Interacting closed string tachyon with modified Chaplygin gas and its stability
aa r X i v : . [ g r- q c ] N ov Interacting closed string tachyon with modified Chaplygin gas and its stability
Ali R. Amani a Faculty of Basic Sciences, Department of Physics, Ayatollah Amoli Branch, Islamic Azad University,P.O. Box 678, Amol, Iran.
Celia Escamilla-Rivera b Fisika Teorikoaren eta Zientziaren Historia Saila,Zientzia eta Teknologia Fakultatea, Euskal Herriko Unibertsitatea,644 Posta Kutxatila, 48080, Bilbao, Spain.
H. R. Faghani c Department of Physics, Central Tehran Branch, Islamic Azad University, Tehran, Iran. (Dated: August 5, 2018)
Abstract
In this paper, we have considered closed string tachyon model with a constant dilaton field and interactedit with Chaplygin gas for evaluating cosmology parameters. The model has been studied in 26-dimensionalthat its 22-dimensional is related to compactification on an internal non-flat space and its other 4-dimensionsis related to FLRW metric. By taking the internal curvature as a negative constant, we obtained the closedstring tachyon potential as a quartic equation. The tachyon field and the scale factor have been achieved asfunctional of time evolution and geometry of curved space where the behaviour of the scale factor describesan accelerated expansion of the universe. Next, we discussed the stability of our model by introducing asound speed factor, which one must be, in our case, a positive function. By drawing sound speed againsttime evolution we investigated stability conditions for non-flat universe in its three stages: early, late andfuture time. As a result we shall see that in these cases remains an instability at early time and a stabilitypoint at late time.
PACS numbers: 04.62.+v; 11.10.Ef; 98.80.-kKeywords: Closed string tachyon; Chaplygin gas; Non-flat space; Equation of State parameter; Stability. a Electronic address: [email protected] b Electronic address: celia [email protected] c Electronic address: [email protected] . INTRODUCTION It is known from so many years ago that the String Theory looks like a highly, good candidateto describe the physical world. At low-energies it clearly gives rise to General Relativity, scalarfields and gauge models. In other words, this theory contains in a generic way all the ingredientsthat overlay our universe. However, at this energy level exist solutions to the effective actionrelated with instabilities, so-called tachyons. Of course, the main reason for discussing thissolutions is that they all carry directly over to the superstring theories, where the most wellknown scenario concerns the presence of a tachyonic mode on the open string spectrum betweenpairs of D-branes and anti-D-branes [1, 2]. In order to be in the minimal state of energy, thetachyon rolls down to the minimum of the potential, and the perturbative approach of the theorybecomes reliable. This process is called tachyon condensation. Notwithstanding, tachyonic modesare not the only solutions on this matter, in fact, in the bosonic string spectrum exist also theclosed string tachyon modes. The fact that this scenarios turns out to sit at an unstable pointis unfortunate, but the positive thing is that we can think in a good minimum elsewhere for thetachyon potential. To get there we know that an expansion of the tachyon potential around T = 0looks like a polynomial and the physics behind can be reliable.All these results leads a wide possibility to construct cosmological models that can reproducethe actual behaviour of the observable universe in which an accelerated expansion is present.Interesting solutions emerged as a result of the following studies in the last few years, for exampleas in [3–7] where the closed string tachyon field drives the collapse of the universe. However, in thisattempts an expansion stage is still absent. To defray this line, in [8] the authors considered withthe above ideas a compactification of a critical bosonic string theory with a tachyonic potentialinto a 4-dimensional flat space-time finding certain conditions in where with an arbitrary closedstring tachyon potential the universe reaches a maximum size and then undergoes to a stage wherecollapses as the tachyon arrives to the minimum of its potential.Regarding to the accelerate stage of the universe, a wide range of research explored thepossibility of introduce certain exotic matter with negative pressure. This acceleration as weknow can be consequence of the dark energy influence, which in some models the ideal candidateto represent it is the extended Chaplygin gas [9–13], a fluid with negative pressure that beginsto dominate the matter content and, at the end, the process of structure formation is driven by2old dark matter without affecting the previous history of the universe. This kind of Chaplygingas cosmology has an interesting connection to String Theory via the Nambu-Goto action fora D-brane moving in a ( D + 2)-dimensional space-time, feature than can be regarded to thetachyonic panorama.Whilst fully realistic models are complicate and have yet to be constructed, this is why thesimplicity of the tachyon model coupled with a Chaplygin gas suggested that it may still findsome use as a model of accelerated universe. This slope is currently under intense scrutiny and inhere we present an attempt to going further.In our present work we want to choose the closure relation between the above ideas by focusingon the particularity of the model to unify the closed string tachyon and the Chaplygin gas. Accord-ing to this point of view, the model seems to be an interesting lead since the acceleration stage ispreserved. For details of the effects of relaxing these assumptions, we refer the lector to the litera-ture cited below. In Section II we explain briefly the system of equations that represent the closedstring tachyon scenario considering a gravitational field in a 4-dimensional non-flat Friedmann–Lemaˆıtre-Robertson-Walker (FLRW) metric. We then present the cosmological equations behindthis scenario. In Section III we present the description of the full model in the Hamiltonian formal-ism. We show that, despite the additional freedom in this model, is possible to reconstructed theclosed string tachyon potential as we present in Section IV. In Section V we adopt the case whenwe coupled a Chaplygin gas to the closed string tachyon model [8]. An interesting cosmologicalanalysis was made in Section VI. We conclude in Section VII that this modified model is capableto describe the current acceleration of the present epoch. II. CLOSED STRING TACHYON BACKGROUND
Let us start by considering critical bosonic string theory with a constant dilaton. The corre-sponding action is written for the closed string tachyon field T in 26-dimensional space-time as thefollowing form [3, 7] S = 12 κ Z d x √− g e − (cid:2) R − ( ∇ T ) − V ( T ) (cid:3) , (1)3here Φ , T ( t ) and V ( T ) are the constant dilaton, a rolling tachyon field and the closed stringtachyon potential, respectively. The 26-dimensional metric, g is in the form, ds = g µν dx µ dx ν + h mn dy m dy n , (2)where the first r.h.s term denotes a spatially 3 + 1 dimensional FLRW metric and indices m , n running from 4 to 25. Thus non-flat FLRW background is given by, ds = − dt + a ( t ) (cid:18) dr − kr + r d Ω (cid:19) , (3)where k denotes the curvature of space i.e., k = − , , d space X as S = 12 κ Z d x √− g V ol ( X ) e − (cid:2) R + R − ( ∇ T ) − V ( T ) (cid:3) , (4)where V ol ( X ) is the volume of X , κ = 8 πG is the gravitational strength in 26 dimensionsand R is constant curvature.In this model we will consider a constant volume for the internal compact space. Thus the effective4 d action is written as, S = m p Z d x p − g E h R (cid:0) g E (cid:1) − ( ∇ T ) − e V ( T ) i , (5)where m p and e V are the reduced 4 d mass of Planck and the effective scalar potential respectively,and they are given as the following form, m p = e − V ol ( X ) κ , (6) e V ( T ) = V ( T ) − R . (7)Now by taking a ( t ) = e α ( t ) and m p = 1 we can obtain Friedmann equations for action (5) as thefollowing form, ρ tot = 3 (cid:18) H + ka (cid:19) = 3 (cid:18) ˙ α + ke α (cid:19) , (8) p tot = − (cid:18) H + 3 H + ka (cid:19) = − (cid:18) α + 3 ˙ α + ke α (cid:19) , (9)and the equation of the closed string tachyon field is,¨ T + 3 ˙ α ˙ T + d e VdT = 0 . (10)4 II. CANONICAL HAMILTONIAN BACKGROUND
In this section we are going to consider the canonical Hamiltonian analysis. For this case, wehave to rewrite the lagrangian density of action (5) with respect to generalized parameters ˙ α and˙ T . Then, the lagrangian density becomes, S ( α, T, t ) = Z L dt (11)where L = (cid:18) − α + 12 ˙ T − e V + 3 ke α (cid:19) e α , (12)and L = dS ( α, T, t ) dt = ∂S∂α ˙ α + ∂S∂T ˙ T + ∂S∂t . (13)By using Hamilton’s principle function, we can write canonical Hamiltonian equations andHamiltonian–Jacobi equation in the following form respectively, π α = ∂S∂α , π T = ∂S∂T , (14a)˙ α = ∂H∂π α , ˙ T = ∂H∂π T , ∂H∂t = − ∂ L ∂t (14b) H ( α, T ; π α , π T ; t ) + ∂S ( α, T, t ) ∂t = 0 , (15)the generalized momenta are written by, π α = ∂ L ∂ ˙ α = − αe α , π T = ∂ L ∂ ˙ T = ˙ T e α . (16)On the other hand, the canonical Hamiltonian will obtain by aforesaid equations as, H = π α ˙ α + π T ˙ T − L = (cid:18) − π α
12 + π T e α e V − ke α (cid:19) e − α . (17)Now using Eqs. (15), (16) and (17) we obtain the following Hamilton equation − (cid:18) ∂S∂α (cid:19) + 12 (cid:18) ∂S∂T (cid:19) + e α e V ( T ) − ke α = 0 . (18)As we know, the choice of closed string tachyon potential plays the role of an important in StringTheory. But we are going to perform the current model for an cosmological analysis. Therefore,different suggestions expressed for selecting of the corresponding potential [14, 15]. In that case,5e will extend the job [8] by taking the function S in a non-flat universe by curvature k in thefollowing form, S ( α, T, t ) = e α ( t ) [ W ( T ( t )) + βkZ ( T ( t ))] , (19)where W ( T ) and Z ( T ) are an arbitrary function with dependence on T , and β is a constantcoefficient. Now by substituting (19) into (18) the effective tachyon potential is written in termsof functions W ( T ) and Z ( T ) as, e V ( T ) = 34 ( W + βkZ ) −
12 ( ∂ T W + βk ∂ T Z ) + 3 ke − α . (20) IV. RECONSTRUCTING CLOSED STRING TACHYON POTENTIAL
In this section we are going to describe the cosmological evolution of our model with the closedstring tachyon coupled to a modified Chaplygin gas. Let us remark that the recent superstringcorrections interpreted compactification on internal manifold that internal curvature is everywherenegative [16, 17]. Therefore, from the point of view 4 d geometry, the internal curvature is anegative constant R <
0, i.e., the internal curvature is not a functional of T .In order to obtain effective tachyon potential, we take the function W ( T ) in the form, W ( T ) = C + DT , (21)where C and D are constant coefficients in which they play a role of important for descriptioncosmological solution. The motivation of this choice is based on a ) crossing of Equation of State(EoS) over phantom-divide-line, and b ) achieve to a polynomial function for tachyonic potential asmentioned in Ref. [7, 8].In this model, we simplicity take W = Z , then by inserting (21) into (20), the effective tachyonpotential is yielded as, e V ( T ) = 34 D (1 + βk ) T + D (cid:18) C − D (cid:19) (1 + βk ) T + (cid:18) C (1 + βk ) + 3 ke − α (cid:19) . (22)As we know, the constant curvature of internal manifold R is not a functional of tachyon field.Now with correspondence of Eqs. (22) and (7), the negative curvature is given by, R = − C (1 + βk ) , (23)6herefore, the closed string tachyon potential is reduced to, V ( T ) = 34 D (1 + βk ) T + D (cid:18) C − D (cid:19) (1 + βk ) T + 3 ke − α . (24)Making use of the momentum related to α and T Eqs. (14a) and the ansatz for S Eq. (19) and W Eq. (21), we can obtain the tachyon field solution and the scale factor function in terms of time: T = e D (1+ βk ) t , (25) a ( t ) = exp (cid:20) − C (1 + βk ) t − e D (1+ βk ) t (cid:21) . (26)We note that Eqs. (25) and (26) are strictly constrained to the values of C and D , which ones willplay an important role for the description of the cosmological evolution. In next section, we intentto investigate the effect of obtained parameters with Chaplygin gas. V. INTERACTING CLOSED STRING TACHYON WITH CHAPLYGIN GAS
In this section, we consider an interaction between the closed string tachyon and Chaplygin gas.In connection with string theory, the equation of state of the Chaplygin gas has obtained from theNambu-Goto action for a D-brane moving in a ( D + 2)-dimensional space-time in the light coneparametrization [12, 18, 19]. The equation of state the modified Chaplygin gas is given by, p MCG = Aρ MCG − Bρ γMCG , (27)where p MCG and ρ MCG are the pressure and energy density of modified Chaplygin gas where A and B are positive constants and 0 ≤ γ ≤
1. Therefore, the total energy density and pressure aregiven respectively by, ρ tot = ρ T + ρ MCG , (28) p tot = p T + p MCG . (29)As we know the continuity equation derived from T µν ; ν = 0, then the general form of continuityequation is, ˙ ρ tot + 3 H ( ρ tot + p tot ) = 0 , (30)now, by taking an energy flow between closed string tachyon and Chaplygin gas, we have tointroduce a phenomenological coupling function in terms of product of the Hubble parameter and7he energy density of the Chaplygin gas. In that case, continuity equations of the closed stringtachyon and Chaplygin gas are written respectively by,˙ ρ T + 3 H ( ρ T + p T ) = − Q, (31)˙ ρ MCG + 3 H ( ρ MCG + p MCG ) = Q, (32)where the quantity Q is the interaction term between tachyon field and the Chaplygin gas andone is equivalent to Q = 3 b Hρ MCG , where b is the coupling parameter or transfer strength [20].We note that the interaction term Q has widely described in the literature [21–24]. This choice isbased on positive motivation Q , because from the observational data at the four years WMAPimplies that the coupling parameter must be a small positive value [25, 26].By substituting (27) into (32) we can obtain energy density of modified Chaplygin gas as thefollowing form, ρ MCG = (cid:20) Bη + c a − η ( γ +1) (cid:21) γ +1 . (33)where c is a constant integral, and employing this expression in Eq. (27) we can rewrite thepressure for the Chaplygin gas as: p MCG = A (cid:20) Bη + a − η ( γ +1) (cid:21) γ +1 − B h Bη + a − η ( γ +1) i γγ +1 , (34)where η = 1 − b + A . If η << ρ T = 3 (cid:18) ˙ α + ke α (cid:19) − (cid:20) Bη + a − η ( γ +1) (cid:21) γ +1 , (35) p T = − (cid:18) α + 3 ˙ α + ke α (cid:19) − A (cid:20) Bη + a − η ( γ +1) (cid:21) γ +1 + B h Bη + a − η ( γ +1) i γγ +1 . (36)By reinserting (26) into Eqs. (35) and (36) the EoS of the closed string tachyon is obtained as, ω T = p T ρ T = − (cid:0) α + 3 ˙ α + ke α (cid:1) − A ρ
MCG + Bρ γMCG
3( ˙ α + ke α ) − ρ MCG . (37)8 IG. 1: The energy density and pressure of closed string tachyon in terms of time evolution for B = 2 , C = − , D = − . , b = 0 . , β = − . , γ = 0 . , c = 3 .
25 and A = 0 .
25 in different cases k = ± , B = 2 , C = − , D = − . , b =0 . , β = − . , γ = 0 . , c = 3 .
25 and A = 0 .
25 in different cases k = ± , We can see variation of the cosmological parameters against time evolution by interactingChaplygin gas with the closed string tachyon by geometries ( k = 0 , ±
1) in the Figures 1 and 2.We note that chosen coefficients play the role of an important to plot the cosmological parameterssuch as the energy density and pressure. Then, the motivation of the selections is based on9
IG. 3: The EoS of closed string tachyon in terms of e-folding number for B = 2 , C = − , D = − . , b =0 . , β = − . , γ = 0 . , c = 3 .
25 and A = 0 .
25 in different cases k = ± , crossing EoS over phantom-divide-line, positivity energy density and negativity pressure.Since in this paper we use the natural units as c = ~ = m p = 1, therefore in order to have amore complete discussion, we represent the free parameters of the model in terms of observablequantities. For this purpose, to have an accelerated expansion we draw the EoS of the closedstring tachyon versus the e-folding number, N = ln ( a ) as the time variable in Fig. 3. We cansee the values of EoS of the closed string tachyon in three cases N → −∞ , late time ( N = 0)and N → + ∞ respectively with values 0 . − .
017 and − .
006 for geometry k = +1 in Fig.3(a), and 0 . − .
125 and − .
005 for geometry k = 0 in Fig. 3(b), and 0 . − .
221 and − . k = − − k = 0 , ±
1) in Fig. 3.
VI. CONDITIONS FOR AN ACCELERATED UNIVERSE AND THE STABILITY ANAL-YSIS
In this section, we are going to investigate two issues: first, the condition of an accelerateduniverse in our model and second, the stability analysis of aforesaid proposal.On one hand, we study on the condition of expanding universe accelerating for the closed stringtachyon. For this purpose, we can obtain ρ T , and p T by inserting (26) into Eqs. (35) and (36) as10ollows, ρ T = 34 (1 + β k ) h C + D e D (1+ βk ) t i + 3 k exp (cid:20) C (1 + βk ) t + 14 e D (1+ βk ) t (cid:21) − (cid:18) Bη + exp (cid:20) Cη ( γ + 1)(1 + βk ) t + 38 η ( γ + 1) e D (1+ βk ) t (cid:21)(cid:19) γ +1 , (38) p T = (1 + β k ) h D e D (1+ βk ) t − (cid:0) C + De D (1+ βk ) t (cid:1) i − k exp (cid:2) C (1 + βk ) t + e D (1+ βk ) t (cid:3) − A (cid:16) Bη + exp (cid:2) Cη ( γ + 1)(1 + βk ) t + η ( γ + 1) e D (1+ βk ) t (cid:3)(cid:17) γ +1 + B h Bη +exp [ Cη ( γ +1)(1+ βk ) t + η ( γ +1) e D (1+ βk ) t ] i γγ +1 . (39)Now we can find a constraint for the accelerated universe by weak energy condition ( ρ > p T + ρ T >
0) in terms of current epoch time t , in the following form,34 (1 + β k ) (cid:16) C + D e D (1+ βk ) t (cid:17) + 3 k exp (cid:20) C (1 + βk ) t + 14 e D (1+ βk ) t (cid:21) − (cid:18) Bη + exp (cid:20) Cη ( γ + 1)(1 + βk ) t + 38 η ( γ + 1) e D (1+ βk ) t (cid:21)(cid:19) γ +1 > , (40)and 4 D (1 + β k ) e D (1+ βk ) t + 2 k exp (cid:18) C (1 + βk ) t + 14 e D (1+ βk ) t (cid:19) − ( A + 1) (cid:20) Bη + exp (cid:18) Cη ( γ + 1)(1 + βk ) t + 38 η ( γ + 1) e D (1+ βk ) t (cid:19)(cid:21) γ +1 + B h Bη + exp (cid:2) Cη ( γ + 1)(1 + βk ) t + η ( γ + 1) e D (1+ βk ) t (cid:3)i γγ +1 > , (41)Eqs. (40) and (41) are a constraints for all the coefficients of the model in the current epoch ofthe universe.On the other hand, we will discuss the stability of our model with presence of closed stringtachyon field. In that case, we will describe the corresponding stability with an useful function c s = ∂p T ∂ρ T = ˙ p T ˙ ρ T . The stability condition occurs when the function c s becomes bigger than zero. Ofcourse this function represent sound speed in a perfect fluid.We noted that a general thermodynamic system can be described with adiabatic and non-adiabatic perturbations by three variables, ρ T , p T and S (entropy). If we consider p T = p T ( S, ρ T ),so the pressure perturbation can be written as: δp T = ∂p T ∂S δS + ∂p T ∂ρ T δρ T = ∂p T ∂S δS + c s δρ T . Thefirst term be related to non-adiabatic system, and c s in the second term be related to adiabatic11 IG. 4: Graphs of the c s in terms of time evolution for B = 2 , C = − , D = − . , b = 0 . , β = − . , γ =0 . , c = 3 .
25 and A = 0 .
25 in geometries k = ± , sound speed, i.e., system is when adiabatic in which δS = 0. Therefore, we describe the stabilityof our model just by adiabatic sound speed.In this way, by making derivative Eqs. (35) and (36) with respect to time evolution andnumerical computing the function c s in terms of time evolution, we can plot speed sound functionby various geometries ( k = 0 , ±
1) in Figure 4.We can see the values of c s in three cases N → −∞ , late time ( N = 0) and N → + ∞ respectively with values 0 .
25, 25 .
25 and − .
867 for geometry k = +1 in Fig. 4(a), and 0 .
25, 0 . − .
867 for geometry k = 0 in Fig. 4(b), and 0 .
25, 0 .
154 and − .
867 for geometry k = − c s are positive for every three universe in late time (i.e., N = 0). VII. CONCLUSIONS
In this paper, we have studied closed string tachyon with a constant dilaton field in 26 d space-time for describing something mysterious in the cosmology. To understand this issue,we have considered the corresponding model by interacting with modified Chaplygin gas. Wenoted that the corresponding action has been written as an effective four-dimensional action12y compactification on a non-flat internal 22 d, in which the internal compact space considereda constant volume. The Einstein and field equations have been obtained and by taking aninteraction between the closed string tachyon with modified Chaplygin gas we could find theenergy density and pressure of closed string tachyon.By using canonical Hamiltonian analysis and the corresponding action, we obtained the conti-nuity equation and then the effective tachyon potential have been found in terms of an arbitraryfunction ( W ( T ) and Z ( T )) proportional to tachyon field. In order to reconstruct closed stringtachyon potential, we took arbitrary function W ( T ) such as a quadratic function of tachyon field.In additional, by employing canonical Hamiltonian equations we obtained the tachyon field and thescale factor in terms of time evolution. One of cosmology characteristics that confirm observationaldata is based on crossing the EoS from phantom-divide-line, in which one calculated in terms oftime evolution and e-folding number. Also we plotted the EoS with respect to time evolutionand e-folding number for various geometries. The graph of EoS showed accelerating universe andcross over phantom-divided line. Next we obtained a constraint by weak energy condition. Finallywe have considered stability analysis for the presented model by using an useful function calledthe sound speed. This function is employed in a perfect fluid, in which its value is greater thanzero. We plotted variation of the sound speed versus time evolution and the corresponding graphsshowed stability in late time. The interesting problem here was to consider the model with thecurvature of the internal space in a non-constant internal volume scenario. VIII. ACKNOWLEDGEMENTS
C. Escamilla-Rivera is supported by Fundaci´on Pablo Garc´ıa and FUNDEC, M´exico.Also the authors would like to thank an anonymous referee for crucial remarks and advices. [1] A. Sen, JHEP 0204, 048 (2002) [arXiv:hep-th/0203211].[2] N. Lambert, H. Liu and J. Maldacena, [arXiv:hep-th/0303139].[3] H. Yang and B. Zwiebach, JHEP , 054 (2005) [arXiv:hep-th/0506077].[4] I. Swanson, Phys. Rev. D , 066020 (2008) [arXiv:hep-th/0804.2262].[5] A. Adams, J. Polchinski and E. Silverstein, JHEP , 029 (2001) [arXiv:hep-th/0108075].[6] J. McGreevy and E. Silverstein, JHEP , 090 (2005) [arXiv:hep-th/0506130].[7] H. Yang and B. Zwiebach, JHEP , 046 (2005).
8] C. Escamilla-Rivera, G. Garcia-Jimenez, O. Loaiza-Brito and O. Obregon, CQG , 035005 (2013)[arXiv:gr-qc/1110.6223].[9] A. Kamenshchik, U. Moschella and V. Pasquier, Phys.Lett. B, 511, 265-268 (2001) [arXiv:gr-qc/0103004].[10] M. C. Bento, O. Bertolami and A.A. Sen, PRD, 66, 043507 (2002) [arXiv:gr-qc/0202064].[11] M. C. Bento, O. Bertolami and A.A. Sen, [arXiv:gr-qc/0305086].[12] H. B. Benaoum, [arXiv:hep-th/0205140].[13] L. P. Chimento, PRD, 69, 123517 (2004) [arXiv:astro-ph/0311613].[14] A. Dabholkar and C. Vafa, JHEP 008, 0202 (2002) [arXiv:hep-th/0111155].[15] M. R. Garousi, JHEP 058, 0305 (2003) [arXiv:hep-th/0304145].[16] G. Shiu and Y. Sumitomo, [arXiv:hep-th/1107.2925].[17] M. R. Douglas and R. Kallosh, [arXiv:hep-th/1001.4008].[18] R. Jackiw, http://arxiv.org/abs/physics/0010042.[19] N. Ogawa, PRD, 62, 8, 085023 (2000).[20] Z. K. Guo, Y. Z. Zhang, PRD, 71, 023501 (2005).[21] H. Wei, and R. G. Cai, PRD, 73, 083002 (2006).[22] S. Nojiri and S. D. Odintsov, PRD, 72, 023003 (2005).[23] L. P. Chimento, A. S. Jakubi, D. Pavon, and W. Zimdahl, PRD, 67, 083513 (2003).[24] E. J. Copeland, A. R. Liddle and D. Wands, PRD, 57, 4686 (1998).[25] C. Feng, and et al, Phys. Lett. B 665, 111 (2008).[26] D. N. Spergel and et al, ApJ, 2003.” arXiv preprint astro-ph/0302209., 035005 (2013)[arXiv:gr-qc/1110.6223].[9] A. Kamenshchik, U. Moschella and V. Pasquier, Phys.Lett. B, 511, 265-268 (2001) [arXiv:gr-qc/0103004].[10] M. C. Bento, O. Bertolami and A.A. Sen, PRD, 66, 043507 (2002) [arXiv:gr-qc/0202064].[11] M. C. Bento, O. Bertolami and A.A. Sen, [arXiv:gr-qc/0305086].[12] H. B. Benaoum, [arXiv:hep-th/0205140].[13] L. P. Chimento, PRD, 69, 123517 (2004) [arXiv:astro-ph/0311613].[14] A. Dabholkar and C. Vafa, JHEP 008, 0202 (2002) [arXiv:hep-th/0111155].[15] M. R. Garousi, JHEP 058, 0305 (2003) [arXiv:hep-th/0304145].[16] G. Shiu and Y. Sumitomo, [arXiv:hep-th/1107.2925].[17] M. R. Douglas and R. Kallosh, [arXiv:hep-th/1001.4008].[18] R. Jackiw, http://arxiv.org/abs/physics/0010042.[19] N. Ogawa, PRD, 62, 8, 085023 (2000).[20] Z. K. Guo, Y. Z. Zhang, PRD, 71, 023501 (2005).[21] H. Wei, and R. G. Cai, PRD, 73, 083002 (2006).[22] S. Nojiri and S. D. Odintsov, PRD, 72, 023003 (2005).[23] L. P. Chimento, A. S. Jakubi, D. Pavon, and W. Zimdahl, PRD, 67, 083513 (2003).[24] E. J. Copeland, A. R. Liddle and D. Wands, PRD, 57, 4686 (1998).[25] C. Feng, and et al, Phys. Lett. B 665, 111 (2008).[26] D. N. Spergel and et al, ApJ, 2003.” arXiv preprint astro-ph/0302209.