Noether symmetry approach in f (T, B) teleparallel gravity with a fermionic field

In this work, we consider a homogeneous and isotropic cosmological model of the universe in f (T, B) gravity with non-minimally coupled fermionic field. In order to find the form of the coupling function F(Ψ), the potential function V (Ψ) of the fermionic field and the function f (T, B), we found through the Noether symmetry approach. The results obtain are coincide with the observational data that describe the late-time accelerated expansion of the universe.


Introduction
In modern cosmology, we apply various modified theories of gravity to describe the available observational data. Usually in the literatures we can see modifications of the components of the gravitational field or the field of matter separately, as well as their general modifications in the action of Einstein-Hilbert. The first type includes models where we modify only components of the gravity: f (R) gravity, where R is the Ricci scalar, f (T ) gravity, where T is the torsion scalar, f (G) gravity, where G is the Gauss-Bonnet invariant and etc [1]- [3]. The second type includes models with a matter and their modifications: quintessence, phantom field, tachyon field, k-essence and etc [4]- [7]. There are a number of useful reviews of dark energy that mainly focused on theory [8]- [11]. The interested readers should consult reviews with more complete reference lists, e.g. [12]- [15]. All these models can describe the dynamics of our universe in different ways, but the choice of the best model we can only be shown by future observational data.
The dynamic equations of such models are nonlinear differential equations of a higher order, usually obtaining their exact solutions is a very difficult problem. The Noether symmetry approach is one way of solving such dynamical equations in cosmology. This approach was consider in the following work [16]. The application of the Noether symmetry approach in cosmological models with scalar fields is considered in [17]- [19]. Interesting works are cosmological models with fermionic fields, where was also used this approach [20,21]. Recently, in paper [22], where was used the Noether symmetry approach in f (T, B) teleparallel cosmology.
This work is organized as follows. In Sect. 2, we give the field equations are derived from a point-like Lagrangian in a spatially flat and isotropic Friedman-Robertson-Walker metric, which is obtained from an action including the fermion field non-minimally coupled to the gravitational field in the framework of f (T, B) teleparallel gravity. In Sect. 3, we search the Noether symmetry for the Lagrangian of theory and Sect. 4, we obtain the particle solution of the field equations 2. The action and the equations of motion The Ricci scalar R and the torsion scalar T differs by a boundary term B via here, for simplicity we introduce B = (2/e)∂ µ (eT µ ) = ∇ µ T µ . The action for a fermion field that is non-minimally couled with the torsion scalar T and a boundary term B where e = det(e a µ ) = √ −g that e a µ is tetrad (vierbein) basis, T is a torsion scalar, B is a boundary term, ψ andψ = ψ † γ 0 denote the spinor field and its adjoint, with the dagger representing complex conjugation. F (Ψ) and V (Ψ) are generic functions, representing the coupling with gravity and the self-interaction potential of the fermionic field respectively. We assume that F and V depend on only functions of the bilinear Ψ =ψψ, Γ µ = e µ a γ a are the generalized Dirac-Pauli matrices satisfying the Clifford algebra {γ µ , γ ν } = 2g µν , where the braces denote the anti-commutation relation, the covariant derivatives e a µ are given by and Above, the fermionic connection Ω µ is defined by with Γ ρ µδ denoting the Christoffel symbols. We will consider here the simplest homogeneous and isotropic cosmological model, FRW, whose spatially flat metric is given by where a(t) is the scale factor of the Universe. For this metric, the vierbein is chosen to be (e a µ ) = diag(1, a, a, a), (e µ a ) = diag(1, 1/a, 1/a, 1/a).
The Dirac matrices of curved spacetime Γ µ are Hence we get Finally, we note that the gamma matrices we write in the Dirac basis that is as where I = diag(1, 1) and the σ k are Pauli matrices having the following form In fact selecting suitable Lagrange multipliers and integrating by parts to eliminate higher order derivatives, the Lagrangian L becomes canonical. In physical units, the action is Here the definitions of torsion scalar and a boundary term in FRW metric have been adopted, that is It is worth stressing that the two Lagrange multipliers are comparable. By varying the action with respect to T and B, one obtains then the above action becomes After an integration by parts, the point-like Lagrangian assumes the following form here, because of homogeneity and isotropy of the metric it is assumed that the spinor field depends only on time, i.e. ψ = ψ(t).

The Euler-Lagrange equations arė
2Ḟ or

The Noether Symmetries Approach
Noether symmetry approach tells us that Lie derivative of the Lagrangian with respect to a given vector field X vanishes, i.e.
We will search the Noether symmetries for our model. In terms of the components of the spinor field ψ = (ψ 0 , ψ 1 , ψ 2 , ψ 3 ) T and its adjointψ = (ψ 0 Here a vector field X we can written as where α, β, γ, η i and χ i are unknown functions of the variables a, T, B, ψ i and ψ † i . In general, the Noether symmetry condition leads to an expression of second degree in the velocities (ȧ,Ṫ ,Ḃ,ψ i andψ † i ) with coefficients being partial derivatives of α, β, γ, η i and χ i with respect to the variables a, T , B, ψ i and ψ † i . Thus, the resulting expression is identically equal to zero if and only if these coefficients are zero. This gives us a set of partial differential equations for α, β, γ, η i and χ i . For the Lagrangian (23), the Noether symmetry condition (22) yields the following system of partial differential equations.