1 Introduction

It is almost unanimously believed that the luminosity distance versus redshift curve (non-linear) observed from Sn1a data, is a consequence of recent accelerated expansion of the universe. A wide class of scalar field (dark energy) theories were promoted to explain the fact. Consequently, search for dark energy in the laboratory was initiated almost a decade back [1, 2], and recently it has apparently been ruled out following a laboratory based atom interferometry experiment, performed with incredible precision [3]. As an alternative to the dark energy, initially, modified theories of gravity, such as F(R), F(G), F(R, G) etc., were developed and extensively studied. Due to the fact that these theories suffer from Ostrogradsky instability, lately, alternative theories of gravity are in the lime light. These are called ‘alternatives’, since instead of curvature, which is the building block of ‘general theory of relativity’ (GTR), metric compatible (since \(\nabla _\mu g_{\alpha \beta } = 0\), where \(\nabla \) is the covariant derivative with respect to the affine connection instead of the Levi-Civita connection) and symmetric (constructed from a general affine connection symmetric in lower indices) teleparallel theories are built from the torsion and non-metricity (general affine connection) respectively. Our concern in this manuscript is the generalised teleparallel F(T) theory of gravity.

In the recent years, a generalized version of the aforementioned ’teleparallel gravity’ with torsion [4], namely the F(T) theory of gravity (where, T stands for the trace of the torsion tensor), also dubbed as ‘gravity with torsion’ has been proposed as an alternative to both the dark energy theories and the modified theories of gravity. Factually, F(T) teleparallel theory of gravity was primarily proposed to drive inflation [5, 6]. Later, it was applied to drive the current accelerated expansion of the present universe without considering dark energy [7, 8]. Thereafter, it has drawn a lot of attention and has been explored vastly and exhaustively in different contexts [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]. It is worth mentioning that ’teleparallel equivalent of general relativity’ (TEGR) is established for \(F(T) \propto T\), since dynamically it leads to GTR, apart from a divergent term. Therefore, unlike Einstein–Hilbert action, it is not required to supplement the teleparallel action with a divergent (Gibbons-Hawking-York) term. The very interesting feature of F(T) gravity is that, very much like Lanczos-Lovelock gravity, it gives second-order field equations, as a result of which Ostrogradsky’s instability is avoided. In this manuscript, our aim is to explore a physically reasonable form of F(T) theory of gravity. For the sake of self-standing, we briefly demonstrate the building block of the teleparallel theory and review the applicability of Noether theorem in different contexts.

The modified teleparallel action of F(T) gravity is given by the following action,

$$\begin{aligned} {\mathbb {A}} = \int d^4 x \mid e \mid F(T)+ S_m, \end{aligned}$$
(1)

in the units \(c = 16 \pi G = 1\), where \(|e| = \det {e^{i}_{\mu }}=\sqrt{-g}\) and \(S_m\) is the matter action. Teleparallelism [43] uses a vielbein (alternatively spelled as vierbein) field \({\textbf{e}}_{i} (x^{\mu })\) as dynamical object (it consists of four linearly independent vector fields forming a local basis for the tangent bundle, instead of the coordinate basis), which is an orthonormal basis for the tangent space at each point \(x^{\mu }\) of the manifold \({\textbf{e}}_{i}\), where \({\textbf{e}}_{ij}={\eta }_{ij}\), i runs from 0, 1, 2, 3, and \({\eta }_{ij} = \textrm{diag}(-1,1,1,1)\). The vector \({\textbf{e}}_{i}\) can be described by its components \(e^{\mu }_{i}\), \(\mu \) = 0, 1, 2, 3 in a coordinate basis, i.e. \({\textbf{e}}_{i}=e^{\mu }_{i}\partial \mu \). In the above, Latin indices refer to the tangent space, while Greek indices label coordinates on the manifold. The metric tensor is obtained from the dual vielbein as \(g_{\mu \nu }(x)=\eta _{ij}e^{i}_{\mu }(x) e^{j}_{\nu }(x)\). In contrast to GTR, which uses the torsion-less Levi-Civita connection as already mentioned, the curvature-less Weitzenb\(\ddot{\textrm{o}}\)ck connection is applied in ‘Teleparallelism’ [44], for which the non-vanishing torsion is,

$$\begin{aligned} {T^{\lambda }}_{\mu \nu } \equiv e^{\lambda }_{i}[\partial _{\mu }e^{i}_{\nu }-\partial _{\nu }e^{i}_{\mu }]. \end{aligned}$$
(2)

The above tensor encompasses all the information regarding the gravitational field. The TEGR Lagrangian, is built with the torsion (2), and its dynamical equations for the vielbein imply Einstein’s equations for the metric. The teleparallel Lagrangian given in [45,46,47] is,

$$\begin{aligned} L_T = {S_{\rho }}^{\mu \nu } {T^{\rho }}_{\mu \nu }, \end{aligned}$$
(3)

where, the tensor \({S_{\rho }}^{\mu \nu }\)

$$\begin{aligned} {S_{\rho }}^{\mu \nu } = \frac{1}{2}[{K^{\mu \nu }}_{\rho }+{\delta }^{\mu }_{\rho }{T^{\theta \nu }}_ {\theta }-{\delta }^{\nu }_{\rho }{T^{\theta \mu }}_{\theta }], \end{aligned}$$
(4)

is built from the contorsion tensor, \(K^{\mu \nu }_{\rho }\), given by

$$\begin{aligned} K^{\mu \nu }_{ \rho } = -\frac{1}{2}[{T^{\mu \nu }}_{\rho }-{T^{\nu \mu }}_ {\rho }-{T_{\rho }}^{\mu \nu }], \end{aligned}$$
(5)

which equals the difference between Weitzenb\(\ddot{\textrm{o}}\)ck and Levi-Civita connections. Having briefly reviewed the basics of F(T) telleparallel theory of gravity, we now move on to explain the basic motivation of the present work.

In order to study cosmological consequence (evolution) of the said alternative theory of gravity, a particular form of F(T) is required. Other than setting up a form of F(T) by hand or reconstructing it from the late history of cosmic evolution, it is always desirable to find its form in view of some physical consideration; as for example, invoking Noether symmetry, which had been applied earlier in different modified theories of gravity, with tremendous success. De Ritis et al. [48] had applied Noether symmetry for the first time in scalar–tensor theory of gravity, and found an exponential form of the potential, which can drive inflation in the very early universe. Such an exciting result inspired authors to extensively search for Noether symmetry in different theories of gravity, such as, scalar–tensor theories, higher order theories, Gauss–Bonnet gravity, quantum cosmology and also F(R) theory, and else. In fact, a host of Noether symmetries were found by several authors in F(R) theory of gravity [49,50,51]. However, after scrutinizing thoroughly, only a handful number of symmetries were found to satisfy the field equations [52,53,54,55,56]. Particularly, since F(T) theory of gravity appeared as an alternative to F(R) gravity theory, therefore for the sake of comparison, we enlist in Table 1, the available Noether symmetries for F(R) theory of gravity, in the background of isotropic and homogeneous Robertson–Walker (RW) metric

$$\begin{aligned} {ds}^2 = - {dt}^2 + {a^2(t)}\left[ {dr^2\over 1-kr^2} + r^2 \big (d\theta ^2 + \sin ^2{\theta }~d\phi ^2\big )\right] ,\nonumber \\ \end{aligned}$$
(6)

in different eras [52,53,54,55,56].

Table 1 Available Noether symmetries in RW space-time for F(R) theory of gravity in different eras. Here, \(\Sigma \) stands for conserved currents
Full size table

Clearly, the form \(F(R) \propto R^2\) in vacuum, apparent from Table 1, supports Starobinsky’s curvature induced Inflation [57]. Likewise, application of Noether symmetry to explore suitable forms of F(T) is also available in the literature [58,59,60,61,62,63], for spatially flat (\(k = 0\)) RW metric,

$$\begin{aligned} {ds}^2 = - {dt}^2 + {a^2(t)}\left[ dr^2 + r^2 \big (d\theta ^2 + \sin ^2{\theta }~d\phi ^2\big )\right] , \end{aligned}$$
(7)

which are arranged systematically in Table 2. It is quite apparent that apart from three symmetries available in matter-dominated era, at least five symmetries, admitting different forms of F(T) are also available in vacuum-dominated era. However, it is important to mention that Noether symmetry is not on-shell for a constrained system, unless the constraint is incorporated in the Noether equation. Gravity is particularly a constrained system, due to the diffeomorphic invariance. In fact, the six \((^i_j)\) equations of Einstein are dynamical (containing second derivatives), while the rest four \((^0_0)\) and \((^i_0)\) are constraint equations (containing first order derivatives only). In the above i and j run from \(1-3\). The last four equations are the energy (Hamiltonian) and the momenta constraint equations. Hence, unless these equations are embodied in Noether equation, it is mandatory to scrutinize that every available Noether symmetry of a gravitational theory satisfies the four constraint equations [64,65,66]. It is to be mentioned that for the isotropic and homogeneous model (6), only the energy constraint equation survives, which has to be satisfied by the symmetry equation. Application of (modified) Poisson first theorem happens to be a straight forward technique to examine such consistency in this regard [67, 68]. For F(R) theory of gravity, a host of Noether symmetries were claimed and only a few, presented in Table 1, survived after such consistency check.

Table 2 Claimed Noether symmetries in RW metric for F(T) theory of gravity in different eras
Full size table

The problem associated with F(T) gravity theory is that, apart from the metric coefficients, the configuration space is spanned by (\(T,~\dot{T}\)), and unfortunately, due to the absence of \(\dot{T}\) term in the action, the Hessian determinant vanishes, which turns out the action to be singular. As a result, the phase-space structure remains obscure [69], and Poisson theorem cannot be applied. Although, Hamiltonian has been constructed following Dirac constrained analysis, it has been found to contain momenta in the denominator and hence impossible to handle [69]. In the context of RW metric, it is therefore suggestive either to incorporate lapse function (N) in the symmetry equation, without fixing it a priori [70, 71], or to incorporate the constraint in the Noether equation, through a Lagrange multiplier [72, 73]. However, in both the situations, the lapse function and the Lagrange multiplier remain arbitrary, and one has to follow trial and error method to explore a symmetry, which is essentially a formidable task.

In homogeneous and isotropic model, only the energy constraint equation survives. In the literature, although a host of Noether symmetries for F(T) gravity, together with their associated conserved currents in spatially flat Robertson-Walker metric (6) are available [58,59,60,61,62,63], which are systematically presented in Table 2, the consistency of these symmetries in connection with the (\(^0_0\)) equation of Einstein, had not been inspected. We therefore take up this task in the present manuscript. In the following section, we explore Noether symmetry following the Lagrange multiplier technique for F(T) teleparallel gravity, in the background of spatially flat homogeneous and isotropic Robertson-Walker space-time (6). We also incorporate the available symmetry in the energy (\(^0_0\)) equation of Einstein, for a consistency check. It is unveiled that pure F(T) only produces a divergent term in the vacuum dominated era. The only propitious result found is \(F(T) \propto T^n\), but for odd integral values of n, in the pressure-less dust era. In Sect. 3, we try to construct the form of F(T) in view of a viable (Friedmann-like decelerated) radiation era. In view of all our findings we propose a reasonable form of F(T) in Sect. 4. Section 5 concludes our work.

2 Noether symmetry analysis in F(T) teleparallel gravity

Our aim in this section is to scrutinize the claims [58,59,60,61,62,63] in regard of available Noether symmetries of F(T) theory of gravity, in the background of RW metric enlisted in Table 2, and also to present true symmetries available in different eras. So called true symmetries may be found if the only constraint, viz., the \((^0_0)\) equation of Einstein is satisfied. For this purpose, our starting point is the action (1), in which the matter action \(S_m\) includes both perfect fluid and dark matter associated with a barotropic equation of state \(\omega \). In the spatially flat (\(k = 0\)) RW space-time (7), the vielbein (vierbein) is,

$$\begin{aligned} e^{i}_{\mu }= \textrm{diag}(1,a(t),a(t),a(t)). \end{aligned}$$
(8)

In the above, a(t) is the cosmological scale factor. Now the canonical formulation of F(T) theory of gravity with finite degrees of freedom may be performed following the Lagrange multiplier technique [58,59,60,61,62,63]. There exists two conventions, viz., \(T = -6\,H^2\) and \(T = 6\,H^2\) [74]. Here, we follow the former convention, and treat \(T + 6 {\dot{a}^2\over a^2}=0\) as a constraint in the spatially flat RW metric (7). This constraint is introduced in the action (1) through a Lagrange multiplier \(\lambda \). In the presence of a barotropic fluid, the action may therefore be expressed as,

$$\begin{aligned} \begin{aligned} {\mathbb {A}} = 2{\pi }^2\int \Big [F(T) - \lambda \Big \{T + 6\Big ({\dot{a}^2\over a^2} \Big )\Big \}-\rho _{0} a^{-{3(\omega +1)}}\Big ]a^3 dt, \end{aligned}\nonumber \\ \end{aligned}$$
(9)

where, \(\rho _0\) is the present value of matter density, either for radiation (\(\rho _{r0}\)) or for pressure-less dust (\(\rho _{m0}\)), and \(\omega \) is the equation of state parameter. Now varying the action with respect to T one gets \(\lambda = F'(T)\), where \(F'(T)\) is the derivative of F(T) with respect to T. Substituting the expression of \(\lambda \), one can express the above action (9) in the following form,

$$\begin{aligned} \begin{aligned} {\mathbb {A}} = 2{\pi }^2\int [-6a \dot{a}^2 F'+ a^3(F-F'T)- \rho _{0} a^{-3\omega }]dt. \end{aligned} \end{aligned}$$
(10)

It is important to mention that, being devoid of the time derivative of T, the Hessian determinant vanishes. Therefore the above action, unlike F(R) theory of gravity, is singular [69], although it is often referred to as canonical. Nonetheless, for Noether symmetry analysis, such an action is well-suited. The point Lagrangian is therefore,

$$\begin{aligned} L(a,\dot{a},T,\dot{T}) = \left[ -6a \dot{a}^2 F'+ a^3(F-F'T)- \rho _{0} a^{-3\omega }\right] ,\nonumber \\ \end{aligned}$$
(11)

and the field equations in terms of the Hubble parameter \(H = {\dot{a}\over a}\) are,

$$\begin{aligned}{} & {} 48H^2{\dot{H}}F'' - 4({\dot{H}}+3H^2)F'-F -\omega \rho _{0} a^{-{3(\omega +1)}}= 0 \nonumber \\ \end{aligned}$$
(12)
$$\begin{aligned}{} & {} T = -6H^2 \nonumber \\ \end{aligned}$$
(13)
$$\begin{aligned}{} & {} F {-} 2TF'{-}\rho _{0} a^{{-}{3(\omega {+}1)}}{=}F {+}12H^2F' {-} \rho _{0} a^{{-}{3(\omega +1)}}{=}0 \nonumber \\ \end{aligned}$$
(14)

Note that, the constraint has been retrieved in Eq. (13). One can also trivially check that the field equations of GTR may be recovered for \(F(T) \propto T = -6H^2\). This is known as telleparallel equivalent of general relativity or TEGR, when abbreviated. As mentioned, we explore Noether symmetries corresponding to the point Lagrangian (11) in the following. Noether theorem states that, if there exists a vector field X, for which the Lie derivative of a given Lagrangian L vanishes i.e. \(\pounds _X L =X L = 0 \), the Lagrangian admits Noether symmetry together with a conserved current \(\Sigma = i_X \theta _L\), where \(\theta _L\) is the Cartan one form. We repeat that Noether symmetry is usually on-shell, but not for constrained system, we are dealing with. So once we find a symmetry, we shall check its consistency in connection with Eq. (14), which is essentially the \((^0_0)\) equation of Einstein.

Noether equations: The configuration space of the Lagrangian (11) under consideration is M(a, T) and the corresponding tangent space is \(T_M(a,T,\dot{a}, \dot{T})\). Hence the generic infinitesimal generator of the Noether Symmetry is,

$$\begin{aligned} X = \alpha _1 \frac{\partial }{\partial a}+\beta _1\frac{\partial }{\partial T} +{\dot{\alpha }}_1 \frac{\partial }{\partial \dot{a}}+ {\dot{\beta }}_1\frac{\partial }{\partial \dot{T}}, \end{aligned}$$
(15)

where, \(\alpha _1 = \alpha _1(a,T)\) and \(\beta _1 = \beta _1(a,T)\). Further, the Cartan one form is:

$$\begin{aligned} \theta _{L} = {\partial L\over \partial a} da + {\partial L\over \partial T} dT, \end{aligned}$$
(16)

and the associated constant of motion is, \(\Sigma = i_{X} \theta _{L}\). Now using the existence condition for Noether Symmetry, viz. \(\pounds _{X} L = X L = 0\), Noether equation is obtained. Thereafter equating the coefficients of \({\dot{a}}^2\), \({\dot{T}}^2\), \(\dot{a} \dot{T}\) along with the term free from derivatives respectively to zero as usual, one can obtain the following system of partial differential equations:

$$\begin{aligned}{} & {} \alpha _1 F' + \beta _1 a F''+ 2a F'\alpha _{1,a}= 0, \end{aligned}$$
(17)
$$\begin{aligned}{} & {} \alpha _1' = 0, \end{aligned}$$
(18)
$$\begin{aligned}{} & {} 3\alpha _1 \left( {F-T F'}\right) +3\omega \rho _{0} a^{-{3(\omega +1)}}-a\beta _1 T F'' =0. \end{aligned}$$
(19)

2.1 Available symmetries in pure vacuum era

It is important to mention that, particles are created, once inflation is over, due to the oscillation of the scalar field. Thus, very early vacuum-dominated era might contain a scalar field, but is devoid of barotropic fluid in any of its form. In the absence of the scalar field and for \(\rho = 0 = p\), the field equations (12) and (14) are considerably simplified, and combined to yield

$$\begin{aligned} 48H^2{\dot{H}} F''-4{\dot{H}} F' = 0, \end{aligned}$$
(20)

which may immediately be solved to obtain \(F(T) \propto T^{\frac{1}{2}}\), which is the only allowed form of F(T) theory in vacuum. It is not clear, despite such unique form, why attempts to find forms of F(T) were taken up through Noether symmetry analysis. In fact, authors [58,59,60] attempted to find Noether Symmetry in pure vacuum era and ended up with \(F \propto T^n\). Let us now examine why such uncanny results appeared. The above set of Noether equations (17), (18) and (19) admit the following solutions in pure vacuum era, viz.

$$\begin{aligned} \alpha _1= a^{1-\frac{3}{2n}}, \;\;\;\beta _1 = -\frac{3}{n} {a}^{-\frac{3}{2n}} T,\;\;\; F(T) = F_0 T^{n}, \end{aligned}$$
(21)

along with its associated conserved current, which reads as,

$$\begin{aligned} \Sigma = -12 a^{(2-\frac{3}{2n})}\dot{a} F'(T)=-12 n F_0 a^{(2-\frac{3}{2n})}\dot{a}T^{n-1}. \end{aligned}$$
(22)

In the above and everywhere else, \(F_0 > 0\) is required to ensure retrieval of GTR (\(n = 1\)). One can now immediately solve the above equation for a(t) as,

$$\begin{aligned} a(t) = {\left( \frac{3}{2n}\right) }^\frac{2n}{3} {\left[ \left( -\frac{1}{6}\right) ^{n-1}\left( -\frac{\Sigma _1}{12 n F_0} \right) \right] }^\frac{2n}{3(2n-1)} {(t-t_0)}^\frac{2n}{3}.\nonumber \\ \end{aligned}$$
(23)

Although, the form of \(F(T) \propto T^n\) (21) so obtained via Noether symmetry analysis, in the very early vacuum dominated era, does not admit de-Sitter solution, the solution of the scale factor (23) is clearly found to admit power law inflation for \(n > {3\over 2}\). For example, \(n = 3\) implies \(a(t)\propto (t-t_0)^2\), and \(n > 3\) gives even faster rate of the expansion of the early universe. However, before coming to a conclusion, note that the (\(^0_0\)) equation (14) for \(F(T)=F_0 T^n\) in vacuum dominated era takes the form,

$$\begin{aligned} E_L=-a^3 \left( F - 2TF'\right) =(2n-1)F_0 a^3 T^n = 0, \end{aligned}$$
(24)

and the above equation is satisfied for none other than \(n = {1\over 2}\). Now for \(n=\frac{1}{2}\), the scale factor \(a(t)\propto {(t-t_0)}^\frac{1}{3}\). As a result only decelerated expansion is administered in the very early universe. Clearly the claim that \(F(T) \propto T^n\) [58,59,60], for arbitrary n is patently false. Note that, such a form of F(T) is meaningless, since \(F(T) \propto T^{1\over 2}\) gives only a total derivative term in the action for RW metric under consideration, and hence does not produce any dynamics. This is a major shortfall of gravity with torsion. On the contrary, F(R) theory of gravity admits at least four reasonably viable forms along with their associated conserved currents as depicted in Table 1. Undoubtedly, vacuum era of F(R) gravity has much reacher structure than F(T) teleparallel theory of gravity. It may be mentioned that in the presence of a scalar field on the contrary, Noether symmetry remains obscure, and arbitrary form of F(T) is admissible. It is therefore suggestive to incorporate a scalar field (dilatonic or Higgs field) which would be responsible to drive inflation in the action F(T) [75].

2.2 Available symmetries in radiation-dominated era

In the radiation dominated era \(p = {1\over 3}\rho , ~i.e. ~\omega = {1\over 3}\), no symmetry exists, and indeed there is no such claim in the literature too in this regard. However, in F(R) theory of gravity, \(F(R) = F_0 R^2\) admits Noether symmetry along with an associated conserved current: \(\Sigma = a^3 \dot{R}\) (Table 1). Again we find that F(R) theory of gravity has richer structure than F(T) teleparallel gravity theory, in radiation dominated era too.

2.3 Available symmetries in pressureless dust era

In pressureless dust era \(p = 0 = \omega \), the above Noether Eqs. (17), (18) and (19) significantly admit the same solutions (21) and (22) as already found in vacuum era. Clearly, the scale factor also admits the same solution (23). Apparently therefore, solutions are identical to those already presented in [61, 62]. But then, as repeatedly mentioned, we need to scrutinize the symmetries in the light of the energy constraint Eq. (14). For the available symmetry \(F(T)=F_0 T^n\), the \((^0_0)\) Eq. (14) takes the following form in the matter dominated era,

$$\begin{aligned} E_L= & {} -a^3 \left( F - 2TF'-\rho _{m0} a^{-3}\right) \nonumber \\= & {} (2n-1)F_0 a^3 T^n+\rho _{m0}=0, \end{aligned}$$
(25)

which may again be solved for a(t) to find,

$$\begin{aligned} a(t) ={\left( \frac{3}{2n}\right) }^\frac{2n}{3} {\left[ \left( -\frac{1}{6}\right) ^{n}\left( -\frac{\rho _{m0}}{(2n-1)F_0}\right) \right] }^\frac{1}{3} {(t-t_0)}^\frac{2n}{3}.\nonumber \\ \end{aligned}$$
(26)

Comparing the scale factors found in view of Noether symmetry analysis (23) and the constraint \(E_L = 0\) (26), it is possible to express \(\rho _{mo}\) as,

$$\begin{aligned} \rho _{mo}= & {} -(2n-1)F_0 {\left( -6\right) }^{\frac{n}{2n-1}} {\left[ -\frac{\Sigma _1}{12 nF_0}\right] }^\frac{2n}{(2n-1)}\nonumber \\= & {} \left( -1\right) ^\frac{3n-1}{2n-1} (2n-1)F_0 {\left[ \frac{{\Sigma _1}^2}{24 n^2 {F_0}^2}\right] }^\frac{n}{(2n-1)}. \end{aligned}$$
(27)

The fact that universe has structures, requires \(\rho _{m0} \ge 0\), and this gives rise to some important consequences. Firstly, \(n \ne {1\over 2}\), so that some amount of matter \(\rho _{m0}\) (the structures) remains present in the universe. Note that in the process, the pathological form \(F(T) \propto \sqrt{T}\), which is essentially a total derivative term, is averted. Next, since \(F_0 > 0\) is required to retrieve GTR as already mentioned, therefore to ensure positivity of \(\rho _{m0}\), we must have \(n > {1\over 2}\) and \(3n-1 = 2m\), where \(m > 0\) is an integer (apart from \(m = 0\), i.e. \(n = {1\over 3}\), which leads to the uncanny form \(F(T) \propto T^{1\over 3}\)). As a consequence, indeed one finds admissible Noether symmetry in the form \(F(T) \propto T^n\), but only for the odd integral values of n, such as, \(n = 1, 3, 5....\) etc. This result indicates that apart from TEGR, leading to the standard Friedmann-like decelerated expansion [\(a(t) \propto t^{2\over 3}\), corresponding to \(F(T) \propto T, ~\Sigma \propto \sqrt{a} \dot{a}\)]; Noether symmetry allows forms of F(T), which are suitable for accelerated expansion. These might be, \(F(T) \propto T^3\), resulting in \(\Sigma \propto a^{5\over 2}H^5\) and \(a(t) \propto t^2\); \(F(T) \propto T^5,~\Sigma \propto a^{27\over 10}H^9\) leading to \(a(t) \propto t^{10\over 3}\) etc. A combination of these terms surely can lead to early deceleration followed by late-time accelerated expansion in the present matter dominated era. This result is definitely encouraging.

3 Reconstruction from radiation era

The success of the standard (FLRW) model of cosmology stands on strong footing with regard to the cosmic evolution in the radiation and the early matter (pressure-less dust) dominated eras. Once the seeds of perturbation are formed (in view of a very early inflationary era), the Friedmann-like decelerated radiation era (\(a(t) \propto \sqrt{t}\)) allows to compute the formation of structures (stars, galaxies, clusters and superclusters) with precession, and also can predict the formation of CMBR at a redshift value \(z \approx 1080\), which are well-supported by experiments. It is therefore primarily required to associate an almost Friedman-like decelerated expansion in the radiation dominated era, to envisage a viable evolution history of the universe. It may be worth mentioning that, since the trace of electro-magnetic field tensor (\(\mathrm {T_{em}} = \rho _{em} - 3p_{em}\)) is zero, the contracted GTR equation (\(R \propto \textrm{T}\)), enforces Ricci scalar to vanish (\(R = 0\)), and the Friedmann-like decelerated expansion (\(a(t) \propto \sqrt{t}\)) results in automatically, in the radiation dominated era. However, in the case of torsion, fortunately enough, although \(F(T) \propto T\) leads to GTR, \(\mathrm {T_{em}} = 0\), does not lead to the static solution \(T = 0\). Therefore, even though all the results of GTR hold, the pathology of discontinuous evolution of the Ricci scalar (large initially, almost vanishing in the middle and small at present) is averted. To inspect the situation here, we combine the field Eqs. (12) and (14) to find,

$$\begin{aligned} \rho + p= & {} \rho _0(1+\omega ) a^{-3(\omega +1)} = {4\over 3} \rho _{r0} a^{-4}\nonumber \\= & {} 48H^2\dot{H} F'' - 4 \dot{H} F' = -4H F'' \dot{T} - 4 \dot{H} F'\nonumber \\= & {} -4{d\over dt}(H F'), \end{aligned}$$
(28)

where, we have substituted \(\omega = {1\over 3}\) for radiation era, and \(\rho _{r0}\) now stands for the current value of the amount of radiation present in the universe. Now seeking a solution in the form \(a = a_0 t^n\), where \(a_0\) and n are constants, one can compute, \(H = {n\over t},~\dot{H} = -{n\over t^2},~T = -6\,H^2 = -6{n^2\over t^2}\). Hence upon integration, one finds,

$$\begin{aligned} F'= & {} f_1{t\over n} + {\rho _{r0}\over 3a_0^4 (4n-1) t^{2(2n-1)}} = f_1{\sqrt{-6}\over \sqrt{T}} \nonumber \\{} & {} + {\rho _{r0}\over 3a_0^4(4n-1)n^{4n-1}(-6)^{2n-1}}T^{2n-1}, \end{aligned}$$
(29)

where, \(f_1\) is a constant of integration. Further integration yields,

$$\begin{aligned} F(T) = 2f_1\sqrt{-6T} + {\rho _{r0}\over 6a_0^4(4n-1)n^{4n}(-6)^{2n-1}}T^{2n}, \end{aligned}$$
(30)

apart from a constant of integration, which does not contribute to the field equations. Note that the first term is essentially a divergent term in the RW metric under consideration. Thus, we are left with the second term only. Under the choice \(n = {1\over 2}\), the radiation era evolves exactly like the standard (FLRW) model, and the action reduces to that of GTR \((F(T) \propto T)\), which is TEGR, as already mentioned. This is actually what we are searching for. Note that, one can also consider other forms of F(T), by choosing n judiciously, keeping in mind that \(F(T) > 0\), to recover GTR, \(n < 1\) for decelerated expansion, and \(n \ne {1\over 4}\) to avert divergence. Satisfying all these conditions one can easily check that, for \({1\over 4}< n < 1\), the best option is to choose \(n = {1\over 2}\), while for \(n < {1\over 4}\), the decelerated expansion is too slow to produce CMB at the right epoch.

4 Proposition

It may be mentioned that the Ricci scalar (R) actually measures the difference between the areas of a curved space and the flat space of a sphere (say), formed by the set of all points at a very small geodesic distance. In view of GTR (\(R_{\mu \nu } - {1\over 2}R g_{\mu \nu } = - \kappa \textrm{T}_{\mu \nu }\)), it is found to be proportional to the trace \(\textrm{T}\) of the energy momentum tensor \(\textrm{T}_{\mu \nu }\). For electromagnetic field, as already mentioned, the trace vanishes, enforcing the Ricci scalar to vanish as well. As a result, in the radiation dominated era, which initiated soon after the hot big-bang, the scalar curvature vanishes \((R = 0)\), and a Friedmann-like radiation era (\(a \propto \sqrt{t}\)) of the cosmic scale factor evolves. It is also important to mention that the structure formation, computed from the seeds of perturbation generated during early inflationary era, is primarily based on the standard model of cosmology, which requires \(R = \textrm{T} = 0\), (or very small due to the presence of an insignificant amount of matter in the early universe) in the radiation dominated era. As matter starts competing with the radiation, the matter energy density grows, as well as the Ricci scalar. However as the matter energy density is redshifted away, Ricci scalar diminishes again, and the nature of the Ricci scalar depends on the growth rate of the matter energy density. This implies, contrary to the evolution of the scale factor, the Ricci scalar does not evolve from a very large to an insignificantly small value smoothly, rather it might have kinks in the middle. However, a viable F(R) theory of gravity is still presented in the form \(F(R) = \alpha R + \beta R^2 + \gamma R^{-1}\) [76], or \(F(R) = \alpha R + \beta R^2 + \gamma R^{3\over 2}\) [77], with the preoccupied assumption that R evolves from a very large at the earlier epoch to an insignificantly small value at present in a continuous manner, so that \(R^2\) dominated in the early universe, leading to inflation, R in the middle, to establish standard model, and \(R^{-1}\) or a combination of R and \(R^{\frac{3}{2}}\) at present, to envisage accelerated expansion of the universe. Clearly, this leads to a conceptual problem. On the contrary, when torsion is attributed to gravity, usually a form such as \(F(T) = f_0 T + f_1 T^2 + \cdots \) is chosen to combat early deceleration in the Friedmann form \(\left( a(t) \propto t^{3\over 2}\right) \) followed by late-time cosmic acceleration in the matter (pressure-less dust) dominated era. In view of our analysis, it is clear that \(F(T) \propto T\) gives exactly Friedmann-like radiation dominated era \((a(t) \propto \sqrt{t})\), and pressure-less dust dominated era \(\left( a(t) \propto t^{2\over 3}\right) \). Therefore, unlike the Ricci scalar, it is not required to set \(T = 0\), at any stage. This has a great conceptual advantage over modified theories of gravity.

Current analysis reveals the fact that pure F(T) gravity in vacuum does not give rise to any dynamics, since it only ends up with a total derivative term. Therefore, to drive inflation, and also to avert the pathological behaviour of pure F(T) gravity in the very early vacuum-dominated era, either unimodular F(T) gravity (\(|e| = det({e_\mu }^A) = \sqrt{-g} = 1, d\tau = a^3 dt\)) has to be considered [78], or a scalar field [75] should be associated with F(T) gravity theory. We prefer a scalar field over the unimodular F(T) theory of gravity. Next, we observe that \(F \propto T\), leads to the standard model of cosmology, both in the radiation and early matter dominated era. Finally, in view of Noether analysis, instead of \(T^2\), one should associate \(T^3\) and higher odd integral powers in the action. That is, a viable form that might explain the cosmic evolutionary history may be in the following form, \(F(T) = f_0 T + f_1 T^3 + \cdots \), and the action may be proposed as,

$$\begin{aligned} A {=} \int \left[ f_0 T {+} f_1 T^3 {+} \cdots - {1\over 2}\phi _{,\mu }\phi ^{,\mu } {-} V(\phi )\right] \sqrt{-g}d^4x, \nonumber \\ \end{aligned}$$
(31)

where, the scalar field drives inflation at the very early stage, and decayed to an insignificant value in the process of particle creation. Additionally, T dominates in the middle to envisage the standard model, while odd-integral higher order terms are responsible for late-time cosmic acceleration.

5 Concluding remarks

Noether symmetry has been extensively studied in the literature for teleparallel gravity with torsion F(T), and several claims were made regarding the forms of F(T) and associated conserved currents in vacuum and pressure-less dust era, as presented in Table 2. Nonetheless, it is trivial to note that pure F(T) gravity is not empowered with a meaningful form in vacuum. This is a major shortfall of gravity with torsion, and at least a scalar field (or a unimodular F(T) theory) is required to forestall F(T) gravity theory from such bizarre situation. Hence, unlike F(R) gravity, in which \(R^2\) term can steer inflation in the early universe, the scalar field can only be responsible to drive inflation in the F(T) gravity theory. Obviously, unless, it can be shown that the scalar is almost completely used up in producing particles at the end of inflation, and the rest is redshifted away, it would be a responsibility to find its trace in the present universe. Otherwise, the main objective to explain late-time acceleration without dark energy, would be in vain.

It is also observed that radiation era does not yield any Noether symmetry, and as such there were no such claim too. On the contrary, it is important to mention that, F(R) theory admits symmetry in radiation era, which are enlisted in Table 1. Result of the present analysis also reveals the fact that the radiation and the early pressure-less dust era is best described by \(F(T) \propto T\). This is a non-trivial result, since in GTR, the Ricci scalar is proportional to the trace of the energy-momentum tensor, which vanishes for radiation and as a result Friedmann-like (\(a \propto \sqrt{t}\)) solution is admissible. Although, \(F(T) \propto T\) is TEGR, however, vanishing trace of energy-momentum tensor does not enforce \(T = 0\), but still, Friedmann-like cosmic evolution is admissible.

Finally, we find that Noether symmetry indeed exists in the pressure-less dust era in the form \(F(T) \propto T^n\), but only for odd integral values of n. In view of all these findings, we put forward an action (31), which might possibly foretell the cosmic evolutionary history of the universe. We particularly give emphasis on the fact that, although, F(R) gravity has a much richer structure than the F(T) gravity theory, unlike the Ricci curvature scalar, it is not required to set torsion to vanish at any stage of cosmic evolution. Further, power lower than first degree in R is required to explain late-time cosmic acceleration. Contrarily, in the case of torsion higher degree terms are required for the simple reason that, the Hubble parameter is reduced with expansion, and higher degree terms in Hubble parameter falls off even faster, reducing the torsion considerably, and eventually leading to late-time cosmic acceleration. In a nut-shell, late-time acceleration is an outcome of lesser torsion. In this sense, F(T) gravity is apparently free from conceptual problem.