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Research Article Dynamics of Mixed Dark Energy Domination in Teleparallel Gravity and Phase-Space Analysis

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Research Article Dynamics of Mixed Dark Energy Domination in Teleparallel Gravity and Phase-Space Analysis 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 DE = DE/DE,whereDE and DE 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 DE =−1,butforquintessencethe Ia [1, 2], large scale structure [3, 4], the baryon acoustic parameter DE ≥−1,forphantomDE ≤−1,andfor oscillations [5], and cosmic microwave background radiation nonminimally coupled tachyon with gravity both DE ≤−1 [6–9]. and DE ≥−1[16–18]. In order to explain the late-time accelerated expansion The scenario of DE crossing the cosmological constant of universe, an unknown form of energy, called dark energy, boundary is referred to as a “Quintom” 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.TheequationofstateparameterDE 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 Λ having a constant energy density filling the space energy should be nonminimally coupled to gravity; namely, it homogeneously [10–13]. But cosmological constant is not should interact with the gravity [19–24]. 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 =∫ 4 [ + () constructed by the contraction of torsion tensor, in contrast 22 1 to the Einstein-Hilbert action of classical gravity in which [ contraction of the curvature tensor is used. In teleparallel gravity, the dynamical variable is a set of four tetrad fields ] ] − () √1+ +2() constructing the bases for the tangent space at each point of () space-time [27–29]. The teleparallel gravity Lagrangian with (1) only torsion scalar corresponds to the matter-dominated 1 1 − ] − () + ℎ () + ] universe; namely, it does not accelerate. Therefore, to obtain 2 ] 3 2 ] a universe with an accelerating expansion, we can either replace with a function (), the so-called () gravity ] (teleparallel analogue of () gravity) [30–32], or add an − () , unknown form of energy, so-called dark energy, to the ] teleparallel gravity Lagrangian that allows also nonminimal coupling between dark energy and gravity to overcome the where =det()=√− and are the orthonormal tetrad no-go theorem. The interesting feature of () theories is the components, such that existence of second or higher order derivatives in equations. Therefore, we prefer the second choice; adding extra scalar ] = ], (2) fields of the unknown energy forms as dark energy. where , runover0,1,2,and3forthetangentspaceateach As different dark energy models, interacting teleparallel darkenergystudieshavebeenintroducedintheliterature;for point of the manifold and , ] 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 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 = ] + ] − ]. this model has been studied in [35–37]. Tachyonic teleparallel 4 ] 2 ] ] (3) dark energy is a generalization of the teleparallel quintessence dark energy by introducing a noncanonical tachyon scalar Here ] is the torsion tensor constructed by the Weitzenbock field in place of the canonical quintessence field [18, 38–40]. connection Γ], such that [41] In this study, the main motivation is that we consider a more general dark energy model including three kinds ] =Γ] −Γ] = (] −]). (4) of dark energy models, instead of taking one dark energy model as in [16, 18, 38–40]. In order to explain the expansion All the information about the gravitational field is con- of universe by adding scalar fields as the dark energy tained in the torsion tensor ] 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, =8in (1) and (), (),and dynamical analysis of the model in FRW space-time. Later ℎ() are the functions responsible for nonminimal coupling on, we translate the evolution equations into an autonomous between gravity and tachyon, quintessence, and phantom dynamical system. After that the phase-space analysis of the fields, respectively. 1, 2,and3 are the dimensionless model and the cosmological implications of the critical (or coupling constants and (), (),and() 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 = − () [ + Ω ], (5) Our model consists of three scalar fields as the three- and a tetrad field of the form = diag(1,,,).Thenthe component dark energy domination without background Friedmann equations for FRW metric read as dark matter and baryonic matter. These are the canonical 2 scalar field, quintessence , and two noncanonical scalar 2 = ( + + ), fields, tachyon and phantom , and all these scalar fields are 3 (6) nonminimally coupled to gravity. Since we consider only the 2 dark energy dominated sector without the matter content of ̇ =− ( + + + + +), the universe, the action of the mixed teleparallel dark energy 2 Advances in High Energy Physics 3 where =/̇ is the Hubble parameter, is the scale factor, values for the tachyon, quintessence, and phantom fields read and dot represents the derivative with respect to cosmic time as . and are the energy density and the pressure of the () corresponding scalar field constituents of the dark energy. 2 =−1()6 + , Conservation of energy gives the evolution equations for 2 (12) √1−(̇ / ()) thedarkenergyconstituents,suchas ̇ 2 ̇ +3( + )=0, =()62 − () √1− 1 () (13) ̇ +3( + )=0, (7) ̇ 2 +41 ( ) +4 1 () ̇ , ̇ +3( + )=0. 1 =−()62 + ̇2 + () , 2 2 (14) The total energy density and the pressure of dark energy read 1 =()62 + ̇2 − () + 4 ̇ () 2 2 2 (15) tot =DE = + +, ̇ (8) +42 () , = = + + , tot DE 1 =−ℎ () 62 − ̇2 + () , 3 2 (16) with
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