Eur. Phys. J. C (2012) 72:1959DOI 10.1140/epjc/s10052-012-1959-4

Regular Article - Theoretical Physics

Attractor solutions in f (T ) cosmology

Mubasher Jamil1,2,a, D. Momeni2,b, Ratbay Myrzakulov2,c

1Center for Advanced Mathematics and Physics (CAMP), National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan2Eurasian International Center for Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan

Received: 13 February 2012 / Revised: 5 March 2012 / Published online: 28 March 2012© Springer-Verlag / Società Italiana di Fisica 2012

Abstract In this paper, we explore the cosmological impli-cations of interacting dark energy model in a torsion-basedgravity namely f (T ). Assuming that dark energy interactswith dark matter and radiation components, we examine thestability of this model by choosing different forms of in-teraction terms. We consider three different forms of darkenergy: cosmological constant, quintessence and phantomenergy. We then obtain several attractor solutions for eachdark energy model interacting with the other components.This model successfully explains the coincidence problemvia the interacting dark energy scenario.

1 Introduction

Its now well-accepted in the astrophysics community thatthe observable universe is in a phase of rapid expansionwhose rate of expansion is increasing, so called ‘acceler-ated expansion’. This conclusion has been drawn by nu-merous recent cosmological and astrophysical data, find-ings of supernovae SNe Ia [1, 2], cosmic microwave back-ground radiations via WMAP [3], galaxy redshift surveysvia SDSS [4] and galactic X-ray [5]. This phenomenon iscommonly termed ‘dark energy’ (DE) in the literature, andsuggests that a cosmic dark fluid possessing negative pres-sure and positive energy density. Although the phenomenonof dark energy in cosmic history is very recent, z ∼ 0.7, ithas opened new areas in cosmology research. Two impor-tant problems like ‘fine tuning’ and ‘cosmic coincidence’are related to dark energy. It is thought that the most el-egant solution to DE paradigm is Einstein’s cosmologicalconstant [6, 7] but it cannot resolve the two problems men-tioned above. Hence cosmologists looked for other theoret-ical models by considering the dynamic nature of dark en-ergy like quintessence scalar field [8–10], a phantom energy

a e-mail: mjamil@camp.nust.edu.pkb e-mail: davidmathphys@yahoo.co.uk, d.momeni@yahoo.comc e-mail: rmyrzakulov@csufresno.edu, rmyrzakulov@gmail.com

field [11, 12] and f-essence [13, 14]. Another interestingset of proposals to the DE puzzle is the ‘modified gravity’which was proposed after the failure of general relativity(GR). This new set of gravity theories passes several solarsystem and astrophysical tests successfully [15–17].

For the past few years, models based on DE interactingwith dark matter or any other exotic component have gainedgreat impetus. Such interacting DE models can successfullyexplain numerous cosmological puzzles including dynamicDE, phantom crossing, cosmic coincidence and cosmic age[18–23] and also in good compatibility with the astrophys-ical observations of cosmic microwave background, super-nova type Ia, baryonic acoustic oscillations and galaxy red-shift surveys [24–30]. There is some criticism on interactingDE models for not being favored by observations and thatthe usual ΛCDM model is favorable [31]. However, the ther-mal properties of this model in various gravity models havebeen discussed in the literature [32–34]. The model in whichdark energy interacts with two different fluids has been in-vestigated in literature. In [35, 36], the two fluids were darkmatter and the other was unspecified. However, in anotherinvestigation [37], the third component was taken as radi-ation to address the cosmic-triple-coincidence problem andstudy the generalized second law of thermodynamics. In arecent investigation, the authors investigated the coincidenceproblem in loop quantum gravity with triple interacting flu-ids including DE, dark matter and unparticle [38, 39].

There is no need to construct a gravitational theory on aRiemannian manifold. A manifold can be divided into twoseparate but connected parts; one with Riemannian struc-ture with a definite metric and another part, with a non-Riemannian structure and with torsion or non-metricity. Thepart which has the zero Riemannian tensor but has non-zerotorsion is based on a tetrad basis, and defines a Weitzen-bock spacetime. f (T ) gravity is an alternative theory forGR, defined on the Weitzenbock non-Riemannian manifold,working only with torsion. This model was firstly proposedby Einstein for unifying electromagnetism and gravity. If

Page 2 of 10 Eur. Phys. J. C (2012) 72:1959

f (T ) = T , this theory is called teleparallel gravity [40–42].It has been shown that with linear T , this model has manycommon features like GR and is in good agreement withsome standard tests of the GR in solar system [40, 41].But introduction of a general f (T ) model goes back a fewyears [43, 44]. This model has many features, for exampleBirkhoff’s theorem has been studied in this model of grav-ity [45]. Earlier the authors in [46] investigated perturba-tion in f (T ) and found that the perturbation in f (T ) gravitygrows slower than that in Einstein general relativity. Bambaet al. [47] studied the evolution of equation of state param-eter and phantom crossing in f (T ) model. Emergent uni-verses in the chameleon f (T ) model is investigated in [48].As a thermodynamical view, there are many strange fea-tures. It has been shown in f (T ) gravity that the famousformula of entropy–area of the black hole thermodynamicsis not valid [50]. The reason goes back to the violation ofthe local Lorentz invariance of this theory [50, 51]. It showsthat it is not possible to use the Wald conjecture [52] forcalculating the entropy as a Noether charge in f (T ). TheHamiltonian formulation of f (T ) gravity has been studiedin [49] and it has been shown that there are five degreesof freedom. In this paper we study the triple coincidenceproblem: why we happen to live during this special epochwhen ρΛ ∼ ρm ∼ ρr? [53]. We investigate this problem inthe framework of f (T ) gravity.

We follow the plan: In Sect. 2 we introduce the basicequations as an autonomous dynamical system. In Sect. 3we choose a model for f (T ) gravity. In Sect. 4 we work outa numerical analysis of the stability and the evolution of thefunctions of the model in detail. We conclude and summa-rize in Sect. 5.

2 Basic equations

One suitable form of action for f (T ) gravity in Weitzenbockspacetime is [43, 44]

S = 1

2κ2

∫d4x

√e(T + f (T ) + Lm

)

Here e = det(eiμ), κ2 = 8πG and ei

μ is the tetrad (vierbein)basis. The dynamical quantity of the model is the scalar tor-sion T and Lm is the matter Lagrangian. We start with theFriedmann equation in this form of the f (T ) model [43, 44],

H 2 = 1

1 + 2fT

(κ2

3ρ − f

6

), (1)

where ρ = ρm + ρd + ρr , while ρm, ρd and ρr represent theenergy densities of matter, dark energy and the radiation,respectively.

Another FRW equation is

H = −κ2

2

(ρ + p

1 + fT + 2TfT T

). (2)

For a spatially flat universe (k = 0), the total energy conser-vation equation is

ρ + 3H(ρ + p) = 0, (3)

where H is the Hubble parameter, ρ is the total energy den-sity and p is the total pressure of the background fluid.

We assume a three component fluid containing matter,dark energy and radiation having an interaction. The corre-sponding continuity equations are [38, 39]

ρd + 3H(ρd + pd) = Γ1,

ρm + 3Hρm = Γ2, (4)

ρr + 3H(ρr + pr) = Γ3,

which satisfy collectively (3) such that Γ1 + Γ2 + Γ3 = 0.We define dimensionless density parameters via

x ≡ κ2ρd

3H 2, y ≡ κ2ρm

3H 2, z ≡ κ2ρr

3H 2. (5)

The continuity equations (4) in dimensionless variables re-duce to

dx

dN= 3x

(x + y + z + wdx + wrz

1 + fT + 2TfT T

)

− 3x(1 + wd) + κ2

3H 3Γ1,

dy

dN= 3y

(x + y + z + wdx + wrz

1 + fT + 2TfT T

)

− 3y + κ2

3H 3Γ2,

dz

dN= 3z

(x + y + z + wdx + wrz

1 + fT + 2TfT T

)

− 3z(1 + wr) + κ2

3H 3Γ3,

(6)

where N ≡ lna is called the e-folding parameter. The cou-pling functions Γi , i = 1,2,3, are in general functions ofthe energy densities and the Hubble parameter i.e. Γi(Hρi).The system of equations in (6) is analyzed by first equatingthem to zero to obtain the critical points. Next we perturbequations up to first order about the critical points and checktheir stability. Below, for our computation, we shall assumewm = 0, wr = 1

3 and wd to be a general non-zero but nega-tive parameter. We are interested in stable critical points (i.e.those points for which all eigenvalues of the Jacobian matrixare negative) as these are attractor solutions of the dynami-cal system.

Eur. Phys. J. C (2012) 72:1959 Page 3 of 10

3 f (T ) model

To avoid analytic and computation problems, we choose asuitable f (T ) expression which contains a constant, linearand a non-linear form of torsion, specifically

f (T ) = 2C1√−T + αT + C2, (7)

where α, C1 and C2 are arbitrary constants. The first andthe third terms (excluding the middle term) have correspon-dence with the cosmological constant EoS in f (T ) grav-ity [54]. There are many kinds of such models, reconstructedfrom different kinds of dark energy models. For example theform (7) may be inspired from a model for dark energy froma proposed form of the Veneziano ghost [55]. But the linearterm is needed to show the differences between our results inf (T ) gravity from the Einstein gravity. Here we choose thismodel to simplify our numerical computations and for eas-ier discussion on the difference of our results with the sameresults in GR. It is the minimum model, but our equationshave been written for a general f (T ) action. It is possibleby repeating the numerical steps as we do in this paper todiscuss the stability of other models. Recently Capozzielloet al. [56] investigated the cosmography of f (T ) cosmologyby using data of BAO, Supernovae Ia and WMAP. Followingtheir interesting results, we notice that if we choose C2 = 0,α = Ωm0 and C1 = √

6H0(Ωm0 − 1), then we can estimatethe parameters of our proposed f (T ) model as a function ofHubble parameter H0 and the cosmographic parameters andthe value of matter density parameter.

4 Analysis of stability in phase space

In this section, we will construct four models by choosingdifferent coupling forms Γi and analyze the stability of thecorresponding dynamical systems about the critical points.We shall plot the phase and evolutionary diagrams accord-ingly. For this reason, we must find the critical points of (6),and then we linearize the system near the critical points upto first order.

4.1 Interacting model I

We consider the model with the following interaction terms:

Γ1 = −6bHρd, Γ2 = Γ3 = 3bHρd, (8)

where b is a coupling parameter and we assume it to be apositive real number of order unity. Thus (8) says that bothmatter and radiation have increase in energy density withtime while dark energy loses its energy density. Therefore,it is decay of dark energy into matter and radiation.

Using (8), the system (6) takes the form

dx

dN= − 3x(1 + wd) + 3x

(x + y + z + wdx + wrz

1 + α

)

− 6bx,

dy

dN= − 3y + 3y

(x + y + z + wdx + wrz

1 + α

)+ 3bx,

dz

dN= − 3z(1 + wr) + 3z

(x + y + z + wdx + wrz

1 + α

)

+ 3bx.

(9)

The critical points for this model are obtained by equatingthe left hand sides of (9) to zero. We obtain four criticalpoints:

• Point A1 : ((1+α)(1+wd+2b)

1+wd,0,0).

• Point B1 : (0,0,0).• Point C1 : (0, (1 − b)(1 + α),0).• Point D1 : (0,0, 3

4 (1 + α)( 43 − b)).

The eigenvalues of the Jacobian matrix for these criticalpoints are:

• Point A1 : λ1 = 3(1 + wd + 2b),λ2 = 3(wd + 3b),λ3 =−1 + 3(wd + 3b).

• Point B1 : λ1 = 3(b − 1), λ2 = −3(b − 1), λ3 = −3(1 +2b + wd).

• Point C1 : λ1 = −1, λ2 = 3(1 − b),λ3 = −3(wd + 3b).• Point D1 : λ1 = 1, λ2 = 4 − 3b,λ3 = 1 − 3wd − 9b.

Point A1 is stable when one of these conditions is satisfied:

wd < −3, b < 1/18, (10)

wd ≥ −3, wd < −10/9, b < 1/18, (11)

wd ≥ −10

9, wd < 0, b < −1

2(1 + wd). (12)

B1 is an unstable critical point since if b > 1 then λ1 > 0but λ2 < 0. C1 is stable if b > 1, wd > −3b. Similarly D1 isunstable since λ1 > 0.

In Figs. 1, 2, 3, 4, 5 and 6, we plot the parameters ofModel I. In Fig. 1, we observe that the dimensionless den-sity parameters evolve from their currently observed valuesto vanishing densities. In Fig. 2, we observe similar behav-ior when plotted against e-folding parameter: the density pa-rameters evolve from their current values to zero. In Fig. 1,we deal with phantom energy, in Fig. 3, the quintessencecase while in Fig. 5, the cosmological constant case.

Page 4 of 10 Eur. Phys. J. C (2012) 72:1959

Fig. 1 Model I: Phase space for wd = −1.2, b = 0.5, α = 0.5. Itshows an attractor behavior

Fig. 2 Model I: variation of x, y, z as a function of the N = ln(a).The initial conditions chosen are x(0) = 0.7, y(0) = 0.3, z(0) = 0.01,wd = −1.2 and b = 0.5

4.2 Interacting model II

We study another model with the choice of the interactionterms

Γ1 = −3bHρd, Γ2 = 3bH(ρd − ρm),

Γ3 = 3bHρm.(13)

This model effectively describes the situation when dark en-ergy loses energy density to matter while the radiation den-sity increases due to interaction with the matter.

dx

dN= 3x

(x + y + z + wdx + wrz

1 + α

)

− 3bx − 3x(1 + wd), (14)

Fig. 3 Model I: Phase space for wd = −0.5, b = 0.5, α = 0.5

Fig. 4 Model I: variation of x, y, z as a function of the N = ln(a).The initial conditions chosen are x(0) = 0.7, y(0) = 0.3, z(0) = 0.01,wd = −0.5 and b = 0.5

dy

dN= 3y

(x + y + z + wdx + wrz

1 + α

)

+ 3b(x − y) − 3y, (15)

dz

dN= 3z

(x + y + z + wdx + wrz

1 + α

)

+ 3by − 3z(1 + wr). (16)

There are four critical points:

• Point A2: (0,0,0).• Point B2: (0,0,1 + α).• Point C2: (0, (1 − 3b)(1 + α),3b(1 + α).

Eur. Phys. J. C (2012) 72:1959 Page 5 of 10

• Point D2: (− 3(2+wd)(b−wd−1)(b−wd−7/3)(α+1)

3wd3+(16−3b)wd

2+(−6b+27)wd+14+b2+b,

− b3(α+1)(b−wd−7/3)(b−wd−1)

3wd3+(16−3b)wd

2+(−6b+27)wd+14+b2+b,

3(α+1)(b−wd−1)b2

14+16wd2−6wdb+b2−3wd

2b+3wd3+27wd+b

).

The eigenvalues of the Jacobian matrix for these criticalpoints are:

• Point A2 : λ1 = −4, λ2 = −3(1+b), λ3 = 3(1+wd −b).• Point B2 : λ1 = 4, λ2 = 1 − 3b, λ3 = 7 + 3(wd − b).• Point C2 : λ1 = 3(1 + b), λ2 = 3b − 1, λ3 = 3(wd + 2).• Point D2 : λ1 = −3(wd + 2), λ2 = 3(b − wd − 1), λ3 =

−7 − 3(b − wd).

A2, D2 are conditionally stable if b > 1 + wd (for A2) andwd > −2 and b < 1 + wd (for D2). But B2 and C2 are un-stable since λ1 > 0.

Fig. 5 Model I: Phase space for wd = −1, b = 0.5, α = 0.5

Fig. 6 Model I: variation of x, y, z as a function of the N = ln(a).The initial conditions chosen are x(0) = 0.7, y(0) = 0.3, z(0) = 0.01,wd = −1 and b = 0.5

In Figs. 7, 8, 9, 10, 11 and 12, we show the dynamics ofModel II. Attractor solutions are shown in Figs. 7, 9 and 11.In Fig. 8, we see that the energy density of dark energy de-cays like quintessence. Also the energy density of radiationfirst increases till N ∼ 1.6 and then starts decreasing. Thematter density always decreases and approaches zero nearlyN ∼ 3. In Fig. 10, the energy density of dark energy in-creases rapidly behaving like phantom energy, energy den-sity of matter rises almost exponentially at later times, whileradiation density increases slower compared to both matterand dark energy. These novel behaviors appear on accountof the interaction between three components. In Fig. 12, thedark energy density behaves like quintessence, while matterand radiation density fall with expansion.

Fig. 7 Model II: Phase space for wd = −1.2, b = 0.5, α = 0.5

Fig. 8 Model II: variation of x, y, z as a function of the N = ln(a).The initial conditions chosen are x(0) = 0.7, y(0) = 0.3, z(0) = 0.01,wd = −1.2 and b = 0.5

Page 6 of 10 Eur. Phys. J. C (2012) 72:1959

Fig. 9 Model II: Phase space for wd = −0.5, b = 0.5, α = 0.5

Fig. 10 Model II: variation of x, y, z as a function of the N = ln(a).The initial conditions chosen are x(0) = 0.7, y(0) = 0.3, z(0) = 0.01,wd = −0.5 and b = 0.5

4.3 Interacting model III

Let us take the interaction terms [38, 39]

Γ1 = −6bκ2H−1ρdρr ,

Γ2 = Γ3 = 3bκ2H−1ρdρr .(17)

The system in (6) takes the form

dx

dN= 3x

(x + y + z + wdx + wrz

1 + α

)

− 3x − 3wdx − 18bxz,

dy

dN= 3y

(x + y + z + wdx + wrz

1 + α

)(18)

− 3y + 9bxz,

Fig. 11 Model II: Phase space for wd = −1, b = 0.5, α = 0.5

Fig. 12 Model II: variation of x, y, z as a function of the N = ln(a).The initial conditions chosen are x(0) = 0.7, y(0) = 0.3, z(0) = 0.01,wd = −1 and b = 0.5

dz

dN= 3z

(x + y + z + wdx + wrz

1 + α

)

− 3z − 3wrz + 9bxz.

There are six critical points:

• Point A3: (0,0,0).• Point B3: (0,1 + α,0).• Point C3: (0,0,1 + α).• Point D3: (1 + α,0,0).• Point E3: ( 4

9b,− 2(1+wd)

9b,− 1+wd

6b).

Eur. Phys. J. C (2012) 72:1959 Page 7 of 10

• Point F3: ((wd+6bα+6b− 1

3 )

b(6bα+6b− 13 +2wd)

,

−3bα−3b+36b2α+18b2α2+9wdbα+18b2+wd2−2wd

13 +9wdb+ 1

9

3b(6bα+6b− 13 +2wd)

,

− wd(3bα+3b− 13 +wd)

3b(6bα+6b− 13 +2wd)

).

The eigenvalues of the Jacobian matrix for these criticalpoints are:

• Point A3: λ1 = −3, λ2 = −4, λ3 = −3(1 + wd).• Point B3: λ1 = 3, λ2 = −1, λ3 = −3wd .• Point C3: λ1 = 3wd , λ2 = 3(1 + wd), λ3 = 9bα + 9b −

1 + 3wd .• Point D3 : λ1 = 4, λ2 = 1−9b−9bα, λ3 = −9bα −9b+

1 − 3wd .

Fig. 13 Model III: Phase space for wd = −1.2, b = 0.5, α = 0.5

Fig. 14 Model III: variation of x, y, z as a function of the N = ln(a).The initial conditions chosen are x(0) = 0.7, y(0) = 0.3, z(0) = 0.01,wd = −1.2 and b = 0.5

A3 is stable for wd > −1. B3, D3 are unstable. C3 is condi-tionally stable when wd < −1, b <

1−3wd

9(1+α).

In Figs. 13, 14, 15, 16, 17 and 18, we have plotted the cos-mological parameters of Model III. In Figs. 13, 15 and 17,we show the attractor solutions of the differential equationsequation. In Fig. 14, we observe the oscillatory behavior ofdark energy and other cosmic components. It shows thatwhen DE energy density decays then corresponding den-sities of dark matter and radiation increase and vice versa.Figure 16 shows that all forms of energy densities vanish byN ∼ 4.5 From Fig. 18, the radiation density stays zero whilethe matter energy density decreases and vanish by N ∼ 2.The dark energy density increases by N ∼ 2 while it de-creases and stays constant at later epochs.

Fig. 15 Model III: Phase space for wd = −0.5, b = 0.5, α = 0.5

Fig. 16 Model III: variation of x, y, z as a function of the N = ln(a).The initial conditions chosen are x(0) = 0.7, y(0) = 0.3, z(0) = 0.01,wd = −0.5 and b = 0.5

Page 8 of 10 Eur. Phys. J. C (2012) 72:1959

4.4 Interacting model IV

Consider another model with the interaction terms [38, 39]

Γ1 = −3bκ2H−1ρdρr,

Γ2 = 3bκ2H−1(ρdρr − ρmρr), (19)

Γ3 = 3bκ2H−1ρmρr .

The system in (6) takes the form

dx

dN= 3x

(x + y + z + wdx + wrz

1 + α

)

− 3x − 3wdx − 9bxz,

dy

dN= 3y

(x + y + z + wdx + wrz

1 + α

)(20)

Fig. 17 Model III: Phase space for wd = −1, b = 0.5, α = 0.5

Fig. 18 Model III: variation of x, y, z as a function of the N = ln(a).The initial conditions chosen are x(0) = 0.7, y(0) = 0.3, z(0) = 0.01,wd = −1 and b = 0.5

− 3y + 9b(xz − yz),

dz

dN= 3z

(x + y + z + wdx + wrz

1 + α

)

− 3z − 3wrz + 9byz.

There are seven critical points:

• Point A4: (0,0,0).• Point B4: (1 + α,0,0).• Point C4: (0,1 + α,0).• Point D4: (0,0,1 + α).

• Point E4: ( 13

43 wd

b(1+wd), 4

9b,− 1

31+wd

b).

• Point F4: (wd+3b+3bα−1/3

3b,− (− 1

3 +wd)(wd+3b+3bα− 13 )

3b(3b+3bα− 13 )

,

wd(− 13 +wd)

3b(3b+3bα− 13 )

).

• Point G4: (0, 49b

,− 13b

).

For points A4, B4, C4, D4, G4, the eigenvalues of the Jaco-bian matrix are

• Point A4: λ1 = −3, λ2 = −4, λ3 = −3(1 + wd).• Point B4: λ1 = 3wd,λ2 = 3(1 + wd),λ3 = 3wd − 1.• Point C4: λ1 = 3, λ2 = −3wd , λ3 = 9b(1 + α) − 1.• Point D4: λ1 = 3(1 + wd), λ2 = −9b − 9bα + 1, λ3 =

−3wd − 9b − 9bα + 1.• Point G4: λ1 = −3wd ,

λ2 =√

3√

b(1+α)(3bα+3b/3α−4/9+3b+b)b(1+α)

,

λ3 = −√

3√

b(1+α)(3bα+3b/3α−4/9+3b+b)b(1+α)

.

It is observed that, in this model, A4 is stable for wd >

−1, B4 is stable for wd < −1. C4 is unstable. D4 is stablefor wd < −1, b > 1

9(1+α). It is not possible to determine the

stability of the point E4. But G4 is unstable.In Figs. 19, 20, 21, 22 and 23, we give description about

Model IV. Figures 19, 21, 23 represent the phase space dia-grams for different forms of dark energy. Figures 22 and 23

Fig. 19 Model IV: Phase space for wd = −1.2, b = 0.5, α = 0.5

Eur. Phys. J. C (2012) 72:1959 Page 9 of 10

Fig. 20 Model IV: variation of x, y, z as a function of the N = ln(a).The initial conditions chosen are x(0) = 0.7, y(0) = 0.3, z(0) = 0.01,wd = −1.2 and b = 0.5

Fig. 21 Model IV: Phase space for wd = −0.5, b = 0.5, α = 0.5

show that energy density of dark energy decreases while inFig. 20, the density of DE increases. It is the typical behav-ior of the phantom fields. Thus, our model predicts the cor-rect evolutionary scheme for the DE density in the regime ofphantom.

5 Conclusion

f (T ) gravity is a powerful and novel theory for explanationof the acceleration expansion of the universe. In this paperwe discussed the stability of the interactive models of thedark energy, matter and radiation in a FRW model, for a gen-eral f (T ) theory. We derived the equations and show whywe have some attractor solutions for some specific forms ofthe interactions. We numerically integrate the equations and

Fig. 22 Model IV: variation of x, y, z as a function of the N = ln(a).The initial conditions chosen are x(0) = 0.7, y(0) = 0.3, z(0) = 0.01,wd = −0.5 and b = 0.5

Fig. 23 Model IV: Phase space for wd = −1, b = 0.5, α = 0.5

show that the evolution of the dark energy density mimicsthree different behaviors, phantom, quintessence and cos-mological constant in some interactive forms. We like tocomment that this interaction is purely phenomenologicallyrequired to meet some observational consequences. Sincethe phase space of the system of the evolutionary equationshas a definitive end point, trackers exist, and depending onthe initial conditions, there are different kinds of tracker.

Acknowledgements We would like to thank anonymous referees forgiving useful comments to improve this paper. M. Jamil and D. Mo-meni would like to thank the warm hospitality of Eurasian NationalUniversity where this work was started and completed.

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