Spinor quintom cosmology with intrinsic spin

We consider a spinor quintom dark energy model with intrinsic spin, in the framework of Eintein- Cartan-Sciama-Kibble theory. After constructing the mathematical formalism of the model, we obtain the spin contributed total energy-momentum tensor giving the energy density and the pressure of the quintom model, and then we find the equation of state parameter in terms of the spinor potential. Choosing suitable potentials leads to the quintom scenario crossing between quintessence and phantom epochs, or vice versa. Analyzed three quintom scenarios provides stable expansion phases avoiding Big Rip singularities, and yielding matter dominated era through the stabilization of the spinor pressure via spin contribution. The stabilization in spinor pressure leads to neglecting it as compared to the increasing energy density, and constituting a matter dominated stable expansion epoch.


Introduction
The astrophysical observations show that the universe is experiencing an accelerating expansion due to an unknown component of energy, named as the dark energy (DE) which is distributed all over the universe and having a negative pressure in order to drive the acceleration of the universe [1][2][3][4][5][6][7][8][9]. There have been proposed various DE scenarios: Cosmological constant  is the oldest DE model which has a constant energy density filling the space homogeneously [10][11][12][13] Other DE scenarios can be constructed from the dynamical components, such that the quintessence, phantom, K-essence, or quintom [14][15][16]. Quintessence is considered as a DE scenario with Eos boundary according to the no-go theorem [26][27][28][29][30][31]. To overcome no-go theorem and to realize crossing phantom divide line, some modifications can be made to the single scalar field DE models. One can construct a quintom scenario by considering two scalar fields, such as quintessence and quintom [32]. The components cannot cross the 1  boundary alone, but can across it combined together. Another quintom scenario is achieved by constructing a scalar field model with non-linear or higher order derivative kinetic term [33,34] or a phantom model coupled to dark matter [35]. Also the scalar field DE models nonminimally coupled to gravity satisfy the crossing cosmological constant boundary [36][37].
The aforementioned quintom models are constructed from the scalar fields providing various phantom behaviors, but the ghost field may cause some instable solutions. By considering the linearized perturbations in the effective quantum field equation at two-loop order one can obtain an acceleration phase [38][39][40][41][42]. On the other hand, there is another quintom model satisfying the acceleration of the universe, which is constructed from the classical homogeneous spinor field  [43][44][45]. In recent years, there can be found many studies for spinor fields in cosmology [20], such that, for inflation and cyclic universe driven by spinor fields, for spinor matter in Bianchi Type I spacetime, for a DE model with spinor matter [46][47][48][49][50][51].
The consistent quintom cosmology has been proposed by using spinor matter in Friedmann-Robertson-Walker (FRW) geometry, in Einstein's general relativity framework [52]. The spinor quintom scenario allows EoS crossing 1  boundary without using a ghost field. When the derivative of the potential term with respect to the scalar bilinear   becomes negative, the spinor field shows a phantom-like behavior. But the spinor quintom exhibits a quintessence-like behavior for the positive definite potential derivative [20]. In this quintom model, there exist three categories of scenario depending on the choice of the type of potentials; one scenario is that the universe evolve from a quintessence-like phase , another scenario is for the universe evolving from a phantom-like , and the third scenario is that the EoS of spinor quintom DE crosses the 1  boundary more than one time.
In this study, we consider the spinor quintom DE, in the framework of Eintein-Cartan-Sciama-Kibble (ECSK) theory which is a generalization of the metric-affine formulation of the Einstein's general relativity with intrinsic spin [53][54][55][56][57][58][59][60][61][62]. Since the ECKS theory is the simplest theory including the intrinsic spin and avoiding the big-bang singularity [63], it is worth considering the spinor quintom in ESCK theory for investigating the acceleration phase of the universe with the phantom behavior. Therefore, we analyze the spinor quintom model with intrinsic spin in ECSK theory whether it provides the crossing cosmological constant boundary. Then if the model provides the crossing 1  boundary, we will find the suitable conditions on the potential for the crossing 1  boundary.

Algebra of spinor quintom with intrinsic spin
The most complicated example of the quantum field theories lying in curved spacetime is the theory of Dirac spinors. There occurs a conceptual problem related to obtaining the energymomentum tensor of the spinor matter field from the variation of the matter field Lagrangian.
For the scalar or tensor fields, energy-momentum tensor is the quantity describing the reaction of the matter field Lagrangian to the variations of the metric, while the matter field is held constant during the change of the metric. But for the spinor fields, the above procedure does not hold for obtaining the energy-momentum tensor from the variation with respect to metric only, because the spinor fields are the sections of a spinor bundle obtained as an associated vector bundle from the bundle of spin frames. The bundle of spin frames is a double covering of the bundle of oriented and time-oriented orthonormal frames. For spinor fields, when one varies the metric, the components of the spinor fields cannot be held fixed with respect to some fixed holonomic frame induced by a coordinate system, as in the tensor field case [64].
Therefore, the intrinsic spin of matter field in curved space time requires ECSK theory which is the simplest generalization of the metric-affine formulation of general relativity.
According to the metric-affine formulation of the gravity, the dynamical variables are and the contortion tensor notation is used for symmetrization and . Here we call these tensors as metric spin tensor and metric energy-momentum tensor, since the spacetime coordinate indices label these tensors and obtained from the variation of the Lagrangian with respect to the torsion (or contortion) tensor j i k C and the metric tensor j i g , respectively. The metric spin tensor is written as , while the metric energymomentum tensor is given Here, the Lagrangian density of the source matter field is and energy-momentum tensor is Total action of the gravitational field and the source matter in metric-affine ECSK theory is given in the same form with the classical Einstein-Hilbert action, such as . Variation of the total action with respect to the contortion tensor gives Cartan equations and with respect to the metric tensor gives Einstein equations in the form of ) ( where colon denotes the Riemannian covariant derivative with respect to the Levi-Civita connection, such as . Therefore, the total energy-momentum tensor is In metric-affine ECSK formulation of gravity, a spinor quintom field with intrinsic spin has a Lagrangian density of the form the potential of the spinor field  and the adjoint spinor . The covariant derivative of the spinor field is given as . The Riemannian covariant derivative of the spinor and adjoint spinor fields for quintom DE are given: . These covariant derivatives including the contortion tensor and the variation with respect to the spinor itself gives adjoint Dirac equation as Then the total energy-momentum tensor of the spinor quintom field is obtained from . Here the metric energy-momentum is obtained by the variation of spinor quintom Lagrange density with respect to the metric tensor, such as ik ik and the spin contributing metric energy-momentum tensor is obtained by substituting the spin tensor for spinor quintom field in ik U . Then the total metric energy-momentum tensor is found to be ik l l k Here, the semicolon covariant derivatives of the spinor field in (3) is decoupled into colon covariant derivatives in (4) and the contortion tensor containing parts of the decoupled covariant derivatives are suppressed in the spin pseudovector l s by the contribution of ik U . In order to rewrite (4) in a more convenient form for our further calculations, we multiply (1) by adjoint spinor  from the left, and multiply (2) by spinor  from right, such that 0 ) )( ( 8 . By using (5) and writing the symmetrizations explicitly in (4), we obtain the total energy-momentum tensor ik  of the spinor field dark energy in the form of We consider the spinor quintom DE model in a FRW spacetime whose metric is given as and the corresponding tetrad components read Therefore, by performing the Riemannian covariant derivatives explicitly in (7), the timelike components 

00
 and the spacelike components    of the space independent spinor field dark energy energy-momentum tensor can be obtained, such as Here  is the Hubble parameter and it comes from the Levi-Civita connections in the Riemannian covariant derivatives. We now write the ECSK Dirac equation (4) and (5) for a space independent spinor field as 0 ) )( ( 8 3 4 The solution of (12) and (13) by adding them leads . Using (13) in (10) leads to the energy density ) )( ( 16 and similarly using (12) in (11) leads to the pressure of the spinor field dark energy ) )( ( 16 respectively. Then the EoS of the spinor field is given as where 0 0    for a FRW metric. We rewrite the EoS in the form of where

Dynamical evolution of spinor quintom
From (20)  phase reads from (20) as Similarly, 1    boundary occurs for and 1    phantom phase occurs for Since prime denotes the derivative with respect to   , the solution of (22) is found as , which the dynamical evolution of potential goes to the Cosmological constant boundary.
In order to obtain a Quintom-A scenario, we define the potential to be for the early times of the universe. Then the potential leads the EoS from (20) as the term ) quintessence scenario (21) with (15), since the scaling factor a is very small at the beginning of the evolution of the universe.
When   becomes equal to 2 / c , this potential leads the spinor field to approach 1    boundary (22). After that scaling factor evolves to a greater value, then   reaches a value  . From Ref. [52].
For a Quintom-B model, the potential can be defined as then the EoS is obtained, such that Cosmological constant phase (22). As the evolution continues   gets smaller than c and spinor quintom reaches a quintessence scenario in (21). The behavior of the spinor Quintom-B scenario is represented in Figure 2 which states that the spinor field starts the evolution from below 1    Third case Quintom-C scenario can be obtained for the potential which leads the EoS as This potential provides two roots in   which is re-crossing the 1  boundary as a transition from quintessence phase to phantom phase again. This scenario is obviously a Quinton-C scenario and is illustrated in Figure 3. We see from the figure that the EoS of the quintom model . From Ref. [52].
Although considering the phantom scenarios normally leads to the Big Rip singularities due to the unbound of EoS from below 1    , our spinor quintom model with intrinsic spin in ECSK theory avoids from the Big Rip singularities by picking up to a bound value and approaching to a stable value, as seen in Figure 1 and 3. Diverging EoS of a dark fluid from a constant bound toward a lower singularity refers to continuous increase in the pressure of the fluid. This scenario is avoided in spinor quintom with intrinsic spin, which may be interpreted as the intrinsic spin of the fluid quanta leads to a bound pressure value.
The increase of the pressure with the energy density is bounded due to the effect of intrinsic spin, then singular values of EoS are avoided, and the universe enters a stable expansion in the final era.

Conclusion
By using the spinor field dark energy in a FRW geometry, a consistent quintom model in which EoS crosses 1  boundary without using a ghost field, has recently been obtained in the framework of general relativity [52]. Here, we consider the spinor field dark energy with intrinsic spin in the formalism of metric-affine ECSK theory. We first introduce the ECSK formalism, then define the model Lagrangian whose variations with respect to the tetrad field and torsion tensor gives the total energy-momentum tensor consisting of metric and spin contributions. Also from the variation of Lagrangian with respect to the spinor field we obtain the ECSK Dirac equation. By using the total energy momentum tensor and ECSK Dirac equation, the energy density and the pressure values of the spinor quintom DE is obtained, from which the EoS of the model is obtained for an arbitrary potential. The dependence of the potential on the spinor field leads to the evolution of potential with the change of scale factor, since the scale factor is increases by time. Constructing the ECSK spinor potential suitably the quintom scenario is reached, for three different cases as Quintom A, B and C models. increases. After the spinor field reaches a very large energy density value, this allows neglecting the pressure relative to energy density value, which imitates a pressure free matter dominated era with zero EoS.