Statefinder Parameters for Different Dark Energy Models with Variable G Correction in Kaluza-Klein Cosmology
Shuvendu Chakraborty, Ujjal Debnath, Mubasher Jamil, Ratbay Myrzakulov
aa r X i v : . [ phy s i c s . g e n - ph ] F e b Statefinder Parameters for Different Dark Energy Models withVariable G Correction in Kaluza-Klein Cosmology
Shuvendu Chakraborty ∗ , Ujjal Debnath † , Mubasher Jamil ‡ and Ratbay Myrzakulov , § Department of Mathematics, Seacom Engineering College, Howrah, 711 302, India. Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711 103, India. Center for Advanced Mathematics and Physics (CAMP),National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan. Eurasian International Center for Theoretical Physics,Eurasian National University, Astana 010008, Kazakhstan. Department of Physics, California State University, Fresno, CA 93740 USA.
In this work, we have calculated the deceleration parameter, statefinder parameters and EoS pa-rameters for different dark energy models with variable G correction in homogeneous, isotropic andnon-flat universe for Kaluza-Klein Cosmology. The statefinder parameters have been obtained interms of some observable parameters like dimensionless density parameter, EoS parameter and Hub-ble parameter for holographic dark energy, new agegraphic dark energy and generalized Chaplygingas models. Contents
I. Introduction II. Kaluza-Klein Model III. Holographic Dark Energy IV. New Agegraphic Dark Energy V. Generalized Chaplygin gas VI. Conclusions
References I. INTRODUCTION
Recent cosmological observations obtained by SNe Ia [1], WMAP [2], SDSS [3] and X-ray [4] indicate thatthe observable universe experiences an accelerated expansion. To explain this phenomena the notion known asdark energy (DE) with large negative pressure is proposed. At present there are a lot of theoretical models ofDE. But the most suitable models of DE is the cosmological constant. According of the modern observationalcosmology, the present value of cosmological constant is 10 − cm − . At the same time, the particle physicstells us that its value must be 10 times greater than this factor. It is one main problem modern cosmologyand known as the cosmological constant problem. In order to solve this problem, some authors considered thecosmological constant as a varying parameter (see e.g. [5–9]). Here we can mention that Dirac showed thatsome fundamental constants do not remain constant forever rather they vary with time due to some causalconnection between micro and macro physics [10] that is known as Large Number Hypothesis (LNH). Thefield equations of General Relativity (GR) involve two physical constants, namely, the gravitational constant G ∗ [email protected] † [email protected], [email protected] ‡ [email protected] § [email protected], [email protected] (couples the geometry and matter) and cosmological constant Λ (vacuum energy in space). According to theLNH, the gravitational constant should also vary with time. In [11] LNH was extended by taking cosmologicalconstant as Λ = πG m p h , where m p is the mass of proton and h is the Plancks constant. It was showed that Λproduces the same gravitational effects in vacuum as that produced by matter [11]. As result, this cosmologicalterm must be included in the physical part of the field equations. In [11] also defined gravitational energy of thevacuum as the interactions of virtual particles separated by a distance hm p c , where c is the speed of light. It isalso interesting to note that a time varying gravitational constant also appears in the entropic interpretationsof gravity [12].In the literature, many modifications of cosmological constant have been proposed for the better descriptionand understanding of DE (see e.g. [13]). For example, in [14] was studied the field equations by using threedifferent forms of the cosmological constant, i.e., Λ ∼ (cid:0) ˙ aa (cid:1) , Λ ∼ (cid:0) ¨ aa (cid:1) and Λ ∼ ρ and shown that these modelsyield equivalent results to the FRW spacetime. From these investigations follow that an investigation aboutthe scale factor and other cosmological parameters with varying G and Λ may be interesting especially fordescription the accelerated expansion of the universe.According modern point of views, multidimensional gravity theories may play important role to explain mainproblems of cosmology and astrophysics in particular DE. One of classical examples of such theories is thetheory of Kaluza–Klein (KK) [15, 16]. It is a 5 dimensional GR in which extra dimension is used to couple thegravity and electromagnetism (see e.g., the review [17–19] and references therein). In the context of our interest- DE, recently it was studied [20] that the non-compact, non-Ricci KK theory and coupled the flat universewith non-vacuum states of the scalar field. For the suitable choose of the equation of state (EoS), the reducedfield equations describe the early inflation and late time acceleration. Moreover, the role played by the scalarfield along the 5th coordinate in the 5D metric is in general very impressed by the role of scale factor over the4D universe.In recent years, the holographic dark energy (HDE) has been studied as a possible candidate for DE. Itis motivated from the holographic principle which might lead to the quantum gravity to explain the eventsinvolving high energy scale. Another interesting models of DE are the so-called new-agegraphic dark energywhich is originated from the uncertainty relation of quantum mechanics together with the gravitational effectof GR. In general, the agegraphic DE model assumes that the observed DE effect comes from spacetime andmatter field fluctuations in the universe.In the interesting paper [21] it was introduced a new cosmological diagnostic pair { r, s } called statefinder whichallows one to explore the properties of DE independent of model. This pair depends on the third derivativeof the scale factor, a ( t ), just like the dependence of Hubble and deceleration parameter on first and secondderivative of respectively. It is used to distinguish flat models of the DE and this pair has been evaluated fordifferent models [22–30]. In [30] it was solved the field equations of the FRW universe with variable G and Λ(see also [31] where was considered the flat KK universe with variable Λ but keeping G fixed). There are manyworks on higher dimensional space-time also [32].In this work, we have calculated the statefinder parameters for different dark energy models with variable G correction in Kaluza-Klein cosmology. We evaluate different cosmological parameters with the assumption thatour universe is filled with different types of matter. The scheme of the paper is as follows. In the next section,the KK model and its field equations are presented. In section III, solution of the field equations for the HDEare presented and section IV the new-agegraphic dark energy case is considered. Generalized Chaplygin Gasmodel is studied in the section V. In section VI, we summarize the results. II. KALUZA-KLEIN MODEL
The metric of a homogeneous and isotropic universe in the Kaluza-Klein model is ds = dt − a ( t ) (cid:20) dr − kr + r ( dθ + sin θdφ ) + (1 − kr ) dψ (cid:21) (1)where a ( t ) is the scale factor, k = − , , T µν = ( ρ m + ρ x + p x ) u µ u ν − p x g µν (2)where u µ is the five velocities satisfying u µ u µ = 1. ρ m and ρ x are the energy densities of matter and darkenergy respectively and p x is the pressure of the dark energy. We consider here the pressure of the matter as zero.The Einstein’s field equations are given by R µν − g µν R = 8 πG ( t ) T µν (3)where R µν , g µν and R are Ricci tensor, metric tensor and Ricci scalar respectively. Here we consider gravitationalconstant G as a function of cosmic time t . Now from the equations (1), (2) and (3) we have the Einstein’s fieldequations for the isotropic Kaluza-Klein space time (1) are H + ka = 4 πG ( t )3 ( ρ m + ρ x ) (4)˙ H + 2 H + ka = − πG ( t )3 p x (5)Let the dark energy obeying the equation of state p x = ωρ x . Equation (4) givesΩ = Ω m + Ω x − Ω k (6)where Ω m , Ω x and Ω k are dimensionless density parameters representing the contribution in the total energydensity. The deceleration parameter q in terms of these parameters are given by q = Ω m + (1 + 2 ω )Ω x where ω = q − Ω − Ω k x (7)The trajectories in the { r, s } plane [33] corresponding to different cosmological models depict qualitativelydifferent behaviour. The statefinder diagnostic along with future SNAP observations may perhaps be used todiscriminate between different dark energy models. The above statefinder diagnostic pair for cosmology areconstructed from the scale factor a . The statefinder parameters are given by r = a ··· aH , s = r − q − / r , we have a relation between r and q is given by r = q + 2 q − ˙ qH (8)From (7) we have ˙ q = ˙Ω m + (1 + 2 ω ) ˙Ω x + 2 ˙ ω Ω x (9)Also we have Ω = ρρ cr − ka H which gives ˙Ω = ˙ ρρ cr − kqa H − ρ ˙ ρ cr ( ρ cr ) (10)where ρ cr = 3 H πG ( t ) which gives after differentiation ˙ ρ cr = ρ cr HH − ˙ GG ! (11)which implies ˙ ρ cr = − Hρ cr (2(1 + q ) + △ G ) (12)where, △ G ≡ G ′ G , ˙ G = HG ′ . Now from equation (10) we have˙Ω = ˙ ρρ cr + Ω k H (2 + △ G ) + Ω H (2(1 + q ) + △ G ) (13)We assume that matter and dark energy are separately conserved. For matter, ˙ ρ m + 4 Hρ m = 0. So from (13)˙Ω m = Ω m H ( − q + △ G ) + Ω k H (2 + △ G ) (14)For dark energy, ˙ ρ x + 4 H (1 + ω ) ρ x = 0. So from (13)˙Ω x = Ω x H ( − − ω + 2 q + △ G ) + Ω k H (2 + △ G ) (15)From (8), (9), (14), (15) we have expression for r and s given by r = 3Ω m + (3 + 10 ω + 8 ω )Ω x − ω )Ω k − △ G (Ω m + (1 + 2 ω )Ω x + 2(1 + ω )Ω k ) − ω Ω x H (16) s = 3Ω m + (3 + 10 ω + 8 ω )Ω x − ω )Ω k − △ G (Ω m + (1 + 2 ω )Ω x + 2(1 + ω )Ω k ) − ω Ω x H − − / m + Ω x + 2 ω Ω x ) (17) III. HOLOGRAPHIC DARK ENERGY
To study the dark energy models from the holographic principle it is important to mention that the numberof degrees of freedom is directly related to the entropy scale with the enclosing area of the system, not with thevolume [34]. Where as Cohen et al [35] suggest a relation between infrared (IR) and the ultraviolet (UV) cutoffin such a way that the total energy of the system with size L must not exceed the mass of the same size blackhole. The density of holographic dark energy is ρ x = 3 c πG L (18)Here c is the holographic parameter of order unity. Considering L = H − one can found the energy densitycompatible with the current observational data. However, if one takes the Hubble scale as the IR cutoff, theholographic dark energy may not capable to support an accelerating universe [36]. The first viable versionof holographic dark energy model was proposed by Li [37], where the IR length scale is taken as the eventhorizon of the universe. The holographic dark energy has been explored in various gravitational frameworks [38]The time evolution is ˙ ρ x = − ρ x H (2 − √ x c cosy + △ G ) (19)where L is defined as L = ar ( t ) with a is the scale factor. Also r ( t ) can be obtained from the relation R H = a ∞ R t dta = R r ( t )0 dr √ − kr .where R H is the event horizon. When R H is the radial size of the event horizon measured in the r direction, L is the radius of the event horizon measured on the sphere of the horizon.For closed (or open) universe we have r ( t ) = √ k siny , where y = √ kR H a .using the definition Ω x = ρ x ρ cr and ρ cr = H πG ( t ) we have HL = c √ x .And using all these we ultimately obtain the relation ˙ L = HL + a ˙ r ( t ) = c √ x − cosy , by which we find theequation (19).From the energy conservation equation and the equation (19) we have the holographic energy equation ofstate given by ω = 14 (cid:18) − − √ x c cosy + △ G (cid:19) (20)where, Ω k = ka H , Ω x = c L H are usual fractional densities in KK model.From the ration of the fractional densities we have, sin y = c Ω k x and naturally, cosy = q x − c Ω k x .Now differentiating (20) and using (15) we have˙ ωH = 16Ω x ( − x ) + c Ω x (3 △ ′ G + Ω k (2 − x )) − c √− c Ω k + 2Ω x ((2 + △ G )Ω k + Ω x (2Ω m + △ G Ω x ))12 c Ω x (21)Now putting (21) in (16) and (17), we have r = 16 c (cid:2) − x )Ω x − c (3(2( − △ G )Ω m + ( −△ G + △ ′ G )Ω x ) + Ω k (3(2 + △ G ) + 14Ω x − x ))+2 c p − c Ω k + 2Ω x (5(2 + △ G )Ω k + Ω x ( − m + △ G ( − x ))) i (22) s = 19 c ( − x √− c Ω k + 2Ω x + c ( − m + △ G Ω x )) (cid:2) − x )Ω x − c (3(2 + 2( − △ G )Ω m + ( −△ G + △ ′ G )Ω x )+Ω k (3(2 + △ G ) + 14Ω x − x )) + 2 c p − c Ω k + 2Ω x (5(2 + △ G )Ω k + Ω x ( − m + △ G ( − x ))) i (23)This is the expressions for { r, s } parameters in terms of fractional densities of holographic dark energy modelin Kaluza-klein cosmology for closed (or open) universe. IV.
NEW AGEGRAPHIC DARK ENERGY
There are another version of the holographic dark energy model called, the new agegraphic dark energy model[39], where the time scale is chosen to be the conformal time. The new agegraphic dark energy is more acceptablethan the original agegraphic dark ennergy, where the time scale is choosen to be the age of the universe. Theoriginal ADE suffers from the difficulty to describe the matter-dominated epoch while the NADE resolved thisissue. The density of new agegraphic dark energy is ρ x = 3 n πG η (24)where n is a constant of order unity. where the conformal time is given by η = R a daHa .If we consider η to be a definite integral, the will be a integral constant and we have ˙ η = a .Considering KK cosmology and using the definition Ω x = ρ x ρ cr and ρ cr = H πG ( t ) we have Hη = n √ x .After introducing the fractional energy densities we have the time evolution of NADE as˙ ρ x = − ρ x H (cid:18) √ x na + △ G (cid:19) (25)From the energy conservation equation and the equation (25) we have the new agegraphic energy equation ofstate given by ω = 14 (cid:18) − √ x na + △ G (cid:19) (26)where, Ω k = ka H , Ω x = n η H are usual fractional densities in KK model.Differentiating (26) and using (15) we have˙ ωH = a △ ′ Gn √ x + 4( − x )Ω / x + √ an ((2 + △ G )Ω k + Ω x (2Ω m + ( − △ G )Ω x ))4 a n √ Ω x (27)Now putting (27) in (16) and (17), we have the expression for r, s as r = − a n h − x )Ω x + √ an p Ω x (3(2 + △ G )Ω k + (2(3 + Ω m − Ω x ) + △ G ( − x ))Ω x )+ a n ( △ G Ω k − m + ( − △ ′ G )Ω x + △ G (2(Ω k + Ω m ) + Ω x )) (cid:3) (28) s = − an (2 √ / x + an ( − m + ( − △ G )Ω x )) h − x )Ω x + √ an p Ω x (3(2 + △ G )Ω k + (2(3 + Ω m − Ω x )+ △ G ( − x ))Ω x ) + a n (2 + △ G Ω k − m + ( − △ ′ G )Ω x + △ G (2(Ω k + Ω m ) + Ω x )) (cid:3) (29)This is the expressions for { r, s } parameters in terms of fractional densities of new agegraphic dark energymodel in Kaluza-klein cosmology for closed (or open) universe. V. GENERALIZED CHAPLYGIN GAS
It is well known to everyone that Chaplygin gas provides a different way of evolution of the universe andhaving behaviour at early time as presureless dust and as cosmological constant at very late times, an advantageof GCG, that is it unifies dark energy and matter into a single equation of state. This model can be obtainedfrom generalized version of the Born-Infeld action. The equation of state for generalized Chaplygin gas is [40] p x = − Aρ αx (30)where 0 < α < A > ρ x = (cid:20) A + Ba α +1) (cid:21) α +1 (31)where B is an integrating constant. ω = − A (cid:18) A + Ba α ) (cid:19) − (32)Differentiating (32) and using (15) we have˙ ωH = − AB (1 + α ) 1 a α ) (cid:18) A + Ba α ) (cid:19) − (33)Now putting (33) in (16) and (17), we have r = 3Ω m − △ G Ω m + Ω x + △ G Ω x − B ((2 + △ G )Ω k + Ω x ( − △ G − α ))( a α A + B ) − B Ω x α ( Aa α + B ) (34) s = 3Ω m − △ G Ω m + Ω x + △ G Ω x − B ((2+ △ G )Ω k +Ω x ( − △ G − α ))( a α A + B ) − B Ω x α ( Aa α + B ) (cid:16) − / m + Ω x − A Ω x A + a − α ) B (cid:17) (35)This is the expressions for { r, s } parameters in terms of fractional densities of generalized Chaplygin gas modelin Kaluza-klein cosmology for closed (or open) universe. VI. CONCLUSIONS
In this work, we have considered the homogeneous, isotropic and non-flat universe in 5D Kaluza-Klein Cos-mology. We have calculated the corrections to statefinder parameters due to variable gravitational constantin Kaluza-Klein Cosmology. These corrections are relevant because several astronomical observations provideconstraints on the variability of G . We have investigated three multipromising models of DE such as the Holo-graphic dark energy, the new-agegraphic dark energy and generalized Chaplygin gas. These dark energies derivethe accelerating phase of the Kaluza-Klein model of the universe. We have assumed that the dark energies donot interact with matter. In this case, the deceleration parameter and equation state parameter for dark en-ergy candidates have been found. The statefinder parameters have been found in terms of the dimensionlessdensity parameters as well as EoS parameter ω and the Hubble parameter. An important thing to note is thatthe G -corrected statefinder parameters are still geometrical since the parameter △ G is a pure number and isindependent of the geometry. Acknowledgments
Special thanks to the referees for numerous comments to improve the quality of this work. [1] Riess A.G. et al.: Astron. J. (1998)1009;Perlmutter, S. et al.: Astrophys. J. (1999)565.[2] Tegmark M. et al.: Phys. Rev.
D69 (2004)103501.[3] Allen S.W. et al.: Mon. Not. Roy. Astron. Soc. (2004)457.[4] Spergel D.N. et al.: Astrophys. J. Suppl. (2003)175;Komatsu E. et al.: Astrophys. J. Suppl. 180(2009)330.[5] Ratra B. and Peebles, P.J.E.: Phys. Rev.
D37 (1988)3406.[6] Dolgov A.D.: Phys. Rev.
D55 (1997)5881.[7] Sahni V. and Starobinsky, A.: Int. J. Mod. Phys. D9 (2000)373.[8] Padmanabhan T.: Phys. Rep. (2003)235.[9] Peebles P.J.E.: Rev. Mod. Phys. (2003)599.[10] P.A.M. Dirac, Proc. R. Soc. Lond. A (1938) 199;A. Beesham, Int. J. Theor. Phys. (1994) 1383;Ray S. et al.: Large Number Hypothesis , arXiv:0705.1836v1;M.R. Setare, D. Momeni, Commun. Theor. Phys. (2011) 691.[11] Zeldovich Ya.B.: Usp. Nauk. (1968)209. [12] D. Momeni , Int. J. Theor. Phys. (2011) 2582;M.R. Setare, D. Momeni, Commun.Theor.Phys. 56 (2011) 691.[13] Overduin J.M. and Cooperstock, F.I.: Phys. Rev. D58 (1998)043506.[14] Ray S. and Mukhopadhyay U.: Grav. Cosmol. (2007) 142;M.S. Berman, Phys. Rev. Phys. Rev. D 43 , 1075 (1991);H. Liu, P. Wesson, (2001) ApJ (2000) 109;A. Pradhan, P. Pandey, Astrophys. Space Sci. (2006) 127;A.I. Arbab, Chin. Phys. Lett. K1 (1921)966.[16] Klein O.: Zeits. Phys. (1926)895.[17] Overduin J.M. and Wesson P.S.: Phys. Rept. (1997)303.[18] Lee H.C.: An Introdution to Kaluza Klein Theories (World Scientific, 1984).[19] Appelquist T., Chodos A. and Freund P.G.O.:
Modern Kaluza-Klein Theories (Addison-Wesley, 1987).[20] Darabi F.:
Dark Pressure in Non-compact and Non-Ricci Flat 5D Kaluza-Klein Cosmology , arXiv/1101.0666v1.[21] Sahni V. et al.: JETP. Lett. (2003)201.[22] Zhang X.: Int. J. Mod. Phys. D14 (2005)1597.[23] Wei H. and Cai, R.G.: Phys. Lett.
B655 (2007)1.[24] Zhang X.: Phys. Lett.
B611 (2005)1.[25] Huang J.Z. et al.: Astrophys. Space Sci. (2008)175.[26] Zhao W.: Int. J . Mod. Phys.
D17 (2008)1245.[27] Hu M. and Meng, X.H.: Phys. Lett.
B635 (2006)186.[28] Zimdahl, W. and Pavon D.: Gen. Relativ. Gravit. (2004)1483.[29] Shao Y. and Gui Y.: Mod. Phys. Lett. A23 (2008)65.[30] Jamil M. and Debnath U.: Int. J. Theor. Phys.
209 (2011);Jamil, M., Int. J. Theor. Phys.
545 (2011);M.U. Farooq et al, Astrophys. Space Sci.
243 (2011);Reddy, D. R. K. and Naidu, R. L., Int. J. Theor. Phys. (2008) 1751;M. Jamil et al, Eur. Phys. J. C
149 (2009);Ozel C., Kayhan H. and Khadekar G.S.: Adv. Studies. Theor. Phys. (2010)117.[32] R. A. El-Nebulsi, Research in Astron. Astrophys.
759 (2011);Tiwari, R. K., Rahaman, F. and Ray, S., Int. J. Theor. Phys. , 111 (2010);Canfora, F., Giacomimi, A. and Zerwekh, A. R., Phys. Rev. D (2003) 201.[34] Susskind L.: J. Math. Phys. (1995) 6377;’t Hooft G: arXiv:9310026 [gr-qc].[35] Cohen A.etal.: Phys. Rev. Lett. (1999) 4971.[36] S. D. H. Hsu: Phys. Lett. B (2004) 13.[37] Li M.: Phys. Lett. B (2004) 1.[38] M.R. Setare, Phys. Lett. B642 (2006) 421;M.R. Setare, Phys. Lett.
B648 (2007) 329;M. R. Setare, J. Zhang, X. Zhang, JCAP (2007) 007;M. Jamil, M.U. Farooq, M.A. Rashid, Eur. Phys. J. C
471 (2009);M. Jamil, M.U.Farooq, Int. J. Theor. Phys. (2010) 42;M.R. Setare, M. Jamil, JCAP (2010) 010;M. Jamil, M.U. Farooq, JCAP (2010) 001;M. Jamil, A. Sheykhi, M.U. Farooq, Int. J. Mod. Phys. D 19 (2010) 1831;H.M. Sadjadi, M. Jamil, Gen. Rel. Grav. (2012) 604;M.R. Setare, M. Jamil, Gen. Relativ. Gravit. , (2011) 293[39] H. Wei and R. G. Cai: Phys. Lett. B (2008) 113;H. Wei and R. G. Cai, Phys. Lett. B (2008) 1;Zhang J. etal.:Eur. Phys. J. C (2008) 303. [40] Gorini V. etal.:Phys. Rev. D (2003) 063509;Alam U. etal.:Mon. Not. Roy. Astron. Soc. (2003) 1057;Bento M. C.:Phys. Rev. D66