Unlike the Noether symmetry, a metric independent general conserved current exists for non-minimally coupled scalar-tensor theory of gravity if the trace of the energymomentum tensor vanishes. Thus, in the context of cosmology, a symmetry exists both in the early vacuum and radiation dominated era. For slow roll, symmetry is sacrificed, but at the end of early inflation, such a symmetry leads to a Friedmann-like radiation era. Late-time cosmic acceleration in the matter dominated era is realized in the absence of symmetry, in view of the same decayed and redshifted scalar field. Thus, unification of early inflation with late-time cosmic acceleration with a single scalar field may be realized.

As has been done before, we study an unknown coupling function, i.e. F(ϕ), together with a function of torsion and also curvature, i.e. f (T ) and f (R), generally depending upon a scalar field. In the f (R) case, it comes from quantum correlations and other sources. Now, what if beside this term in f (T ) gravity context, we enhance the action through another term which depends upon both scalar field and its derivatives? In this paper, we have added such an unprecedented term in the generic common action of f (T ) gravity such that in this new term, an unknown function of torsion has coupled with an unknown function of both scalar field and its derivatives. We explain in detail why we can append such a term. By the Noether symmetry approach, we consider its behavior and effect. We show that it does not produce an anomaly, but rather it works successfully, and numerical analysis of the exact solutions of field equations coincides with all most important observational data, particularly late-time-accelerated expansion. So, this new term may be added to the gravitational actions of f (T ) gravity.

In this paper, we present the Noether symmetries of flat FRW spacetime in the context of a new action in teleparallel gravity which we construct based on the f (R) version. This modified action contains a coupling between the scalar field potential and magnetism. Also, we introduce an innovative approach, the beyond Noether symmetry (B.N.S.) approach, for exact solutions which carry more conserved currents than the Noether approach. By data analysis of the exact solutions, obtained from the Noether approach, late-time acceleration and phase crossing are realized, and some deep connections with observational data such as the age of the universe, the present value of the scale factor as well as the state and deceleration parameters are observed. In the B.N.S. approach, we consider the dark energy dominated era.

A technique for the reconstruction of the potential for a scalar field in cosmological models based on induced gravity has recently been developed by Alexander Y. Kamenshchik, Alessandro Tronconi, and Giovanni Venturi [73]. In this paper, this extended reconstruction method is utilized to investigate the nature of the scalar potentials of the most extended action of F (T ) -gravity context. First, a general formalism is formulated and then it is utilized for considering some well-known special cases including 'Barotropic Fluid', 'Cosmological Constant', and 'Modified Chaplygin Gas'. The analysis of the results is carried out by the use of the B-function method which has recently been suggested by the author [74]. As we know, assuming a proportional relation between the scale factor of the x-direction and the scale factors of y and z directions (i.e. A = B m ) in dealing with LRS Bianchi-I background is prevalent. Pursuant to observational data, the correct physical range of m is extracted. It is demonstrated that, unlike several papers, m is very close to 1. Some interesting discussions about the modified and generalized Chaplygin gases are performed. The ranges of the amounts of free parameters of the various types of Chaplygin gases are corrected according to observational data. It is demonstrated that the generalized Chaplygin gas model namely P = −σ 2 /ρ ν may be developed asUltimately, by combining ref.[73] and the current paper, a general prescription is recommended for the reconstruction of the potentials of alternative theories of gravity (especially F (R) and F (T ) ), and their relevant analysis.

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