Noether gauge symmetry of Dirac field in $2+1$ dimensional gravity

In this paper, we consider a gravitational theory including a Dirac field that is non-minimal coupled to gravity in $2+1$ dimensions. Noether gauge symmetry approach can be used to fix the form of coupling function $F(\Psi)$ and the potential $V(\Psi)$ of the Dirac field and to obtain a constant of motion for the dynamical equations. In the context of $2+1$ dimensions gravity, we investigate cosmological solutions of the field equations using these forms obtained by the existent of Noether gauge symmetry. In this picture, it is shown that for the non-minimal coupling case, the cosmological solutions indicate both an early-time inflation and late-time acceleration for the universe.


Introduction
The general theory of gravitation describes the physical phenomena coming about in a strong gravitational field and the physical behaviour of the Universe in a large-scale. At the same time, it has a remarkable mathematical and physical perspective. However, as our knowledge about the universe increases by new observational data, some new modified gravitation theories based on the theory are appeared to understand the reality of the universe. Moreover, the quantisation of the theory still keeps standing the most important problem in physics. Therefore, the general theory of gravitation is one of the most active research areas.
Giving the observational data [6][7][8][9], the universe had been accelerated in early time which is called inflation. To investigate the cosmic inflation of the universe or the beginning of expansion, the standard cosmological models were used [10,11]. Another cosmic acceleration of the universe occurs in the late-time universe and confirms various observational evidences which are the observations of supernovae Type Ia (SNe Ia) [12,13], cosmic microwave background radiation (CMB) [14,15], and large-scale structure [16]. To clarify such an accelerated expansion, many authors introduced mysterious cosmic fluid, the so-called dark energy. It has been proposed severel models in the literature such as quintessence [17], phantom [18], F (R) [19][20][21][22] and F (T ) gravity [23][24][25][26]. Recently, to understand the problem in 3+1 dimensional spacetime, it has been realized that fermionic or Dirac fields as a gravitational source which cause inflationary period in early universe and dark energy in old universe are being considered [27][28][29][30][31][32][33].In this connection, Sousa and Kremer [46] have analysed a model with Dirac field that is non-minimally coupled to the gravity in the 3 + 1 dimensional. They have utilized Noether symmetry approach to determine the forms of the potential and the coupling function to show dependence on Dirac fields in the model. Noether symmetry approach firstly introduced by Cappoziello et al. [34] to find new cosmological solutions in the 3 + 1 dimensional gravity has been used to determine the shape of the potential and the coupling function dynamically in the scalar-tensor gravity theory. It is important to note that this approach gives us constants of motion (first integral) for the dynamical equation. Recently, this approach has been extensively studied in various cosmological models i.e., in scalar field cosmology [35][36][37][38], f (R) metric and Palatini theory [39][40][41][42], f (T ) theory [43,44], telleparalel dark energy model [45]. All of these studies, it has been considered the Noether symmetry approach without a gauge term. If one consider the gauge term, then he may expect extra symmetries which yield a new constants of motion, but the term makes the calculations more complicating.
The general theory of gravitation in 3 + 1 dimensions is very difficult because it has complicated calculations. However, in 2 + 1 dimensional spacetime, it takes much more simple form because Weyl tensor vanishes and Riemann tensor is reduced to Ricci tensor. Moreover, the version in the 2 + 1 dimensions also contains the physical results in the 3 + 1 dimensional theory [1][2][3][4]. Therefore, the 2 + 1 dimensional spacetime becomes a perfect theoretical laboratory to construct new modified gravitation theories. In particular, if one wants to probe the mathematical and physical properties of the universe in the 2 + 1 dimensional spacetime by using Dirac spinor fields as a gravitational source, one can simply describe the properties of the universe according to its analogous 3 + 1 dimensional spacetime because Dirac spinor fields with 4-components in the 3 + 1 dimensional spacetime are also reduced to the 2-components spinor fields corresponding one negative and one positive energy states [5]. Furthermore, Dirac spinorial fields can give us useful information about inflation in the early time of the universe because Dirac theory has vacuum which includes zitterbewegung oscillations between positive and negative energy states. Also, the theory perfectly describe an interaction between Dirac particles and mattes. Therefore, Dirac spinorial fields can probe how to expand the universe in late time.
From these points of view, we want to study the Dirac fields as a source of early-time inflation and late-time acceleration in 2 + 1 dimensional gravity for Friedmann-Robertson-Walker (FRW) background by using Noether gauge symmetry approach. The results to be performed in this study are important in consequence of the 2 + 1 dimensional gravity and Dirac theory, because they give information about the expansion of the universe in early and late time. Also, it will be first example carried out, in regarding gravity and Dirac fields by using Noether gauge symmetry method. Therefore, the study will satisfy a motivations for new studies in the 2 + 1 and 3 + 1 dimensional gravity.
This paper is organized as follows. In the following section, we give the field equations of a theory in which the Dirac field is non-minimal coupled to the gravity in the 2+1 dimensions. In Section 3, we search the Noether gauge symmetry for the Lagrangian of the theory with the Dirac field. In Section 4, we obtain the solutions of the field equations by using Noether gauge symmetry approach. Finally, in the Section 5, we conclude with a summary of the obtained results. Throughout the paper, we use c = G = = 1.

The action and field equations
In 2 + 1 dimensional curved spacetime, the action for a Dirac field which is non-minimally coupled to scalar curvature, is given by (2.1) where F (Ψ) and V (Ψ) generic functions, representing the coupling with gravity and the selfinteraction potential of the Dirac field respectively, and they depend on only functions of the bilinear Ψ =ψψ; g is the determinant of the metric tensor g µν ; R is the Ricci scalar; ψ is two components, particle and anti-particle, Dirac field;ψ is adjoint of the ψ andψ = ψ † σ 3 . In this action, Ω µ (x) are spin connection and are given as where Γ α νµ is Christoffell symbol, and g µν is given in term of triads, e (i) µ (x), as follows, where µ and ν are curved spacetime indices running from 0 to 2. i and j are flat spacetime indices running from 0 to 2 and η ij is the 2+1 dimensional Minkowskian metric with signature (1,-1,-1). The s λν (x), spin operators, are given by whereσ µ (x) are the spacetime dependent Dirac matrices in the 2 + 1 dimensional. Thanks to triads, e µ (i) (x),σ µ (x) are related to the flat spacetime Dirac matrices, σ i , as follows (2.6) σ 1 , σ 2 and σ 3 are Pauli matrices [5]. In this representation, the Dirac equation gives an important information about the curved spacetime [47][48][49]. To analyse the expansion of the universe, we will consider the spatially flat spacetime background in 2 + 1 dimensional which is analogous of the 3+1 dimensional Friedmann-Robertson-Walker metric as follows, where a(t) is the scale factor of the Universe. The scalar curvature corresponding to the FRW metric (2.7) takes the form R = −2 2ä a +ȧ 2 a 2 , where the dot represents differentiation with respect to cosmic time t. Given the background in Eq.(2.7), it is possible to obtain the point-like Lagrangian from action (2.1) in the following form Here the prime denotes the derivative with respect to the bilinear Ψ. Because of homogeneity and isotropy of the metric, it is assumed that the spinor field only depends on time t, i.e. ψ = ψ(t). The Dirac's equations for the spinor field ψ and its adjointψ are obtained from the point-like Lagrangian (2.8) such that the Euler-Lagrange equations for ψ andψ arė which yields the Friedmann equation as follows In Eqs. (2.11) and (2.13), ρ f and p f are the effective energy density and pressure of the fermion field, respectively, so that they have the following form 14) In order to solve the field equations, we have to choose a form for the coupling function and for the potential density. To do this, in the following section we will use the Noether gauge symmetry approach.

The Noether gauge symmetry approach
Thanks to Pauli matrices, in terms of the components of the spinor field ψ = (ψ 1 , ψ 2 ) T and its adjointψ = (ψ 1 † , −ψ 2 † ), the Lagrangian (2.8) can be rewritten as follows Noether theorem is useful tool in theoretical physics which states that any differentiable symmetry of an action of a physical system leads to a corresponding conserved quantity [50]. The idea to use Noether symmetry approach without gauge term in generalized theories of gravity studies is not new and was first introduced by Cappoziello et al. to find new cosmological solutions. An another technique is related with the more general symmetries known as Noether gauge symmetries which include non-zero gauge term [51,52]. Taking into account a gauge term in Noether symmetry equation gives a more general definition of the Noether symmetry.
A vector field X for the point Lagrangian (3.1) is where α, β j and γ j are depend on t, a, ψ j and ψ † j and they are determined from the Noether gauge symmetry condition. The first prolongation of X is given by in which The vector field X is a Noether gauge symmetry corresponding to the Lagrangian L(t, a, ψ j , ψ † j ,ȧ,ψ j ,ψ † j ), if the condition holds, where B(t, a, ψ j , ψ † j ,ȧ,ψ j ,ψ † j ) is a gauge function and D t is the operator of total differentiation with respect to t The significance of Noether gauge symmetry is clearly comes from the fact that if the vector field X is the Noether gauge symmetry corresponding to the Lagrangian L(t, a, ψ j , ψ † j ,ȧ,ψ j ,ψ † j ), then is a first integral or a conserved quantity associated with X. Hence the Noether gauge symmetry condition (3.5) for the Lagrangian (3.1) leads to the following the over-determined system of differential equations we will neglect the case F ′ = 0 that corresponds to the fermionic field minimal coupled to gravity. From the equations (3.10), one can immediately sees τ = τ (t). From the rest Noether gauge symmetry equations, the complete solution is obtained as follows 17) and the coupling and the potential function are power law forms of the function of the bilinear Ψ, i.e., where c l , λ, f 0 and n (n = 1) are integration constants. From the vector field (3.17), the Lagrangian (3.1) admits three Noether gauge symmetries which are

21)
These generators constitute a well-known three-dimensional Lie algebra with commutation relations The three first integrals (or conserved quantities) associated with the Noether gauge symmetries are Here the constant parameter c 4 is assumed to be zero in the gauge function B. We note that the first integral (3.24) is related with the energy function (2.12), so that the first integral I 1 vanishes.

The solutions of field equations
Since the coupling function F depends on the bilinear function Ψ, from Dirac's equations (2.9) and (2.10) one getsΨ + 2ȧ a Ψ = 0, where a 0 is an integration constant. Using the solution (4.4) in the Friedmann equation (2.13) with (2.14) and the acceleration equation (2.11) with (2.15), we obtain a constraint relations between the constants as λ = I 2 2 (n−1) 2 2Ψ 2 an accelerated power law expansion while for n < 1/2 a decelerated expansion occurs. For n = 1/2, we have This solution correspond to the matter dominant universe with the pressure of the Dirac field p f = 0 in the General Relativity in 2 + 1 dimensions. Therefore, this solution shows that the Dirac field behaves as a standard pressureless matter field in the 2 + 1 dimensions. For the our model, we search that whether the fermionic field can provide alternative for dark energy or not. For this purpose we can define the equation of state parameter of the fermionic field by using the energy density (2.14) and pressure (2.15) as ω f = p f /ρ f . Considering the equations (3.18), (3.19), (4.2) and (4.4), we obtain (4.6) According to astrophysical data, the equation of state parameter tend to value −1. For the equation of state parameter less than −1 the dark energy is described by phantom, for −1 < ω < −1/3 the quintessence dark energy is observed and the case ω = −1 corresponds to the cosmological constant. For the our model, if 2/3 < n < 1, we obtain the quintessence phase, if n > 1, we have the phantom phase. In both cases, the universe is both expanding and accelerating. Therefore, the results show that the fermionic field may behave like both quintessence and phantom dark energy field in the late-time universe. Now, we return the case n = 1 so that the coupling and potential functions have linear forms of Ψ from the equation (3.18) and (3.19) as follows V (Ψ) = λΨ.  which has the solution and a 0 is a constant. It is clear that this solution describes an inflationary period, where the cosmic scale factor increases exponentially with the cosmic time. Therefore, one can say that the Dirac field can be behaved as inflaton field in the 2 + 1 dimensions. From Eqs. (2.14) and (2.15), the energy density and the pressure of the Dirac field are given by with the equation of state parameter ω f = −1.

Concluding remarks
The study of cosmologically models in the 2 + 1 dimensional gravitation theories provide mathematical simplicity in understanding the physical models. In the present study, we have considered a theory including the Dirac field which is non-minimally coupled to the gravity in 2 + 1 dimension. Using the Noether gauge symmetry approach, we have determined the explicit forms of the coupling function and potential as a power-law functions of the bilinear Ψ given by (3.18) and (3.19), respectively. The solutions of the field equations for FRW spacetime are presented by using the results obtained from the Noether gauge symmetry approach. It is shown that in the general case n, the Dirac field play role of the dark energy in the late-time universe. For the special case n = 1/2, the solution of dynamical equation describes a decelerated universe with a matter dominated behavior in 2 + 1 dimension. We also consider a model where the coupling function and the potential have linear forms of Ψ (i.e. the case n = 1). For such a model, the cosmological solution describes an inflationary period for the early-time universe. Therefore, we may conclude that the Dirac field behaves as an inflation field.