Late-time-accelerated expansion arisen from gauge fields in an anisotropic background and a fruitful trick for Noether's approach

In this paper, a modified teleparallel gravity action containing a coupling between a scalar field potential and magnetism, in anisotropic and homogeneous backgrounds, is investigated through Noether symmetry approach. The focus of this work is to describe late-time-accelerated expansion. Since finding analytical solutions carrying all conserved currents emerged by Noether symmetry approach, is very difficult, hence regularly in the literature, the authors split the total symmetry into sub-symmetries and then select, usually, some of them to be carried by the solutions. This manner limits the forms of unknown functions obtained. However, in ref. [68], B.N.S. approach was proposed in order to solve such problems but its main motivation was carrying more conserved currents by solutions. In this paper, by eliminating the aforementioned limitation as much as possible, a trick leading to some graceful forms of unknown functions is suggested. Through this fruitful approach, the solutions may carry more conserved currents than usual ways and maybe new forms of symmetries. I named this new approach to be CSSS-trick (Combination of Sub-symmetries through Special Selections). With this approach, it is demonstrated that the unified dark matter potential is deduced by the gauge fields. Utilizing the $\mathfrak{B}\text{-function}$ method, a detailed data analysis of results obtained yielding perfect agreements with recent observational data are performed. And finally, the Wheeler-De Witt (WDW) equation is discussed to demonstrate recovering the Hartle criterion due to the oscillating feature of the wave function of the universe.


Introduction
One of the major challenges for physicists is the explanation of the essence and mechanism of the acceleration of our universe. Accelerated expansion of the universe has been confirmed by several astrophysical observations including supernova type Ia [1, 2], weak lensing [3], CMB studies [4], baryon acoustic oscillations [5], and largescale structure [6]. This discovery is inconsistent with the standard Einstein's general relativity. In general, two main classes of ideas (solutions) to understand the late-time-accelerated expansion have been proposed. Since this acceleration needs to negative pressure to occur, hence the first approach is the existence of an exotic liquid, so-called dark energy that about 70% of the universe is made up of it. The most probable solution to dark energy was thought that is Einstein's cosmological constant [7], but it failed because it cannot resolve 'fine tuning' and 'cosmic coincidence' problems. Hence, other theoretical models such as the phantom field [8][9][10][11][12][13], quintom [14][15][16][17], quintessence [18][19][20], and tachyon field [21] have been suggested. The second possibility is to modify Einstein's general relativity [22,23] making the action of the theory dependent upon a function of the curvature scalar R . In a certain limit of the parameters, as we expect, the theory reduces to general relativity. Recently, various novel gravitational modification theories like scalar-tensor theories, f (R) -gravity, f (T ) -gravity, f (T ) -gravity with boundary term ( f (T, B) ) [24], f (T ) -gravity with an unusual term [25] and etcetera have been suggested.
In the actions of extended theories of gravity, there are unknown functions. The choice of the unknown functions, somewhat arbitrary, has given rise to the objection of fine-tuning, the very problem whose solutions have been set out through inflationary theories. Therefore, it is desirable to have a standard path to extract unknown functions (especially the potential) of extended theories of gravity. One such approach is based upon the celebrated Noether symmetry approach and it was applied by many authors (for example, see refs. ). Noether symmetry approach enables one to find out conserved quantities from the presence of variational symmetries [69]. However, some hidden conserved currents may not be obtained by the Noether symmetry approach [70,71]. Furthermore, this approach may fail to get the purpose (Finding solutions whose carry all conserved currents or at least more of those which obtained by Noether symmetry approach), hence the B.N.S. approach has recently been suggested [68].
Before terminating this section, let us present a short review of the Noether symmetry approach from Prof. S. Capozziello's papers on this subject (For more and complete information see, for example, refs. [72,73]).
Let L(q i ,q i ) be a canonical, non degenerate point-like Lagrangian satisfying where H ij is the Hessian matrix of the Lagrangian and a dot indicates derivative with respect to the affine parameter λ which usually corresponds to the cosmic time t . The Lagrangian in analytic mechanics is of the form where E k and V are the positive definite quadratic kinetic energy and potential energy, respectively. The Hamiltonian associated with L is: it coincides with the total energy E k + V , and is a constant of motion. Note that this constant of motion is a fruit of (complete) Noether symmetry approach when the point-like Lagrangian does not explicitly depend upon time and its generator is ∂/∂t . The prevalent cosmological problems have a finite number degrees of freedom, hence the point transformations are considered. Any smooth and invertible transformation of the generalized coordinates q i → Q i (q) induces a transformation of the generalized velocitieṡ the matrix J = ∂Q i /∂q j is the Jacobian of the transformation on the positions, and it is assumed to be nonzero. A point transformation Q i = Q i (q) may depend upon one or more parameters. One may suppose that a point transformation depends upon a parameter, therefore, the transformation is then generated by a vector field. In general, an infinitesimal point transformation is represented by a generic vector field on Q The induced transformation (4) is then represented by A function F (q,q) is invariant under the transformation X (c) if where L X (c) is the Lie derivative of F along X (c) . If L X (c) = 0 , X (c) is then a symmetry for the dynamics derived by L . Now, we consider a Lagrangian L leading to the Euler-Lagrange equations d dλ and the vector field (6) which is called the complete lift of X . Contracting (8) with α i 's yields Using the total derivative relation as one obtains from equation (9) that Noether theorem is a straightforward consequence of this equation. The Noether theorem states that if L X (c) L = 0 , then the function is a constant of motion. It is worth noting that eq. (12) may be expressed in a coordinate-independent way as the contraction of X with the Cartan one-form Given a generic vector field Y = y i ∂/∂x i and 1-form β = β i dx i it is, by definition, i Y β = y i β i , and eq. (12) can then be expressed as By a point-transformation, the vector field X (c) becomes X (c) is still the lift of a vector field defined on the space of positions (configuration space). If X is a symmetry and we choose a point transformation such that we getX Therefore, Q 1 is a cyclic coordinate and the dynamics can be reduced. The coordinate transformation (16) is not unique and a clever choice is very important part of this procedure as it can be so advantageous. In general, the solution of equation (16) is not defined on the whole space, rather, it is local. The important point which is also used in this paper is that in the case of multiple vector fields X , say X 1 and X 2 , if these commute, i.e.
[X 1 , X 2 ] = 0 , then two cyclic coordinates can be found by solving the following system Hence, ∂/∂Q 1 and ∂/∂Q 2 would be the transformed fields. Because in the current problem of study, our symmetry generators commute with each other, hence we do not review what we should do if they do not commute.

The model and field equations
We start with the gravitational action of the form [68] where e = det(e i ν ) = √ −g with e i ν being a vierbein (tetrad) basis, M Pl is the reduced Planck mass, T is the torsion scalar, ϕ µ stands for the components of the gradient of ϕ(t) , V (ϕ) is the scalar field potential, and f 2 (ϕ) is the gauge kinetic function that has been coupled to the strength tensor F µν . The electromagnetic field tensor F is generated by the vector potential A of electromagnetic theory through the geometric relation F = −(antisymmetric part of ∇A ) . Hence, for a given 4-potential A µ , the field strength of the vector field is defined by In several papers such as refs. [68,[74][75][76][77], the action (19) was investigated 1 , the studies of which led to satisfactory results especially describing early inflation and late-time-accelerated expansion. The action (19) is the most generic action for single field inflation. However, there is room to make some 'trivial' generalizations like adding a scalar field coupling function to torsion and etcetera, but the basic is the action of the form (19). The success of the action (19) in the elucidation of the inflation era is due to the fact that the gauge fields are the main driving force for the inflationary background. It is worth mentioning that there are several fields, such as the vector fields and the nonlinear electromagnetic fields, which are able to produce negative pressure effects. On the other hand, the accelerating picture of the expanding nature of the universe requires a negative pressure. Hence, it is a substantial motivation to obtain a unified model (with a single scalar field) by action (19) which describes the stages of cosmic evolution. In this paper, we want to answer the question of whether or not this model may describe the late-time-accelerated expansion in the anisotropic and homogeneous background, namely Locally Rotationally Symmetric Bianchi type I (LRS B-I).
The LRS B-I line element is given by where the expansion radii a and b are functions of time t . Therefore, the torsion scalar for this background turns out to be where the dot denotes a differentiation with respect to time and H 1 , and H 2 are the directional Hubble parameters ( H 1 along x direction while H 2 along y and z directions).
In a spatially homogeneous model the ratio of shear scalar σ , to expansion scalar Θ , is constant (i.e. σ/Θ = constant ). This compels the condition a = b m with m = 0 . Manifestly, m = 0 is nonphysical because it means that one of the scale factors is constant (i.e. a = 1 ), and m = 1 is flat FRW space-time. It has been demonstrated in ref. [80] that according to recent observational data, m is very close to 1. Utilizing this well-known reasonable condition, a = b m , the expansion scalar (21) takes the form Regarding (20), we introduce the homogeneous and anisotropic vector field as whence we get One may choose the gauge A 0 = χ(t) = 0 , by using the gauge invariance [77]. For simplicity, let us assume that the direction of the vector field does not change in time. Pursuant to the background geometry (20), generally The special case k 1 = k 2 / √ 2 is for FRW space-time. Furthermore, it is readily observed that one may take one of k 1 or k 2 equal to zero. But, we prefer to keep both. However, in section (3), it is indicated that 1 Note that both actions d 4 x √ −g[R + · · · ] and d 4 x e[T + · · · ] lead to the same field equations.
the Noether symmetry approach does not allow to maintain both, hence it generates two classes. Writing the action (19) in the canonical form S = dt L(Q,Q) + Σ 0 down, the point-like Lagrangian would be 2 where the reduced Planck mass M Pl has been set equal to 1 . The Euler-Lagrange equations for a dynamical system are given by in which q i are the generalized positions in the corresponding configuration space Q = {q i } . According to (27), our configuration space reads Q = {b, ϕ, A} and consequently, its tangent space would be T Q = {b,ḃ, ϕ,φ, A,Ȧ} . Pursuant to the point-like Lagrangian (27), the corresponding Euler-Lagrange equation for the scale factor b reads For the scalar field ϕ, the Euler-Lagrange equation becomes which is the Klein-Gordon equation. The prime indicates the derivative with respect to ϕ . For the vector potential A, the Euler-Lagrange equation takes the following form: The energy function associated with a Lagrangian is given by Therefore the Hamiltonian constraint or total energy E L corresponding to the 0 0 -Einstein equation becomes According to (29) and (33), the effective Equation of State (EoS) parameter turns out to be: The dynamic of our system is given by these four equations (i.e. 29, 30, 31, and 33). The Noether approach is used in the next section to obtain exact solutions with symmetries of the extended theory of gravity (19).

Nother symmetry approach and CSSS-trick
In this section, solving field equations (i.e. 29, 30, 31, and 33) in order to investigate the circumstances of some important cosmological events like late-time accelerated expansion, phase crossing, and etcetera, are desired. Finding suitable forms of the unknown functions of the action (19) to reach the aforementioned purpose are challenging, hence exploring their forms through a 'standard way' seems necessary. Furthermore, it would be very beautiful if the solutions carry some conserved currents (Symmetries) as well. To this end, we utilize the Noether symmetry approach which exactly does this job. Pursuant to our tangent space of the configuration space, T Q = {b,ḃ, ϕ,φ, A,Ȧ}, the existence of the Noether symmetry implies the existence of a vector field X as where such that This equation yields the following system of linear partial differential equations: The 4-dimensional configuration space Q = {a, b, ϕ, A} was reduced to the 3-dimensional one Q = {b, ϕ, A} due to the physical assumption a = b m , hence we have seven partial differential equations instead of eleven numbers.
Solving this system of linear partial differential equations, one may obtain where and As is observed, in the special case m = 1 (FRW), both are equal: n 1 = n 2 = 1/3 . It is important to mention that if one wants to examine FRW-case, then he must take k 1 = k 2 / √ 2 in (25). In this stage, we encounter a bifurcation in equations due to n , and therefore two classes are separated by it. Indeed, the suitable solutions obtained by Noether symmetry approach, do not allow to have a four-potential of the form (25), hence, it is readily observed that (25) must be split into two independent cases: Therefore, according to (45) and (46), the symmetry generator, (35), turns out to be For convenience, let us, from now on, write the symmetry generators on the configuration space Q = {b, ϕ, A} , not on T Q. Hence, (53) splits into three independent generators: because (53) may be taken as: Consequently, the corresponding conserved currents are found to be respectively. Note that there is no Einstein summation convention over the subscript j in (60) 3 . For underlining this point, let us use I 3j instead of I 3 . It may easily be indicated that all the symmetries commute with each other, therefore the Lie algebra is satisfied. The corresponding constants of motion also close the same algebra in terms of Poisson bracket: The relation (61) is very important for us, since in what follows this point is used to obtain further suitable solutions, especially the form of interest for the potential (i.e. The unified dark matter potential). The conserved current I 3 is automatically carried by (31), hence we put it aside and therefore, two conserved currents, I 1 and I 2 , remain. Now, if we act as usual, then there are three possibilities: 1 . {c 1 = 0, c 2 = 0} ; 2 . {c 2 = 0, c 1 = 0} ; 3 . {c 1 = 0, c 2 = 0}. The cases 1 and 2 are easy to be considered, but the third option is not an easy task, since its system of cyclic equations which will contain two cyclic variables cannot be solved easily. Let us do different work.
As we know, regularly, the forms of unknown functions of extended theories of gravity are specified by the Noether symmetry approach in which the symmetries are also obtained. But in almost all cases in the literature, we cannot obtain the solutions which carry all conserved currents or at least more of those. In order to solve this problem and also some further reasons, the B.N.S. approach was proposed (See ref. [68]). In this paper, I suggest a new approach which may be more interesting for cosmologists: "Combination of Sub-symmetries through Special Selections" (CSSS-trick). In this way, not only the solutions carry more/new conserved currents, but also this approach leads to graceful results; for example, in our case of study, the unified dark matter potentials of the forms V = V 0 cosh 2 (µϕ) and also V = V 0 sinh 2 (µϕ) are produced which are highly rewarding.
• CSSS-Trick (Combination of Sub-symmetries through Special Selections): In the Noether symmetry approach, it is usual that after finding the symmetry generator and its corresponding conserved current of the forms where c i are constants, we split this total symmetry X tot. into sub-symmetries X i because X tot. is a sum of D independent symmetry generators and then search for the cases leading to analytical solutions which carry some of the sub-conserved currents I i . The reason for acting in such manner is that in almost all cases we cannot find analytical solutions which carry the total conserved current. I propose that instead of this work which surely leads to graceful results, we may also notice to the forms of unknown functions and then select the constant parameters in a way that they yield interesting forms for them. More precisely, first of all, after writing the total symmetry generator of the form (63), make sure that your chosen sub-symmetries, X i s, are independent from each other by considering all commutator of each two members of the set of sub-symmetries {X 1 , X 2 , · · · , X D }. When all these commutators vanish -i.e, the Lie algebra [X ς , X τ ] = 0; ς, τ ∈ {1, 2, · · · , D} is satisfied -, then your selections in (63) are true. Otherwise, any nonzero commutator is also a symmetry and the procedure is repeated until the vector fields close the Lie algebra. If we did this at first, then it guarantees that no new symmetry will produce after any combination of symmetries. It will be advantageous and highly rewarding if, with tuning the constant parameters in (63) (for example c 1 and c 2 here; see (47)-(48) and (57)), we act on a way that some interesting and well-known forms of unknown functions be achieved. Hence, we must back to the forms of unknown functions and first tune their constant parameters. Tuning the constant parameters nontrivially through special selections imply special combinations of symmetry generators and consequently the conserved currents. This trick covers the results of the usual approach. Let me clarify this trick by an example: In our case of study, we obtained: As already noticed, I 3 is carried automatically by the field equations. However, its constant factor namely c 4 has not appeared in (65) and (66), hence the forms of V and f are not affected by it. Leave it aside. Now, if we act on the usual way, we split X into X 1 , X 2 , and X 3 . But since finding a set of a solution in which both I 1 and I 2 are carried, is a very hard task, hence we must put one of c 1 and c 2 equal to zero. But either we set {c 1 = 0, c 2 = 0} (leading to V ≈ exp(−2µϕ) and f ≈ exp(−nµϕ) ) or {c 1 = 0, c 2 = 0} (yielding V ≈ exp(+2µϕ) and f ≈ exp(+nµϕ) ), the forms of functions V and f are limited. The CSSS-trick suggests that instead of this work, we follow the following prescription: First of all, we must notice the forms of the potential and coupling function. We should select the constant parameters appeared in the obtained forms of the potential and coupling function in a way that they lead to well-known forms for them. Then according to these selections for constant parameters, we combine the subsymmetries and consequently sub-conserved currents. In our case, based, at least, on (65), there are at least four well-known options: 1. (Usual) Selection: {c 1 = 0, c 2 = 0} : Therefore, between X 1 and X 2 , only X 1 is the symmetry of the system and consequently, I 1 will be carried by the solutions.
2. (Usual) Selection: {c 1 = 0, c 2 = 0} : Therefore, between X 1 and X 2 , only X 2 is the symmetry of the system and consequently, I 2 will be carried by the solutions.

(Unusual) Selection
Therefore, instead of X 1 and X 2 , the new symmetry X new1 = (X 1 − X 2 )/2 is the symmetry of the system and consequently, I new1 = (I 1 − I 2 )/2 will be carried by the solutions.

(Unusual) Selection
Therefore, instead of X 1 and X 2 , the new symmetry X new2 = (X 1 + X 2 )/2 is the symmetry of the system and consequently, I new2 = (I 1 + I 2 )/2 will be carried by the solutions.
It must be noted that, in the CSSS-trick process, the commutators of symmetries which we want to be carried, after combination must be considered. The remain of Noether symmetry approach namely cyclic variable process should be performed as usual with the difference that you will work with new set of symmetries. However, I think that it is better after achieving the desired forms of unknown functions, we proceed with the minimum number of symmetries, because when your system has a number of symmetries, indeed its behavior is restricted by these disciplines (symmetries) and thereby, finding analytical solution would be hard and in the most cases of interest, it is impossible. In our case of study, for the third selection above we have: The same relations hold for the fourth selection above. Indeed, (74) is a result of (73). Vanishing all commutators before combination guarantee vanishing the new ones after CSSS process, otherwise, it should be considered. Even though in each aforementioned selection, two symmetries exist for the system ( X 3 is common among them), but since I 3 is carried automatically by the field equations, hence for each of four cases mentioned above, only one cyclic variable will exist. In sub-section (3 3.2), these points are clarified. In this sub-section, two cases {c 2 = 0 , c 1 = 0} (C1) and {c 1 = 0 , c 2 = 0} (C2) are considered. It must be noted that since in each case, I 3j is carried automatically by field equations, hence we take c 4 = 0 in throughout this paper.
In order to simplify the system of equations, we use cyclic variables associated with the Noether symmetry generators X 1 and X 2 for cases {c 2 = 0 , c 1 = 0} (C1-class) and {c 1 = 0 , c 2 = 0} (C2-class), respectively. The existence of the Noether symmetry ensures the presence of cyclic variables, say where w = w(b, ϕ, A) , u = u(b, ϕ, A) , and v = v(b, ϕ, A) , such that the Lagrangian becomes cyclic in one of them ( w in our case). By defining a transformation i : (b, ϕ, A) → (w, u, v) as an interior product such that and be held for C1 and C2, respectively, the cyclic variables may then be found. Solving eqs. (76) and (77) independently, leads to and for C1 and C2, respectively. It is worthwhile mentioning that the choices in eqs. (78) and (79) the corresponding inverse transformations would be where the subscripts 1 and 2 correspond to C1 and C2 classes. As is clear from (47) and (48), the Noether symmetry approach gives the forms of the scalar field potential V (ϕ) and the coupling function f (ϕ) as and where for both C1 and C2. Therefore, according to (81) and (82), they are translated as follows: for both C1 and C2. Therefore, the point-like Lagrangian (27) The subscripts 1 and 2 refer to the type of class. As already noticed, the Noether symmetry approach does not allow to keep both k 1 and k 2 nonzero simultaneously, hence the point-like Lagrangians (90) must be decomposed into following Lagrangians: where the subscript j can only take j = 1 and j = 2 corresponding to {k 1 = 0 & k 2 = 0} and {k 1 = 0 & k 2 = 0} , respectively, and we have defined in which we have used (50). Note that there is no Einstein's summation rule over the subscript j in (91). Both point-like Lagrangians (91) lead to the following Euler-Lagrange equations with respect to w , A, and u, respectively:ü The corresponding conserved currents, (58) and (59), turn out to be where {c i ; i = 1, · · · , 6} are constants of integration. After taking integrations we arrive at where ϕ 1j,2j (t) = −λ 2µ ln c 2 + c 1 t + c 8 t 2 + c 9 t 3 (c 5 t + c 6 ) −1 + c 10 (c 5 t + c 6 ) −2nj , In the section (5), these solutions are analyzed to demonstrate the most events of the universe evolution. Note that because of t 2 term in the parentheses of eq. (109), the solutions obtained earlier in other papers for FRW background such as in refs. [75]- [76] are not recovered when the special case m = 1 (namely FRW) is investigated, and consequently, other things are also different. In this sub-section, two cases {c 1 = c 2 = 1/2} (C3) and {c 1 = −c 2 = 1/2} (C4) are investigated. It is remembered that since I 3 is carried automatically by field equations, hence we take c 4 = 0 in throughout this paper. Regarding (61), if we want both I 1 and I 2 -generated by X 1 and X 2 , respectively -are carried by field equations, a further symmetry does not produce. Hence, as mentioned in CSSS-trick, they may be combined in some suitable ways to lead to graceful results. Now, by assuming that and are symmetries (symmetry generators) and conserved currents that are carried by C3-class and C4-class, respectively, we seek point transformations on the vector fields X 1+2 and X 1−2 for C3-class and C4-class respectively, such that and i X1−2 dw = 1, i X1−2 du = 0, i X1−2 dv = 0 (120) ϕ, A) . In each case, w is a cyclic variable. Note that since we keep both X 1 and X 2 , hence, two cyclic coordinates may practically be found by solving a system like but, because both were combined in special ways and therefore, now, we have 'one' (new/combined) symmetry for each class, then, for each class, (121) must be recast to where X is a mixed symmetry generator of X 1 and X 2 . Therefore we have one cyclic variable for each class again. Indeed, (121) must be considered when one does not choose specified values for constants c 1 and c 2 and he wants two symmetries to be carried by the system independently and simultaneously. In order to write down the equations and solutions of both classes in a unified (closed) forms, let us define some useful parameters: Therefore one has: With these definitions at hand, the solutions (45)-(48) for both cases are now written in unified forms: and the symmetry generators and conserved currents turn also out to be One of the fruits of our different taken procedure is cleared here: Further interesting forms of the potentials namely V C3 (ϕ) = V 0 sinh 2 (µϕ) and V C4 (ϕ) = V 0 cosh 2 (µϕ) were acquired. In section (5), some interesting discussions about the obtained forms of potentials are performed. Solving eqs. (119) and (120) leads to So, the corresponding inverse transformations are Therefore, the potential, (132), and coupling function, (133), would be Again, it must be noted that the coordinate transformation is not unique, but our choices are very advantageous. Now, like the previous cases in sub-section (3 3.1), the point-like Lagrangian (27) is split into two Lagrangians for each class and they may be rewritten in terms of cyclic variables as where: Indeed, (145) represents four types of different Lagrangians (i.e L 31 , L 32 , L 41 , and L 42 ). The Lagrangians (145) lead to the following field equations with respect to u, w , and A, respectively: 4n j l 6j u 2nj −1uȦ + 2l 6j u 2njÄ = 0.
Again, in these cases, the conserved currents (135) in terms of the cyclic variables do not add new equations to our systems. The above system yields the following solutions: in which .
The forms of the 4-vector potentials are given by (152). When the value of n j is not exactly clear, taking this integral is somewhat complicated, hence we kept it in the form (152). In section (5), by singling suitable values of constant parameters out, the analysis of all solutions are easily carried out. The conserved currents {I 3j , I = (I 1 + I 2 )/2} and {I 3j , I = (I 1 − I 2 )/2} are carried by {b 3j , ϕ 3j , A 3j } (C3-class) and {b 4j , ϕ 4j , A 4j } (C4-class), respectively.
• An important point.
Perhaps, it seems that there are degeneracies in the solutions of all cases studied in this chapter, due to the existence of n j , when one has n 1 = n 2 , but it is completely wrong idea because n 1 = n 2 holds only for m = 1 , namely FRW case. On the other hand, in FRW case, we do not have permission to adopt one of the 4-vector potentials (51) and (52) because of the background geometry. Indeed, in FRW space-time, we must take the 4-vector potential of the form A µ = (χ(t); k 1 A(t), k 1 A(t), k 1 A(t)) (i.e. we must take k 1 = k 2 / √ 2 ) since both (51) and (52) violate the cosmological principle on which FRW metric is based.
turn out to be and respectively. In our case, eqs. (161) and (162) may be recast the same equation, viz. In general, there is an Einstein summation convention over the subscript j , nonetheless, it also holds true for each of indices j = 1 ( k 1 = 0 , k 2 = 0 ) and j = 2 ( k 1 = 0 , k 2 = 0 ). According to (163), k j b (2+m)nj f 2Ȧ is a time-independent term, so it is a constant of motion, as it emerged by the use of Noether symmetry approach (See (60); I 3 = k j b (2+m)nj f 2Ȧ ). As we observe, eq. (163) is equivalent to the third field equation namely eq. (31) and hence Maxwell's equations are satisfied automatically. And also it is needless to consider X 3 (or I 3 ) because I 3 is carried automatically. Before terminating this section, let us define the electric E and magnetic B fields covariantly, which are seen by an observer who is characterized by the 4-velocity vector u µ . For the components of these fields, one has [79] where the tensor ε µνκ is defined by the relation in which η µνκλ is an antisymmetric permutation tensor of spacetime with η 0123 = 1/ √ −g or η 0123 = √ −g . In cosmic time for a comoving observer with u µ = (1, 0, 0, 0) , we obtain where ijk is the well-known Levi-Civita symbol with 123 = 1 . Therefore, after specifying the form of the 4-vector potential, forms of the electric and magnetic fields would be achievable.

Data Analysis
In section (3), four classes (i.e. C1, C2, C3, and C4) of solutions were obtained. Due to (51) and (52) or equivalently because of the existences of n j and k j in the solutions, each of these classes of solutions has two sub-classes. Therefore, eight sets of solutions were acquired. In this section, data analysis of these solutions to illustrate the descriptions of late-time-accelerated expansion from the perspective of the studied model, are carried out. But, due to some reasonable reasons which are presented in what follows, we excerpt two sets to perform this interesting work. As is clear, our solutions are more general than ref. [68], but since the data analysis done in ref.            [68] was for the set of solutions which have the potential of the form V ∼ V 0 exp(−ϕ) which is about similar to C2-class (See (80) and (84)-(85)), hence between C1 and C2 classes, we analyze C1-class only. Between C3 and C4 classes, we select C4-class because it seems easier to tune than C3, because we must have a real scale factor and as is observed from (157), there is a power (1/(m + 2)) ≈ 1/3 hence the term under it must be positive, on the other hand, sinh 3/2 (t) -function grows so quickly than t 2 -function, hence tuning the constant parameters by choosing θ = −1 (i.e. C3-class) in order to have a real scale factor is more easier than θ = +1 (i.e. C4-class). Note that both are doable, but only for making our work easy we act on this manner. Therefore, up to now, we singled out four sets of solutions among eight sets. Based on the recent observational data, in ref. [80], it has been demonstrated that m is very close to one 4 . The case m = 1 is FRW. Indeed taking m = 1 causes that k 1 = k 2 / √ 2 which implies only one form for the 4-vector (i.e. A µ = (χ(t); k 1 A(t), k 1 A(t), k 1 A(t)) ) instead of (51) and (52). Note that our solutions for this case holds true and taking each admissible value for m excluding one, makes no considerable changes in the values of parameters and also in plots. Hence, without loss of generality, let us take m = 1 and consequently n 1 = n 2 = n = 1/3 . Indeed, with this choice, four remained cases reduced to two cases. Therefore, our analysis in what follows would be on C1 and C4 classes by taking m = 1 . Roughly speaking of constant parameters, the forms of the scale factors for C1 and C2 are totally the same and also the same situation exists between C3 and C4 classes (See (109) and (156)). However, other things like the forms of the potentials, coupling functions, scalar fields are different among the obtained cases, but the role of the form of the scale factor is very important than others when we focus on the elaborations of the recent discoveries like late-time-accelerated expansion and phase crossing. Therefore, our selections are completely justifiable and reasonable. Moreover, for these two selected cases, we present thirteen figures ( 26 plots), hence if we analyze all the obtained cases, then 104 plots should be presented which is completely unusual. First of all, let us review the important values of parameters especially from Planck data 2018 [82]: • The present value of the scale factor = 1 , • The present value of the redshift = 0 , • The age of the universe = 13.801 ± 0.024 Gyr , • The present value of the Hubble parameter = 67.4 ± 0.5 Km.s −1 .Mpc −1 , • The present value of the EoS parameter = −1.03 ± 0.03 , • The present value of the temperature of our universe = 2.7255 ± 0.0006 Kelvin .
• The onset of acceleration around the redshift z = 0.6 .
Using B-function method which has recently been suggested by the author [81], the amounts of the constants parameters are singled out as follows: 1. For C1-class (λ = +1 , c 2 = 0 , c 1 = 0 ): With these selections, thirteen figures (Twenty-six plots) are presented for data analysis. In all figures, the lefthand side plots (P 1 s) are of C1-class while the right-hand side plots (P 2 s) are of C4-class. Both plots in figure in which N 0 is about 10 −10 . Hence, m has a very narrow bound around one.
(1) indicate the scale factors, of increasing characters, expressing first the decelerated and then the accelerated expansion of the universe. According to these, if as usual one sets the present amount of the scale factor to one, b 0 = 1 , then the age of the universe is found to be t 0 = 13.801 Gyr in both cases. The scale factor versus redshift plots presented in figure (2), confirm that the present value of the scale factor and redshift are exactly 1 and 0 , respectively (i.e. (b 0 , z 0 ) = (1, 0) ) for both classes. Also, figure (2) indicates that the redshifts go down, while the scale factors increase with time. As usual, ignoring a small variation of the prefactor, we consider that the CMBR temperature falls as b −1 , then, according to figure (3), its present value in both cases would be T 0 = 2.7255 Kelvin (i.e. (T 0 , z 0 ) = (2.7255, 0) ). However, as is clear from figure (3), getting colder process in P 2 -plot is faster than P 1 -plot at the same redshift interval. It means that the scale factor of the C4-class grows faster than the scale factor of C1-class at the same time/redshift interval. This point is readily observed from figures (1) and (2) as well.
One may also learn this point by exploiting the Taylor expansion for the forms of the scale factors. The evolutions of the Hubble parameters shown in figure (4) (5) indicates passing from positive to negative values, which corresponds first to the decelerating universe, q > 0 , then to the accelerating universe, q < 0 , and its present value is q 0 = −1.045 for both cases. Obviously, q = 0 renders the inflection point (i.e. shifting from decelerated to accelerated expansion). According to these plots, the onset of acceleration for C1 class is z acc. = 0.5921 or equivalently t acc. = 6.09770 Gyr and for the C4 one is z acc. = 0.6183 which corresponds to t acc. = 6.96689 Gyr . Therefore, both estimate that the time of the start of acceleration has been at about half the age of the universe. As is clearly observed from figure (5), the deceleration parameter of C1-class falls quicker than the deceleration parameter of C4-class. This may be learned from figure (4) as well because, as is seen, the speed of the Hubble parameters in P 2 -plot of figure (4) varies faster than the Hubble parameter of P 1 -plot. The low speed of the variation of the Hubble parameter of C1-class causes that its related deceleration parameter survives longer time in the accelerated era than the deceleration parameter of C4-class and it is the reason of the differences between the depths of the holes in graphs before the present time (i.e. The fast speed of the variation of the deceleration parameter, or equivalently, the low speed of the variation of the Hubble parameter −→ Deep hole and high curvature in graph). The redshift corresponded to the onset of the acceleration of the universe expansion is also recovered by figure (6) (7) and (8) indicate that the crossing of the phantom divide line W eff. = −1 occurs for both classes from the quintessence phase W eff. > −1 to the phantom phase W eff. < −1 . These phase transitions occur at about (t, z) = (7.128Gyr, 0.526) and (t, z) = (9.578Gyr, 0.332) for C1 and C4 classes, respectively and hence our universe is currently in phantom phase. Therefore, the phase transition of C1-class is befallen sooner than C4-class. Moreover, these transitions occur at redshift-distances ∆z = 0.066 and ∆z = 0.286 after the onset of acceleration for C1 and C4, respectively. The present value of the EoS parameter is calculated to be W eff. = −1.03 for both classes of study. The reason for the deeper holes and higher curvatures in P 1 -plots than P 2 -plots of figures (7) and (8) may be argued in the same way as above discussed. When (dH/dt) > 0 , then the type of the acceleration of our universe is so-called "super-acceleration", since the universe not only accelerates but the Hubble parameter is also increasing. According to figure (9), after the redshifts z = 0.526 and z = 0.332 , for C1 and C4 classes, respectively, the type of the expansion of our universe from the point of view of the model of study is super-accelerated expansion. Hence, as we expect, the phase of our universe is phantom in throughout of evolution of the type 'super-accelerated-expansion'. Furthermore, the current phase and acceleration of our universe are phantom and super-acceleration. It is worthwhile mentioning that the redshift corresponded to the onset of super-acceleration for each class is exactly equal to the redshift of the interring to phantom phase, as it must be. All the important things which we learned up to now, are also recovered by figure (10). The red and green lines in them are the line of inflection points namely shifting from decelerated expansion to accelerated expansion, and Phantom divide line (Phase transition line), respectively. The era B[t, 0; a] > 0 represents the expansion of the universe. The region in which B[t, 0; H] > 0 is related to super-accelerated expansion. Hence, the blue curves in figure (10), demonstrating the evolution of our universe, provide all the important events deduced above through several plots. According to figure (11), both scalar fields first increase and then decrease as the universe ages. A difference between the two is that the amounts of the scalar field of C1-class are negative while for the C4-class they are positive. The plots of figure (12) indicate the manners of the scalar potentials versus time. As we observe, the behavior of the potential of C1-class first is detractive and then is additive while the potential of the C4-class behaves inverse of C1-class. And finally, the plots of the gauge kinetic functions ( f 2 s) with respect to time are presented in figure (13). As is observed their graphs are similar to their own scalar field plots (i.e. first increasing and then decreasing). In a nutshell, without assuming m = 1 , four types of well-known potentials were obtained (Note that µ > 0 ): where ν is the number of symmetries, l are the directions where symmetries do not exist, ϑ is the total dimension of the minisuperspace, we obtain a unified form of the wave function for all cases of study: It is worthwhile mentioning that the appearance of the exponential functions is due to the separation of variables in eqs. (178) and (179) where b 1 = −îl 2 l 3 /6 , b 2 = −îl 3 /(4l 1 k 2 j ) , and b 0 is an integration constant. As is clearly observed, the oscillating feature of the wave function of the universe recovers the so-called Hartle criterion [84]. Unfortunately, inserting the solution (182) into eq. (179) does not give an analytical solution. Nonetheless, by the numerical methods it may easily be demonstrated that it also leads to solutions that have oscillating feature, hence the Hartle criterion is also recovered.

Conclusion
In this paper, a modified teleparallel gravity action, containing gauge fields as a substantial part, in the framework of teleparallel gravity, was investigated. The background geometry of the studied model was anisotropic and homogeneous which covers FRW as well. By the use of the Noether approach, eight sets of solutions, leading to late-time-accelerated expansion and phase crossing from quintessence to phantom, were acquired. A usual way in dealing with total symmetry generator produced by Noether symmetry approach is that we split it into sub-symmetries and then select some of them, yielding suitable analytical solutions, to be carried by the solutions. But this approach puts limitation on the forms of the unknown functions obtained by Noether symmetry approach. In order to have more suitable solutions and interesting forms for unknown functions, an approach, CSSS-trick, was suggested. Using this trick, in our case of study, the unified dark matter potentials of the forms V = V 0 cosh 2 (µϕ) and also V = V 0 sinh 2 (µϕ) were produced which are highly rewarding. Utilizing the B-function method , data analysis of the results was performed. It was demonstrated that our solutions completely conform with all important events and astrophysical and observational data and consequently, the resulting cosmological model accommodate all the important events and data. For example, some of our findings and astrophysical data are compared in Table (I). Finally, using the Wheeler-De Witt (WDW) equation, it was indicated that the Hartle criterion due to the oscillating feature of the wave function of the universe is recovered.
Pursuant to some papers, the studied model is successful in the description of the early inflation, hence we conclude that gauge fields are able to produce both early inflation and late-time-accelerated expansion and consequently, through this term, a unified model describing all stages of the universe may be achieved.