Research Article Interacting Dark Matter and -Deformed ...downloads.hindawi.com/journals/ahep/2016/7380372.pdf · dark energy and the dark matter which transfers the energy and momentum

Research ArticleInteracting Dark Matter and 119902-Deformed DarkEnergy Nonminimally Coupled to Gravity

Emre Dil

Department of Physics Sinop University Korucuk 57000 Sinop Turkey

Correspondence should be addressed to Emre Dil emredilsakaryaedutr

Received 5 October 2016 Revised 10 November 2016 Accepted 17 November 2016

Academic Editor Sergei D Odintsov

Copyright copy 2016 Emre Dil This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

In this paper we propose a new approach to study the dark sector of the universe by considering the dark energy as an emerging119902-deformed bosonic scalar field which is not only interacting with the dark matter but also nonminimally coupled to gravity in theframework of standard Einsteinian gravity In order to analyze the dynamic of the system we first give the quantum field theoreticaldescription of the 119902-deformed scalar field dark energy and then construct the action and the dynamical structure of this interactingand nonminimally coupled dark sector As a second issue we perform the phase-space analysis of the model to check the reliabilityof our proposal by searching the stable attractor solutions implying the late-time accelerating expansion phase of the universe

1 Introduction

The dark energy is accepted as the effect of causing the late-time accelerated expansion of universe which is experiencedby the astrophysical observations such as Supernova Ia [1 2]large-scale structure [3 4] the baryon acoustic oscillations[5] and cosmic microwave background radiation [6ndash9]According to the standardmodel of cosmological data 70 ofthe content of the universe consists of dark energy Moreoverthe remaining 25 of the content is an unknown form ofmatter having a mass but in nonbaryonic form that is calleddark matter and the other 5 of the energy content of theuniverse belongs to ordinary baryonic matter [10] While thedark energy spread all over the intergalactic media of the uni-verse and produces a gravitational repulsion by its negativepressure to drive the accelerating expansion of the universethe dark matter is distributed over the inner galactic mediainhomogeneously and it contributes to the total gravitationalattraction of the galactic structure and fixes the estimatedmotion of galaxies and galactic rotation curves [11 12]

Miscellaneous dark models have been proposed toexplain a better mechanism for the accelerated expansion ofthe universeThesemodels include interactions between darkenergy dark matter and the gravitational field The couplingbetween dark energy and dark matter seems possible due to

the equivalence of order of themagnitudes in the present time[13ndash22] On the other hand there are also models in whichthe dark energy nonminimally couples to gravity in orderto provide quantum corrections and renormalizability of thescalar field in the curved spacetime Also the crossing of thedark energy from the quintessence phase to phantom phaseknown as the Quintom scenario can be possible in the mod-els where the dark energy interactswith the gravity If the darkenergy minimally couples to gravity the equation of stateparameter of the dark energy cannot cross the cosmologicalconstant boundary 120596 = 1 in the Friedmann-Robertson-Walker (FRW) geometry therefore it is possible to emerge theQuintom scenario in the model where the dark energy non-minimally couples to gravity [23ndash37]

The constitution of the dark energy can be alternativelythe cosmological constant Λ with a constant energy densityfilling the space homogeneously [38ndash41] As the varyingenergy density dark energy models instead of the cosmo-logical constant quintessence phantom and tachyon fieldscan be considered However all these different dark energymodels are the same in terms of the nondeformedfield consti-tuting the dark energy There is no reason to prevent us fromassuming that the dark energy is a deformed scalar field hav-ing a negative pressure too as expected from the dark energyTherefore we propose that the dark energy considered in

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 7380372 17 pageshttpdxdoiorg10115520167380372

2 Advances in High Energy Physics

this study is formed of the deformed scalar field whose fieldequations are defined by the deformed oscillator algebras

The quantum algebra and quantum group structurewere firstly introduced by Kulish et al [42ndash44] during theinvestigations of integrable systems in quantum field theoryand statistical mechanics Quantum groups and deformedboson algebras are closely related terms It is known that thedeformation of the standard boson algebra is first proposedby Arik-Coon [45] Later on Macfarlane and Biedenharnhave realized the deformation of boson algebra in a differentmanner from Arik-Coon [46 47] The relation betweenquantum groups and the deformed oscillator algebras canbe constructed obviously with this study by expressing thedeformed boson operators in terms of the 119904119906119902(2) Lie algebraoperatorsTherefore the construction of the relation betweenquantum groups and deformed algebras leads the deformedalgebras of great interest with many different applicationsThe deformed version of Bardeen-Cooper-Schrieffer (BCS)many-body formalism in nuclear force deformed creationand annihilation operators are used to study the quantumoccupation probabilities [48] As another study in Nambu-Jona-Lasinio (NJL) model the deformed fermion operatorsare used instead of standard fermion operators and this leadsto an increase in the NJL four-fermion coupling force andthe quark condensation related to the dynamical mass [49]The statistical mechanical studies of the deformed bosonand fermion systems have been familiar in recent years [50ndash60] Moreover the investigations on the internal structure ofcomposite particles involve the deformed fermions or bosonsas the building block of the composite structures [61 62]There are also applications of the deformed particles in blackhole physics [63ndash66] The range of the deformed boson andfermion applications diverses from atomic-molecular physicsto solid state physics in a widespread manner [67ndash72]

The ideas on considering the dark energy as the deformedscalar field have become common in the literature [73ndash76] Inthis study we then take into account the deformed bosonsas the scalar field dark energy interacting with the darkmatter and also nonminimally coupled to gravity In order toconfirm our proposal that the dark energy can be consideredas a deformed scalar field we firstly introduce the dynamics ofthe interacting and nonminimally coupled dark energy darkmatter and gravity model in a spatially flat FRW backgroundand then perform the phase-space analysis to check whetherit will provide the late-time stable attractor solutions implyingthe accelerated expansion phase of the universe

2 Dynamics of the Model

The field equations of the scalar field dark energy areconsidered to be defined by the 119902-deformed boson fields inour model Constructing a 119902-deformed quantum field theoryafter the idea of 119902-deformation of the single particle quantummechanics [45ndash47] has naturally been nonsurprising [77ndash79]The bosonic part of the deformed particle fields correspondsto the deformed scalar field and the fermionic counterpartcorresponds to the deformed vector field In this study weconsider the 119902-deformed bosonic scalar field as the 119902-deformed

dark energy under consideration In our model the 119902-deformed dark energy interacts with the darkmatter and alsononminimally couples to gravity

Early Universe scenarios can be well understood bystudying the quantum field theory in curved spacetime Thebehavior of the classical scalar field near the initial singularitycan be translated to the quantumfield regime by constructingthe coherent states in quantummechanics for anymode of thescalar field It is now impossible to determine the quantumstate of the scalar field near the initial singularity by anobserver at the present universe In order to overcome theundeterministic nature Hawking proposes to take the ran-dom superposition of all possible states in that spacetimeIt has been realized by Berger with taking random super-position of coherent states Also the particle creation inan expanding universe with a nonquantized gravitationalmetric has been investigated by Parker It has been stated byGoodison and Toms that if the field quanta obey the Bose orFermi statistics when considering the evolution of the scalarfield in an expanding universe then the particle creation doesnot occur in the vacuum state Their result gives significationto the possibility of the existence of the deformed statistics incoherent or squeezed states in the Early Universe [79ndash84]

Motivated by this significant possibility we propose thatthe dark energy consists of a 119902-deformed scalar field whoseparticles obey the 119902-deformed algebras Therefore we nowdefine the 119902-deformed scalar field constructing the darkenergy in our model The field operator of the 119902-deformedscalar field dark energy can be given as [79]120601119902 (119909) = int 1198893119896(2120587)32 1(2119908119896)12 [119886119902 (119896) 119890119894119896119909 + 119886lowast119902 (119896) 119890minus119894119896119909] (1)

The following commutation relations for the deformed anni-hilation operator 119886119902(119896) and creations operator 119886lowast119902 (119896) in 119902-bosonic Fock space are given by [45]119886119902 (119896) 119886lowast119902 (1198961015840) minus 1199022119886lowast119902 (1198961015840) 119886119902 (119896) = 120575 (119896 minus 1198961015840) 119886119902 (119896) 119886119902 (1198961015840) minus 1199022119886119902 (1198961015840) 119886119902 (119896) = 0 (2)

where 119902 is a real deformation parameter in interval 0 lt 119902 lt infinand [(119896)] = 119886lowast119902 (119896)119886119902(119896) is the deformed number operator of119896th mode whose eigenvalue spectrum is given as[119873 (119896)] = 1 minus 1199022119873(119896)1 minus 1199022 (3)

Here (119896) = 119886lowast119904 (119896)119886119904(119896) is the standard nondeformed num-ber operator By using (2) in (1) we can obtain the commu-tation relations and planewave expansion of the 119902-deformedscalar field 120601119902(119909) as follows120601119902 (119909) 120601lowast119902 (1199091015840) minus 1199022120601lowast119902 (1199091015840) 120601119902 (119909) = 119894Δ (119909 minus 1199091015840) (4)

where Δ (119909 minus 1199091015840) = minus1(2120587)3 int 1198893119896119908119896 sin119908119896 (119909 minus 1199090) (5)

Advances in High Energy Physics 3

Themetric of the spatially flat FRW spacetime in which the 119902-oscillator algebra represents the 119902-deformed scalar field darkenergy is defined by1198891199042 = minus1198891199052 + 1198862 (119905) [1198891199032 + 11990321198891205792 + 1199032sin21205791198891206012] (6)

and for a FRWmetric1199082119896 = 119892(sum119894

11989621198941198862 + 119898) (7)

where 119892 = det119892120583] Also the relation between deformed andstandard annihilation operators 119886119902 and 119886119904 [85] is given as

119886119902 = 119886119904radic [] (8)

which is used to obtain the relation between deformed andstandard bosonic scalar fields by using (3) in (8) and (1)

120601119902 = 120601radic 1 minus 1199022(1 minus 1199022) (9)

Here we have used theHermiticity of the number operator Now the Friedmann equations will be derived for

our interacting dark matter and nonminimally coupled 119902-deformed dark energy model in a FRW spacetime by usingthe scale factor 119886(119905) in Einsteinrsquos equations In order to obtainthese equations we relate the scale factor to the energy-momentum tensor of the objects in the model under con-sideration We use the fluid description of the objects in ourmodel by considering energy and matter as a perfect fluidwhich are dark energy and matter in our model An isotropicfluid in one coordinate frame leads to an isotropic metric inanother frame coinciding with the frame of the fluid Thismeans that the fluid is at rest in commoving coordinatesThen the four velocities of the fluid are given as [52]119880120583 = (1 0 0 0) (10)

and the energy-momentum tensor follows as

119879120583] = (120588 + 119901)119880120583119880] + 119901119892120583] =(120588 0 0 000 1198921198941198951199010 ) (11)

A more suitable form can be obtained by raising one suchthat 119879120583] = diag (minus120588 119901 119901 119901) (12)

Since we have two constituents 119902-deformed dark energy andthe darkmatter in ourmodel the total energy density and thepressure are given by 120588tot = 120588119902 + 120588119898119901tot = 119901119902 + 119901119898 (13)

where 120588119902 and 119901119902 are the energy density and the pressure ofthe 119902-deformed dark energy and 120588119898 and 119901119898 are the energydensity and the pressure of the dark matter respectively Theequation of state of the energy-momentum carrying cosmo-logical fluid component under consideration in the FRWuniverse is given by119901 = 120596120588which relates the pressure and theenergy density and120596 is called the equation of state parameterWe then express the total equation of state parameter suchthat 120596tot = 119901tot120588tot = 120596119902Ω119902 + 120596119898Ω119898 (14)

where Ω119902 = 120588119902120588tot and Ω119898 = 120588119898120588tot are the density param-eters for the 119902-deformed dark energy and the dark matterrespectively Then the total density parameter is defined asΩtot = Ω119902 + Ω119898 = 1205812120588tot31198672 = 1 (15)

We now turn to Einsteinrsquos equations of the form 119877120583] =1205812(119879120583] minus (12)119892120583]119879) Then by using the components of theRicci tensor for a FRWspacetime (6) and the energy-momen-tum tensor in (12) we rewrite Einsteinrsquos equations for120583] = 00and 120583] = 119894119895 as follows minus3 119886 = 12058122 (120588 + 3119901) (16)119886 + 2 ( 119886)2 = 12058122 (120588 minus 119901) (17)

respectively Here dot also represents the derivative withrespect to cosmic time 119905 Using (16) and (17) gives theFriedmann equations for the FRWmetric as1198672 = 12058123 (120588119902 + 120588119898) = minus12058122 (120588119902 + 119901119902 + 120588119898 + 119901119898) (18)

where119867 = 119886 is the Hubble parameter From the conserva-tion of energy we can obtain the continuity equations for the119902-deformed dark energy and the dark matter constituents inthe form of evolution equations such as119902 + 3119867(120588119902 + 119901119902) = minus119876 (19)119898 + 3119867 (120588119898 + 119901119898) = 119876 (20)

where 119876 is an interaction current between the 119902-deformeddark energy and the dark matter which transfers the energyand momentum from the dark matter to dark energy andvice versa 119876 vanishes for the models having no interactionbetween the dark energy and the dark matter

Now we will define the Dirac-Born-Infeld type actionintegral of the interacting dark matter and 119902-deformed darkenergy nonminimally coupled to gravity in the framework ofEisteinian general relativity [86ndash88] After that we will obtainthe energy-momentum tensor 119879120583] for the 119902-deformed dark


energy and the dark matter in order to get the energy density120588 and pressure 119901 of these dark objects explicitly Then theaction is given as119878 = int1198894119909radicminus119892 [ 11987721205812 minus 12119892120583]120597120583120601119902120597]120601119902 minus 119881minus 120585119891 (120601119902) 119877 + 119871119898] (21)

where 120585 is a dimensionless coupling constant between 119902-deformed dark energy and the gravity so 120585119891(120601119902)119877 denotesthe explicit nonminimal coupling between energy and thegravity Also 119871119902 = minus(12)119892120583]120597120583120601119902120597]120601119902 minus 119881 minus 120585119891(120601119902)119877 and119871119898 are the Lagrangian densities of the 119902-deformed darkenergy and the dark matter respectively Then the energy-momentum tensors of the dark energy constituent of ourmodel can be calculated as follows [89]119879119902120583] = minus2 120597119871119902120597119892120583] + 119892120583]119871119902= 120597120583120601119902120597]120601119902 + 2120585119891 (120601119902) 120597119877120597119892120583]minus 12119892120583] [119892120572120573120597120572120601119902120597120573120601119902 + 2119881] minus 119892120583]120585119891 (120601119902) 119877

(22)

In order to find the derivative of the Ricci scalar with respectto themetric tensor we use the variation of the contraction ofthe Ricci tensor identity 120575119877 = 119877120583]120575119892120583] + 119892120583]120575119877120583] This leadsus to finding the variation of the contraction of the Riemanntensor identity as follows 120575119877120583] = 120575119877120588120583120588] = nabla120588(120575Γ120588]120583) minusnabla](120575Γ120588120588120583) Here nabla120583 represents the covariant derivative and Γ120588]120583represents the Christoffel connection By using the metriccompatibility and the tensor nature of 120575Γ120588]120583 we finally obtain120575119877120575119892120583] = 119877120583] + 119892120583]◻ minus nabla120583nabla] (23)

where ◻ = 119892120572120573nabla120572nabla120573 is the covariant drsquoAlembertian Using(23) in (22) gives119879119902120583] = 120597120583120601119902120597]120601119902 minus 12119892120583] [119892120572120573120597120572120601119902120597120573120601119902] minus 119892120583]119881+ 2120585 [119877120583] minus 12119892120583]119877]119891 (120601119902) + 2120585◻119891 (120601119902)minus 2120585nabla120583nabla]119891 (120601119902)

(24)

Then the 120583] = 0 0 component of the energy-momentumtensor leads to the energy density 120588119902120588119902 = 11987911990200 = 12 2119902 + 119881 + 61205851198672119891 (120601119902) + 61205851198671198911015840 (120601119902) 119902 (25)

where prime refers to derivative with respect to the field 120601119902and we use ◻ = minus12059720 minus 31198671205970 because of the homogeneityand the isotropy for 120601119902 in space Also 11987700 = minus3119886 and

119877 = 6[119886 + 21198862] is used for the FRW geometry The120583] = 119894 119894 components of 119879119902120583] also give the pressure 119901119902 as119901119902 = 119892119894119894119879119902119894119894 = 12 2119902 minus 119881 minus 2120585 [2119891 (120601119902) + 31198672119891 (120601119902)+ 11989110158401015840 (120601119902) 2119902 + 1198911015840 (120601119902) 119902 + 21198671198911015840 (120601119902) 119902] (26)

wherewe use (nabla119894nabla119894)119891(120601119902) = (120597119894nabla119894minusΓ120582119894119894nabla120582)119891(120601119902) = minusΓ0111205970119891(120601119902)with Γ011 = 119886 for the FRW spacetime We can now obtain theequation ofmotion for the 119902-deformed dark energy by insert-ing (25) and (26) into the evolution equation (19) such that119902 + 3119867119902 + 120597119881120597120601119902 + 1205851198771198911015840 (120601119902) = minus 120601119902 (27)

The usual assumption in the literature is to consider the cou-pling function as 119891(120601119902) = 12060121199022 [90] and the potential as 119881 =1198810119890minus120581120582120601119902 [91ndash93] In order to find the energy density pressureand equation of motion in terms of the deformation parame-ter 119902 we use the above coupling function and potential withthe rearrangement of equation (9) as 120601119902 = Δ(119902)120601 in theequations (25)ndash(27) and obtain120588119902 = 12Δ22 + 119890minus120581120582Δ120601 + 31205851198672Δ21206012 + 6120585119867Δ2120601+ 12Δ21206012 + ΔΔ120601 + 6120585119867ΔΔ1206012 (28)

119901119902 = 12Δ22 minus 119890minus120581120582Δ120601minus 2120585Δ2 [1206012 + 3211986721206012 + 2 + 120601 + 2119867120601]+ (1 minus 8120585) ΔΔ120601 + (12 minus 2120585)Δ21206012 minus 2120585ΔΔ1206012minus 4120585119867120601ΔΔ1206012(29)

Δ + 3Δ119867 minus 120581120582119890minus120581120582Δ120601 + 120585Δ119877 + 2Δ + Δ120601+ 3119867Δ120601 = minus120573120581120588119898 (30)

Here we consider that the particles in each modecan vary by creation or annihilation in time forΔ = radic(1 minus 1199022119873)(1 minus 1199022)119873 therefore its time derivatives arenonvanishing On the other hand the common interactioncurrent in the literature 119876 = 120573120581120588119898119902 is used here [17]

Now the phase-space analysis for our interacting darkmatter and nonminimally coupled 119902-deformed dark energymodel will be performed whether the late-time stable attrac-tor solutions can be obtained in order to confirm our model

3 Phase-Space and Stability Analysis

The cosmological properties of the proposed 119902-deformeddark energy model can be investigated by performing the


phase-space analysis Therefore we first transform the equa-tions of the dynamical system into its autonomous form byintroducing the auxiliary variables [15 94ndash98] such as119909 = 120581Δradic6119867 = Δ119909119904

119910 = 120581radic119890minus120581120582Δ120601radic3119867 = radic119890minus120581120582120601(Δminus1)119910119904119911 = 120581Δ120601radic6119867 119911119904 = 0119906 = 120581Δ120601 = Δ119906119904

(31)

where 119909119904 119910119904 119911119904 and 119906119904 are the standard form of the auxiliaryvariables in 119902 rarr 1 limitWe nowwrite the density parametersfor the dark matter and 119902-deformed scalar field dark energyin the autonomous system by using (28) with (36)Ω119898 = 120581212058811989831198672 (32)

Ω119902 = 120581212058811990231198672= 1199092 + 1199102 + 1205851199062 + 2radic6120585119909119906 + 1199112 + 2119909119911+ 2radic6120585119911119906 (33)

Then the total density parameter readsΩtot = 1205812120588tot31198672= 1199092 + 1199102 + 1205851199062 + 2radic6120585119909119906 + 1199112 + 2119909119911+ 2radic6120585119911119906 + Ω119898 = 1 (34)

We should also obtain the 120581211990111990231198672 in the autonomous formto write the equation of state parameters such that120581211990111990231198672 = (1 minus 4120585) 1199092 minus 1199102 + (23120585 + 41205852) 1199041199062+ (81205852 minus 120585) 1199062 + 2radic63 120585119909119906 + (1 minus 4120585) 1199112

+ 2 (1 minus 4120585) 119909119911 + 2radic63 120585119911119906 + 2120585120573119906Ω119898minus 21205851205821199102119906(35)

where 119904 = minus1198672 Using (33) and (35) we find the equationof state parameter for the dark energy as120596119902 = 119901119902120588119902 = [(1 minus 4120585) 1199092 minus 1199102 + (23120585 + 41205852) 1199041199062+ (81205852 minus 120585) 1199062 + 2radic63 120585119909119906 + (1 minus 4120585) 1199112

+ 2 (1 minus 4120585) 119909119911 + 2radic63 120585119911119906 + 2120585120573119906Ω119898 minus 21205851205821199102119906]sdot [1199092 + 1199102 + 1205851199062 + 2radic6120585119909119906 + 1199112 + 2119909119911+ 2radic6120585119911119906]minus1 (36)

Also from (33) and (36) the total equation of state parametercan be obtained as120596tot = 120596119902Ω119902 + 120596119898Ω119898= (1 minus 4120585) 1199092 minus 1199102 + (23120585 + 41205852) 1199041199062+ (81205852 minus 120585) 1199062 + 2radic63 120585119909119906 + (1 minus 4120585) 1199112

+ 2 (1 minus 4120585) 119909119911 + 2radic63 120585119911119906 + 2120585120573119906Ω119898minus 21205851205821199102119906 + (120574 minus 1)Ω119898(37)

where 120574 = 1 + 120596119898 is defined to be the barotropic index Weneed to give the junk parameter 119904 in the autonomous formsuch that

119904 = minus 1198672 = 32 (1 + 120596tot) = 32 [1 + (1 minus 4120585) 1199092 minus 1199102+ (23120585 + 41205852) 1199041199062 + (81205852 minus 120585) 1199062 + 2radic63 120585119909119906+ (1 minus 4120585) 1199112 + 2 (1 minus 4120585) 119909119911 + 2radic63 120585119911119906+ 2120585120573119906Ω119898 minus 21205851205821199102119906 + (120574 minus 1)Ω119898]

(38)

Pulling 119904 from the right-hand side of (38) to the left-hand sidegives

119904 = [1 + (1 minus 4120585) 1199092 minus 1199102 + (81205852 minus 120585) 1199062 + 2radic63 120585119909119906+ (1 minus 4120585) 1199112 + 2 (1 minus 4120585) 119909119911 + 2radic63 120585119911119906+ 2120585120573119906Ω119898 minus 21205851205821199102119906 + (120574 minus 1)Ω119898] [23 minus 231205851199062minus 412058521199062]minus1

(39)


Table 1 Critical points and existence conditions

Label 119909 + 119911 119910 119906 120596tot 119902119863 Existence119860 0 1 0 minus1 minus1 120582 = 0 Ω119898 = 0119861 0 minus1 0 minus1 minus1 120582 = 0 Ω119898 = 0119862 0 radic(4120585120582) (minus2120582 + radic41205822 + 1120585) (minus2120582 + radic41205822 + 1120585) minus1 minus1 120582 = 0 Ω119898 = 0119863 0 minusradic(4120585120582) (minus2120582 + radic41205822 + 1120585) (minus2120582 + radic41205822 + 1120585) minus1 minus1 120582 = 0 Ω119898 = 0While 119904 is a junk parameter alone it gains physical meaningin the deceleration parameter 119902119863 such that119902119863 = minus1 + 119904 = minus1 + [1 + (1 minus 4120585) 1199092 minus 1199102

+ (81205852 minus 120585) 1199062 + 2radic63 120585119909119906 + (1 minus 4120585) 1199112+ 2 (1 minus 4120585) 119909119911 + 2radic63 120585119911119906 + 2120585120573119906Ω119898 minus 21205851205821199102119906+ (120574 minus 1)Ω119898] [23 minus 231205851199062 minus 412058521199062]minus1

(40)

Nowwe convert the Friedmann equations (18) the continuityequation (20) and the equation of motion (30) into theautonomous system by using the auxiliary variables in (31)and their derivatives with respect to 119873 = ln 119886 For anyquantity 119865 this derivative has the relation with the timederivative as = 119867(119889119865119889119873) = 1198671198651015840 Then we will obtain1198831015840 = 119891(119883) where 119883 is the column vector including theauxiliary variables and 119891(119883) is the column vector of theautonomous equations We then find the critical points 119883119888of 119883 by setting 1198831015840 = 0 We then expand 1198831015840 = 119891(119883) around119883 = 119883119888 + 119880 where 119880 is the column vector of perturbationsof the auxiliary variables such as 120575119909 120575119910 120575119911 and 120575119906 for eachconstituent in our model Thus we expand the perturbationequations up to the first order for each critical point as 1198801015840 =119872119880 where 119872 is the matrix of perturbation equations Theeigenvalues of perturbationmatrix119872 determine the type andstability of each critical point [99ndash108]Then the autonomousform of the cosmological system is1199091015840 = minus3119909 minus 3119911 + 119904119909 minus 1199111015840 + 119904119911 + radic6120585119904119906 minus 2radic6120585119906+ radic62 1205821199102 minus radic62 120573Ω119898 (41)

1199101015840 = 119904119910 minus radic62 120582119910119909 minus radic62 120582119910119911 (42)1199111015840 = minus3119909 minus 3119911 + 119904119909 minus 1199091015840 + 119904119911 + radic6120585119904119906 minus 2radic6120585119906+ radic62 1205821199102 minus radic62 120573Ω119898 (43)

1199061015840 = radic6119909 + radic6119911 (44)

Here (41) and (43) in fact give the same autonomous equa-tions which means that the variables 119909 and 119911 do not form an

orthonormal basis in the phase-space However +119911 119910 and 119906form a complete orthonormal set for the phase-space There-fore we set (41) and (43) in a single autonomous equation as1199091015840 + 1199111015840 = minus3119909 minus 3119911 + 119904119909 + 119904119911 + radic6120585119904119906 minus 2radic6120585119906+ radic62 1205821199102 minus radic62 120573Ω119898 (45)

The autonomous equation system (42) (44) and (45) repre-sents three invariant submanifolds +119911 = 0 119910 = 0 and 119906 = 0which by definition cannot be intersected by any orbit Thismeans that there is no global attractor in the deformed darkenergy cosmology [109] We will make finite analysis of thephase space The finite fixed points are found by setting thederivatives of the invariant submanifolds of the auxiliary vari-ables We can also write these autonomous equations in 119902 rarr1 limit in terms of the standard auxiliary variables such as

1199091015840119904 = minus3119909119904 + 119904119904119909119904 + radic6120585119904119904119906119904 minus 2radic6120585119906119904 + radic62 1205821199102119904minus radic62 120573Ω1198981199101015840119904 = 119904119904119910119904 minus radic62 1205821199101199041199091199041199061015840119904 = radic6119909119904(46)

Here we need to get the finite fixed points (critical points)of the autonomous system in (41)ndash(45) in order to performthe phase-space analysis of the model We will obtain thesepoints by equating the left-hand sides of the equations (42)(44) and (45) to zero by using Ωtot = 1 in (34) and also byassuming 120596tot = minus1 and 119902119889 = minus1 in (37) and (40) for eachcritical point After some calculations four sets of solutionsare found as the critical points which are listed in Table 1with the existence conditions The same critical points arealso valid for 119909119904 119910119904 and 119906119904 instead of 119909 + 119911 119910 and 119906 in the119902 rarr 1 standard dark energy model limit

Now we should find 120575119904 from (39) which will exist in theperturbations 1205751199091015840 + 1205751199111015840 1205751199101015840 and 1205751199061015840 such that

120575119904 = [2 (1 minus 4120585) (119909 + 119911) + 2radic63 120585119906] 1119875 (120575119909 + 120575119911)+ [minus2119910 minus 4120585120582119910119906] 1119875120575119910 + [(81205852 minus 120585) 2119906


+ 2radic63 120585 (119909 + 119911) + 2120585120573Ω119898 minus 21205851205821199102+ (43120585 + 81205852) 119904119906] 1119875120575119906(47)

where 119875 = 23 minus (23)1205851199062 minus 412058521199062 Then the perturbations1205751199091015840 +1205751199111015840 1205751199101015840 and 1205751199061015840 for each phase-space coordinate in ourmodel can be found by using the variations of (42) (44) and(45) such that1205751199091015840 + 1205751199111015840 = [2 (1 minus 4120585) (119909 + 119911)2 + (minus119904 + 3) 119875 + 412058521199062

+ (83120585 minus 81205852)radic6 (119909 + 119911) 119906] 1119875 (120575119909 + 120575119911)+ [(minus2 (119909 + 119911) + radic6120582119875) 119910 minus 4120585120582 (119909 + 119911) 119910119906minus 2radic6120585119910119906 minus 4radic612058521205821199101199062] 1119875120575119910+ [(101205852 minus 120585) 2 (119909 + 119911) 119906 + 2radic63 120585 (119909 + 119911)2+ 2120585120573 (119909 + 119911)Ω119898 minus 2120585120582 (119909 + 119911) 1199102+ (43120585 + 81205852) (119909 + 119911) 119904119906 + (81205852 minus 120585) 2radic61205851199062+ 2radic61205852120573119906Ω119898 minus 2radic612058521205821199102119906+ (43120585 + 81205852)radic61205851199041199062 + radic6120585 (119904 minus 2) 119875] 11198751205751199061205751199101015840 = [2 (1 minus 4120585) (119909 + 119911) 119910 + 2radic63 120585119906119910 minus radic62 120582119910119875]sdot 1119875 (120575119909 + 120575119911) + [minus21199102 minus 41205851205821199102119906 + 119904119875minus radic62 120582 (119909 + 119911) 119875] 1119875120575119910 + [(81205852 minus 120585) 119906119910+ 2radic63 120585 (119909 + 119911) 119910 + 2120585120573119910Ω119898 minus 21205851205821199103+ (43120585 + 81205852) 119904119910119906] 11198751205751199061205751199061015840 = radic6 (120575119909 + 120575119911)

(48)

From (48) we find the 3 times 3 perturbation matrix119872 whoseelements are given as

11987211 = [2 (1 minus 4120585) (119909 + 119911)2 + (minus119904 + 3) 119875 + 412058521199062+ (83120585 minus 81205852)radic6 (119909 + 119911) 119906] 1119875 11987212 = [(minus2 (119909 + 119911) + radic6120582119875) 119910 minus 4120585120582 (119909 + 119911) 119910119906minus 2radic6120585119910119906 minus 4radic612058521205821199101199062] 1119875 11987213 = [(101205852 minus 120585) 2 (119909 + 119911) 119906 + 2radic63 120585 (119909 + 119911)2+ 2120585120573 (119909 + 119911)Ω119898 minus 2120585120582 (119909 + 119911) 1199102+ (43120585 + 81205852) (119909 + 119911) 119904119906 + (81205852 minus 120585) 2radic61205851199062+ 2radic61205852120573119906Ω119898 minus 2radic612058521205821199102119906+ (43120585 + 81205852)radic61205851199041199062 + radic6120585 (119904 minus 2) 119875] 1119875 11987221 = [2 (1 minus 4120585) (119909 + 119911) 119910 + 2radic63 120585119906119910 minus radic62 120582119910119875]sdot 1119875 11987222 = [minus21199102 minus 41205851205821199102119906 minus +119904119875 minus radic62 120582 (119909 + 119911) 119875] 1119875 11987223 = [(81205852 minus 120585) 119906119910 + 2radic63 120585 (119909 + 119911) 119910 + 2120585120573119910Ω119898minus 21205851205821199103 + (43120585 + 81205852) 119904119910119906] 1119875 11987231 = radic611987232 = 11987233 = 0

(49)

We insert the linear perturbations (119909 + 119911) rarr (119909119888 + 119911119888) +(120575119909 + 120575119911) 119910 rarr 119910119888 + 120575119910 and 119906 rarr 119906119888 + 120575119906 about thecritical points in the autonomous system (42) (44) and(45) in order to calculate the eigenvalues of perturbationmatrix 119872 for four critical points given in Table 1 with thecorresponding existing conditionsTherefore we first give thefour perturbation matrices for the critical points119860 119861 119862 and119863 with the corresponding existing conditions such that

119872119860 = 119872119861 = (minus3 0 minus2radic61205850 minus3 0radic6 0 0 ) (50)

8 Advances in High Energy Physics119872119862=((

412058521199062119862119875 minus 3 minus4radic612058211990621198621205852119875 minus 2radic6120585119910119862119906119862119875 + radic6119910119862120582 minusradic6 (21205852 minus 161205853) 1199062119862119875 minus 2radic612058521205821199102119862119906119862119875 minus 2radic61205852radic61205851199101198621199061198623119875 minus radic61205821199101198622 minus21199102119862119875 minus 41205851205821199102119862119906119862119875 minus21205851205821199103119862119875 minus (2120585 minus 161205852) 119906119862119910119862119875radic6 0 0))

(51)

where 119910119862 = radic4120585120582(minus2120582 + radic41205822 + 1120585) and 119906119862 = minus2120582+radic41205822 + 1120585119872119863=((

412058521199062119863119875 minus 3 4radic612058211990621198631205852119875 + 2radic6120585119910119863119906119863119875 minus radic6119910119863120582 minusradic6 (21205852 minus 161205853) 1199062119863119875 minus 2radic612058521205821199102119863119906119863119875 minus 2radic6120585radic61205821199101198632 minus 2radic61205851199101198631199061198633119875 minus21199102119863119875 minus 41205851205821199102119863119906119863119875 minus21205851205821199103119863119875 minus (2120585 minus 161205852) 119906119863119910119863119875radic6 0 0))

(52)

where 119910119863 = minusradic4120585120582(minus2120582 + radic41205822 + 1120585) and 119906119863 = minus2120582 +radic41205822 + 1120585 Also by using 119909119904 119910119904 and 119906119904 instead of 119909 + 119911 119910and 119906 in the perturbation matrix elements above we obtainthe standard perturbation matrix elements in 119902 rarr 1 limitThen substituting the standard critical points we again obtainthe same matrices 119872119860 119872119861 119872119862 and 119872119863 Therefore thestability of the standard model agrees with the stability of thedeformed model

We need to obtain the four sets of eigenvalues and inves-tigate the sign of the real parts of eigenvalues so that we candetermine the type and stability of critical points If all the realparts of the eigenvalues are negative the critical point is saidto be stableThephysicalmeaning of the stable critical point isa stable attractor namely the Universe keeps its state foreverin this state and thus it can attract the universe at a late timeHere an accelerated expansion phase occurs because 120596tot =minus1 lt minus13 However if the suitable conditions are satisfiedthere can even exist an accelerated contraction for 120596tot =minus1 lt minus13 value Eigenvalues of the four119872matrices and thestability conditions are represented in Table 2 for each criticalpoint119860119861119862 and119863 FromTable 2 the first two critical points119860 and 119861 have the same eigenvalues as 119862 and 119863 have thesame eigenvalues too Here the eigenvalues and the stabilityconditions of the perturbation matrices for critical points119862 and 119863 have been obtained by the numerical methodsdue to the complexity of the matrices (51) and (52) Thestability conditions of each critical point are listed in Table 2according to the sign of the real part of the eigenvalues

Now we will study the cosmological behavior of eachcritical point by considering the attractor solutions in scalarfield cosmology [110] We know that the energy density of ascalar field has a role in the determination of the evolution of

Universe Cosmological attractors provide the understandingof evolution and the factors affecting on this evolution suchthat from the dynamical conditions the evolution of scalarfield approaches a particular type of behavior without usingthe initial fine tuning conditions [111ndash121] We know that theattractor solutions imply a behavior in which a collection ofphase-space points evolve into a particular region and neverleave from there In order to solve the differential equationsystem (42) (44) and (45) we use adaptive Runge-Kuttamethod of 4th and 5th order in MATLAB programmingWe use the present day values for the dark matter densityparameter Ω119898 = 03 interaction parameter 120573 = 145 and0 lt 120574 lt 2 values in solving the differential equation system[94 122] Then the solutions with the stability conditions ofcritical points are plotted for each pair of the solution setbeing the auxiliary variables 119909 + 119911 119910 and 119906Critical Point A This point exists for 120582 = 0 which meansthat the potential 119881 is constant Acceleration occurs at thispoint because of 120596tot = minus1 lt minus13 and it is an expansionphase since119910 is positive so119867 is positive too Point119860 is stablemeaning that Universe keeps its further evolution for 0 lt120585 le 316 with 120582 120573 isin R but it is a saddle point meaning theuniverse evolves between different states for 120585 lt 0 In Figure 1we illustrate the 2-dimensional projections of 4-dimensionalphase-space trajectories for the stability condition 120585 = 015and for the present day values120573 = 145 120574 = 15 andΩ119898 = 03and three auxiliary 120582 valuesThis state corresponds to a stableattractor starting from the critical point 119860 = (0 1 0) as seenfrom the plots in Figure 1

Critical Point B Point 119861 also exists for 120582 = 0meaning that thepotential119881 is constant Acceleration phase is again valid here


Table 2 Eigenvalues and stability of critical points

Criticalpoints Eigenvalues 120585 120582 Stability

A and B minus30000 minus(12)radic9 minus 48120585 minus 32 (12)radic9 minus 48120585 minus 32 Stable point for 0 lt 120585 le 316with 120582 120573 isin R

Saddle point for 120585 lt 0 with120582 120573 isin R

C and D

minus10642 minus15576 minus55000 01000 10000

Stable point for 0 lt 120585 120582 = 1 and120573 isin RSaddle point if 120585 lt 0 and 120582 = 1

minus10193 minus10193 minus72507 10000 10000minus08407 minus08407 minus77519 20000 10000minus07080 minus07080 minus80701 30000 10000minus06014 minus06014 minus83107 40000 10000minus05121 minus05121 minus85060 50000 10000minus04353 minus04353 minus86709 60000 10000minus03680 minus03680 minus88136 70000 10000minus03082 minus03082 minus89395 80000 10000minus02544 minus02544 minus90520 90000 10000minus02055 minus02055 minus91535 100000 10000

A

0506070809

11112131415

y

minus07 minus06 minus05 minus04 minus03 minus02 minus01 0minus08

x + z

(a)

A


x + z

minus15

minus1

minus05

0

u

(b)

A

minus15

minus1

minus05

0

u

109 1311 12 1407 08y

(c)

Figure 1 Two-dimensional projections of the phase-space trajectories for stability condition 120585 = 015 and for present day values 120573 = 145120574 = 15 and Ω119898 = 03 All plots begin from the critical point 119860 being a stable attractor

since 120596tot = minus1 lt minus13 but this point refers to contractionphase because 119910 is negative here For the stability of the point119861 it is again stable for 0 lt 120585 le 316 with 120582 120573 isin R but it is asaddle point for 120585 lt 0 Therefore the stable attractor behavioris represented for contraction starting from the critical point119861 = (0 minus1 0) as seen from the graphs in Figure 2 We plotphase-space trajectories for the stability condition 120585 = 015

and for the present day values120573 = 145 120574 = 15 andΩ119898 = 03and three auxiliary 120582 valuesCritical Point C Critical point 119862 occurs for 120582 = 0 meaninga field dependent potential 119881 The cosmological behavior isagain an acceleration phase since 120596tot lt minus13 and an expan-sion phase since119910 is positive Point119862 is stable for 0 lt 120585120582 = 1


B


x + z

minus14minus13minus12minus11minus1

minus09minus08minus07minus06minus05minus04

y

(a)

B


x + z

minus15

minus1

minus05

0

u

(b)

B

minus12minus1

minus08minus06minus04minus02

0

u

minus115 minus105 minus1minus11 minus095 minus085minus125 minus12minus13 minus08minus09y

(c)


and120573 isin R and saddle point if 120585 lt 0 and 120582 = 1 2-dimensionalprojections of phase-space are represented in Figure 3 for thestability conditions 120585 = 2 120582 = 1 and for the present day values120573 = 145Ω119898 = 03 and three auxiliary 120574 values in the presentday value rangeThe stable attractor starting from the criticalpoint 119862 can be inferred from the plots in Figure 3

Critical Point D This point exists for 120582 = 0 meaning a fielddependent potential 119881 Acceleration phase is again valid dueto 120596tot lt minus13 but this point refers to a contraction phasebecause 119910 is negative Point 119863 is also stable for 0 lt 120585 120582 = 1and 120573 isin R However it is a saddle point while 120585 lt 0 and120582 = 1 2-dimensional plots of phase-space trajectories areshown in Figure 4 for the stability conditions 120585 = 2 120582 = 1and for the present day values 120573 = 145 Ω119898 = 03 and threeauxiliary 120574 values in the present day value range This stateagain corresponds to a stable attractor starting from the point119863 as seen from the plots in Figure 4

All the plots in Figures 1ndash4 have the structure of stableattractor since each of them evolves to a single point whichis in fact one of the critical points in Table 1 The three-dimensional plots of the evolution of phase-space trajectoriesfor the stable attractors are given in Figure 5These evolutionsto the critical points are the attractor solutions of ourcosmological model interacting darkmatter and 119902-deformeddark energy nonminimally coupled to gravity which implyan expanding universe On the other hand the constructionof the model in the 119902 rarr 1 limit reproduces the results ofthe phase-space analysis for the nondeformed standard darkenergy caseThe critical points and perturbationmatrices are

the same for the deformed and standard dark energy modelswith the equivalence of the auxiliary variables as 119909 + 119911 = 119909119904119910 = 119910119904 and 119906 = 119906119904 Therefore it is confirmed that thedark energy in our model can be defined in terms of the 119902-deformed scalar fields obeying the 119902-deformed boson algebrain (2) According to the stable attractor behaviors it makessense to consider the dark energy as a scalar field defined bythe 119902-deformed scalar field with a negative pressure

We know that the deformed dark energy model is aconfirmed model since it reproduces the same stabilitybehaviors critical points and perturbation matrices withthe standard dark energy model but the auxiliary variablesof deformed and standard models are not the same Therelation between deformed and standard dark energy can berepresented regarding auxiliary variable equations in (31)119909 = radic 1 minus 1199022119873(1 minus 1199022)119873119909119904

119910 = radicexp(minus119888(radic 1 minus 1199022119873(1 minus 1199022)119873 minus 1))119910119904119906 = radic 1 minus 1199022119873(1 minus 1199022)119873119906119904

(53)

where 119888 is a constant From the equations (53) we now illus-trate the behavior of the deformed and standard dark energyauxiliary variables with respect to the deformation parameter119902 in Figure 6 We infer from the figure that the value of the


C

minus06 minus05 minus04 minus03 minus02 02minus01 010

x + z

02

04

06

08

1

12

14

y

(a)

C

minus07 minus06 minus05 minus04 minus03 minus02 minus01 0

x + z

minus01

minus005

0

005

01

015

u

(b)

C

minus01

minus005

0

005

01

015

u

109 11 12 13 14 1508y

(c)

Figure 3 Two-dimensional projections of the phase-space trajectories for stability conditions 120585 = 2120582 = 1 and for present day values120573 = 145Ω119898 = 03 All plots begin from the critical point 119862 being a stable attractor

D


x + z

minus14

minus08

minus02

y

(a)

D


x + z

0

minus01

minus005

005

01

015

u

(b)

D

0

minus01

minus005

005

01

015

u

minus1 minus09minus13 minus08minus14 minus12 minus07minus11

y

(c)

Figure 4 Two-dimensional projections of the phase-space trajectories for stability conditions 120585 = 2120582 = 1 and for present day values120573 = 145Ω119898 = 03 All plots begin from the critical point119863 being a stable attractor


minus08minus06

minus04minus02

0

minus0200204060811214minus18minus14


y

u

A

x+z

(a)

minus07minus05

minus03minus01

01minus14 minus12 minus1 minus08 minus06 minus04 minus02 0 02

minus2minus15

minus1minus05

0

y

u

B

x+z

(b)

minus06 minus05 minus04 minus03 minus02 minus01 0 01 02

002

0406

081

1214

minus02minus01

001

y

u

C

x + z

(c)

minus06minus04

minus020

02

minus14minus12minus1minus08minus06minus04minus020

y

D

minus02minus01

001

u

x + z

(d)

Figure 5 Three-dimensional plots of the phase-space trajectories for the critical points 119860 119861 119862 and119863 being the stable attractors

0 01 02 03 04 05 06 07 08 09 1

12345678910040506070809

1

qN

xx

s(o

ruu

s)

(a)1 11 12 13 14 15 16 17 18 19 2

11522533541

152

253

354

455

qN

xx

s(o

ruu

s)

(b)

01 02 03 04 0506 07 08 09

0

1

1234567891007

07508

08509

0951

qN

yy

s

(c)

11 12 13 14 15 16 17 18 19

1

2

115225

33541234567

qN

yy

s

(d)

Figure 6 Behavior of the auxiliary variables 119909 119910 and 119906 with respect to the deformation parameter 119902 and the particle number119873


deformed119909119910 and119906 decreaseswith decreasing 119902 for the 119902 lt 1interval for large particle number and the decrease in thevariables119909119910 and 119906 refers to the decrease in deformed energydensity Also we conclude that the value of the auxiliaryvariables 119909 119910 and 119906 increases with increasing 119902 for the 119902 gt 1interval for large particle number In 119902 rarr 1 limit deformedvariables goes to standard ones

4 Conclusion

In this study we propose that the dark energy is formed of thenegative-pressure 119902-deformed scalar field whose field equa-tion is defined by the 119902-deformed annihilation and creationoperators satisfying the deformed boson algebra in (2) sinceit is known that the dark energy has a negative pressuremdashlike the deformed bosonsmdashacting as a gravitational repulsionto drive the accelerated expansion of universe We consideran interacting dark matter and 119902-deformed dark energynonminimally coupled to the gravity in the framework ofEinsteinian gravity in order to confirm our proposal Thenwe investigate the dynamics of the model and phase-spaceanalysiswhether it will give stable attractor solutionsmeaningindirectly an accelerating expansion phase of universeThere-fore we construct the action integral of the interacting darkmatter and 119902-deformed dark energy nonminimally coupledto gravity model in order to study its dynamics With this theHubble parameter and Friedmann equations of themodel areobtained in the spatially flat FRW geometry Later on we findthe energy density and pressure with the evolution equationsfor the 119902-deformed dark energy and dark matter from thevariation of the action and the Lagrangian of the modelAfter that we translate these dynamical equations into theautonomous form by introducing the suitable auxiliary vari-ables in order to perform the phase-space analysis of themodel Then the critical points of autonomous system areobtained by setting each autonomous equation to zero andfour perturbation matrices are obtained for each criticalpoint by constructing the perturbation equations We thendetermine the eigenvalues of four perturbation matrices toexamine the stability of the critical points We also calculatesome important cosmological parameters such as the totalequation of state parameter and the deceleration parameter tocheck whether the critical points satisfy an accelerating uni-verse We obtain four stable attractors for the model depend-ing on the coupling parameter 120585 interaction parameter120573 andthe potential constant120582 An accelerating universe exists for allstable solutions due to120596tot lt minus13The critical points119860 and119861are late-time stable attractors for 0 lt 120585 le 316 and 120582 120573 isin Rwith the point119860 referring to an expansion with a stable accel-eration while the point 119861 refers to a contraction Howeverthe critical points 119862 and 119863 are late-time stable attractors for0 lt 120585 120582 = 1 and 120573 isin R with the point 119862 referring to anexpansion with a stable acceleration while the point119863 refersto a contraction The stable attractor behavior of the modelat each critical point is demonstrated in Figures 1ndash4 In orderto solve the differential equation system (42) (44) and (45)with the critical points and plot the graphs in Figures 1ndash4we use adaptive Runge-Kutta method of 4th and 5th order in

MATLAB programmingThen the solutions with the stabilityconditions of critical points are plotted for each pair of thesolution set being the auxiliary variables in 119909 + 119911 119910 and 119906

These figures show that by using the convenient param-eters of the model according to the existence and stabilityconditions and the present day values we can obtain thestable attractors as 119860 119861 119862 and119863

The 119902-deformed dark energy is a generalization of thestandard scalar field dark energy As seen from (9) in the119902 rarr 1 limit the behavior of the deformed energy densitypressure and scalar field functions with respect to thestandard functions all approach the standard correspondingfunction values Consequently 119902-deformation of the scalarfield dark energy gives a self-consistent model due to theexistence of standard case parameters of the dark energy inthe 119902 rarr 1 limit and the existence of the stable attractorbehavior of the accelerated expansion phase of universe forthe considered interacting and nonminimally coupled darkenergy and dark matter model Although the deformed darkenergy model is confirmed through reproducing the samestability behaviors critical points and perturbation matriceswith the standard dark energy model the auxiliary variablesof deformed and standard models are of course differentBy using the auxiliary variable equations in (31) we findthe relation between deformed and standard dark energyvariables From these equations we represent the behaviorof the deformed and standard dark energy auxiliary variableswith respect to the deformation parameter for 119902 lt 1 and 119902 gt 1intervals in Figure 6 Then the value of the deformed 119909 119910and 119906 or equivalently deformed energy density decreases withdecreasing 119902 for the 119902 lt 1 interval for large particle numberAlso the value of the auxiliary variables 119909 119910 and 119906 increaseswith increasing 119902 for the 119902 gt 1 interval for large particlenumber In 119902 rarr 1 limit all the deformed variables transformto nondeformed variables

The consistency of the proposed 119902-deformed scalar fielddark energy model is confirmed by the results since it givesthe expected behavior of the universeThe idea of consideringthe dark energy as a 119902-deformed scalar field is a very recentapproach There are more deformed particle algebras inthe literature which can be considered as other and maybemore suitable candidates for the dark energy As a furtherstudy for the confirmation of whether the dark energy canbe considered as a general deformed scalar field the otherinteractions and couplings between deformed dark energymodels dark matter and gravity can be investigated in thegeneral relativity framework or in the framework of othermodified gravity theories such as teleparallelism

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ and 120582 from 42 high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 pp 565ndash586 1999


[2] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquoAstronomical Journal vol 116 no 3 pp1009ndash1038 1998

[3] U Seljak A Makarov P McDonald et al ldquoCosmologicalparameter analysis including SDSS Ly120572 forest and galaxybias constraints on the primordial spectrum of fluctuationsneutrino mass and dark energyrdquo Physical Review D vol 71 no10 Article ID 103515 2005

[4] M Tegmark M A Strauss M R Blanton et al ldquoCosmologicalparameters from SDSS andWMAPrdquo Physical Review D vol 69no 10 Article ID 103501 2004

[5] D J Eisenstein I Zehavi D W Hogg et al ldquoDetection of thebaryon acoustic peak in the large-scale correlation function ofSDSS luminous red galaxiesrdquo Astrophysical Journal Letters vol633 no 2 pp 560ndash574 2005

[6] D N Spergel L Verde H V Peiris et al ldquoFirst-year wilkinsonmicrowave anisotropy probe (WMAP)lowast observations deter-mination of cosmological parametersrdquo Astrophysical JournalSupplement Series vol 148 no 1 pp 175ndash194 2003

[7] E Komatsu K M Smith J Dunkley et al ldquoSeven-yearwilkinson microwave anisotropy probe (WMAP) observa-tions cosmological interpretationrdquo The Astrophysical JournalSupplement Series vol 192 no 2 2011

[8] G Hinshaw D Larson E Komatsu et al ldquoNine-year wilkinsonmicrowave anisotropy probe (WMAP) observations cosmolog-ical parameter resultsrdquoAstrophysical Journal Supplement Seriesvol 208 no 2 article 19 2013

[9] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XVI Cosmological parametersrdquo Astronomy ampAstrophysics vol 571 article A16 66 pages 2013

[10] P A R Ade N AghanimM Arnaud et al ldquoPlanck 2015 resultsXIII Cosmological parametersrdquoAstronomyampAstrophysics vol594 article A13 2015

[11] F Zwicky ldquoOn themasses of nebulae and of clusters of nebulaerdquoThe Astrophysical Journal vol 86 pp 217ndash246 1937

[12] H W Babcock ldquoThe rotation of the Andromeda nebulardquo LickObservatory Bulletin vol 498 pp 41ndash51 1939

[13] C Wetterich ldquoCosmology and the fate of dilatation symmetryrdquoNuclear Physics Section B vol 302 no 4 pp 668ndash696 1988

[14] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 p 3406 1988

[15] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal ofModern Physics D GravitationAstrophysics Cosmology vol 15 no 11 pp 1753ndash1935 2006

[16] M Li X-D Li S Wang and Y Wang ldquoDark energyrdquo Commu-nications inTheoretical Physics vol 56 no 3 pp 525ndash604 2011

[17] C Wetterich ldquoThe cosmon model for an asymptotically van-ishing time dependent cosmological lsquoconstantrsquordquo Astronomy ampAstrophysics vol 301 pp 321ndash328 1995

[18] L Amendola ldquoCoupled quintessencerdquo Physical Review D vol62 no 4 Article ID 043511 2000

[19] N Dalal K Abazajian E Jenkins and A V Manohar ldquoTestingthe cosmic coincidence problem and the nature of dark energyrdquoPhysical Review Letters vol 87 no 14 Article ID 141302 2001

[20] W Zimdahl D Pavon and L P Chimento ldquoInteractingquintessencerdquo Physics Letters Section B Nuclear ElementaryParticle and High-Energy Physics vol 521 no 3-4 pp 133ndash1382001

[21] B Gumjudpai T Naskar M Sami and S Tsujikawa ldquoCoupleddark energy towards a general description of the dynamicsrdquoJournal of Cosmology and Astroparticle Physics vol 506 article7 2005

[22] A P Billyard and A A Coley ldquoInteractions in scalar fieldcosmologyrdquo Physical Review D vol 61 no 8 Article ID 0835032000

[23] J-Q Xia Y-F Cai T-T Qiu G-B Zhao and X ZhangldquoConstraints on the sound speed of dynamical dark energyrdquoInternational Journal of Modern Physics D vol 17 no 8 pp1229ndash1243 2008

[24] A Vikman ldquoCan dark energy evolve to the phantomrdquo PhysicalReview D vol 71 no 2 Article ID 023515 2005

[25] W Hu ldquoCrossing the phantom divide dark energy internaldegrees of freedomrdquo Physical Review DmdashParticles Fields Grav-itation and Cosmology vol 71 no 4 Article ID 047301 2005

[26] R R Caldwell and M Doran ldquoDark-energy evolution acrossthe cosmological-constant boundaryrdquo Physical Review DmdashParticles Fields Gravitation and Cosmology vol 72 no 4Article ID 043527 pp 1ndash6 2005

[27] G-B Zhao J-Q Xia M Li B Feng and X Zhang ldquoPertur-bations of the quintom models of dark energy and the effectson observationsrdquo Physical Review D vol 72 no 12 Article ID123515 2005

[28] M Kunz and D Sapone ldquoCrossing the phantom dividerdquoPhysical Review D vol 74 no 12 Article ID 123503 2006

[29] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1-3 pp 39ndash43 1984

[30] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 no 2 Article ID023507 2000

[31] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 no 4 Article ID043539 2004

[32] K Bamba S Capozziello S Nojiri and S D Odintsov ldquoDarkenergy cosmology the equivalent description via differenttheoretical models and cosmography testsrdquo Astrophysics andSpace Science vol 342 no 1 pp 155ndash228 2012

[33] O Hrycyna andM Szydowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 no 4 article no 26 2009

[34] O Hrycyna and M Szydłowski ldquoExtended quintessence withnonminimally coupled phantom scalar fieldrdquo Physical ReviewD vol 76 no 12 Article ID 123510 2007

[35] R C de Souza andGMKremer ldquoConstraining non-minimallycoupled tachyon fields by the Noether symmetryrdquo Classical andQuantum Gravity vol 26 no 13 Article ID 135008 2009

[36] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled k-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010

[37] E Dil and E Kolay ldquoDynamics of mixed dark energy domina-tion in teleparallel gravity and phase-space analysisrdquo Advancesin High Energy Physics vol 2015 Article ID 608252 20 pages2015

[38] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[39] S M Carroll W H Press and E L Turner ldquoThe cosmologicalconstantrdquoAnnual Review ofAstronomyandAstrophysics vol 30no 1 pp 499ndash542 1992


[40] L M Krauss and M S Turner ldquoThe cosmological constant isbackrdquoGeneral Relativity andGravitation vol 27 no 11 pp 1137ndash1144 1995

[41] G Huey L Wang R Dave R R Caldwell and P J SteinhardtldquoResolving the cosmological missing energy problemrdquo PhysicalReview D - Particles Fields Gravitation and Cosmology vol 59no 6 pp 1ndash6 1999

[42] P Kulish and N Reshetiknin ldquoQuantum linear problem for theSine-Gordon equation and higher representationsrdquo Journal ofSoviet Mathematics vol 23 no 4 pp 2435ndash2441 1981

[43] E Sklyanin L Takhatajan and L Faddeev ldquoQuantum inverseproblem method Irdquo Theoretical and Mathematical Physics vol40 no 2 pp 688ndash706 1979

[44] L C Biedenharn and M A Lohe Quantum Group SymmetryandQ-Tensor Algebras World Scientific Publishing River EdgeNJ USA 1995

[45] M Arik and D D Coon ldquoHilbert spaces of analytic func-tions and generalized coherent statesrdquo Journal of MathematicalPhysics vol 17 no 4 pp 524ndash527 1976

[46] A J Macfarlane ldquoOn q-analogues of the quantum harmonicoscillator and the quantum group SUq(2)rdquo Journal of Physics Avol 22 no 21 pp 4581ndash4588 1989

[47] L C Biedenharn ldquoThequantumgroup SU119902(2) and a q-analogueof the boson operatorsrdquo Journal of Physics A Mathematical andGeneral vol 22 no 18 pp L873ndashL878 1989

[48] L Tripodi and C L Lima ldquoOn a q-covariant form of the BCSapproximationrdquoPhysics Letters B vol 412 no 1-2 pp 7ndash13 1997

[49] V S Timoteo and C L Lima ldquoEffect of q-deformation in theNJL gap equationrdquo Physics Letters B vol 448 no 1-2 pp 1ndash51999

[50] C R Lee and J P Yu ldquoOn q-analogues of the statisticaldistributionrdquo Physics Letters A vol 150 no 2 pp 63ndash66 1990

[51] J A Tuszynski J L Rubin J Meyer and M Kibler ldquoStatisticalmechanics of a q-deformed boson gasrdquo Physics Letters A vol175 no 3-4 pp 173ndash177 1993

[52] P N Swamy ldquoInterpolating statistics and q-deformed oscillatoralgebrasrdquo International Journal of Modern Physics B vol 20 no6 pp 697ndash713 2006

[53] M R Ubriaco ldquoThermodynamics of a free SU119902(2) fermionicsystemrdquo Physics Letters A vol 219 no 3-4 pp 205ndash211 1996

[54] M R Ubriaco ldquoHigh and low temperature behavior of aquantum group fermion gasrdquoModern Physics Letters A vol 11no 29 pp 2325ndash2333 1996

[55] M R Ubriaco ldquoAnyonic behavior of quantum group gasesrdquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 55 no 1 part A pp 291ndash296 1997

[56] A Lavagno and P N Swamy ldquoThermostatistics of a q-deformedboson gasrdquo Physical Review E vol 61 no 2 pp 1218ndash1226 2000

[57] A Lavagno and P N Swamy ldquoGeneralized thermodynamicsof q-deformed bosons and fermionsrdquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 65 no 3Article ID 036101 2002

[58] A Algin and M Baser ldquoThermostatistical properties of a two-parameter generalised quantum group fermion gasrdquo Physica AStatistical Mechanics and Its Applications vol 387 no 5-6 pp1088ndash1098 2008

[59] A Algin M Arik and A S Arikan ldquoHigh temperaturebehavior of a two-parameter deformed quantum group fermiongasrdquo Physical Review E vol 65 no 2 Article ID 026140 p0261405 2002

[60] A Algin A S Arikan and E Dil ldquoHigh temperature thermo-statistics of fermionic Fibonacci oscillators with intermediatestatisticsrdquo Physica A Statistical Mechanics and its Applicationsvol 416 pp 499ndash517 2014

[61] O W Greenberg ldquoParticles with small violations of Fermi orBose statisticsrdquo Physical Review D vol 43 no 12 pp 4111ndash41201991

[62] A M Gavrilik I I Kachurik and Y A Mishchenko ldquoQuasi-bosons composed of two q-fermions realization by deformedoscillatorsrdquo Journal of Physics A Mathematical and Theoreticalvol 44 no 47 Article ID 475303 2011

[63] P Pouliot ldquoFinite number of states de Sitter space and quantumgroups at roots of unityrdquoClassical and QuantumGravity vol 21no 1 pp 145ndash162 2004

[64] A Strominger ldquoBlack hole statisticsrdquo Physical Review Lettersvol 71 no 21 pp 3397ndash3400 1993

[65] E Dil ldquoQ-deformed einstein equationsrdquo Canadian Journal ofPhysics vol 93 no 11 pp 1274ndash1278 2015

[66] E Dil and E Kolay ldquoSolution of deformed Einstein equationsand quantum black holesrdquoAdvances in High Energy Physics vol2016 Article ID 3973706 7 pages 2016

[67] D Bonatsos E N Argyres and P P Raychev ldquoSU119902(11)description of vibrational molecular spectrardquo Journal of PhysicsA Mathematical and General vol 24 no 8 pp L403ndashL4081991

[68] D Bonatsos P P Raychev and A Faessler ldquoQuantum algebraicdescription of vibrational molecular spectrardquo Chemical PhysicsLetters vol 178 no 2-3 pp 221ndash226 1991

[69] D Bonatsos and C Daskaloyannis ldquoGeneralized deformedoscillators for vibrational spectra of diatomic moleculesrdquo Phys-ical Review A vol 46 no 1 pp 75ndash80 1992

[70] P P Raychev R P Roussev and Y F Smirnov ldquoThe quantumalgebra SUq(2) and rotational spectra of deformed nucleirdquoJournal of Physics G Nuclear and Particle Physics vol 16 no8 pp L137ndashL141 1990

[71] A I Georgieva K D Sviratcheva M I Ivanov and J PDraayer ldquoq-Deformation of symplectic dynamical symmetriesin algebraic models of nuclear structurerdquo Physics of AtomicNuclei vol 74 no 6 pp 884ndash892 2011

[72] M R-Monteiro L M C S Rodrigues and SWulck ldquoQuantumalgebraic nature of the phonon spectrum in4Herdquo PhysicalReview Letters vol 76 no 7 1996

[73] A E Shalyt-Margolin ldquoDeformed quantum field theory ther-modynamics at low and high energies and gravity II Defor-mation parameterrdquo International Journal of Theoretical andMathematical Physics vol 2 no 3 pp 41ndash50 2012

[74] A E Shalyt-Margolin and V I Strazhev ldquoDark energy anddeformed quantum theory in physics of the early universeIn non-eucleden geometry in modern physicsrdquo in Proceedingsof the 5th Intentional Conference of Bolyai-Gauss-Lobachevsky(BGL-5 rsquo07) Y Kurochkin and V Redrsquokov Eds pp 173ndash178Minsk Belarus 2007

[75] Y J Ng ldquoHolographic foam dark energy and infinite statisticsrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 657 no 1ndash3 pp 10ndash14 2007

[76] Y J Ng ldquoSpacetime foamrdquo International Journal of ModernPhysics D vol 11 no 10 pp 1585ndash1590 2002

[77] M Chaichian R G Felipe and C Montonen ldquoStatistics of q-oscillators quons and relations to fractional statisticsrdquo Journalof Physics AMathematical andGeneral vol 26 no 16 pp 4017ndash4034 1993


[78] J Wess and B Zumino ldquoCovariant differential calculus on thequantum hyperplanerdquo Nuclear Physics BmdashProceedings Supple-ments vol 18 no 2 pp 302ndash312 1991

[79] G Vinod Studies in quantum oscillators and q-deformed quan-tum mechanics [PhD thesis] Cochin University of Science andTechnology Department of Physics Kochi India 1997

[80] B K Berger ldquoScalar particle creation in an anisotropic uni-verserdquo Physical Review D vol 12 no 2 pp 368ndash375 1975

[81] S W Hawking ldquoBreakdown of predictability in gravitationalcollapserdquo Physical Review D vol 14 no 10 pp 2460ndash2473 1976

[82] B K Berger ldquoClassical analog of cosmological particle cre-ationrdquo Physical Review D vol 18 no 12 pp 4367ndash4372 1978

[83] L Parker ldquoQuantized fields and particle creation in expandinguniverses Irdquo Physical Review vol 183 no 5 article 1057 1969

[84] J W Goodison and D J Toms ldquoNo generalized statistics fromdynamics in curved spacetimerdquo Physical Review Letters vol 71no 20 1993

[85] D Bonatsos and C Daskaloyannis ldquoQuantum groups and theirapplications in nuclear physicsrdquo Progress in Particle and NuclearPhysics vol 43 no 1 pp 537ndash618 1999

[86] R G Leigh ldquoDirac-born-infeld action from dirichlet 120590-modelrdquoModern Physics Letters A vol 04 no 28 pp 2767ndash2772 1989

[87] N A Chernikov and E A Tagirov ldquoQuantum theory ofscalar field in de Sitter space-timerdquo Annales de lrsquoIHP PhysiqueTheorique vol 9 pp 109ndash141 1968

[88] C G Callan Jr S Coleman and R Jackiw ldquoA new improvedenergy-momentum tensorrdquo Annals of Physics vol 59 pp 42ndash73 1970

[89] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical ReviewD vol 62 no 2 Article ID 023504 2000

[90] S C Park and S Yamaguchi ldquoInflation by non-minimalcouplingrdquo Journal of Cosmology and Astroparticle Physics vol2008 no 8 p 9 2008

[91] A de Felice and S Tsujikawa ldquof(R) theoriesrdquo Living Reviews inRelativity vol 13 article no 3 2010

[92] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[93] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1pp 115ndash146 2007

[94] A Banijamali ldquoDynamics of interacting tachyonic teleparalleldark energyrdquoAdvances in High Energy Physics vol 2014 ArticleID 631630 14 pages 2014

[95] B Fazlpour and A Banijamali ldquoTachyonic teleparallel darkenergy in phase spacerdquo Advances in High Energy Physics vol2013 Article ID 279768 9 pages 2013

[96] P G Ferreira and M Joyce ldquoStructure formation with a self-tuning scalar fieldrdquo Physical Review Letters vol 79 no 24 pp4740ndash4743 1997

[97] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998

[98] X-M Chen Y Gong and EN Saridakis ldquoPhase-space analysisof interacting phantom cosmologyrdquo Journal of Cosmology andAstroparticle Physics vol 2009 no 4 article no 001 2009

[99] Z-K Guo Y-S Piao X Zhang and Y-Z Zhang ldquoCosmo-logical evolution of a quintom model of dark energyrdquo PhysicsLetters Section B Nuclear Elementary Particle and High-EnergyPhysics vol 608 no 3-4 pp 177ndash182 2005

[100] R Lazkoz andG Leon ldquoQuintom cosmologies admitting eithertracking or phantom attractorsrdquo Physics Letters B vol 638 no4 pp 303ndash309 2006

[101] Z-K Guo Y-S Piao X Zhang and Y-Z Zhang ldquoTwo-fieldquintom models in the 119908 minus 1199081015840 planerdquo Physical Review D vol74 no 12 Article ID 127304 4 pages 2006

[102] R Lazkoz G Leon and I Quiros ldquoQuintom cosmologies witharbitrary potentialsrdquo Physics Letters B vol 649 no 2-3 pp 103ndash110 2007

[103] M Alimohammadi ldquoAsymptotic behavior of 120596 in generalquintommodelrdquo General Relativity and Gravitation vol 40 no1 pp 107ndash115 2008

[104] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters Section B vol 668 no3 pp 177ndash181 2008

[105] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics DGravitation Astrophysics Cosmology vol 18 no 4 pp 549ndash5572009

[106] M R Setare and E N Saridakis ldquoThe quintom modelwith O(N) symmetryrdquo Journal of Cosmology and AstroparticlePhysics vol 2008 no 9 article no 026 2008

[107] M R Setare and E N Saridakis ldquoQuintom dark energy modelswith nearly flat potentialsrdquo Physical Review D vol 79 no 4Article ID 043005 2009

[108] G Leon R Cardenas and J L Morales ldquoEquilibrium setsin quintom cosmologies the past asymptotic dynamicsrdquohttpsarxivorgabs08120830

[109] S Carloni E Elizalde and P J Silva ldquoAn analysis of the phasespace of HoravandashLifshitz cosmologiesrdquo Classical and QuantumGravity vol 27 no 4 Article ID 045004 2010

[110] Z-K Guo Y-S Piao R-G Cai and Y-Z Zhang ldquoInflationaryattractor from tachyonic matterrdquo Physical Review D vol 68 no4 Article ID 043508 2003

[111] L A Urena-Lopez and M J Reyes-Ibarra ldquoOn the dynamicsof a quadratic scalar field potentialrdquo International Journal ofModern Physics D vol 18 no 4 pp 621ndash634 2009

[112] V A Belinsky L P Grishchuk I M Khalatnikov and Y BZeldovich ldquoInflationary stages in cosmological models with ascalar fieldrdquo Physics Letters B vol 155 no 4 pp 232ndash236 1985

[113] T Piran and R M Williams ldquoInflation in universes with amassive scalar fieldrdquo Physics Letters B vol 163 no 5-6 pp 331ndash335 1985

[114] A R Liddle and D H Lyth Cosmological inflation andlarge-scale structure Cambridge University Press CambridgeEngland 2000

[115] V V Kiselev and S A Timofeev ldquoQuasiattractor dynamics of1205821206014-inflationrdquo httpsarxivorgabs08012453[116] S Downes B Dutta and K Sinha ldquoAttractors universality and

inflationrdquo Physical Review D vol 86 no 10 Article ID 1035092012

[117] J Khoury and P J Steinhardt ldquoGenerating scale-invariantperturbations from rapidly-evolving equation of staterdquo PhysicalReview DmdashParticles Fields Gravitation and Cosmology vol 83no 12 Article ID 123502 2011

[118] S Clesse C Ringeval and J Rocher ldquoFractal initial conditionsand natural parameter values in hybrid inflationrdquo PhysicalReview D vol 80 no 12 Article ID 123534 2009

[119] V V Kiselev and S A Timofeev ldquoQuasiattractor in models ofnew and chaotic inflationrdquo General Relativity and Gravitationvol 42 no 1 pp 183ndash197 2010


[120] G F R Ellis R Maartens andM A HMacCallum Relativisticcosmology Cambridge University Press Cambridge UK 2012

[121] YWang JM Kratochvil A Linde andM Shmakova ldquoCurrentobservational constraints on cosmic doomsdayrdquo Journal ofCosmology and Astroparticle Physics vol 412 article 6 2004

[122] L Amendola and D Tocchini-Valentini ldquoStationary darkenergy the present universe as a global attractorrdquo PhysicalReview D vol 64 no 4 Article ID 043509 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


this study is formed of the deformed scalar field whose fieldequations are defined by the deformed oscillator algebras

The quantum algebra and quantum group structurewere firstly introduced by Kulish et al [42ndash44] during theinvestigations of integrable systems in quantum field theoryand statistical mechanics Quantum groups and deformedboson algebras are closely related terms It is known that thedeformation of the standard boson algebra is first proposedby Arik-Coon [45] Later on Macfarlane and Biedenharnhave realized the deformation of boson algebra in a differentmanner from Arik-Coon [46 47] The relation betweenquantum groups and the deformed oscillator algebras canbe constructed obviously with this study by expressing thedeformed boson operators in terms of the 119904119906119902(2) Lie algebraoperatorsTherefore the construction of the relation betweenquantum groups and deformed algebras leads the deformedalgebras of great interest with many different applicationsThe deformed version of Bardeen-Cooper-Schrieffer (BCS)many-body formalism in nuclear force deformed creationand annihilation operators are used to study the quantumoccupation probabilities [48] As another study in Nambu-Jona-Lasinio (NJL) model the deformed fermion operatorsare used instead of standard fermion operators and this leadsto an increase in the NJL four-fermion coupling force andthe quark condensation related to the dynamical mass [49]The statistical mechanical studies of the deformed bosonand fermion systems have been familiar in recent years [50ndash60] Moreover the investigations on the internal structure ofcomposite particles involve the deformed fermions or bosonsas the building block of the composite structures [61 62]There are also applications of the deformed particles in blackhole physics [63ndash66] The range of the deformed boson andfermion applications diverses from atomic-molecular physicsto solid state physics in a widespread manner [67ndash72]

The ideas on considering the dark energy as the deformedscalar field have become common in the literature [73ndash76] Inthis study we then take into account the deformed bosonsas the scalar field dark energy interacting with the darkmatter and also nonminimally coupled to gravity In order toconfirm our proposal that the dark energy can be consideredas a deformed scalar field we firstly introduce the dynamics ofthe interacting and nonminimally coupled dark energy darkmatter and gravity model in a spatially flat FRW backgroundand then perform the phase-space analysis to check whetherit will provide the late-time stable attractor solutions implyingthe accelerated expansion phase of the universe

2 Dynamics of the Model

The field equations of the scalar field dark energy areconsidered to be defined by the 119902-deformed boson fields inour model Constructing a 119902-deformed quantum field theoryafter the idea of 119902-deformation of the single particle quantummechanics [45ndash47] has naturally been nonsurprising [77ndash79]The bosonic part of the deformed particle fields correspondsto the deformed scalar field and the fermionic counterpartcorresponds to the deformed vector field In this study weconsider the 119902-deformed bosonic scalar field as the 119902-deformed

dark energy under consideration In our model the 119902-deformed dark energy interacts with the darkmatter and alsononminimally couples to gravity

Early Universe scenarios can be well understood bystudying the quantum field theory in curved spacetime Thebehavior of the classical scalar field near the initial singularitycan be translated to the quantumfield regime by constructingthe coherent states in quantummechanics for anymode of thescalar field It is now impossible to determine the quantumstate of the scalar field near the initial singularity by anobserver at the present universe In order to overcome theundeterministic nature Hawking proposes to take the ran-dom superposition of all possible states in that spacetimeIt has been realized by Berger with taking random super-position of coherent states Also the particle creation inan expanding universe with a nonquantized gravitationalmetric has been investigated by Parker It has been stated byGoodison and Toms that if the field quanta obey the Bose orFermi statistics when considering the evolution of the scalarfield in an expanding universe then the particle creation doesnot occur in the vacuum state Their result gives significationto the possibility of the existence of the deformed statistics incoherent or squeezed states in the Early Universe [79ndash84]

Motivated by this significant possibility we propose thatthe dark energy consists of a 119902-deformed scalar field whoseparticles obey the 119902-deformed algebras Therefore we nowdefine the 119902-deformed scalar field constructing the darkenergy in our model The field operator of the 119902-deformedscalar field dark energy can be given as [79]120601119902 (119909) = int 1198893119896(2120587)32 1(2119908119896)12 [119886119902 (119896) 119890119894119896119909 + 119886lowast119902 (119896) 119890minus119894119896119909] (1)

The following commutation relations for the deformed anni-hilation operator 119886119902(119896) and creations operator 119886lowast119902 (119896) in 119902-bosonic Fock space are given by [45]119886119902 (119896) 119886lowast119902 (1198961015840) minus 1199022119886lowast119902 (1198961015840) 119886119902 (119896) = 120575 (119896 minus 1198961015840) 119886119902 (119896) 119886119902 (1198961015840) minus 1199022119886119902 (1198961015840) 119886119902 (119896) = 0 (2)

where 119902 is a real deformation parameter in interval 0 lt 119902 lt infinand [(119896)] = 119886lowast119902 (119896)119886119902(119896) is the deformed number operator of119896th mode whose eigenvalue spectrum is given as[119873 (119896)] = 1 minus 1199022119873(119896)1 minus 1199022 (3)

Here (119896) = 119886lowast119904 (119896)119886119904(119896) is the standard nondeformed num-ber operator By using (2) in (1) we can obtain the commu-tation relations and planewave expansion of the 119902-deformedscalar field 120601119902(119909) as follows120601119902 (119909) 120601lowast119902 (1199091015840) minus 1199022120601lowast119902 (1199091015840) 120601119902 (119909) = 119894Δ (119909 minus 1199091015840) (4)

where Δ (119909 minus 1199091015840) = minus1(2120587)3 int 1198893119896119908119896 sin119908119896 (119909 minus 1199090) (5)




11989621198941198862 + 119898) (7)


119886119902 = 119886119904radic [] (8)






119879120583] = (120588 + 119901)119880120583119880] + 119901119892120583] =(120588 0 0 000 1198921198941198951199010 ) (11)













(22)



(24)















(31)


Ω119902 = 120581212058811990231198672= 1199092 + 1199102 + 1205851199062 + 2radic6120585119909119906 + 1199112 + 2119909119911+ 2radic6120585119911119906 (33)










(38)



(39)





(40)



1199061015840 = radic6119909 + radic6119911 (44)









+ 2radic63 120585 (119909 + 119911) + 2120585120573Ω119898 minus 21205851205821199102+ (43120585 + 81205852) 119904119906] 1119875120575119906(47)



(48)



(49)





(51)



(52)











C and D




A

0506070809

11112131415

y


x + z

(a)

A


x + z

minus15

minus1

minus05

0

u

(b)

A

minus15

minus1

minus05

0

u

109 1311 12 1407 08y

(c)





B


x + z



y

(a)

B


x + z

minus15

minus1

minus05

0

u

(b)

B

minus12minus1


0

u


(c)








(53)



C


x + z

02

04

06

08

1

12

14

y

(a)

C


x + z

minus01

minus005

0

005

01

015

u

(b)

C

minus01

minus005

0

005

01

015

u

109 11 12 13 14 1508y

(c)


D


x + z

minus14

minus08

minus02

y

(a)

D


x + z

0

minus01

minus005

005

01

015

u

(b)

D

0

minus01

minus005

005

01

015

u


y

(c)



minus08minus06

minus04minus02

0



y

u

A

x+z

(a)

minus07minus05

minus03minus01


minus2minus15

minus1minus05

0

y

u

B

x+z

(b)


002

0406

081

1214

minus02minus01

001

y

u

C

x + z

(c)

minus06minus04

minus020

02


y

D

minus02minus01

001

u

x + z

(d)


0 01 02 03 04 05 06 07 08 09 1

12345678910040506070809

1

qN

xx

s(o

ruu

s)

(a)1 11 12 13 14 15 16 17 18 19 2

11522533541

152

253

354

455

qN

xx

s(o

ruu

s)

(b)

01 02 03 04 0506 07 08 09

0

1

1234567891007

07508

08509

0951

qN

yy

s

(c)

11 12 13 14 15 16 17 18 19

1

2

115225

33541234567

qN

yy

s

(d)




4 Conclusion






Competing Interests


References




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






11989621198941198862 + 119898) (7)


119886119902 = 119886119904radic [] (8)






119879120583] = (120588 + 119901)119880120583119880] + 119901119892120583] =(120588 0 0 000 1198921198941198951199010 ) (11)













(22)



(24)















(31)


Ω119902 = 120581212058811990231198672= 1199092 + 1199102 + 1205851199062 + 2radic6120585119909119906 + 1199112 + 2119909119911+ 2radic6120585119911119906 (33)










(38)



(39)





(40)



1199061015840 = radic6119909 + radic6119911 (44)









+ 2radic63 120585 (119909 + 119911) + 2120585120573Ω119898 minus 21205851205821199102+ (43120585 + 81205852) 119904119906] 1119875120575119906(47)



(48)



(49)





(51)



(52)











C and D




A

0506070809

11112131415

y


x + z

(a)

A


x + z

minus15

minus1

minus05

0

u

(b)

A

minus15

minus1

minus05

0

u

109 1311 12 1407 08y

(c)





B


x + z



y

(a)

B


x + z

minus15

minus1

minus05

0

u

(b)

B

minus12minus1


0

u


(c)








(53)



C


x + z

02

04

06

08

1

12

14

y

(a)

C


x + z

minus01

minus005

0

005

01

015

u

(b)

C

minus01

minus005

0

005

01

015

u

109 11 12 13 14 1508y

(c)


D


x + z

minus14

minus08

minus02

y

(a)

D


x + z

0

minus01

minus005

005

01

015

u

(b)

D

0

minus01

minus005

005

01

015

u


y

(c)



minus08minus06

minus04minus02

0



y

u

A

x+z

(a)

minus07minus05

minus03minus01


minus2minus15

minus1minus05

0

y

u

B

x+z

(b)


002

0406

081

1214

minus02minus01

001

y

u

C

x + z

(c)

minus06minus04

minus020

02


y

D

minus02minus01

001

u

x + z

(d)


0 01 02 03 04 05 06 07 08 09 1

12345678910040506070809

1

qN

xx

s(o

ruu

s)

(a)1 11 12 13 14 15 16 17 18 19 2

11522533541

152

253

354

455

qN

xx

s(o

ruu

s)

(b)

01 02 03 04 0506 07 08 09

0

1

1234567891007

07508

08509

0951

qN

yy

s

(c)

11 12 13 14 15 16 17 18 19

1

2

115225

33541234567

qN

yy

s

(d)




4 Conclusion






Competing Interests


References




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






(22)



(24)















(31)


Ω119902 = 120581212058811990231198672= 1199092 + 1199102 + 1205851199062 + 2radic6120585119909119906 + 1199112 + 2119909119911+ 2radic6120585119911119906 (33)










(38)



(39)





(40)



1199061015840 = radic6119909 + radic6119911 (44)









+ 2radic63 120585 (119909 + 119911) + 2120585120573Ω119898 minus 21205851205821199102+ (43120585 + 81205852) 119904119906] 1119875120575119906(47)



(48)



(49)





(51)



(52)











C and D




A

0506070809

11112131415

y


x + z

(a)

A


x + z

minus15

minus1

minus05

0

u

(b)

A

minus15

minus1

minus05

0

u

109 1311 12 1407 08y

(c)





B


x + z



y

(a)

B


x + z

minus15

minus1

minus05

0

u

(b)

B

minus12minus1


0

u


(c)








(53)



C


x + z

02

04

06

08

1

12

14

y

(a)

C


x + z

minus01

minus005

0

005

01

015

u

(b)

C

minus01

minus005

0

005

01

015

u

109 11 12 13 14 1508y

(c)


D


x + z

minus14

minus08

minus02

y

(a)

D


x + z

0

minus01

minus005

005

01

015

u

(b)

D

0

minus01

minus005

005

01

015

u


y

(c)



minus08minus06

minus04minus02

0



y

u

A

x+z

(a)

minus07minus05

minus03minus01


minus2minus15

minus1minus05

0

y

u

B

x+z

(b)


002

0406

081

1214

minus02minus01

001

y

u

C

x + z

(c)

minus06minus04

minus020

02


y

D

minus02minus01

001

u

x + z

(d)


0 01 02 03 04 05 06 07 08 09 1

12345678910040506070809

1

qN

xx

s(o

ruu

s)

(a)1 11 12 13 14 15 16 17 18 19 2

11522533541

152

253

354

455

qN

xx

s(o

ruu

s)

(b)

01 02 03 04 0506 07 08 09

0

1

1234567891007

07508

08509

0951

qN

yy

s

(c)

11 12 13 14 15 16 17 18 19

1

2

115225

33541234567

qN

yy

s

(d)




4 Conclusion






Competing Interests


References




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






(31)


Ω119902 = 120581212058811990231198672= 1199092 + 1199102 + 1205851199062 + 2radic6120585119909119906 + 1199112 + 2119909119911+ 2radic6120585119911119906 (33)










(38)



(39)





(40)



1199061015840 = radic6119909 + radic6119911 (44)









+ 2radic63 120585 (119909 + 119911) + 2120585120573Ω119898 minus 21205851205821199102+ (43120585 + 81205852) 119904119906] 1119875120575119906(47)



(48)



(49)





(51)



(52)











C and D




A

0506070809

11112131415

y


x + z

(a)

A


x + z

minus15

minus1

minus05

0

u

(b)

A

minus15

minus1

minus05

0

u

109 1311 12 1407 08y

(c)





B


x + z



y

(a)

B


x + z

minus15

minus1

minus05

0

u

(b)

B

minus12minus1


0

u


(c)








(53)



C


x + z

02

04

06

08

1

12

14

y

(a)

C


x + z

minus01

minus005

0

005

01

015

u

(b)

C

minus01

minus005

0

005

01

015

u

109 11 12 13 14 1508y

(c)


D


x + z

minus14

minus08

minus02

y

(a)

D


x + z

0

minus01

minus005

005

01

015

u

(b)

D

0

minus01

minus005

005

01

015

u


y

(c)



minus08minus06

minus04minus02

0



y

u

A

x+z

(a)

minus07minus05

minus03minus01


minus2minus15

minus1minus05

0

y

u

B

x+z

(b)


002

0406

081

1214

minus02minus01

001

y

u

C

x + z

(c)

minus06minus04

minus020

02


y

D

minus02minus01

001

u

x + z

(d)


0 01 02 03 04 05 06 07 08 09 1

12345678910040506070809

1

qN

xx

s(o

ruu

s)

(a)1 11 12 13 14 15 16 17 18 19 2

11522533541

152

253

354

455

qN

xx

s(o

ruu

s)

(b)

01 02 03 04 0506 07 08 09

0

1

1234567891007

07508

08509

0951

qN

yy

s

(c)

11 12 13 14 15 16 17 18 19

1

2

115225

33541234567

qN

yy

s

(d)




4 Conclusion






Competing Interests


References




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







(40)



1199061015840 = radic6119909 + radic6119911 (44)









+ 2radic63 120585 (119909 + 119911) + 2120585120573Ω119898 minus 21205851205821199102+ (43120585 + 81205852) 119904119906] 1119875120575119906(47)



(48)



(49)





(51)



(52)











C and D




A

0506070809

11112131415

y


x + z

(a)

A


x + z

minus15

minus1

minus05

0

u

(b)

A

minus15

minus1

minus05

0

u

109 1311 12 1407 08y

(c)





B


x + z



y

(a)

B


x + z

minus15

minus1

minus05

0

u

(b)

B

minus12minus1


0

u


(c)








(53)



C


x + z

02

04

06

08

1

12

14

y

(a)

C


x + z

minus01

minus005

0

005

01

015

u

(b)

C

minus01

minus005

0

005

01

015

u

109 11 12 13 14 1508y

(c)


D


x + z

minus14

minus08

minus02

y

(a)

D


x + z

0

minus01

minus005

005

01

015

u

(b)

D

0

minus01

minus005

005

01

015

u


y

(c)



minus08minus06

minus04minus02

0



y

u

A

x+z

(a)

minus07minus05

minus03minus01


minus2minus15

minus1minus05

0

y

u

B

x+z

(b)


002

0406

081

1214

minus02minus01

001

y

u

C

x + z

(c)

minus06minus04

minus020

02


y

D

minus02minus01

001

u

x + z

(d)


0 01 02 03 04 05 06 07 08 09 1

12345678910040506070809

1

qN

xx

s(o

ruu

s)

(a)1 11 12 13 14 15 16 17 18 19 2

11522533541

152

253

354

455

qN

xx

s(o

ruu

s)

(b)

01 02 03 04 0506 07 08 09

0

1

1234567891007

07508

08509

0951

qN

yy

s

(c)

11 12 13 14 15 16 17 18 19

1

2

115225

33541234567

qN

yy

s

(d)




4 Conclusion






Competing Interests


References




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




+ 2radic63 120585 (119909 + 119911) + 2120585120573Ω119898 minus 21205851205821199102+ (43120585 + 81205852) 119904119906] 1119875120575119906(47)



(48)



(49)





(51)



(52)











C and D




A

0506070809

11112131415

y


x + z

(a)

A


x + z

minus15

minus1

minus05

0

u

(b)

A

minus15

minus1

minus05

0

u

109 1311 12 1407 08y

(c)





B


x + z



y

(a)

B


x + z

minus15

minus1

minus05

0

u

(b)

B

minus12minus1


0

u


(c)








(53)



C


x + z

02

04

06

08

1

12

14

y

(a)

C


x + z

minus01

minus005

0

005

01

015

u

(b)

C

minus01

minus005

0

005

01

015

u

109 11 12 13 14 1508y

(c)


D


x + z

minus14

minus08

minus02

y

(a)

D


x + z

0

minus01

minus005

005

01

015

u

(b)

D

0

minus01

minus005

005

01

015

u


y

(c)



minus08minus06

minus04minus02

0



y

u

A

x+z

(a)

minus07minus05

minus03minus01


minus2minus15

minus1minus05

0

y

u

B

x+z

(b)


002

0406

081

1214

minus02minus01

001

y

u

C

x + z

(c)

minus06minus04

minus020

02


y

D

minus02minus01

001

u

x + z

(d)


0 01 02 03 04 05 06 07 08 09 1

12345678910040506070809

1

qN

xx

s(o

ruu

s)

(a)1 11 12 13 14 15 16 17 18 19 2

11522533541

152

253

354

455

qN

xx

s(o

ruu

s)

(b)

01 02 03 04 0506 07 08 09

0

1

1234567891007

07508

08509

0951

qN

yy

s

(c)

11 12 13 14 15 16 17 18 19

1

2

115225

33541234567

qN

yy

s

(d)




4 Conclusion






Competing Interests


References




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





(51)



(52)











C and D




A

0506070809

11112131415

y


x + z

(a)

A


x + z

minus15

minus1

minus05

0

u

(b)

A

minus15

minus1

minus05

0

u

109 1311 12 1407 08y

(c)





B


x + z



y

(a)

B


x + z

minus15

minus1

minus05

0

u

(b)

B

minus12minus1


0

u


(c)








(53)



C


x + z

02

04

06

08

1

12

14

y

(a)

C


x + z

minus01

minus005

0

005

01

015

u

(b)

C

minus01

minus005

0

005

01

015

u

109 11 12 13 14 1508y

(c)


D


x + z

minus14

minus08

minus02

y

(a)

D


x + z

0

minus01

minus005

005

01

015

u

(b)

D

0

minus01

minus005

005

01

015

u


y

(c)



minus08minus06

minus04minus02

0



y

u

A

x+z

(a)

minus07minus05

minus03minus01


minus2minus15

minus1minus05

0

y

u

B

x+z

(b)


002

0406

081

1214

minus02minus01

001

y

u

C

x + z

(c)

minus06minus04

minus020

02


y

D

minus02minus01

001

u

x + z

(d)


0 01 02 03 04 05 06 07 08 09 1

12345678910040506070809

1

qN

xx

s(o

ruu

s)

(a)1 11 12 13 14 15 16 17 18 19 2

11522533541

152

253

354

455

qN

xx

s(o

ruu

s)

(b)

01 02 03 04 0506 07 08 09

0

1

1234567891007

07508

08509

0951

qN

yy

s

(c)

11 12 13 14 15 16 17 18 19

1

2

115225

33541234567

qN

yy

s

(d)




4 Conclusion






Competing Interests


References




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








C and D




A

0506070809

11112131415

y


x + z

(a)

A


x + z

minus15

minus1

minus05

0

u

(b)

A

minus15

minus1

minus05

0

u

109 1311 12 1407 08y

(c)





B


x + z



y

(a)

B


x + z

minus15

minus1

minus05

0

u

(b)

B

minus12minus1


0

u


(c)








(53)



C


x + z

02

04

06

08

1

12

14

y

(a)

C


x + z

minus01

minus005

0

005

01

015

u

(b)

C

minus01

minus005

0

005

01

015

u

109 11 12 13 14 1508y

(c)


D


x + z

minus14

minus08

minus02

y

(a)

D


x + z

0

minus01

minus005

005

01

015

u

(b)

D

0

minus01

minus005

005

01

015

u


y

(c)



minus08minus06

minus04minus02

0



y

u

A

x+z

(a)

minus07minus05

minus03minus01


minus2minus15

minus1minus05

0

y

u

B

x+z

(b)


002

0406

081

1214

minus02minus01

001

y

u

C

x + z

(c)

minus06minus04

minus020

02


y

D

minus02minus01

001

u

x + z

(d)


0 01 02 03 04 05 06 07 08 09 1

12345678910040506070809

1

qN

xx

s(o

ruu

s)

(a)1 11 12 13 14 15 16 17 18 19 2

11522533541

152

253

354

455

qN

xx

s(o

ruu

s)

(b)

01 02 03 04 0506 07 08 09

0

1

1234567891007

07508

08509

0951

qN

yy

s

(c)

11 12 13 14 15 16 17 18 19

1

2

115225

33541234567

qN

yy

s

(d)




4 Conclusion






Competing Interests


References




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




B


x + z



y

(a)

B


x + z

minus15

minus1

minus05

0

u

(b)

B

minus12minus1


0

u


(c)








(53)



C


x + z

02

04

06

08

1

12

14

y

(a)

C


x + z

minus01

minus005

0

005

01

015

u

(b)

C

minus01

minus005

0

005

01

015

u

109 11 12 13 14 1508y

(c)


D


x + z

minus14

minus08

minus02

y

(a)

D


x + z

0

minus01

minus005

005

01

015

u

(b)

D

0

minus01

minus005

005

01

015

u


y

(c)



minus08minus06

minus04minus02

0



y

u

A

x+z

(a)

minus07minus05

minus03minus01


minus2minus15

minus1minus05

0

y

u

B

x+z

(b)


002

0406

081

1214

minus02minus01

001

y

u

C

x + z

(c)

minus06minus04

minus020

02


y

D

minus02minus01

001

u

x + z

(d)


0 01 02 03 04 05 06 07 08 09 1

12345678910040506070809

1

qN

xx

s(o

ruu

s)

(a)1 11 12 13 14 15 16 17 18 19 2

11522533541

152

253

354

455

qN

xx

s(o

ruu

s)

(b)

01 02 03 04 0506 07 08 09

0

1

1234567891007

07508

08509

0951

qN

yy

s

(c)

11 12 13 14 15 16 17 18 19

1

2

115225

33541234567

qN

yy

s

(d)




4 Conclusion






Competing Interests


References




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




C


x + z

02

04

06

08

1

12

14

y

(a)

C


x + z

minus01

minus005

0

005

01

015

u

(b)

C

minus01

minus005

0

005

01

015

u

109 11 12 13 14 1508y

(c)


D


x + z

minus14

minus08

minus02

y

(a)

D


x + z

0

minus01

minus005

005

01

015

u

(b)

D

0

minus01

minus005

005

01

015

u


y

(c)



minus08minus06

minus04minus02

0



y

u

A

x+z

(a)

minus07minus05

minus03minus01


minus2minus15

minus1minus05

0

y

u

B

x+z

(b)


002

0406

081

1214

minus02minus01

001

y

u

C

x + z

(c)

minus06minus04

minus020

02


y

D

minus02minus01

001

u

x + z

(d)


0 01 02 03 04 05 06 07 08 09 1

12345678910040506070809

1

qN

xx

s(o

ruu

s)

(a)1 11 12 13 14 15 16 17 18 19 2

11522533541

152

253

354

455

qN

xx

s(o

ruu

s)

(b)

01 02 03 04 0506 07 08 09

0

1

1234567891007

07508

08509

0951

qN

yy

s

(c)

11 12 13 14 15 16 17 18 19

1

2

115225

33541234567

qN

yy

s

(d)




4 Conclusion






Competing Interests


References




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus08minus06

minus04minus02

0



y

u

A

x+z

(a)

minus07minus05

minus03minus01


minus2minus15

minus1minus05

0

y

u

B

x+z

(b)


002

0406

081

1214

minus02minus01

001

y

u

C

x + z

(c)

minus06minus04

minus020

02


y

D

minus02minus01

001

u

x + z

(d)


0 01 02 03 04 05 06 07 08 09 1

12345678910040506070809

1

qN

xx

s(o

ruu

s)

(a)1 11 12 13 14 15 16 17 18 19 2

11522533541

152

253

354

455

qN

xx

s(o

ruu

s)

(b)

01 02 03 04 0506 07 08 09

0

1

1234567891007

07508

08509

0951

qN

yy

s

(c)

11 12 13 14 15 16 17 18 19

1

2

115225

33541234567

qN

yy

s

(d)




4 Conclusion






Competing Interests


References




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





4 Conclusion






Competing Interests


References




































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics












FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article Interacting Dark Matter and -Deformed ...downloads.hindawi.com/journals/ahep/2016/7380372.pdf · dark energy and the dark matter which transfers the energy and momentum