Hindawi Publishing Corporation Advances in High Energy Physics Volume 2015, Article ID 608252, 20 pages http://dx.doi.org/10.1155/2015/608252
Research Article Dynamics of Mixed Dark Energy Domination in Teleparallel Gravity and Phase-Space Analysis
Emre Dil1 and Erdinç Kolay2
1 Department of Physics, Sinop University, Korucuk, 57000 Sinop, Turkey 2Department of Statistics, Sinop University, Korucuk, 57000 Sinop, Turkey
Correspondence should be addressed to Emre Dil; [email protected]
Received 16 October 2015; Revised 26 November 2015; Accepted 29 November 2015
Academic Editor: Frank Filthaut
Copyright © 2015 E. Dil and E. Kolay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.
We consider a novel dark energy model to investigate whether it will provide an expanding universe phase. Here we propose a mixed dark energy domination which is constituted by tachyon, quintessence, and phantom scalar fields nonminimally coupled to gravity, in the absence of background dark matter and baryonic matter, in the framework of teleparallel gravity. We perform the phase-space analysis of the model by numerical methods and find the late-time accelerated attractor solutions implying the acceleration phase of universe.
1. Introduction quantity of dark energy; its equation of state parameter đDE = đDE/đDE,wheređDE and đDE are the pressure and energy The universe is known to be experiencing an accelerating density of the dark energy, respectively. For cosmological expansion by astrophysical observations such as Supernova constant boundary đDE =â1,butforquintessencethe Ia [1, 2], large scale structure [3, 4], the baryon acoustic parameter đDE â„â1,forphantomđDE â€â1,andfor oscillations [5], and cosmic microwave background radiation nonminimally coupled tachyon with gravity both đDE â€â1 [6â9]. and đDE â„â1[16â18]. In order to explain the late-time accelerated expansion The scenario of đDE crossing the cosmological constant of universe, an unknown form of energy, called dark energy, boundary is referred to as a âQuintomâ scenario. The explicit is proposed. This unknown component of energy possesses construction of Quintom scenario has a difficulty, due to a some interesting properties; for instance, it is not clustered no-gotheorem.TheequationofstateparameterđDE of a butspreadallovertheuniverseanditspressureisnegativefor scalar field cannot cross the cosmological constant boundary driving the current acceleration of the universe. Observations according to no-go theorem, if the dark energy described by show that the dark energy occupies 70% of our universe. the scalar field is minimally coupled to gravity in Friedmann- What is the constitution of the dark energy? One can- Robertson-Walker (FRW) geometry. The requirement for didate for the answer of this question is the cosmological crossing the cosmological constant boundary is that the dark constant Î having a constant energy density filling the space energy should be nonminimally coupled to gravity; namely, it homogeneously [10â13]. But cosmological constant is not should interact with the gravity [19â24]. There are also mod- well accepted since the cosmological problem [14] and the els in which possible coupling between dark energy and dark age problem [15]. For this reason, many other dark energy matter can occur [25, 26]. In this paper, we consider a mixed models have been proposed instead of the cosmological dark energy model constituted by tachyon, quintessence, and constant. Other candidates for dark energy constitution are phantom scalar fields nonminimally coupled to gravity. quintessence,phantom,andtachyonfields.Wecanbriefly The mixed dark energy model in this study is considered classifythedarkenergymodelsintermsofthemostpowerful in the framework of teleparallel gravity instead of classical 2 Advances in High Energy Physics gravity. The teleparallel gravity is the equivalent form of with a nonminimal coupling to the gravity can be written as the classical gravity, but in place of torsion-less Levi-Civita [16, 38, 39] connection, curvature-less Weitzenbock one is used. The Lagrangian of teleparallel gravity contains torsion scalar đ đ đ=â« đ4đ„đ [ +đ đ(đ)đ constructed by the contraction of torsion tensor, in contrast 2đ 2 1 to the Einstein-Hilbert action of classical gravity in which [ contraction of the curvature tensor đ is used. In teleparallel đ đđ đ gravity, the dynamical variable is a set of four tetrad fields đ] đ ] âđđ (đ) â1+đ +đ2đ(đ)đ constructing the bases for the tangent space at each point of đđ (đ) space-time [27â29]. The teleparallel gravity Lagrangian with (1) only torsion scalar đ corresponds to the matter-dominated 1 1 â đđ]đ đđ đâđ (đ) + đ â (đ) đ+ đđ]đ đđ đ universe; namely, it does not accelerate. Therefore, to obtain 2 đ ] đ 3 2 đ ] a universe with an accelerating expansion, we can either replace đ with a function đ(đ), the so-called đ(đ) gravity ] (teleparallel analogue of đ(đ ) gravity) [30â32], or add an âđđ (đ) , unknown form of energy, so-called dark energy, to the ] teleparallel gravity Lagrangian that allows also nonminimal đ đ coupling between dark energy and gravity to overcome the where đ=det(đđ)=ââđ and đđ are the orthonormal tetrad no-go theorem. The interesting feature of đ(đ) theories is the components, such that existence of second or higher order derivatives in equations. đ đ Therefore, we prefer the second choice; adding extra scalar đđ] =đđđ đđđ], (2) fields of the unknown energy forms as dark energy. where đ, đ runover0,1,2,and3forthetangentspaceateach As different dark energy models, interacting teleparallel đ darkenergystudieshavebeenintroducedintheliterature;for point đ„ of the manifold and đ, ] take the values 0, 1, 2, and instance, Geng et al. [33, 34] consider a quintessence scalar 3 and are the coordinate indices of the manifold. While đ is field with a nonminimal coupling between quintessence and the torsion scalar, it is defined as gravity in the context of teleparallel gravity. The dynamics of 1 1 đ= đđđ]đ + đđđ]đ âđđ đ]đ. this model has been studied in [35â37]. Tachyonic teleparallel 4 đđ] 2 ]đđ đ] đ (3) dark energy is a generalization of the teleparallel quintessence đ dark energy by introducing a noncanonical tachyon scalar Here đđ] is the torsion tensor constructed by the Weitzenbock đ field in place of the canonical quintessence field [18, 38â40]. connection Îđ], such that [41] In this study, the main motivation is that we consider đ đ đ đ đ đ a more general dark energy model including three kinds đđ] =Î]đ âÎđ] =đđ (đđđ] âđ]đđ). (4) of dark energy models, instead of taking one dark energy model as in [16, 18, 38â40]. In order to explain the expansion All the information about the gravitational field is con- đ of universe by adding scalar fields as the dark energy tained in the torsion tensor đđ] in teleparallel gravity. The constituents, there has never been assumed a cosmological Lagrangian of the theory is set up according to the condi- model including three kinds of dark energy models. We tions of invariance under general coordinate transformations, assume tachyon, quintessence, and phantom fields as a global Lorentz transformations, and the parity operations mixed dark energy model which is nonminimally coupled to [42]. 2 gravity in the framework of teleparallel gravity. We make the Furthermore, đ =8đđșin (1) and đ(đ), đ(đ),and dynamical analysis of the model in FRW space-time. Later â(đ) are the functions responsible for nonminimal coupling on, we translate the evolution equations into an autonomous between gravity đ and tachyon, quintessence, and phantom dynamical system. After that the phase-space analysis of the fields, respectively. đ1, đ2,andđ3 are the dimensionless model and the cosmological implications of the critical (or coupling constants and đđ(đ), đđ(đ),andđđ(đ) are the fixed)pointsofthemodelwillbestudiedfromthestability potentials for tachyon, quintessence, and phantom fields, behavior of the critical points. Finally, we will make a brief respectively. summary of the results. We consider a spatially flat FRW metric
2 2 2 2 2 2 2. Dynamics of the Model đđ =đđĄ âđ (đĄ) [đđ +đ đΩ ], (5) đ Our model consists of three scalar fields as the three- and a tetrad field of the form đđ = diag(1,đ,đ,đ).Thenthe component dark energy domination without background Friedmann equations for FRW metric read as dark matter and baryonic matter. These are the canonical đ đ 2 scalar field, quintessence , and two noncanonical scalar đ»2 = (đ +đ +đ ), fields, tachyon đ and phantom đ, and all these scalar fields are 3 đ đ đ (6) nonminimally coupled to gravity. Since we consider only the 2 dark energy dominated sector without the matter content of Ì đ đ»=â (đđ +đđ +đđ +đđ +đđ +đđ), the universe, the action of the mixed teleparallel dark energy 2 Advances in High Energy Physics 3 where đ»=đ/đÌ is the Hubble parameter, đ is the scale factor, values for the tachyon, quintessence, and phantom fields read and dot represents the derivative with respect to cosmic time as đĄ. đ and đ are the energy density and the pressure of the đ (đ) corresponding scalar field constituents of the dark energy. 2 đ đđ =âđ1đ(đ)6đ» + , Conservation of energy gives the evolution equations for 2 (12) â1â(đÌ /đđ (đ)) thedarkenergyconstituents,suchas đÌ 2 đÌ +3đ»(đ +đ )=0, đ =đđ(đ)6đ»2 âđ (đ) â1â đ đ đ đ 1 đ đ (đ) đ (13) đÌ +3đ»(đ +đ )=0, đ đ đ (7) Ì óž 2 +4đ1đ»đ ( đ ) +4đ 1đ»đ (đ) đÌ , đÌ +3đ»(đ +đ )=0. đ đ đ 1 đ =âđđ(đ)6đ»2 + đÌ2 +đ (đ) , đ 2 2 đ (14) The total energy density and the pressure of dark energy read 1 đ =đđ(đ)6đ»2 + đÌ2 âđ (đ) + 4đ đ»đÌ (đ) đ 2 2 đ 2 (15) đtot =đDE =đđ +đđ +đđ, óž Ì (8) +4đ2đ»đ (đ) đ, đ =đ =đ +đ +đ , tot DE đ đ đ 1 đ =âđâ (đ) 6đ»2 â đÌ2 +đ (đ) , đ 3 2 đ (16) with the total equation of state parameter 1 đ =đâ (đ) 6đ»2 â đÌ2 âđ (đ) +4đ đ»âÌ (đ) đ đ 3 2 đ 3 đ = tot =đ Ω +đ Ω +đ Ω , (17) tot đ đ đ đ đ đ đ (9) óž tot +4đ3đ»â (đ) đ,Ì
2 where đđ =đđ/đđ, đđ =đđ/đđ,andđđ =đđ/đđ are the respectively.Herewehaveusedtherelationđ=â6đ»,and equation of state parameters and Ωđ =đđ/đtot, Ωđ =đđ/đtot, prime denotes the derivative of related coupling functions and Ωđ =đđ/đtot are the density parameters for the tachyon, withrespecttotherelatedfieldvariables.Now,wecanfindthe quintessence, and phantom fields, respectively. Then the total equation of motions for three scalar fields from the variation density parameter is defined as of the field Lagrangians (11) with respect to the field variables đ, đ,andđ,suchthat đ 2đ Ω =Ω +Ω +Ω = tot =1, đÌ 2 3 đÌ 2 tot đ đ đ 2 (10) 3đ» đ+3đ»Ì đ(1âÌ )+(1â )đđ (đ) đđ (đ) 2 đđ (đ)
3/2 where we assume that three kinds of scalar fields constitute đÌ 2 the dark energy with an equal proportion of density parame- +6đ (1 â ) đ»2đóž (đ) = 0, 1 đ (đ) (18) ter such that Ωđ =Ωđ =Ωđ =1/3. đ The Lagrangian of the scalar fields is reexpressed from the đ+đÌ óž (đ) + 6đ đ»2đóž (đ) + 3đ»đ=0,Ì action in (1), as đ 2 óž 2 óž đâđÌ đ (đ) â6đ3đ» â (đ) +3đ»đ=0.Ì đÌ 2 đż =đđ(đ)đâđ (đ) â1â , đ 1 đ These equations of motions are for the tachyon, quintessence, đđ (đ) and phantom constituents of dark energy, respectively. Here, 1 2 (11) the prime of potentials denotes the derivative of related field đż =đđ(đ)đ+ đÌ âđ (đ) , đ 2 2 đ potentialswithrespecttotherelatedfieldvariables.Allthese evolution equations in (18) can also be obtained by using the 1 đż =đâ (đ) đâ đÌ2 âđ (đ) . relations (12)â(17) in the continuity equations (7). đ 3 2 đ We now perform the phase-space analysis of the model in order to investigate the late-time solutions of the universe Then the energy density and pressure values for three scalar considered here. fieldscanbefoundbythevariationofthetotalLagrangian đđ in (1) with respect to the tetrad field đ .Afterthevariation 3. Phase-Space and Stability Analysis of Lagrangian, there come contributions from the geometric terms so the (0, 0)-component and (đ, đ)-component of the We study the properties of the constructed dark energy stress-energy tensor give the energy density and the pressure, model by performing the phase-space analysis. Therefore respectively. Accordingly the energy density and pressure we transform the aforementioned dynamical system into its 4 Advances in High Energy Physics autonomous form [38, 39, 43â46]. To proceed we introduce form by using (12)â(17) in đ=đ/đfor every scalar field, such the auxiliary variables as
đđ đÌ đđ = đ„đ = , đđ âđđ (đ) â1 2 â âđ đŠ +2đ1đąđ [(2 3/3) đŒđđ„đđŠđ +đąđ (1â(2/3) đ )] = đ , 2 2 đđŠđ â2đ1đąđ đ âđđ (đ) đŠ = , đ đ â3đ» đ đđ = đđ (22) đą =đ âđ(đ), 2 2 â đ đ„ âđŠ +2đ2đąđ [(2 6/3) đŒđđ„đ +đąđ (1â(2/3) đ )] = đ đ , 2 2 2 đ„đ +đŠđ â2đ2đąđ đ đÌ đ„đ = , đđ â6đ» đđ = đđ đ âđ (đ) (19) âđ„2 âđŠ2 +2đ đą [(2â6/3) đŒ đ„ +đą (1â(2/3) đ )] đ = đ đ 3 đ đ đ đ , đŠđ = , âđ„2 +đŠ2 â2đ đą2 â3đ» đ đ 3 đ óž óž where đŒđ =đ(đ)/âđ(đ), đŒđ =đ(đ)/âđ(đ), đŒđ = đą =đ âđ(đ), óž 2 đ â (đ)/ââ(đ) and đ =âđ»/đ»Ì . From (9) and (20) and (22), we obtain đtot in the autonomous system, as đ đÌ đ„đ = , â1 2 â6đ» đtot =âđ đŠđ 2â3 2 đ âđđ (đ) 2 đŠ = , +2đ1đąđ [ đŒđđ„đđŠđ +đąđ (1 â đ )] +đ đ„ đ â3đ» 3 3
2 2â6 2 đą =đ ââ (đ), âđŠ +2đ đą [ đŒ đ„ +đą (1 â đ )] (23) đ đ 2 đ 3 đ đ đ 3 đ= đ đč together with ln and for any quantity ,thetime âđ„2 âđŠ2 derivative is đč=đ»(đđč/đđ)Ì . đ đ We rewrite the density parameters for the fields đ, đ,and 2â6 2 đ +2đ đą [ đŒ đ„ +đą (1 â đ )] . in the autonomous system by using (10), (12), (14), and (16) 3 đ 3 đ đ đ 3 with (19) We can express đ in the autonomous system by using (6) and đ 2đ Ω = đ =đđŠ2 â2đ đą2 , (23), such that đ 3đ»2 đ 1 đ đ»Ì 3 3 3 đ =â = (1 + đ )= â đâ1đŠ2 đ 2đ đ»2 2 tot 2 2 đ Ω = đ =đ„2 +đŠ2 â2đ đą2 , (20) đ 3đ»2 đ đ 2 đ 3 3 +đ đą [2â3đŒ đ„ đŠ +đą (3â2đ )]+ đ„2 â đŠ2 1 đ đ đ đ đ 2 đ 2 đ đ 2đ Ω = đ =âđ„2 +đŠ2 â2đ đą2 , 3 3 đ 3đ»2 đ đ 3 đ +đ đą [2â6đŒ đ„ +đą (3â2đ )]â đ„2 â đŠ2 2 đ đ đ đ 2 đ 2 đ and the total density parameter is â +đ3đąđ [2 6đŒđđ„đ +đąđ (3â2đ )], (24) đ 2đ 3 3 Ω = tot =Ω +Ω +Ω đ ={ â đâ1đŠ2 +đ đą [2â3đŒ đ„ đŠ +3đą ] tot 3đ»2 đ đ đ 2 2 đ 1 đ đ đ đ đ 2 2 2 2 2 2 2 (21) 3 2 3 2 3 2 =đđŠđ â2đ1đąđ +đ„đ +đŠđ â2đ2đąđ âđ„đ +đŠđ + đ„ â đŠ +đ đą [2â6đŒ đ„ +3đą ]â đ„ 2 đ 2 đ 2 đ đ đ đ 2 đ 2 â2đ3đąđ =1, 3 â đŠ2 +đ đą [2â6đŒ đ„ +3đą ]} [1 + 2đ đą2 2 đ 3 đ đ đ đ 1 đ Ì 2 2 where đ=1/â1â(đ /đđ(đ)) = 1/â1âđ„đ.Thentheequa- â1 +2đ đą2 +2đ đą2 ] . tion of state parameters can be written in the autonomous 2 đ 3 đ Advances in High Energy Physics 5
đ â6 Here is only a jerk parameter used in other equations of đąóž = đŒ đ„ , cosmological parameters. Then the deceleration parameter đ đ 2 đ đ is óž 2 2 2 â đ„đ =â3đ„đ [1 + đ„đ âđ„đ + đ2 (đ đąđ â 6đŒđđ„đ)đąđ đ»Ì 1 3 3 đ=â1â = + đtot 2 đ đ»2 2 2 + đ (đ đą â â6đŒ đ„ )đą â đ„2 đŠ2 3 3 đ đ đ đ 2 đ đ 1 3 â1 2 â = â đ đŠ +đ1đąđ [2 3đŒđđ„đđŠđ +đąđ (3â2đ )] 2 â6 2 2 đ + đ (đ đą â â6đŒ đ„ )đą ]+đ đŠ2 (25) 3 1 đ đ đ đ đ 2 đ 3 3 + đ„2 â đŠ2 +đ đą [2â6đŒ đ„ +đą 3â2đ ] đ đ 2 đ đ đ đ ( ) â 2 2 + 6đ3đŒđđąđ, 3 3 2 2 â óž đ 2 2 2 2 â đ„ â đŠ +đ3đąđ [2 6đŒđđ„đ +đąđ (3â2đ )]. đŠ =3đŠ [ đ„ đŠ â đ (đ đą â â6đŒ đ„ )đą +đ„ 2 đ 2 đ đ đ 2 đ đ 3 1 đ đ đ đ đ Now we transform the equations of motions (6) and 2 â đ (đ đą â â6đŒ đ„ )đą âđ„2 (18) into the autonomous system containing the auxiliary 3 2 đ đ đ đ đ variables in (19) and their derivatives with respect to đ=ln đ. đóž = đ(đ) đ 2 â6 Thus we obtain ,where is the column vector â đ (đ đą â â6đŒ đ„ )đą âđ đ„2 ], including the auxiliary variables and đ(đ) is the column 3 3 đ đ đ đ đ 6 đ óž vector of the autonomous equations. After writing đ ,wefind óž the critical points đđ of đ,bysettingđ =0. We expand óž â6 óž đą = đŒđđ„đ, đ = đ(đ) around đ=đđ +đ,wheređ isthecolumnvector đ 2 đżđ„ đżđŠ of perturbations of the auxiliary variables, such as , ,and (26) đżđą for each scalar field. Thus, we expand the perturbation óž óž óž óž equations up to the first order for each critical point as đ = where đđ =âđđ/đ đđ, đđ =âđđ/đ đđ,andđđ =âđđ/đ đđ. đđ,wheređ is the matrix of perturbation equations. For Henceforth, we assume the nonminimal coupling functions đ 2 2 2 each critical point, the eigenvalues of perturbation matrix đ(đ) âđ , đ(đ) âđ ,andâ(đ) â đ ;thusđŒđ, đŒđ,and determine the type and stability of the critical points [47â50]. đŒđ are constant. Also the usual assumption in the literature âđđđđđ âđđđđđ Particularly, the autonomous form of the cosmological is to take the potentials đđ =đđ0đ , đđ =đđ0đ , system in (6) and (18) is [51â60] âđđđđđ and đđ =đđ0đ [43, 61â63]. Such potentials give also â3 constants đđ, đđ,andđđ. đ„óž = [đ đ„2 đŠ +đ (2 â 3đ„2 )đŠ đ 2 đ đ đ đ đ đ Now we perform the phase-space analysis of the model by finding the critical points of the autonomous system in (26). â3 â1 â â2 â4đŒđđ1đąđđ đŠđ â2 3đ„đđ ], To obtain these points, we set the left hand sides of (26) to zero. After some calculations, four critical points are found by â3 assuming đtot and đ as â1 for each critical point. The critical đŠóž =[â đ đ„ đŠ +đ ]đŠ , đ 2 đ đ đ đ points are listed in Table 1 with the existence conditions. Then we insert linear perturbations đ„âđ„đ +đżđ„, đŠâ â3 đŠđ +đżđŠ,andđąâđąđ +đżđąabout the critical points for three đąóž = đŒ đ„ đŠ , đ 2 đ đ đ scalar fields đ, đ,andđ in the autonomous system (26). Thus we obtain a 9Ă9perturbation matrix đ whose elements are 2 đ„óž =â3đ„ [1 + đ„2 âđ„2 + đ (đ đą â â6đŒ đ„ )đą given as đ đ đ đ 3 2 đ đ đ đ đ11 =â3, 2 đ + đ (đ đą â â6đŒ đ„ )đą â đ„2 đŠ2 3 đ đ đ đ đ đ â â â2 3 2 đ12 = 3đđ +2 3đŒđđ1đŠđ đąđ,
2 â6 2 â â1 + đ (đ đą â â6đŒ đ„ )đą ]+đ đŠ đ13 =â2 3đŒđđ1đŠđ , 3 1 đ đ đ đ đ 2 đ 2â3đŒ đ â đ 1 2 â3 2 â 6đ2đŒđđąđ, đ = đą đŠ â đ đŠ , 21 đ đ đ 2 đ đ đ 2 óž 2 2 â 2 3 2 đŠđ =3đŠđ [ đ„đđŠđ â đ1 (đ đąđ â 6đŒđđ„đ)đąđ +đ„đ đ =â đŠ , 2 3 22 đ đ 2 6đ â đ (đ đą â â6đŒ đ„ )đą âđ„2 đ = 1 đą đŠ , 3 2 đ đ đ đ đ 23 đ đ đ 2 â6 â â 2 2 6đŒđđ2 â đ3 (đ đąđ â 6đŒđđ„đ)đąđ âđđ đ„đ], đ = đą đŠ , 3 6 24 đ đ đ 6 Advances in High Energy Physics
0.65 0.35 0.25 0.6 0.3 0.2 0.25 0.55 0.15 0.2 0.5 0.15 0.1 đ đ đ x y 0.45 u 0.1 0.05 0.05 0.4 0 0 0.35 â0.05 â0.05 â0.1 â0.1 â0.2 â0.1 0 0.1 0.2 0.3 0.4 0.5 â0.2 â0.1 0 0.1 0.2 0.3 0.4 0.5 â0.2 â0.1 00.1 0.2 0.30.4 0.5
xđ xđ xđ 0.6 0.5 0.25 0.58 0.4 0.2 0.56 0.3 0.15 0.54 0.2 0.1 đ đ 0.52 đ y u 0.1 x 0.05 0.5 0.48 0 0 0.46 â0.1 â0.05 0.44 â0.2 â0.1 â0.2 â0.1 0 0.1 0.2 0.3 0.4 0.5 â0.2â0.1 0 0.1 0.2 0.30.4 0.5 â0.2â0.1 0 0.1 0.2 0.30.4 0.5 x xđ xđ đ 0.7 0 0.35 0.3 0.65 â0.05 â0.1 0.25 0.6 0.2 â0.15 0.55 0.15 đ đ đ â0.2 u y 0.5 u 0.1 â0.25 0.05 0.45 â0.3 0 0.4 â0.35 â0.05 0.35 â0.4 â0.1 â0.2 â0.1 0 0.1 0.2 0.3 0.4 0.5 â0.2 â0.1 0 0.1 0.2 0.3 0.4 0.5 0.35 0.4 0.45 0.5 0.55 0.6 0.65 y xđ xđ đ 0.25 0.6 0.5 0.2 0.58 0.4 0.15 0.56 0.3 0.54 0.1 0.2 đ đ đ 0.52 u x 0.05 y 0.1 0.5 0 0 0.48 â0.05 0.46 â0.1 â0.1 0.44 â0.2 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.35 0.4 0.45 0.5 0.55 0.60.65 0.35 0.4 0.45 0.5 0.55 0.6 0.65
yđ yđ yđ 0.25 0.7 0 0.2 0.65 â0.05 â0.1 0.15 0.6 â0.15 0.1 0.55 đ đ đ â0.2 u y x 0.05 0.5 â0.25 0 0.45 â0.3 â0.05 0.4 â0.35 â0.1 0.35 â0.4 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.35 0.4 0.45 0.5 0.55 0.6 0.65 yđ yđ yđ
Figure 1: Continued. Advances in High Energy Physics 7
0.25 0.6 0.5 0.2 0.58 0.4 0.15 0.56 0.3 0.54 0.1 0.2 đ đ đ 0.52 x u 0.05 y 0.1 0.5 0 0.48 0 â0.05 0.46 â0.1 â0.1 0.44 â0.2 â0.05 0.05 0.15 0.25 0.35 â0.05 0.05 0.15 0.25 0.35 â0.05 0.05 0.15 0.25 0.35
uđ uđ uđ 0.25 0.7 0 0.2 0.65 â0.05 0.15 0.6 â0.1 â0.15 0.1 0.55 đ đ đ â0.2 x u 0.05 y 0.5 â0.25 0 0.45 â0.3 â0.05 0.4 â0.35 â0.1 0.35 â0.4 â0.05 0.05 0.15 0.25 0.35 â0.05 0.05 0.15 0.25 0.35 â0.05 0.05 0.15 0.25 0.35
uđ uđ uđ 0.6 0.5 0.25 0.58 0.4 0.2 0.56 0.3 0.15 0.54 0.2 0.1 đ đ đ 0.52 u y 0.1 x 0.05 0.5 0.48 0 0 0.46 â0.1 â0.05 0.44 â0.2 â0.1 â0.1â0.05 0 0.05 0.1 0.15 0.2 0.25 â0.1â0.05 0 0.05 0.1 0.15 0.2 0.25 â0.1â0.05 0 0.05 0.1 0.15 0.2 0.25
xđ xđ xđ 0.7 0 0.5 0.65 â0.05 0.4 0.6 â0.1 0.3 â0.15 0.55 0.2 đ đ đ â0.2 y u 0.5 u 0.1 â0.25 0.45 â0.3 0 0.4 â0.35 â0.1 0.35 â0.4 â0.2 â0.1 â0.05 0 0.05 0.1 0.15 0.2 0.25 â0.1 â0.05 0 0.05 0.1 0.15 0.2 0.25 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6
xđ xđ yđ 0.25 0.7 0 0.2 0.65 â0.05 0.15 0.6 â0.1 â0.15 0.1 0.55 đ đ đ â0.2 u x 0.05 y 0.5 â0.25 0 0.45 â0.3 â0.05 0.4 â0.35 â0.1 0.35 â0.4 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6
yđ yđ yđ
Figure 1: Continued. 8 Advances in High Energy Physics
0.25 0.7 0 0.2 0.65 â0.05 0.15 0.6 â0.1 â0.15 0.1 0.55 đ đ đ â0.2 x u 0.05 y 0.5 â0.25 0 0.45 â0.3 â0.05 0.4 â0.35 â0.1 0.35 â0.4 â0.2 â0.1 0 0.1 0.2 0.3 0.4 0.5 â0.2â0.10 0.1 0.2 0.3 0.4 0.5 â0.2â0.1 0 0.1 0.2 0.3 0.4 0.5
uđ uđ uđ 0.7 0 0 0.65 â0.05 â0.05 0.6 â0.1 â0.1 â0.15 â0.15 0.55 đ đ â0.2 đ â0.2 y u 0.5 u â0.25 â0.25 0.45 â0.3 â0.3 0.4 â0.35 â0.35 0.35 â0.4 â0.4 â0.1 â0.05 0 0.05 0.10.15 0.2 0.25 â0.1 â0.05 0 0.05 0.1 0.15 0.2 0.25 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
xđ xđ yđ
Figure 1: Two-dimensional projections of the phase-space trajectories for đ1 =đ2 =âđ3 = 0.5, đŒđ =đŒđ =âđŒđ = 1.5.Allplotsbeginfrom the critical point đŽ being a stable attractor.
Table 1: Critical points and existence conditions.
Label đ„đ đŠđ đąđ đ„đ đŠđ đąđ đ„đ đŠđ đąđ đtot đ Existence 1 1 1 đŽ0 00 00 0â1â1đ =đ =đ =0 â3 â3 â3 đ đ đ â1 â1 â1 đ”0 00 00 0â1â1đ =đ =đ =0 â3 â3 â3 đ đ đ
đŒđ =đŒđ =đŒđ =0 1 1 1 đ¶00 00 00 â1 â1 and ó”š ó”š ó”š ó”š ó”š ó”š â ó”š ó”š â ó”š ó”š â ó”š ó”š đ1,đ2,đ3 <0, 6 ó”šđ1ó”š 6 ó”šđ2ó”š 6 ó”šđ3ó”š đđ,đđ,đđ <0
đŒđ =đŒđ =đŒđ =0 â1 â1 â1 đ·00 00 00 â1 â1 and ó”š ó”š ó”š ó”š ó”š ó”š â ó”š ó”š â ó”š ó”š â ó”š ó”š đ1,đ2,đ3 <0, 6 ó”šđ1ó”š 6 ó”šđ2ó”š 6 ó”šđ3ó”š đđ,đđ,đđ >0
3 đ =â3, đ =â đŠ đŠ , 44 25 đ đ đ 6đ đ = â6đ đŠ , đ = 2 đą đŠ , 45 đ đ 26 đ đ đ â 2â6đŒ đ đ46 =â 6đŒđđ2, đ = đ 3 đą đŠ , 27 đ đ đ 4â3đŒ đ 3 â đ 1 đâ1 đ =â đŠ đŠ , đ51 =2 6đŒđđ1đąđđŠđ â ( )đąđđŠđđŠđ, 28 đ đ đ đ 2 6đ 6 đâ1 đ = 3 đą đŠ , đ = ( )đŠ đŠ , 29 đ đ đ 52 đ 2 đ đ 12đ đâ1 â3 đ =â 1 ( )đą đŠ , đ = đŒ đŠ , 53 đ 2 đ đ 31 2 đ đ Advances in High Energy Physics 9
0 6 0.25 â0.1 5 0.2 â0.2 4 â0.3 0.15 3 â0.4 0.1 đ đ đ â0.5 2 x u y 0.05 â0.6 1 â0.7 0 0 â0.8 â0.05 â0.9 â1 â1 â2 â0.1 â1â0.5 0 0.5 â1 â0.5 0 0.5 1 â1 â0.5 0 0.5 1 xđ xđ xđ 0.1 0.25 1 0.8 0 0.2 0.6 0.15 â0.1 0.4 0.1 â0.2 0.2 đ đ đ 0.05 0 u y â0.3 x 0 â0.2 â0.4 â0.4 â0.05 â0.6 â0.5 â0.1 â0.8 â0.6 â0.15 â1 â1 â0.5 0 0.5 1 â1 â0.5 0 0.5 1 â1â0.6 â0.2 0.2 0.6
xđ xđ xđ 1 1 3 0.8 2.5 0.6 0.5 2 0.4 0 0.2 1.5 đ đ 0 â0.5 đ 1 u y u â0.2 0.5 â0.4 â1 0 â0.6 â1.5 â0.8 â0.5 â1 â2 â1 â1 â0.5 0 0.5 â1 â0.5 0 0.5 1 â2 â1.5 â1 â0.5 0
xđ xđ yđ 0.3 0 0.3 0.25 â0.1 0.25 0.2 0.2 â0.2 0.15 0.15 â0.3 0.1 đ đ 0.1 đ â0.4 0.05 u y x 0.05 â0.5 0 â0.05 0 â0.6 â0.1 â0.05 â0.7 â0.15 â0.1 â0.8 â0.2 â2â1.5 â1 â0.5 0 â2â1.5 â1 â0.5 0 â2â1.5 â1 â0.5 0
yđ yđ yđ 1.5 0 0 â0.5 â0.05 1 â1 â0.1 â1.5 â0.15 â2 â0.2 đ đ 0.5 đ u y x â2.5 â0.25 â3 â0.3 0 â3.5 â0.35 â4 â0.4 â0.5 â4.5 â0.45 â2 â1.5â1 â0.5 0 â2 â1.5â1 â0.5 0 â2 â1.5â1 â0.5 0
yđ yđ yđ
Figure 2: Continued. 10 Advances in High Energy Physics
0.25 0 0.25 0.2 0.2 â0.1 0.15 0.15 â0.2 0.1 0.1 đ đ đ â0.3 0.05 u y x 0.05 0 â0.4 0 â0.05 â0.05 â0.5 â0.1 â0.1 â0.15 â2 â1 031 2 â2 â1.5 â1 â0.5 0 0.5 1 1.5 2 â2 â1 01234 56
uđ uđ uđ 2 0 0 â0.2 1.5 â0.5 â0.4 â1 â0.6 1 â0.8 đ đ đ â1.5 â1 x u 0.5 y â1.2 â2 â1.4 0 â2.5 â1.6 â1.8 â0.5 â3 â2 â2 â1.5 â1 â0.5 0 0.5 1 1.5 2 â1 â0.5 0 0.5 1 1.5 2 â2 â1 031 2
uđ uđ uđ 0.1 0.25 20 18 0 0.2 16 0.15 â0.1 14 0.1 â0.2 12 đ đ đ 0.05 10 x u y â0.3 0 8 â0.4 6 â0.05 4 â0.5 â0.1 2 â0.6 â0.15 0 â0.1 â0.05 00.05 0.10.15 0.2 â0.1 â0.05 00.05 0.10.15 0.2 0.25 â0.1 â0.05 0 0.05 0.1 0.15 0.2
xđ xđ xđ 0 0 0.25 â0.2 0.2 â5 â0.4 0.15 â10 â0.6 â0.8 0.1 đ đ â15 â1 đ 0.05 u y u â1.2 0 â20 â1.4 â0.05 â25 â1.6 â1.8 â0.1 â30 â2 â0.15 â0.1 â0.05 0 0.05 0.1 0.15 0.2 â0.1 â0.05 0 0.05 0.1 0.15 0.2 â0.6 â0.5 â0.4 â0.3 â0.2 â0.1 0 0.1
xđ xđ yđ 30 0 0 â0.2 25 â10 â20 â0.4 20 â30 â0.6 â40 â0.8 đ đ
15 đ
x â50 u y â1 10 â60 â70 â1.2 5 â80 â1.4 â90 â1.6 0 â100 â1.8 â0.5 â0.4 â0.3 â0.2 â0.1 0 0.1 â0.5 â0.4 â0.3 â0.2 â0.1 0 0.1 â0.6 â0.5 â0.4 â0.3 â0.2 â0.1 0 0.1
yđ yđ yđ
Figure 2: Continued. Advances in High Energy Physics 11
3 0 0 â0.2 â0.05 2.5 â0.4 â0.1 2 â0.6 â0.15 â0.8 â0.2 đ đ 1.5 đ â1 â0.25 x u y â1.2 â0.3 1 â1.4 â0.35 0.5 â1.6 â0.4 â1.8 â0.45 0 â2 â0.5 â0.2 â0.1 0 0.1 0.2 0.3 â0.2 â0.1 00.1 0.2 0.3 â0.2 â0.1 00.1 0.2 0.3
uđ uđ uđ 0 0 0.1 â0.1 â0.1 â0.2 0 â0.2 â0.3 â0.1 â0.4 â0.3 đ đ â0.5 â0.4 đ â0.2 y u u â0.6 â0.5 â0.7 â0.3 â0.6 â0.8 â0.4 â0.9 â0.7 â1 â0.8 â0.5 â0.1â0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.5 1 1.5 2 â3 â2.5 â2 â1.5 â1 â0.5 0
xđ xđ yđ
Figure 2: Two-dimensional projections of the phase-space trajectories for đ1 =đ2 =âđ3 = 0.5, đŒđ =đŒđ =âđŒđ = 1.5.Allplotsbeginfrom the critical point đ” being a stable attractor.
â 12đ đâ1 4 6đŒđđ2 đâ1 1 đ =2â6đŒ đ đą đŠ â ( )đą đŠ đ83 =â ( )đąđđŠđ, 54 đ 2 đ đ đ 2 đ đ đ 2 4â6đŒ đ â đ 2 đâ1 6đđ đ =2â6đŒ đ đą đŠ â ( )đą đŠ , â đŠ , 84 đ 2 đ đ đ 2 đ đ 2 đ 6 đâ1 6 đâ1 đ85 = ( )đŠđđŠđ, đ = ( )đŠ2, đ 2 55 đ 2 đ 12đ2 đâ1 đ86 =â ( )đąđđŠđ, 12đ2 đâ1 đ 2 đ56 =â ( )đąđđŠđ, đ 2 4â6đŒ đ đâ1 đ =2â6đŒ đ đą đŠ â đ 3 ( )đą đŠ 4â6đŒ đ đâ1 87 đ 3 đ đ đ 2 đ đ đ =2â6đŒ đ đą đŠ â đ 3 ( )đą đŠ , 57 đ 3 đ đ đ đ â đ 2 6đđ â đŠđ, 6 đâ1 2 đ58 = ( )đŠđđŠđ, 6 đâ1 đ 2 đ = ( )đŠ2, 88 đ 2 đ 12đ3 đâ1 đ59 =â ( )đąđđŠđ, 12đ đâ1 đ 2 đ =â 3 ( )đą đŠ , 89 đ 2 đ đ â6 đ = đŒ , â6 64 2 đ đ = đŒ , 97 2 đ đ =â3, 77 (27) đ =ââ6đ đŠ , 78 đ đ and all the other matrix elements except those given above đ=1+2đđą2 +2đđą2 + â in (27) are zero. Here also 1 đ 2 đ đ79 = 6đŒđđ3, 2 2đ3đąđ. Then we calculate the eigenvalues of perturbation 4â3đŒ đ đâ1 matrix đ for four critical points given in Table 1 with the đ =2â6đŒ đ đą đŠ â đ 1 ( )đą đŠ đŠ , 81 đ 1 đ đ đ 2 đ đ đ corresponding existing conditions. We obtain four sets of eigenvalues for four perturbation matrices for each of the four 6 đâ1 đ = ( )đŠ đŠ , critical points. To determine the type and stability of critical 82 đ 2 đ đ points, we examine the sign of the real parts of eigenvalues. 12 Advances in High Energy Physics
0.6 0.25 0.5 0.2 0.6 0.4 0.15 0.5 0.3 0.1 0.4 0.2 0.05 đ đ đ 0.1 0 x y 0.3 u 0 â0.05 0.2 â0.1 â0.1 â0.2 â0.15 0.1 â0.3 â0.2 0 â0.4 â0.25 â1 â0.5 0 0.5 1 â1 â0.5 0 0.5 1 â1.5â1â0.5 0 0.5 1 1.5
xđ xđ xđ
1 0.25 0.8 0.2 0.5 0.6 0.15 0.4 0.4 0.2 0.1 đ đ đ 0.3 0 x u y â0.2 0.05 0.2 â0.4 0 â0.6 0.1 â0.05 â0.8 0 â1 â0.1 â1 â0.5 0 0.5 1 â1 â0.5 0 0.5 1 â1 â0.5 0 0.5 1
xđ xđ xđ
0.8 0 0.25 0.7 â0.05 0.2 0.6 â0.1 0.15 0.1 0.5 â0.15 0.05 đ đ đ 0.4 â0.2 0 u y u â0.05 0.3 â0.25 â0.1 0.2 â0.3 â0.15 0.1 â0.35 â0.2 â0.25 0 â0.4 â1 â0.5 0 0.5 1 â1 â0.5 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
xđ xđ yđ
0.25 0.5 0.2 0.5 0.4 0.15 0.3 0.1 0.4 0.2 0.05 0.1 đ 0 đ 0.3 x y đ 0
â0.05 u â0.1 0.2 â0.1 â0.15 â0.2 â0.2 0.1 â0.3 â0.25 â0.4 0 â0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
yđ yđ yđ 0.2 0.8 0 0.7 â0.01 0.15 â0.02 0.6 0.1 â0.03 0.5 â0.04 đ đ 0.05 đ 0.4 â0.05 y u x 0.3 â0.06 0 â0.07 0.2 â0.05 â0.08 0.1 â0.09 â0.1 0 â0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.2 0.250.3 0.35 0.4 0.45 0.5 0.55 0.6
yđ yđ yđ
Figure 3: Continued. Advances in High Energy Physics 13
0.25 0.5 0.2 0.5 0.4 0.15 0.3 0.1 0.4 0.2 0.05 0.1 đ đ đ
0 0.3 u 0 x â0.05 y â0.1 â0.1 0.2 â0.2 â0.15 â0.3 â0.2 0.1 â0.4 â0.25 â0.5 0 â0.2 â0.1 0 0.1 0.2 0.3 â0.4 â0.3 â0.2 â0.1 0 0.1 0.2 0.3 â0.5 0 0.5
uđ uđ uđ 0.3 0.8 0 0.25 â0.01 0.7 â0.02 0.2 0.6 â0.03 0.15 â0.04 đ đ đ 0.1 0.5 â0.05 y u x 0.05 â0.06 0.4 â0.07 0 0.3 â0.08 â0.05 â0.09 â0.1 0.2 â0.1 â0.4â0.3 â0.2 â0.1 0 0.1 0.2 0.3 â0.2 â0.1 0 0.1 0.2 0.3 â0.2 â0.1 0 0.1 0.2
uđ uđ uđ
0.6 0.5 0.55 0.6 0.4 0.5 0.4 0.3 0.45 0.2 0.2 0.4 0.1 đ đ 0.35 0 đ 0 y u 0.3 x â0.1 â0.2 0.25 â0.2 0.2 â0.4 â0.3 0.15 â0.6 â0.4 0.1 â0.5 â0.2 â0.1 0 0.1 0.2 0.3 â0.2 â0.1 0 0.1 0.2 0.3 â0.2 â0.1 0 0.1 0.2
xđ xđ xđ 0.8 0.01 0 0.6 0.7 â0.01 0.4 0.6 â0.02 â0.03 0.2 0.5 â0.04 đ đ đ 0 y u 0.4 â0.05 u â0.06 â0.2 0.3 â0.07 â0.4 0.2 â0.08 â0.09 â0.6 0.1 â0.1 â0.2 â0.1 0 0.1 0.2 0.3 â0.2 â0.1 0 0.1 0.2 0.3 â0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
xđ xđ yđ 0.2 1 0 0.9 â0.01 0.15 0.8 â0.02 0.1 0.7 â0.03 0.6 â0.04 đ đ 0.05 đ 0.5 â0.05 y u x 0.4 â0.06 0 0.3 â0.07 â0.05 0.2 â0.08 0.1 â0.09 â0.1 0 â0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
yđ yđ yđ
Figure 3: Continued. 14 Advances in High Energy Physics
0.2 0.8 0 0.7 â0.01 0.15 â0.02 0.6 0.1 â0.03 0.5 â0.04 đ đ 0.05 đ 0.4 â0.05 y u x 0.3 â0.06 0 â0.07 0.2 â0.05 â0.08 0.1 â0.09 â0.1 0 â0.1 â0.4 â0.2 0 0.2 0.4 0.6 â0.4 â0.2 00.2 0.4 0.6 â0.2 â0.1 0 0.1 0.2
uđ uđ uđ 1 0 0 â0.01 0.9 â0.02 â0.02 0.8 â0.04 â0.03 0.7 â0.04 đ â0.06 đ đ u â0.05 y 0.6 u â0.08 â0.06 0.5 â0.07 â0.1 â0.08 0.4 â0.12 â0.09 0.3 â0.1 â0.1â0.05 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 0.5 0.55 0.6 0.65 0.7 x đ xđ yđ
Figure 3: Two-dimensional projections of the phase-space trajectories for đ1 =đ2 =đ3 = â0.5, đŒđ =đŒđ =đŒđ = â1.5.Allplotsbeginfrom the critical point đ¶ being a stable attractor.
Table 2: Eigenvalues of the perturbation matrix for each critical Table 3: Stability of the critical points. point. Critical points Stability 3 Critical points Eigenvalues 0< <đ 1 3 Stable point, if 2 1, 2 4đŒđ â9â12đ1đŒ â , 2 đ 2 3 1 3 0< <đ â â9â12đ đŒ2 â đŽ đ” 4đŒ2 2 , 2 1 đ 2 , and đ 1 3 â3 2 đ < <0 â9â12đ2đŒđ â , and 3 2 2 2 4đŒđ đŽ đ” 1 3 đ ,đ <0 and â â9â12đ đŒ2 â Saddle point, if 1 2 2 2 đ 2 , 1 3 đ¶ and đ· Stable point for all đ, đŒ,andđ â12đ đŒ2 +9â 2 3 đ 2 , 1 3 â â12đ đŒ2 +9â 2 3 đ 2 , â1 In order to analyze the cosmological behavior of each critical point, the attractor solutions in scalar field cosmology đ¶ đ· and â3 should be noted [64]. In modern theoretical cosmology it is common that the energy density of one or more scalar fields exerts a crucial influence on the evolution of the universe and some certain conditions or behaviors naturally affect this The critical point is stable for negative real part of eigenvalues. evolution. The evolution and the affecting factors on this evo- The physical meaning of negative eigenvalue is always a late- lution meet in the term of cosmological attractors: the scalar time stable attractor; namely, if the universe reaches this field evolution approaches a certain kind of behavior by the solution, it keeps its state forever and thus it can attract the dynamical conditions without finely tuned initial conditions universe at a late-time. Accelerated expansion occurs here, [65â77], either in inflationary cosmology or in late-time dark because đtot =â1<â1/3. If a convenient condition is energy models. Attractor behavior is a situation in which a provided, an accelerated contraction can even exist for đtot = collection of phase-space points evolve into a certain region â1 < â1/3 value. Eigenvalues of the matrix đ are represented and never leave. in Table 2 for each of the critical points đŽ, đ”, đ¶,andđ·. As seen in Table 2, the first two critical points đŽ and đ” Critical Point A. This point exists for all values of đ1, đ2, đ3, havethesameeigenvaluesasđ¶ and đ· have same eigenvalues. đŒđ, đŒđ,andđŒđ while đđ =đđ =đđ =0which means the The stability conditions of each critical point are listed in potentials đđ, đđ,andđđ are constant. Acceleration occurs at Table 3, according to the sign of the eigenvalues. this point since đtot =â1<â1/3and this is an expansion Advances in High Energy Physics 15
â0.5 0.4 0.4 â0.55 0.3 0.35 â0.6 0.2 0.3 â0.65 0.1 0.25 â0.7 0 0.2 đ đ đ â0.75 â0.1 0.15 x y â0.8 u â0.2 0.1 â0.85 â0.3 0.05 â0.9 â0.4 0 â0.95 â0.5 â0.05 â1 â0.1 â0.4 â0.2 0 0.2 0.4 0.6 â0.4 â0.2 0 0.2 0.4 0.6 â0.6 â0.4 â0.2 0
xđ xđ xđ â0.5 0.4 0.25 0.35 â0.55 0.2 0.15 0.3 0.1 0.25 â0.6 0.05 đ đ đ 0 0.2 u y x â0.65 â0.05 0.15 â0.1 â0.15 0.1 â0.7 â0.2 0.05 â0.25 â0.75 0 â0.4 â0.2 0 0.2 0.4 0.6 â1â0.5 0 0.5 1 â0.4 â0.2 0 0.2 0.4 0.6
xđ xđ xđ
â0.5 0 0.4 â0.55 â0.1 0.3 â0.6 0.2 â0.65 â0.2 0.1 â0.7 â0.3 đ đ â0.75 đ 0 y u â0.4 u â0.8 â0.1 â0.85 â0.5 â0.2 â0.9 â0.6 â0.95 â0.3 â1 â0.4 â0.8â0.6â0.4 â0.2 0 0.2 0.4 0.6 0.8 â1â0.5 0 0.5 1 â1 â0.9 â0.8 â0.7 â0.6 â0.5
xđ xđ yđ
0.01 â0.5 0.008 â0.51 0.25 0.2 0.006 â0.52 0.15 0.004 â0.53 0.1 0.002 â0.54 0.05 đ đ 0 đ â0.55 0 u y x â0.002 â0.56 â0.05 â0.004 â0.57 â0.1 â0.15 â0.006 â0.58 â0.2 â0.008 â0.59 â0.25 â0.01 â0.6 â0.6 â0.59 â0.58 â0.57 â0.56 â1 â0.9â0.8â0.7 â0.6 â0.5 â1 â0.9 â0.8 â0.7 â0.6 â0.5
yđ yđ yđ
0.4 â0.5 0 0.35 â0.55 â0.05 â0.6 0.3 â0.1 â0.65 0.25 â0.7 â0.15 đ đ 0.2 đ â0.75 â0.2 y u x 0.15 â0.8 â0.25 â0.85 0.1 â0.3 â0.9 0.05 â0.95 â0.35 0 â1 â0.4 â1 â0.9â0.8 â0.7 â0.6 â0.5 â1 â0.9 â0.8â0.7 â0.6 â0.5 â1 â0.9 â0.8 â0.7 â0.6 â0.5
yđ yđ yđ
Figure 4: Continued. 16 Advances in High Energy Physics
â0.5 0.5 0.25 0.4 â0.55 0.2 0.3 0.15 0.2 â0.6 0.1 0.1 0.05 đ 0 â0.65 đ 0 y u đ â0.1 â0.05 x â0.2 â0.7 â0.1 â0.3 â0.15 â0.4 â0.75 â0.2 â0.5 â0.25 â0.8 â0.5 0 0.5 â0.5 0 0.5 â0.4 â0.2 0 0.2 0.4 0.6
uđ uđ uđ 1 â0.5 0 0.9 â0.55 0.8 â0.6 â0.1 0.7 â0.65 â0.2 0.6 â0.7 đ đ 0.5 đ â0.75 â0.3 y u x 0.4 â0.8 0.3 â0.85 â0.4 0.2 â0.9 â0.5 0.1 â0.95 0 â1 â0.5 0 0.5 â0.4 â0.3 â0.2 â0.1 0 0.1 0.2 0.3 â0.6 â0.4 â0.2 0 0.2 0.4 0.6
uđ uđ uđ â0.5 1 â0.51 0.25 0.9 â0.52 0.2 0.8 0.15 â0.53 0.1 0.7 â0.54 0.05 0.6 đ đ â0.55 đ 0 0.5 y x â0.56 u â0.05 0.4 â0.57 â0.1 0.3 â0.15 â0.58 â0.2 0.2 â0.59 â0.25 0.1 â0.6 0 â0.2 â0.1 0 0.1 0.2 â0.5 0 0.5 â0.5 0 0.5
xđ xđ xđ â0.5 0 â0.55 0.25 â0.6 â0.1 0.2 0.15 â0.65 â0.2 0.1 â0.7 0.05 đ đ â0.75 â0.3 đ 0 y u â0.8 u â0.05 â0.85 â0.4 â0.1 â0.15 â0.9 â0.5 â0.2 â0.95 â0.25 â1 â0.2 â0.10 0.1 0.2 0.3 â0.4 â0.2 0 0.2 0.4 0.6 â0.6 â0.58 â0.56 â0.54 â0.52 â0.5
xđ xđ yđ 1 â0.5 0 0.9 â0.55 â0.05 0.8 â0.6 â0.1 0.7 â0.65 â0.15 0.6 â0.7 â0.2 đ đ 0.5 đ â0.75 â0.25 y u x 0.4 â0.8 â0.3 0.3 â0.85 â0.35 0.2 â0.9 â0.4 0.1 â0.95 â0.45 0 â1 â0.5 â0.8 â0.7 â0.6â0.5 â0.4 â0.65 â0.6 â0.55 â0.5 â0.8 â0.75â0.7 â0.65â0.6 â0.55 â0.5
yđ yđ yđ
Figure 4: Continued. Advances in High Energy Physics 17
1 â0.5 0 0.9 â0.55 â0.1 0.8 â0.6 â0.2 0.7 â0.65 â0.3 0.6 â0.7 â0.4 đ đ 0.5 đ â0.75 â0.5 y u x 0.4 â0.8 â0.6 0.3 â0.85 â0.7 0.2 â0.9 â0.8 0.1 â0.95 â0.9 0 â1 â1 â0.2 â0.10 0.1 0.2 0.3 â0.2 â0.1 0 0.1 0.2 â0.4 â0.3 â0.2 â0.1 0 0.1 0.2 0.3
uđ uđ uđ
â0.5 0 0 â0.55 â0.05 â0.6 â0.1 â0.05 â0.65 â0.15 â0.1 â0.7 â0.2 đ đ đ â0.75 â0.25 â0.15 y u â0.8 â0.3 u â0.85 â0.35 â0.2 â0.9 â0.4 â0.25 â0.95 â0.45 â1 â0.5 0 0.05 0.10.15 0.20.25 0.3 0 0.2 0.4 0.6 0.8 1 â1 â0.9 â0.8â0.7 â0.6 â0.5
xđ xđ yđ
Figure 4: Two-dimensional projections of the phase-space trajectories for đ1 =đ2 =đ3 = â0.5, đŒđ =đŒđ =đŒđ = â1.5.Allplotsbeginfrom the critical point đ· being a stable attractor.
phase because đŠđ, đŠđ,andđŠđ have positive values; therefore đŒđ = â1.5,andthreeauxiliaryđ values for each field đ, đ,and đ» is positive, too. Point đŽ is stable (meaning that universe đ. A stable attractor starting from the critical point đ¶ is seen 2 2 keeps its evolution), if 0 < 3/(4đŒđ)<đ1, 0 < 3/(4đŒđ)<đ2 and from the plots in Figure 3. 2 đ3 < â3/(4đŒđ)<0, but it is a saddle point (meaning that the Critical Point D. This point exists for đ1,đ2,đ3 <0and universe evolves between different states) for values đ1,đ2 <0. đđ,đđ,đđ >0while đŒđ =đŒđ =đŒđ =0implying constant In Figure 1, we represent the 2-dimensional projections of 9- đ(đ), đ(đ), â(đ) coupling functions. Acceleration phase is dimensional phase-space trajectories for the values of đ1 = valid due to đtot <â1/3.Pointđ· is also stable for all đ, đŒ, đ2 =âđ3 = 0.5, đŒđ =đŒđ =âđŒđ = 1.5 and three auxiliary đ and đ values of fields đ, đ,andđ.Two-dimensionalplotsof values for each field đ, đ,andđ. This state corresponds to a phase-space trajectories are shown in Figure 4 for đ1 =đ2 = stable attractor starting from the critical point đŽ,asseenfrom đ3 = â0.5, đŒđ =đŒđ =đŒđ = â1.5,andthreeauxiliaryđ values the plots in Figure 1. for each field đ, đ,andđ.Thisstateagaincorrespondstoa stable attractor starting from the point đ¶,asinFigure4. đ” đ đ đ Critical Point B.Point also exists for any values of 1, 2, 3, AlltheplotsinFigures1â4havethestructureofstable đŒ đŒ đŒ đ =đ =đ =0 đ, đ,and đ while đ đ đ which means the attractor, since each of them evolves to a single point which đ đ đ potentials đ, đ,and đ are constant. Acceleration phase is is in fact one of the critical points in Table 1. These evolutions đ =â1<â1/3 again valid here since tot , but this point refers to the critical points are the attractor solutions in mixed dark đŠ đŠ đŠ to contraction phase because đ, đ,and đ have negative energy domination cosmology of our model which imply an sign. Stability of the point đ” is same with the point đŽ for expanding universe. same conditions. Therefore the stable attractor behavior for contraction is represented in Figure 2. We plot 2-dimensional projections of phase-space trajectories for same values as 4. Conclusion point đŽ, đ1 =đ2 =âđ3 = 0.5, đŒđ =đŒđ =âđŒđ = 1.5,and auxiliary đ values. Mixed dark energy is a generalized combination of tachyon, quintessence, and phantom fields nonminimally coupled to Critical Point C. Critical point đ¶ occurs for đ1,đ2,đ3 <0and gravity[33,34,38,39].Thesethreescalarfieldsarethe đđ,đđ,đđ <0while đŒđ =đŒđ =đŒđ =0meaning constant constituents of mixed dark energy. Firstly, the action integral coupling functions đ(đ), đ(đ),andâ(đ).Thecosmological of nonminimally coupled mixed dark energy model is set behavior is again an acceleration phase since đtot <â1/3. up to study its dynamics. Here we consider that our dark Point đ¶ is stable for any values of đ, đŒ,andđ of fields đ, energy constituents interact only with the gravity. There đ,andđ. Two-dimensional projections of phase-space are could also be chosen some interactions between the dark represented in Figure 3 for đ1 =đ2 =đ3 = â0.5, đŒđ =đŒđ = constituents, but in order to start from a pedagogical order we 18 Advances in High Energy Physics prefer to exclude the dark interactions. We obtain the Hubble a cosmological constant,â The Astronomical Journal,vol.116,no. parameter and Friedmann equations of model in spatially 3, pp. 1009â1038, 1998. flat FRW geometry. Energy density and pressure values [3] U. Seljak, A. Makarov, P. McDonald et al., âCosmological with the evolution equations for tachyon, quintessence, and parameter analysis including SDSS LyđŒ forest and galaxy phantom fields are obtained from the variation of action bias: constraints on the primordial spectrum of fluctuations, and Lagrangian of model. Then we translate these dynam- neutrino mass, and dark energy,â Physical Review D,vol.71,no. ical expressions into the autonomous form by introducing 10, Article ID 103515, 20 pages, 2005. suitable auxiliary variables in order to perform the phase- [4]M.Tegmark,M.Strauss,M.Blantonetal.,âCosmological space analysis. We find the critical points of autonomous parameters from SDSS and WMAP,â Physical Review D,vol.69, system by setting each autonomous equation to zero. By Article ID 103501, 2004. constructing the perturbation equations, we find four per- [5]D.J.Eisenstein,I.Zehavi,D.W.Hoggetal.,âDetectionofthe turbation matrices for each critical point. The eigenvalues of baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies,â The Astrophysical Journal,vol.633, four perturbation matrices are determined to examine the no. 2, pp. 560â574, 2005. stability of critical points. We also calculate some important [6]D.N.Spergel,L.Verde,H.V.Peirisetal.,âFirst-yearWilkinson cosmological parameters, for instance, the total equation of Microwave Anisotropy Probe (WMAP) observations: determi- state parameter and deceleration parameter to see whether nation of cosmological parameters,â The Astrophysical Journal the critical points correspond to an accelerating universe Supplement Series,vol.148,no.1,pp.175â194,2003. or not. There are four stable attractors of model depending đ đ đ đŒ [7]E.Komatsu,K.M.Smith,J.Dunkleyetal.,âSeven-year on the nonminimal coupling parameters 1, 2, 3 and đ, Wilkinson microwave anisotropy probe (WMAP) observations: đŒđ, đŒđ values. All of the stable solutions correspond to an cosmological interpretation,â The Astrophysical Journal Supple- accelerating universe due to đtot <â1/3.Forconstant ment Series, vol. 192, no. 2, article 18, 2011. potentials đđ, đđ,andđđ the critical points đŽ and đ” are late- [8] G. Hinshaw, D. Larson, E. Komatsu et al., âNine-year Wilkinson 2 2 time stable attractors for 0 < 3/(4đŒđ)<đ1, 0 < 3/(4đŒđ)<đ2, microwave anisotropy probe (WMAP) observations: cosmolog- 2 ical parameter results,â The Astrophysical Journal Supplement and đ3 < â3/(4đŒ )<0. While point đŽ refers to an expansion đ Series,vol.208,no.2,p.19,2013. with a stable acceleration, point đ” refers to a contraction. [9] P. A. R. Ade, N. Aghanim, C. Armitage-Caplan et al., âPlanck However, for constant coupling functions đ(đ), đ(đ),and â(đ) đ¶ đ· 2013 results. XVI. Cosmological parameters,â Astronomy & the critical points and are stable attractors for any Astrophysics,vol.571,articleA16,2013. values of đ1, đ2, đ3, đŒđ, đŒđ,andđŒđ.Thebehaviorofthemodel [10] S. Weinberg, âThe cosmological constant problem,â Reviews of at each critical point being a stable attractor is demonstrated Modern Physics,vol.61,no.1,pp.1â23,1989. in Figures 1â4. In order to plot the graphs in Figures 1â4, [11] S. M. Carroll, W. H. Press, and E. L. Turner, âThe cosmological we use adaptive Runge-Kutta method of 4th and 5th order constant,â Annual Review of Astronomy and Astrophysics,vol.30, to solve differential equations (26) in Matlab. Solutions for no. 1, pp. 499â542, 1992. the equations with stability conditions of critical points are [12] L. M. Krauss and M. S. Turner, âThe cosmological constant is plotted for each pair of the solution being the auxiliary back,â General Relativity and Gravitation, vol. 27,no. 11, pp. 1137â variables in (19). 1144, 1995. These figures show that, by choosing parameters of the [13] G. Huey, L. Wang, R. Dave, R. R. Caldwell, and P. J. Steinhardt, model depending on the existence conditions of critical âResolving the cosmological missing energy problem,â Physical points đŽ, đ”, đ¶,andđ·, we obtain the attractors of the model as Review D,vol.59,ArticleID063005,1999. đŽ, đ”, đ¶,andđ·. Then depending on the stability conditions, [14] S. M. Carroll, âThe cosmological constant,â Living Reviews in suitable parameters give stable behavior for each attractor Relativity,vol.4,no.1,2001. ofthemodel.Theresultsareconsistentwiththeobserved [15] R.-J. Yang and S. N. Zhang, âThe age problem in the ÎCDM and expected behavior of the universe in which some epochs model,â Monthly Notices of the Royal Astronomical Society,vol. correspond to an accelerating expansion phase, and some will 407, no. 3, pp. 1835â1841, 2010. correspond to an accelerating contraction in future times [1â [16] Y.-F. Cai, E. N. Saridakis, M. R. Setare, and J.-Q. Xia, âQuintom 9, 76, 77]. cosmology: theoretical implications and observations,â Physics Reports,vol.493,no.1,pp.1â60,2010. [17]K.Bamba,S.Capozziello,S.Nojiri,andS.D.Odintsov,âDark Conflict of Interests energy cosmology: the equivalent description via different theoretical models and cosmography tests,â Astrophysics and The authors declare that there is no conflict of interests Space Science,vol.342,no.1,pp.155â228,2012. regarding the publication of this paper. [18] A. Banijamali and B. Fazlpour, âTachyonic teleparallel dark energy,â Astrophysics and Space Science,vol.342,no.1,pp.229â References 235, 2012. [19] J.-Q. Xia, Y.-F. Cai, T.-T. Qiu, G.-B. Zhao, and X. Zhang, [1] S. Perlmutter, G. Aldering, G. Goldhaber et al., âMeasurements âConstraints on the sound speed of dynamical dark energy,â of Ω and Î from 42 high-redshift supernovae,â The Astrophysical International Journal of Modern Physics D,vol.17,no.8,pp. Journal,vol.517,no.2,pp.565â586,1999. 1229â1243, 2008. [2]A.G.Riess,A.V.Filippenko,P.Challisetal.,âObservational [20] A. Vikman, âCan dark energy evolve to the phantom?â Physical evidence from supernovae for an accelerating universe and Review D,vol.71,no.2,ArticleID023515,14pages,2005. Advances in High Energy Physics 19
[21]W.Hu,âCrossingthephantomdivide:darkenergyinternal [42] K. Hayashi and T. Shirafuji, âNew general relativity,â Physical degrees of freedom,â Physical Review D,vol.71,ArticleID Review D,vol.19,article3524,1979. 047301, 2005. [43] E. J. Copeland, M. Sami, and S. Tsujikawa, âDynamics of dark [22] R. R. Caldwell and M. Doran, âDark-energy evolution across energy,â International Journal of Modern Physics. D,vol.15,no. the cosmological-constant boundary,â Physical Review D,vol. 11,pp.1753â1935,2006. 72,no.4,ArticleID043527,6pages,2005. [44] P. G. Ferreira and M. Joyce, âStructure formation with a self- [23]G.-B.Zhao,J.-Q.Xia,M.Li,B.Feng,andX.Zhang,âPertur- tuning scalar field,â Physical Review Letters,vol.79,no.24,pp. bations of the quintom models of dark energy and the effects 4740â4743, 1997. on observations,â Physical Review D,vol.72,no.12,ArticleID [45] E. J. Copeland, A. R. Liddle, and D. Wands, âExponential 123515, 2005. potentials and cosmological scaling solutions,â Physical Review [24] M. Kunz and D. Sapone, âCrossing the phantom divide,â D,vol.57,no.8,pp.4686â4690,1998. Physical Review D,vol.74,no.12,ArticleID123503,2006. [46] X. M. Chen, Y. G. Gong, and E. N. Saridakis, âPhase-space anal- [25] L. Amendola, âCoupled quintessence,â Physical Review D,vol. ysis of interacting phantom cosmology,â Journal of Cosmology 62, no. 4, Article ID 043511, 10 pages, 2000. and Astroparticle Physics,vol.2009,no.4,article001,2009. [26] B. Gumjudpai, T. Naskar, M. Sami, and S. Tsujikawa, âCoupled [47] A.R.Liddle,P.Parsons,andJ.D.Barrow,âFormalizingtheslow- dark energy: towards a general description of the dynamics,â roll approximation in inflation,â Physical Review D,vol.50,no. JournalofCosmologyandAstroparticlePhysics,vol.2005,no.6, 12, pp. 7222â7232, 1994. article 007, 2005. [48] P. G. Ferreira and M. Joyce, âCosmology with a primordial [27] A. Einstein, âRiemanngeometrie mit Aufrechterhaltung scaling field,â Physical Review D,vol.58,no.2,ArticleID023503, des Begriffes des Fern-Parallelismus,â in Sitzungsberichte 1998. der Preussischen Akademie der Wissenschaften,pp.217â221, [49] Y. Gong, A. Wang, and Y.-Z. Zhang, âExact scaling solutions Physikalisch-Mathematische Klasse, 1928. andfixedpointsforgeneralscalarfield,âPhysics Letters B,vol. [28] K. Hayashi and T. Shirafuji, âNew general relativity,â Physical 636, no. 5, pp. 286â292, 2006. Review D,vol.19,no.12,pp.3524â3553,1979. [50] X.-M. Chen and Y. Gong, âFixed points in interacting dark [29] K. Hayashi and T. Shirafuji, âAddendum to ânew general energy models,â Physics Letters B,vol.675,no.1,pp.9â13,2009. relativityâ,â Physical Review D,vol.24,no.12,p.3312,1981. [51]Z.-K.Guo,Y.-S.Piao,X.M.Zhang,andY.-Z.Zhang,âCosmo- [30] R. Ferraro and F.Fiorini, âModified teleparallel gravity: inflation logical evolution of a quintom model of dark energy,â Physics without an inflaton,â Physical Review D,vol.75,ArticleID Letters B,vol.608,no.3-4,pp.177â182,2005. 084031, 2007. [52] R. Lazkoz and G. Leon,ÂŽ âQuintom cosmologies admitting either [31] G. R. Bengochea and R. Ferraro, âDark torsion as the cosmic tracking or phantom attractors,â Physics Letters B,vol.638,no. speed-up,â Physical Review D,vol.79,no.12,ArticleID124019, 4, pp. 303â309, 2006. 2009. [53]Z.-K.Guo,Y.-S.Piao,X.Zhang,andY.-Z.Zhang,âTwo-field [32] E. V.Linder, âEinsteinâs other gravity and the acceleration of the óž quintom models in the đ€âđ€ plane,â Physical Review D,vol.74, Universe,â Physical Review D,vol.81,no.12,ArticleID127301, no. 12, Article ID 127304, 2006. 2010. [33] C.-Q. Geng, C.-C. Lee, E. N. Saridakis, and Y.-P. Wu, [54] R. Lazkoz, G. Leon,ÂŽ and I. Quiros, âQuintom cosmologies with ââTeleparallelâ dark energy,â Physics Letters B,vol.704,no.5,pp. arbitrary potentials,â Physics Letters B,vol.649,no.2-3,pp.103â 384â387, 2011. 110, 2007. đ [34] C.-Q. Geng, C.-C. Lee, and E. N. Saridakis, âObservational [55] M. Alimohammadi, âAsymptotic behavior of in general constraints on teleparallel dark energy,â Journal of Cosmology quintom model,â General Relativity and Gravitation,vol.40,no. and Astroparticle Physics,vol.2012,no.1,article002,2012. 1, pp. 107â115, 2008. [35]C.Xu,E.N.Saridakis,andG.Leon,âPhase-spaceanalysisof [56]M.R.SetareandE.N.Saridakis,âCoupledoscillatorsasmodels teleparallel dark energy,â Journal of Cosmology and Astroparticle of quintom dark energy,â Physics Letters B,vol.668,no.3,pp. Physics,vol.2012,no.7,article005,2012. 177â181, 2008. [36] H. Wei, âDynamics of teleparallel dark energy,â Physics Letters [57] M. R. Setare and E. N. Saridakis, âQuintom cosmology with B,vol.712,no.4-5,pp.430â436,2012. general potentials,â International Journal of Modern Physics D, vol.18,no.4,pp.549â557,2009. [37] G. Otalora, âScaling attractors in interacting teleparallel dark energy,â JournalofCosmologyandAstroparticlePhysics,vol. [58]M.R.SetareandE.N.Saridakis,âQuintommodelwithO(N) 2013, no. 7, article 44, 2013. symmetry,â Journal of Cosmology and Astroparticle Physics,vol. [38] A. Banijamali, âDynamics of interacting tachyonic teleparallel 2008,no.9,article026,2008. dark energy,â Advances in High Energy Physics,vol.2014,Article [59] M. R. Setare and E. N. Saridakis, âQuintom dark energy models ID 631630, 14 pages, 2014. with nearly flat potentials,â Physical Review D,vol.79,no.4, [39] B. Fazlpour and A. Banijamali, âTachyonic teleparallel dark Article ID 043005, 2009. energy in phase space,â Advances in High Energy Physics,vol. [60]G.Leon,R.Cardenas,andJ.L.Morales,âEquilibriumsets 2013, Article ID 279768, 9 pages, 2013. in quintom cosmologies: the past asymptoticdynamics,â http:// [40] G. Otalora, âCosmological dynamics of tachyonic teleparallel arxiv.org/abs/0812.0830. dark energy,â Physical Review D,vol.88,no.6,ArticleID [61] A. De Felice and S. Tsujikawa, âf (R) theories,â Living Reviews in 063505, 8 pages, 2013. Relativity,vol.13,p.3,2010. [41] R. Weitzenbock,š Invarianten Theorie, Nordho, Groningen, The [62] T. P. Sotiriou and V. Faraoni, âđ(đ ) theories of gravity,â Reviews Netherlands, 1923. of Modern Physics,vol.82,no.1,pp.451â497,2010. 20 Advances in High Energy Physics
[63] S. Tsujikawa, âModified gravity models of dark energy,â in Lectures on Cosmology,vol.800ofLecture Notes in Physics,pp. 99â145, Springer, 2010. [64] G. N. Remmen and S. M. Carroll, âAttractor solutions in scalar- field cosmology,â Physical Review D,vol.88,no.8,ArticleID 083518, 2013. [65] Z.-K. Guo, Y.-S. Piao, R.-G. Cai, and Y.-Z. Zhang, âInflationary attractor from tachyonic matter,â Physical Review D,vol.68,no. 4, Article ID 043508, 2003. [66] L. A. Urena-LË opezÂŽ and M. J. Reyes-Ibarra, âOn the dynamics of a quadratic scalar field potential,â International Journal of Modern Physics D,vol.18,no.4,pp.621â634,2009. [67]V.A.Belinsky,L.P.Grishchuk,I.M.Khalatnikov,andY.B. Zeldovich, âInflationary stages in cosmological models with a scalar field,â Physics Letters B,vol.155,no.4,pp.232â236,1985. [68] T. Piran and R. M. Williams, âInflation in universes with a massive scalar field,â Physics Letters B,vol.163,no.5-6,pp.331â 335, 1985. [69] B. Ratra and P. J. E. Peebles, âCosmological consequences of a rolling homogeneous scalar field,â Physical Review D,vol.37,no. 12, pp. 3406â3427, 1988. [70] A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large- Scale Structure, Cambridge University Press, Cambridge, UK, 2000. [71] V. V. Kiselev and S. A. Timofeev, âQuasiattractor dynamics of 4 đđ -inflation,â http://arxiv.org/abs/0801.2453. [72]S.Downes,B.Dutta,andK.Sinha,âAttractors,universality,and inflation,â Physical Review D,vol.86,no.10,ArticleID103509, 14 pages, 2012. [73] J. Khoury and P. J. Steinhardt, âGenerating scale-invariant perturbations from rapidly-evolving equation of state,â Physical Review D, vol. 83, no. 12, Article ID 123502, 2011. [74] S. Clesse, C. Ringeval, and J. Rocher, âFractal initial conditions and natural parameter values in hybrid inflation,â Physical Review D,vol.80,no.12,ArticleID123534,2009. [75] V. V. Kiselev and S. A. Timofeev, âQuasiattractor in models of new and chaotic inflation,â General Relativity and Gravitation, vol.42,no.1,pp.183â197,2010. [76] G. F. R. Ellis, R. Maartens, and M. A. H. MacCallum, Relativistic Cosmology, Cambridge University Press, Cambridge, UK, 2012. [77] Y.Wang,J.M.Kratochvil,A.Linde,andM.Shmakova,âCurrent observational constraints on cosmic doomsday,â Journal of Cosmology and Astroparticle Physics,vol.2004,no.12,article 006, 2004. Journal of Journal of The Scientific Journal of Advances in Gravity Photonics World Journal Soft Matter Condensed Matter Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014
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