Research ArticleDynamics of Generalized Tachyon Field in Teleparallel Gravity
Behnaz Fazlpour1 and Ali Banijamali2
1Department of Physics Islamic Azad University Babol Branch Babol Iran2Department of Basic Sciences Babol University of Technology Babol 47148-71167 Iran
Correspondence should be addressed to Behnaz Fazlpour bfazlpourumzacir
Received 11 October 2014 Accepted 15 January 2015
Academic Editor Chao-Qiang Geng
Copyright copy 2015 B Fazlpour and A Banijamali This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited The publication of this article was funded by SCOAP3
We study dynamics of generalized tachyon scalar field in the framework of teleparallel gravity This model is an extension oftachyonic teleparallel dark energy model which has been proposed by Banijamali and Fazlpour (2012) In contrast with tachyonicteleparallel dark energymodel that has no scaling attractors here we find some scaling attractors whichmeans that the cosmologicalcoincidence problem can be alleviated Scaling attractors are presented for both interacting and noninteracting dark energy anddark matter cases
1 Introduction
The usual proposal to explain the late-time acceleratedexpansion of our universe is an unknown energy componentdubbed as dark energyThenatural choice andmost attractivecandidate for dark energy is the cosmological constant butit is not well accepted because of the cosmological constantproblem [1 2] as well as the age problem [3] Thus manydynamical dark energy models as alternative possibilitieshave been proposed Quintessence phantom k-essencequintom and tachyon field are the most familiar dark energymodels in the literature (for reviews on dark energy modelssee [4 5]) The tachyon field arising in the context of stringtheory [4 5] and its application in cosmology both as a sourceof early inflation and late-time cosmic acceleration have beenextensively studied [6ndash9]
The so-called ldquoteleparallel equivalent of general relativityrdquoor teleparallel gravity was first constructed by Einstein [10ndash13] In this formulation one uses the curvature-less Weitzen-bock connection instead of the torsion-less Levi-Civita con-nection The relevant Lagrangian in teleparallel gravity is thetorsion scalar 119879 which is constructed by contraction of thetorsion tensorWe recall that the Einstein-Hilbert Lagrangian119877 is constructed by contraction of the curvature tensor Sinceteleparallel gravitywith torsion scalar as Lagrangian density iscompletely equivalent to a matter-dominated universe in the
framework of general relativity it cannot be acceleratedThusone should generalize teleparallel gravity either by replacing119879 with an arbitrary function the so-called 119891(119879) gravity[14ndash16] or by adding dark energy into teleparallel gravityallowing also a nonminimal coupling between dark energyand gravity Note that both approaches are inspired by thesimilarmodifications of general relativity that is119891(119877) gravity[17 18] and nonminimally coupled dark energymodels in theframework of general relativity [19ndash26]
Recently Geng et al [27 28] have included a nonminimalcoupling between quintessence and gravity in the context ofteleparallel gravity This theory has been called ldquoteleparalleldark energyrdquo and its dynamics was studied in [29ndash31] Tachy-onic teleparallel dark energy is a generalization of teleparalleldark energy by inserting a noncanonical scalar field insteadof quintessence in the action [32] Phase-space analysis of thismodel has been investigated in [33] On the other hand thereis no physical argument to exclude the interaction betweendark energy and dark matter The interaction between thesecompletely different components of our universe has the sameimportant consequences such as addressing the coincidenceproblem [34] There are also observational evidences of theinteraction in dark sector [35ndash40] In [35ndash38] Bertolami et alhave shown that this interaction might imply a violation ofthe equivalence principle By using optical X-ray and weaklensing data from the relaxed galaxy clusters Abdalla et al
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 283273 12 pageshttpdxdoiorg1011552015283273
2 Advances in High Energy Physics
[39 40] have found the signature of interaction between darkenergy and dark matter An interacting scenario providessolution to a number of cosmological problems in bothcanonical scalar field models [41] and nonminimally coupledscalar field models such as Brans-Dick field [42 43]
In this paper we consider generalized tachyon field asresponsible for dark energy in the framework of teleparallelgravity We will be interested in performing a dynamicalanalysis of such a model in FRW space time In such astudy we investigate our model for both interacting andnoninteracting cases We consider the nonminimal couplingfunction of the form 119891(120601) prop 120601
2 and choose 119899 = 2 in (2)The basic equations are presented in Section 2 In Section 3the evolution equations are translated in the language of theautonomous dynamical system by suitable transformation ofthe basic variables Section 31 deals with phase-space analysisas well as the cosmological implications of the equilibriumpoints of the model in noninteracting dark energy darkmatter case In Section 32 an interaction between darkenergy and dark matter has been considered and criticalpoints and their behavior have been extracted Section 4 isdevoted to a short summary of our results
2 Basic Equations
Our model is described by the following action as a general-ization of tachyon teleparallel dark energy model [32]
119878 = int1198894119909119890 [L
119879+L120593+L119898]
L119879=
119879
21205812
(1)
L120593= 120585119891 (120593) 119879 minus 119881 (120593) (1 minus 2119883)
119899 (2)
where 119890 = det(119890119894120583) = radicminus119892 (119890119894
120583are the orthonormal comp-
onents of the tetrad) while11987921205812 is the Lagrangian of telepar-allelism with 119879 as the torsion scalar (for an introductoryreview of teleparallelism see [12]) L
120593shows nonminimal
coupling of generalized tachyon field 120593 with gravity in theframework of teleparallel gravity and 119883 = (12)120597
120583120593120597120583120593 The
second part in L120593is the Lagrangian density of the general-
ized tachyon field which has been studied in [44 45] 119891(120593)is the nonminimal coupling function 120585 is a dimensionlessconstant measuring the nonminimal coupling and L
119898is
the matter Lagrangian For 119899 = 12 our model reduced totachyonic teleparallel dark energy discussed in [32] Here weconsider the case 119899 = 2 for two reasons The first is that forarbitrary 119899 our equations will be very complicated and onecannot solve themanalytically and the second is that for 119899 = 2wewill obtain interesting physical results as we will see below
Furthermore due to complexity of tachyon dynamics[46] has proposed an approach based on the redefinition ofthe tachyon field as follows
120593 997888rarr 120601 = int119889120593radic119881 (120593) lArrrArr 120597120593 =120597120601
radic119881 (120601)
(3)
In order to obtain a closed autonomous system and performthe phase-space analysis of the model we apply (3) in (2) for119899 = 2 that leads to the following action
119878 = int1198894119909119890 [
119879
21205812+ 120585119891 (120601) 119879 minus 119881 (120601)(1 minus
2119883
119881 (120601))
2
+L119898]
(4)
In a spatially flat FRW space-time
1198891199042= 1198891199052minus 1198862(119905) (119889119903
2+ 1199032119889Ω2) (5)
and a vierbein choice of the form 119890119894
120583= diag(1 119886 119886 119886) the
corresponding Friedmann equations are given by
1198672=1
3(120588120601+ 120588119898)
= minus1
2(120588120601+ 119875120601+ 120588119898+ 119875119898)
(6)
where 119867 = 119886119886 is the Hubble parameter and a dot standsfor the derivative with respect to the cosmic time 119905 In theseequations 120588
119898and 119875
119898are the matter energy density and
pressure respectivelyThe effective energy density and pressure of generalized
tachyon dark energy read
120588120601= 119881 (120601) + 2 120601
2minus 3
1206014
119881 (120601)minus 6120585119867
2119891 (120601)
119875120601= minus119881 (120601) + 2120585 (3119867
2+ 2) 119891 (120601) + 10120585119867119891
120601120601
+ 1206012(2 minus
1206012
119881 (120601))
(7)
where 119891120601= 119889119891119889120601
The equation of motion of the scalar field can be obtainedby variation of the action (4) with respect to 120601
120601 + 3120583minus2]2119867 120601 +
1
4]2 (1 +
3 1206014
1198812 (120601))119881120601
+ 6120585]21198672119891120601= minus
]21198764 120601
(8)
with 119876 a general interaction coupling term between darkenergy and dark matter 120583 = 1radic1 minus 2119883119881 and ] =
1radic1 minus 6119883119881 In (7) and (8) we have used the useful relation
119879 = minus61198672 (9)
which simply arises from the calculation of torsion scalar forthe FRW metric (5) The scalar field evolution (8) expressesthe continuity equation for the field and matter as follows
120588120601+ 3119867(1 + 120596
120601) 120588120601= minus119876
120588119898+ 3119867 (1 + 120596
119898) 120588119898= 119876
(10)
Advances in High Energy Physics 3
Table 1 Location of the critical points and the corresponding values of the dark energy density parameter Ω120601and equation of state 120596
120601and
the condition required for an accelerating universe for119876 = 0 Here 1205821= 120572(4120572120585+radic1612057221205852 minus 21205822120585)120582
2 and 1205822= 120572(4120572120585minusradic1612057221205852 minus 21205822120585)120582
2
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198601
0 2radic12058211205821205821
212057212058541205821(1 minus
1205822
812058512057221205821)
1612058521205724
(12058221205822
1+ 21205851205722) (1205822120582
1minus 81205851205722)
1205822gt1205851205722(41205821minus 1)
1205822
1
(1 + radic1 minus321205821
(1 minus 41205821)2)
or
1205822lt1205851205722(41205821minus 1)
1205822
1
(1 minus radic1 minus321205821
(1 minus 41205821)2)
1198602
0 2radic12058221205821205822
212057212058541205822(1 minus
1205822
812058512057221205822)
1612058521205724
(12058221205822
2+ 21205851205722) (1205822120582
2minus 81205851205722)
1205822gt1205851205722(41205822minus 1)
1205822
2
(1 + radic1 minus321205822
(1 minus 41205822)2)
or
1205822lt1205851205722(41205822minus 1)
1205822
2
(1 minus radic1 minus321205822
(1 minus 41205822)2)
Table 2 Stability and existence conditions of the critical points ofthe model for 119876 = 0
Label Stability Existence
1198601
Saddle pointif 120572 gt 0 and 120585120582 gt 0 and
stable pointif 120585 gt 0 120572 lt 0 and 120582 lt 0
For all 120585 lt 0or
120585 ge 120582281205722
1198602
Saddle pointif 120572 lt 0 and 120585120582 lt 0 and
stable pointif 120585 gt 0 120572 gt 0 and 120582 gt 0
For all 120585 lt 0or
120585 ge 120582281205722
where 120596120601= 119875120601120588120601is the equation of state parameter of dark
energy which is attributed to the scalar field 120601The barotropicindex is defined by 120574 equiv 1 + 120596
119898with 0 lt 120574 lt 2
Although dynamics of tachyonic teleparallel dark energyhas been studied in [33] no scaling attractors has been foundHere we are going to perform a phase-space analysis ofgeneralized tachyonic teleparallel dark energy and as we willsee below that some interesting scaling attractors appear insuch theory
3 Cosmological Dynamics
In order to perform phase-space and stability analysis of themodel we introduce the following auxiliary variables
119909 equiv
120601
radic119881 119910 equiv
radic119881
radic3119867 119906 equiv radic119891 (11)
The auxiliary variables allow us to straightforwardly obtainthe density parameter of dark energy and dark matter
Ω120601equiv
120588120601
31198672= 120583minus21199102(1 + 3119909
2) minus 2120585119906
2 (12)
Ω119898equiv120588119898
31198672= 1 minus Ω
120601 (13)
while the equation of state of the field reads
120596120601equiv
119875120601
120588120601
=
minus120583minus41199102+ 2120585119906 [(5radic33) 120572119909119910 + 119906 (1 minus (23) 119904)]
120583minus21199102 (1 + 31199092) minus 21205851199062
(14)
where 120572 equiv 119891120601radic119891 and
119904 = minus
1198672= (2120585119906
2+ 1)minus1
sdot [5radic3120572120585119906119909119910 + 6120583minus211990921199102
minus3
2120574120583minus21199102(1 + 3119909
2)] +
3120574
2
(15)
From now we concentrate on exponential scalar field poten-tial of the form 119881 = 119881
0119890minus119896120582120601 and the nonminimal coupling
function of the form 119891(120601) prop 1206012 These choices lead to
constant 120582 and 120572 respectivelyAnother quantities with great physical significance
namely the total equation of state parameter and the decel-eration parameter are given by
120596tot equiv119875120601+ 119875119898
120588120601+ 120588119898
= 120583minus21199102(41199092minus 120574 (1 + 3119909
2))
+ 2120585119906 [5radic3
3120572119909119910 + 119906(120574 minus
2
3119904)]
+ 120574 minus 1
(16)
119902 equiv minus1 minus
1198672=1
2+3
2120596tot
=3
2120583minus21199102(41199092minus 120574 (1 + 3119909
2))
+ 120585119906 [5radic3120572119909119910 + 119906 (3120574 minus 2119904)] +3120574
2minus 1
(17)
4 Advances in High Energy Physics
110
109
108
107
106
105
104
103
102
101
y(t)
0 005 010 015 020
x(t)
(a)
0 005 010 015 020
x(t)
003
002
004
005
006
007
008
u(t)
(b)
110
109
108
107
106
105
104
103
102
101
y(t)
003002 004 005 006 007 008
u(t)
(c)
Figure 1 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 and 120572 = minus2for 119876 = 0 For these values of the parameters point 119860
1is a stable attractor of the model
Using auxiliary variables (11) the evolution equations (6)and (8) can be recast as a dynamical system of ordinarydifferential equations
1199091015840=radic3
2[1205821199092119910 +
1
2120582]2 (1 + 31199094) 119910
minus 4120572120585]2119906119910minus1 minus 2radic3120583minus2]2119909] minus 119876
1199101015840= (minus
radic3
2120582119909119910 + 119904)119910
1199061015840=radic3120572119909119910
2
(18)
where 119876 = 119876 120601119867radic119881(120601) 120582 equiv minus119881120601119896119881 and prime in (18)
denotes differentiation with respect to the so-called e-foldingtime119873 = ln 119886
Advances in High Energy Physics 5
110
109
108
107
106
105
104
103
102
101
y(t)
minus005 0 005 010 015 020
x(t)
(a)
007
009
008
011
010
013
012
015
014
u(t)
minus005 0 005 010 015 020
x(t)
(b)
110
109
108
107
106
105
104
103
102
101
y(t)
007 008 010 012 014009 011 013 015
u(t)
(c)
Figure 2 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 and 120572 = 2 for119876 = 0 For these values of the parameters point 119860
2is a stable attractor of the model
The next step is the introduction of interaction term 119876
to obtain an autonomous system out of (18) The fixed points(119909119888 119910119888 119906119888) for which 1199091015840 = 1199101015840 = 1199061015840 = 0 depend on the choice
of the interaction term 119876 and two general possibilities willbe treated in the sequel The stability of the system at a fixedpoint can be obtained from the analysis of the determinantand trace of the perturbation matrix 119872 Such a matrix canbe constructed by substituting linear perturbations 119909 rarr
119909119888+ 120575119909 119910 rarr 119910
119888+ 120575119910 and 119906 rarr 119906
119888+ 120575119906 about the
critical point (119909119888 119910119888 119906119888) into the autonomous system (18)The
3 times 3 matrix119872 of the linearized perturbation equations ofthe autonomous system is shown in the appendix Thereforefor each critical point we examine the sign of the real partof the eigenvalues of 119872 According to the usual dynamicalsystem analysis if the eigenvalues are real and have oppositesigns the corresponding critical point is a saddle point Afixed point is unstable if the eigenvalues are positive and itis stable for negative real part of the eigenvalues
In the following subsections we will study the dynamicsof generalized tachyon field with different interaction term
6 Advances in High Energy Physics
109
107105
103
101 y(t)
x(t)
u(t)
008
007
006
005
004
003
002
0005
010015
020
(a)
110
107105 y
(t)
u(t)
minus005 0 005 010 015 020
x(t)
015
014
013
012
011
010
009
008
(b)
Figure 3 Three-dimensional phase-space trajectories of the model for 119876 = 0 with stable attractors 1198601(a) and 119860
2(b) The values of the
parameters are those mentioned in Figures 1 and 2 respectively
119876 Although one can work with a general 120574 (120574 = 1 and 43correspond to dust matter and radiation resp) without lossof generality we assume 120574 = 1 for simplicity
31 The Case for 119876 = 0 The first case 119876 = 0 clearlymeans that there is no interaction between dark energy andbackground matter In this case there are two critical pointspresented in Table 1 From (12) and (14) one can obtain thecorresponding values of density parameter Ω
120601and equation
of state of dark energy120596120601at each point Also using (17)we can
find the condition required for acceleration (119902 lt 0) at eachpoint These parameters and conditions have been shownin Table 1 The stability and existence conditions of criticalpoints 119860
1and 119860
2are presented in Table 2 We mention that
the corresponding eigenvalues of perturbation matrix 119872 atcritical points 119860
1and 119860
2are considerably involved and here
we do not present their explicit expressions but we can findsign of them numerically
Critical Point 1198601 This critical point is a scaling attractor if
120585 gt 0 120572 lt 0 and 120582 lt 0 Therefore cosmological coincidenceproblem can be alleviated at this point 119860
1is a saddle point
for 120572 gt 0 and 120585120582 gt 0
Critical Point 1198602 1198602can also be a scaling attractor of the
model or a saddle point under the same conditions as for1198601
In Figure 1 we have chosen the values of the parameters120585 120582 and 120572 such that 119860
1became a stable attractor of the
model Plots in Figure 1 show the phase-space trajectories on119909-119910 119909-119906 and 119906-119910 planes from (a) to (c) respectively Thesame plots are shown in Figure 2 for critical point 119860
2 Note
that the values of the parameter have been chosen in the waythat 119860
2became a stable point of the model In Figure 3 the
corresponding 3-dimensional phase-space trajectories of themodel have been presented One can see that 119860
1and 119860
2are
stable attractors of themodel in (a) and (b) plots respectively
32The Case for119876 = 120573120581120588119898120601 This deals with the most famil-
iar interaction term extensively considered in the literature(see eg [31 47ndash51]) Here 119876 in terms of auxiliary variablesis 119876 = radic3120573119910minus1Ω
119898 Inserting such an interaction term in (18)
and setting the left hand sides of the equations to zero leadto the critical points 119861
1 1198612 1198613 and 119861
4presented in Table 3
In the same table we have provided the corresponding valuesofΩ120601and 120596
120601as well as the condition needed for accelerating
universe at each fixed pointsThe stability and existence conditions for each point are
presented in Table 4 Since the corresponding eigenvalues ofthe fixed points are complicated we do not give them herebut one can obtain their signs numerically and then examinethe stability properties of the critical points
Critical Point 1198611This point exists for 120582 gt 0 and 120585 ge 120582281205722
However it is an unstable saddle point
Critical Point 1198612The critical point 119861
2exists for 120582 gt 0 and 120585 lt
0 or 120585 ge 120582281205722 This point is a scaling attractor of the modelif 120572 gt 0 and 120585 gt 0 Figure 4 shows clearly such a behavior ofthe model for suitable choices of 120585 120582 and 120572
Critical Point 1198613This point exists for negative values of 120582 and
120585 or when 120585 ge 120582281205722 Also it is a stable point if 120572 lt 0
and 120585 gt 0 and a saddle point if 120572 gt 0 and 120585 lt 0 Thevalues of parameters have been chosen in Figure 5 such that
Advances in High Energy Physics 7
110
109
108
107
106
105
104
103
102
101
y(t)
0 01 02
x(t)
minus01
(a)
0 01 02
x(t)
minus01
024
022
020
018
016
014
012
010
008
u(t)
(b)
110107105103101 102 104 106 108 109
y(t)
024
022
020
018
016
014
012
010
008
u(t)
(c)
Figure 4 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 120572 = 2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
2is a stable attractor of the model
1198613became a attractor of the model as it is clear from phase-
space trajectories
Critical Point 1198614Thepoint119861
4exists for120582 lt 0 and 120585 ge 120582281205722
It is a stable point if 120572 lt 0 and 120585 gt 0 In Figure 6 values of theparameters 120585 and 120572 are those satisfying these constraints andso 1198614becomes a attractor point for phase-plane trajectories
The corresponding 3-dimensional phase-space trajectories ofthe model for attractor points 119861
2(a) 119861
3(b) and 119861
4(c) are
plotted in Figure 7
4 Conclusion
A model of dark energy with nonminimal coupling ofquintessence scalar field with gravity in the framework ofteleparallel gravity was called teleparallel dark energy [27] Ifone replaces quintessence by tachyon field in such a modelthen tachyonic teleparallel dark energy will be constructed[32]
Moreover although dark energy and dark matter scaledifferently with the expansion of our universe according to
8 Advances in High Energy Physics
110
108
106
104
102
100
y(t)
minus005 0 005 010 015 020
x(t)
(a)u(t)
minus005 0 005 010 015 020
x(t)
minus002
minus004
008
006
004
002
0
(b)
110108106104102100
y(t)
u(t)
minus002
minus004
008
006
004
002
0
(c)
Figure 5 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
3is a stable attractor of the model
the observations [52ndash54] we are living in an epoch in whichdark energy and dark matter densities are comparable andthis is the well-known cosmological coincidence problem[30] This problem can be alleviated in most dark energymodels via the method of scaling solutions in which thedensity parameters of dark energy and dark matter are bothnonvanishing over there
In this paper we investigated the phase-space analysisof generalized tachyon cosmology in the framework ofteleparallel gravity Our model was described by action (2)
which generalizes tachyonic teleparallel dark energy modelproposed in [32] We found some scaling attractors in ourmodel for the case 119899 = 2 and the coupling function119891(120601) prop 120601
2 These scaling attractors are 1198601and 119860
2when
there is no interaction between dark energy and dark matter1198612 1198613 and 119861
4are scaling attractors in the case that dark
energy interacts with dark matter through the interactingterm 119876 = 120573120581120588
119898120601 As it is shown in [29] original teleparallel
dark energy model also has a late-time attractor in whichdark energy behaves like a cosmological constant and so
Advances in High Energy Physics 9
y(t)
x(t)
323
322
321
320
319
318
317
minus0004 minus0002 0 000600040002
(a)u(t)
x(t)
3025
3020
3015
3010
3005
minus0001 0001 0003 0005 0007
(b)
y(t)
u(t)
3025
3020
3015
3010
3170 3175 3180 3185 3190
3005
(c)
Figure 6 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus05and 120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
4is a stable attractor of the model
it provides a natural way for the stabilization of the darkenergy equation of state to the cosmological constant valuewithout the need for parameter tuning Also the authors in[55] by using power-law potentials and nonminimal couplingfunctions have shown that teleparallel gravity has no stablefuture solutions for positive nonminimal coupling and thescalar field results always in oscillationsOur results show thatgeneralized tachyon field represents interesting cosmologicalbehavior in comparison with ordinary tachyon fields in the
framework of teleparallel gravity because there is no scalingattractor in the latter model Note that however in [33]Otalora has considered a dynamically changing coefficient120572 = 119891
120601radic119891 and found a field matter dominated solution
in which it has some portions of the dark energy density inthe matter dominated era So generalized tachyon field givesus the hope that cosmological coincidence problem can bealleviated without fine-tunings One can study our model fordifferent kinds of potential and other famous interaction termbetween dark energy and dark matter
10 Advances in High Energy Physics
u(t)
minus01 0 01 02
x(t)
110
107
105104103
101
y(t)
024
022
020
018
016
014
012
010
008
(a)u(t)
minus002
minus004
008
006
004
002
0
minus005 0 005 010 015 020
x(t)
110
108
106
104
102
100
y(t)
(b)
y(t)
3170
3175
31803185
3190
minus00020
00020004
0006
x(t)
u(t)
3025
3020
3015
3010
3005
(c)
Figure 7Three-dimensional phase-space trajectories of the model for119876 = 120573120581120588119898120601with stable attractors 119861
2(a) 119861
3(b) and 119861
4(c)The values
of the parameters are those mentioned in Figures 4 5 and 6 respectively
Table 3 Location of the critical points and the corresponding values of the dark energy density parameterΩ120601and equation of state120596
120601and the
condition required for an accelerating universe for119876 = 120573120581120588119898120601 Here 120579
1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579
2= (4120572120585minusradic1612057221205852 minus 21205822120585)2120582120585
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198611
02radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198612
02radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572(120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
1198613
0 minus2radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198614
0 minus2radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572 (120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
Advances in High Energy Physics 11
Table 4 Stability and existence conditions of the critical points ofthe model for 119876 = 120573120581120588
119898120601
Label Stability Existence
1198611
Saddle pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 and 120585 ge 120582281205722
1198612
Saddle pointif 120572 lt 0 and 120585 lt 0 and
stable pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198613
Saddle pointif 120572 gt 0 and 120585 lt 0 and
stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198614
Stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 and 120585 ge 120582281205722
Appendix
Perturbation Matrix Elements
The elements of 3 times 3 matrix 119872 of the linearized per-turbation equations for the real and physically meaningfulcritical points (119909
119888 119910119888 119906119888) of the autonomous system (18) read
as follows
11987211= 3]2119888(radic3
2120582119909119888119910119888(21199092
119888+ ]2119888(1 + 3119909
4
119888)) minus 6120583
minus2
119888]21198881199092
119888
minus 4radic3120572120585119906119888119909119888]2119888119910minus1
119888) + radic3120582119909
119888119910119888minus 3 +M
11
11987212=radic3
4(120582 (2119909
2
119888+ ]2119888(1 + 3119909
4
119888)) + 8120572120585119906
119888]2119888119910minus2
119888) +M
12
11987213= minus2radic3120572120585]2
119888119910minus1
119888+M13
11987221=
21199102
119888(radic3120572120585119906
119888+ 3119909119888119910119888]minus2119888)
(21205851199062119888+ 1)
minusradic3120582119910
2
119888
2
11987222=
2119910119888(minus (94) 120583
minus4
119888119910119888+ 5radic3120572120585119909
119888119906119888)
(21205851199062119888+ 1)
minus radic3120582119909119888119910119888+3
2
11987223=
61205851199061198881199102
119888(minus (10radic33) 120572120585119906
119888119909119888+ 120583minus4
1198881199102
119888)
(21205851199062119888+ 1)2
+5radic3120572120585119909
1198881199102
119888
21205851199062119888+ 1
11987231=radic3120572119910
119888
2
11987232=radic3120572119909
119888
2
11987233= 0
(A1)
where in the case of 119876 = 0 we have M11= M12= M13= 0
and in the case of 119876 = 120573120581120588119898120601 we have
M11= 4radic3120573]minus2
119888119909119888119910119888
M12= 2radic3120573120583
minus2
119888(1 + 3119909
2
119888)
M13= minus4radic3120573120585119906
119888119910minus1
119888
(A2)
Examining the eigenvalues of the matrix119872 for each criticalpoint one determines its stability conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989
[2] S M Carroll ldquoThe cosmological constantrdquo Living Reviews inRelativity vol 4 article 1 2001
[3] R-J Yang and S N Zhang ldquoThe age problem in the ΛCDMmodelrdquo Monthly Notices of the Royal Astronomical Society vol407 no 3 pp 1835ndash1841 2010
[4] Y-F Cai E N Saridakis M R Setare and J-Q Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010
[5] 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
[6] A Sen ldquoRemarks on tachyon driven cosmologyrdquo PhysicaScripta vol 2005 no 117 article 70 2005
[7] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002
[8] M R Garousi ldquoTachyon couplings on non-BPS D-branes andDiracndashBornndashInfeld actionrdquo Nuclear Physics B vol 584 no 1-2pp 284ndash299 2000
[9] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002
[10] A Unzicker and T Case ldquoTranslation of Einsteinrsquos attempt ofa unified field theory with teleparallelismrdquo httparxivorgabsphysics0503046
[11] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 pp 3524ndash3553 1979 Addendum in PhysicalReview D vol 24 pp 3312-3314 1982
[12] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction Springer Dordrecht Netherlands 2013
[13] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013
[14] R Ferraro andF Fiorini ldquoModified teleparallel gravity inflationwithout an inflatonrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 75 no 8 Article ID 084031 2007
[15] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D Particles Fields Gravitation andCosmology vol 79 Article ID 124019 2009
[16] E V Linder ldquoEinsteins other gravity and the acceleration of theUniverserdquo Physical Review D vol 81 Article ID 127301 2010
12 Advances in High Energy Physics
[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010
[18] 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
[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984
[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999
[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000
[22] 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 Article ID 0435392004
[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009
[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007
[25] 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
[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010
[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011
[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012
[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012
[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012
[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013
[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012
[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013
[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014
[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007
[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence
principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009
[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007
[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008
[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009
[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010
[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005
[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006
[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008
[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008
[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012
[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010
[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998
[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999
[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005
[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011
[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003
[52] 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
[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008
[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011
[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
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Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
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AerodynamicsJournal of
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PhotonicsJournal of
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Journal of
Biophysics
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ThermodynamicsJournal of
2 Advances in High Energy Physics
[39 40] have found the signature of interaction between darkenergy and dark matter An interacting scenario providessolution to a number of cosmological problems in bothcanonical scalar field models [41] and nonminimally coupledscalar field models such as Brans-Dick field [42 43]
In this paper we consider generalized tachyon field asresponsible for dark energy in the framework of teleparallelgravity We will be interested in performing a dynamicalanalysis of such a model in FRW space time In such astudy we investigate our model for both interacting andnoninteracting cases We consider the nonminimal couplingfunction of the form 119891(120601) prop 120601
2 and choose 119899 = 2 in (2)The basic equations are presented in Section 2 In Section 3the evolution equations are translated in the language of theautonomous dynamical system by suitable transformation ofthe basic variables Section 31 deals with phase-space analysisas well as the cosmological implications of the equilibriumpoints of the model in noninteracting dark energy darkmatter case In Section 32 an interaction between darkenergy and dark matter has been considered and criticalpoints and their behavior have been extracted Section 4 isdevoted to a short summary of our results
2 Basic Equations
Our model is described by the following action as a general-ization of tachyon teleparallel dark energy model [32]
119878 = int1198894119909119890 [L
119879+L120593+L119898]
L119879=
119879
21205812
(1)
L120593= 120585119891 (120593) 119879 minus 119881 (120593) (1 minus 2119883)
119899 (2)
where 119890 = det(119890119894120583) = radicminus119892 (119890119894
120583are the orthonormal comp-
onents of the tetrad) while11987921205812 is the Lagrangian of telepar-allelism with 119879 as the torsion scalar (for an introductoryreview of teleparallelism see [12]) L
120593shows nonminimal
coupling of generalized tachyon field 120593 with gravity in theframework of teleparallel gravity and 119883 = (12)120597
120583120593120597120583120593 The
second part in L120593is the Lagrangian density of the general-
ized tachyon field which has been studied in [44 45] 119891(120593)is the nonminimal coupling function 120585 is a dimensionlessconstant measuring the nonminimal coupling and L
119898is
the matter Lagrangian For 119899 = 12 our model reduced totachyonic teleparallel dark energy discussed in [32] Here weconsider the case 119899 = 2 for two reasons The first is that forarbitrary 119899 our equations will be very complicated and onecannot solve themanalytically and the second is that for 119899 = 2wewill obtain interesting physical results as we will see below
Furthermore due to complexity of tachyon dynamics[46] has proposed an approach based on the redefinition ofthe tachyon field as follows
120593 997888rarr 120601 = int119889120593radic119881 (120593) lArrrArr 120597120593 =120597120601
radic119881 (120601)
(3)
In order to obtain a closed autonomous system and performthe phase-space analysis of the model we apply (3) in (2) for119899 = 2 that leads to the following action
119878 = int1198894119909119890 [
119879
21205812+ 120585119891 (120601) 119879 minus 119881 (120601)(1 minus
2119883
119881 (120601))
2
+L119898]
(4)
In a spatially flat FRW space-time
1198891199042= 1198891199052minus 1198862(119905) (119889119903
2+ 1199032119889Ω2) (5)
and a vierbein choice of the form 119890119894
120583= diag(1 119886 119886 119886) the
corresponding Friedmann equations are given by
1198672=1
3(120588120601+ 120588119898)
= minus1
2(120588120601+ 119875120601+ 120588119898+ 119875119898)
(6)
where 119867 = 119886119886 is the Hubble parameter and a dot standsfor the derivative with respect to the cosmic time 119905 In theseequations 120588
119898and 119875
119898are the matter energy density and
pressure respectivelyThe effective energy density and pressure of generalized
tachyon dark energy read
120588120601= 119881 (120601) + 2 120601
2minus 3
1206014
119881 (120601)minus 6120585119867
2119891 (120601)
119875120601= minus119881 (120601) + 2120585 (3119867
2+ 2) 119891 (120601) + 10120585119867119891
120601120601
+ 1206012(2 minus
1206012
119881 (120601))
(7)
where 119891120601= 119889119891119889120601
The equation of motion of the scalar field can be obtainedby variation of the action (4) with respect to 120601
120601 + 3120583minus2]2119867 120601 +
1
4]2 (1 +
3 1206014
1198812 (120601))119881120601
+ 6120585]21198672119891120601= minus
]21198764 120601
(8)
with 119876 a general interaction coupling term between darkenergy and dark matter 120583 = 1radic1 minus 2119883119881 and ] =
1radic1 minus 6119883119881 In (7) and (8) we have used the useful relation
119879 = minus61198672 (9)
which simply arises from the calculation of torsion scalar forthe FRW metric (5) The scalar field evolution (8) expressesthe continuity equation for the field and matter as follows
120588120601+ 3119867(1 + 120596
120601) 120588120601= minus119876
120588119898+ 3119867 (1 + 120596
119898) 120588119898= 119876
(10)
Advances in High Energy Physics 3
Table 1 Location of the critical points and the corresponding values of the dark energy density parameter Ω120601and equation of state 120596
120601and
the condition required for an accelerating universe for119876 = 0 Here 1205821= 120572(4120572120585+radic1612057221205852 minus 21205822120585)120582
2 and 1205822= 120572(4120572120585minusradic1612057221205852 minus 21205822120585)120582
2
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198601
0 2radic12058211205821205821
212057212058541205821(1 minus
1205822
812058512057221205821)
1612058521205724
(12058221205822
1+ 21205851205722) (1205822120582
1minus 81205851205722)
1205822gt1205851205722(41205821minus 1)
1205822
1
(1 + radic1 minus321205821
(1 minus 41205821)2)
or
1205822lt1205851205722(41205821minus 1)
1205822
1
(1 minus radic1 minus321205821
(1 minus 41205821)2)
1198602
0 2radic12058221205821205822
212057212058541205822(1 minus
1205822
812058512057221205822)
1612058521205724
(12058221205822
2+ 21205851205722) (1205822120582
2minus 81205851205722)
1205822gt1205851205722(41205822minus 1)
1205822
2
(1 + radic1 minus321205822
(1 minus 41205822)2)
or
1205822lt1205851205722(41205822minus 1)
1205822
2
(1 minus radic1 minus321205822
(1 minus 41205822)2)
Table 2 Stability and existence conditions of the critical points ofthe model for 119876 = 0
Label Stability Existence
1198601
Saddle pointif 120572 gt 0 and 120585120582 gt 0 and
stable pointif 120585 gt 0 120572 lt 0 and 120582 lt 0
For all 120585 lt 0or
120585 ge 120582281205722
1198602
Saddle pointif 120572 lt 0 and 120585120582 lt 0 and
stable pointif 120585 gt 0 120572 gt 0 and 120582 gt 0
For all 120585 lt 0or
120585 ge 120582281205722
where 120596120601= 119875120601120588120601is the equation of state parameter of dark
energy which is attributed to the scalar field 120601The barotropicindex is defined by 120574 equiv 1 + 120596
119898with 0 lt 120574 lt 2
Although dynamics of tachyonic teleparallel dark energyhas been studied in [33] no scaling attractors has been foundHere we are going to perform a phase-space analysis ofgeneralized tachyonic teleparallel dark energy and as we willsee below that some interesting scaling attractors appear insuch theory
3 Cosmological Dynamics
In order to perform phase-space and stability analysis of themodel we introduce the following auxiliary variables
119909 equiv
120601
radic119881 119910 equiv
radic119881
radic3119867 119906 equiv radic119891 (11)
The auxiliary variables allow us to straightforwardly obtainthe density parameter of dark energy and dark matter
Ω120601equiv
120588120601
31198672= 120583minus21199102(1 + 3119909
2) minus 2120585119906
2 (12)
Ω119898equiv120588119898
31198672= 1 minus Ω
120601 (13)
while the equation of state of the field reads
120596120601equiv
119875120601
120588120601
=
minus120583minus41199102+ 2120585119906 [(5radic33) 120572119909119910 + 119906 (1 minus (23) 119904)]
120583minus21199102 (1 + 31199092) minus 21205851199062
(14)
where 120572 equiv 119891120601radic119891 and
119904 = minus
1198672= (2120585119906
2+ 1)minus1
sdot [5radic3120572120585119906119909119910 + 6120583minus211990921199102
minus3
2120574120583minus21199102(1 + 3119909
2)] +
3120574
2
(15)
From now we concentrate on exponential scalar field poten-tial of the form 119881 = 119881
0119890minus119896120582120601 and the nonminimal coupling
function of the form 119891(120601) prop 1206012 These choices lead to
constant 120582 and 120572 respectivelyAnother quantities with great physical significance
namely the total equation of state parameter and the decel-eration parameter are given by
120596tot equiv119875120601+ 119875119898
120588120601+ 120588119898
= 120583minus21199102(41199092minus 120574 (1 + 3119909
2))
+ 2120585119906 [5radic3
3120572119909119910 + 119906(120574 minus
2
3119904)]
+ 120574 minus 1
(16)
119902 equiv minus1 minus
1198672=1
2+3
2120596tot
=3
2120583minus21199102(41199092minus 120574 (1 + 3119909
2))
+ 120585119906 [5radic3120572119909119910 + 119906 (3120574 minus 2119904)] +3120574
2minus 1
(17)
4 Advances in High Energy Physics
110
109
108
107
106
105
104
103
102
101
y(t)
0 005 010 015 020
x(t)
(a)
0 005 010 015 020
x(t)
003
002
004
005
006
007
008
u(t)
(b)
110
109
108
107
106
105
104
103
102
101
y(t)
003002 004 005 006 007 008
u(t)
(c)
Figure 1 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 and 120572 = minus2for 119876 = 0 For these values of the parameters point 119860
1is a stable attractor of the model
Using auxiliary variables (11) the evolution equations (6)and (8) can be recast as a dynamical system of ordinarydifferential equations
1199091015840=radic3
2[1205821199092119910 +
1
2120582]2 (1 + 31199094) 119910
minus 4120572120585]2119906119910minus1 minus 2radic3120583minus2]2119909] minus 119876
1199101015840= (minus
radic3
2120582119909119910 + 119904)119910
1199061015840=radic3120572119909119910
2
(18)
where 119876 = 119876 120601119867radic119881(120601) 120582 equiv minus119881120601119896119881 and prime in (18)
denotes differentiation with respect to the so-called e-foldingtime119873 = ln 119886
Advances in High Energy Physics 5
110
109
108
107
106
105
104
103
102
101
y(t)
minus005 0 005 010 015 020
x(t)
(a)
007
009
008
011
010
013
012
015
014
u(t)
minus005 0 005 010 015 020
x(t)
(b)
110
109
108
107
106
105
104
103
102
101
y(t)
007 008 010 012 014009 011 013 015
u(t)
(c)
Figure 2 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 and 120572 = 2 for119876 = 0 For these values of the parameters point 119860
2is a stable attractor of the model
The next step is the introduction of interaction term 119876
to obtain an autonomous system out of (18) The fixed points(119909119888 119910119888 119906119888) for which 1199091015840 = 1199101015840 = 1199061015840 = 0 depend on the choice
of the interaction term 119876 and two general possibilities willbe treated in the sequel The stability of the system at a fixedpoint can be obtained from the analysis of the determinantand trace of the perturbation matrix 119872 Such a matrix canbe constructed by substituting linear perturbations 119909 rarr
119909119888+ 120575119909 119910 rarr 119910
119888+ 120575119910 and 119906 rarr 119906
119888+ 120575119906 about the
critical point (119909119888 119910119888 119906119888) into the autonomous system (18)The
3 times 3 matrix119872 of the linearized perturbation equations ofthe autonomous system is shown in the appendix Thereforefor each critical point we examine the sign of the real partof the eigenvalues of 119872 According to the usual dynamicalsystem analysis if the eigenvalues are real and have oppositesigns the corresponding critical point is a saddle point Afixed point is unstable if the eigenvalues are positive and itis stable for negative real part of the eigenvalues
In the following subsections we will study the dynamicsof generalized tachyon field with different interaction term
6 Advances in High Energy Physics
109
107105
103
101 y(t)
x(t)
u(t)
008
007
006
005
004
003
002
0005
010015
020
(a)
110
107105 y
(t)
u(t)
minus005 0 005 010 015 020
x(t)
015
014
013
012
011
010
009
008
(b)
Figure 3 Three-dimensional phase-space trajectories of the model for 119876 = 0 with stable attractors 1198601(a) and 119860
2(b) The values of the
parameters are those mentioned in Figures 1 and 2 respectively
119876 Although one can work with a general 120574 (120574 = 1 and 43correspond to dust matter and radiation resp) without lossof generality we assume 120574 = 1 for simplicity
31 The Case for 119876 = 0 The first case 119876 = 0 clearlymeans that there is no interaction between dark energy andbackground matter In this case there are two critical pointspresented in Table 1 From (12) and (14) one can obtain thecorresponding values of density parameter Ω
120601and equation
of state of dark energy120596120601at each point Also using (17)we can
find the condition required for acceleration (119902 lt 0) at eachpoint These parameters and conditions have been shownin Table 1 The stability and existence conditions of criticalpoints 119860
1and 119860
2are presented in Table 2 We mention that
the corresponding eigenvalues of perturbation matrix 119872 atcritical points 119860
1and 119860
2are considerably involved and here
we do not present their explicit expressions but we can findsign of them numerically
Critical Point 1198601 This critical point is a scaling attractor if
120585 gt 0 120572 lt 0 and 120582 lt 0 Therefore cosmological coincidenceproblem can be alleviated at this point 119860
1is a saddle point
for 120572 gt 0 and 120585120582 gt 0
Critical Point 1198602 1198602can also be a scaling attractor of the
model or a saddle point under the same conditions as for1198601
In Figure 1 we have chosen the values of the parameters120585 120582 and 120572 such that 119860
1became a stable attractor of the
model Plots in Figure 1 show the phase-space trajectories on119909-119910 119909-119906 and 119906-119910 planes from (a) to (c) respectively Thesame plots are shown in Figure 2 for critical point 119860
2 Note
that the values of the parameter have been chosen in the waythat 119860
2became a stable point of the model In Figure 3 the
corresponding 3-dimensional phase-space trajectories of themodel have been presented One can see that 119860
1and 119860
2are
stable attractors of themodel in (a) and (b) plots respectively
32The Case for119876 = 120573120581120588119898120601 This deals with the most famil-
iar interaction term extensively considered in the literature(see eg [31 47ndash51]) Here 119876 in terms of auxiliary variablesis 119876 = radic3120573119910minus1Ω
119898 Inserting such an interaction term in (18)
and setting the left hand sides of the equations to zero leadto the critical points 119861
1 1198612 1198613 and 119861
4presented in Table 3
In the same table we have provided the corresponding valuesofΩ120601and 120596
120601as well as the condition needed for accelerating
universe at each fixed pointsThe stability and existence conditions for each point are
presented in Table 4 Since the corresponding eigenvalues ofthe fixed points are complicated we do not give them herebut one can obtain their signs numerically and then examinethe stability properties of the critical points
Critical Point 1198611This point exists for 120582 gt 0 and 120585 ge 120582281205722
However it is an unstable saddle point
Critical Point 1198612The critical point 119861
2exists for 120582 gt 0 and 120585 lt
0 or 120585 ge 120582281205722 This point is a scaling attractor of the modelif 120572 gt 0 and 120585 gt 0 Figure 4 shows clearly such a behavior ofthe model for suitable choices of 120585 120582 and 120572
Critical Point 1198613This point exists for negative values of 120582 and
120585 or when 120585 ge 120582281205722 Also it is a stable point if 120572 lt 0
and 120585 gt 0 and a saddle point if 120572 gt 0 and 120585 lt 0 Thevalues of parameters have been chosen in Figure 5 such that
Advances in High Energy Physics 7
110
109
108
107
106
105
104
103
102
101
y(t)
0 01 02
x(t)
minus01
(a)
0 01 02
x(t)
minus01
024
022
020
018
016
014
012
010
008
u(t)
(b)
110107105103101 102 104 106 108 109
y(t)
024
022
020
018
016
014
012
010
008
u(t)
(c)
Figure 4 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 120572 = 2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
2is a stable attractor of the model
1198613became a attractor of the model as it is clear from phase-
space trajectories
Critical Point 1198614Thepoint119861
4exists for120582 lt 0 and 120585 ge 120582281205722
It is a stable point if 120572 lt 0 and 120585 gt 0 In Figure 6 values of theparameters 120585 and 120572 are those satisfying these constraints andso 1198614becomes a attractor point for phase-plane trajectories
The corresponding 3-dimensional phase-space trajectories ofthe model for attractor points 119861
2(a) 119861
3(b) and 119861
4(c) are
plotted in Figure 7
4 Conclusion
A model of dark energy with nonminimal coupling ofquintessence scalar field with gravity in the framework ofteleparallel gravity was called teleparallel dark energy [27] Ifone replaces quintessence by tachyon field in such a modelthen tachyonic teleparallel dark energy will be constructed[32]
Moreover although dark energy and dark matter scaledifferently with the expansion of our universe according to
8 Advances in High Energy Physics
110
108
106
104
102
100
y(t)
minus005 0 005 010 015 020
x(t)
(a)u(t)
minus005 0 005 010 015 020
x(t)
minus002
minus004
008
006
004
002
0
(b)
110108106104102100
y(t)
u(t)
minus002
minus004
008
006
004
002
0
(c)
Figure 5 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
3is a stable attractor of the model
the observations [52ndash54] we are living in an epoch in whichdark energy and dark matter densities are comparable andthis is the well-known cosmological coincidence problem[30] This problem can be alleviated in most dark energymodels via the method of scaling solutions in which thedensity parameters of dark energy and dark matter are bothnonvanishing over there
In this paper we investigated the phase-space analysisof generalized tachyon cosmology in the framework ofteleparallel gravity Our model was described by action (2)
which generalizes tachyonic teleparallel dark energy modelproposed in [32] We found some scaling attractors in ourmodel for the case 119899 = 2 and the coupling function119891(120601) prop 120601
2 These scaling attractors are 1198601and 119860
2when
there is no interaction between dark energy and dark matter1198612 1198613 and 119861
4are scaling attractors in the case that dark
energy interacts with dark matter through the interactingterm 119876 = 120573120581120588
119898120601 As it is shown in [29] original teleparallel
dark energy model also has a late-time attractor in whichdark energy behaves like a cosmological constant and so
Advances in High Energy Physics 9
y(t)
x(t)
323
322
321
320
319
318
317
minus0004 minus0002 0 000600040002
(a)u(t)
x(t)
3025
3020
3015
3010
3005
minus0001 0001 0003 0005 0007
(b)
y(t)
u(t)
3025
3020
3015
3010
3170 3175 3180 3185 3190
3005
(c)
Figure 6 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus05and 120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
4is a stable attractor of the model
it provides a natural way for the stabilization of the darkenergy equation of state to the cosmological constant valuewithout the need for parameter tuning Also the authors in[55] by using power-law potentials and nonminimal couplingfunctions have shown that teleparallel gravity has no stablefuture solutions for positive nonminimal coupling and thescalar field results always in oscillationsOur results show thatgeneralized tachyon field represents interesting cosmologicalbehavior in comparison with ordinary tachyon fields in the
framework of teleparallel gravity because there is no scalingattractor in the latter model Note that however in [33]Otalora has considered a dynamically changing coefficient120572 = 119891
120601radic119891 and found a field matter dominated solution
in which it has some portions of the dark energy density inthe matter dominated era So generalized tachyon field givesus the hope that cosmological coincidence problem can bealleviated without fine-tunings One can study our model fordifferent kinds of potential and other famous interaction termbetween dark energy and dark matter
10 Advances in High Energy Physics
u(t)
minus01 0 01 02
x(t)
110
107
105104103
101
y(t)
024
022
020
018
016
014
012
010
008
(a)u(t)
minus002
minus004
008
006
004
002
0
minus005 0 005 010 015 020
x(t)
110
108
106
104
102
100
y(t)
(b)
y(t)
3170
3175
31803185
3190
minus00020
00020004
0006
x(t)
u(t)
3025
3020
3015
3010
3005
(c)
Figure 7Three-dimensional phase-space trajectories of the model for119876 = 120573120581120588119898120601with stable attractors 119861
2(a) 119861
3(b) and 119861
4(c)The values
of the parameters are those mentioned in Figures 4 5 and 6 respectively
Table 3 Location of the critical points and the corresponding values of the dark energy density parameterΩ120601and equation of state120596
120601and the
condition required for an accelerating universe for119876 = 120573120581120588119898120601 Here 120579
1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579
2= (4120572120585minusradic1612057221205852 minus 21205822120585)2120582120585
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198611
02radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198612
02radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572(120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
1198613
0 minus2radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198614
0 minus2radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572 (120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
Advances in High Energy Physics 11
Table 4 Stability and existence conditions of the critical points ofthe model for 119876 = 120573120581120588
119898120601
Label Stability Existence
1198611
Saddle pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 and 120585 ge 120582281205722
1198612
Saddle pointif 120572 lt 0 and 120585 lt 0 and
stable pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198613
Saddle pointif 120572 gt 0 and 120585 lt 0 and
stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198614
Stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 and 120585 ge 120582281205722
Appendix
Perturbation Matrix Elements
The elements of 3 times 3 matrix 119872 of the linearized per-turbation equations for the real and physically meaningfulcritical points (119909
119888 119910119888 119906119888) of the autonomous system (18) read
as follows
11987211= 3]2119888(radic3
2120582119909119888119910119888(21199092
119888+ ]2119888(1 + 3119909
4
119888)) minus 6120583
minus2
119888]21198881199092
119888
minus 4radic3120572120585119906119888119909119888]2119888119910minus1
119888) + radic3120582119909
119888119910119888minus 3 +M
11
11987212=radic3
4(120582 (2119909
2
119888+ ]2119888(1 + 3119909
4
119888)) + 8120572120585119906
119888]2119888119910minus2
119888) +M
12
11987213= minus2radic3120572120585]2
119888119910minus1
119888+M13
11987221=
21199102
119888(radic3120572120585119906
119888+ 3119909119888119910119888]minus2119888)
(21205851199062119888+ 1)
minusradic3120582119910
2
119888
2
11987222=
2119910119888(minus (94) 120583
minus4
119888119910119888+ 5radic3120572120585119909
119888119906119888)
(21205851199062119888+ 1)
minus radic3120582119909119888119910119888+3
2
11987223=
61205851199061198881199102
119888(minus (10radic33) 120572120585119906
119888119909119888+ 120583minus4
1198881199102
119888)
(21205851199062119888+ 1)2
+5radic3120572120585119909
1198881199102
119888
21205851199062119888+ 1
11987231=radic3120572119910
119888
2
11987232=radic3120572119909
119888
2
11987233= 0
(A1)
where in the case of 119876 = 0 we have M11= M12= M13= 0
and in the case of 119876 = 120573120581120588119898120601 we have
M11= 4radic3120573]minus2
119888119909119888119910119888
M12= 2radic3120573120583
minus2
119888(1 + 3119909
2
119888)
M13= minus4radic3120573120585119906
119888119910minus1
119888
(A2)
Examining the eigenvalues of the matrix119872 for each criticalpoint one determines its stability conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989
[2] S M Carroll ldquoThe cosmological constantrdquo Living Reviews inRelativity vol 4 article 1 2001
[3] R-J Yang and S N Zhang ldquoThe age problem in the ΛCDMmodelrdquo Monthly Notices of the Royal Astronomical Society vol407 no 3 pp 1835ndash1841 2010
[4] Y-F Cai E N Saridakis M R Setare and J-Q Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010
[5] 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
[6] A Sen ldquoRemarks on tachyon driven cosmologyrdquo PhysicaScripta vol 2005 no 117 article 70 2005
[7] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002
[8] M R Garousi ldquoTachyon couplings on non-BPS D-branes andDiracndashBornndashInfeld actionrdquo Nuclear Physics B vol 584 no 1-2pp 284ndash299 2000
[9] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002
[10] A Unzicker and T Case ldquoTranslation of Einsteinrsquos attempt ofa unified field theory with teleparallelismrdquo httparxivorgabsphysics0503046
[11] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 pp 3524ndash3553 1979 Addendum in PhysicalReview D vol 24 pp 3312-3314 1982
[12] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction Springer Dordrecht Netherlands 2013
[13] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013
[14] R Ferraro andF Fiorini ldquoModified teleparallel gravity inflationwithout an inflatonrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 75 no 8 Article ID 084031 2007
[15] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D Particles Fields Gravitation andCosmology vol 79 Article ID 124019 2009
[16] E V Linder ldquoEinsteins other gravity and the acceleration of theUniverserdquo Physical Review D vol 81 Article ID 127301 2010
12 Advances in High Energy Physics
[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010
[18] 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
[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984
[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999
[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000
[22] 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 Article ID 0435392004
[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009
[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007
[25] 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
[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010
[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011
[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012
[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012
[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012
[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013
[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012
[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013
[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014
[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007
[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence
principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009
[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007
[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008
[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009
[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010
[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005
[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006
[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008
[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008
[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012
[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010
[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998
[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999
[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005
[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011
[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003
[52] 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
[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008
[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011
[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD
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Advances in High Energy Physics 3
Table 1 Location of the critical points and the corresponding values of the dark energy density parameter Ω120601and equation of state 120596
120601and
the condition required for an accelerating universe for119876 = 0 Here 1205821= 120572(4120572120585+radic1612057221205852 minus 21205822120585)120582
2 and 1205822= 120572(4120572120585minusradic1612057221205852 minus 21205822120585)120582
2
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198601
0 2radic12058211205821205821
212057212058541205821(1 minus
1205822
812058512057221205821)
1612058521205724
(12058221205822
1+ 21205851205722) (1205822120582
1minus 81205851205722)
1205822gt1205851205722(41205821minus 1)
1205822
1
(1 + radic1 minus321205821
(1 minus 41205821)2)
or
1205822lt1205851205722(41205821minus 1)
1205822
1
(1 minus radic1 minus321205821
(1 minus 41205821)2)
1198602
0 2radic12058221205821205822
212057212058541205822(1 minus
1205822
812058512057221205822)
1612058521205724
(12058221205822
2+ 21205851205722) (1205822120582
2minus 81205851205722)
1205822gt1205851205722(41205822minus 1)
1205822
2
(1 + radic1 minus321205822
(1 minus 41205822)2)
or
1205822lt1205851205722(41205822minus 1)
1205822
2
(1 minus radic1 minus321205822
(1 minus 41205822)2)
Table 2 Stability and existence conditions of the critical points ofthe model for 119876 = 0
Label Stability Existence
1198601
Saddle pointif 120572 gt 0 and 120585120582 gt 0 and
stable pointif 120585 gt 0 120572 lt 0 and 120582 lt 0
For all 120585 lt 0or
120585 ge 120582281205722
1198602
Saddle pointif 120572 lt 0 and 120585120582 lt 0 and
stable pointif 120585 gt 0 120572 gt 0 and 120582 gt 0
For all 120585 lt 0or
120585 ge 120582281205722
where 120596120601= 119875120601120588120601is the equation of state parameter of dark
energy which is attributed to the scalar field 120601The barotropicindex is defined by 120574 equiv 1 + 120596
119898with 0 lt 120574 lt 2
Although dynamics of tachyonic teleparallel dark energyhas been studied in [33] no scaling attractors has been foundHere we are going to perform a phase-space analysis ofgeneralized tachyonic teleparallel dark energy and as we willsee below that some interesting scaling attractors appear insuch theory
3 Cosmological Dynamics
In order to perform phase-space and stability analysis of themodel we introduce the following auxiliary variables
119909 equiv
120601
radic119881 119910 equiv
radic119881
radic3119867 119906 equiv radic119891 (11)
The auxiliary variables allow us to straightforwardly obtainthe density parameter of dark energy and dark matter
Ω120601equiv
120588120601
31198672= 120583minus21199102(1 + 3119909
2) minus 2120585119906
2 (12)
Ω119898equiv120588119898
31198672= 1 minus Ω
120601 (13)
while the equation of state of the field reads
120596120601equiv
119875120601
120588120601
=
minus120583minus41199102+ 2120585119906 [(5radic33) 120572119909119910 + 119906 (1 minus (23) 119904)]
120583minus21199102 (1 + 31199092) minus 21205851199062
(14)
where 120572 equiv 119891120601radic119891 and
119904 = minus
1198672= (2120585119906
2+ 1)minus1
sdot [5radic3120572120585119906119909119910 + 6120583minus211990921199102
minus3
2120574120583minus21199102(1 + 3119909
2)] +
3120574
2
(15)
From now we concentrate on exponential scalar field poten-tial of the form 119881 = 119881
0119890minus119896120582120601 and the nonminimal coupling
function of the form 119891(120601) prop 1206012 These choices lead to
constant 120582 and 120572 respectivelyAnother quantities with great physical significance
namely the total equation of state parameter and the decel-eration parameter are given by
120596tot equiv119875120601+ 119875119898
120588120601+ 120588119898
= 120583minus21199102(41199092minus 120574 (1 + 3119909
2))
+ 2120585119906 [5radic3
3120572119909119910 + 119906(120574 minus
2
3119904)]
+ 120574 minus 1
(16)
119902 equiv minus1 minus
1198672=1
2+3
2120596tot
=3
2120583minus21199102(41199092minus 120574 (1 + 3119909
2))
+ 120585119906 [5radic3120572119909119910 + 119906 (3120574 minus 2119904)] +3120574
2minus 1
(17)
4 Advances in High Energy Physics
110
109
108
107
106
105
104
103
102
101
y(t)
0 005 010 015 020
x(t)
(a)
0 005 010 015 020
x(t)
003
002
004
005
006
007
008
u(t)
(b)
110
109
108
107
106
105
104
103
102
101
y(t)
003002 004 005 006 007 008
u(t)
(c)
Figure 1 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 and 120572 = minus2for 119876 = 0 For these values of the parameters point 119860
1is a stable attractor of the model
Using auxiliary variables (11) the evolution equations (6)and (8) can be recast as a dynamical system of ordinarydifferential equations
1199091015840=radic3
2[1205821199092119910 +
1
2120582]2 (1 + 31199094) 119910
minus 4120572120585]2119906119910minus1 minus 2radic3120583minus2]2119909] minus 119876
1199101015840= (minus
radic3
2120582119909119910 + 119904)119910
1199061015840=radic3120572119909119910
2
(18)
where 119876 = 119876 120601119867radic119881(120601) 120582 equiv minus119881120601119896119881 and prime in (18)
denotes differentiation with respect to the so-called e-foldingtime119873 = ln 119886
Advances in High Energy Physics 5
110
109
108
107
106
105
104
103
102
101
y(t)
minus005 0 005 010 015 020
x(t)
(a)
007
009
008
011
010
013
012
015
014
u(t)
minus005 0 005 010 015 020
x(t)
(b)
110
109
108
107
106
105
104
103
102
101
y(t)
007 008 010 012 014009 011 013 015
u(t)
(c)
Figure 2 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 and 120572 = 2 for119876 = 0 For these values of the parameters point 119860
2is a stable attractor of the model
The next step is the introduction of interaction term 119876
to obtain an autonomous system out of (18) The fixed points(119909119888 119910119888 119906119888) for which 1199091015840 = 1199101015840 = 1199061015840 = 0 depend on the choice
of the interaction term 119876 and two general possibilities willbe treated in the sequel The stability of the system at a fixedpoint can be obtained from the analysis of the determinantand trace of the perturbation matrix 119872 Such a matrix canbe constructed by substituting linear perturbations 119909 rarr
119909119888+ 120575119909 119910 rarr 119910
119888+ 120575119910 and 119906 rarr 119906
119888+ 120575119906 about the
critical point (119909119888 119910119888 119906119888) into the autonomous system (18)The
3 times 3 matrix119872 of the linearized perturbation equations ofthe autonomous system is shown in the appendix Thereforefor each critical point we examine the sign of the real partof the eigenvalues of 119872 According to the usual dynamicalsystem analysis if the eigenvalues are real and have oppositesigns the corresponding critical point is a saddle point Afixed point is unstable if the eigenvalues are positive and itis stable for negative real part of the eigenvalues
In the following subsections we will study the dynamicsof generalized tachyon field with different interaction term
6 Advances in High Energy Physics
109
107105
103
101 y(t)
x(t)
u(t)
008
007
006
005
004
003
002
0005
010015
020
(a)
110
107105 y
(t)
u(t)
minus005 0 005 010 015 020
x(t)
015
014
013
012
011
010
009
008
(b)
Figure 3 Three-dimensional phase-space trajectories of the model for 119876 = 0 with stable attractors 1198601(a) and 119860
2(b) The values of the
parameters are those mentioned in Figures 1 and 2 respectively
119876 Although one can work with a general 120574 (120574 = 1 and 43correspond to dust matter and radiation resp) without lossof generality we assume 120574 = 1 for simplicity
31 The Case for 119876 = 0 The first case 119876 = 0 clearlymeans that there is no interaction between dark energy andbackground matter In this case there are two critical pointspresented in Table 1 From (12) and (14) one can obtain thecorresponding values of density parameter Ω
120601and equation
of state of dark energy120596120601at each point Also using (17)we can
find the condition required for acceleration (119902 lt 0) at eachpoint These parameters and conditions have been shownin Table 1 The stability and existence conditions of criticalpoints 119860
1and 119860
2are presented in Table 2 We mention that
the corresponding eigenvalues of perturbation matrix 119872 atcritical points 119860
1and 119860
2are considerably involved and here
we do not present their explicit expressions but we can findsign of them numerically
Critical Point 1198601 This critical point is a scaling attractor if
120585 gt 0 120572 lt 0 and 120582 lt 0 Therefore cosmological coincidenceproblem can be alleviated at this point 119860
1is a saddle point
for 120572 gt 0 and 120585120582 gt 0
Critical Point 1198602 1198602can also be a scaling attractor of the
model or a saddle point under the same conditions as for1198601
In Figure 1 we have chosen the values of the parameters120585 120582 and 120572 such that 119860
1became a stable attractor of the
model Plots in Figure 1 show the phase-space trajectories on119909-119910 119909-119906 and 119906-119910 planes from (a) to (c) respectively Thesame plots are shown in Figure 2 for critical point 119860
2 Note
that the values of the parameter have been chosen in the waythat 119860
2became a stable point of the model In Figure 3 the
corresponding 3-dimensional phase-space trajectories of themodel have been presented One can see that 119860
1and 119860
2are
stable attractors of themodel in (a) and (b) plots respectively
32The Case for119876 = 120573120581120588119898120601 This deals with the most famil-
iar interaction term extensively considered in the literature(see eg [31 47ndash51]) Here 119876 in terms of auxiliary variablesis 119876 = radic3120573119910minus1Ω
119898 Inserting such an interaction term in (18)
and setting the left hand sides of the equations to zero leadto the critical points 119861
1 1198612 1198613 and 119861
4presented in Table 3
In the same table we have provided the corresponding valuesofΩ120601and 120596
120601as well as the condition needed for accelerating
universe at each fixed pointsThe stability and existence conditions for each point are
presented in Table 4 Since the corresponding eigenvalues ofthe fixed points are complicated we do not give them herebut one can obtain their signs numerically and then examinethe stability properties of the critical points
Critical Point 1198611This point exists for 120582 gt 0 and 120585 ge 120582281205722
However it is an unstable saddle point
Critical Point 1198612The critical point 119861
2exists for 120582 gt 0 and 120585 lt
0 or 120585 ge 120582281205722 This point is a scaling attractor of the modelif 120572 gt 0 and 120585 gt 0 Figure 4 shows clearly such a behavior ofthe model for suitable choices of 120585 120582 and 120572
Critical Point 1198613This point exists for negative values of 120582 and
120585 or when 120585 ge 120582281205722 Also it is a stable point if 120572 lt 0
and 120585 gt 0 and a saddle point if 120572 gt 0 and 120585 lt 0 Thevalues of parameters have been chosen in Figure 5 such that
Advances in High Energy Physics 7
110
109
108
107
106
105
104
103
102
101
y(t)
0 01 02
x(t)
minus01
(a)
0 01 02
x(t)
minus01
024
022
020
018
016
014
012
010
008
u(t)
(b)
110107105103101 102 104 106 108 109
y(t)
024
022
020
018
016
014
012
010
008
u(t)
(c)
Figure 4 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 120572 = 2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
2is a stable attractor of the model
1198613became a attractor of the model as it is clear from phase-
space trajectories
Critical Point 1198614Thepoint119861
4exists for120582 lt 0 and 120585 ge 120582281205722
It is a stable point if 120572 lt 0 and 120585 gt 0 In Figure 6 values of theparameters 120585 and 120572 are those satisfying these constraints andso 1198614becomes a attractor point for phase-plane trajectories
The corresponding 3-dimensional phase-space trajectories ofthe model for attractor points 119861
2(a) 119861
3(b) and 119861
4(c) are
plotted in Figure 7
4 Conclusion
A model of dark energy with nonminimal coupling ofquintessence scalar field with gravity in the framework ofteleparallel gravity was called teleparallel dark energy [27] Ifone replaces quintessence by tachyon field in such a modelthen tachyonic teleparallel dark energy will be constructed[32]
Moreover although dark energy and dark matter scaledifferently with the expansion of our universe according to
8 Advances in High Energy Physics
110
108
106
104
102
100
y(t)
minus005 0 005 010 015 020
x(t)
(a)u(t)
minus005 0 005 010 015 020
x(t)
minus002
minus004
008
006
004
002
0
(b)
110108106104102100
y(t)
u(t)
minus002
minus004
008
006
004
002
0
(c)
Figure 5 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
3is a stable attractor of the model
the observations [52ndash54] we are living in an epoch in whichdark energy and dark matter densities are comparable andthis is the well-known cosmological coincidence problem[30] This problem can be alleviated in most dark energymodels via the method of scaling solutions in which thedensity parameters of dark energy and dark matter are bothnonvanishing over there
In this paper we investigated the phase-space analysisof generalized tachyon cosmology in the framework ofteleparallel gravity Our model was described by action (2)
which generalizes tachyonic teleparallel dark energy modelproposed in [32] We found some scaling attractors in ourmodel for the case 119899 = 2 and the coupling function119891(120601) prop 120601
2 These scaling attractors are 1198601and 119860
2when
there is no interaction between dark energy and dark matter1198612 1198613 and 119861
4are scaling attractors in the case that dark
energy interacts with dark matter through the interactingterm 119876 = 120573120581120588
119898120601 As it is shown in [29] original teleparallel
dark energy model also has a late-time attractor in whichdark energy behaves like a cosmological constant and so
Advances in High Energy Physics 9
y(t)
x(t)
323
322
321
320
319
318
317
minus0004 minus0002 0 000600040002
(a)u(t)
x(t)
3025
3020
3015
3010
3005
minus0001 0001 0003 0005 0007
(b)
y(t)
u(t)
3025
3020
3015
3010
3170 3175 3180 3185 3190
3005
(c)
Figure 6 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus05and 120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
4is a stable attractor of the model
it provides a natural way for the stabilization of the darkenergy equation of state to the cosmological constant valuewithout the need for parameter tuning Also the authors in[55] by using power-law potentials and nonminimal couplingfunctions have shown that teleparallel gravity has no stablefuture solutions for positive nonminimal coupling and thescalar field results always in oscillationsOur results show thatgeneralized tachyon field represents interesting cosmologicalbehavior in comparison with ordinary tachyon fields in the
framework of teleparallel gravity because there is no scalingattractor in the latter model Note that however in [33]Otalora has considered a dynamically changing coefficient120572 = 119891
120601radic119891 and found a field matter dominated solution
in which it has some portions of the dark energy density inthe matter dominated era So generalized tachyon field givesus the hope that cosmological coincidence problem can bealleviated without fine-tunings One can study our model fordifferent kinds of potential and other famous interaction termbetween dark energy and dark matter
10 Advances in High Energy Physics
u(t)
minus01 0 01 02
x(t)
110
107
105104103
101
y(t)
024
022
020
018
016
014
012
010
008
(a)u(t)
minus002
minus004
008
006
004
002
0
minus005 0 005 010 015 020
x(t)
110
108
106
104
102
100
y(t)
(b)
y(t)
3170
3175
31803185
3190
minus00020
00020004
0006
x(t)
u(t)
3025
3020
3015
3010
3005
(c)
Figure 7Three-dimensional phase-space trajectories of the model for119876 = 120573120581120588119898120601with stable attractors 119861
2(a) 119861
3(b) and 119861
4(c)The values
of the parameters are those mentioned in Figures 4 5 and 6 respectively
Table 3 Location of the critical points and the corresponding values of the dark energy density parameterΩ120601and equation of state120596
120601and the
condition required for an accelerating universe for119876 = 120573120581120588119898120601 Here 120579
1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579
2= (4120572120585minusradic1612057221205852 minus 21205822120585)2120582120585
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198611
02radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198612
02radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572(120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
1198613
0 minus2radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198614
0 minus2radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572 (120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
Advances in High Energy Physics 11
Table 4 Stability and existence conditions of the critical points ofthe model for 119876 = 120573120581120588
119898120601
Label Stability Existence
1198611
Saddle pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 and 120585 ge 120582281205722
1198612
Saddle pointif 120572 lt 0 and 120585 lt 0 and
stable pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198613
Saddle pointif 120572 gt 0 and 120585 lt 0 and
stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198614
Stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 and 120585 ge 120582281205722
Appendix
Perturbation Matrix Elements
The elements of 3 times 3 matrix 119872 of the linearized per-turbation equations for the real and physically meaningfulcritical points (119909
119888 119910119888 119906119888) of the autonomous system (18) read
as follows
11987211= 3]2119888(radic3
2120582119909119888119910119888(21199092
119888+ ]2119888(1 + 3119909
4
119888)) minus 6120583
minus2
119888]21198881199092
119888
minus 4radic3120572120585119906119888119909119888]2119888119910minus1
119888) + radic3120582119909
119888119910119888minus 3 +M
11
11987212=radic3
4(120582 (2119909
2
119888+ ]2119888(1 + 3119909
4
119888)) + 8120572120585119906
119888]2119888119910minus2
119888) +M
12
11987213= minus2radic3120572120585]2
119888119910minus1
119888+M13
11987221=
21199102
119888(radic3120572120585119906
119888+ 3119909119888119910119888]minus2119888)
(21205851199062119888+ 1)
minusradic3120582119910
2
119888
2
11987222=
2119910119888(minus (94) 120583
minus4
119888119910119888+ 5radic3120572120585119909
119888119906119888)
(21205851199062119888+ 1)
minus radic3120582119909119888119910119888+3
2
11987223=
61205851199061198881199102
119888(minus (10radic33) 120572120585119906
119888119909119888+ 120583minus4
1198881199102
119888)
(21205851199062119888+ 1)2
+5radic3120572120585119909
1198881199102
119888
21205851199062119888+ 1
11987231=radic3120572119910
119888
2
11987232=radic3120572119909
119888
2
11987233= 0
(A1)
where in the case of 119876 = 0 we have M11= M12= M13= 0
and in the case of 119876 = 120573120581120588119898120601 we have
M11= 4radic3120573]minus2
119888119909119888119910119888
M12= 2radic3120573120583
minus2
119888(1 + 3119909
2
119888)
M13= minus4radic3120573120585119906
119888119910minus1
119888
(A2)
Examining the eigenvalues of the matrix119872 for each criticalpoint one determines its stability conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989
[2] S M Carroll ldquoThe cosmological constantrdquo Living Reviews inRelativity vol 4 article 1 2001
[3] R-J Yang and S N Zhang ldquoThe age problem in the ΛCDMmodelrdquo Monthly Notices of the Royal Astronomical Society vol407 no 3 pp 1835ndash1841 2010
[4] Y-F Cai E N Saridakis M R Setare and J-Q Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010
[5] 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
[6] A Sen ldquoRemarks on tachyon driven cosmologyrdquo PhysicaScripta vol 2005 no 117 article 70 2005
[7] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002
[8] M R Garousi ldquoTachyon couplings on non-BPS D-branes andDiracndashBornndashInfeld actionrdquo Nuclear Physics B vol 584 no 1-2pp 284ndash299 2000
[9] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002
[10] A Unzicker and T Case ldquoTranslation of Einsteinrsquos attempt ofa unified field theory with teleparallelismrdquo httparxivorgabsphysics0503046
[11] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 pp 3524ndash3553 1979 Addendum in PhysicalReview D vol 24 pp 3312-3314 1982
[12] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction Springer Dordrecht Netherlands 2013
[13] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013
[14] R Ferraro andF Fiorini ldquoModified teleparallel gravity inflationwithout an inflatonrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 75 no 8 Article ID 084031 2007
[15] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D Particles Fields Gravitation andCosmology vol 79 Article ID 124019 2009
[16] E V Linder ldquoEinsteins other gravity and the acceleration of theUniverserdquo Physical Review D vol 81 Article ID 127301 2010
12 Advances in High Energy Physics
[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010
[18] 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
[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984
[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999
[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000
[22] 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 Article ID 0435392004
[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009
[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007
[25] 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
[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010
[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011
[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012
[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012
[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012
[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013
[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012
[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013
[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014
[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007
[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence
principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009
[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007
[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008
[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009
[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010
[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005
[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006
[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008
[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008
[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012
[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010
[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998
[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999
[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005
[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011
[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003
[52] 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
[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008
[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011
[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Superconductivity
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ThermodynamicsJournal of
4 Advances in High Energy Physics
110
109
108
107
106
105
104
103
102
101
y(t)
0 005 010 015 020
x(t)
(a)
0 005 010 015 020
x(t)
003
002
004
005
006
007
008
u(t)
(b)
110
109
108
107
106
105
104
103
102
101
y(t)
003002 004 005 006 007 008
u(t)
(c)
Figure 1 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 and 120572 = minus2for 119876 = 0 For these values of the parameters point 119860
1is a stable attractor of the model
Using auxiliary variables (11) the evolution equations (6)and (8) can be recast as a dynamical system of ordinarydifferential equations
1199091015840=radic3
2[1205821199092119910 +
1
2120582]2 (1 + 31199094) 119910
minus 4120572120585]2119906119910minus1 minus 2radic3120583minus2]2119909] minus 119876
1199101015840= (minus
radic3
2120582119909119910 + 119904)119910
1199061015840=radic3120572119909119910
2
(18)
where 119876 = 119876 120601119867radic119881(120601) 120582 equiv minus119881120601119896119881 and prime in (18)
denotes differentiation with respect to the so-called e-foldingtime119873 = ln 119886
Advances in High Energy Physics 5
110
109
108
107
106
105
104
103
102
101
y(t)
minus005 0 005 010 015 020
x(t)
(a)
007
009
008
011
010
013
012
015
014
u(t)
minus005 0 005 010 015 020
x(t)
(b)
110
109
108
107
106
105
104
103
102
101
y(t)
007 008 010 012 014009 011 013 015
u(t)
(c)
Figure 2 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 and 120572 = 2 for119876 = 0 For these values of the parameters point 119860
2is a stable attractor of the model
The next step is the introduction of interaction term 119876
to obtain an autonomous system out of (18) The fixed points(119909119888 119910119888 119906119888) for which 1199091015840 = 1199101015840 = 1199061015840 = 0 depend on the choice
of the interaction term 119876 and two general possibilities willbe treated in the sequel The stability of the system at a fixedpoint can be obtained from the analysis of the determinantand trace of the perturbation matrix 119872 Such a matrix canbe constructed by substituting linear perturbations 119909 rarr
119909119888+ 120575119909 119910 rarr 119910
119888+ 120575119910 and 119906 rarr 119906
119888+ 120575119906 about the
critical point (119909119888 119910119888 119906119888) into the autonomous system (18)The
3 times 3 matrix119872 of the linearized perturbation equations ofthe autonomous system is shown in the appendix Thereforefor each critical point we examine the sign of the real partof the eigenvalues of 119872 According to the usual dynamicalsystem analysis if the eigenvalues are real and have oppositesigns the corresponding critical point is a saddle point Afixed point is unstable if the eigenvalues are positive and itis stable for negative real part of the eigenvalues
In the following subsections we will study the dynamicsof generalized tachyon field with different interaction term
6 Advances in High Energy Physics
109
107105
103
101 y(t)
x(t)
u(t)
008
007
006
005
004
003
002
0005
010015
020
(a)
110
107105 y
(t)
u(t)
minus005 0 005 010 015 020
x(t)
015
014
013
012
011
010
009
008
(b)
Figure 3 Three-dimensional phase-space trajectories of the model for 119876 = 0 with stable attractors 1198601(a) and 119860
2(b) The values of the
parameters are those mentioned in Figures 1 and 2 respectively
119876 Although one can work with a general 120574 (120574 = 1 and 43correspond to dust matter and radiation resp) without lossof generality we assume 120574 = 1 for simplicity
31 The Case for 119876 = 0 The first case 119876 = 0 clearlymeans that there is no interaction between dark energy andbackground matter In this case there are two critical pointspresented in Table 1 From (12) and (14) one can obtain thecorresponding values of density parameter Ω
120601and equation
of state of dark energy120596120601at each point Also using (17)we can
find the condition required for acceleration (119902 lt 0) at eachpoint These parameters and conditions have been shownin Table 1 The stability and existence conditions of criticalpoints 119860
1and 119860
2are presented in Table 2 We mention that
the corresponding eigenvalues of perturbation matrix 119872 atcritical points 119860
1and 119860
2are considerably involved and here
we do not present their explicit expressions but we can findsign of them numerically
Critical Point 1198601 This critical point is a scaling attractor if
120585 gt 0 120572 lt 0 and 120582 lt 0 Therefore cosmological coincidenceproblem can be alleviated at this point 119860
1is a saddle point
for 120572 gt 0 and 120585120582 gt 0
Critical Point 1198602 1198602can also be a scaling attractor of the
model or a saddle point under the same conditions as for1198601
In Figure 1 we have chosen the values of the parameters120585 120582 and 120572 such that 119860
1became a stable attractor of the
model Plots in Figure 1 show the phase-space trajectories on119909-119910 119909-119906 and 119906-119910 planes from (a) to (c) respectively Thesame plots are shown in Figure 2 for critical point 119860
2 Note
that the values of the parameter have been chosen in the waythat 119860
2became a stable point of the model In Figure 3 the
corresponding 3-dimensional phase-space trajectories of themodel have been presented One can see that 119860
1and 119860
2are
stable attractors of themodel in (a) and (b) plots respectively
32The Case for119876 = 120573120581120588119898120601 This deals with the most famil-
iar interaction term extensively considered in the literature(see eg [31 47ndash51]) Here 119876 in terms of auxiliary variablesis 119876 = radic3120573119910minus1Ω
119898 Inserting such an interaction term in (18)
and setting the left hand sides of the equations to zero leadto the critical points 119861
1 1198612 1198613 and 119861
4presented in Table 3
In the same table we have provided the corresponding valuesofΩ120601and 120596
120601as well as the condition needed for accelerating
universe at each fixed pointsThe stability and existence conditions for each point are
presented in Table 4 Since the corresponding eigenvalues ofthe fixed points are complicated we do not give them herebut one can obtain their signs numerically and then examinethe stability properties of the critical points
Critical Point 1198611This point exists for 120582 gt 0 and 120585 ge 120582281205722
However it is an unstable saddle point
Critical Point 1198612The critical point 119861
2exists for 120582 gt 0 and 120585 lt
0 or 120585 ge 120582281205722 This point is a scaling attractor of the modelif 120572 gt 0 and 120585 gt 0 Figure 4 shows clearly such a behavior ofthe model for suitable choices of 120585 120582 and 120572
Critical Point 1198613This point exists for negative values of 120582 and
120585 or when 120585 ge 120582281205722 Also it is a stable point if 120572 lt 0
and 120585 gt 0 and a saddle point if 120572 gt 0 and 120585 lt 0 Thevalues of parameters have been chosen in Figure 5 such that
Advances in High Energy Physics 7
110
109
108
107
106
105
104
103
102
101
y(t)
0 01 02
x(t)
minus01
(a)
0 01 02
x(t)
minus01
024
022
020
018
016
014
012
010
008
u(t)
(b)
110107105103101 102 104 106 108 109
y(t)
024
022
020
018
016
014
012
010
008
u(t)
(c)
Figure 4 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 120572 = 2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
2is a stable attractor of the model
1198613became a attractor of the model as it is clear from phase-
space trajectories
Critical Point 1198614Thepoint119861
4exists for120582 lt 0 and 120585 ge 120582281205722
It is a stable point if 120572 lt 0 and 120585 gt 0 In Figure 6 values of theparameters 120585 and 120572 are those satisfying these constraints andso 1198614becomes a attractor point for phase-plane trajectories
The corresponding 3-dimensional phase-space trajectories ofthe model for attractor points 119861
2(a) 119861
3(b) and 119861
4(c) are
plotted in Figure 7
4 Conclusion
A model of dark energy with nonminimal coupling ofquintessence scalar field with gravity in the framework ofteleparallel gravity was called teleparallel dark energy [27] Ifone replaces quintessence by tachyon field in such a modelthen tachyonic teleparallel dark energy will be constructed[32]
Moreover although dark energy and dark matter scaledifferently with the expansion of our universe according to
8 Advances in High Energy Physics
110
108
106
104
102
100
y(t)
minus005 0 005 010 015 020
x(t)
(a)u(t)
minus005 0 005 010 015 020
x(t)
minus002
minus004
008
006
004
002
0
(b)
110108106104102100
y(t)
u(t)
minus002
minus004
008
006
004
002
0
(c)
Figure 5 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
3is a stable attractor of the model
the observations [52ndash54] we are living in an epoch in whichdark energy and dark matter densities are comparable andthis is the well-known cosmological coincidence problem[30] This problem can be alleviated in most dark energymodels via the method of scaling solutions in which thedensity parameters of dark energy and dark matter are bothnonvanishing over there
In this paper we investigated the phase-space analysisof generalized tachyon cosmology in the framework ofteleparallel gravity Our model was described by action (2)
which generalizes tachyonic teleparallel dark energy modelproposed in [32] We found some scaling attractors in ourmodel for the case 119899 = 2 and the coupling function119891(120601) prop 120601
2 These scaling attractors are 1198601and 119860
2when
there is no interaction between dark energy and dark matter1198612 1198613 and 119861
4are scaling attractors in the case that dark
energy interacts with dark matter through the interactingterm 119876 = 120573120581120588
119898120601 As it is shown in [29] original teleparallel
dark energy model also has a late-time attractor in whichdark energy behaves like a cosmological constant and so
Advances in High Energy Physics 9
y(t)
x(t)
323
322
321
320
319
318
317
minus0004 minus0002 0 000600040002
(a)u(t)
x(t)
3025
3020
3015
3010
3005
minus0001 0001 0003 0005 0007
(b)
y(t)
u(t)
3025
3020
3015
3010
3170 3175 3180 3185 3190
3005
(c)
Figure 6 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus05and 120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
4is a stable attractor of the model
it provides a natural way for the stabilization of the darkenergy equation of state to the cosmological constant valuewithout the need for parameter tuning Also the authors in[55] by using power-law potentials and nonminimal couplingfunctions have shown that teleparallel gravity has no stablefuture solutions for positive nonminimal coupling and thescalar field results always in oscillationsOur results show thatgeneralized tachyon field represents interesting cosmologicalbehavior in comparison with ordinary tachyon fields in the
framework of teleparallel gravity because there is no scalingattractor in the latter model Note that however in [33]Otalora has considered a dynamically changing coefficient120572 = 119891
120601radic119891 and found a field matter dominated solution
in which it has some portions of the dark energy density inthe matter dominated era So generalized tachyon field givesus the hope that cosmological coincidence problem can bealleviated without fine-tunings One can study our model fordifferent kinds of potential and other famous interaction termbetween dark energy and dark matter
10 Advances in High Energy Physics
u(t)
minus01 0 01 02
x(t)
110
107
105104103
101
y(t)
024
022
020
018
016
014
012
010
008
(a)u(t)
minus002
minus004
008
006
004
002
0
minus005 0 005 010 015 020
x(t)
110
108
106
104
102
100
y(t)
(b)
y(t)
3170
3175
31803185
3190
minus00020
00020004
0006
x(t)
u(t)
3025
3020
3015
3010
3005
(c)
Figure 7Three-dimensional phase-space trajectories of the model for119876 = 120573120581120588119898120601with stable attractors 119861
2(a) 119861
3(b) and 119861
4(c)The values
of the parameters are those mentioned in Figures 4 5 and 6 respectively
Table 3 Location of the critical points and the corresponding values of the dark energy density parameterΩ120601and equation of state120596
120601and the
condition required for an accelerating universe for119876 = 120573120581120588119898120601 Here 120579
1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579
2= (4120572120585minusradic1612057221205852 minus 21205822120585)2120582120585
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198611
02radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198612
02radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572(120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
1198613
0 minus2radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198614
0 minus2radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572 (120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
Advances in High Energy Physics 11
Table 4 Stability and existence conditions of the critical points ofthe model for 119876 = 120573120581120588
119898120601
Label Stability Existence
1198611
Saddle pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 and 120585 ge 120582281205722
1198612
Saddle pointif 120572 lt 0 and 120585 lt 0 and
stable pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198613
Saddle pointif 120572 gt 0 and 120585 lt 0 and
stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198614
Stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 and 120585 ge 120582281205722
Appendix
Perturbation Matrix Elements
The elements of 3 times 3 matrix 119872 of the linearized per-turbation equations for the real and physically meaningfulcritical points (119909
119888 119910119888 119906119888) of the autonomous system (18) read
as follows
11987211= 3]2119888(radic3
2120582119909119888119910119888(21199092
119888+ ]2119888(1 + 3119909
4
119888)) minus 6120583
minus2
119888]21198881199092
119888
minus 4radic3120572120585119906119888119909119888]2119888119910minus1
119888) + radic3120582119909
119888119910119888minus 3 +M
11
11987212=radic3
4(120582 (2119909
2
119888+ ]2119888(1 + 3119909
4
119888)) + 8120572120585119906
119888]2119888119910minus2
119888) +M
12
11987213= minus2radic3120572120585]2
119888119910minus1
119888+M13
11987221=
21199102
119888(radic3120572120585119906
119888+ 3119909119888119910119888]minus2119888)
(21205851199062119888+ 1)
minusradic3120582119910
2
119888
2
11987222=
2119910119888(minus (94) 120583
minus4
119888119910119888+ 5radic3120572120585119909
119888119906119888)
(21205851199062119888+ 1)
minus radic3120582119909119888119910119888+3
2
11987223=
61205851199061198881199102
119888(minus (10radic33) 120572120585119906
119888119909119888+ 120583minus4
1198881199102
119888)
(21205851199062119888+ 1)2
+5radic3120572120585119909
1198881199102
119888
21205851199062119888+ 1
11987231=radic3120572119910
119888
2
11987232=radic3120572119909
119888
2
11987233= 0
(A1)
where in the case of 119876 = 0 we have M11= M12= M13= 0
and in the case of 119876 = 120573120581120588119898120601 we have
M11= 4radic3120573]minus2
119888119909119888119910119888
M12= 2radic3120573120583
minus2
119888(1 + 3119909
2
119888)
M13= minus4radic3120573120585119906
119888119910minus1
119888
(A2)
Examining the eigenvalues of the matrix119872 for each criticalpoint one determines its stability conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989
[2] S M Carroll ldquoThe cosmological constantrdquo Living Reviews inRelativity vol 4 article 1 2001
[3] R-J Yang and S N Zhang ldquoThe age problem in the ΛCDMmodelrdquo Monthly Notices of the Royal Astronomical Society vol407 no 3 pp 1835ndash1841 2010
[4] Y-F Cai E N Saridakis M R Setare and J-Q Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010
[5] 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
[6] A Sen ldquoRemarks on tachyon driven cosmologyrdquo PhysicaScripta vol 2005 no 117 article 70 2005
[7] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002
[8] M R Garousi ldquoTachyon couplings on non-BPS D-branes andDiracndashBornndashInfeld actionrdquo Nuclear Physics B vol 584 no 1-2pp 284ndash299 2000
[9] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002
[10] A Unzicker and T Case ldquoTranslation of Einsteinrsquos attempt ofa unified field theory with teleparallelismrdquo httparxivorgabsphysics0503046
[11] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 pp 3524ndash3553 1979 Addendum in PhysicalReview D vol 24 pp 3312-3314 1982
[12] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction Springer Dordrecht Netherlands 2013
[13] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013
[14] R Ferraro andF Fiorini ldquoModified teleparallel gravity inflationwithout an inflatonrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 75 no 8 Article ID 084031 2007
[15] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D Particles Fields Gravitation andCosmology vol 79 Article ID 124019 2009
[16] E V Linder ldquoEinsteins other gravity and the acceleration of theUniverserdquo Physical Review D vol 81 Article ID 127301 2010
12 Advances in High Energy Physics
[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010
[18] 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
[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984
[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999
[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000
[22] 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 Article ID 0435392004
[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009
[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007
[25] 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
[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010
[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011
[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012
[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012
[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012
[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013
[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012
[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013
[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014
[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007
[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence
principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009
[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007
[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008
[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009
[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010
[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005
[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006
[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008
[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008
[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012
[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010
[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998
[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999
[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005
[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011
[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003
[52] 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
[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008
[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011
[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
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GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
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Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
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Soft MatterJournal of
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AerodynamicsJournal of
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PhotonicsJournal of
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Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 5
110
109
108
107
106
105
104
103
102
101
y(t)
minus005 0 005 010 015 020
x(t)
(a)
007
009
008
011
010
013
012
015
014
u(t)
minus005 0 005 010 015 020
x(t)
(b)
110
109
108
107
106
105
104
103
102
101
y(t)
007 008 010 012 014009 011 013 015
u(t)
(c)
Figure 2 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 and 120572 = 2 for119876 = 0 For these values of the parameters point 119860
2is a stable attractor of the model
The next step is the introduction of interaction term 119876
to obtain an autonomous system out of (18) The fixed points(119909119888 119910119888 119906119888) for which 1199091015840 = 1199101015840 = 1199061015840 = 0 depend on the choice
of the interaction term 119876 and two general possibilities willbe treated in the sequel The stability of the system at a fixedpoint can be obtained from the analysis of the determinantand trace of the perturbation matrix 119872 Such a matrix canbe constructed by substituting linear perturbations 119909 rarr
119909119888+ 120575119909 119910 rarr 119910
119888+ 120575119910 and 119906 rarr 119906
119888+ 120575119906 about the
critical point (119909119888 119910119888 119906119888) into the autonomous system (18)The
3 times 3 matrix119872 of the linearized perturbation equations ofthe autonomous system is shown in the appendix Thereforefor each critical point we examine the sign of the real partof the eigenvalues of 119872 According to the usual dynamicalsystem analysis if the eigenvalues are real and have oppositesigns the corresponding critical point is a saddle point Afixed point is unstable if the eigenvalues are positive and itis stable for negative real part of the eigenvalues
In the following subsections we will study the dynamicsof generalized tachyon field with different interaction term
6 Advances in High Energy Physics
109
107105
103
101 y(t)
x(t)
u(t)
008
007
006
005
004
003
002
0005
010015
020
(a)
110
107105 y
(t)
u(t)
minus005 0 005 010 015 020
x(t)
015
014
013
012
011
010
009
008
(b)
Figure 3 Three-dimensional phase-space trajectories of the model for 119876 = 0 with stable attractors 1198601(a) and 119860
2(b) The values of the
parameters are those mentioned in Figures 1 and 2 respectively
119876 Although one can work with a general 120574 (120574 = 1 and 43correspond to dust matter and radiation resp) without lossof generality we assume 120574 = 1 for simplicity
31 The Case for 119876 = 0 The first case 119876 = 0 clearlymeans that there is no interaction between dark energy andbackground matter In this case there are two critical pointspresented in Table 1 From (12) and (14) one can obtain thecorresponding values of density parameter Ω
120601and equation
of state of dark energy120596120601at each point Also using (17)we can
find the condition required for acceleration (119902 lt 0) at eachpoint These parameters and conditions have been shownin Table 1 The stability and existence conditions of criticalpoints 119860
1and 119860
2are presented in Table 2 We mention that
the corresponding eigenvalues of perturbation matrix 119872 atcritical points 119860
1and 119860
2are considerably involved and here
we do not present their explicit expressions but we can findsign of them numerically
Critical Point 1198601 This critical point is a scaling attractor if
120585 gt 0 120572 lt 0 and 120582 lt 0 Therefore cosmological coincidenceproblem can be alleviated at this point 119860
1is a saddle point
for 120572 gt 0 and 120585120582 gt 0
Critical Point 1198602 1198602can also be a scaling attractor of the
model or a saddle point under the same conditions as for1198601
In Figure 1 we have chosen the values of the parameters120585 120582 and 120572 such that 119860
1became a stable attractor of the
model Plots in Figure 1 show the phase-space trajectories on119909-119910 119909-119906 and 119906-119910 planes from (a) to (c) respectively Thesame plots are shown in Figure 2 for critical point 119860
2 Note
that the values of the parameter have been chosen in the waythat 119860
2became a stable point of the model In Figure 3 the
corresponding 3-dimensional phase-space trajectories of themodel have been presented One can see that 119860
1and 119860
2are
stable attractors of themodel in (a) and (b) plots respectively
32The Case for119876 = 120573120581120588119898120601 This deals with the most famil-
iar interaction term extensively considered in the literature(see eg [31 47ndash51]) Here 119876 in terms of auxiliary variablesis 119876 = radic3120573119910minus1Ω
119898 Inserting such an interaction term in (18)
and setting the left hand sides of the equations to zero leadto the critical points 119861
1 1198612 1198613 and 119861
4presented in Table 3
In the same table we have provided the corresponding valuesofΩ120601and 120596
120601as well as the condition needed for accelerating
universe at each fixed pointsThe stability and existence conditions for each point are
presented in Table 4 Since the corresponding eigenvalues ofthe fixed points are complicated we do not give them herebut one can obtain their signs numerically and then examinethe stability properties of the critical points
Critical Point 1198611This point exists for 120582 gt 0 and 120585 ge 120582281205722
However it is an unstable saddle point
Critical Point 1198612The critical point 119861
2exists for 120582 gt 0 and 120585 lt
0 or 120585 ge 120582281205722 This point is a scaling attractor of the modelif 120572 gt 0 and 120585 gt 0 Figure 4 shows clearly such a behavior ofthe model for suitable choices of 120585 120582 and 120572
Critical Point 1198613This point exists for negative values of 120582 and
120585 or when 120585 ge 120582281205722 Also it is a stable point if 120572 lt 0
and 120585 gt 0 and a saddle point if 120572 gt 0 and 120585 lt 0 Thevalues of parameters have been chosen in Figure 5 such that
Advances in High Energy Physics 7
110
109
108
107
106
105
104
103
102
101
y(t)
0 01 02
x(t)
minus01
(a)
0 01 02
x(t)
minus01
024
022
020
018
016
014
012
010
008
u(t)
(b)
110107105103101 102 104 106 108 109
y(t)
024
022
020
018
016
014
012
010
008
u(t)
(c)
Figure 4 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 120572 = 2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
2is a stable attractor of the model
1198613became a attractor of the model as it is clear from phase-
space trajectories
Critical Point 1198614Thepoint119861
4exists for120582 lt 0 and 120585 ge 120582281205722
It is a stable point if 120572 lt 0 and 120585 gt 0 In Figure 6 values of theparameters 120585 and 120572 are those satisfying these constraints andso 1198614becomes a attractor point for phase-plane trajectories
The corresponding 3-dimensional phase-space trajectories ofthe model for attractor points 119861
2(a) 119861
3(b) and 119861
4(c) are
plotted in Figure 7
4 Conclusion
A model of dark energy with nonminimal coupling ofquintessence scalar field with gravity in the framework ofteleparallel gravity was called teleparallel dark energy [27] Ifone replaces quintessence by tachyon field in such a modelthen tachyonic teleparallel dark energy will be constructed[32]
Moreover although dark energy and dark matter scaledifferently with the expansion of our universe according to
8 Advances in High Energy Physics
110
108
106
104
102
100
y(t)
minus005 0 005 010 015 020
x(t)
(a)u(t)
minus005 0 005 010 015 020
x(t)
minus002
minus004
008
006
004
002
0
(b)
110108106104102100
y(t)
u(t)
minus002
minus004
008
006
004
002
0
(c)
Figure 5 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
3is a stable attractor of the model
the observations [52ndash54] we are living in an epoch in whichdark energy and dark matter densities are comparable andthis is the well-known cosmological coincidence problem[30] This problem can be alleviated in most dark energymodels via the method of scaling solutions in which thedensity parameters of dark energy and dark matter are bothnonvanishing over there
In this paper we investigated the phase-space analysisof generalized tachyon cosmology in the framework ofteleparallel gravity Our model was described by action (2)
which generalizes tachyonic teleparallel dark energy modelproposed in [32] We found some scaling attractors in ourmodel for the case 119899 = 2 and the coupling function119891(120601) prop 120601
2 These scaling attractors are 1198601and 119860
2when
there is no interaction between dark energy and dark matter1198612 1198613 and 119861
4are scaling attractors in the case that dark
energy interacts with dark matter through the interactingterm 119876 = 120573120581120588
119898120601 As it is shown in [29] original teleparallel
dark energy model also has a late-time attractor in whichdark energy behaves like a cosmological constant and so
Advances in High Energy Physics 9
y(t)
x(t)
323
322
321
320
319
318
317
minus0004 minus0002 0 000600040002
(a)u(t)
x(t)
3025
3020
3015
3010
3005
minus0001 0001 0003 0005 0007
(b)
y(t)
u(t)
3025
3020
3015
3010
3170 3175 3180 3185 3190
3005
(c)
Figure 6 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus05and 120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
4is a stable attractor of the model
it provides a natural way for the stabilization of the darkenergy equation of state to the cosmological constant valuewithout the need for parameter tuning Also the authors in[55] by using power-law potentials and nonminimal couplingfunctions have shown that teleparallel gravity has no stablefuture solutions for positive nonminimal coupling and thescalar field results always in oscillationsOur results show thatgeneralized tachyon field represents interesting cosmologicalbehavior in comparison with ordinary tachyon fields in the
framework of teleparallel gravity because there is no scalingattractor in the latter model Note that however in [33]Otalora has considered a dynamically changing coefficient120572 = 119891
120601radic119891 and found a field matter dominated solution
in which it has some portions of the dark energy density inthe matter dominated era So generalized tachyon field givesus the hope that cosmological coincidence problem can bealleviated without fine-tunings One can study our model fordifferent kinds of potential and other famous interaction termbetween dark energy and dark matter
10 Advances in High Energy Physics
u(t)
minus01 0 01 02
x(t)
110
107
105104103
101
y(t)
024
022
020
018
016
014
012
010
008
(a)u(t)
minus002
minus004
008
006
004
002
0
minus005 0 005 010 015 020
x(t)
110
108
106
104
102
100
y(t)
(b)
y(t)
3170
3175
31803185
3190
minus00020
00020004
0006
x(t)
u(t)
3025
3020
3015
3010
3005
(c)
Figure 7Three-dimensional phase-space trajectories of the model for119876 = 120573120581120588119898120601with stable attractors 119861
2(a) 119861
3(b) and 119861
4(c)The values
of the parameters are those mentioned in Figures 4 5 and 6 respectively
Table 3 Location of the critical points and the corresponding values of the dark energy density parameterΩ120601and equation of state120596
120601and the
condition required for an accelerating universe for119876 = 120573120581120588119898120601 Here 120579
1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579
2= (4120572120585minusradic1612057221205852 minus 21205822120585)2120582120585
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198611
02radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198612
02radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572(120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
1198613
0 minus2radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198614
0 minus2radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572 (120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
Advances in High Energy Physics 11
Table 4 Stability and existence conditions of the critical points ofthe model for 119876 = 120573120581120588
119898120601
Label Stability Existence
1198611
Saddle pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 and 120585 ge 120582281205722
1198612
Saddle pointif 120572 lt 0 and 120585 lt 0 and
stable pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198613
Saddle pointif 120572 gt 0 and 120585 lt 0 and
stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198614
Stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 and 120585 ge 120582281205722
Appendix
Perturbation Matrix Elements
The elements of 3 times 3 matrix 119872 of the linearized per-turbation equations for the real and physically meaningfulcritical points (119909
119888 119910119888 119906119888) of the autonomous system (18) read
as follows
11987211= 3]2119888(radic3
2120582119909119888119910119888(21199092
119888+ ]2119888(1 + 3119909
4
119888)) minus 6120583
minus2
119888]21198881199092
119888
minus 4radic3120572120585119906119888119909119888]2119888119910minus1
119888) + radic3120582119909
119888119910119888minus 3 +M
11
11987212=radic3
4(120582 (2119909
2
119888+ ]2119888(1 + 3119909
4
119888)) + 8120572120585119906
119888]2119888119910minus2
119888) +M
12
11987213= minus2radic3120572120585]2
119888119910minus1
119888+M13
11987221=
21199102
119888(radic3120572120585119906
119888+ 3119909119888119910119888]minus2119888)
(21205851199062119888+ 1)
minusradic3120582119910
2
119888
2
11987222=
2119910119888(minus (94) 120583
minus4
119888119910119888+ 5radic3120572120585119909
119888119906119888)
(21205851199062119888+ 1)
minus radic3120582119909119888119910119888+3
2
11987223=
61205851199061198881199102
119888(minus (10radic33) 120572120585119906
119888119909119888+ 120583minus4
1198881199102
119888)
(21205851199062119888+ 1)2
+5radic3120572120585119909
1198881199102
119888
21205851199062119888+ 1
11987231=radic3120572119910
119888
2
11987232=radic3120572119909
119888
2
11987233= 0
(A1)
where in the case of 119876 = 0 we have M11= M12= M13= 0
and in the case of 119876 = 120573120581120588119898120601 we have
M11= 4radic3120573]minus2
119888119909119888119910119888
M12= 2radic3120573120583
minus2
119888(1 + 3119909
2
119888)
M13= minus4radic3120573120585119906
119888119910minus1
119888
(A2)
Examining the eigenvalues of the matrix119872 for each criticalpoint one determines its stability conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989
[2] S M Carroll ldquoThe cosmological constantrdquo Living Reviews inRelativity vol 4 article 1 2001
[3] R-J Yang and S N Zhang ldquoThe age problem in the ΛCDMmodelrdquo Monthly Notices of the Royal Astronomical Society vol407 no 3 pp 1835ndash1841 2010
[4] Y-F Cai E N Saridakis M R Setare and J-Q Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010
[5] 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
[6] A Sen ldquoRemarks on tachyon driven cosmologyrdquo PhysicaScripta vol 2005 no 117 article 70 2005
[7] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002
[8] M R Garousi ldquoTachyon couplings on non-BPS D-branes andDiracndashBornndashInfeld actionrdquo Nuclear Physics B vol 584 no 1-2pp 284ndash299 2000
[9] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002
[10] A Unzicker and T Case ldquoTranslation of Einsteinrsquos attempt ofa unified field theory with teleparallelismrdquo httparxivorgabsphysics0503046
[11] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 pp 3524ndash3553 1979 Addendum in PhysicalReview D vol 24 pp 3312-3314 1982
[12] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction Springer Dordrecht Netherlands 2013
[13] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013
[14] R Ferraro andF Fiorini ldquoModified teleparallel gravity inflationwithout an inflatonrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 75 no 8 Article ID 084031 2007
[15] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D Particles Fields Gravitation andCosmology vol 79 Article ID 124019 2009
[16] E V Linder ldquoEinsteins other gravity and the acceleration of theUniverserdquo Physical Review D vol 81 Article ID 127301 2010
12 Advances in High Energy Physics
[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010
[18] 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
[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984
[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999
[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000
[22] 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 Article ID 0435392004
[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009
[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007
[25] 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
[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010
[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011
[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012
[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012
[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012
[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013
[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012
[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013
[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014
[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007
[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence
principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009
[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007
[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008
[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009
[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010
[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005
[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006
[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008
[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008
[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012
[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010
[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998
[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999
[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005
[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011
[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003
[52] 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
[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008
[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011
[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Soft MatterJournal of
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AerodynamicsJournal of
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PhotonicsJournal of
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Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
6 Advances in High Energy Physics
109
107105
103
101 y(t)
x(t)
u(t)
008
007
006
005
004
003
002
0005
010015
020
(a)
110
107105 y
(t)
u(t)
minus005 0 005 010 015 020
x(t)
015
014
013
012
011
010
009
008
(b)
Figure 3 Three-dimensional phase-space trajectories of the model for 119876 = 0 with stable attractors 1198601(a) and 119860
2(b) The values of the
parameters are those mentioned in Figures 1 and 2 respectively
119876 Although one can work with a general 120574 (120574 = 1 and 43correspond to dust matter and radiation resp) without lossof generality we assume 120574 = 1 for simplicity
31 The Case for 119876 = 0 The first case 119876 = 0 clearlymeans that there is no interaction between dark energy andbackground matter In this case there are two critical pointspresented in Table 1 From (12) and (14) one can obtain thecorresponding values of density parameter Ω
120601and equation
of state of dark energy120596120601at each point Also using (17)we can
find the condition required for acceleration (119902 lt 0) at eachpoint These parameters and conditions have been shownin Table 1 The stability and existence conditions of criticalpoints 119860
1and 119860
2are presented in Table 2 We mention that
the corresponding eigenvalues of perturbation matrix 119872 atcritical points 119860
1and 119860
2are considerably involved and here
we do not present their explicit expressions but we can findsign of them numerically
Critical Point 1198601 This critical point is a scaling attractor if
120585 gt 0 120572 lt 0 and 120582 lt 0 Therefore cosmological coincidenceproblem can be alleviated at this point 119860
1is a saddle point
for 120572 gt 0 and 120585120582 gt 0
Critical Point 1198602 1198602can also be a scaling attractor of the
model or a saddle point under the same conditions as for1198601
In Figure 1 we have chosen the values of the parameters120585 120582 and 120572 such that 119860
1became a stable attractor of the
model Plots in Figure 1 show the phase-space trajectories on119909-119910 119909-119906 and 119906-119910 planes from (a) to (c) respectively Thesame plots are shown in Figure 2 for critical point 119860
2 Note
that the values of the parameter have been chosen in the waythat 119860
2became a stable point of the model In Figure 3 the
corresponding 3-dimensional phase-space trajectories of themodel have been presented One can see that 119860
1and 119860
2are
stable attractors of themodel in (a) and (b) plots respectively
32The Case for119876 = 120573120581120588119898120601 This deals with the most famil-
iar interaction term extensively considered in the literature(see eg [31 47ndash51]) Here 119876 in terms of auxiliary variablesis 119876 = radic3120573119910minus1Ω
119898 Inserting such an interaction term in (18)
and setting the left hand sides of the equations to zero leadto the critical points 119861
1 1198612 1198613 and 119861
4presented in Table 3
In the same table we have provided the corresponding valuesofΩ120601and 120596
120601as well as the condition needed for accelerating
universe at each fixed pointsThe stability and existence conditions for each point are
presented in Table 4 Since the corresponding eigenvalues ofthe fixed points are complicated we do not give them herebut one can obtain their signs numerically and then examinethe stability properties of the critical points
Critical Point 1198611This point exists for 120582 gt 0 and 120585 ge 120582281205722
However it is an unstable saddle point
Critical Point 1198612The critical point 119861
2exists for 120582 gt 0 and 120585 lt
0 or 120585 ge 120582281205722 This point is a scaling attractor of the modelif 120572 gt 0 and 120585 gt 0 Figure 4 shows clearly such a behavior ofthe model for suitable choices of 120585 120582 and 120572
Critical Point 1198613This point exists for negative values of 120582 and
120585 or when 120585 ge 120582281205722 Also it is a stable point if 120572 lt 0
and 120585 gt 0 and a saddle point if 120572 gt 0 and 120585 lt 0 Thevalues of parameters have been chosen in Figure 5 such that
Advances in High Energy Physics 7
110
109
108
107
106
105
104
103
102
101
y(t)
0 01 02
x(t)
minus01
(a)
0 01 02
x(t)
minus01
024
022
020
018
016
014
012
010
008
u(t)
(b)
110107105103101 102 104 106 108 109
y(t)
024
022
020
018
016
014
012
010
008
u(t)
(c)
Figure 4 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 120572 = 2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
2is a stable attractor of the model
1198613became a attractor of the model as it is clear from phase-
space trajectories
Critical Point 1198614Thepoint119861
4exists for120582 lt 0 and 120585 ge 120582281205722
It is a stable point if 120572 lt 0 and 120585 gt 0 In Figure 6 values of theparameters 120585 and 120572 are those satisfying these constraints andso 1198614becomes a attractor point for phase-plane trajectories
The corresponding 3-dimensional phase-space trajectories ofthe model for attractor points 119861
2(a) 119861
3(b) and 119861
4(c) are
plotted in Figure 7
4 Conclusion
A model of dark energy with nonminimal coupling ofquintessence scalar field with gravity in the framework ofteleparallel gravity was called teleparallel dark energy [27] Ifone replaces quintessence by tachyon field in such a modelthen tachyonic teleparallel dark energy will be constructed[32]
Moreover although dark energy and dark matter scaledifferently with the expansion of our universe according to
8 Advances in High Energy Physics
110
108
106
104
102
100
y(t)
minus005 0 005 010 015 020
x(t)
(a)u(t)
minus005 0 005 010 015 020
x(t)
minus002
minus004
008
006
004
002
0
(b)
110108106104102100
y(t)
u(t)
minus002
minus004
008
006
004
002
0
(c)
Figure 5 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
3is a stable attractor of the model
the observations [52ndash54] we are living in an epoch in whichdark energy and dark matter densities are comparable andthis is the well-known cosmological coincidence problem[30] This problem can be alleviated in most dark energymodels via the method of scaling solutions in which thedensity parameters of dark energy and dark matter are bothnonvanishing over there
In this paper we investigated the phase-space analysisof generalized tachyon cosmology in the framework ofteleparallel gravity Our model was described by action (2)
which generalizes tachyonic teleparallel dark energy modelproposed in [32] We found some scaling attractors in ourmodel for the case 119899 = 2 and the coupling function119891(120601) prop 120601
2 These scaling attractors are 1198601and 119860
2when
there is no interaction between dark energy and dark matter1198612 1198613 and 119861
4are scaling attractors in the case that dark
energy interacts with dark matter through the interactingterm 119876 = 120573120581120588
119898120601 As it is shown in [29] original teleparallel
dark energy model also has a late-time attractor in whichdark energy behaves like a cosmological constant and so
Advances in High Energy Physics 9
y(t)
x(t)
323
322
321
320
319
318
317
minus0004 minus0002 0 000600040002
(a)u(t)
x(t)
3025
3020
3015
3010
3005
minus0001 0001 0003 0005 0007
(b)
y(t)
u(t)
3025
3020
3015
3010
3170 3175 3180 3185 3190
3005
(c)
Figure 6 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus05and 120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
4is a stable attractor of the model
it provides a natural way for the stabilization of the darkenergy equation of state to the cosmological constant valuewithout the need for parameter tuning Also the authors in[55] by using power-law potentials and nonminimal couplingfunctions have shown that teleparallel gravity has no stablefuture solutions for positive nonminimal coupling and thescalar field results always in oscillationsOur results show thatgeneralized tachyon field represents interesting cosmologicalbehavior in comparison with ordinary tachyon fields in the
framework of teleparallel gravity because there is no scalingattractor in the latter model Note that however in [33]Otalora has considered a dynamically changing coefficient120572 = 119891
120601radic119891 and found a field matter dominated solution
in which it has some portions of the dark energy density inthe matter dominated era So generalized tachyon field givesus the hope that cosmological coincidence problem can bealleviated without fine-tunings One can study our model fordifferent kinds of potential and other famous interaction termbetween dark energy and dark matter
10 Advances in High Energy Physics
u(t)
minus01 0 01 02
x(t)
110
107
105104103
101
y(t)
024
022
020
018
016
014
012
010
008
(a)u(t)
minus002
minus004
008
006
004
002
0
minus005 0 005 010 015 020
x(t)
110
108
106
104
102
100
y(t)
(b)
y(t)
3170
3175
31803185
3190
minus00020
00020004
0006
x(t)
u(t)
3025
3020
3015
3010
3005
(c)
Figure 7Three-dimensional phase-space trajectories of the model for119876 = 120573120581120588119898120601with stable attractors 119861
2(a) 119861
3(b) and 119861
4(c)The values
of the parameters are those mentioned in Figures 4 5 and 6 respectively
Table 3 Location of the critical points and the corresponding values of the dark energy density parameterΩ120601and equation of state120596
120601and the
condition required for an accelerating universe for119876 = 120573120581120588119898120601 Here 120579
1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579
2= (4120572120585minusradic1612057221205852 minus 21205822120585)2120582120585
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198611
02radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198612
02radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572(120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
1198613
0 minus2radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198614
0 minus2radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572 (120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
Advances in High Energy Physics 11
Table 4 Stability and existence conditions of the critical points ofthe model for 119876 = 120573120581120588
119898120601
Label Stability Existence
1198611
Saddle pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 and 120585 ge 120582281205722
1198612
Saddle pointif 120572 lt 0 and 120585 lt 0 and
stable pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198613
Saddle pointif 120572 gt 0 and 120585 lt 0 and
stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198614
Stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 and 120585 ge 120582281205722
Appendix
Perturbation Matrix Elements
The elements of 3 times 3 matrix 119872 of the linearized per-turbation equations for the real and physically meaningfulcritical points (119909
119888 119910119888 119906119888) of the autonomous system (18) read
as follows
11987211= 3]2119888(radic3
2120582119909119888119910119888(21199092
119888+ ]2119888(1 + 3119909
4
119888)) minus 6120583
minus2
119888]21198881199092
119888
minus 4radic3120572120585119906119888119909119888]2119888119910minus1
119888) + radic3120582119909
119888119910119888minus 3 +M
11
11987212=radic3
4(120582 (2119909
2
119888+ ]2119888(1 + 3119909
4
119888)) + 8120572120585119906
119888]2119888119910minus2
119888) +M
12
11987213= minus2radic3120572120585]2
119888119910minus1
119888+M13
11987221=
21199102
119888(radic3120572120585119906
119888+ 3119909119888119910119888]minus2119888)
(21205851199062119888+ 1)
minusradic3120582119910
2
119888
2
11987222=
2119910119888(minus (94) 120583
minus4
119888119910119888+ 5radic3120572120585119909
119888119906119888)
(21205851199062119888+ 1)
minus radic3120582119909119888119910119888+3
2
11987223=
61205851199061198881199102
119888(minus (10radic33) 120572120585119906
119888119909119888+ 120583minus4
1198881199102
119888)
(21205851199062119888+ 1)2
+5radic3120572120585119909
1198881199102
119888
21205851199062119888+ 1
11987231=radic3120572119910
119888
2
11987232=radic3120572119909
119888
2
11987233= 0
(A1)
where in the case of 119876 = 0 we have M11= M12= M13= 0
and in the case of 119876 = 120573120581120588119898120601 we have
M11= 4radic3120573]minus2
119888119909119888119910119888
M12= 2radic3120573120583
minus2
119888(1 + 3119909
2
119888)
M13= minus4radic3120573120585119906
119888119910minus1
119888
(A2)
Examining the eigenvalues of the matrix119872 for each criticalpoint one determines its stability conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989
[2] S M Carroll ldquoThe cosmological constantrdquo Living Reviews inRelativity vol 4 article 1 2001
[3] R-J Yang and S N Zhang ldquoThe age problem in the ΛCDMmodelrdquo Monthly Notices of the Royal Astronomical Society vol407 no 3 pp 1835ndash1841 2010
[4] Y-F Cai E N Saridakis M R Setare and J-Q Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010
[5] 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
[6] A Sen ldquoRemarks on tachyon driven cosmologyrdquo PhysicaScripta vol 2005 no 117 article 70 2005
[7] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002
[8] M R Garousi ldquoTachyon couplings on non-BPS D-branes andDiracndashBornndashInfeld actionrdquo Nuclear Physics B vol 584 no 1-2pp 284ndash299 2000
[9] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002
[10] A Unzicker and T Case ldquoTranslation of Einsteinrsquos attempt ofa unified field theory with teleparallelismrdquo httparxivorgabsphysics0503046
[11] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 pp 3524ndash3553 1979 Addendum in PhysicalReview D vol 24 pp 3312-3314 1982
[12] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction Springer Dordrecht Netherlands 2013
[13] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013
[14] R Ferraro andF Fiorini ldquoModified teleparallel gravity inflationwithout an inflatonrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 75 no 8 Article ID 084031 2007
[15] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D Particles Fields Gravitation andCosmology vol 79 Article ID 124019 2009
[16] E V Linder ldquoEinsteins other gravity and the acceleration of theUniverserdquo Physical Review D vol 81 Article ID 127301 2010
12 Advances in High Energy Physics
[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010
[18] 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
[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984
[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999
[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000
[22] 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 Article ID 0435392004
[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009
[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007
[25] 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
[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010
[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011
[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012
[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012
[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012
[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013
[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012
[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013
[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014
[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007
[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence
principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009
[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007
[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008
[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009
[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010
[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005
[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006
[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008
[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008
[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012
[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010
[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998
[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999
[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005
[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011
[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003
[52] 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
[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008
[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011
[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
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AerodynamicsJournal of
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PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 7
110
109
108
107
106
105
104
103
102
101
y(t)
0 01 02
x(t)
minus01
(a)
0 01 02
x(t)
minus01
024
022
020
018
016
014
012
010
008
u(t)
(b)
110107105103101 102 104 106 108 109
y(t)
024
022
020
018
016
014
012
010
008
u(t)
(c)
Figure 4 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 120572 = 2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
2is a stable attractor of the model
1198613became a attractor of the model as it is clear from phase-
space trajectories
Critical Point 1198614Thepoint119861
4exists for120582 lt 0 and 120585 ge 120582281205722
It is a stable point if 120572 lt 0 and 120585 gt 0 In Figure 6 values of theparameters 120585 and 120572 are those satisfying these constraints andso 1198614becomes a attractor point for phase-plane trajectories
The corresponding 3-dimensional phase-space trajectories ofthe model for attractor points 119861
2(a) 119861
3(b) and 119861
4(c) are
plotted in Figure 7
4 Conclusion
A model of dark energy with nonminimal coupling ofquintessence scalar field with gravity in the framework ofteleparallel gravity was called teleparallel dark energy [27] Ifone replaces quintessence by tachyon field in such a modelthen tachyonic teleparallel dark energy will be constructed[32]
Moreover although dark energy and dark matter scaledifferently with the expansion of our universe according to
8 Advances in High Energy Physics
110
108
106
104
102
100
y(t)
minus005 0 005 010 015 020
x(t)
(a)u(t)
minus005 0 005 010 015 020
x(t)
minus002
minus004
008
006
004
002
0
(b)
110108106104102100
y(t)
u(t)
minus002
minus004
008
006
004
002
0
(c)
Figure 5 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
3is a stable attractor of the model
the observations [52ndash54] we are living in an epoch in whichdark energy and dark matter densities are comparable andthis is the well-known cosmological coincidence problem[30] This problem can be alleviated in most dark energymodels via the method of scaling solutions in which thedensity parameters of dark energy and dark matter are bothnonvanishing over there
In this paper we investigated the phase-space analysisof generalized tachyon cosmology in the framework ofteleparallel gravity Our model was described by action (2)
which generalizes tachyonic teleparallel dark energy modelproposed in [32] We found some scaling attractors in ourmodel for the case 119899 = 2 and the coupling function119891(120601) prop 120601
2 These scaling attractors are 1198601and 119860
2when
there is no interaction between dark energy and dark matter1198612 1198613 and 119861
4are scaling attractors in the case that dark
energy interacts with dark matter through the interactingterm 119876 = 120573120581120588
119898120601 As it is shown in [29] original teleparallel
dark energy model also has a late-time attractor in whichdark energy behaves like a cosmological constant and so
Advances in High Energy Physics 9
y(t)
x(t)
323
322
321
320
319
318
317
minus0004 minus0002 0 000600040002
(a)u(t)
x(t)
3025
3020
3015
3010
3005
minus0001 0001 0003 0005 0007
(b)
y(t)
u(t)
3025
3020
3015
3010
3170 3175 3180 3185 3190
3005
(c)
Figure 6 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus05and 120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
4is a stable attractor of the model
it provides a natural way for the stabilization of the darkenergy equation of state to the cosmological constant valuewithout the need for parameter tuning Also the authors in[55] by using power-law potentials and nonminimal couplingfunctions have shown that teleparallel gravity has no stablefuture solutions for positive nonminimal coupling and thescalar field results always in oscillationsOur results show thatgeneralized tachyon field represents interesting cosmologicalbehavior in comparison with ordinary tachyon fields in the
framework of teleparallel gravity because there is no scalingattractor in the latter model Note that however in [33]Otalora has considered a dynamically changing coefficient120572 = 119891
120601radic119891 and found a field matter dominated solution
in which it has some portions of the dark energy density inthe matter dominated era So generalized tachyon field givesus the hope that cosmological coincidence problem can bealleviated without fine-tunings One can study our model fordifferent kinds of potential and other famous interaction termbetween dark energy and dark matter
10 Advances in High Energy Physics
u(t)
minus01 0 01 02
x(t)
110
107
105104103
101
y(t)
024
022
020
018
016
014
012
010
008
(a)u(t)
minus002
minus004
008
006
004
002
0
minus005 0 005 010 015 020
x(t)
110
108
106
104
102
100
y(t)
(b)
y(t)
3170
3175
31803185
3190
minus00020
00020004
0006
x(t)
u(t)
3025
3020
3015
3010
3005
(c)
Figure 7Three-dimensional phase-space trajectories of the model for119876 = 120573120581120588119898120601with stable attractors 119861
2(a) 119861
3(b) and 119861
4(c)The values
of the parameters are those mentioned in Figures 4 5 and 6 respectively
Table 3 Location of the critical points and the corresponding values of the dark energy density parameterΩ120601and equation of state120596
120601and the
condition required for an accelerating universe for119876 = 120573120581120588119898120601 Here 120579
1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579
2= (4120572120585minusradic1612057221205852 minus 21205822120585)2120582120585
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198611
02radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198612
02radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572(120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
1198613
0 minus2radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198614
0 minus2radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572 (120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
Advances in High Energy Physics 11
Table 4 Stability and existence conditions of the critical points ofthe model for 119876 = 120573120581120588
119898120601
Label Stability Existence
1198611
Saddle pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 and 120585 ge 120582281205722
1198612
Saddle pointif 120572 lt 0 and 120585 lt 0 and
stable pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198613
Saddle pointif 120572 gt 0 and 120585 lt 0 and
stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198614
Stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 and 120585 ge 120582281205722
Appendix
Perturbation Matrix Elements
The elements of 3 times 3 matrix 119872 of the linearized per-turbation equations for the real and physically meaningfulcritical points (119909
119888 119910119888 119906119888) of the autonomous system (18) read
as follows
11987211= 3]2119888(radic3
2120582119909119888119910119888(21199092
119888+ ]2119888(1 + 3119909
4
119888)) minus 6120583
minus2
119888]21198881199092
119888
minus 4radic3120572120585119906119888119909119888]2119888119910minus1
119888) + radic3120582119909
119888119910119888minus 3 +M
11
11987212=radic3
4(120582 (2119909
2
119888+ ]2119888(1 + 3119909
4
119888)) + 8120572120585119906
119888]2119888119910minus2
119888) +M
12
11987213= minus2radic3120572120585]2
119888119910minus1
119888+M13
11987221=
21199102
119888(radic3120572120585119906
119888+ 3119909119888119910119888]minus2119888)
(21205851199062119888+ 1)
minusradic3120582119910
2
119888
2
11987222=
2119910119888(minus (94) 120583
minus4
119888119910119888+ 5radic3120572120585119909
119888119906119888)
(21205851199062119888+ 1)
minus radic3120582119909119888119910119888+3
2
11987223=
61205851199061198881199102
119888(minus (10radic33) 120572120585119906
119888119909119888+ 120583minus4
1198881199102
119888)
(21205851199062119888+ 1)2
+5radic3120572120585119909
1198881199102
119888
21205851199062119888+ 1
11987231=radic3120572119910
119888
2
11987232=radic3120572119909
119888
2
11987233= 0
(A1)
where in the case of 119876 = 0 we have M11= M12= M13= 0
and in the case of 119876 = 120573120581120588119898120601 we have
M11= 4radic3120573]minus2
119888119909119888119910119888
M12= 2radic3120573120583
minus2
119888(1 + 3119909
2
119888)
M13= minus4radic3120573120585119906
119888119910minus1
119888
(A2)
Examining the eigenvalues of the matrix119872 for each criticalpoint one determines its stability conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989
[2] S M Carroll ldquoThe cosmological constantrdquo Living Reviews inRelativity vol 4 article 1 2001
[3] R-J Yang and S N Zhang ldquoThe age problem in the ΛCDMmodelrdquo Monthly Notices of the Royal Astronomical Society vol407 no 3 pp 1835ndash1841 2010
[4] Y-F Cai E N Saridakis M R Setare and J-Q Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010
[5] 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
[6] A Sen ldquoRemarks on tachyon driven cosmologyrdquo PhysicaScripta vol 2005 no 117 article 70 2005
[7] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002
[8] M R Garousi ldquoTachyon couplings on non-BPS D-branes andDiracndashBornndashInfeld actionrdquo Nuclear Physics B vol 584 no 1-2pp 284ndash299 2000
[9] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002
[10] A Unzicker and T Case ldquoTranslation of Einsteinrsquos attempt ofa unified field theory with teleparallelismrdquo httparxivorgabsphysics0503046
[11] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 pp 3524ndash3553 1979 Addendum in PhysicalReview D vol 24 pp 3312-3314 1982
[12] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction Springer Dordrecht Netherlands 2013
[13] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013
[14] R Ferraro andF Fiorini ldquoModified teleparallel gravity inflationwithout an inflatonrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 75 no 8 Article ID 084031 2007
[15] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D Particles Fields Gravitation andCosmology vol 79 Article ID 124019 2009
[16] E V Linder ldquoEinsteins other gravity and the acceleration of theUniverserdquo Physical Review D vol 81 Article ID 127301 2010
12 Advances in High Energy Physics
[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010
[18] 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
[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984
[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999
[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000
[22] 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 Article ID 0435392004
[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009
[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007
[25] 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
[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010
[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011
[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012
[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012
[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012
[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013
[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012
[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013
[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014
[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007
[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence
principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009
[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007
[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008
[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009
[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010
[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005
[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006
[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008
[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008
[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012
[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010
[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998
[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999
[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005
[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011
[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003
[52] 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
[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008
[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011
[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
8 Advances in High Energy Physics
110
108
106
104
102
100
y(t)
minus005 0 005 010 015 020
x(t)
(a)u(t)
minus005 0 005 010 015 020
x(t)
minus002
minus004
008
006
004
002
0
(b)
110108106104102100
y(t)
u(t)
minus002
minus004
008
006
004
002
0
(c)
Figure 5 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus2 and120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
3is a stable attractor of the model
the observations [52ndash54] we are living in an epoch in whichdark energy and dark matter densities are comparable andthis is the well-known cosmological coincidence problem[30] This problem can be alleviated in most dark energymodels via the method of scaling solutions in which thedensity parameters of dark energy and dark matter are bothnonvanishing over there
In this paper we investigated the phase-space analysisof generalized tachyon cosmology in the framework ofteleparallel gravity Our model was described by action (2)
which generalizes tachyonic teleparallel dark energy modelproposed in [32] We found some scaling attractors in ourmodel for the case 119899 = 2 and the coupling function119891(120601) prop 120601
2 These scaling attractors are 1198601and 119860
2when
there is no interaction between dark energy and dark matter1198612 1198613 and 119861
4are scaling attractors in the case that dark
energy interacts with dark matter through the interactingterm 119876 = 120573120581120588
119898120601 As it is shown in [29] original teleparallel
dark energy model also has a late-time attractor in whichdark energy behaves like a cosmological constant and so
Advances in High Energy Physics 9
y(t)
x(t)
323
322
321
320
319
318
317
minus0004 minus0002 0 000600040002
(a)u(t)
x(t)
3025
3020
3015
3010
3005
minus0001 0001 0003 0005 0007
(b)
y(t)
u(t)
3025
3020
3015
3010
3170 3175 3180 3185 3190
3005
(c)
Figure 6 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus05and 120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
4is a stable attractor of the model
it provides a natural way for the stabilization of the darkenergy equation of state to the cosmological constant valuewithout the need for parameter tuning Also the authors in[55] by using power-law potentials and nonminimal couplingfunctions have shown that teleparallel gravity has no stablefuture solutions for positive nonminimal coupling and thescalar field results always in oscillationsOur results show thatgeneralized tachyon field represents interesting cosmologicalbehavior in comparison with ordinary tachyon fields in the
framework of teleparallel gravity because there is no scalingattractor in the latter model Note that however in [33]Otalora has considered a dynamically changing coefficient120572 = 119891
120601radic119891 and found a field matter dominated solution
in which it has some portions of the dark energy density inthe matter dominated era So generalized tachyon field givesus the hope that cosmological coincidence problem can bealleviated without fine-tunings One can study our model fordifferent kinds of potential and other famous interaction termbetween dark energy and dark matter
10 Advances in High Energy Physics
u(t)
minus01 0 01 02
x(t)
110
107
105104103
101
y(t)
024
022
020
018
016
014
012
010
008
(a)u(t)
minus002
minus004
008
006
004
002
0
minus005 0 005 010 015 020
x(t)
110
108
106
104
102
100
y(t)
(b)
y(t)
3170
3175
31803185
3190
minus00020
00020004
0006
x(t)
u(t)
3025
3020
3015
3010
3005
(c)
Figure 7Three-dimensional phase-space trajectories of the model for119876 = 120573120581120588119898120601with stable attractors 119861
2(a) 119861
3(b) and 119861
4(c)The values
of the parameters are those mentioned in Figures 4 5 and 6 respectively
Table 3 Location of the critical points and the corresponding values of the dark energy density parameterΩ120601and equation of state120596
120601and the
condition required for an accelerating universe for119876 = 120573120581120588119898120601 Here 120579
1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579
2= (4120572120585minusradic1612057221205852 minus 21205822120585)2120582120585
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198611
02radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198612
02radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572(120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
1198613
0 minus2radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198614
0 minus2radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572 (120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
Advances in High Energy Physics 11
Table 4 Stability and existence conditions of the critical points ofthe model for 119876 = 120573120581120588
119898120601
Label Stability Existence
1198611
Saddle pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 and 120585 ge 120582281205722
1198612
Saddle pointif 120572 lt 0 and 120585 lt 0 and
stable pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198613
Saddle pointif 120572 gt 0 and 120585 lt 0 and
stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198614
Stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 and 120585 ge 120582281205722
Appendix
Perturbation Matrix Elements
The elements of 3 times 3 matrix 119872 of the linearized per-turbation equations for the real and physically meaningfulcritical points (119909
119888 119910119888 119906119888) of the autonomous system (18) read
as follows
11987211= 3]2119888(radic3
2120582119909119888119910119888(21199092
119888+ ]2119888(1 + 3119909
4
119888)) minus 6120583
minus2
119888]21198881199092
119888
minus 4radic3120572120585119906119888119909119888]2119888119910minus1
119888) + radic3120582119909
119888119910119888minus 3 +M
11
11987212=radic3
4(120582 (2119909
2
119888+ ]2119888(1 + 3119909
4
119888)) + 8120572120585119906
119888]2119888119910minus2
119888) +M
12
11987213= minus2radic3120572120585]2
119888119910minus1
119888+M13
11987221=
21199102
119888(radic3120572120585119906
119888+ 3119909119888119910119888]minus2119888)
(21205851199062119888+ 1)
minusradic3120582119910
2
119888
2
11987222=
2119910119888(minus (94) 120583
minus4
119888119910119888+ 5radic3120572120585119909
119888119906119888)
(21205851199062119888+ 1)
minus radic3120582119909119888119910119888+3
2
11987223=
61205851199061198881199102
119888(minus (10radic33) 120572120585119906
119888119909119888+ 120583minus4
1198881199102
119888)
(21205851199062119888+ 1)2
+5radic3120572120585119909
1198881199102
119888
21205851199062119888+ 1
11987231=radic3120572119910
119888
2
11987232=radic3120572119909
119888
2
11987233= 0
(A1)
where in the case of 119876 = 0 we have M11= M12= M13= 0
and in the case of 119876 = 120573120581120588119898120601 we have
M11= 4radic3120573]minus2
119888119909119888119910119888
M12= 2radic3120573120583
minus2
119888(1 + 3119909
2
119888)
M13= minus4radic3120573120585119906
119888119910minus1
119888
(A2)
Examining the eigenvalues of the matrix119872 for each criticalpoint one determines its stability conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989
[2] S M Carroll ldquoThe cosmological constantrdquo Living Reviews inRelativity vol 4 article 1 2001
[3] R-J Yang and S N Zhang ldquoThe age problem in the ΛCDMmodelrdquo Monthly Notices of the Royal Astronomical Society vol407 no 3 pp 1835ndash1841 2010
[4] Y-F Cai E N Saridakis M R Setare and J-Q Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010
[5] 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
[6] A Sen ldquoRemarks on tachyon driven cosmologyrdquo PhysicaScripta vol 2005 no 117 article 70 2005
[7] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002
[8] M R Garousi ldquoTachyon couplings on non-BPS D-branes andDiracndashBornndashInfeld actionrdquo Nuclear Physics B vol 584 no 1-2pp 284ndash299 2000
[9] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002
[10] A Unzicker and T Case ldquoTranslation of Einsteinrsquos attempt ofa unified field theory with teleparallelismrdquo httparxivorgabsphysics0503046
[11] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 pp 3524ndash3553 1979 Addendum in PhysicalReview D vol 24 pp 3312-3314 1982
[12] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction Springer Dordrecht Netherlands 2013
[13] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013
[14] R Ferraro andF Fiorini ldquoModified teleparallel gravity inflationwithout an inflatonrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 75 no 8 Article ID 084031 2007
[15] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D Particles Fields Gravitation andCosmology vol 79 Article ID 124019 2009
[16] E V Linder ldquoEinsteins other gravity and the acceleration of theUniverserdquo Physical Review D vol 81 Article ID 127301 2010
12 Advances in High Energy Physics
[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010
[18] 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
[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984
[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999
[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000
[22] 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 Article ID 0435392004
[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009
[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007
[25] 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
[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010
[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011
[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012
[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012
[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012
[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013
[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012
[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013
[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014
[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007
[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence
principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009
[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007
[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008
[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009
[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010
[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005
[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006
[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008
[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008
[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012
[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010
[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998
[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999
[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005
[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011
[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003
[52] 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
[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008
[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011
[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 9
y(t)
x(t)
323
322
321
320
319
318
317
minus0004 minus0002 0 000600040002
(a)u(t)
x(t)
3025
3020
3015
3010
3005
minus0001 0001 0003 0005 0007
(b)
y(t)
u(t)
3025
3020
3015
3010
3170 3175 3180 3185 3190
3005
(c)
Figure 6 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus05and 120573 = 15 for 119876 = 120573120581120588
119898120601 For these values of the parameters point 119861
4is a stable attractor of the model
it provides a natural way for the stabilization of the darkenergy equation of state to the cosmological constant valuewithout the need for parameter tuning Also the authors in[55] by using power-law potentials and nonminimal couplingfunctions have shown that teleparallel gravity has no stablefuture solutions for positive nonminimal coupling and thescalar field results always in oscillationsOur results show thatgeneralized tachyon field represents interesting cosmologicalbehavior in comparison with ordinary tachyon fields in the
framework of teleparallel gravity because there is no scalingattractor in the latter model Note that however in [33]Otalora has considered a dynamically changing coefficient120572 = 119891
120601radic119891 and found a field matter dominated solution
in which it has some portions of the dark energy density inthe matter dominated era So generalized tachyon field givesus the hope that cosmological coincidence problem can bealleviated without fine-tunings One can study our model fordifferent kinds of potential and other famous interaction termbetween dark energy and dark matter
10 Advances in High Energy Physics
u(t)
minus01 0 01 02
x(t)
110
107
105104103
101
y(t)
024
022
020
018
016
014
012
010
008
(a)u(t)
minus002
minus004
008
006
004
002
0
minus005 0 005 010 015 020
x(t)
110
108
106
104
102
100
y(t)
(b)
y(t)
3170
3175
31803185
3190
minus00020
00020004
0006
x(t)
u(t)
3025
3020
3015
3010
3005
(c)
Figure 7Three-dimensional phase-space trajectories of the model for119876 = 120573120581120588119898120601with stable attractors 119861
2(a) 119861
3(b) and 119861
4(c)The values
of the parameters are those mentioned in Figures 4 5 and 6 respectively
Table 3 Location of the critical points and the corresponding values of the dark energy density parameterΩ120601and equation of state120596
120601and the
condition required for an accelerating universe for119876 = 120573120581120588119898120601 Here 120579
1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579
2= (4120572120585minusradic1612057221205852 minus 21205822120585)2120582120585
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198611
02radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198612
02radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572(120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
1198613
0 minus2radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198614
0 minus2radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572 (120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
Advances in High Energy Physics 11
Table 4 Stability and existence conditions of the critical points ofthe model for 119876 = 120573120581120588
119898120601
Label Stability Existence
1198611
Saddle pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 and 120585 ge 120582281205722
1198612
Saddle pointif 120572 lt 0 and 120585 lt 0 and
stable pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198613
Saddle pointif 120572 gt 0 and 120585 lt 0 and
stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198614
Stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 and 120585 ge 120582281205722
Appendix
Perturbation Matrix Elements
The elements of 3 times 3 matrix 119872 of the linearized per-turbation equations for the real and physically meaningfulcritical points (119909
119888 119910119888 119906119888) of the autonomous system (18) read
as follows
11987211= 3]2119888(radic3
2120582119909119888119910119888(21199092
119888+ ]2119888(1 + 3119909
4
119888)) minus 6120583
minus2
119888]21198881199092
119888
minus 4radic3120572120585119906119888119909119888]2119888119910minus1
119888) + radic3120582119909
119888119910119888minus 3 +M
11
11987212=radic3
4(120582 (2119909
2
119888+ ]2119888(1 + 3119909
4
119888)) + 8120572120585119906
119888]2119888119910minus2
119888) +M
12
11987213= minus2radic3120572120585]2
119888119910minus1
119888+M13
11987221=
21199102
119888(radic3120572120585119906
119888+ 3119909119888119910119888]minus2119888)
(21205851199062119888+ 1)
minusradic3120582119910
2
119888
2
11987222=
2119910119888(minus (94) 120583
minus4
119888119910119888+ 5radic3120572120585119909
119888119906119888)
(21205851199062119888+ 1)
minus radic3120582119909119888119910119888+3
2
11987223=
61205851199061198881199102
119888(minus (10radic33) 120572120585119906
119888119909119888+ 120583minus4
1198881199102
119888)
(21205851199062119888+ 1)2
+5radic3120572120585119909
1198881199102
119888
21205851199062119888+ 1
11987231=radic3120572119910
119888
2
11987232=radic3120572119909
119888
2
11987233= 0
(A1)
where in the case of 119876 = 0 we have M11= M12= M13= 0
and in the case of 119876 = 120573120581120588119898120601 we have
M11= 4radic3120573]minus2
119888119909119888119910119888
M12= 2radic3120573120583
minus2
119888(1 + 3119909
2
119888)
M13= minus4radic3120573120585119906
119888119910minus1
119888
(A2)
Examining the eigenvalues of the matrix119872 for each criticalpoint one determines its stability conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989
[2] S M Carroll ldquoThe cosmological constantrdquo Living Reviews inRelativity vol 4 article 1 2001
[3] R-J Yang and S N Zhang ldquoThe age problem in the ΛCDMmodelrdquo Monthly Notices of the Royal Astronomical Society vol407 no 3 pp 1835ndash1841 2010
[4] Y-F Cai E N Saridakis M R Setare and J-Q Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010
[5] 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
[6] A Sen ldquoRemarks on tachyon driven cosmologyrdquo PhysicaScripta vol 2005 no 117 article 70 2005
[7] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002
[8] M R Garousi ldquoTachyon couplings on non-BPS D-branes andDiracndashBornndashInfeld actionrdquo Nuclear Physics B vol 584 no 1-2pp 284ndash299 2000
[9] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002
[10] A Unzicker and T Case ldquoTranslation of Einsteinrsquos attempt ofa unified field theory with teleparallelismrdquo httparxivorgabsphysics0503046
[11] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 pp 3524ndash3553 1979 Addendum in PhysicalReview D vol 24 pp 3312-3314 1982
[12] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction Springer Dordrecht Netherlands 2013
[13] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013
[14] R Ferraro andF Fiorini ldquoModified teleparallel gravity inflationwithout an inflatonrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 75 no 8 Article ID 084031 2007
[15] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D Particles Fields Gravitation andCosmology vol 79 Article ID 124019 2009
[16] E V Linder ldquoEinsteins other gravity and the acceleration of theUniverserdquo Physical Review D vol 81 Article ID 127301 2010
12 Advances in High Energy Physics
[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010
[18] 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
[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984
[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999
[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000
[22] 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 Article ID 0435392004
[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009
[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007
[25] 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
[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010
[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011
[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012
[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012
[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012
[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013
[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012
[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013
[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014
[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007
[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence
principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009
[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007
[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008
[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009
[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010
[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005
[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006
[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008
[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008
[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012
[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010
[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998
[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999
[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005
[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011
[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003
[52] 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
[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008
[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011
[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
10 Advances in High Energy Physics
u(t)
minus01 0 01 02
x(t)
110
107
105104103
101
y(t)
024
022
020
018
016
014
012
010
008
(a)u(t)
minus002
minus004
008
006
004
002
0
minus005 0 005 010 015 020
x(t)
110
108
106
104
102
100
y(t)
(b)
y(t)
3170
3175
31803185
3190
minus00020
00020004
0006
x(t)
u(t)
3025
3020
3015
3010
3005
(c)
Figure 7Three-dimensional phase-space trajectories of the model for119876 = 120573120581120588119898120601with stable attractors 119861
2(a) 119861
3(b) and 119861
4(c)The values
of the parameters are those mentioned in Figures 4 5 and 6 respectively
Table 3 Location of the critical points and the corresponding values of the dark energy density parameterΩ120601and equation of state120596
120601and the
condition required for an accelerating universe for119876 = 120573120581120588119898120601 Here 120579
1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579
2= (4120572120585minusradic1612057221205852 minus 21205822120585)2120582120585
Label (119909119888 119910119888 119906119888) Ω
120601120596120601
Acceleration
1198611
02radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198612
02radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572(120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
1198613
0 minus2radic2120582120572120585120579
1
120582 1205791
21205851205791(4120572
120582minus 1205791)
2 (1205851205793
1+ 2120572120582)
(1205791minus 4120572120582) (1 + 2120585120579
2
1)
120582 gt8120572 (120585120579
2
1minus 1)
1205791(81205851205792
1+ 1)
1198614
0 minus2radic2120582120572120585120579
2
120582 1205792
21205851205792(4120572
120582minus 1205792)
2 (1205851205793
2+ 2120572120582)
(1205792minus 4120572120582) (1 + 2120585120579
2
2)
120582 gt8120572 (120585120579
2
2minus 1)
1205792(81205851205792
2+ 1)
Advances in High Energy Physics 11
Table 4 Stability and existence conditions of the critical points ofthe model for 119876 = 120573120581120588
119898120601
Label Stability Existence
1198611
Saddle pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 and 120585 ge 120582281205722
1198612
Saddle pointif 120572 lt 0 and 120585 lt 0 and
stable pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198613
Saddle pointif 120572 gt 0 and 120585 lt 0 and
stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198614
Stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 and 120585 ge 120582281205722
Appendix
Perturbation Matrix Elements
The elements of 3 times 3 matrix 119872 of the linearized per-turbation equations for the real and physically meaningfulcritical points (119909
119888 119910119888 119906119888) of the autonomous system (18) read
as follows
11987211= 3]2119888(radic3
2120582119909119888119910119888(21199092
119888+ ]2119888(1 + 3119909
4
119888)) minus 6120583
minus2
119888]21198881199092
119888
minus 4radic3120572120585119906119888119909119888]2119888119910minus1
119888) + radic3120582119909
119888119910119888minus 3 +M
11
11987212=radic3
4(120582 (2119909
2
119888+ ]2119888(1 + 3119909
4
119888)) + 8120572120585119906
119888]2119888119910minus2
119888) +M
12
11987213= minus2radic3120572120585]2
119888119910minus1
119888+M13
11987221=
21199102
119888(radic3120572120585119906
119888+ 3119909119888119910119888]minus2119888)
(21205851199062119888+ 1)
minusradic3120582119910
2
119888
2
11987222=
2119910119888(minus (94) 120583
minus4
119888119910119888+ 5radic3120572120585119909
119888119906119888)
(21205851199062119888+ 1)
minus radic3120582119909119888119910119888+3
2
11987223=
61205851199061198881199102
119888(minus (10radic33) 120572120585119906
119888119909119888+ 120583minus4
1198881199102
119888)
(21205851199062119888+ 1)2
+5radic3120572120585119909
1198881199102
119888
21205851199062119888+ 1
11987231=radic3120572119910
119888
2
11987232=radic3120572119909
119888
2
11987233= 0
(A1)
where in the case of 119876 = 0 we have M11= M12= M13= 0
and in the case of 119876 = 120573120581120588119898120601 we have
M11= 4radic3120573]minus2
119888119909119888119910119888
M12= 2radic3120573120583
minus2
119888(1 + 3119909
2
119888)
M13= minus4radic3120573120585119906
119888119910minus1
119888
(A2)
Examining the eigenvalues of the matrix119872 for each criticalpoint one determines its stability conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989
[2] S M Carroll ldquoThe cosmological constantrdquo Living Reviews inRelativity vol 4 article 1 2001
[3] R-J Yang and S N Zhang ldquoThe age problem in the ΛCDMmodelrdquo Monthly Notices of the Royal Astronomical Society vol407 no 3 pp 1835ndash1841 2010
[4] Y-F Cai E N Saridakis M R Setare and J-Q Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010
[5] 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
[6] A Sen ldquoRemarks on tachyon driven cosmologyrdquo PhysicaScripta vol 2005 no 117 article 70 2005
[7] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002
[8] M R Garousi ldquoTachyon couplings on non-BPS D-branes andDiracndashBornndashInfeld actionrdquo Nuclear Physics B vol 584 no 1-2pp 284ndash299 2000
[9] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002
[10] A Unzicker and T Case ldquoTranslation of Einsteinrsquos attempt ofa unified field theory with teleparallelismrdquo httparxivorgabsphysics0503046
[11] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 pp 3524ndash3553 1979 Addendum in PhysicalReview D vol 24 pp 3312-3314 1982
[12] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction Springer Dordrecht Netherlands 2013
[13] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013
[14] R Ferraro andF Fiorini ldquoModified teleparallel gravity inflationwithout an inflatonrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 75 no 8 Article ID 084031 2007
[15] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D Particles Fields Gravitation andCosmology vol 79 Article ID 124019 2009
[16] E V Linder ldquoEinsteins other gravity and the acceleration of theUniverserdquo Physical Review D vol 81 Article ID 127301 2010
12 Advances in High Energy Physics
[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010
[18] 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
[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984
[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999
[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000
[22] 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 Article ID 0435392004
[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009
[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007
[25] 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
[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010
[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011
[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012
[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012
[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012
[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013
[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012
[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013
[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014
[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007
[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence
principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009
[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007
[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008
[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009
[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010
[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005
[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006
[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008
[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008
[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012
[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010
[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998
[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999
[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005
[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011
[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003
[52] 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
[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008
[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011
[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 11
Table 4 Stability and existence conditions of the critical points ofthe model for 119876 = 120573120581120588
119898120601
Label Stability Existence
1198611
Saddle pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 and 120585 ge 120582281205722
1198612
Saddle pointif 120572 lt 0 and 120585 lt 0 and
stable pointif 120572 gt 0 and 120585 gt 0
120582 gt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198613
Saddle pointif 120572 gt 0 and 120585 lt 0 and
stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 andfor all 120585 lt 0
or120585 ge 120582
281205722
1198614
Stable pointif 120572 lt 0 and 120585 gt 0
120582 lt 0 and 120585 ge 120582281205722
Appendix
Perturbation Matrix Elements
The elements of 3 times 3 matrix 119872 of the linearized per-turbation equations for the real and physically meaningfulcritical points (119909
119888 119910119888 119906119888) of the autonomous system (18) read
as follows
11987211= 3]2119888(radic3
2120582119909119888119910119888(21199092
119888+ ]2119888(1 + 3119909
4
119888)) minus 6120583
minus2
119888]21198881199092
119888
minus 4radic3120572120585119906119888119909119888]2119888119910minus1
119888) + radic3120582119909
119888119910119888minus 3 +M
11
11987212=radic3
4(120582 (2119909
2
119888+ ]2119888(1 + 3119909
4
119888)) + 8120572120585119906
119888]2119888119910minus2
119888) +M
12
11987213= minus2radic3120572120585]2
119888119910minus1
119888+M13
11987221=
21199102
119888(radic3120572120585119906
119888+ 3119909119888119910119888]minus2119888)
(21205851199062119888+ 1)
minusradic3120582119910
2
119888
2
11987222=
2119910119888(minus (94) 120583
minus4
119888119910119888+ 5radic3120572120585119909
119888119906119888)
(21205851199062119888+ 1)
minus radic3120582119909119888119910119888+3
2
11987223=
61205851199061198881199102
119888(minus (10radic33) 120572120585119906
119888119909119888+ 120583minus4
1198881199102
119888)
(21205851199062119888+ 1)2
+5radic3120572120585119909
1198881199102
119888
21205851199062119888+ 1
11987231=radic3120572119910
119888
2
11987232=radic3120572119909
119888
2
11987233= 0
(A1)
where in the case of 119876 = 0 we have M11= M12= M13= 0
and in the case of 119876 = 120573120581120588119898120601 we have
M11= 4radic3120573]minus2
119888119909119888119910119888
M12= 2radic3120573120583
minus2
119888(1 + 3119909
2
119888)
M13= minus4radic3120573120585119906
119888119910minus1
119888
(A2)
Examining the eigenvalues of the matrix119872 for each criticalpoint one determines its stability conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989
[2] S M Carroll ldquoThe cosmological constantrdquo Living Reviews inRelativity vol 4 article 1 2001
[3] R-J Yang and S N Zhang ldquoThe age problem in the ΛCDMmodelrdquo Monthly Notices of the Royal Astronomical Society vol407 no 3 pp 1835ndash1841 2010
[4] Y-F Cai E N Saridakis M R Setare and J-Q Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010
[5] 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
[6] A Sen ldquoRemarks on tachyon driven cosmologyrdquo PhysicaScripta vol 2005 no 117 article 70 2005
[7] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002
[8] M R Garousi ldquoTachyon couplings on non-BPS D-branes andDiracndashBornndashInfeld actionrdquo Nuclear Physics B vol 584 no 1-2pp 284ndash299 2000
[9] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002
[10] A Unzicker and T Case ldquoTranslation of Einsteinrsquos attempt ofa unified field theory with teleparallelismrdquo httparxivorgabsphysics0503046
[11] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 pp 3524ndash3553 1979 Addendum in PhysicalReview D vol 24 pp 3312-3314 1982
[12] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction Springer Dordrecht Netherlands 2013
[13] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013
[14] R Ferraro andF Fiorini ldquoModified teleparallel gravity inflationwithout an inflatonrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 75 no 8 Article ID 084031 2007
[15] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D Particles Fields Gravitation andCosmology vol 79 Article ID 124019 2009
[16] E V Linder ldquoEinsteins other gravity and the acceleration of theUniverserdquo Physical Review D vol 81 Article ID 127301 2010
12 Advances in High Energy Physics
[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010
[18] 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
[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984
[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999
[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000
[22] 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 Article ID 0435392004
[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009
[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007
[25] 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
[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010
[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011
[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012
[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012
[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012
[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013
[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012
[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013
[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014
[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007
[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence
principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009
[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007
[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008
[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009
[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010
[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005
[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006
[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008
[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008
[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012
[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010
[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998
[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999
[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005
[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011
[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003
[52] 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
[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008
[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011
[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
12 Advances in High Energy Physics
[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010
[18] 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
[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984
[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999
[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000
[22] 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 Article ID 0435392004
[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009
[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007
[25] 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
[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010
[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011
[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012
[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012
[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012
[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013
[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012
[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013
[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014
[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007
[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence
principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009
[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007
[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008
[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009
[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010
[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005
[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006
[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008
[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008
[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012
[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010
[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998
[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999
[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005
[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011
[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003
[52] 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
[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008
[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011
[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
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