the dynamic evolutions of the generalized hobbit model and structure formation

Gen Relativ Gravit (2009) 41:1677–1693DOI 10.1007/s10714-008-0737-y

RESEARCH ARTICLE

The dynamic evolutions of the generalized Hobbitmodel and structure formation

Ya Bo Wu · Ming Hui Fu · Fang Yuan Cheng ·Huan Huan Fu

Received: 24 February 2008 / Accepted: 30 November 2008 / Published online: 19 December 2008© Springer Science+Business Media, LLC 2008

Abstract In this paper a more general phenomenological model is proposed onthe basis of the Hobbit model in Cardone et al. (Phys Rev D 69:083517, 2004). Themain purpose of constructing our model is to make this model can not only mimicthe �CDM model, but also describe a four-phase smooth transition of the universe,namely the inflationary phase, a radiation-dominated phase, a matter-dominated phaseand a de Sitter-like phase. In order to check whether this model is a viable one, theevolutions of the universe are respectively discussed in the two cases, and the possiblephysical interpretations for this model are also respectively shown by using the scalarfields. Finally, the structure formations in our model are simply discussed for the bothcases, and the results given by us reconfirm that our model can be regarded as a fit tothe �CDM model, if we choose the proper conditions.

Keywords Dark matter · Dark energy · Structure formation

1 Introduction

In recent years, many observational evidences from Type Ia supernovae(SNe Ia) [1,2]indicate that the expansion of the universe is currently accelerating, also along withthe observations of CMBR anisotropy spectrum [3] and large scale structure (LSS)[4]. It is now widely accepted that an exotic energy component with negative pressure

Y. B. Wu · F. Y. Cheng · H. H. FuDepartment of Physics, Liaoning Normal University, 116029 Dalian, People’s Republic of Chinae-mail: [email protected]

M. H. Fu (B)Department of Mathematics and Physics, Shenyang Command College,110113 Shenyang, People’s Republic of Chinae-mail: [email protected]

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1678 Y. B. Wu et al.

often called dark energy may exist, which dominates the present universe and affectsthe recent expansion [5]. According to the latest joint observations from the set ofESSENCE SNe Ia and the SuperNovae Legacy Survey SNe Ia [6], dark energy occu-pies about 73% of the energy budget in the universe, the components of dark matterand baryon occupies about 27%, and the current SNe Ia data are fully consistent witha cosmological constant.

At present, many dark energy models have been established, such as quintessence[7], phantom [8], unified dark energy (UDE) [9–11], holographic dark energy [12,13]and so on. But the mystery of dark energy is far to be solved. In order to understandthe nature of dark energy and reproduce the universe that we observe, Cardone et al.[14] proposed a phenomenological model, that is the so-called Hobbit model, whichdescribes the evolution of the universe from the perspective of a three-phase smoothtransition, i.e., a radiation-dominated phase, a matter-dominated phase and finally a deSitter-like phase. But as we know, our universe undergoes an inflation stage in the veryearly time. Hence we hope to describe the generally recognized evolution of the uni-verse in a broader redshift range. Also, the relations between inflation and the currentacceleration have become the concerned topic by researchers. These considerationslead us to extend the theory of Cardone [14] to a more general phenomenologicalmodel, which is designed to describe a four-phase transition process, namely theinflationary phase, a radiation-dominated phase, a matter-dominated phase and a deSitter-like phase. In order to check this idea, we will respectively discuss the evolutionsof the universe in two different cases and present possible physical interpretations forthe both cases. Furthermore, we will use Newtonian treatment to discuss the structureformations for the both cases in our model by comparing the growth variable andgrowth index with those of �CDM model.

The paper is organized as follows. In Sect. 2, we describe the evolution of thegeneralized Hobbit as a unified dark sector model. In Sect. 3, the cosmic evolution ofthe generalized Hobbit model as a single fluid is discussed. In Sect. 4, the structureformations for the both cases are respectively discussed. Finally, the conclusions arepresented in Sect. 5.

2 The evolution of the generalized Hobbit as a unified dark sector model

It is widely accepted that the universe is made up of three components which arebaryon matter, dark matter and dark energy. However, in our model we ignore thecomponent of baryon matter in order to simplify the discussions, and suppose that theuniverse consists of a two-component fluid. Concretely, we consider our model as aunified dark sector model and the following expression of the total energy density isintroduced as:

ρ = N(

1 + aI

a

)β (1 + aR

a

)α [1 +

(aX

a

)γ ]. (1)

Here, N is a normalization constant. aI , aR and aX are three invariant scaling fac-tors satisfying aI � aR � aX . α, β, γ are all constant parameters. When we choose(α, β, γ ) = (1, 0, 3), our model can recover the Hobbit model. In order to reproduce

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The dynamic evolutions of the generalized Hobbit model and structure formation 1679

the universe that we observe, we choose (α, β, γ ) = (1,−4, 3) here. Then Eq. (1)becomes:

ρ = N(

1 + aI

a

)−4 (1 + aR

a

) [1 +

(aX

a

)3]

. (2)

It is quite easy to see that: (C1 � C2 and C1, C2 are constants)

ρ ∼ C1 for a � aI , ρ ∼ a−4 for aI � a � aR,

ρ ∼ a−3 for aR � a � aX , ρ ∼ C2 for a � aX .

The energy density ρ can also be rewritten as a function of redshift z, which is

ρ = N

(1 + 1 + z

1 + zI

)−4 (1 + 1 + z

1 + zR

) [1 +

(1 + z

1 + zX

)3]

(3)

having defined zI = 1aI

−1, zR = 1aR

−1, zX = 1aX

−1, where zI = 108, zR = 3454.Here zX is a constant parameter, which can be constrained by some observationalquantities. (zI is arbitrarily chosen. Here we claim that it must satisfy aI � aR � aX

in order to fit the actual evolution, and we take zR = 3454 according to zeq = 3454in [15]).

The equation of state is defined as ω ≡ pρ

, thus the expression of the unified stateparameter ω is:

ω = −1 − 4

3

(1 + z

2 + z + zI

)+ 1

3

(1 + z

2 + z + zR

)+ (1 + z)3

(1 + zX )3 + (1 + z)3 . (4)

From Eq. (4), some useful information can be obtained, which is

ω � −1 for z � zI , ω � 1

3for zI � z � zR,

ω � 0 for zR � z � zX , ω � −1 for z � zX .

For z � zI , the total energy density remains constant, which behaves like the usualcosmological constant �. Thus the above expressions confirm that the early uni-verse undergoes inflation, as intended. For zI � z � zR , the results of ρ ∼ a−4 andω � 1

3 indicate that the universe undergoes a radiation-dominated expansion. ForzR � z � zX , ρ ∼ a−3 and ω � 0, which shows the picture of a matter-dominateduniverse. While for z � zX , we find that the universe would ultimately approach a deSitter-like phase. Note that the energy density ρ of inflationary epoch is much largerthan that of a de Sitter-like phase, and hence the universe may expand much fasterwhen inflation occurs. In Fig. 1, we plot the evolutional trajectory of state parameterω in Eq. (4).

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Fig. 1 The evolution of state parameter ω. a The solid line shows the recent evolution in our model, whichoverlaps with that in �CDM model denoted by the dotted line; while b describes the evolution of the veryearly stage. We choose zX = 0.393

The total energy density ρ can be decomposed as ρ = ρ1 + ρ2 with:

ρ1 = N(aX

a

)3 (1 + aI

a

)−4 (1 + aR

a

)(5)

ρ2 = N(

1 + aI

a

)−4 (1 + aR

a

). (6)

We can easily get:

ρ1 ∼ const, ρ2 ∼ 0 for a � aI

ρ1 ∼ a−4, ρ2 ∼ a−1,ρ2

ρ1∼ 0 for aI � a � aR

ρ1 ∼ a−3, ρ2 ∼ const,ρ2

ρ1∼ 0 for aR � a � aX

ρ1 ∼ a−3, ρ2 ∼ const,ρ1

ρ2∼ 0 for a � aX .

Suppose that there is no interaction between the two components, the expressionsof state parameter ω1 and ω2 can be derived as follows:

ω1 = −1 + 1

3

[3 − 4(1 + z)

2 + z + zI+ 1 + z

2 + z + zR

](7)

ω2 = −1 + 1

3

[ −4(1 + z)

2 + z + zI+ 1 + z

2 + z + zR

]. (8)

It is easy to see that: for z � zI , ω1 � −1; for zI � z � zR , ω1 � 13 ; for z � zR ,

ω1 � 0, ω2 � −1.

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For a � aI , ρ1 occupies most of the total energy density and the behavior of thecomponent with ρ1 is similar to that of vacuum energy. It follows that the componentwith ρ1 drives inflation. For aI � a � aR , ρ2 is small and the universe is radiation-dominated. After a � aR , the behaviors of the two components do not change anymore, but the proportions of ρ1 and ρ2 would change, the component with ρ1 domi-nated first and then the component with ρ2 dominated. Moreover, from Fig. 1a as wellas ρ1 ∼ a−3, ρ2 ∼ const and ω2 � −1 when z � zR , it is obvious that our modelcould mimic the �CDM model to describe the evolution of the universe as intended.

Also, the fractional density parameters can be easily obtained

�1 = (1 + z)3

[(1 + zX )3 + (1 + z)3] (9)

�2 = (1 + zX )3

[(1 + zX )3 + (1 + z)3] . (10)

In order to fit the observations, we respectively choose �01 � 0.27 and �02 � 0.73according to the latest data �0m � 0.27, �0de � 0.73 [6]. Thus Eq. (9) or Eq. (10)gives zX � 0.393. Using ρ1(zc) = ρ2(zc), the coincidence redshift zc � 0.393 isobtained, which is consistent with the data measured by SDSS-II [16]. Figure 2 showsthe evolution of the fractional density parameters.

Furthermore, the deceleration parameter q can be expressed as

q =−1 − H

H2 =−1 + 1

2

[ −4(1 + z)

2 + zI + z+ 1 + z

2 + zR + z+ 3(1 + z)3

(1 + zX )3+(1 + z)3

]. (11)

From Eq. (11), the transition redshift zT and the current value of q could be respec-tively obtained, zT � 0.755 and q0 = −0.595, which are fit to those of �CDM for�0m � 0.27 and the joint analysis of SNe and CMB data q0 = −0.63 ± 0.12 [17,18]and zT = 0.69 ± 0.08 [19]. In Fig. 3, we plot the evolutional trajectory of q.

There are many ways to reconstruct the scalar field and the potential [20,21]. Below,following the approach of Cardone et al. [14], we will give a possible physical inter-pretation in the framework of the universe made out of two components. We mayconsider the two components with ρ1 and ρ2 as two different scalar fields (φ1, φ2) andthey could roll down their potentials (V1, V2), respectively. The energy density andpressure for the scalar field φ are ρ = 1

2 φ2 + V (φ) and p = 12 φ2 − V (φ), so that in

this framework we may recover our model, if we reconstruct the following forms ofφ1, φ2 and V1, V2:

φ1 =√

3

4πG

a∫

0

[a3

X

a′3 + a3X

(1 + 1 − ω1

1 + ω1

)−1] 1

2 da′

a′ ,

V1 =(

1 + 1 − ω1

1 + ω1

)−1 1 − ω1

1 + ω1ρ1

(12)

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Fig. 2 The evolutions of fractional density parameters for �1 and �2

Fig. 3 The evolution of the deceleration parameter q(z)

φ2 =√

3

4πG

a∫

0

[a′3

a′3 + a3X

(1 + 1 − ω2

1 + ω2

)−1] 1

2 da′

a′ ,

V2 =(

1 + 1 − ω2

1 + ω2

)−1 1 − ω2

1 + ω2ρ2.

(13)

From Eq. (12), when a � aI , the results can be get: 12 φ1

2 → 0, V1 → constant and

φ12 � V1, φ1 � V1, from which we see that the condition (slow-roll approximation)

for inflation could be satisfied. While a � aR , 12 φ1

2 � V1 and p1 � 0 are obtained.According to the above analyses, we think that the component with ρ1 may be a specialscalar field since it could drive the inflation in the very early time and when a � aR ,

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Fig. 4 Relation between the potential V2 and the scalar field φ2

it has the feature of pressureless matter. Furthermore, in Fig. 4, the relation betweenV2 and φ2 is shown by numerically solving Eq. (13). From the figure, we see that thepotential V2 is a monotonically decreasing function of the scalar field φ2 as expectedand the variation of V2 with respect to V20 is quite small, less than 5% over the fullredshift range. (Here, φ20 and V20 are the current values of the scalar field φ2 andthe potential V2, respectively.) It follows that the component with ρ2 is similar to thecosmological constant when a � aR and drives the late-time cosmic acceleration.

3 The cosmic evolution of the generalized Hobbit model with a single fluid

In the following, we will investigate the generalized Hobbit model with a single fluid,which could be considered as the so-called dark energy component only. Thus the totalenergy density of unverse is ρ = ρde +ρm , where ρde and ωde can be described as thesame as those in Eqs. (2)–(4), the only difference is the value of the parameter zX , i.e.,its value is here chosen as 60 so that ωde � −1 can be kept for much long time. Andρm denotes a standard dust component. According to the features of ρde and ωde, thismodel could also describe a four-phase smooth transition of the universe and mimicthe �CDM model for a � aX , if we add to a standard dust component. Making useof the relativistic energy conservation equation, we can get

ρm = ρ0m(1 + z)3. (14)

Also, the expression of the unified state parameter ω can be obtained:

ω = ωdeρde

ρde + ρm. (15)

From ρm(zc) = ρde(zc), the coincidence redshift zc � 0.393 is also obtained. We plotthe recent evolutional trajectory of the unified state parameter ω in Fig. 5 and show

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Fig. 5 The recent evolution of the unified state parameter ω. The trajectory in our model is denoted by thesolid line, which overlaps with that in �CDM model denoted by the dotted line

Fig. 6 The evolution of the state parameter ωde of dark energy in the very early stage

the evolution of the state parameter ωde of dark energy in Fig. 6 (Since the recentevolution of state parameter ωde is much closed to −1 for a long time, we will notpresent its recent evolution image.).

The fractional density parameters can be derived as

�m = �0m(1 + z)3

E21

, (16)

�de = E21 − �0m(1 + z)3

E21

, (17)

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Fig. 7 The evolutions of fractional density parameters for �m and �de

where E21 = �0m(1+ z)3 +�0de

(2+z+zI )−4(2+z+zR)[(1+zX )3+(1+z)3]

(2+zI )−4(2+zR)[(1+zX )3+1] . The evolution of

the fractional energy densities in this model is plotted in Fig. 7. The current parametersadopted here are �0m = 0.27 and �0de = 0.73.

Furthermore, the deceleration parameter can be expressed as

q = −1 + 1

2

3�0m(1+z)3 +[

E21−�0m(1 + z)3

] [−4(1+z)2+zI +z + 1+z

2+zR+z + 3(1+z)3

(1+zX )3+(1+z)3

]

E21

.

(18)

Although the expression of the deceleration parameter q, i.e., Eq. (18) is different fromEq. (11), its evolution is as the same as that in Fig. 3. Hence, according to Eq. (18),we can give the present value of the deceleration parameter and the transition redshift:q0 = −0.595 and zT � 0.755, which are fit to the observations.

In the same way as above, we reconstruct the scalar field φ and V in order to givea possible physical interpretation to ρde. The expressions of φ and V are

φ =√

3

4πG

a∫

0

[ρde

ρde + ρm

(1 + 1 − ωde

1 + ωde

)−1] 1

2 da′

a′ ,

(19)

V =(

1 + 1 − ωde

1 + ωde

)−1 1 − ωde

1 + ωdeρde

According to Eq. (19), we can give the following results: when a � aI , 12 φ2 → 0,

V → constant and φ2 � V, φ � V, from which we see that the condition (slow-roll approximation) for inflation could be satisfied. While when a � aX , the relationbetween V and φ is shown in Fig. 8, from which we see that the potential V is a mono-tonically decreasing function of the scalar field φ, the variation of V with respect to V0

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Fig. 8 Relation between the potential V and the scalar field φ

is quite small less than 2.5% over the range of 0.05 ≤ a ≤ 1 and the scalar field φ hasalmost no change. (Here, φ0 and V0 are the corresponding current values.) Therefore,the component with ρde could be approximately seen as the cosmological constantwhen a � aX . It follows that the component with ρde could lead to both inflation andthe current accelerated expansion of the universe.

4 Structure formation

As we know, investigating the structure formation is an important aspect for a darkenergy model. Below we will discuss the structure formation in our model by thecomparison with the �CDM model. Newtonian treatment is used here to investigatethe growth rate of matter perturbations in the linear regime within the framework ofFRW cosmology.

Denoting the matter density contrast with δ ≡ δρmρm

, the perturbation equation reads:

δ + 2H δ − 4πGρmδ = 0. (20)

Firstly, we discuss the first case (i.e., the generalized Hobbit model as the unified darksector scenario). As mentioned above, when a � aR the component with ρ1 has thefeature of pressureless matter. Therefore, ρm may be replaced with ρ1 in Eq. (20) andEq. (20) may finally be rewritten as:

δ′′ +[

3

a+ (lnE2)′

2

]δ′ − 3�01

2E2a2

(a + aI )−4(a + aR)

(1 + aI )−4(1 + aR)δ = 0, (21)

where the prime denotes the derivation with respect to the scale factor a and E2 =(2+z+zI )

−4(2+z+zR)[(1+zX )3+(1+z)3](2+zI )

−4(2+zR)[(1+zX )3+1] . The evolution of δ in the linear regime is illustrated

in Fig. 9.

123


Fig. 9 The evolution of δ in the linear regime

In the following, we compare the growth variable and growth index of our modelwith those of �CDM. The growth variable is defined as T ≡ δ

a [22]. From Eq. (21),it is easy to determine the equation:

T ′′ +[

5

a+ (lnE2)′

2

]T ′ +

{3

a

[1 − �01

2E2

(a + aI )−4(a + aR)

(1 + aI )−4(1 + aR)

]+ (lnE2)′

2

}T

a=0.

(22)

Rather than looking at T(a) directly, it is more convenient to consider the quantity�T ≡ (T − T�)/T�, which represents the percentage deviation of the growth vari-able in our model with respect to that in �CDM model. We plot the evolution trajectoryof �T in Fig. 10, which mainly illustrates the deviations of the growth variable fora � aR . From the figure, the positive deviations can be received and keep about 10%,which implies that the growth variable of our model is similar to that �CDM model.This result shows that in our model the assembly of galaxies and clusters of galaxiesare likely to have taken place in a similar way to that in �CDM model. It is notedthat we use the same boundary conditions in our model as those in �CDM model,and choose a = aL S = (1 + zL S)−1 as the initial condition, where zL S is the redshiftof the last-scattering surface. Although we do not accurately calculate the redshiftcorresponding to maximum of �T , from Fig. 10 we may see that �T is relativelylarge when a → 0. Therefore, we could infer that the largest deviation may occur nearthe regime a = aR .

The other important quantity is the growth index, whose definition is f ≡ dlnδdlna =

aδ

dδda . Still making use of Eq. (24), the growth index f satisfies the following equation:

f ′ + f 2

a+

[2

a+ (lnE2)′

2

]f − 3�01

2E2a

(a + aI )−4(a + aR)

(1 + aI )−4(1 + aR)= 0. (23)

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Fig. 10 The trajectory of �T as a function of the scale factor a. The boundary conditions are T = 1 andT ′ = 0, when a → 0

Fig. 11 a describes the evolution trajectory of the growth index f and the dot locates the valuef � 0.56 when z = 0.15. b shows the evolution trajectory of η = � f . The boundary condition isf = 1 when a → 0

In Fig. 11a, we give the evolution trajectory of the growth index f . By calculations, wecan give the value of f at z = 0.15 (namely a � 0.87), i.e. f � 0.56. According to thedocuments [23,24], one may estimate f = 0.51 ± 0.1 or f = 0.58 ± 0.11 at z = 0.15measured by the 2dF galaxy redshift survey (2dFGRS). It follows that the value of fgiven by us is fit to the observations. In addition, the quantity η = � f ≡ ( f − f�)/ f�is also considered, and the curve of � f is plotted in Fig. 11b. It is easy to see thatthe deviations are quite small and the value of � f is relatively large when a → 0.Similarly, we also use aL S as the initial condition and infer that the largest deviationmay occur near the regime a = aR .

Next, we discuss the second case (i.e., the generalized Hobbit model with a singlefluid). In this case Eq. (20) may be rewritten as:

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δ′′ +[

3

a+ (lnE2

1)′

2

]δ′ − 3�0m

2E21a5

δ = 0, (24)

where the prime still denotes the derivation with respect to the scale factor a. Figure 12demonstrates the evolution of δ in the linear regime.

Now we still compare the growth variable and growth index of our model withthose of �CDM model. From Eq. (24), it is easy to derive the following equations:

T ′′ +[

5

a+ (lnE2

1)′

2

]T ′ +

{3

a

[1 − �0m

2E21a3

]+ (lnE2

1)′

2

}T

a= 0, (25)

f ′ + f 2

a+

[2

a+ (lnE2

1)′

2

]f − 3�0m

2E21a4

= 0. (26)

In Figs. 13 and 14, we describe the evolutions of the growth variable T and growthindex f , respectively. The trajectories of T and f in our model both overlap withthose in �CDM model, which make us confident that the assembly of galaxies andclusters of galaxies would have taken place in the same way as that in �CDM model.Furthermore, the value of f at z = 0.15 can be calculated, i.e. f � 0.568, which isalso fit to the data measured by (2dFGRS).

Finally, it should be stressed that for the first case (i.e., the generalized Hobbitmodel as the unified dark sector scenario), if ρm in Eq. (20) could be replaced withρ in Eq. (2), the corresponding perturbation equation, growth variable equation andgrowth index equation can be derived as follows:

δ′′ +[

3

a+ (lnE2)′

2

]δ′ − 3

2a2 δ = 0, (27)

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Fig. 13 The evolutions of the growth variable T . The boundary conditions are T = 1 and T ′ = 0 whena → 0

Fig. 14 The evolutions of the growth index f and the dot locates the value f � 0.568 when z = 0.15.The boundary condition is f = 1 when a → 0

T ′′ +[

5

a+ (lnE2)′

2

]T ′ +

[3

2a+ (lnE2)′

2

]T

a= 0, (28)

and

f ′ + f 2

a+

[2

a+ (lnE2)′

2

]f − 3

2a= 0, (29)

where E2 = (2+z+zI )−4(2+z+zR)[(1+zX )3+(1+z)3]

(2+zI )−4(2+zR)[(1+zX )3+1] .

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Fig. 16 The trajectory of �T as a function of the scale factor a

Thus, the corresponding evolution trajectories of δ,�T, f and � f are illustratedin Figs. 15, 16, 17 and 18, respectively.

According to Figs. 15, 16, 17 and 18, we find that, in comparison with �CDMmodel, except the evolution trajectory of δ those of �T (when a > 0.5), f (whenz = 0.15, i.e., a = 0.87, f = 0.79) and η = � f have larger deviations, which showsthat in the treatment of the growth of structure in the unified case, if the full fluid isconsidered, the results given in our model are not consistent with observations. Thisimplies that dark energy could not cluster in this case.

5 Conclusions

In summary, we have extended the theory of Cardone [14] to a more general phenom-enological model. By discussing the evolution of the universe, we have confirmed that

123


Fig. 17 The evolution trajectory of the growth index f and the dot locates the value f � 0.79 whenz = 0.15

Fig. 18 The evolution trajectory of η = � f

the two different cases of the generalized Hobbit model can describe a four-phasesmooth transition of the universe. For the first case (i.e., the generalized Hobbit modelas the unified dark sector scenario), the results show that our model could mimic the�CDM model for a � aR , and the component with ρ1 could drive the early inflationwhile the other one with ρ2 could drive the present accelerated expansion. Further-more, the possible physical interpretation for this scenario is shown by the two differentscalar fields. While for the second case (i.e., the generalized Hobbit model with a sin-gle fluid ), the results show that this model could also mimic the �CDM model but thecondition is a � aX , and the component with ρde drives both inflation and the currentacceleration. In addition, the structure formations for the both cases are respectivelydiscussed by comparing the growth variable and growth index of our model with thoseof �CDM model, and the results given by us reconfirm that our model can be regardedas a fit to the �CDM model, if we choose the proper conditions.

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Acknowledgments The research work is supported by the National Natural Science Foundation of China(Grant No 10875056) and the Scientific Research Foundation of the Higher Education Institute of LiaoningProvince, China (Grant No 05L215 and 2007T087).

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the dynamic evolutions of the generalized hobbit model and structure formation

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