm {\left[1+16C_5^2-520C_5\right]}^{1/2}\right]\approx {\displaystyle \frac{1}{12}}\left[-\left(1+4C_5\right)\pm \left(1-260C_5\right)\right].\hfill \end{array}$$

(57)

Under the $C_5\to 0$ limit: $r_{+}\approx -22C_5\to 0$ and $r_{-}\approx -{\displaystyle \frac{1}{6}}+{\displaystyle \frac{64}{3}}{C}_5\to -{\displaystyle \frac{1}{6}}$. In this case, Equation (52) leads to $F\left(T\right)\to -{\mathrm{\Lambda }}_0+F_{-}{T}^{-1/6}$ with the cosmological constant ${\mathrm{\Lambda }}_0$. This last solution is similar to those of ref. [62].

4.

$\mathbf{n}\to \infty$: Under this limit, Equation (50) simplifies as a simple homogeneous equation for $P_0\ne =-{\rho }_0^2$:

$$\begin{array}{cc}\hfill & T{F}_T\approx {\displaystyle \frac{F}{2}}\Rightarrow F\left(T\right)\approx F_0\sqrt{T},\hfill \end{array}$$

(58)

as we can find for any pure flat background cosmological simple solutions [49,55,56].

3.4. $k=+1$ Cosmological Solutions

The Equation (43) with $P\left(\rho \right)={\displaystyle \frac{P_0}{\rho }}$ is by substituting Equation (45) solution:

$$\begin{array}{cc}\hfill & \left[-{\displaystyle \frac{F}{2}}+6n^2{t}^{-2}\left(T\right)F_T\right]\left({\displaystyle \frac{\left(P_0+{\rho }_0^2\right)t^{-6n}\left(T\right)+2P_0}{\left(P_0+{\rho }_0^2\right)t^{-6n}\left(T\right)-P_0}}\right)\hfill \\ \hfill & =F-\left(6n(n-1)t^{-2}\left(T\right)+T\right)F_T+12n^2{t}^{-2}\left(T\right)\left(T-6n(n-1)t^{-2}\left(T\right)\right)F_{TT}.\hfill \end{array}$$

(59)

The possible solution to Equation (59) are by using the $F\left(T\right)=F_0{T}^r$ ansatz:

1.

$\mathbf{n}={\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$: By using the approximation ${\displaystyle \frac{a_0^4}{6k^2}}T\ll 1$ and setting the − root, Equation (59) simplifies as:

$$\begin{array}{cc}\hfill 0\approx & C_7{T}^{2}F+\left(3C_{6}T-2C_6{C}_7{T}^4-1\right)F_T+2C_6\left(1+C_{6}T\right)T^2{F}_{TT},\hfill \end{array}$$

(60)

where $C_6={\displaystyle \frac{a_0^4}{24k^2}}$ and $C_7=\left(1+{\displaystyle \frac{{\rho }_0^2}{P_0}}\right){\displaystyle \frac{a_0^6}{216k^3}}\ll 1$. The solution of Equation (60) is Equation (52) with $r_{\pm }$ roots:

$$\begin{array}{cc}\hfill r_{\pm }=& {\displaystyle \frac{1}{4C_6(1+C_6)}}[-\left(13C_6-2C_6{C}_7+6C_6^2-1\right)\pm [{\left(13C_6-2C_6{C}_7+6C_6^2-1\right)}^2\hfill \\ \hfill & -8C_6(1+C_6)\left(4C_6^2+21C_6-4\right){]}^{1/2}],\hfill \end{array}$$

(61)

where $C_6\ne \left\{-1,0\right\}$.

2.

$\mathbf{n}=\mathbf{1}$: Equation (59) becomes:

$$\begin{array}{cc}\hfill 0\approx & C_9{T}^{2}F+\left(1-{\displaystyle \frac{2}{C_8}}\left(1+C_9{T}^3\right)\right)F_T-{\displaystyle \frac{2T{F}_{TT}}{C_8}},\hfill \end{array}$$

(62)

where $C_8=1-{\displaystyle \frac{k}{a_0^2}}$ and $C_9={\displaystyle \frac{\left(1+\frac{{\rho }_0^2}{P_0}\right)}{216{\left(1-\frac{k}{a_0^2}\right)}^3}}\ll 1$. The solution of Equation (62) is Equation (52) with $r_{\pm }$ roots:

$$\begin{array}{cc}\hfill r_{\pm }=& -{\displaystyle \frac{1}{2}}\left(C_9+{\displaystyle \frac{C_8}{2}}+6\right)\pm {\left[{\displaystyle \frac{1}{4}}{\left(C_9+{\displaystyle \frac{C_8}{2}}+6\right)}^2-\left(9+{\displaystyle \frac{3C_8}{2}}-{\displaystyle \frac{C_8{C}_9}{2}}\right)\right]}^{1/2}.\hfill \end{array}$$

(63)

3.

$\mathbf{n}=\mathbf{2}$: By using the approximation ${\displaystyle \frac{k}{24a_0^2}}T\ll 1$ and setting the − root, Equation (59) simplifies under the $C_{10}\ll 1$ approximation as:

$$\begin{array}{cc}\hfill 0\approx & C_{10}{T}^{5}F-\left({\displaystyle \frac{1}{2}}+2C_{10}{T}^6\right)F_T-T{F}_{TT},\hfill \end{array}$$

(64)

where $C_{10}=\left(1+{\displaystyle \frac{{\rho }_0^2}{P_0}}\right){\left({\displaystyle \frac{1}{24}}\right)}^6\ll 1$. The solution of Equation (64) is Equation (52) with $r_{\pm }$ roots:

$$\begin{array}{cc}\hfill r_{\pm }=& -\left(C_{10}+{\displaystyle \frac{23}{4}}\right)\pm {\left[{\left(C_{10}+{\displaystyle \frac{23}{4}}\right)}^2-33+C_{10}\right]}^{1/2}\approx -C_{10}-{\displaystyle \frac{23}{4}}\pm \left({\displaystyle \frac{1}{4}}+25C_{10}\right).\hfill \end{array}$$

(65)

Under the $C_{10}\to 0$ limit: $r_{+}\to -{\displaystyle \frac{11}{2}}$ and $r_{-}\to -6$. Then Equation (52) becomes $F\left(T\right)\to F_{+}{T}^{-{\displaystyle \frac{11}{2}}}+F_{+}{T}^{-6}$.

4.

$\mathbf{n}\to \infty$: Once again for $P_0\ne -{\rho }_0^2$, we find under this limit the same differential equation and solution as Equation (58), i.e. $F\left(T\right)\approx F_0\sqrt{T}$.

4. General Polytropic Gas Teleparallel Field Equation Solutions

4.1. Conservation Law Solutions and Energy Conditions

A polytropic gas is defined as $\rho =\rho \left(t\right)$ and $P=-P_0{\rho }^{1+{\displaystyle \frac{1}{p}}}$ where ${\displaystyle \frac{1}{3}}<P_0\le 1$ and $0<p<\infty$[103,104,105,106,107,108,109,110]. Under the $p\to \infty$ limit, we find that $P=-P_0\rho$, a linear DE PF. However the $p\to 0$ limit will lead to an infinitely huge pressure universe looking like the very early universe. The Equation (15) for this general type of gas is by using the $a\left(t\right)=a_0{t}^n$ ansatz:

$$\begin{array}{cc}\hfill & \dot{\rho }+{\displaystyle \frac{3n}{t}}\rho \left(1-P_0{\rho }^{{\displaystyle \frac{1}{p}}}\right)=0,\hfill \\ \hfill & \Rightarrow \rho \left(t\right)={\left[P_0-\left(P_0-{\rho }_0^{-1/p}\right)t^{3n/p}\right]}^{-p}.\hfill \end{array}$$

(66)

We will apply the ECs defined by Equations (44) for Equation (66) CL solution as:

$$\begin{array}{cc}\hfill & \left(P_0-{\rho }_0^{-1/p}\right)t^{3n/p}\le 0,\left(P_0-{\rho }_0^{-1/p}\right)t^{3n/p}\le -2P_0,\left(P_0-{\rho }_0^{-1/p}\right)t^{3n/p}\le P_0-\left|P_0\right|,\hfill \\ \hfill & \left(P_0-{\rho }_0^{-1/p}\right)t^{3n/p}\le P_0.\hfill \end{array}$$

(67)

The dominating energy conditions is exactly $t^{3n/p}\le {\displaystyle \frac{-2|P_0|}{\left(P_0-{\rho }_0^{-1/p}\right)}}$.

4.2. $k=0$ Cosmological Solutions

The unified FEs using Equation (18), $P=-P_0{\rho }^{1+{\displaystyle \frac{1}{p}}}$ and Equation (66) is [56]:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{3}{2}}\left(n-1+{\displaystyle \frac{n{P}_0}{\left[{\beta }_p{T}^{-3n/2p}-P_0\right]}}\right)G_T=T{G}_{TT},\hfill \\ \hfill & \Rightarrow G_T\left(T\right)=G_T\left(0\right)T^{{\displaystyle \frac{3}{2}}(n-1)}{\left[P_0{T}^{3n/2p}-{\beta }_p\right]}^{-2p/3},\hfill \end{array}$$

(68)

where ${\beta }_p=\left(P_0-{\rho }_0^{-1/p}\right){\left(6n^2\right)}^{3n/2p}$. The chaplygin gas case corresponds to the $p=-{\displaystyle \frac{1}{2}}$, i.e. ${\beta }_{1/2}=\beta$ in Section 3. The general teleparallel $F\left(T\right)$ solution is exactly:

$$\begin{array}{cc}\hfill F\left(T\right)=& G\left(0\right)\sqrt{T}+G_T\left(0\right)\sqrt{T}\left[\int d{T}^{\prime }{T}^{\prime 3(n-1)/2}{\left(P_0{T}^{3n/2p}-{\beta }_p\right)}^{-2p/3}\right],\hfill \\ \hfill =& G\left(0\right)\sqrt{T}+G_T\left(0\right)\sqrt{T}{I}_p\left(T\right),\hfill \end{array}$$

(69)

where $I_p\left(T\right)$ is a new integral-based special function. Some values of $I_p\left(T\right)$ are displayed in Table 1. The $p\ge 0$ solutions yield to hypergeometric functions and $p<0$ solution yield to power-law superposition solutions. We confirm in part the refs [55,56] where $k=0$ flat cosmological case still yields to analytical teleparallel $F\left(T\right)$ solution for each value of p and n.

4.3. $k=-1$ Cosmological Solutions

The Equation (32) for $P=-P_0{\rho }^{1+{\displaystyle \frac{1}{p}}}$ is by substituting Equation (66) solution:

$$\begin{array}{cc}\hfill & \left[-{\displaystyle \frac{F}{2}}+\sqrt{6}n{\delta }_1{t}^{-1}\left(T\right)\sqrt{T}{F}_T\right]\left({\displaystyle \frac{\left(P_0-{\rho }_0^{-1/p}\right)t^{3n/p}\left(T\right)+2P_0}{\left(P_0-{\rho }_0^{-1/p}\right)t^{3n/p}\left(T\right)-P_0}}\right)\hfill \\ \hfill & =F-\left(6n(n-1)t^{-2}\left(T\right)+T\right)F_T+12n{t}^{-1}\left(T\right)\left((1-n)t^{-1}\left(T\right)+{\delta }_1\sqrt{{\displaystyle \frac{T}{6}}}\right)T{F}_{TT}.\hfill \end{array}$$

(70)

The possible solution to Equation (70) are by using the $F\left(T\right)=F_0{T}^r$ ansatz:

1.

$\mathbf{n}={\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$: By using the approximation ${\displaystyle \frac{a_0^2{\delta }_1}{k}}\sqrt{{\displaystyle \frac{2T}{3}}}\ll 1$ and the + root, Equation (70) becomes:

$$\begin{array}{cc}\hfill 0\approx & {\displaystyle \frac{3F}{2}}+\left(C_2^{2}T+C_2{T}^{1/2}-1\right)T{F}_T+2C_2\left(C_2\sqrt{T}-1\right)T^{5/2}{F}_{TT},\hfill \end{array}$$

(71)

where $C_{11}=\left(1-{\displaystyle \frac{1}{P_0{\rho }_0^{1/p}}}\right){\displaystyle \frac{{(-6k)}^{3/2p}}{{\left(\delta {\delta }_1{a}_0\right)}^{3/p}}}\gg 1$ and then ${\displaystyle \frac{C_{11}{T}^{-3/p}-2}{C_{11}{T}^{-3/p}+1}}\to 1$ under the same limit. Using the same power-law ansatz, we find the Equation (52) as solution with the roots:

$$\begin{array}{c}\hfill r_{\pm }=-{\displaystyle \frac{1}{4C_2}}\pm {\left[{\displaystyle \frac{1}{16C_2^2}}-{\displaystyle \frac{1+2C_2}{4C_2(C_2-1)}}\right]}^{1/2},\end{array}$$

(72)

where $C_2\ne \left\{0,1\right\}$.

2.

$\mathbf{n}=\mathbf{1}$: We find:

$$\begin{array}{cc}\hfill 0\approx & {\displaystyle \frac{3F}{2}}-\left(1+{\displaystyle \frac{1}{C_4}}\right)T{F}_T+{\displaystyle \frac{2}{C_4}}{T}^2{F}_{TT},\hfill \end{array}$$

(73)

where $C_{12}=\left(1-{\displaystyle \frac{1}{P_0{\rho }_0^{1/p}}}\right){\left(\sqrt{6}{\delta }_1\left({\displaystyle \frac{\delta \sqrt{-k}}{a_0}}+1\right)\right)}^{3/p}\gg 1$ and then ${\displaystyle \frac{2+C_{12}{T}^{-3/2p}}{1-C_{12}{T}^{-3/2p}}}\to -1$. We find the Equation (52) as solution with the roots $r_{+}={\displaystyle \frac{C_4}{2}}$ and $r_{-}={\displaystyle \frac{3}{2}}$.

3.

$\mathbf{n}=\mathbf{2}$: By using ${\displaystyle \frac{\delta {\delta }_1\sqrt{-k}}{a_0\sqrt{6}}}\sqrt{T}\ll 1$ and the + root, Equation (70) simplifies as:

$$\begin{array}{cc}\hfill 0\approx & {\displaystyle \frac{3F}{2}}-{\displaystyle \frac{1}{2}}T{F}_T-3T^2{F}_{TT},\hfill \end{array}$$

(74)

where $C_{13}=\left(1-{\displaystyle \frac{1}{P_0{\rho }_0^{1/p}}}\right){\left(24\right)}^{3/p}\gg 1$ and then ${\displaystyle \frac{2+C_{13}{T}^{-3/p}}{1-C_{13}{T}^{-3/p}}}\to -1$. We find the Equation (52) as solution with the roots $r_{\pm }={\displaystyle \frac{5\pm \sqrt{97}}{12}}$.

4.

$\mathbf{n}\to \infty$ limit: For $P_0\ne {\rho }_0^{-1/p}$, we obtain the Equation (58) with $F\left(T\right)\approx F_0\sqrt{T}$ as solution.

4.4. $k=+1$ Cosmological Solutions

The Equation (43) for $P=-P_0{\rho }^{1+{\displaystyle \frac{1}{p}}}$ is by substituting Equation (66) solution:

$$\begin{array}{cc}\hfill & \left[-{\displaystyle \frac{F}{2}}+6n^2{t}^{-2}\left(T\right)F_T\right]\left({\displaystyle \frac{\left(P_0-{\rho }_0^{-1/p}\right)t^{3n/p}\left(T\right)+2P_0}{\left(P_0-{\rho }_0^{-1/p}\right)t^{3n/p}\left(T\right)-P_0}}\right)\hfill \\ \hfill & =F-\left(6n(n-1)t^{-2}\left(T\right)+T\right)F_T+12n^2{t}^{-2}\left(T\right)\left(T-6n(n-1)t^{-2}\left(T\right)\right)F_{TT}.\hfill \end{array}$$

(75)

The possible solution to Equation (75) are by using the $F\left(T\right)=F_0{T}^r$ ansatz:

1.

$\mathbf{n}={\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$: By using the approximation ${\displaystyle \frac{a_0^4}{6k^2}}T\ll 1$ and the − root, Equation (75) simplifies as:

$$\begin{array}{cc}\hfill 0\approx & {\displaystyle \frac{3F}{2}}-T{F}_T+2C_6\left(1+C_{6}T\right)T^3{F}_{TT},\hfill \end{array}$$

(76)

where $C_{14}=\left(1-{\displaystyle \frac{1}{P_0{\rho }_0^{1/p}}}\right){\left(-{\displaystyle \frac{6k}{a_0^2}}\right)}^{3/2p}\gg 1$. We find the Equation (52) as solution with the roots:

$$\begin{array}{c}\hfill r_{\pm }={\displaystyle \frac{\left(1-2C_6+2C_6^2\right)}{4C_6(1+C_6)}}\pm {\left[{\displaystyle \frac{{\left(1-2C_6+2C_6^2\right)}^2}{16C_6^2{(1+C_6)}^2}}+{\displaystyle \frac{1}{2}}\right]}^{1/2},\end{array}$$

(77)

where $C_6\ne \left\{-1,0\right\}$.

2.

$\mathbf{n}=\mathbf{1}$: With the approximation ${\displaystyle \frac{a_0^4}{6k^2}}T\ll 1$, we find that:

$$\begin{array}{cc}\hfill 0\approx & {\displaystyle \frac{3F}{2}}-\left(1+{\displaystyle \frac{1}{C_8}}\right)T{F}_T+{\displaystyle \frac{2T^2{F}_{TT}}{C_8}},\hfill \end{array}$$

(78)

where $C_{15}=\left(1-{\displaystyle \frac{1}{P_0{\rho }_0^{1/p}}}\right){\left(6\left(1-{\displaystyle \frac{k}{a_0^2}}\right)\right)}^{3/2p}\gg 1$. We find the Equation (52) as solution with the roots $r_{+}={\displaystyle \frac{C_8}{2}}$ and $r_{-}={\displaystyle \frac{3}{2}}$

3.

$\mathbf{n}=\mathbf{2}$: By using ${\displaystyle \frac{k}{24a_0^2}}T\ll 1$, $C_{13}\gg 1$ and − root, Equation (75) simplifies as:

$$\begin{array}{cc}\hfill 0\approx & {\displaystyle \frac{3F}{2}}-{\displaystyle \frac{5}{2}}T{F}_T+T^2{F}_{TT}.\hfill \end{array}$$

(79)

We find the Equation (52) as solution with the roots $r_{+}=3$ and $r_{-}={\displaystyle \frac{1}{2}}$.

4.

$\mathbf{n}\to \infty$ limit: Here again for $P_0\ne {\rho }_0^{-1/p}$, we obtain the Equation (58) with $F\left(T\right)\approx F_0\sqrt{T}$ as solution.

5. Comparison Between Chaplygin and Polytropic Gases in Teleparallel $F\left(T\right)$ Gravity

We compare the teleparallel $F\left(T\right)$ solutions for flat cosmological ($k=0$) Chaplygin and polytropic gases cases. We have plotted Equation (49) (Chaplygin gas), $p\to 0$ and 3 subcases of Equation (69) on the Figure 1. The top left subfigure compares for $n={\displaystyle \frac{1}{2}}$ case the Chaplygin and polytropic p-type gases solutions. We removed the cosmological geometry (homogeneous) $G\left(0\right)\sqrt{T}$ term (i.e. the $n\to \infty$ solution) and kept only the source contribution for making the distinction between polytropic and Chaplygin gases for $n={\displaystyle \frac{1}{2}}$, 1 and 2. The Chaplygin gas is a specific case of the general polytropic gas where we can consider the parameter $p=-{\displaystyle \frac{1}{2}}$ in the polytropic EoS (even if the EoS is defined for positive values of p). The Chaplygin gas case leads to some characteristic curves as we can also see on the top-right graphs of Figure 1. This feature makes the comparison more relevant and expresses the real nature of Chaplygin and polytropic gases.

We found for spatially curved $k=\pm 1$ cases some approximated teleparallel $F\left(T\right)$ solutions, all described by double power-law from defined by the Equation (52) form. We only found the power $r_{\pm }$ in each cases after approximating the Equations (50) and (59) for Chaplygin gas, and Equations (70) and (75) for general polytropic gas. A first feature is that the polytropic p-type teleparallel $F\left(T\right)$ solutions do not contain any p-dependent term as seen in Section 4.3 and Section 4.4 for $k=\pm 1$ solutions. The approximated polytropic teleparallel solutions are greatly simplified by some parameter approximations.

In general, there is no cosmological constant ${\mathrm{\Lambda }}_0$ in the teleparallel solutions in almost all the cases. The only exception to this rule is Equations (56)–(57) under the $C_5\to 0$ limit of $n=2$ subcase, where $r_{+}=0$ leading to $F_{+}=-{\mathrm{\Lambda }}_0$, and then a simple polynomial teleparallel $F\left(T\right)$ solution.

6. Concluding Remarks

We will first conclude the current paper by stating that the flat cosmological $k=0$ teleparallel $F\left(T\right)$ solutions are described by special functions, especially by hypergeometric functions plotted on the Figure 1 for Chaplygin ($p=-{\displaystyle \frac{1}{2}}$) and polytropic $p\to 0$ limit and 3 gases. For the spatially curved cosmological $k=\pm 1$ teleparallel $F\left(T\right)$ solutions are practically all approximated by a double power-law function described by the Equation (52). The $k=\pm 1$ general polytropic solutions found in Section 4.3 and Section 4.4 are not p-dependent because of some dominating terms approximations. There is an important feature in the new solutions: no cosmological constant ${\Lambda }_0$ term in almost all the cases. The only exception is the Equations (56)-(57) under the $C_5\to 0$ limit leading to a simple polynomial with ${\Lambda }_0$ term. Under all previous considerations, we can claim that non-flat $k=\pm 1$ teleparallel $F\left(T\right)$ solutions have some common points with those found for cosmological teleparallel $F\left(T\right)$ and $F(T,B)$ solutions found in refs [55,56,62], because we use the same coframe/spin-connection pair and ansatz.

Therefore, there are some perspective of future works going further than the current works. We can proceed for example with the same type of polytropic fluid sources for KS and static SS teleparallel spacetimes for new additional classes of solutions by using the same process and coframe ansatz approaches as in refs. [48,49,51,52]. We can also use the same TRW geometry, replace polytropic sources by an electromagnetic source, proceed by the same approach, and finally find the corresponding classes of teleparallel $F\left(T\right)$ solutions. A last insight is that we will be able to study the Anti-deSitter (AdS) spacetimes in teleparallel gravity by using polytropic sources. However, we also need to investigate the polytropic principles in teleparallel gravity framework for possible symmetries and additional KVs. More physically, we will be able to elaborate a pure teleparallel quartessence models explaining the dark energy and the dark matter under an unified model. All these mentioned suggestions are feasible in a near future and will allow to study the AdS wormholes and BHs solutions in teleparallel gravity.