Dynamics of interacting phantom and quintessence dark energies
aa r X i v : . [ phy s i c s . g e n - ph ] A p r Dynamics of interacting phantom and quintessence darkenergies
M. Umar Farooq • Mubasher Jamil • Ujjal Debnath Abstract
We present models, in which phantom en-ergy interacts with two different types of dark energiesincluding variable modified Chaplygin gas (VMCG)and new modified Chaplygin gas (NMCG). We thenconstruct potentials for these cases. It has been shownthat the potential of the phantom field decreases froma higher value with the evolution of the Universe.
Keywords
Dark energy; Chaplygin gas; quintessence;phantom energy.
One of the outstanding developments in cosmologicalphysics in the past decade is the discovery of the ac-celerated expansion of the universe, supposedly drivenby some exotic dark energy (Perlmutter et al 1999;Riess et al 1998; Spergel et al 2003, 2007; Copeland et al2006). Surprisingly, the energy density of the dark en-ergy is two-third of the critical density (Ω Λ ≃ . m ≃ . z ≃ . M. Umar FarooqMubasher JamilUjjal Debnath Center for Advanced Mathematics and Physics, National Uni-versity of Sciences and Technology, H-12, Islamabad, Pakistan.Email: m [email protected] Center for Advanced Mathematics and Physics, National Uni-versity of Sciences and Technology, H-12, Islamabad, Pakistan.Email: [email protected] , [email protected] Department of Mathematics, Bengal Engineering and ScienceUniversity, Shibpur, Howrah-711 103, India. Email: [email protected] , [email protected] and composition of dark energy is still an open prob-lem. With the thermodynamical studies of dark energy,it is conjectured that the constituents of dark energymay be massless particles (bosons or fermions) whosecollective behavior resembles a kind of radiation fluidwith negative pressure. Moreover, the temperature ofthe universe filled with dark energy will increase as theuniverse expands (Lima & Alacaniz 2004). The earli-est proposal to explain the recent accelerated expan-sion was the cosmological constant Λ represented bythe equation of state (EoS) p = − ρ (or w = −
1) havinga negative pressure. In order to comply with the data,the cosmological constant has to be fine tuned up to120 orders of magnitude (Doglov 2004), which requiresextreme fine tuning of several cosmological parameters.The cosmological constant also poses a famous cosmiccoincidence problem (the question of explaining whythe vacuum energy came to dominate the universe veryrecently) (Bento et al 2002). The coincidence prob-lem is tackled with the use of a homogeneous andtime dependent scalar field φ , in which the scalar fieldrolls down a potential V ( Q ) according to an attractor-like solution to the equations of motion (Zlatev et al1999). But here the field has difficulties in reaching w < − .
7, while current observations favor w < − . cardassian term in the Friedmann-Robertson-Walker(FRW) equations (Freese & Lewis 2002), a generalizedChaplygin gas (GCG) (Bento et al 2006; Setare 2006,2007) and a phantom energy ( w < −
1) arising fromthe violation of energy conditions (Caldwell et al 2003;Babichev et al 2004; Nesseris & Perivolaropoulos 2004;Setare 2007). Another possibility is the ‘geometricdark energy’ based on the Ricci scalar R representedby ℜ = R/ H , where H is the Hubble parameter(Linder 2005). Notice that ℜ > / ℜ > ℜ = 1 / w > − w < − H ( z ) datarules out the occurrence of any such interaction andfavors the possibility of either more exotic couplingsor no interaction at all (Wei & Zhang 2007; Umar et al2010, 2009). The consideration of interaction betweenquintessence and phantom dark energies can be moti-vated from the quintom models (Zhang 2005). In thiscontext, we have investigated the interaction of the darkenergy with dark matter by using a more general inter-action term. We have focused on the inhomogeneousEoS for dark energy as these are phenomenologicallyrelevant.The outline of the paper is as follows. In the sectionII, we present a general interacting model for our dy-namical system. Following (Chattopadhyay & Debnath2010), we consider the two interacting dark energy models like variable modified Chaplygin gas (VMCG)and new modified Chaplygin gas (NMCG) interactwith phantom field in sections III and IV. We foundthe phantom potential in these scenarios. Finally, wepresent our conclusion. We assume the background to be a spatially flatisotropic and homogeneous FRW spacetime, given by ds = dt − a ( t )[ dr + r ( dθ + sin θdφ )] , (1)where a ( t ) is the scale factor. The corresponding Ein-stein field equations are3 H = ρ tot (2)and6( ˙ H + H ) = − ( ρ tot + 3 p tot ) . (3)Here ρ tot and p tot represent the total energy density andisotropic pressure respectively (8 πG = c = 1). More-over, the energy conservation for our gravitational sys-tem is given by˙ ρ tot + 3 H ( ρ tot + p tot ) = 0 . (4)Suppose we have a two-component model of the form ρ tot = ρ + ρ , (5)and p tot = p + p . (6)Here ρ and p denote the energy density and pressureof quintessence and ρ , p denote the energy densityand pressure of phantom dark energy. The stress energytensor for matter-energy is T µυ = − ∂ µ Φ ∂ υ Φ − g µυ h σ g βδ ∂ β ∂ δ Φ + V (Φ) i . (7)By assuming that the phantom field is evolving inan isotropic homogenous universe and that Φ is merelyfunction of time, from Eq. (7) one can extract energydensity and pressure as ρ = σ + V (Φ) , (8) p = σ − V (Φ) . (9)Here σ = − σ = +1 represents the standard scalar field which rep-resents the quintessence field, also V (Φ) is the potential. In this case, the equation of state w is given by w = p ρ = σ ˙Φ − V (Φ) σ ˙Φ + 2 V (Φ) . (10)We observe that it results in the violation of the nullenergy condition ρ + p = σ ˙Φ >
0, if σ = −
1. Sincethe null energy condition is the basic condition, its vi-olation yields other standard energy conditions to beviolated likewise dominant energy condition ( ρ > ρ ≥ | p | ) and the strong energy condition ( ρ + p > ρ + 3 p > = 1 σ (1 + ω ) ρ , (11) V (Φ) = 12 (1 − ω ) ρ . (12) Firstly, let us suppose that we have variable modifiedChaplygin gas (VMCG) representing the dark energyand is given by (Debnath 2007) p = A ρ − B a ( t ) − n ρ α , (13)where 0 ≤ α ≤ , ≤ A ≤ , B and n are constantparameters. The Chaplygin gas behaves like dust in theearly evolution of the universe and subsequently growsto an asymptotic cosmological constant at late timewhen the universe is sufficiently large. In the cosmolog-ical context, the Chaplygin gas was first suggested as analternative to quintessence (Curbelo et al 2006). Lateron, the Chaplygin gas state equation was extended to amodified form by adding a barotropic term (Benaoum2002; Debnath et al 2004; Jamil et al 2009). Recentsupernovae data also favor the two-fluid cosmologicalmodel with Chaplygin gas and matter (Panotopoulos2008). Suppose that the phantom field interacts with(VMCG), so under this interaction (supposing the in-teraction term is Q ) the continuity equations can bewritten as˙ ρ + 3 H ( ρ + p ) = Q, (14)˙ ρ + 3 H ( ρ + p ) = − Q. (15) In case of Q = 0 , we arrive at the non-interacting sit-uation while Q > ρ to other fluid of density ρ . In order to solve the above continuity equations dif-ferent forms of Q have been considered. Here we willproceed to solve the continuity equation (15) by taking Q = 3 δHρ ( δ is a coupling constant), so we get ρ = h B (1 + α )[3 A (1 + α ) + 3(1 + α )(1 + δ ) − n ] a n + Ca α )(1+ δ + A ) i α , (16)where C is the constant of integration. One can beseen that if n = 0 and A, B approache to zero, then ρ ∼ a − δ ) . Now for simplicity, we choose V = m ˙Φ ,where m is a positive constant. So using (14) and (16),we obtain the kinetic term as˙Φ = C a − σσ +2 m + 6 σ (1 + α )( − mn + (6 − n + 6 α ) σ ) × (cid:16) B (1 + α ) − n + 3(1 + α )(1 + δ + A ) (cid:17) α a − n α × F h x, −
11 + α , x, − Y a n − α )(1+ δ + A ) i , (17)and the potential energy has the form V = mC a − σσ +2 m + 6 mσ (1 + α )( − mn + (6 − n + 6 α ) σ ) × (cid:16) B (1 + α ) − n + 3(1 + α )(1 + δ + A ) (cid:17) α a − n α × F h x, −
11 + α , x, − Y a n − α )(1+ δ + A ) i , (18)where Y = C ( − n + 3(1 + α )(1 + δ + A ))3 B (1 + α ) ,x = 2 mn + ( n − α )) σ (1 + α )(2 m + σ )( − n + 3(1 + α )(1 + δ + A )) , and C is the constant of integration. From the ex-pression (18) it is clear that the potential energy is afunction of scale factor a . The graphs represented byFig. 1 and Fig. 2 show that φ increases with the pas-sage of time while the V decreases with the increase ofcosmic time t . The model which behaves as a dark matter (radia-tion) at the early stage and X- type dark energy at late stage is the New Modified Chaplygin Gas (NMCG)(Zhang et al 2006; Chattopadhyay & Debnath 2008) p = βρ + wA a − w )(1+ α ) ρ α , A > , β > . (19)In view of (second energy eq.) , the energy density ofthe (NMCG) can be expressed as ρ = (cid:20) A wa − w )(1+ α ) w − δ − β + C a − δ + β )(1+ α ) (cid:21) α (20)Now for simplicity, we again choose V = m ˙Φ , where m is a positive constant. So using (14) and (20), weobtain the kinetic term as˙Φ = C a − σσ +2 m − a − w ) σ (2 m (1 + w ) + σ ( − w )) × F h x , −
11 + α , x , Y a α )( w − β − δ ) i , (21)and the potential energy has the form V = mC a − σσ +2 m − ma − w ) σ (2 m (1 + w ) + σ ( − w )) × F h x , −
11 + α , x , Y a α )( w − β − δ ) i , (22)where Y = C ( − w + β + δ ) A w ,x = − m (1 + w ) + σ (1 − w )(1 + α )(2 m + σ )( w − β − δ ) , and C is the constant of integration. In this work, we have considered the interacting sce-nario of the universe, in which phantom energy inter-acts with two different types of dark energies includingvariable modified Chaplygin gas (VMCG), new modi-fied Chaplygin gas (NMCG). By considering some par-ticular form of interaction term, we have constructedthe potential of the phantom field. By looking at theenergy conservation equations (14) and (15) it is foundthat the energies of the (VMCG) and (NMCG) are get-ting transferred to the phantom field. With the helpof graphs we studied the variations of V and φ withthe variation of the cosmic time. From the figures we see that the potential decreases from the lower valuewith the evolution of the universe. Thus in the pres-ence of an interaction, the potential decreases and thefield decreases with the evolution of the universe. Φ Fig. 1
The variation of Φ against cosmic time t with partic-ular values of parameters A = 0 . , B = 0 . , δ = 0 . , α =0 . , C = 1 , C = 1 , n = 0 . , m = 0 . V Fig. 2
The variation of V against cosmic time t σ = − A = 0 . , B = 0 . , δ = 0 . , α = 0 . , C = 1 , C =1 , n = 0 . , m = 0 . Φ V Fig. 3
The variation of V against Φ for σ = − A = 0 . , B = 0 . , δ = 0 . , α = 0 . , C = 1 , C = 1 , n =0 . , m = 0 . Φ Fig. 4
The variations of Φ and t against cosmic time t respectively and Fig. 6 represents the variation of V againstΦ for σ = − A = 0 . , β = 0 . , δ = 0 . , α =0 . , C = 1 , C = 1 , w = − . , m = 0 . V Fig. 5
The variation of V against t for σ = − A = 0 . , β = 0 . , δ = 0 . , α = 0 . , C = 1 , C = 1 , w = − . , m = 0 . Φ V Fig. 6
The variation of V against Φ for σ = − A = 0 . , β = 0 . , δ = 0 . , α = 0 . , C = 1 , C = 1 , w = − . , m = 0 . References
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