Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 3740957, 10 pages http://dx.doi.org/10.1155/2016/3740957

Research Article Spinor Quintom Cosmology with Intrinsic Spin

Emre Dil

DepartmentofPhysics,SinopUniversity,Korucuk,57000Sinop,Turkey

Correspondence should be addressed to Emre Dil; [email protected]

Received 5 October 2016; Revised 5 November 2016; Accepted 7 November 2016

Academic Editor: Dandala R. K. Reddy

Copyright © 2016 Emre Dil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We consider a spinor quintom dark energy model with intrinsic spin, in the framework of Einstein-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, Hubble parameter, deceleration parameter, state finder parameter, and some distance parameter in terms of the spinor potential. Choosing suitable potentials leads to the quintom scenario crossing between quintessence and phantom epochs, or vice versa. Analyzing 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.

1. Introduction which is also named as a ghost [17]. This model experiences the shortcoming, such that its energy state is unbounded The astrophysical observations show that the universe is from below and this leads to the quantum instability problem experiencing an accelerating expansion due to an unknown [18]. If the potential value is not bounded from above, this component of energy, named as the dark energy (DE) which scenario is even instable at the classical regime known as the is distributed all over the universe and having a negative Big Rip singularity [19]. Another DE scenario, K-essence, is pressure in order to drive the acceleration of the universe [1– constructed by using kinetic term in the domain of a general 9]. Various DE scenarios have been proposed: cosmological function in the field Lagrangian. This model can realize both constant Λ is the oldest DE model which has a constant 𝜔>−1and 𝜔<−1due to the existence of a positive energy density filling the space homogeneously [10–13]. energy density but cannot allow a consistent crossing of the −1 Equation of state (EoS) parameter of a cosmological fluid is cosmological constant boundary [20]. 𝜔=𝑝/𝜌,where𝑝 and 𝜌 are the pressure and energy density ThetimevariationoftheEoSofDEhasbeenrestrictedby and the DE scenario formed by the cosmological constant the data obtained by Supernovae Ia (SNIa). According to the SNIa data, some attempts have come out to estimate the band refers to a perfect fluid with EoS 𝜔Λ =−1. power and density of the DE EoS as a function of the redshift OtherDEscenarioscanbeconstructedfromthedynam- [21].ThereoccurtwomainmodelsforthevariationofEoS ical components, such as the quintessence, phantom, K- with respect to time; Model A and Model B. While Model essence, or quintom [14–16]. Quintessence is considered as a A is valid in low redshift, Model B suffers in low redshift; 𝜔>−1 DE scenario with EoS . Such a model can be described therefore, it works in high redshift values. These models imply by using a canonical scalar field. Recent observational data that the evolution of the DE EoS begins from 𝜔>−1in the presents the idea that the EoS of DE can be in a region past to 𝜔<−1in the present time, namely, the observational where 𝜔<−1.Themostcommonscenariogeneralizing andtheoreticalresultsallowtheEoS𝜔 of DE crossing the this regime is to use a scalar field with a negative kinetic cosmological constant boundary or phantom divide during term. This DE model is known as the phantom scenario theevolutionoftheuniverse[22–25]. 2 Advances in High Energy Physics

Crossing of the DE EoS the cosmological constant bound- if the model provides the crossing −1 boundary, we will find ary is named as the quintom scenario and this can be the suitable conditions on the potential for the crossing −1 constructed in some special DE models. For instance, if boundary. we consider a single perfect fluid or single scalar field DE constituent, this model does not allow the DE EoS to cross the 𝜔=−1boundary according to the no-go theorem [26–31]. 2. Algebra of Spinor Quintom with To overcome no-go theorem and to realize crossing phantom Intrinsic Spin divide line, some modifications can be made to the single The most complicated example of the quantum field theories scalar field DE models. One can construct a quintom scenario lying in curved spacetime is the theory of Dirac spinors. by considering two scalar fields, such as quintessence and There occurs a conceptual problem related to obtaining the quintom [22]. The components cannot cross the −1 boundary alone but can cross it combined together. Another quintom energy-momentum tensor of the spinor matter field from scenario is achieved by constructing a scalar field model with the variation of the matter field Lagrangian. For the scalar nonlinear or higher order derivative kinetic term [27, 32] or a or tensor fields, energy-momentum tensor is the quantity phantom model coupled to dark matter [33]. Also the scalar describing the reaction of the matter field Lagrangian to field DE models nonminimally coupled to gravity satisfy the the variations of the metric, while the matter field is held crossing cosmological constant boundary [34, 35]. constant during the change of the metric. But for the spinor The aforementioned quintom models are constructed fields, the above procedure does not hold for obtaining the from the scalar fields providing various phantom behaviors, energy-momentum tensor from the variation with respect buttheghostfieldmaycausesomeinstablesolutions.By to metric only, because the spinor fields are the sections considering the linearized perturbations in the effective of a spinor bundle obtained as an associated vector bundle quantum field equation at two-loop order one can obtain from the bundle of spin frames. The bundle of spin frames is an acceleration phase [36–40]. On the other hand, there is double covering of the bundle of oriented and time-oriented another quintom model satisfying the acceleration of the uni- orthonormal frames. For spinor fields, when one varies the verse, which is constructed from the classical homogeneous metric, the components of the spinor fields cannot be held spinor field 𝜓 [41–44]. In recent years, many studies for spinor fixed with respect to some fixed holonomic frame induced by fields in cosmology can be found [20], such as those for a coordinate system, as in the tensor field case [63]. Therefore, inflationandcyclicuniversedrivenbyspinorfields,forspinor the intrinsic spin of matter field in curved spacetime requires matter in Bianchi Type I spacetime, and for a DE model with ECSK theory which is the simplest generalization of the spinor matter [45–50]. metric-affine formulation of general relativity. The consistent quintom cosmology has been proposed by According to the metric-affine formulation of the gravity, usingspinormatterinFriedmann-Robertson-Walker(FRW) the dynamical variables are the tetrad (vierbein or frame) field 𝑖 𝑎 𝑎 𝑗 𝑎 𝑗 𝑗 𝑖 geometry, in Einstein’s general relativity framework [51]. The 𝑒𝑎 and the spin connection 𝜔𝑏𝑘 =𝑒𝑗 𝑒 =𝑒𝑗 (𝑒 +Γ 𝑒𝑏). −1 𝑏;𝑘 𝑏,𝑘 𝑖𝑘 spinor quintom scenario allows EoS crossing boundary Here comma denotes the partial derivative with respect to 𝑘 without using a ghost field. When the derivative of the the 𝑥 coordinate, while the semicolon refers to the covariant potential term with respect to the scalar bilinear 𝜓𝜓 becomes 𝑖 derivativewithrespecttotheaffineconnectionΓ𝑗𝑘.The negative, the spinor field shows a phantom-like behavior. But antisymmetric lower indices of the affine connection give the the spinor quintom exhibits a quintessence-like behavior for 𝑆𝑖 =Γ𝑖 the positive definite potential derivative [20]. In this quintom torsion tensor 𝑗𝑘 [𝑗𝑘]. The tetrad gives the rela- model, there exist three categories of scenario depending on tion between spacetime coordinates denoted by the indices 𝑖, 𝑗, 𝑘, . the choice of the type of potentials; one scenario is that the and local Lorentz coordinates denoted by the indices 𝑎, 𝑏, 𝑐, . 𝑉𝑎 =𝑉𝑖𝑒𝑎 𝑉𝑎 universe evolves from a quintessence-like phase 𝜔>−1to ,suchthat 𝑖 ,where is a Lorentz vector 𝑖 a phantom-like phase 𝜔<−1, another scenario is for the and 𝑉 is a usual vector. Covariant derivative of a Lorentz vec- universe evolving from a phantom-like phase 𝜔<−1to a tor is defined with respect to the spin connection and denoted 𝑎 𝑎 𝑎 𝑏 𝑏 quintessence-like phase 𝜔>−1, and the third scenario is that by a bar: 𝑉|𝑖 =𝑉,𝑖 +𝜔𝑏𝑖𝑉 and 𝑉𝑎|𝑖 =𝑉𝑎,𝑖 −𝜔𝑎𝑖𝑉𝑏.Also the EoS of spinor quintom DE crosses the −1 boundary more the covariant derivative of a vector is defined in terms of the 𝑘 𝑘 𝑘 𝑙 than one time. affine connection: 𝑉;𝑖 =𝑉,𝑖 +Γ𝑙𝑖𝑉 and 𝑉𝑘;𝑖 =𝑉𝑘,𝑖 − 𝑙 In this study, we consider the spinor quintom DE, in the Γ𝑘𝑖𝑉𝑙. Local Lorentz coordinates are lowered or raised by the framework of Einstein-Cartan-Sciama-Kibble (ECSK) theory Minkowski metric 𝜂𝑎𝑏 of the flat spacetime, while the space- which is a generalization of the metric-affine formulation time coordinates are lowered or raised by the metric tensor of Einstein’s general relativity with intrinsic spin [52–61]. 𝑔𝑖𝑘.Metriccompatibilitycondition𝑔𝑖𝑗;𝑘 =0leads to the 𝑘 𝑘 𝑘 Since the ECSK theory is the simplest theory including the definition of affine connection Γ𝑖𝑗 ={𝑖𝑗 }+𝐶𝑖𝑗 in terms of the intrinsic spin and avoiding the big-bang singularity [62], it 𝑘 𝑘𝑚 Christoffel symbols { 𝑖𝑗 } = (1/2)𝑔 (𝑔𝑚𝑖,𝑗 +𝑔𝑚𝑗,𝑖 −𝑔𝑖𝑗,𝑚 ) and is worth considering the spinor quintom in ECSK theory 𝐶𝑖 =𝑆𝑖 +2𝑆𝑖 for investigating the acceleration phase of the universe with the contortion tensor 𝑗𝑘 𝑗𝑘 (𝑗𝑘).Throughoutthe 𝐴 = (1/2)(𝐴 +𝐴 ) the phantom behavior. Therefore, we analyze the spinor paper, the (𝑗𝑘) 𝑗𝑘 𝑘𝑗 notation is used for 𝐴 =𝐴 −𝐴 quintom model with intrinsic spin in ECSK theory whether it symmetrization and [𝑗𝑘] 𝑗𝑘 𝑘𝑗 is used for the anti- 𝑎 𝑏 𝑗 provides the crossing cosmological constant boundary. Then symmetrization. With the definitions 𝑔𝑖𝑘 =𝜂𝑎𝑏𝑒𝑖 𝑒𝑘 and 𝑆𝑖𝑘 = Advances in High Energy Physics 3

𝑗 𝑎 𝑗 𝑘 𝑘 𝜔[𝑖𝑘] +𝑒[𝑖,𝑘]𝑒𝑎, the metric tensor and the torsion tensor can the form ℓ𝑚 = 𝑒(𝑖/2)(𝜓𝛾 𝜓;𝑘 − 𝜓;𝑘𝛾 𝜓) − 𝑒𝑉,where𝑉 is be considered as the dynamical variable instead of the tetrad the potential of the spinor field 𝜓 and the adjoint spinor + 0 and spin connection. 𝜓=𝜓𝛾 . The covariant derivative of the spinor field is 𝑖𝑗⋅⋅⋅ 𝜓 =𝜓 −Γ𝜓 𝜓 = 𝜓 −Γ𝜓 Γ = A tensor density Α 𝑘𝑙⋅⋅⋅ is given in terms of the corre- given as ;𝑘 ,𝑘 𝑘 and ;𝑘 ,𝑘 𝑘 ,where 𝑘 𝑖𝑗⋅⋅⋅ 𝑖𝑗⋅⋅⋅ 𝑖𝑗⋅⋅⋅ 𝑎 −(1/4)𝜔 𝛾𝑎𝛾𝑏 sponding tensor 𝐴𝑘𝑙⋅⋅⋅ as Α 𝑘𝑙⋅⋅⋅ =𝑒𝐴𝑘𝑙⋅⋅⋅,where𝑒=det 𝑒𝑖 = 𝑎𝑏𝑘 is the Fock-Ivanenko spin connection, and 𝛾𝑘 𝛾𝑎 √−det 𝑔𝑖𝑘. Therefore, we represent the spin density and then and are the metric and dynamical Dirac gamma 𝛾𝑘 =𝑒𝑘𝛾𝑎 𝛾(𝑘𝛾𝑚) =𝑔𝑘𝑚𝐼 𝛾(𝑎𝛾𝑏) = the energy-momentum density, such as 𝜎𝑖𝑗𝑘 =𝑒𝑠𝑖𝑗𝑘 and matrices satisfying 𝑎 , ,and 𝑎𝑏 Τ𝑖𝑘 =𝑒𝑇𝑖𝑘. Here we call these tensors metric spin tensor and 𝜂 𝐼. The covariant derivative of the spinor can be decom- metric energy-momentum tensor, since the spacetime coor- posed into the Riemannian covariant derivative plus a dinate indices label these tensors and are obtained from the contortion tensor 𝐶𝑖𝑗𝑘 containing term, such as 𝜓;𝑘 = variation of the Lagrangian with respect to the torsion (or [𝑖 𝑗] [𝑖 𝑗] 𝜓:𝑘 + (1/4)𝐶𝑖𝑗𝑘 𝛾 𝛾 𝜓 and 𝜓;𝑘 = 𝜓:𝑘 − (1/4)𝐶𝑖𝑗𝑘 𝜓𝛾 𝛾 .The 𝐶𝑖𝑗 𝑔𝑖𝑗 contortion) tensor 𝑘 and the metric tensor ,respectively. Riemannian covariant derivative of the spinor and adjoint 𝑘 𝑖𝑗 𝜓 =𝜓+ The metric spin tensor is written as 𝑠𝑖𝑗 = (2/𝑒)(𝛿ℓ𝑚/𝛿𝐶𝑘 )= spinor fields for quintom DE are given: :𝑘 ,𝑘 𝑖𝑗 𝑖 𝑗 𝑚 𝑖 𝑗 𝑚 (1/4)𝑔𝑖𝑘 { 𝑗𝑚 }𝛾 𝛾 𝜓 and 𝜓 = 𝜓 − (1/4)𝑔𝑖𝑘 { 𝑗𝑚 }𝛾 𝛾 𝜓. (2/𝑒)(𝜕ℓ𝑚/𝜕𝐶𝑘 ), while the metric energy-momentum tensor :𝑘 ,𝑘 𝑖𝑗 𝑖𝑗 These covariant derivatives including the contortion tensor is given by 𝑇𝑖𝑗 = (2/𝑒)(𝛿ℓ𝑚/𝛿𝑔 ) = (2/𝑒)[𝜕ℓ𝑚/𝜕𝑔 − 𝑖𝑗 𝐶𝑖𝑗𝑘 are embedded in the spinor quintom Lagrange density. 𝜕 (𝜕ℓ /𝜕(𝑔 ))] 𝑘 𝑚 ,𝑘 . Here, the Lagrangian density of the source However, the explicit form of the contortion tensor which ℓ =𝑒𝐿 matter field is 𝑚 𝑚. When the local Lorentz coordinates can be obtained from the Cartan equations is needed. Since 𝜎𝑖 =𝑒𝑠𝑖 Τ𝑎 =𝑒𝑇𝑎 are also used in these tensors as 𝑎𝑏 𝑎𝑏 and 𝑖 𝑖 ,we the right hand side of Cartan equations contains the spin 𝑖 𝑎 call 𝑠𝑎𝑏 and 𝑇𝑖 dynamical spin tensor and dynamical energy- tensor density, we obtain the spin tensor from the variation of momentum tensor, respectively, and they are obtained from the spinor Lagrangian with respect to the contortion tensor, 𝑖𝑗𝑘 𝑖𝑗𝑘 𝑖𝑗𝑘𝑙 𝑖𝑗𝑘𝑙 the variation of the Lagrangian with respect to the tetrad such that 𝑠 = (1/𝑒)𝜎 = −(1/𝑒)𝜀 𝑠𝑙,where𝜀 𝑖 𝑎𝑏 𝑖 𝑖 5 𝑒𝑎 and the spin connection 𝜔𝑖 . The dynamical spin tensor is the Levi-Civita symbol, 𝑠 = (1/2) 𝜓𝛾 𝛾 𝜓 is the spin 𝑖 𝑎𝑏 𝑎𝑏 5 0 1 2 3 is 𝑠𝑎𝑏 = (2/𝑒)(𝛿ℓ𝑚/𝛿𝜔𝑖 ) = (2/𝑒)(𝜕ℓ𝑚/𝜕𝜔𝑖 ), and energy- pseudovector, and 𝛾 =𝑖𝛾𝛾 𝛾 𝛾 . Inserting the spin tensor 𝑎 𝑖 𝑖 momentum tensor is 𝑇𝑖 = (1/𝑒)(𝛿ℓ𝑚/𝛿𝑒𝑎) = (1/𝑒)[𝜕ℓ𝑚/𝜕𝑒𝑎 − for spinor quintom field in the Cartan equations gives the 𝑖 𝑆 =𝐶 = (1/2)𝜅𝜀 𝑠𝑙 𝜕𝑗(𝜕ℓ𝑚/𝜕(𝑒𝑎,𝑗))]. torsion tensor 𝑖𝑗𝑘 𝑖𝑗𝑘 𝑖𝑗𝑘𝑙 which will take place Total action of the gravitational field and the source in the spinor quintom Lagrange density [52–62]. matter in metric-affine ECSK theory is given in the same The variation of the spinor quintom matter Lagrangian form with the classical Einstein-Hilbert action, such as 𝑆= density with respect to the adjoint spinor gives the ECSK 4 𝜅 ∫(ℓ𝑔 +ℓ𝑚)𝑑 𝑥,where𝜅=8𝜋𝐺and ℓ𝑔 = −(1/2𝜅)𝑒𝑅 is Dirac equation, such as the gravitational Lagrangian density. Here Ricci scalar is 𝑘 𝜕𝑉 3 𝑘 5 5 constructed from the spin connection containing curvature 𝑖𝛾 𝜓:𝑘 − + 𝜅(𝜓𝛾 𝛾 𝜓)𝑘 𝛾 𝛾 𝜓=0, 𝑏 𝑗 𝑏 𝑏𝑐 𝑘 𝜕𝜓 8 (1) tensor, such that 𝑅=𝑅𝑗𝑒𝑏,where𝑅𝑗 =𝑅𝑗𝑘𝑒𝑐 is the Ricci 𝑅𝑏𝑐 tensor obtained from the curvature tensor 𝑗𝑘 and finally this and the variation with respect to the spinor itself gives adjoint curvature tensor is expressed in terms of the spin connection, 𝑎 𝑎 𝑎 𝑎 𝑐 𝑎 𝑐 Dirac equation as such that 𝑅𝑏𝑖𝑗 =𝜔𝑏𝑗,𝑖 −𝜔𝑏𝑖,𝑗 +𝜔𝑐𝑖𝜔𝑏𝑗 −𝜔𝑐𝑗 𝜔𝑏𝑖. Variation of the total action with respect to the contortion tensor gives Cartan 𝑘 𝜕𝑉 3 𝑘 5 5 𝑗 𝑗 𝑗 𝑗 𝑖𝜓:𝑘𝛾 + − 𝜅(𝜓𝛾 𝛾 𝜓) 𝜓𝛾𝑘𝛾 =0. (2) equations 𝑆𝑖𝑘 −𝑆𝑖𝛿𝑘 +𝑆𝑘𝛿𝑖 = −(𝜅/2𝑒)𝜎𝑖𝑘 and with respect to 𝜕𝜓 8 the metric tensor gives Einstein equations in the form of 𝐺𝑖𝑘 = 𝑗 𝑙𝑚 𝜅(𝑇𝑖𝑘 +𝑈𝑖𝑘),where𝐺𝑖𝑘 =𝑃 − (1/2)𝑃𝑙𝑚 𝑔𝑖𝑘 is the Einstein Then the total energy-momentum tensor of the spinor quin- 𝑖𝑗𝑘 Θ =𝑇 +𝑈 𝑃𝑗 tom field is obtained from 𝑖𝑘 𝑖𝑘 𝑖𝑘.Herethemetric tensor and 𝑖𝑗𝑘 is the Riemann curvature tensor satisfying energy-momentum tensor is obtained by the variation of 𝑖 𝑖 𝑖 𝑖 𝑗 𝑖 𝑗 𝑖 𝑅𝑘𝑙𝑚 =𝑃𝑘𝑙𝑚 +𝐶𝑘𝑚:𝑙 −𝐶𝑘𝑙:𝑚 +𝐶𝑘𝑚𝐶𝑗𝑙 −𝐶𝑘𝑙𝐶𝑗𝑚,wherecolon spinor quintom Lagrange density with respect to the metric denotes the Riemannian covariant derivative with respect tensor, such as 𝑘 𝑘 𝑘 𝑙 to the Levi-Civita connection, such as 𝑉:𝑖 =𝑉,𝑖 +{𝑙𝑖 }𝑉 𝑉 =𝑉 −{𝑙 }𝑉 2 𝜕ℓ /𝜕𝑔𝑖𝑘 and 𝑘:𝑖 𝑘,𝑖 𝑘𝑖 𝑙. Also for torsion-free general 𝑇 = [ 𝑚 ] relativity theory, curvature tensor turns out to be the Rie- 𝑖𝑘 𝑒 𝑖𝑘 −𝜕𝑗 (𝜕ℓ𝑚/𝜕(𝑔 ,𝑗 )) mann tensor. Right hand side of Einstein equations contains [ ] an extra term 𝑈𝑖𝑘 which is the contribution to the energy- 𝑖 = (𝜓𝛿𝑗 𝛾 𝜓 − 𝜓 𝛿𝑗 𝛾 𝜓) (3) momentum tensor from the torsion and it is quadratic in 2 (𝑖 𝑘) ;𝑗 ;𝑗 (𝑖 𝑘) 𝑈 = 𝜅(−𝑠𝑖𝑗 𝑠𝑘𝑙 − (1/2)𝑠𝑖𝑗𝑙 𝑠𝑘 +(1/ the spin tensor, such as 𝑖𝑘 [𝑙 𝑗] 𝑗𝑙 𝑖 𝑗𝑙𝑖 𝑘 𝑖𝑘 𝑙 𝑗𝑚 𝑗𝑙𝑚 𝑗 𝑗 4)𝑠 𝑠 + (1/8)𝑔 (−4𝑠 𝑠 +𝑠 𝑠 )) − (𝜓𝛾 𝜓;𝑗 − 𝜓 𝛾 𝜓)𝑖𝑘 𝑔 +𝑉𝑔𝑖𝑘, 𝑗𝑙 𝑗[𝑚 𝑙] 𝑗𝑙𝑚 . Therefore, the total 2 ;𝑗 energy-momentum tensor is Θ𝑖𝑘 =𝑇𝑖𝑘 +𝑈𝑖𝑘. In metric-affine ECSK formulation of gravity, a spinor and the spin contributing metric energy-momentum tensor quintom field with intrinsic spin has a Lagrangian density of is obtained by substituting the spin tensor for spinor quintom 4 Advances in High Energy Physics

field in 𝑈𝑖𝑘. Then the total metric energy-momentum tensor covariant derivatives. We now write ECSK Dirac equations is found to be (4) and (5) for a space independent spinor field as

𝑖 𝑗 𝑗 3 𝑙 0 3𝑖 0 󸀠 Θ = (𝜓𝛿 𝛾 𝜓 − 𝜓 𝛿 𝛾 𝜓) + 𝜅𝑠 𝑠 𝑔 . (4) 𝑖𝜓𝛾 𝜓+̇ 𝐻𝜓𝛾 𝜓−𝑉𝜓𝜓 𝑖𝑘 2 (𝑖 𝑘) :𝑗 :𝑗 (𝑖 𝑘) 4 𝑙 𝑖𝑘 4 (12) 3 + 𝜅(𝜓𝛾0𝛾5𝜓) (𝜓𝛾 𝛾5𝜓) = 0, Here, the semicolon covariant derivatives of the spinor field 8 0 in (3) are decoupled into colon covariant derivatives in (4) 3𝑖 and the contortion tensor containing parts of the decoupled 𝑖𝜓𝛾̇ 0𝜓+ 𝐻𝜓𝛾0𝜓+𝑉󸀠𝜓𝜓 covariant derivatives are suppressed in the spin pseudovector 4 𝑙 (13) 𝑠 by the contribution of 𝑈𝑖𝑘. In order to rewrite (4) in a more 3 − 𝜅(𝜓𝛾0𝛾5𝜓) (𝜓𝛾 𝛾5𝜓) = 0. convenient form for our further calculations, we multiply (1) 8 0 by adjoint spinor 𝜓 fromtheleftandmultiply(2)byspinor𝜓 from right, such that The solution of (12) and (13) by adding them leads to

𝑘 󸀠 3 𝑘 5 5 3 𝑖𝜓𝛾 𝜓 −𝑉𝜓𝜓 + 𝜅(𝜓𝛾 𝛾 𝜓)𝜓𝛾 ( 𝛾 𝜓) = 0, (5) 𝜓𝜓+̇ 𝜓𝜓̇ + 𝐻𝜓𝜓 = 0, :𝑘 8 𝑘 2 (14)

𝑘 󸀠 3 𝑘 5 5 𝑁 𝑖𝜓:𝑘𝛾 𝜓+𝑉𝜓𝜓− 𝜅 (𝜓𝛾 𝛾 𝜓)(𝜓𝛾𝑘𝛾 𝜓) =0, (6) 𝜓𝜓 = . (15) 8 𝑎3/2

󸀠 𝑁 where 𝑉 = 𝜕𝑉/𝜕(𝜓𝜓) for which 𝜓(𝜕𝑉/𝜕𝜓) = (𝜕𝑉/𝜕𝜓)𝜓 = Here is the integration constant, and then, by using the 󸀠 𝛽𝑡 𝑉 𝜓𝜓. By using (5) and writing the symmetrizations explicitly scale factor 𝑎∝𝑒 for a cosmological fluid [64], we can also −3𝛽𝑡/2 in (4), we obtain the total energy-momentum tensor Θ𝑖𝑘 of obtain 𝜓𝜓 = 𝑁𝑒 . Using (13) in (10) leads to the energy the spinor field dark energy in the form of density

𝑖 0 󸀠 3 0 5 5 Θ = (𝜓𝛾 𝜓 + 𝜓𝛾 𝜓 − 𝜓 𝛾 𝜓−𝜓 𝛾 𝜓) 𝜌=Θ =𝑉𝜓𝜓 − 𝜅(𝜓𝛾 𝛾 𝜓)𝜓𝛾 ( 0𝛾 𝜓) , (16) 𝑖𝑘 4 𝑖 :𝑘 𝑘 :𝑖 :𝑖 𝑘 :𝑘 𝑖 0 16 (7) 1 󸀠 𝑙 and similarly using (12) in (11) leads to the pressure of the + (𝑉 𝜓𝜓𝜓𝛾 −𝑖 𝜓:𝑙)𝑔𝑖𝑘. 2 spinor field dark energy

Weconsider the spinor quintom DE model in FRW spacetime 𝜄 3 0 5 5 𝑝=−Θ =− 𝜅 (𝜓𝛾 𝛾 𝜓)(𝜓𝛾 𝛾 𝜓) , (17) whose metric is given as 𝜄 16 0

2 2 2 2 2 2 respectively. Then the EoS of the spinor field is given as 𝑑𝑠 =𝑑𝑡 −𝑎 (𝑡) [𝑑𝑥 +𝑑𝑦 +𝑑𝑧 ], (8) 2 𝑝 (3/16) 𝜅(𝜓𝛾0𝛾5𝜓) and the corresponding tetrad components read 𝜔= = , (18) 𝜌 (3/16) 𝜅(𝜓𝛾0𝛾5𝜓)2 −𝑉󸀠𝜓𝜓 𝑒𝑖 =𝛿𝑖 , 0 0 0 where 𝛾 =𝛾0 for a FRW metric. We rewrite the EoS in the 1 (9) 𝑒𝑖 = 𝛿𝑖 . form of 𝑎 𝑎 (𝑡) 𝑎 𝜔=−1+𝛼, (19) Therefore, by performing the Riemannian covariant deriva- 𝜓 tives explicitly in (7), the time-like components Θ00 and the where Θ𝜓 space-like components 𝜄𝜄 of the space independent spinor 2 field dark energy energy-momentum tensor can be obtained, 6𝜅 (𝜓𝛾0𝛾5𝜓) − 16𝑉󸀠𝜓𝜓 𝛼= . (20) such as 3𝜅 (𝜓𝛾0𝛾5𝜓)2 −16𝑉󸀠𝜓𝜓

𝑖 ̇ 1 󸀠 3𝑖 Θ =− 𝜓𝛾 𝜓+ 𝑉 𝜓𝜓 − 𝐻𝜓𝛾 𝜓, 𝛼=4/3 𝜔= 00 2 0 2 8 0 (10) It is known that for the EoS of the spinor field is 1/3 and it behaves like radiation, but for 𝛼=1, 𝜔=0,andit 𝑖 1 󸀠 3𝑖 is normal matter. On the other hand, if 𝛼<2/3,theEoS𝜔< Θ𝜄𝜄 =− 𝜓𝛾0𝜓𝑔̇ 𝜄𝜄 + 𝑉 𝜓𝜓𝑔𝜄𝜄 − 𝐻𝜓𝛾0𝜓𝑔𝜄𝜄. (11) 2 2 8 −1/3 meaning that the spinor field behaves like a DE leading to the acceleration of universe. The 𝛼<2/3region allows us Here 𝐻=𝑎(𝑡)/𝑎(𝑡)̇ is the Hubble parameter and it to investigate the dynamical evolution of the spinor quintom comes from the Levi-Civita connections in the Riemannian DE described in ECSK formalism with intrinsic spin. Advances in High Energy Physics 5

3. Dynamical Evolution of Spinor Quintom 10 8 From (20) we deduce that the spinor field can have an EoS of −1<𝜔<−1/3for 0<𝛼<2/3and shows a quintessence- 6 like behavior, but it has 𝜔=−1cosmological constant value if 4 𝛼=0 𝜔<−1 𝛼< ,andthenitbehaveslikeaphantomfor if 2 0. Therefore, the spinor field exhibits a quintom picture by crossing the cosmological constant boundary 𝜔=−1from 𝜔 0 above or below this boundary depending on the sign of 𝛼 in −2 (20). −4 There exist three categories of spinor quintom evolution depending on the behavior of the potential 𝑉.Thequintom −6 −1 < 𝜔 scenario may exhibit an evolution starting from −8 quintessence phase to 𝜔<−1phantom phase, called 𝜔<−1 −10 Quintom-A. Another scenario may evolve from to −1 012345 −1 < 𝜔 ,Quintom-Bscenario.Thelastquintomscenario t contains the evolution for crossing 𝜔=−1more than one time, and it is called Quintom-C model. Figure 1: Evolution of 𝜔 withpotential(24)asafunctionoftime. Considering the quintom scenario in which the spinor For the numerical analysis we assume 𝑁=3, 𝑐=6,and𝛽=1. field comes from −1 < 𝜔 quintessence phase to 𝜔<−1 From [51]. phantom phase, we first need to find the condition 0<𝛼. 󸀠 Since the energy density (16) must be positive definite, 𝑉 is 𝜔 positive. Therefore, the condition of occurring −1 < 𝜔 phase by numerical analysis. According to the figure, starts its −1 −1 reads from (20) as evolution from above to below .Wesetthecrossing cosmological constant boundary as at 𝑡=0.Aftercrossing 󸀠 0 5 2 −1 16𝑉 𝜓𝜓 > 6𝜅𝜓𝛾 ( 𝛾 𝜓) . (21) the boundary, spinor quintom picks up and is avoided from a Big Rip singularity and then enters a stable matter 𝜔=0 Similarly, 𝜔=−1boundary occurs for dominated expansion with value. Wecan also find other important cosmological quantities, 󸀠 0 5 2 such as luminosity distance, Hubble parameter, deceleration 16𝑉 𝜓𝜓=6𝜅(𝜓𝛾 𝛾 𝜓) , (22) parameter, and jerk and state finder parameters. For this we use the Friedmann equations and 𝜔<−1phantom phase occurs for 𝑎̈ 4𝜋𝐺 2 =− (𝜌 + 3𝑝) , (26) 16𝑉󸀠𝜓𝜓 < 6𝜅𝜓𝛾 ( 0𝛾5𝜓) . (23) 𝑎 3 𝑎̇ 2 8𝜋𝐺 Since prime denotes the derivative with respect to 𝜓𝜓,the ( ) = 𝜌. (27) 0 5 2 𝑎 3 solution of (22) is found as 𝑉Λ =(6𝜅/16)(𝜓𝛾 𝛾 𝜓) ln 𝜓𝜓, in which the dynamical evolution of potential goes to the By using (16) for the potential (24), we obtain spinor energy cosmological constant boundary. density as In order to obtain Quintom-A scenario, we define the 3𝜅 2 𝜌=( )(𝜓𝛾0𝛾5𝜓) + (2𝜓𝜓−𝑐) 𝜓𝜓, potential to be 16 (28)

6𝜅 0 5 2 𝑉=( )(𝜓𝛾 𝛾 𝜓) ln 𝜓𝜓 − (𝑐 − 𝜓𝜓) 𝜓𝜓, (24) and substituting (15) and (28) in (27) we obtain 16 𝑎̇ 𝐻= fortheearlytimesoftheuniverse.Thenthepotentialleads 𝑎 the EoS from (20) as (29) 8𝜋𝐺 3𝜅𝑁2 = √ √(− +2𝑁2)𝑒−3𝛽𝑡 −𝑐𝑁𝑒−3𝛽𝑡/2. 16 (2𝜓𝜓 − 𝑐) 𝜓𝜓 3 8 𝜔=−1+ , (25) 16 (2𝜓𝜓 − 𝑐) 𝜓𝜓 + (3𝜅/16) (𝜓𝛾0𝛾5𝜓)2 Solving the differential equation in (29) gives the scale factor as and the term (2𝜓𝜓 − 𝑐) in this potential satisfies −1 < 𝜔 quintessence scenario (21) with (15), since the scaling factor { 8𝜋𝐺 3𝜅𝑁2 𝑎 𝑎= √ [𝐶 +𝑡√2𝑁2 − −𝑐𝑁 is very small at the beginning of the evolution of the exp { 3 1 8 universe. When 𝜓𝜓 becomes equal to 𝑐/2, this potential leads { [ the spinor field to approach 𝜔=−1boundary (22). After (30) 𝜓𝜓 3𝛽 (4𝑁2 −3𝜅𝑁2/4 − 𝑐𝑁) } that scaling factor evolves to a greater value, then reaches a 2 ] 𝑐/2 +𝑡 +⋅⋅⋅ } , value smaller than .Thisgivesthecondition(23)phantom √2𝑁2 −3𝜅𝑁2/8 − 𝑐𝑁 phase 𝜔<−1.WeillustratethisbehaviorinFigure1 ]} 6 Advances in High Energy Physics from which we obtain the redshift 0 −1 { 8𝜋𝐺 𝑧=−1+ −√ [𝐶 exp { 3 1 −2 { [ −3 3𝜅𝑁2 +𝑡√2𝑁2 − −𝑐𝑁 (31) −4 8 𝜔 −5 2 2 3𝛽 (4𝑁 −3𝜅𝑁 /4 − 𝑐𝑁) } −6 2 ] +𝑡 +⋅⋅⋅ } . √2𝑁2 −3𝜅𝑁2/8 − 𝑐𝑁 ]} −7 −8 Therefore, we can find the Luminosity distance −9 −1 1 2 −1 012345 𝑑𝐿 =𝐻0 [𝑧 + (1 − 𝑞0)𝑧 +⋅⋅⋅], (32) 2 t in terms of the measurable quantities present time Hubble Figure 2: Evolution of 𝜔 with potential (38) as a function of time. 𝑁=3 𝑐=3 𝛽=1 parameter 𝐻0 and deceleration parameter 𝑞0.Moreover,to For the numerical analysis we assume , ,and .From find the deceleration parameter, we use (16) and (17) for the [51]. potential (24) and obtain the term in (26), such that

3𝜅 2 𝜌+3𝑝=(− )(𝜓𝛾0𝛾5𝜓) +(2𝜓𝜓 − 𝑐) 𝜓𝜓, For a Quintom-B model, the potential can be defined as 8 (33) 6𝜅 2 𝑉=( )(𝜓𝛾0𝛾5𝜓) 𝜓𝜓 − (𝑐 − 𝜓𝜓)2 , and inserting (15) and (33) in (26) we find 16 ln (38) 𝑎̈ 4𝜋𝐺 3𝜅𝑁2 =− [( +2𝑁2)𝑒−3𝛽𝑡 −𝑐𝑁𝑒−3𝛽𝑡/2]. 𝑎 3 4 (34) andthentheEoSisobtained,suchthat

Then, the deceleration parameter is obtained from (29) and 32 (𝑐 − 𝜓𝜓) 𝜓𝜓 𝜔=−1+ , (39) (34), such as 32 (𝑐 − 𝜓𝜓) 𝜓𝜓 + (3𝜅/16) (𝜓𝛾0𝛾5𝜓)2 𝑎𝑎̈ 𝑎̈ 𝑞=− =−𝐻−2 𝑎̇2 𝑎 which lead to 𝜔<−1phantom phase (23) at the beginning of the evolution of universe. With the increasing of the scale 2 2 −3𝛽𝑡 −3𝛽𝑡/2 (35) 1 (3𝜅𝑁 /4 + 2𝑁 )𝑒 −𝑐𝑁𝑒 factor, 𝜓𝜓 decreases to 𝑐 and the term 2(𝑐 − 𝜓𝜓) becomes = . 𝜔=−1 2 (−3𝜅𝑁2/8 + 2𝑁2)𝑒−3𝛽𝑡 −𝑐𝑁𝑒−3𝛽𝑡/2 zero. This gives cosmological constant phase (22). As the evolution continues 𝜓𝜓 gets smaller than 𝑐 and spinor Also,wefinallyobtainthestatefinderparametersforspinor quintom reaches a quintessence scenario −1 < 𝜔 in (21). The Quintom-A by taking time derivative of (34), such that behavior of the spinor Quintom-B scenario is represented in Figure 2 which states that the spinor field starts the evolution 𝑎̈ ∙ 𝜔=−1 𝜔=−1 ( ) from below to above . Crossing from phantom 𝑎 to quintessence phase continues in this phase with an EoS (36) value of −1 < 𝜔 < −1/3 which imitates a stable de Sitter 3𝜅𝑁2 𝑐𝑁 =12𝜋𝛽𝐺[( +2𝑁2)𝑒−3𝛽𝑡 − 𝑒−3𝛽𝑡/2]. accelerated expansion for a scalar field dark energy model. 4 2 We proceed to find the other cosmological quantities; by using (16) for the potential (38), we get the quintom energy Then the state finder parameter is density ... ∙ 𝑎 𝑎̈ −3 𝑟= =( ) 𝐻 −𝑞 3𝜅 2 𝑎𝐻3 𝑎 𝜌=( )(𝜓𝛾0𝛾5𝜓) +2(𝑐−𝜓𝜓) 𝜓𝜓, 16 (40) 9𝛽 (3𝜅𝑁2/4 + 2𝑁2)𝑒−3𝛽𝑡 − (𝑐𝑁/2) 𝑒−3𝛽𝑡/2 = √32𝜋𝐺 ((−3𝜅𝑁2/8 + 2𝑁2)𝑒−3𝛽𝑡 −𝑐𝑁𝑒−3𝛽𝑡/2)3/2 (37) and inserting (15) and (40) in (27) we find

2 2 −3𝛽𝑡 −3𝛽𝑡/2 𝑎̇ 1 (3𝜅𝑁 /4 + 2𝑁 )𝑒 −𝑐𝑁𝑒 𝐻= − , 𝑎 2 (−3𝜅𝑁2/8 + 2𝑁2)𝑒−3𝛽𝑡 −𝑐𝑁𝑒−3𝛽𝑡/2 (41) 8𝜋𝐺 3𝜅𝑁2 and one can obtain the second state finder parameter 𝑠=2(𝑟− = √ √(− −2𝑁2)𝑒−3𝛽𝑡 +2𝑐𝑁𝑒−3𝛽𝑡/2. 3 8 1)/3(2𝑞 − 1). Advances in High Energy Physics 7

Bysolvingthedifferentialequationin(41),weobtainthescale 30 factor as 25 { 8𝜋𝐺 3𝜅𝑁2 𝑎= √ [𝐶 +𝑡√2𝑐𝑁 − 2𝑁2 − exp { 3 2 8 20 { [ (42) 15 3𝛽 (2𝑐𝑁 −4𝑁2 −3𝜅𝑁2/4) } 2 ] 𝜔 +𝑡 +⋅⋅⋅ } , √2𝑐𝑁 − 2𝑁2 −3𝜅𝑁2/8 10 ]} which gives the redshift as 5 8𝜋𝐺 0 𝑧=−1+ {−√ [𝐶 exp 3 2 −5 −1 0123456789 +𝑡√2𝑐𝑁 − 2𝑁2 −3𝜅𝑁2/8 (43) t 𝜔 3𝛽 (2𝑐𝑁 −4𝑁2 −3𝜅𝑁2/4) Figure 3: Evolution of with potential (49) as a function of time. +𝑡2 +⋅⋅⋅]}. For the numerical analysis we assume 𝑁=3, 𝑐=3,and𝛽=1.From √2𝑐𝑁 − 2𝑁2 −3𝜅𝑁2/8 [51].

−1 ThusweobtaintheLuminositydistance𝑑𝐿 =𝐻0 [𝑧 + (1/2)(1−𝑞 )𝑧2 +⋅⋅⋅] 𝐻 0 in terms of the measurable quantities 0 from which we can obtain the second state finder parameter 𝑞 and 0. Then to find the deceleration parameter, we use (16) 𝑠 = 2(𝑟 − 1)/3(2𝑞 −1). and (17) for the potential (24) and obtain the term in (26), Third case Quintom-C scenario can be obtained for the such that potential 3𝜅 0 5 2 𝜌+3𝑝=(− )(𝜓𝛾 𝛾 𝜓) +2(𝑐−𝜓𝜓) 𝜓𝜓, 6𝜅 2 8 (44) 𝑉=( )(𝜓𝛾0𝛾5𝜓) 𝜓𝜓 − (𝑐 − 𝜓𝜓)2 𝜓𝜓, 16 ln (49) and inserting (15) and (44) in (26) we obtain which leads the EoS as 𝑎̈ 4𝜋𝐺 3𝜅𝑁2 =− [( −2𝑁2)𝑒−3𝛽𝑡 +2𝑐𝑁𝑒−3𝛽𝑡/2]. 𝑎 3 4 (45) 𝜔=−1 16 (𝑐 − 3𝜓𝜓)𝜓𝜓 ( − 𝑐) 𝜓𝜓 (50) The deceleration parameter is obtained from (41) and (45) as + . 2 𝑎𝑎̈ 𝑎̈ 16 (𝑐 − 3𝜓𝜓)𝜓𝜓 ( − 𝑐) 𝜓𝜓 + (3𝜅/16) (𝜓𝛾0𝛾5𝜓) 𝑞=− =−𝐻−2 2 𝑎̇ 𝑎 󸀠 This potential provides two roots in 𝑉 𝜓𝜓 for crossing the 2 2 −3𝛽𝑡 −3𝛽𝑡/2 (46) 𝜔=−1 1 (3𝜅𝑁 /4 − 2𝑁 )𝑒 +𝑐𝑁𝑒 boundary. The term coming from the derivative = . 2 2 2 2 −3𝛽𝑡 −3𝛽𝑡/2 of 𝑉 is −3(𝜓𝜓) +4𝑐𝜓𝜓 −𝑐 which determines the sign of 2 (−3𝜅𝑁 /8 − 2𝑁 )𝑒 +𝑐𝑁𝑒 󸀠 16𝑉 𝜓𝜓 in (21)–(23). During the evolution of universe with We can obtain the state finder parameters for spinor the increase in scale factor, 𝜓𝜓 decreases firstly to the value Quintom-B from the time derivative of (45) as 𝑐 which is the bigger root. This is a transition from phantom phasetoquintessencephasebycrossing−1 boundary. After 𝑎̈ ∙ ( ) continuing the evolution 𝜓𝜓 decreases to the second root 𝑎 𝑐/3 which is recrossing the −1 boundary as a transition from (47) 3𝜅𝑁2 quintessence phase to phantom phase again. This scenario is =12𝜋𝛽𝐺[( −2𝑁2)𝑒−3𝛽𝑡 +𝑐𝑁𝑒−3𝛽𝑡/2], 4 obviously a Quinton-C scenario and is illustrated in Figure 3. We see from the figure that the EoS of the quintom model 𝜔=−1 𝜔= andthestatefinderparameteris crosses the boundary twice, firstly from below ... −1 to above 𝜔=−1and secondly from above to below 𝑎 𝑎̈ ∙ 9𝛽 𝑟= =( ) 𝐻−3 −𝑞= 𝜔=−1, then it picks up and then is avoided from Big Rip 𝑎𝐻3 𝑎 √32𝜋𝐺 singularities, and finally it asymptotically evolves to a stable 𝜔=0 2 2 −3𝛽𝑡 −3𝛽𝑡/2 matter dominated expansion epoch with a value of . (3𝜅𝑁 /4 − 2𝑁 )𝑒 +𝑐𝑁𝑒 1 ⋅ − Although considering the phantom scenarios normally ((−3𝜅𝑁2/8 − 2𝑁2)𝑒−3𝛽𝑡 +2𝑐𝑁𝑒−3𝛽𝑡/2)3/2 2 (48) leads to the Big Rip singularities due to the unbound of EoS from below 𝜔=−1, our spinor quintom model with intrinsic (3𝜅𝑁2/4 − 2𝑁2)𝑒−3𝛽𝑡 +𝑐𝑁𝑒−3𝛽𝑡/2 spin in ECSK theory is avoided from the Big Rip singularities ⋅ , bypickinguptoaboundvalueandapproachingastable (−3𝜅𝑁2/8 − 2𝑁2)𝑒−3𝛽𝑡 +𝑐𝑁𝑒−3𝛽𝑡/2 value, as seen in Figures 1 and 3. Diverging EoS of a dark fluid 8 Advances in High Energy Physics

from a constant bound toward a lower singularity refers to Therefore, we can find the luminosity distance as 𝑑𝐿 = −1 2 continuous increase in the pressure of the fluid. This scenario 𝐻0 [𝑧+(1/2)(1−𝑞0)𝑧 +⋅⋅⋅]in terms of the quantities 𝐻0 is avoided in spinor quintom with intrinsic spin, which may and 𝑞0. Here the deceleration parameter is again obtained by be interpreted as the intrinsic spin of the fluid quanta leads using (16) and (17) for the potential (49) and obtaining the to a bound pressure value. The increase of the pressure with term in (26), such that the energy density is bounded due to the effect of intrinsic spin, then singular values of EoS are avoided, and the universe 3𝜅 2 enters a stable expansion in the final era. 𝜌+3𝑝=(− )(𝜓𝛾0𝛾5𝜓) 8 We now obtain other cosmological quantities; by using (55) (16) for the potential (49), we get the Quintom-C energy +(𝑐−3𝜓𝜓)𝜓𝜓 ( − 𝑐) 𝜓𝜓, density as

3𝜅 2 𝜌=( )(𝜓𝛾0𝛾5𝜓) +(𝑐−3𝜓𝜓) (𝜓𝜓 − 𝑐) 𝜓𝜓, and by substituting (15) and (55) in (26) we obtain 16 (51) 𝑎̈ 4𝜋𝐺 3𝜅𝑁2 andinserting(15)and(51)in(27)wefind =− [(− +4𝑐𝑁2)𝑒−3𝛽𝑡 −𝑐2𝑁𝑒−3𝛽𝑡/2 𝑎 3 4 𝑎̇ 𝐻= 𝑎 (56) 3 −9𝛽𝑡/2 8𝜋𝐺 3𝜅𝑁2 (52) −3𝑁 𝑒 ]. = √ √(− +4𝑐𝑁2)𝑒−3𝛽𝑡 −𝑐2𝑁𝑒−3𝛽𝑡/2 −3𝑁3𝑒−9𝛽𝑡/2. 3 8 The deceleration parameter is obtained from (52) and (56) as We now solve the differential equation in (52) to obtain the scale factor, such that 𝑎𝑎̈ 𝑎̈ 1 𝑞=− =−𝐻−2 = 𝑎̇2 𝑎 2 8𝜋𝐺 𝑎= {√ [𝐶 𝑡√4𝑐𝑁2 −𝑐2𝑁−3𝑁3 −3𝜅𝑁2/8 (57) exp 3 3+ (−3𝜅𝑁2/4 + 4𝑐𝑁2)𝑒−3𝛽𝑡 −𝑐2𝑁𝑒−3𝛽𝑡/2 −3𝑁3𝑒−9𝛽𝑡/2 ⋅ . (53) (−3𝜅𝑁2/8 + 4𝑐𝑁2)𝑒−3𝛽𝑡 −𝑐2𝑁𝑒−3𝛽𝑡/2 −3𝑁3𝑒−9𝛽𝑡/2 3𝛽 (8𝑐𝑁2 −𝑐2𝑁+9𝑁3 −3𝜅𝑁2/4) +𝑡2 +⋅⋅⋅]}, √4𝑐𝑁2 −𝑐2𝑁−3𝑁3 −3𝜅𝑁2/8 By using the time derivative of (56), we obtain the state finder andthisgivestheredshift,suchas parameters for Quintom-C model as

8𝜋𝐺 2 √ 𝑎̈ ∙ 3𝜅𝑁 𝑧=−1+exp {− [𝐶3 ( ) = 12𝜋𝛽𝐺 [(− +4𝑐𝑁2)𝑒−3𝛽𝑡 3 𝑎 4 (58) +𝑡√4𝑐𝑁2 −𝑐2𝑁−3𝑁3 −3𝜅𝑁2/8 (54) 𝑐2𝑁 9𝑁3 − 𝑒−3𝛽𝑡/2 + 𝑒−9𝛽𝑡/2], 2 2 3𝛽 (8𝑐𝑁2 −𝑐2𝑁+9𝑁3 −3𝜅𝑁2/4) +𝑡2 +⋅⋅⋅]}. √4𝑐𝑁2 −𝑐2𝑁−3𝑁3 −3𝜅𝑁2/8 andthestatefinderparameteris

... 2 2 −3𝛽𝑡 2 −3𝛽𝑡/2 3 −9𝛽𝑡/2 𝑎 9𝛽 (−3𝜅𝑁 /4 + 4𝑐𝑁 )𝑒 −(𝑐 𝑁/2) 𝑒 + (9𝑁 /2) 𝑒 𝑟= = 𝑎𝐻3 √32𝜋𝐺 ((−3𝜅𝑁2/8 + 4𝑐𝑁2)𝑒−3𝛽𝑡 −𝑐2𝑁𝑒−3𝛽𝑡/2 −3𝑁3𝑒−9𝛽𝑡/2)3/2 (59) 2 2 −3𝛽𝑡 2 −3𝛽𝑡/2 3 −9𝛽𝑡/2 1 (−3𝜅𝑁 /4 + 4𝑐𝑁 )𝑒 −𝑐 𝑁𝑒 −3𝑁 𝑒 − , 2 (−3𝜅𝑁2/8 + 4𝑐𝑁2)𝑒−3𝛽𝑡 −𝑐2𝑁𝑒−3𝛽𝑡/2 −3𝑁3𝑒−9𝛽𝑡/2

from which we can obtain the second state finder parameter without using a ghost field, has recently been obtained in the 𝑠 = 2(𝑟 − 1)/3(2𝑞 −1). framework of general relativity [51]. Here, we consider the spinor field dark energy with intrinsic spin in the formalism 4. Conclusion of metric-affine ECSK theory. We first introduce the ECSK formalism and then define the model Lagrangian whose By using the spinor field dark energy in a FRW geometry, a variations with respect to the tetrad field and torsion tensor consistent quintom model, in which EoS crosses −1 boundary give the total energy-momentum tensor consisting of metric Advances in High Energy Physics 9 and spin contributions. Also from the variation of Lagrangian cosmological constant,” Astronomical Journal,vol.116,no.3,pp. with respect to the spinor field we obtain the ECSK Dirac 1009–1038, 1998. equation. By using the total energy-momentum tensor and [3] U. Seljak, A. Makarov, P. McDonald et al., “Cosmological ECSK Dirac equation, the energy density and the pressure parameter analysis including SDSS Ly𝛼 forest and galaxy values of the spinor quintom DE are obtained, from which the bias: constraints on the primordial spectrum of fluctuations, EoS of the model is obtained for an arbitrary potential. The neutrino mass, and dark energy,” Physical Review D,vol.71, dependence of the potential on the spinor field leads to the Article ID 103515, 2005. evolution of potential with the change of scale factor, since the [4]M.Tegmark,M.Strauss,M.Blantonetal.,“Cosmological scale factor increases by time. Constructing the ECSK spinor parameters from SDSS and WMAP,” Physical Review D,vol.69, potential suitably the quintom scenario is reached, for three no. 10, Article ID 103501, 2004. different cases as Quintom-A, Quintom-B, and Quintom- [5]D.J.Eisenstein,I.Zehavi,D.W.Hoggetal.,“Detectionofthe C models. We also obtained the redshift values for three baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies,” The Astrophysical Journal,vol.633, quintom scenarios from the scale factors of each quintom no.2,p.560,2005. model. Then, we find other cosmological parameters, such as the Hubble parameter, deceleration parameter, and state [6]D.N.Spergel,L.Verde,H.V.Peirisetal.,“First-yearWilkinson Microwave Anisotropy Probe (WMAP) observations: deter- finder parameters for three different potential values of each mination of cosmological parameters,” Astrophysical Journal, quintom scenario, respectively. Supplement Series,vol.148,no.1,pp.175–194,2003. The Quintom-A case exhibits the transition of EoS from [7] D. Larson, J. Dunkley, G. Hinshaw et al., “Seven-year wilkinson quintessence phase to phantom phase, evolving to a stable microwave anisotropy probe (WMAP*) observations: power 𝜔→0 matter dominated expansion with . This scenario is spectra and WMAP-derived parameters,” The Astrophysical avoided from the Big Rip singularities due to the balancing Journal, Supplement Series, vol. 192, no. 2, 2011. of energy density and pressure of spinor DE by intrinsic spin. [8] G. Hinshaw, D. Larson, E. Komatsu et al., “Nine-year wilkinson Similarly, in the Quintom-B scenario, the EoS of the model microwave anisotropy probe (WMAP) observations: cosmolog- evolves from phantom region 𝜔<−1to quintessence region ical parameter results,” Astrophysical Journal, Supplement Series, 𝜔>−1and approaches an EoS value of −1<𝜔<−1/3 vol. 208, no. 2, 2013. referring to a stable de Sitter accelerated expansion for a scalar [9]K.Bamba,S.Capozziello,S.Nojiri,andS.D.Odintsov,“Dark field dark energy model. 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