Higher Dimensional Cosmology with Some Dark Energy Models in Emergent, Logamediate and Intermediate Scenarios of the Universe
aa r X i v : . [ phy s i c s . g e n - ph ] A ug Higher Dimensional Cosmology with Some Dark Energy Models inEmergent, Logamediate and Intermediate Scenarios of the Universe
Chayan Ranjit ∗ Shuvendu Chakraborty † and Ujjal Debnath ‡ Department of Mathematics, Seacom Engineering College, Howrah - 711 302, India. Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711 103, India. (Dated: November 9, 2018)We have considered N -dimensional Einstein field equations in which four-dimensional space-timeis described by a FRW metric and that of extra dimensions by an Euclidean metric. We havechosen the exponential forms of scale factors a and d numbers of b in such a way that there is nosingularity for evolution of the higher dimensional Universe. We have supposed that the Universeis filled with K-essence, Tachyonic, Normal Scalar Field and DBI-essence. Here we have found thenature of potential of different scalar field and graphically analyzed the potentials and the fields forthree scenario namely Emergent Scenario, Logamediate Scenario and Intermediate Scenario. Alsographically we have depicted the geometrical parameters named statefinder parameters and slow-rollparameters in the higher dimensional cosmology with the above mentioned scenarios. PACS numbers: I. INTRODUCTION
From recent observations it is strongly believed that the most interesting problems of particle physics cosmol-ogy are the origin due to accelerated expansion of the present Universe. The observation from type Ia supernovae[1,2] in associated with Large scale Structure [3] and Cosmic Microwave Background anisotropies(CMB) [4] haveshown the evidences to support cosmic acceleration. The theory of Dark energy is the main responsible candi-date for this scenario. From recent cosmological observations including supernova data [5] and measurements ofcosmic microwave background radiation(CMBR) [4] it is evident that our present Universe is made up of about4% ordinary matter, about 74% dark energy and about 22% dark matter. Several interesting mechanisms havebeen suggested to explain this feature of this Universe, such as Loop Quantum Cosmology (LQC) [6], modifiedgravity [7], Higher dimensional phenomena [8], Brans-Dicke theory [9], brane-world model [10] and many others.Recently many cosmological models have been constructed by introducing dark energies such as Phantom[11], Tachyon scalar field [12], Hessence [13], Dilaton scalar field [14], K-essence scalar field [15], DBI essencescalar field [16], and many others. After realizing that many interesting of particle interactions need more thanfour dimensions for their formulation, the study of higher dimensional theory has been revived. The modelof higher dimensions was proposed by Kaluza and Klein [17,18] who tried to introducing an extra dimensionwhich is basically an extension of Einstein general relativity in 5D. The activities of extra dimensions alsoverified from the STM theory [19] proposed recently by Wesson et al [20]. As our space-time is explicitly fourdimensional in nature so the ‘hidden’ dimensions must be related to the dark matter and dark energy whichare also ‘invisible’ in nature.Form the cosmological observation the present phase of acceleration of the Universe is not clearly understood.Standard Big Bang cosmology with perfect fluid assumption fails to accommodate the observational fact.Recently, Ellis and Maartens [21] have considered a cosmological model where inflationary cosmologies existin which the horizon problem is solved before inflation begins, no big-bang singularity exist, no exotic physicsis involved and quantum gravity regime can even be avoided. An emergent Universe model if developed in aconsistent way is capable of solving the conceptual problems of the big-bang model. Actually the Universestarts out in the infinite past as an almost static Universe and expands slowly, eventually evolving into a hotbig-bang era. An interesting example of this scenario is given by Ellis, Murugan and Tsagas [22], for a closedUniverse model with a minimally coupled scalar field φ , which has a special form of interaction potential V ( φ ). There are several features for the emergent Universe [21,23] viz. (i) the Universe is almost static at the ∗ [email protected] † [email protected] ‡ [email protected], [email protected] finite past, (ii) there is no time like singularity, (iii) the Universe is always large enough so that the classicaldescription of space time is adequate, (iv) the Universe may contains exotic matter so that the energy conditionmay be violated, (v) the Universe is accelerating etc.Here we also consider another two scenarios: (i) “intermediate scenario” and (ii) “logamediate scenario”[24-27] to study of the expanding anisotropic Universe in the presence of different scalar fields. In the first casethe scale factors evolves separately as a ( t ) = exp( At f ) and b ( t ) = exp( Bt f ) where A > B >
0, 0 < f < < f <
1. So the expansion of the Universe is slower than standard de Sitter inflation (arises when f = f = 1) but faster than power law inflation with power greater than 1. The Harrison - Zeldovich spectrumof fluctuation arises when f = f = 1 and f = f = 2 /
3. In the second case we analyze the inflation withscale factors separately of the form a ( t ) = exp( A (ln t ) λ ) and b ( t ) = exp( B (ln t ) λ ) with A > B > λ > λ >
1. When λ = λ = 1 this model reduces to power law inflation. The logamediate inflationary form ismotivated by considering a class of possible cosmological solutions with indefinite expansion which result fromimposing weak general conditions on the cosmological model.In this work, we have considered N-dimensional Einstein field equations in which 4-dimensional space-time isdescribed by a FRW metric and that of the extra d -dimensions by an Euclidean metric. We also consider theUniverse is filled with K-essence scalar field, normal scalar field, tachyonic field and DBI essence and investigatethe natures of the dark energy candidates for Emergent, Intermediate and Logamediate scenarios of the Universe.Here in extra dimensional phenomenon we have shown the change of the potential V ( φ ) corresponding to thefield φ for the dark energies mentioned above and also analyze the anisotropic Universe using the “ slow roll ’parameters in Hamilton-Jacobi formalism and in terms of above mentioned scalar field φ and they are given by[24] ǫ = 2 ˙ H H ˙ φ and η = 2 H " ˙ φ ¨ H − ˙ H ¨ φ ˙ φ (1)Sahni et al [28] proposed the trajectories in the { r, s } plane corresponding to different cosmological modelsto depict qualitatively different behavior. The statefinder diagnostic along with future SNAP observations mayperhaps be used to discriminate between different dark energy models. The above statefinder diagnostic pairfor higher dimensional anisotropic cosmology are constructed from the scale factors a ( t ) and b ( t ) as follows: r = 1 + 3 ˙ HH + ¨ HH and s = r − q − ) (2)where q is the deceleration parameter defined by q = − − ˙ HH and H is the Hubble parameter. Since thisparameters are dimensionless so they allow us to characterize the properties of dark energy in a model inde-pendently. Finally we graphically analyzed geometrical parameters r, s in the higher dimensional anisotropicUniverse in emergent, logamediate and intermediate scenarios of the universe. II.
BASIC EQUATIONS
We consider homogeneous and anisotropic N -dimensional space-time model described by the line element[29,30] ds = ds F RW + d X i =1 b ( t ) dx i (3)where d is the number of extra dimensions ( d = N −
4) and ds F RW represents the line element of the FRWmetric in four dimensions is given by ds F RW = − dt + a ( t ) (cid:20) dr − kr + r ( dθ + sin θdφ ) (cid:21) (4)where a ( t ) and b ( t ) are the functions of t alone represent the scale factors of 4-dimensional space time andextra d -dimensions respectively. Here k (= 0 , ±
1) is the curvature index of the corresponding 3-space, so thatthe above Universe is described as flat, closed and open respectively.The Einstein’s field equations for the above non-vacuum higher dimensional space-time symmetry are3 (cid:18) ˙ a + ka (cid:19) = ¨ DD − d b b + d b b + ρ (5)2 ¨ aa + ˙ a + ka = ˙ aa ˙ DD + d b b − d b b − p (6)and ¨ bb + 3 ˙ aa ˙ bb + ˙ DD ˙ bb − ˙ b b = − p ρ and p are energy density and isotropic pressure respectively. Here we choose here 8 πG = c = 1 and D = b d ( t ), so we have ˙ DD = d bb and ¨ DD = d bb + d − d b b . Also in this model we define the Hubble parameter as H = d +3 (3 ˙ aa + d ˙ bb ) . III.
EMERGENT SCENARIO
At first we consider Emergent scenario, where the scale factors a ( t ) and b ( t ) are consider as the power ofcosmic time t are given by [23, 31, 32] a = a ( β + e αt ) m and b = b ( µ + e νt ) n (8)where a , b , α , β , µ , ν , m and n are positive constants. So the field equations (5), (6) and (7) become3 m α e αt ( β + e αt ) + 3 ka ( β + e αt ) − m = dnν e νt µ + e νt ) ( dn + n + 4 µe − νt ) + ρ (9) mα e αt ( β + e αt ) (3 m + 2 βe − αt ) + ka ( β + e αt ) − m = dmnανe ( α + ν ) t β + e αt )( µ + e νt ) + d ( d − n ν e νt µ + e νt ) − p (10)and nµν e νt ( µ + e νt ) + 3 mnανe ( α + ν ) t ( β + e αt )( µ + e νt ) + dn ν e νt µ + e νt ) + p r and s . • K-essence Field:
The energy density and pressure due to K-essence field φ are given by [15] ρ = V ( φ )( − χ + χ ) (12)and p = V ( φ )( − χ + 3 χ ) (13) Φ V Α= Ν= Α= Ν= Α= Ν=
50 2 4 6 8 10 12 - - - Ε Η Fig.1 Fig.2Fig. 1 shows the variations of V against φ , for a = 2 , α = 1 . , β = 1 . , µ = 3 , ν = 1 , m = 1 , n = 3 , k = 1 , d = 5 and Fig.2 shows the variation of the slow roll parameters ǫ against η for m = 30 , n = 10 , β = 15 , a = 1 , µ = 1 , d = 40 ,k = − , , where χ = ˙ φ and V ( φ ) is the relevant potential for K-essence Scalar field φ .Using equations (5)-(7), we can find the expressions for V ( φ ) and φ as V ( φ ) = (cid:16) − m α e αt ( β + e αt ) − mα βe αt ( β + e αt ) + dmnανe ( α + ν ) t ( β + e αt )( µ + e νt ) + d ( − d ) n ν e νt ( µ + e νt ) + dnµν e νt ( µ + e νt ) − k ( β + e αt ) − m a (cid:17) (cid:16) − m α e αt ( β + e αt ) − mα βe αt ( β + e αt ) + dmnανe ( α + ν ) t ( β + e αt )( µ + e νt ) + d n ν e νt µ + e νt ) + dnµν e νt ( µ + e νt ) − k ( β + e αt ) − m a (cid:17) (14)and φ = Z − m α e αt ( β + e αt ) − mα βe αt ( β + e αt ) + dmnανe ( α + ν ) t ( β + e αt )( µ + e νt ) + d n ν e νt µ + e νt ) + dnµν e νt ( µ + e νt ) − k ( β + e αt ) − m a − m α e αt ( β + e αt ) − mα βe αt ( β + e αt ) + dmnανe ( α + ν ) t ( β + e αt )( µ + e νt ) + d ( − d ) n ν e νt ( µ + e νt ) + dnµν e νt ( µ + e νt ) − k ( β + e αt ) − m a dt (15)From above forms of V and φ , we see that V can not be expressed explicitly in terms of φ . • Tachyonic field:
The energy density ρ and pressure p due to the Tachyonic field φ are given by [12] ρ = V ( φ ) q − ˙ φ (16)and p = − V ( φ ) q − ˙ φ (17)where V ( φ ) is the relevant potential for the Tachyonic field φ . Using equations (5)-(7), we can find theexpressions for V ( φ ) and φ as V ( φ ) = s(cid:18) mα e αt (2 β + 3 me αt )( β + e αt ) − dmnανe ( α + ν ) t β + e αt )( µ + e νt ) − d ( d − n ν e νt µ + e νt ) + k ( β + e αt ) − m a (cid:19) × s(cid:18) m α e αt ( β + e αt ) − dnν e νt (4 µ + (1 + d ) ne νt )8( µ + e νt ) + 3 k ( β + e αt ) − m a (cid:19) (18)and φ = Z vuuut − mα e αt (2 β +3 me αt )( β + e αt ) + dmnανe ( α + ν ) t ( β + e αt )( µ + e νt ) + d ( d − n ν e νt ( µ + e νt ) − k ( β + e αt ) − m a (cid:16) m α e αt ( β + e αt ) − dnν e νt (4 µ +(1+ d ) ne νt )8( µ + e νt ) + k ( β + e αt ) − m a (cid:17) dt (19)From above forms of V and φ , we see that V can not be expressed explicitly in terms of φ . • Normal Scalar field:
The energy density ρ and pressure p due to the Normal Scalar field φ are given by [37] ρ = 12 ˙ φ + V ( φ ) (20)and p = 12 ˙ φ − V ( φ ) (21)where V ( φ ) is the relevant potential for the Normal Scalar field φ . Using equations (5)-(7), we can find theexpressions for V ( φ ) and φ as V ( φ ) = 3 m α e αt ( β + e αt ) + mα βe αt ( β + e αt ) − dmnανe ( α + ν ) t β + e αt )( µ + e νt ) − d n ν e νt µ + e νt ) − dnµν e νt µ + e νt ) + 2 k ( β + e αt ) − m a (22)and φ = 12 Z (cid:18) − mα βe αt ( β + e αt ) + 2 dmnανe ( α + ν ) t ( β + e αt )( µ + e νt ) − dn ν e νt ( µ + e νt ) − dnµν e νt ( µ + e νt ) + 8 k ( β + e αt ) − m a (cid:19) dt (23) • DBI-essence:
The energy density ρ and pressure p due to the DBI-essence field φ are given by [35,36] ρ = ( γ − T ( φ ) + V ( φ ) (24)and p = γ − γ T ( φ ) − V ( φ ) (25) Φ V Β= Μ= Β= Μ= Β= Μ=
20 50 100 150 200 250 - - Ε Η Fig.3 Fig.4Fig. 3 shows the variations of V against φ , for a = 2 , α = 1 . , β = 1 . , µ = 1 . , ν = 1 . , m = 1 , n = 2 , k = 1 , d = 5 andFig. 4 shows the variation of the slow roll parameters ǫ against η for m = 30 , n = 6 , α = 1 , a = 1 , , ν = 5 , d = 15 ,k = − , , where γ is given by γ = 1 q − ˙ φ T ( φ ) (26)and V ( φ ) is the relevant potential for the DBI-essence field φ .The energy conservation equation is given by˙ ρ φ + 3 H ( ρ φ + p φ ) = 0 (27)where H is the Hubble parameter in terms of scale factor as H = 13 aa + d ˙ bb ! (28)From energy conservation equation we have the wave equation for φ as¨ φ − T ′ ( φ )2 T ( φ ) ˙ φ + T ′ ( φ ) + aa + d ˙ bb ! ˙ φγ + 1 γ [ V ′ ( φ ) − T ′ ( φ )] = 0 (29)where ′ is the derivative with respect to φ . Now for simplicity of calculations, we consider two particular cases: γ = constant and γ = constant. Case I: γ = constant.In this case, for simplicity, we assume T ( φ ) = σ ˙ φ ( σ >
1) and V ( φ ) = δ ˙ φ ( δ > γ = q σσ − .In these choices we have the following solutions for V ( φ ), T ( φ ) and φ from equation (29)as φ = Z φ a F b G ( β + e αt ) mF + nG dt (30) Φ V Β= Μ= Β= Μ= Β= Μ= - - - - - - - Ε Η Fig.5 Fig.6Fig. 5 shows the variations of V against φ , for a = 0 . , α = 0 . , β = 0 . , µ = 0 . , ν = 0 . , m = 1 , n = 8 , k = 1 , d = 15and Fig. 6 shows the variation of the slow roll parameters ǫ against η for m = 5 , n = 4 , α = 3 , a = 1 , , ν = 7 , d = 15 ,k = − , , from V ( φ ) = δφ a F b G ( β + e αt ) mF +2 nG (31)and T ( φ ) = σφ a F b G ( β + e αt ) mF +2 nG (32)where E = γ [2( δ − σ ) + 2( σ − γ ], γ = q σσ − , F = − E , G = − dE and φ is an integrating constants. FromFig. 7 and Fig. 8, we see that V ( φ ) and T ( φ ) are both exponentially decreasing with DBI scalar field φ . Case II: γ = constant.In this case, we consider γ = ˙ φ − and V ( φ ) = T ( φ ). Using equations (5)-(7) and (24)-(26), we can find theexpressions for V ( φ ), T ( φ ) and φ as [34] V ( φ ) = T ( φ ) = (cid:18) ln [ C a b d ( β + e αt ) (3 m + dn ) ] (cid:19) vuuut − C a b d ( β + e αt ) (3 m + dn ) ] (33)and φ = Z − C a b d ( β + e αt ) (3 m + dn ) ] dt (34)where C is an integrating constant. • Statefinder parameters:
The geometrical parameters { r, s } for higher dimensional anisotropic cosmology in emergent scenario can beconstructed from the scale factors a ( t ) and b ( t ) as r = 1 + (3 + d ) (cid:16) (cid:16) mαe αt β + e αt + dnνe νt µ + e νt (cid:17) (cid:16) mα βe αt ( β + e αt ) + dnµν e νt ( µ + e νt ) (cid:17) + (3 + d ) (cid:16) mα βe αt ( β − e αt )( β + e αt ) + dnµν e νt ( µ − e νt ) ( µ + e νt ) (cid:17)(cid:17) ( mαe αt β + e αt + dnνe νt µ + e νt ) (35) Φ V Φ T Fig.7 Fig.8 Ε Η Fig.9Fig. 7 shows the variations of V against φ , and Fig. 8 shows the variations of T against φ for a = 0 . , b = . , α = 0 . , β = 0 . , m = 15 , n = 3 , k = 1 , d = 15 , σ = 3 , δ = 2 , φ = 2 Fig. 9 shows the variation of theslow roll parameters ǫ against η for a = 0 . , b = 0 . , α = 0 . , β = 0 . , m = 2 , n = 5 , d = 5 , σ = 2 , δ = 5 respectivelyin the 1 st case of DBI-essence Scalar field scenario. s = (cid:16) (3 + d ) (cid:16) (cid:16) mαe αt β + e αt + dnνe νt µ + e νt (cid:17) (cid:16) mα βe αt ( β + e αt ) + dnµν e νt ( µ + e νt ) (cid:17) + (3 + d ) (cid:16) mα βe αt ( β − e αt )( β + e αt ) + dnµν e νt ( µ − e νt )( µ + e νt ) (cid:17)(cid:17)(cid:17) (cid:16) mαe αt β + e αt + dnνe νt µ + e νt (cid:17) − − d ) (cid:0) mα βeαt ( β + eαt )2 + dnµν eνt ( µ + eνt )2 (cid:1)(cid:0) mαeαtβ + eαt + dnνeνtµ + eνt (cid:1) !! (36)The relation between r and s has been shown in Fig.12. From Fig.12, we see that s is negative when r ≥ CDM model ( r = 1 , s = 0). Φ V - Ε Η Fig.10 Fig.11Fig. 10 shows the variations of V against φ , for a = 0 . , b = 0 . , C = 2 , α = 5 , β = 6 , m = 0 . , n = 0 . , d = 5 andFig. 11 shows the variation of the slow roll parameters ǫ against η for a = 0 . , b = 0 . , C = 2 , α = 5 , β = 6 , m = 0 . , n = 0 . , d = 5 respectively in the 2 nd case of DBI-essence Scalarfield for Emergent Scenario. - - - - - r s Fig.12Fig. 12 shows the variations of r against s , for α = 0 . , β = 10 , µ = 10 , ν = 75 , m = 10 , n = . , d = 15 in EmergentScenario. Φ V Λ = Λ = Λ = Λ = Λ = Λ =
60 100 200 300 400 500 600 700050100150200250300 Ε Η Fig.13 Fig.14Fig. 13 shows the variations of V against φ , for A = 1 , B = 2 , k = 1 , λ = 2 , λ = 3 , d = 5 and Fig. 14 shows thevariation of the slow roll parameters ǫ against η for λ = 3 , λ = 6 , d = 5 k = − , , IV.
LOGAMEDIATE SCENARIO
Now we consider Logamediate scenario, where the scale factors a ( t ) and b ( t ) are consider as the power ofcosmic time t are given by [24] a ( t ) = e A (ln t ) λ and b ( t ) = e B (ln t ) λ (37)where A , B , m and n are positive constants. So the field equations (5), (6) and (7) become3 A λ (ln t ) λ − t + 3 ke − A (ln t ) λ = Bdλ (ln t ) λ − t (cid:0) λ −
1) + Bλ ( d + 1)(ln t ) λ − t (cid:1) + ρ (38) Aλ (ln t ) λ − (2( λ −
1) + 3 Aλ (ln t ) λ ) t + ke − A (ln t ) λ = Aλ (ln t ) λ − (4 + Bdλ (ln t ) λ − )2 t + B λ d ( d − t ) λ − t − p (39) Bλ (ln t ) λ − (6 Aλ (ln t ) λ ) + Bdλ (ln t ) λ − t + 2 λ − t + p = 0 (40)We now consider K-essence field, Tachyonic field, normal scalar field and DBI essence field. For these fourcases we analyze the behavior of the Logamediate Universe in extra dimension and finally we analyze thebehavior of the state finder parameters r and s . • K-essence Field:
The energy density and pressure due to K-essence field φ are given by the equations (12) and (13). Usingequations (38)-(40), we can find the expressions for V ( φ ) and φ as V ( φ ) = e − A (ln t ) λ h B dλ (2 d − t ) λ e A (ln t ) λ + 2 Bdλ (ln t ) λ e A (ln t ) λ (cid:0) Aλ (ln t ) λ − ln t + λ − (cid:1) t (ln t ) (cid:0) − kt (ln t ) − Aλ e A (ln t ) λ ( λ − − ln t )(ln t ) λ − A λ (ln t ) λ e A (ln t ) λ − (cid:16) kt (ln t ) + Aλ e A (ln t ) λ ( λ − − ln t )(ln t ) λ + 2 A λ (ln t ) λ e A (ln t ) λ (cid:17)i + B d λ (ln t ) λ e A (ln t ) λ + 2 Bdλ (ln t ) λ e A (ln t ) λ ( Aλ (ln t ) λ − ln t + λ − (cid:1) (41)and φ = Z " − B d λ (ln t ) λ e A (ln t ) λ − Bdλ (ln t ) λ e A (ln t ) λ (cid:0) Aλ (ln t ) λ − ln t + λ − (cid:1) + 16 (cid:0) kt (ln t ) − B dλ (2 d − t ) λ e A (ln t ) λ − Bdλ (ln t ) λ e A (ln t ) λ (3 Aλ (ln t ) λ − ln t + λ − Aλ e A (ln t ) λ ( λ − − ln t )(ln t ) λ + 3 A λ (ln t ) λ e A (ln t ) λ (cid:17) +24 (cid:0) kt (ln t ) + Aλ e A (ln t ) λ ( λ − − ln t )(ln t ) λ + 2 A λ (ln t ) λ e A (ln t ) λ (cid:1) dt (42) • Tachyonic field:
The energy density ρ and pressure p due to the Tachyonic field φ are given by the equations (16) and (17).Using equations (38)-(40), we can find the expressions for V ( φ ) and φ as V ( φ ) = s (cid:0) kt (ln t ) e − A (ln t ) λ + 24 A λ (ln t ) λ − Bdλ ( λ − t ) λ − B d ( d + 1) λ (ln t ) λ + 4 Bdλ (ln t ) λ +1 (cid:1) t (ln t ) × q(cid:2) kt (ln t ) e − A (ln t ) λ + 16 Aλ ( λ − t ) λ + 24 A λ (ln t ) λ − Aλ (ln t ) λ +1 − B d ( d − λ (ln t ) λ − ABdλ λ (ln t ) λ + λ ] (43)and φ = Z s(cid:20) − kt (ln t ) e − A (ln t ) λ + 16 Aλ ( λ − t ) λ + 24 A λ (ln t ) λ − Aλ (ln t ) λ +1 − B d ( d − λ (ln t ) λ kt (ln t ) e − A (ln t ) λ + 24 A λ (ln t ) λ − Bdλ ( λ − t ) λ − B d ( d + 1) λ (ln t ) λ − ABdλ λ (ln t ) λ + λ +4 Bdλ (ln t ) λ +1 (cid:21) dt (44) • Normal Scalar field:
The energy density ρ and pressure p due to the Normal Scalar field φ are given by the equations (20) and(21). Using equations (38)-(40), we can find the expressions for V ( φ ) and φ as V ( φ ) = 18 t (ln t ) h kt (ln t ) e − A (ln t ) λ + 8 Aλ ( λ − t ) λ + 24 A λ (ln t ) λ − Aλ (ln t ) λ +1 − Bdλ ( λ − t ) λ − B d λ (ln t ) λ + 2 Bdλ (ln t ) λ +1 − ABdλ λ (ln t ) λ + λ (cid:3) (45)and φ = Z s t (ln t ) (cid:2) kt (ln t ) e − A (ln t ) λ + 8 Aλ ( λ − t ) λ + 8 Aλ (ln t ) λ +1 − Bdλ ( λ − t ) λ − B d λ (ln t ) λ + 2 Bdλ (ln t ) λ +1 − ABdλ λ (ln t ) λ + λ ] dt (46)2 Φ V Λ = Λ = Λ = Λ = Λ = Λ =
60 5 10 15 2051015202530 Ε Η Fig.15 Fig.16Fig. 15 shows the variations of V against φ , for A = 1 , B = 2 , k = 1 , λ = 5 , λ = 4 , d = 15 and Fig. 16 shows thevariation of the slow roll parameters ǫ against η for λ = 3 , λ = 7 , d = 5 k = − , , Φ V Λ = Λ = Λ = Λ = Λ = Λ =
950 100 150 200 250020406080100 Ε Η Fig.17 Fig.18Fig. 17 shows the variations of V against φ , for A = 0 . , B = 0 . , k = 1 , λ = 5 , λ = 4 , d = 15 and Fig. 18 shows thevariation of the slow roll parameters ǫ against η for λ = 2 , λ = 3 , d = 5 k = − , , Φ V Φ T Fig.19 Fig.20
40 60 80 100 120 1403040506070 Ε Η Fig.21Fig. 19 shows the variations of V against φ and Fig. 20 shows the variations of T against φ , for A = 0 . , B = 0 . , λ = 4 , λ = 3 , d = 15 , σ = 3 , δ = 2 , φ = 2 and Fig. 21 shows the variation of the slow roll parameters ǫ against η for A = 0 . , B = 0 . , λ = 3 , λ = 8 , σ = 8 , δ = 2 , φ = 1 , d = 20 in the 1 st case of DBI-essence Scalar fieldfor Logamediate Scenario. • DBI-essence:
The energy density ρ and pressure p due to the DBI-essence field φ are given by the equations (24) and (25). Case I: γ = constant.Using equations (38)-(40), we can find the expressions for V ( φ ), T ( φ ) and φ as V ( φ ) = δφ e (2 AF (ln t ) λ +2 BG (ln t ) λ ) (47) T ( φ ) = σφ e (2 AF (ln t ) λ +2 BG (ln t ) λ ) (48)and φ = Z φ e ( AF (ln t ) λ + BG (ln t ) λ ) dt (49)where E = γ [2( δ − σ ) + 2( σ − γ ], γ = q σσ − , F = − E , G = − dE and φ is an integrating constants. Fromabove, we see that V ( φ ) and T ( φ ) are both exponentially decreasing with DBI scalar field φ .4 Φ V Ε Η Fig.22 Fig.23Fig. 22 shows the variations of V against φ , for A = 0 . , B = 0 . , C = 3 , λ = 2 , λ = 3 , d = 5 and Fig. 23 shows thevariation of the slow roll parameters ǫ against η for A = 2 , B = 3 , C = 1 , λ = 3 , λ = 4 , d = 5 in the 2 nd case ofDBI-essence Scalar field for Logamediate Scenario. Case II: γ = constant.Using equations (38)-(40), we can find the expressions for V ( φ ) and φ as V ( φ ) = ln [ C e (3 A (ln t ) λ + dB (ln t ) λ ) ] s − C e (3 A (ln t ) λ dB (ln t ) λ ] (50)and φ = Z − C e (3 A (ln t ) λ dB (ln t ) λ ] ! dt (51)Where C is an integrating constant. • Statefinder parameters:
The geometrical parameters { r, s } for higher dimensional anisotropic cosmology in Logamediate scenario canbe constructed from the scale factors a ( t ) and b ( t ) as r = 1 + (3( d + 3)(3 Aλ ( − ln t + λ − t ) λ + Bdλ ( − ln t + λ − t ) λ ))(3 Aλ (ln t ) λ + Bdλ (ln t ) λ ) + ( d + 3) (3 Aλ (ln t ) λ (2 + λ ( λ −
3) + ln t (2 ln t − λ + 3)) + Bdλ (ln t ) λ (2 + λ ( λ −
3) + ln t (2 ln t − λ + 3)))(3 Aλ (ln t ) λ + Bdλ (ln t ) λ ) (52) s = ( d + 3)3(3 Aλ (ln t ) λ + Bdλ (ln t ) λ )(3 Aλ ( − ln t + λ − t ) λ + Bdλ ( − ln t + λ − t ) λ )[3(3 Aλ (ln t ) λ + Bdλ (ln t ) λ ) +( d + 3)(3 Aλ (ln t ) λ (2 + λ ( λ −
3) + ln t (2 ln t − λ + 3)) + Bdλ (ln t ) λ (2 + λ ( λ −
3) + ln t (2 ln t − λ + 3))) (cid:3) ( − + (( d +3)( − Aλ ( − ln t + λ − t ) λ − Bdλ ( − ln t + λ − t ) λ ))(3 Aλ (ln t ) λ + Bdλ (ln t ) λ ) ) i (53)5 - - - - r s Fig.24Fig. 24 shows the variations of r against s , for A = 1 . , B = 3 . , λ = 2 , λ = 2 , d = 3 in Logamediate Scenario. V. INTERMEDIATE SCENARIO
Finally we consider Intermediate scenario, where the scale factors a ( t ) and b ( t ) are consider as the power ofcosmic time t are given by [24] a ( t ) = e At f and b ( t ) = e Bt f (54)where A , B , m and n are positive constants. So the field equations (3), (4) and (5) become18 t e − At f (24 kt + e At f (24 A f t f − Bdf t f ( − f (4 + B ( d + 1) t f )))) − ρ = 0 (55)18 t e − At f ( − kt + e At f ( − A f t f + B ( d − df t f − Af t f (4 f − Bdf t f − − p = 0 (56) Bf t f − (6 Af t f + n ( Bdt f + 2) −
2) + p = 0 (57)We now consider K-essence field, Tachyonic field, normal scalar field and DBI essence field. For these fourcases we analyze the behavior of the Intermediate Universe in extra dimension and finally we analyze thebehavior of the state finder parameters r and s . • K-essence Field:
The energy density and pressure due to K-essence field φ are given by the equations (12) and (13). Usingequations (55)-(57), we can find the expressions for V ( φ ) and φ as V ( φ ) = − (cid:18) e − At f (cid:16) kt + e At f (cid:0) − B d (2 d − f t f + 24 Af t f (2 Af t f + f − − Bdf t f (cid:0) Af t f + f − (cid:1)(cid:1)(cid:17) (cid:19)(cid:0) t (cid:0) kt + e At f ( − B d f t f − Bdf t f ( Af t f + f −
1) + 8 Af t f (3 Af t f + f − (cid:1)(cid:1) (58)and φ = Z s (cid:0) kt + e At f ( − B d f t f − Bdf t f ( Af t f + f −
1) + 8 Af t f (3 Af t f + f − (cid:1)(cid:0) kt + e At f ( − B d (2 d − f t f + 24 Af t f (2 Af t f + f − − Bdf t f (3 Af t f + f − (cid:1) dt (59) • Tachyonic field: Φ V f = f = f = f = f = f = Ε Η Fig.25 Fig.26Fig. 25 shows the variations of V against φ , for A = 20 , B = 25 , k = 1 , f = 0 . , f = 0 . , d = 5 and Fig. 26 shows thevariation of the slow roll parameters ǫ against η for A = 400 , B = 300 , d = 75 and k = − , , The energy density ρ and pressure p due to the Tachyonic field φ are given by the equations (16) and (17).Using equations (55)-(57), we can find the expressions for V ( φ ) and φ as V ( φ ) = 18 (cid:20) t e − At f (cid:16) kt + e At f (cid:0) − B d ( d − f t f − ABdf f t f + f + 8 Af t f (cid:0) − f (cid:0) At f + 2 (cid:1)(cid:1)(cid:1)(cid:17)(cid:16) kt + e At f (cid:0) A f t f − Bdf t f (cid:0) f − (cid:0) B ( d + 1) t f (cid:1)(cid:1)(cid:1)(cid:17)i (60)and φ = Z s (cid:0) kt + e At f ( − B d ( d − f t f − ABdf f t f + f + 8 Af t f ( − f (3 At f + 2))) (cid:1)(cid:0) kt + e At f (24 A f t f − Bdf t f ( f − B ( d + 1) t f ))) (cid:1) dt (61) • Normal Scalar field:
The energy density ρ and pressure p due to the Normal Scalar field φ are given by the equations (20) and(21). Using equations (55)-(57), we can find the expressions for V ( φ ) and φ as V ( φ ) = 18 t e − At f (cid:16) kt + e At f (cid:0) − B d f t f − Bdf t f ( Af t f + f −
1) + 8 Af t f (3 Af t f + f − (cid:1)(cid:17) (62)and φ = Z s(cid:18) t e − At f (cid:0) kt − e At f (2 Af t f ( − Bdf t f + 4 f −
4) +
Bdf t f ( − f (2 + Bt f ))) (cid:1)(cid:19) dt (63) • DBI-essence:
The energy density ρ and pressure p due to the DBI-essence field φ are given by the equations (24) and (25). Case I: γ = constant.7 Φ V f = f = f = f = f = f = Ε Η Fig.27 Fig.28Fig. 27 shows the variations of V against φ , for A = 0 . , B = 0 . , k = 1 , f = 0 . , f = 0 . , d = 5 and Fig. 28 shows thevariation of the slow roll parameters ǫ against η for A = 400 , B = 200 , d = 5 and k = − , , Φ V f = .02, f = f = .007, f = .09 f = .001, f =
130 10 20 30 40510152025303540 Ε Η Fig.29 Fig.30Fig. 29 shows the variations of V against φ , for A = 1 , B = 0 . , k = 1 , f = 0 . , f = 0 . , d = 5 and Fig. 30 shows thevariation of the slow roll parameters ǫ against η for A = . , B = . , d = 5 and k = − , , Using equations (55)-(57), we can find the expressions for V ( φ ), T ( φ ) and φ as V ( φ ) = δφ e (2 AF t f +2 BGt f ) (64) T ( φ ) = σφ e (2 AF t f +2 BGt f ) (65)and φ = Z φ e ( AF t f + BGt f ) dt (66)where E = γ [2( δ − σ ) + 2( σ − γ ], γ = q σσ − , F = − E , G = − dE and φ is an integrating constants. Fromabove, we see that V ( φ ) and T ( φ ) are both exponentially decreasing with DBI scalar field φ .8 Φ V Φ T Fig.31 Fig.32 Ε Η Fig.33Fig. 31 shows the variations of V against φ and Fig. 32 shows the variations of T against φ , for A = 1 , B = 1 , f = 0 . , f = 0 . , σ = 9 , δ = 10 , d = 5 , φ = 2 and Fig. 33 shows the variation of the slow roll parameters ǫ against η for A = 1 , B = 2 , σ = 5 , δ = 2 , f = 0 . , f = 0 . , φ = 2 , d = 5 in the 1 st case of DBI-essence Scalar field forIntermediate Scenario. Case II: γ = constant.Again using equations (55)-(57), we can find the expressions for V ( φ ) and φ as V ( φ ) = ln [ C e (3 At f + dBt f ) ] s − C e (3 Atf dBtf ] (67)and φ = Z − C e (3 Atf dBtf ] ! dt (68)where C is an integrating constant.9 Φ V Ε Η Fig.34 Fig.35Fig. 34 shows the variations of V against φ , for A = 1 , B = 1 , f = 0 . , f = 0 . , C = 1000 , d = 5 and Fig. 35 showsthe variation of the slow roll parameters ǫ against η for A = 1 , B = 1 , f = 0 . , f = 0 . , C = 3 , d = 15 in the 2 nd case of DBI-essence Scalar field for Intermediate Scenario. r s Fig.36Fig. 36 shows the variations of r against s , for A = 0 . , B = 40 , f = 0 . , f = 0 . , d = 5in Intermediate Scenario. • Statefinder parameters:
The geometrical parameters { r, s } for higher dimensional anisotropic cosmology in Intermediate scenario canbe constructed from the scale factors a ( t ) and b ( t ) as r = 1+ 3( d + 3)(3 A ( f − f t f + Bdf ( f − t f )(3 Af t f + Bdf t f ) + (3 + d ) (3 Af (2 + f ( f − t f + Bdf (2 + f ( f − t f )(3 Af t f + Bdf t f ) (69) s = − [( d + 3) (cid:0) Af t f + Bdf t f )(3 A ( f − f t f + Bdf ( f − t f ) + (3 + d ) (cid:0) Af (2 + f ( f − t f [3(3 Af t f + Bdf t f ) ( + ( d +3)(3 A ( f − f t f + Bdf ( f − t f )(3 Af t f + Bdf t f ) )]+ Bdf (2 + f ( f − t f (cid:1)(cid:1)(cid:3) (70)0 VI.
DISCUSSIONS
In this work, we have considered N (= 4 + d )-dimensional Einstein’s field equations in which 4-dimensionalspace-time is described by a FRW metric and that of the extra d -dimensions by an Euclidean metric. We haveconsidered three scenarios, namely, Emergent, Intermediate and Logamediate scenarios where the universe isfilled with K-essence, Tachyonic, Normal Scalar Field and DBI-essence types dark energy models. The naturesof the potentials as well as dynamics of scalar fields for the dark energy models have been analyzed. Thestatefinder and slow-roll parameters have been considered and their natures have been investigated for all darkenergy models due to three scenarios of the universe.In the case of Emergent scenario, we have considered a particular forms of scale factors a and b in such a waythat there is no singularity for evolution of the anisotropic Universe. We have found φ and potential V in termsof cosmic time t for K-essence, Tachyonic, Normal Scalar Field and DBI-essence models. Here we have shownthat the emergent scenario is possible for open, closed or flat Universe if the Universe contains K-essence,Tachyonic, Normal Scalar Field and DBI-essence field. From figures 1, 3, 5, 7, 8, 10 it has been seen that thepotential always increases with K-essence, Tachyonic and decreases with Normal Scalar Field and also withDBI-essence field when γ = constant and γ = constant and also the figures 2, 4, 6, 9, 11 shows the variationof slow-roll parameters ǫ and η in above dark energy model for open,closed and flat universe where they areincrease with all dark energy field except Normal scalar field where it increases 1st then decreases. The { r, s } diagram (fig.12) shows that the evolution of the emergent Universe starts from asymptotic Einstein’s static era( r → ∞ , s → −∞ ) and goes to ΛCDM model ( r = 1 , s = 0). It is also observed that r, s are independentof the dimension d . So, from statefinder parameters, the behavior of different stages of the evolution of theemergent Universe have been generated.In the case of Logamediate scenario, we have considered a particular forms of scale factors a and b in sucha way that there is no singularity for evolution of the anisotropic Universe. We have found φ and potential V in terms of cosmic time t for K-essence, Tachyonic, Normal Scalar Field and DBI-essence models. Here wehave shown that the logamediate scenario is possible for open, closed or flat Universe if the Universe containsK-essence, Tachyonic, Normal Scalar Field and DBI-essence field. From figures 13, 15, 17, 19, 20, 22 it hasbeen seen that the potential are increases with K-essence, Tachyonic and decreases with Normal Scalar Fieldand also with DBI-essence field when γ = constant and γ = constant and also the figures 14, 16, 18, 21, 23shows the variation of slow-roll parameters ǫ and η in above dark energy models for open, closed and flatuniverse where they are increasing with all dark energy field except DBI essence scalar field where it increases1st then decreases then again increases. The { r, s } diagram (fig.24) shows that the evolution of the Universestarts from asymptotic Einstein static era ( r → ∞ , s → −∞ ) and goes to ΛCDM model ( r = 1 , s = 0). It isalso observed that r, s are independent of the dimension d . So, from statefinder parameters, the behavior ofdifferent stages of the evolution of the Logamediate Universe have been generated.In the case of Intermediate scenario, we have considered a particular forms of scale factors a and b in sucha way that there is no singularity for evolution of the anisotropic Universe. We have found φ and potential V in terms of cosmic time t for K-essence, Tachyonic, Normal Scalar Field and DBI-essence models. Here wehave shown that the intermediate scenario is possible for open, closed or flat Universe if the Universe containsK-essence, Tachyonic, Normal Scalar Field and DBI-essence field. From figures 25, 27, 29, 31, 32, 34 it has beenseen that the potential are decreases with K-essence, DBI-essence field when γ = constant and γ = constantand 1st increases then decreases with Normal Scalar Field and with Tachyonic and also the figures 26, 28, 30,33, 35 shows the variation of slow-roll parameters ǫ and η in above dark energy model for open,closed and flatuniverse where they are increasing with all dark energy field except DBI essence scalar field where it increases1st then decreases then again increases. The { r, s } diagram (fig.36) shows that the evolution of the Universestarts from asymptotic Einstein static era ( r → ∞ , s → −∞ ) and goes to ΛCDM model ( r = 1 , s = 0). It isalso observed that r, s are independent of the dimension d . So, from statefinder parameters, the behavior ofdifferent stages of the evolution of the Intermediate Universe have been generated. References: [1] A.G.Riess et al,
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