Physics Letters B 704 (2011) 255–259

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Physics Letters B

www.elsevier.com/locate/physletb

The most general cosmological dynamics for ELKO matter fields

Luca Fabbri

INFN, Sezione di Bologna & Dipartimento di Fisica, Università di Bologna, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 July 2011Received in revised form 7 September 2011Accepted 7 September 2011Available online 14 September 2011Editor: S. Dodelson

Not long ago, the definition of eigenspinors of charge-conjugation belonging to a special Wigner classhas lead to the unexpected theoretical discovery of a form of matter with spin 1

2 and mass dimension 1,called ELKO matter field; ELKO matter fields defined in flat spacetimes have been later extended tocurved and twisted spacetimes, in order to include in their dynamics the coupling to gravitational fieldspossessing both metric and torsional degrees of freedom: the inclusion of non-commuting spinorialcovariant derivatives allows for the introduction of more general dynamical terms influencing thebehaviour of ELKO matter fields. In this Letter, we shall solve the theoretical problem of finding themost general dynamics for ELKO matter, and we will face the phenomenological issue concerning howthe new dynamical terms may affect the behavior of ELKO matter; we will see that new effects will arisefor which the very existence of ELKO matter will be endangered, due to the fact that ELKOs will turnincompatible with the cosmological principle. Thus we have that anisotropic universes must be takeninto account if ELKOs are to be considered in their most general form.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Quite recently, a new form of matter called ELKO has been de-fined; this form of matter gets its name from the acronym of theGerman Eigenspinoren des LadungsKonjugationsOperators, designat-ing spinors that are eigenstates of the charge conjugation oper-ator: ELKO fields are spin- 1

2 fermions λ verifying the conditionsγ 2λ∗ = ±λ for self- and antiself-conjugated fields, and thereforethey are a type of Majorana fields [1]. This definition spells thatELKOs are topologically neutral fermions, and because a topologicalcharge is what keeps localized the otherwise extended field, thenthe absence of such charges ensures that nothing protects the fieldfrom spreading, allowing them to display non-locality; as a fur-ther consequence, according to the Wigner prescription, for whichfundamental fields are classified in terms of irreducible representa-tions of the Poincaré group, ELKOs belong to non-standard Wignerclasses [2].

As it turns out, ELKO fields have mass dimension 1; this impliesthat they are dynamically described by second-order field equa-tions, so that the problems regarding the Majorana mass term,usually met when dealing with first-order field equations, are inthis case circumvented [3,4]. On the other hand, still referring tothe Wigner scheme, fundamental fields are labelled in terms ofboth mass and spin quantum numbers, implying that matter fields

E-mail address: luca.fabbri@bo.infn.it.

0370-2693/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2011.09.024

have both energy and spin density tensors, thus requiring both cur-vature and torsion, if ELKOs are extended to include the couplingwith gravitational backgrounds [5,6].

That ELKOs have second-order differential field equations hastwo important consequences: first, the ELKO field equations havekinetic term with two derivatives, and secondly, their spin den-sity tensor is differentially related to the torsion of the space-time; these two consequences taken together imply that, withinthe ELKO field equations, there is the appearance of derivativesof torsion, which is itself containing derivatives of the ELKOs, andthus additional second-order derivatives of the ELKOs do arise. Thiscircumstance tells that in the ELKO field equations, the second-order derivatives are given by the standard d’Alembertian ∇2λ

plus additional terms of the type Cαβ∇α∇βλ for some matrix Cαβ

that depends on torsion: the occurrence of this latter term givesrise to a situation in which, on the one hand, ELKOs may sufferacausal propagation, and on the other hand, as the field densitytends to increase ELKOs may undergo singularity formation. Nev-ertheless, amazingly enough, all torsional fermionic back-reactionscancel exactly, so that ELKO causal propagation is preserved [7,8],while as their density increases the ELKO gravitational pull van-ishes, resulting in a gravitational asymptotic freedom for whichtheir topological non-locality is extended as to include a dynam-ical non-locality [9].

All these positive results constitute a successful ground uponwhich to develop further consequences of the ELKO theory, someof them coming from the specific features of the field and some

256 L. Fabbri / Physics Letters B 704 (2011) 255–259

coming from their peculiar dynamical character; the applicationsof ELKO models range from particle physics to cosmology, fromtheir coupling to the Higgs field and supersymmetric extensionsto candidates for dark matter and inflationary universes. All theseresults are discussed in references [10–24].

Now, because the ELKO matter field is relatively new, it still hasseveral open problems, especially in the foundations; one of thefundamental open problems is to find the most general dynam-ics for the ELKO matter field: for instance, due to the presence ofcurvature and torsion, we have lost the commutation of the ELKOspinorial covariant derivatives, and so dynamical terms that werebefore reduced to the divergence of a vector, irrelevant in the ac-tion, may now be added to the Lagrangian, generalizing the ELKOaction. For example, one of these terms is the one given by

Sadditional =∫

Dμ

¬λ σμν DνλdΩ

≡∫ [

−Q μ

(¬λ σμν Dνλ

) − 1

2Q αμν(

¬λ σμν Dαλ)

− 1

4Gαβμν

(¬λ σμνσαβλ

)]dΩ (1)

in terms of the tensors Q αμν and Gαβμν , irrelevant when torsionand curvature are not present; but nevertheless, in general thisterm does contribute to the ELKO dynamics. We may now ask ifthis is the only dynamical term possible, addressing the problemof finding the most general ELKO dynamics.

Since a common trend in physics is to look for generalizationsof existing physical theories, this problem is already much inter-esting in itself; however, it also carries a consequent issue: that ishow the additional dynamical terms may change the ELKO dynam-ics, in applications to cosmology.

In this Letter, we are going to pursue a double task, first con-sidering the fundamental problem of finding the most general dy-namics of ELKO and then investigating the consequences of thisgeneralization by studying an application to the standard model ofcosmology.

2. The most general ELKO

In this Letter, the Riemann–Cartan geometry is defined in termsof spacetime connections given by Γ

μασ and used to define the Rie-

mann curvature tensor

Gρημν = ∂μΓ

ρην − ∂νΓ

ρημ + Γ

ρσμΓ σ

ην − ΓρσνΓ σ

ημ (2)

which has one independent contraction given by Gρηρν = Gην

whose contraction is given by Gην gην = G as usual; then we de-fine Cartan torsion tensor

Q ρμν = Γ

ρμν − Γ

ρνμ (3)

and the contorsion tensor is defined by

Kρμν = 1

2

(Q ρ

μν + Q μνρ + Q νμ

ρ)

(4)

with one independent contraction given by Kνρρ = Q ρ

ρν = Q ν =Kν as convention: by using these tensors it is possible to build thecovariant derivatives, the commutator of covariant derivatives andthe cyclic permutations of covariant derivatives, known as Jacobi–Bianchi identities, useful in the following.

Our matter fields are ELKO and ELKO dual defined by

γ 2λ∗±∓ = ηλ±∓¬λ

∗±∓γ 2 = −η¬λ±∓ (5)

for which the explicit relationship between ELKO and ELKO dual isdefined as

¬λ∓±= ±iλ†

±∓γ 0 (6)

where η = ±1 for self- or antiself-conjugate fields and with thelabel ±∓ indicating that the fields decompose into irreducible chi-ral projections that are eigenstates with positive/negative or nega-tive/positive eigenvalues of the helicity operator: if we want to giveto the ELKO and ELKO dual an explicit decomposition in terms ofthe irreducible chiral projections we may write

¬λ±∓= (±iηLT∓σ 2 ∓iL†

∓)

λ±∓ =(

L±−ησ 2L∗±

)(7)

with the label ± designating the eigenstate with positive or nega-tive eigenvalue of the helicity operator respectively given by

¬λ+−= (−ηq 0 0 −iq∗ ) λ+− =

⎛⎜⎝

d00

−iηd∗

⎞⎟⎠ (8)

¬λ−+= (0 −ηp ip∗ 0 ) λ−+ =

⎛⎜⎝

0b

iηb∗0

⎞⎟⎠ (9)

when their spin and momentum have the same direction, as dis-cussed in Refs. [1,2]. This definition of ELKO and ELKO dual endowsthem with specific properties under discrete transformations, asdiscussed in [3,4]. Being ELKO spin- 1

2 spinor fields, their spinorialcovariant derivatives are defined as usual, and the covariant deriva-tives with torsion Dμ are decomposable in terms of the covariantderivatives without torsion ∇μ plus torsional contributions as

Dμ

¬λ= ∇μ

¬λ −1

2Kαβ

μ

¬λ σαβ

Dμλ = ∇μλ + 1

2Kαβ

μσαβλ (10)

in terms of the sigma matrices σαβ along with contorsion; theircommutator is

[Dμ, Dν ] ¬λ= Q ρ

μν Dρ

¬λ −1

2Gκι

μν

¬λ σκι

[Dμ, Dν ]λ = Q ρμν Dρλ + 1

2Gκι

μνσκιλ (11)

in terms of the sigma matrices σκι = 14 [γκ ,γι] along with tor-

sion and curvature, where the gamma matrices γι belong to theClifford algebra and from which the parity-odd gamma matrixγ = iγ 0γ 1γ 2γ 3 is defined as usual. These properties of the defi-nition of ELKO and ELKO dual derivatives are discussed in [5,6].

At this point it is possible to employ the curvature tensors andthe derivatives of the ELKO to construct invariants that will en-ter in the action, defining the dynamics of the model: its variationyields the field equations describing the coupling between curva-ture and energy density T μν and between contorsion and spindensity Sρμν as the system of gravitational field equations givenby

Gμν − 1

2gμνG = 1

2T μν (12)

Kμαβ − Kμβα + Kα gβμ − Kβ gαμ = −Sμαβ (13)

where the gravitational constant has been normalized away,known as Einstein–Sciama–Kibble field equations, and the ELKO

L. Fabbri / Physics Letters B 704 (2011) 255–259 257

field equations. In the scheme provided by the Riemann–Cartangeometry the Jacobi–Bianchi identities above are used to see thatthe Einstein–Sciama–Kibble field equations are converted into con-servation laws of the form

Dρ Sρμν + Q ρ Sρμν + 1

2

(T μν − T νμ

) = 0 (14)

DμT μρ + Q μT μρ − Tμβ Q βμρ − Sβμκ Gμκβρ = 0 (15)

which have to be verified, once the ELKO matter field equationsare considered.

And now we are in the position to face the problem of find-ing the most general dynamics of ELKO matter fields, that is wehave to find all dynamical terms in the ELKO matter fields thatmay appear in the ELKO action: to this purpose, the first thingwe have to consider is that, because we do not want to get in-teractions of external potentials with the ELKO fields nor self-interactions between ELKO fields themselves but we want to getonly dynamical terms for ELKO fields, then we cannot have any-thing else but derivatives of the ELKO fields; secondly, becauseof the fact that ELKOs have mass dimension 1, then we have toconsider two derivatives of the ELKOs. As a consequence we havethat the most general dynamical term for the ELKOs is obtainedby placing between the derivatives of the ELKO and ELKO dualthe most general rank-2 matrix given by Mij: as any matrix iswritable as product of gamma matrices, then the most generalrank-2 matrix is the most general product of two gamma matricesgiven by Mαβ = (pI + rγ )γαγβ + (qI + sγ )γβγα alone; the invari-ance under parity requires the absence of the parity-odd gammamatrix γ therefore leaving the matrix Mαβ = pγαγβ + qγβγα

solely. It is possible to rewrite this matrix in the following formMαβ = 2(p − q)σαβ + (p + q)Iηαβ equivalently written by renam-ing the parameters as Mαβ = aσαβ + bIηαβ where the parameterb may be normalized away with a redefinition of the fields: in theend, the most general dynamical term for ELKO matter fields isgiven by the action

S =∫ [

G − Dα

¬λ

(ηαβ

I + aσαβ)

Dβλ + m2 ¬λ λ

]dΩ (16)

in terms of the most general rank-2 matrix as the most generalproduct of two gamma matrices depending on a single a param-eter. Its variation with respect to the independent fields gives theEinstein–Sciama–Kibble field equations with conserved quantitiesgiven by: the spin and energy densities

Sμαβ = 1

2(Dμ

¬λ σαβλ− ¬

λ σαβ Dμλ)

+ a

2

(Dρ

¬λ σρμσαβλ− ¬

λ σαβσμρ Dρλ)

(17)

Tμν = (Dμ

¬λ Dνλ + Dν

¬λ Dμλ − gμν Dρ

¬λ Dρλ

)

+ a(

Dν

¬λ σμρ Dρλ + Dρ

¬λ σρμDνλ − gμν Dρ

¬λ σρσ Dσ λ

)

+ gμνm2 ¬λ λ (18)

which is no longer symmetric because of the terms proportionalto the constant a of the generalized model; and the matter fieldequations

(D2λ + KμDμλ

) + a(σρμDρ Dμλ + KρσρμDμλ

) + m2λ = 0

(19)

where again the terms proportional to the constant a accounts forthe generalization of the present model. When the matter field

equations

D2λ + KμDμλ + aσρμDρ Dμλ + aKρσρμDμλ + m2λ = 0 (20)

are taken into account, the geometry-matter coupling field equa-tions given by the curvature-energy and spin-torsion field equa-tions

2Gμν − gμνG

= Dμ

¬λ Dνλ + Dν

¬λ Dμλ − gμν Dρ

¬λ Dρλ

+ aDν

¬λ σμρ Dρλ + aDρ

¬λ σρμDνλ

− agμν Dρ

¬λ σρσ Dσ λ + gμνm2 ¬

λ λ (21)

2(Kμβα − Kα gβμ − Kμαβ + Kβ gαμ)

= Dμ

¬λ σαβλ− ¬

λ σαβ Dμλ

+ aDρ¬λ σρμσαβλ − a

¬λ σαβσμρ Dρλ (22)

become the Jacobi–Bianchi identities, therefore showing that thismodel is still consistently defined. As it is widely known, the en-ergy is not symmetric and the spin is not conserved for spinorfields in general, and as it was pointed out in [5,6], compared toother spinor fields the ELKO fields have a more complex spinorialstructure: thus one should expect that the energy is not symmetricand the spin not conserved a fortiori for ELKO fields, but this occursonly in the present generalization of the ELKO model. Therefore toensure the properties ELKO fields are expected to have, we regardthis generalization as needed.

One of the first problems that now needs to be solved is theproblem of the inversion of contorsion: as it is clear from Eq. (22),the contorsion tensor is given by the spin, which is itself writ-ten in terms of derivatives containing contorsion; once in Eq. (22)the derivatives containing contorsion are written in terms of thederivatives without contorsion plus contorsional contributions, weare left with an equation that defines contorsion only implicitly,and making contorsion explicit in this expression means to solvethe contorsion inversion problem. Since the relationship that im-plicitly defines contorsion is linear in the contorsion itself, thenthis problem always admits a solution in principle, although find-ing it may be rather difficult in practice: in [7–9] for the simplesttheory with vanishing a the problem of the inversion of contorsionhas been solved completely, and here for the most general casewhere a has general values it is possible to follow the same proce-dure, getting for the completely antisymmetric irreducible decom-position the expression

Kαβμεαβμρ[(

8 + (1 + a)¬λ λ

)2 + (1 + a)2(i¬λ γ λ)2]

= 4(1 + a)2(i¬λ γ λ)

(∇μ

¬λ σμρλ− ¬

λ σμρ∇μλ)

− 4(1 + a)(8 + (1 + a)

¬λ λ

)i(∇μ

¬λ γ σμρλ− ¬

λ σμργ ∇μλ)

+ 3a(1 + a)(i¬λ γ λ)∇ρ(

¬λ λ)

− 3a(8 + (1 + a)

¬λ λ

)∇ρ(i¬λ γ λ) (23)

and for the trace irreducible decomposition the expression

Kρ[(

8 + (1 + a)¬λ λ

)2 + (1 + a)2(i¬λ γ λ)2]

= 2(1 + a)(8 + (1 + a)

¬λ λ

)(∇μ

¬λ σμρλ− ¬

λ σμρ∇μλ)

+ 2(1 + a)2(i¬λ γ λ)i

(∇μ

¬λ γ σμρλ− ¬

λ σμργ ∇μλ)

258 L. Fabbri / Physics Letters B 704 (2011) 255–259

+ 3a

2(1 + a)(i

¬λ γ λ)∇ρ(i

¬λ γ λ)

+ 3a

2

(8 + (1 + a)

¬λ λ

)∇ρ(¬λ λ) (24)

with which the entire contorsion can be made explicit, and thecontorsion inversion problem would be solved completely as well.However, obtaining a complete solution of this problem would leadus far from the central idea of the present Letter, and thereforewe will not try to get the inversion of contorsion, but rather, weare going to look for circumstances where (23) and (24) are theonly non-vanishing irreducible decompositions of contorsion, sothat contorsion would automatically be inverted, and the problembypassed.

3. The most general cosmological ELKO matter

Actually, we do not need to look around very much to finda simple application to a relevant physical situation in which (23)and (24) are the only non-vanishing irreducible decomposition ofcontorsion, as in the standard model of cosmology one may em-ploy the cosmological principle for the contorsion, as well as forthe metric and for the matter fields, and consequently we havethat considerable simplifications do occur: indeed, when both ho-mogeneity and isotropy are implemented, we have that actuallytorsion is decomposable only in terms of the two vectorial irre-ducible parts as it was desired, while the metric is described interms of the Friedmann–Lemaître–Robertson–Walker spatially flatspacetime

g00 = 1

g jj = −e2σ j = x, y, z (25)

and where we can always choose the ELKO and ELKO dual as

¬λ= eϕ (0 1 −i 0 ) λ = eϕ

⎛⎜⎝

01i0

⎞⎟⎠ (26)

in terms of functions of the time; notice that as an additional sim-plification we have that the completely antisymmetric part of con-torsion happens to vanish when this particular form of the ELKO istaken. Substituting these expressions into the field equations, andthen the inverted form of the torsion trace vector into all remain-ing field equations, we get a field equations for the curvature andfor the ELKO field in terms of the σ and ϕ functions of the timealone.

At this point it is possible to solve the problem of the prop-agation of the ELKO field: by following the general procedurethat has been outlined in Refs. [7,8], we may now easily seethat the ELKO field equation yields a characteristic equation ofthe form n2 ≡ 0 showing that causality is always maintained;so all results obtained in [7,8] are here recovered for this mostgeneral dynamical ELKO at least in the case of cosmological sys-tems.

In the following, we will focus on the case of the ultravioletregime, where the ELKO density field has values that are extremelyhigh and masslessness is well approximated: in this case we havethat the torsion is

Q 0 = 3σ̇ +(

3a

1 + a

)ϕ̇ (27)

while the curvature has form

G0 j0 j =

(a

1 + a

)(σ̇ ϕ̇ + ϕ̈) j = x, y, z

G jkjk = −

(a

1 + a

)2

ϕ̇2 j,k = x, y, z (28)

and with ELKO and ELKO dual field derivatives

D0¬λ=¬

λ ϕ̇ D0λ = ϕ̇λ

D j¬λ=¬

λ σ j0ϕ̇

(a

1 + a

)D jλ =

(a

1 + a

)ϕ̇σ0 jλ

j = x, y, z (29)

where the upper dot designates the partial derivative with respectto time.

And at this point it is possible to solve also the problem of sin-gularity formation: first, we notice that, following [9], we no longerhave the situation for which the ELKO fields exhibit gravitationalasymptotic freedom, as seen from the fact that the curvatures donot vanish, because the energy does not vanish in the first place;then, we also remark that, following [13], we do not have the cir-cumstance for which the ELKO fields energy and spin densitiesare compatible with the cosmological principle, as seen form thefact that the spatial spinorial derivatives do not vanish, becauserelation Dμλ = −i Pμγ λ tells that the spatial components of themomentum do not vanish themselves: both instances are due tothe fact that the parameter a does not vanish, and therefore bothare genuine features of this generalization. On the other hand how-ever, if we do insist that the ELKO energy and spin densities arecompatible with the cosmological principle, then the spatial spino-rial derivatives must vanish, and if a is allowed to have a generalvalue, then ϕ̇ = 0 and consequently Dμλ ≡ 0 identically; then wehave that Gαβμν ≡ 0 identically as well, and so in the ultravioletlimit we have the flatness of the torsion-metric spacetime. In thiscase the ELKO matter field equation and the energy-curvature fieldequations will be verified automatically, although the spin-torsionfield equations are verified only if σ̇ = 0, vanishing the torsion andgiving the flatness of the metric spacetime. Therefore, the ELKOfield will have zero energy and spin densities, and henceforth theywill displaying asymptotic gravitational freedom being compatiblewith the cosmological principle. In this way, the results obtainedin [9] and [13] will be recovered for this most general dynami-cal ELKO in the case of cosmological symmetries in the ultravioletlimit, but only in a case in which the ELKOs have no dynamics.

To draw a parallel with previous works about ELKO matter incosmological applications we consider that in [13] Boehmer andBurnett take into account ELKO matter finding that the model iscompatible with the cosmological principle while here we haveshown that ELKO matter in the most general model is incompati-ble with the cosmological principle: as it is widely known, the spindensity is associated to rotational degrees of freedom that shouldallow only axial symmetries for the correspondingly anisotropicbackgrounds, but this happens only in the present generalizationof the ELKO model; to ensure the symmetries ELKO fields are ex-pected to have we are again lead to regard this generalization asa necessary one. Correspondingly, if we want to maintain ELKOs intheir utmost generality then anisotropic universes have to be con-sidered.

Finally we recall that in [18] Ahluwalia and Horvath discusshow the ELKO fields are local along the preferred axis of theVery Special Relativity while here we have suggested a general-ization in which anisotropies arise, and therefore the ELKO localityaxis may be dynamically justified within a generalization in whichanisotropies are naturally present. However as the results of [18]are not fully encompassed in this Letter we are not here in the

L. Fabbri / Physics Letters B 704 (2011) 255–259 259

position to discuss this issue any further, and we shall defer it tofuture works.

4. Conclusion

In this Letter we have discussed the ELKO matter in the mostgeneral cosmological dynamics; we have discussed the problem ofthe inversion of contorsion indicating the way to solve it in generaland seeing that it may be solved for in particular homogeneous–isotropic universes: we have seen that causality is alwaysmain-tained, but the asymptotic gravitational freedom and the com-patibility with the cosmological principle are lost if a is generic.We have discussed that ELKO fields should have non-symmetricenergy with non-conserved spin possessing rotational degrees offreedom that should allow only axial symmetries for anisotropicbackgrounds, and we have pointed out that this occurs only if thepresent generalization of the ELKO model is considered. These aretwo fundamental reasons that lead us to regard this generalizationas necessary.

The ELKO field is certainly one of the best candidates to solvemany problems of the standard model of cosmology, and havingELKO fields described in terms of the most general theory is thesimplest way we have to increase our explanatory power, so toembrace into a unique theory as many of the problems of the stan-dard model of cosmology as we can. On the other hand however,anisotropic universes is what we have to pay to buy ELKOs gener-ality.

Acknowledgement

I am deeply grateful to Prof. Dharam Vir Ahluwalia for his valu-able suggestions and kind encouragement.

References

[1] D.V. Ahluwalia, D. Grumiller, JCAP 0507 (2005) 012.[2] D.V. Ahluwalia, D. Grumiller, Phys. Rev. D 72 (2005) 067701.[3] D.V. Ahluwalia, C.Y. Lee, D. Schritt, Phys. Lett. B 687 (2010) 248.[4] D.V. Ahluwalia, C.Y. Lee, D. Schritt, Phys. Rev. D 83 (2011) 065017.[5] C.G. Boehmer, Annalen Phys. 16 (2007) 38.[6] C.G. Boehmer, Annalen Phys. 16 (2007) 325.[7] L. Fabbri, Mod. Phys. Lett. A 25 (2010) 151.[8] L. Fabbri, Mod. Phys. Lett. A 25 (2010) 2483.[9] L. Fabbri, Gen. Rel. Grav. 43 (2011) 1607.

[10] D.V. Ahluwalia, Int. J. Mod. Phys. D 15 (2006) 2267.[11] C.G. Böhmer, Phys. Rev. D 77 (2008) 123535.[12] C.G. Böhmer, D.F. Mota, Phys. Lett. B 663 (2008) 168.[13] C. Boehmer, J. Burnett, Phys. Rev. D 78 (2008) 104001.[14] C. Boehmer, J. Burnett, Mod. Phys. Lett. A 25 (2010) 101.[15] C. Boehmer, J. Burnett, D. Mota, D.J. Shaw, JHEP 1007 (2010) 53.[16] S. Shankaranarayanan, arXiv:0905.2573 [astro-ph.CO].[17] S. Shankaranarayanan, arXiv:1002.1128 [astro-ph.CO].[18] D.V. Ahluwalia, S.P. Horvath, JHEP 1011 (2010) 078.[19] R. da Rocha, W.A.J. Rodrigues, Mod. Phys. Lett. A 21 (2006) 65.[20] R. da Rocha, J.M. Hoff da Silva, Adv. Appl. Clif. Alg. 20 (2010) 847.[21] R. da Rocha, J.M. Hoff da Silva, J. Math. Phys. 48 (2007) 123517.[22] J. Hoff da Silva, R. da Rocha, Int. J. Mod. Phys. A 24 (2009) 3227.[23] M. Dias, F. de Campos, J.M.H. da Silva, arXiv:1012.4642 [hep-ph].[24] K.E. Wunderle, R. Dick, arXiv:1010.0963 [hep-th].