Brane induced cosmological acceleration and crossing of w_{eff}=-1
aa r X i v : . [ h e p - t h ] D ec December, 2014
Brane induced cosmological acceleration andcrossing of w ef f = − Nobuyuki MOTOYUI ∗ Department of Physics, Faculty of Sciences,Ibaraki University, Bunkyo 2-1-1, Mito, 310-8512, Japan
Abstract
The cosmological observation indicates that the effective equation of state pa-rameter w eff varies with z : It changes from w eff > − w eff < − z ∼ . w eff are quite different between the two cases. Keywords: Brane universe; dark energy; effective equation of state parameter.PACS numbers: 95.36.+x, 11.25.Mj. ∗ [email protected] Introduction
The cosmological observations indicate that our universe have not only experienced thelarge amount of accelerating expansion in its infancy but it is still undergoing a smallrate of accelerating expansion [1, 2]. The present cosmological data also indicates thatmore than 70% of the total energy density is attributed to a component of dark energydensity [3, 4]. The source of driving force to the present accelerating expansion is thoughtto be dark energy. Although the physical origin of dark energy is still unknown, manycosmological candidates such as the cosmological constant or several kinds exotic matterlike phantom fields [5, 6], quintessence [7–9], or modifications of gravitational theory havebeen proposed. If we define the ratio of the pressure of the universe P to its energydensity ρ as w ≡ P/ρ , then the cosmological constant is characterized by w = − w < −
1. If the cosmological constant is identifiedas dark energy, w is a constant. On the other hand, w varies with time if we adoptthe assumptions of exotic matter. Although the simplest candidate of dark energy is thecosmological constant Λ, there is a well known ’fine tuning problem’ between the energydensity of cosmological constant ρ Λ and the radiation density ρ r : The ratio of ρ Λ and ρ r depends on the energy scale and ρ Λ /ρ r ∼ − at the Planck scale, on the other hand ρ Λ /ρ r ∼ − at the electroweak scale because ρ Λ ≃ − GeV is a constant.However the effective equation of state parameter is not necessarily a constant. Alter-native ways for dark energy model in which either dark energy or its effective equationof state parameter is a function of time. The cosmological data indicates that the timevarying dark energy model is a better fitting than a cosmological constant: the effectiveequation of state parameter of dark energy w eff ∼ − .
21 at z = 0 and it changes from w eff > − w eff < − z ∼ . w eff behaves as phantom-likeat lower redshifts z . . w eff < − w eff crosses w eff = − w eff of dark energy. In section 4, weexamine the evolutions of w eff and investigate under which condition it exhibits crossingof w eff = − z ∼ . w eff ∼ − .
21 at z = 0 in two cases: (i) there is energyexchange between the four dimensional universe and the fifth dimension, and (ii) there isno energy exchange between the four dimensional universe and the fifth dimension. Wealso examine the evolutions of deceleration parameter and energy density components.Section 5 is devoted to summary. We consider the five-dimensional action of the following form, S = Z d x p − g (5) " κ (cid:0) R (5) − (5) (cid:1) + L ( m ) B + Z d x √− g b (cid:16) − T b + L ( m ) b (cid:17) δ ( y ) . (1)We denote the coordinate of fifth dimension by y and it takes the values −∞ < y < ∞ .We consider a 3-brane is located at y = 0 and assume a Z symmetry for y around y = 0. In the above expression, R (5) is the five-dimensional Ricci scalar, Λ (5) is the five-dimensional cosmological constant and T b is the tension of the 3-brane. We can includematter content in the bulk L ( m ) B or on the brane L ( m ) b . We denote the five-dimensionalmetric as g (5) AB , the four dimensional metric as g µν and the four dimensional metric onthe brane as g b . We define the signature of g (5) AB as ( − , + , + , + , +) and that of g µν as( − , + , + , +). The line element is described as ds = g (5) AB dx A dx B = g µν dx µ dx ν + b ( t, y ) dy . (2)The capital Latin indices indicate (0 , · · · ,
4) and the Greek indices indicate (0 , · · · , κ (5) is related to the five-dimensional Newton constant G (5) and the five-dimensional Planck mass M (5) by the relation κ = 8 πG (5) = M − . We assume that thefive-dimensional metric is described as follows, ds = − n ( t, y ) dt + a ( t, y ) γ ij dx i dx j + b ( t, y ) dy , (3)where γ ij is a maximally symmetric FLRW metric. Its spatial curvature is parametrizedby K which takes the values K = − , ,
1. 3he energy-momentum tensor T AB is given as T AB = T ( B ) AB + T ( b ) AB − T b √− g b √− g (5) g µν δ µA δ νB δ ( y ) , (4)where T ( B ) AB is the component which results from L ( m ) B and T ( b ) AB is the componentwhich results from L ( m ) b . We assume the bulk energy-momentum tensor T ( B ) AB as T ( B ) AB = − ρ B Q P B δ ij − n b Q P T . (5)The brane energy-momentum tensor T ( b ) AB is generally expressed as T ( b ) AB = δ ( y ) b − ρ b P b δ ij
00 0 0 . (6)In the above expression, Q is responsible for the energy exchange between the four dimen-sional spacetime and the extra dimension. We generally allow the anisotropic choice ofthe bulk pressure: P B = P T . The dynamics of the five-dimensional universe is governedby the five-dimensional Einstein equations. They take the usual form, G (5) AB ≡ R (5) AB − g (5) AB R (5) = − Λ (5) g AB + κ T AB . (7)Substituting (3), (5) and (6) into (7), we obtain the following equations,3 ( ˙ aa ˙ aa + ˙ bb ! − n b (cid:18) a ′′ a + a ′ a (cid:18) a ′ a − b ′ b (cid:19)(cid:19) + K n a ) = n Λ (5) + κ ρ B + κ b ( ρ b + T b ) δ ( y ) ! , (8) a b (cid:26) a ′ a (cid:18) a ′ a + 2 n ′ n (cid:19) − b ′ b (cid:18) n ′ n + 2 a ′ b (cid:19) + 2 a ′′ a + n ′′ n (cid:27) + a n ( ˙ aa (cid:18) − ˙ aa + 2 ˙ nn (cid:19) − aa + 2 ˙ bb (cid:18) − ˙ aa + ˙ nn (cid:19) − ¨ bb ) + K = a − Λ (5) + κ P B + κ b ( P b − T b ) δ ( y ) ! , (9)3 n ′ n ˙ aa + a ′ a ˙ bb − ˙ a ′ a ! = − n κ Q, (10)3 (cid:26) a ′ a (cid:18) a ′ a + n ′ n (cid:19) − b n (cid:18) ˙ aa (cid:18) ˙ aa − ˙ nn (cid:19) + ¨ aa (cid:19) − K b a (cid:27) = b (cid:0) − Λ (5) + κ P T (cid:1) , (11)where dots stand for differentiations with respect to t and primes stand for differentiationswith respect to y .We make two assumptions to simplify these equations:4i) the scale factor of extra dimension is normalized to b ( t, y ) = 1,(ii) the scale factor of temporal dimension at y = 0 is normalized to n ( t,
0) = 1.After these simplifications, Einstein equations on the brane are expressed as follows,3 ((cid:18) ˙ a b a b (cid:19) − a ′′ b a b − (cid:18) a ′ b a b (cid:19) + K a b ) = Λ (5) + κ ρ B + κ ( ρ b + T b ) δ ( y ) , (12) (cid:26) a ′ b a b (cid:18) a ′ b a b + 2 n ′ b (cid:19) + 2 a ′′ b a b + n ′′ b (cid:27) − (cid:18) ˙ a b a b (cid:19) − a b a b − K a b = − Λ (5) + κ P B + κ ( P b − T b ) δ ( y ) , (13)3 (cid:18) n ′ b ˙ a b a b − ˙ a ′ b a b (cid:19) = − κ Q, (14)3 ( a ′ b a b (cid:18) a ′ b a b + n ′ b (cid:19) − (cid:18) ˙ a b a b (cid:19) − ¨ a b a b − K a b ) = − Λ (5) + κ P T , (15)where the subscripts b denotes that these functions are evaluated at y = 0.We have to take into account of the junction conditions [14] when we solve the Einsteinequations on the brane. Because the first derivatives of the metric with respect to y canbe discontinuous at y = 0 while the metric is required to be continuous across the brane,the delta functions appear in the second derivatives of the metric. We define the jumpfactor ‡ f ‡ and the mean value ♯f ♯ of a function f by the following equations, ‡ f ‡ ≡ f ( t, − f ( t, − ) , (16) ♯f ♯ ≡ f ( t, f ( t, − )2 . (17)We obtain the junction conditions as follows, ‡ a ′ ‡ a b = − κ ρ b + T b ) b b , (18) ‡ n ′ ‡ n b = κ ρ b + 3 P b − T b ) b b . (19)Taking into account the Z symmetry around y = 0, we obtain a ′ ( t, +0) a b = − κ ρ b + T b ) , (20) n ′ ( t, +0) = κ ρ b + 3 P b − T b ) . (21)Taking the jump of (15), we also obtain a relation between mean values ♯a ′ ♯ and ♯n ′ ♯ , ♯a ′ ♯a b ( P b − T b ) = 13 ( ρ b + T b ) ♯n ′ ♯n b . (22)5ubstituting (20) and (21) in (15), we obtain the effective Friedmann equation in the limit y → +0, ¨ a b a b + (cid:18) ˙ a b a b (cid:19) + Ka b = − κ
36 (1 + 3 w b ) ρ b + κ
36 (1 − w b ) T b ρ b + 13 Λ (5) + κ T b ! − κ P T . (23)In the above expression, we used the equation of state for the brane: P b = w b ρ b . Notethat the 55-component of the bulk energy-momentum tensor P T and the quadratic termof the brane energy density ρ b appear on the right hand side of this equation. They willaffect the cosmological evolution of the 3-brane.The equations of Energy-momentum conservation are, ∇ A T AB = ∂ A T AB + Γ ADA T DB − Γ DBA T AD = 0 . (24)The 0-component and the 5-component of the above equations are˙ ρ B + n b Q ′ + 3( P B + ρ B ) ˙ aa + 3 Q n b (cid:18) n ′ n + a ′ a (cid:19) − Q n b ′ b +( P T + ρ B ) ˙ bb + δ ( y ) ( ˙ ρ b + 3( P b + ρ b ) ˙ aa + ρ b ˙ bb ) = 0 , (25)˙ Q + P T ′ + Q ˙ nn + 3 ˙ aa + ˙ bb ! + ( P T + ρ B ) n ′ n + 3( P T − P B ) a ′ a = 0 . (26)Integrating (25) around y = 0 and using the Z symmetry, we obtain the conservation ofenergy-momentum on the brane,˙ ρ b + 3(1 + w b ) ρ b ˙ a b a b + 2 Q ( t ) = 0 , (27)where we used two assumptions b ( t, y ) = 1 and n ( t,
0) = 1. Using (20) and (21), weobtain the conservation of energy-momentum in the limit y → +0,˙ ρ B + Q ′ + 3( P B + ρ B ) ˙ a b a b + κ Q { (1 + 3 w b ) ρ b − T b } = 0 , (28)˙ Q + P T ′ + 3 Q ˙ a b a b + κ P T + ρ B ) { (2 + 3 w b ) ρ b − T b } − κ P T − P B )( ρ b + T b ) = 0 , (29)where we used the two assumptions for the scale factors of extra dimension and thetemporal dimension. We obtained an effective Friedmann equation on the 3-brane, theevolution equations for ρ b , ρ B and Q .The time evolution of the bulk energy density ρ B and the flow of energy from/to theextra dimension Q are governed by (28) and (29). The bulk pressure P T contributes6o the effective Friedmann equation on a brane through (23). However we cannot fullydetermine Q and P T since there are unknown functions Q ′ and P ′ T in (28) and (29). Weput two ans¨atze for Q and P T as follows, Q = F (cid:18) ˙ a b ( t ) a b ( t ) (cid:19) a b ( t ) µ , P T = Da b ( t ) ν , (30)where D, F, µ and ν are some constants. A justification of these ans¨atze is found in [17]. To transform the second order equation (23) to the first order equation, we introduce anew variable χ ( t ) which is called dark energy variable [17], χ ≡ (cid:18) ˙ a b a b (cid:19) + Ka b − γρ b − βρ b − λ κ P T , (31)where we redefined some constants β ≡ κ , γ ≡ κ T b , λ ≡ Λ (5) + κ T b ! . (32)Then (23) is rewritten as a pair of the first order equations, (cid:18) ˙ a b a b (cid:19) + Ka b = 2 γρ b + βρ b + χ + λ − κ P T , (33)˙ χ = − a b a b χ + 4 Q ( γ + βρ b ) + κ P T . (34)The equation (33) is analogous to the Hubble equation of standard four dimensionalcosmology. However the quadratic term of brane energy density ρ b , the bulk pressure andthe brane tension appear in the right hand side. The dark energy variable χ ( t ) accountsfor the non-standard contributions to the Friedmann equation (33). The evolution of ρ b is determined by (27) and it is solved as ρ b = − (cid:18) a b (0) a b ( t ) (cid:19) w b ) Z Q ( t ) (cid:18) a b ( t ) a b (0) (cid:19) w b ) dt. (35)We treat w b as a constant in the above expressions. Now we perform the integration of(34). With the ans¨atze (30), ρ b is expressed as, ρ b = e C a b ( t ) − w b ) − F w b ) + µ a b ( t ) µ , (36)7here e C is some integration constant. With the ans¨atze (30), we can perform the inte-gration of (34) and we obtain χ = 4 F γ µ a µb + 4 F β e C µ − w b + 1 a µ − w b ) b − F β (3(1 + w b ) + µ )(2 + µ ) a µb + κ Dν ν ) a νb + C a b , (37)where C is some integration constant. Substituting (36) and (37) into (33), we obtain theHubble equation on the brane, (cid:18) ˙ a b a b (cid:19) = λ − Ka b + C a b − κ D ν ) a νb + 2 γ e C a w b ) b + β e C a w b ) b − F γ (1 − w b )(4 + µ )(3(1 + w b ) + µ ) a µb − F β (1 + 3 w b )(3(1 + w b ) + µ ) (2 + µ ) a µb + 8 F β e C (1 + 3 w b )(1 − w b + µ )(3(1 + w b ) + µ ) a µ − w b ) b . (38)This equation is quite different from the Hubble equation of a standard four dimensionalFLRW cosmology: • The brane energy density appears in a linear and quadratic form, whereas it appearsin a linear form in the standard four dimensional FLRW cosmology. • There is a bulk pressure term Da νb and an energy exchange term F a µb between thefour dimensional spacetime and the extra dimension. • There is a bulk radiation term C /a b .We can write the Hubble equation (38) in the conventional form, (cid:18) ˙ a b a b (cid:19) = − Ka b + Λ (4) πG (4) ρ eff , (39) ρ eff ≡ ρ b + β γ ρ b + χ γ − κ γ P T , (40)where ρ eff is the effective energy density. The four dimensional Newton constant G (4) and the four dimensional cosmological constant Λ (4) are defined as G (4) ≡ γ π = 4 πG T b , (41)Λ (4) ≡ λ = 12 Λ (5) + κ T b ! . (42)Deceleration parameter q is defined by q ≡ − (cid:18) a b ˙ a b (cid:19) (cid:18) ¨ a b a b (cid:19) , (43)8here the acceleration behavior is described by¨ a b a b = λ − C a b + (2 + ν ) κ D ν ) a νb − (1 + 3 w b ) γ e C a w b ) b − (2 + 3 w b ) β e C a w b ) b + 2 F γ ( − − µ + 6 w b + 3 w b µ )(4 + µ )(3(1 + w b ) + µ ) a µb − F β (1 + 3 w b )(1 + µ )(3(1 + w b ) + µ ) (2 + µ ) a µb − F e C β (1 + 3 w b )(1 + 3 w b − µ )(3(1 + w b ) + µ )(1 − w b + µ ) a µ − w b ) b . (44)We assume that K = 0, Λ (4) = 0 and the matter on the brane is all ordinary non-relativistic matter which is taken to be w b = 0, and we obtain H ( t ) = (cid:18) ˙ a b a b (cid:19) = C a b − κ D ν ) a νb + 2 γ e C a b + β e C a b − F γ (4 + µ )(3 + µ ) a µb − F β (3 + µ ) (2 + µ ) a µb + 8 F β e C (1 + µ )(3 + µ ) a µ − b . (45)In the above equation, H ( t ) gives the expansion rate of the four dimensional universe.Within the flat universe, in the presence of dark energy, the expansion rate is given as H ( t ) H = Ω m a b + 1 − Ω m a w ) b , (46)where H ≡ H (0), Ω m is the dimensionless matter density and w is the parameter ofequation of state of the dark energy. The dimensionless dark energy density is given by1 − Ω m . The second term in the right hand side of (46) describes the contribution of thedark energy to the expansion rate of our universe. Following Linder et al. [19], we modifythis equation as δH H ≡ H ( t ) H − Ω m a b , (47) w eff ≡ − − d ln( δH ) d ln a b , (48)where δH /H accounts for any modification to the usual Hubble equation in the fourdimensional spacetime and w eff is the effective equation of state parameter. In our case, w eff is given as w eff ( z ) = − − − C ( z + 1) − κ νD ν ) ( z + 1) − ν − β e C ( z + 1) − γµF (4 + µ )(3 + µ ) ( z + 1) − µ − βµF (3 + µ ) (2 + µ ) ( z + 1) − µ + 8 β ( µ − e C F (1 + µ )(3 + µ ) ( z + 1) − µ ! C ( z + 1) − κ D ν ) ( z + 1) − ν + β e C ( z + 1) − γF (4 + µ )(3 + µ ) ( z + 1) − µ − βF (3 + µ ) (2 + µ ) ( z + 1) − µ + 8 β e C F (1 + µ )(3 + µ ) ( z + 1) − µ ! , (49)where we use the redshift parameter z defined as z ( t ) + 1 ≡ a ( t ) /a ( t ). Decelerationparameter is given by q ( z ) = − −C ( z + 1) + (2 + ν ) κ D ν ) ( z + 1) − ν − γ e C ( z + 1) − β e C ( z + 1) − F γ (2 + µ )(4 + µ )(3 + µ ) ( z + 1) − µ − F β (1 + µ )(3 + µ ) (2 + µ ) ( z + 1) − µ − F e C β (1 − µ )(3 + µ )(1 + µ ) ( z + 1) − µ !, C ( z + 1) − κ D ν ) ( z + 1) − ν + 2 γ e C ( z + 1) + β e C ( z + 1) − F γ (4 + µ )(3 + µ ) ( z + 1) − µ − F β (3 + µ ) (2 + µ ) ( z + 1) − µ + 8 F β e C (1 + µ )(3 + µ ) ( z + 1) − µ ! . (50) w ef f and the crossing w ef f = − We examine the evolution of the effective equation of state parameter w eff and investigateunder which condition it exhibits w eff < −
1. The cosmological data indicates that w eff ∼ − .
21 at z = 0 and it changes from w eff > − w eff < − z ∼ . K = 0 and λ = 0. Each parameters must satisfy the following constraint which is obtainedfrom (45), C H − κ D ν ) H − F γ (4 + µ )(3 + µ ) H − F β (3 + µ ) (2 + µ ) H + 8 F β e C (1 + µ )(3 + µ ) H + 2 γ e C H + β e C H = 1 . (51)In the above expression, 2 γ e C /H corresponds to the dimensionless matter density and weassume that 2 γ e C /H <
1. We consider two cases:10i) there is energy exchange between the four dimensional universe and the fifth dimen-sion,(ii) there is no energy exchange between the four dimensional universe and the fifthdimension.
We consider that there is energy exchange between the four dimensional universe and thefifth dimension. We assume that the brane energy density is much smaller than the branetension: ρ b ≪ T b . In this case we can neglect the quadratic term in ρ b . We also assumethat the bulk radiation is negligible: C = 0. We still assume that the brane matter is allordinary matter ( w b = 0) and we use the ansatz (30). With these assumptions, we obtainthe Hubble equation (38) and the acceleration behavior (44) as follows, (cid:18) ˙ a b a b (cid:19) = − Aa νb − Ba µb + Ca b , (52)¨ a b a b = (2 + ν ) A a νb − (2 + µ ) B a µb − C a b , (53)where we used the notations A ≡ κ w B C B ν ) , B ≡ F γ (4 + µ )(3 + µ ) , C ≡ γ e C . (54)The parameter A corresponds to the contribution to Hubble parameter from the bulkmatter, B corresponds to the contribution from the the energy exchange between theextra dimension and C corresponds to the contribution from the ordinary matter on thebrane. The Hubble equation (52) is expressed as, − A − B + C = H . (55)The first two terms correspond to effective dark energy density.The deceleration parameter q , the effective equation of state parameter w eff and itspresent value are given by q = (2 + ν ) A z + 1) ν − (2 + µ ) B z + 1) µ − C z + 1) A ( z + 1) ν + B ( z + 1) µ − C ( z + 1) , (56) w eff = − − νA ( z + 1) ν + µB ( z + 1) µ A ( z + 1) ν + B ( z + 1) µ , (57) w eff (0) = − − (cid:18) νA + µBA + B (cid:19) . (58)11e look for the set of parameters which satisfy w eff < − < z < . w eff > − z < .
2. The denominators of (57) and (58) correspond to the opposite sign ofeffective dark energy density and it must be negative. We can achieve w eff < − Aν ( z + 1) − ν + Bµ ( z + 1) − µ < , (59) A ( z + 1) − ν + B ( z + 1) − µ < . (60)for z < .
2. We can achieve w eff (0) = − .
21 when A ( ν − .
63) + B ( µ − .
63) = 0 . (61)There is another constraint which comes from w eff (0 .
2) = − νµ = − BA (1 . ν − µ . (62)This equation determines the relative sign of µ and ν . The allowed choices of parametersare found in [18] and they are:(i) A, B < µ > , ν < A, B < µ < , ν > ν = − C = 0 .
04, we obtain A = − . B = − . µ = 1 .
82. Figure 1 show the behaviors of w eff ( z ) and q ( z ) in this case. When weassume ν = − C = 0 .
04, we obtain A = − . B = − .
39 and µ = 2 .
98. Figure2 show the behaviors of w eff ( z ) and q ( z ) in this case. In each cases, w eff ( z ) and q ( z )increase with z and q ( z ) become positive z ≃ . z ≃ . w eff ( z ) and q ( z ) with the parameters ( A, B, C, µ, ν ) = ( − , − , − , , −
2) isfound in [18].We consider the behavior of energy density components. We write the constraint (52)as Ω A ( z ) + Ω B ( z ) + Ω C ( z ) = 1 , (63)whereΩ A ( z ) ≡ − A ( z + 1) ν H ( z ) , Ω B ( z ) ≡ − B ( z + 1) µ H ( z ) , Ω C ( z ) ≡ C ( z + 1) H ( z ) . (64)Ω A corresponds to the contribution from the bulk matter, Ω B corresponds to the contri-bution from the energy exchange between the extra dimension and Ω C corresponds to thecontribution from the brane matter. The behavior of each energy density components areshown in figures 3 and 4. In figure 3, Ω B is dominant in small z and Ω A is dominant inlarge z . In figure 4, Ω A is dominant even in small z . The linear contributions from thebrane matter Ω C are smaller than the other part, but they increase with z . Ω C becomesubdominant at z ∼ . z ∼ . w e ff , q zw eff (z)q(z) Figure 1:
Graph of w eff ( z ) and q ( z ) withparameters A = − . B = − . C =0 . µ = 1 . ν = − -1.4-1.2-1-0.8-0.6-0.4-0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 w e ff , q zw eff (z)q(z) Figure 2:
Graph of w eff ( z ) and q ( z ) withparameters A = − . B = − . C =0 . µ = 2 . ν = − Ω A , Ω B , Ω C z Ω A Ω B Ω C Figure 3:
Graph of Ω A , Ω B , Ω C with pa-rameters A = − . B = − . C = 0 . µ = 1 . ν = − Ω A , Ω B , Ω C z Ω A Ω B Ω C Figure 4:
Graph of Ω A , Ω B , Ω C with pa-rameters A = − . B = − . C = 0 . µ = 2 . ν = − We assume that there is no energy flow to/from the extra dimension: T = T = 0. Itcorresponds to F = 0. We put F = 0 and P T = Da νb in (36) and (37), we obtain ρ b = e C a w b ) b , (65) χ = C a b + κ Dν ν ) a νb , (66)where C and e C are some constants. The equation for the bulk matter (28) is expressed as,˙ ρ B = − P B + ρ B ) ˙ a b a b , (67)13here we assumed P B = w B ρ B for the bulk matter. It is solved as ρ B = C B a w B ) b , (68)where C B is some integration constant. We obtain the relation P T = P B = w B C B a w B ) b , (69)when we assume that the pressure of bulk matter is isotropic. We read ν = − w B )and D = w B C B from (30). Assuming w b = 0, the Hubble equation and the accelerationbehavior is expressed as follows, (cid:18) ˙ a b a b (cid:19) = Aa b − Ba w B ) b + Ca b + Da b , (70)¨ a b a b = − Aa b − B (1 + 3 w B )2 a w B ) b − C a b − Da b . (71)where we used the notation A ≡ C , B ≡ κ w B C B − w B ) , C ≡ γ e C , D ≡ β e C . (72)Each parameters must satisfy the constraint from (70), A − B + C + D = H (73)where C corresponds to the energy density of ordinary matter. The deceleration parameter q , the effective equation of state parameter w eff and its present value are given by q = A (1 + z ) + (1 + 3 w B ) B z ) w B ) + C z ) + 2 D (1 + z ) !, A (1 + z ) − B (1 + z ) w B ) + C (1 + z ) + D (1 + z ) ! , (74) w eff = − ( A ( z + 1) − B (1 + w B )( z + 1) w B ) + 6 D ( z + 1) ),( A ( z + 1) − B ( z + 1) w B ) + D ( z + 1) ) , (75) w eff (0) = − (cid:18) A − B (1 + w B ) + 6 DA − B + D (cid:19) . (76)We look for the set of parameters which satisfy w eff < − < z < . w eff > − z < .
2. The denominators of (75) and (76) correspond to the effective14ark energy density and it must be positive. We can achieve w eff < − z < . A ( z + 1) − B (1 + w B )( z + 1) w B ) + 6 D ( z + 1) < , (77) A ( z + 1) − B ( z + 1) w B ) + D ( z + 1) > . (78)They gives the upper and the lower bound to achieve w eff (0) = − .
21. We can achieve w eff (0) = − .
21 when 4 . A − B (1 .
21 + w B ) + 6 . D = 0 . (79)There is another constraint which comes from w eff (0 .
2) = − A (1 . − B (1 + w B )(1 . w B ) + 6 D (1 . = 0 . (80)We look for the set of parameters which satisfy (79) and (80) under the condition of (73).When we assume w B = − / C = 0 .
04, we obtain A = − . B = − .
19 and D = 0 .
96. Figure 5 show the behaviors of w eff ( z ) and q ( z ) in this case. When we assume w B = − / C = 0 .
04, we obtain A = − . B = − .
83 and D = 0 .
52. Figure6 show the behaviors of w eff ( z ) and q ( z ) in this case. When we assume w B = − C = 0 .
04, we obtain A = − . B = − .
22 and D = 0 .
23. Figure 7 show the behaviorsof w eff ( z ) and q ( z ) in this case. In each cases, q ( z ) become positive z ≃ . z ≃ . z ≃ .
02 in figure 7.We consider the behavior of energy density components. We write the constraint (70)as Ω A ( z ) + Ω B ( z ) + Ω C ( z ) + Ω D ( z ) = 1 , (81)where Ω A ( z ) ≡ C ( z + 1) H ( z ) , Ω B ( z ) ≡ − κ w B C B ( z + 1) w B ) − w B ) H ( z ) , Ω C ( z ) ≡ γ e C (1 + z ) w b ) H ( z ) , Ω D ( z ) ≡ β e C (1 + z ) w b ) H ( z ) . (82)Ω A corresponds to the contribution from the bulk radiation, Ω B corresponds to the con-tribution from the bulk matter, Ω C corresponds to the linear contribution from the branematter and Ω D corresponds to the quadratic contribution from the brane matter. Thebehavior of each energy density components are shown in figures 8–10.There are negative contributions from the bulk radiation to the total energy density in eachcases. The contributions from the bulk matter are dominant in late time and the quadraticcontributions from the brane matter are dominant in early time. The linear contributionsfrom the brane matter are much smaller than the other contributions through the time.15 w e ff , q z w eff (z)q(z) Figure 5:
Graph of w eff ( z ) and q ( z ) withparameters w B = − / A = − . B = − . C = 0 . D = 0 . -1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 w e ff , q z w eff (z)q(z) Figure 6:
Graph of w eff ( z ) and q ( z ) withparameters w B = − / A = − . B = − . C = 0 . D = 0 . -1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 w e ff , q z w eff (z)q(z) Figure 7:
Graph of w eff ( z ) and q ( z ) with pa-rameters w B = − A = − . B = − . C = 0 . D = 0 . The cosmological observation indicates that the effective equation of state parameter w eff varies with z : w eff ∼ − .
21 at z = 0 and it crosses w eff = − z ∼ .
2. We investigatedthat under which condition this behavior occurs based on the five-dimensional braneworldscenario. The Hubble equation on the 3-brane is quite different from that of a standardfour dimensional FLRW cosmology: (i) The brane energy density appears in a linearand quadratic form. (ii) A bulk pressure term, a bulk radiation term, and an energyexchange term between the four dimensional spacetime and the extra dimension appearin the Hubble equation. They contribute to the effective equation of state parameter.We considered two cases: (i) There is energy exchange between the four dimensionaluniverse and the fifth dimension. (ii) There is no energy exchange between the fourdimensional universe and the fifth dimension. In both cases,we obtained that the crossingof w eff = − Ω A , Ω B , Ω C , Ω D z Ω A Ω B Ω C Ω D Figure 8:
Graph of Ω A , Ω B , Ω C , Ω D withparameters w B = − / A = − . B = − . C = 0 . D = 0 . -5-4-3-2-1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ω A , Ω B , Ω C , Ω D z Ω A Ω B Ω C Ω D Figure 9:
Graph of Ω A , Ω B , Ω C , Ω D withparameters w B = − / A = − . B = − . C = 0 . D = 0 . -2-1.5-1-0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ω A , Ω B , Ω C , Ω D z Ω A Ω B Ω C Ω D Figure 10:
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