February 2, 2008 16:1 WSPC/INSTRUCTION FILE power-law International JournalofModernPhysicsD (cid:13)c WorldScientificPublishingCompany 8 0 0 2 Power-law cosmological solution derived from DGP brane with a brane n tachyon field a J 1 YongliPing∗,LixinXu,HongyaLiu† ] School of Physics and Optoelectronic Technology, h Dalian Universityof Technology, Dalian, Liaoning 116024, P.R.China. t - p YingShao e h Department of physics, [ Dalian Maritime University, Dalian, Liaoning 116024, P.R.China. 1 v Bystudyingatachyon field onthe DGP branemodel, inorder to embedthe 4D stan- 8 dard Friedmann equation with a brane tachyon field in5D bulk, the metric of the 5D 6 spacetimeispresented.Then,adoptingtheinversesquarepotentialoftachyonfield,we 2 obtainanexpandinguniversewithpower-lawonthebraneandanexact5D solution. 0 . Keywords:Branecosmology,tachyon field. 1 0 8 1. Introduction 0 v: An increasing number of people believe that extra dimensions can be probed by i gravitonsandeventually non-standardmatter.These models usually yieldthe cor- X rect Newtonian (1/r)-potential at large distances since the gravitational field is r a quenchedonsub-millimetertransversescales.Thisquenchingappearseitherdueto 1–6 finite extension of the transverse dimensions or sub-millimeter transverse cur- 7–12 vature scales derived by negative cosmological constants. A common feature to these models is that they predict deviations from the 4D Einstein gravity at short distances. Over large distances, a model which predicts deviations from the standard 4D gravity is proposedby Dvali, Gabadadze and Porrati(DGP).13,14,15 There are the brane and bulk Einstein terms in the action of DGP model. It was shown that the DGP model allows for an embedding of the standard Friedmann cosmology in the sense that the cosmological evolution of the background metric on the brane can entirely be described by the standard Friedmann equation plus energy conservation on the brane.16,17 Recently, K. Atazadeh and H. R. Sepangi studythe DGPbranewithascalarfieldandproposecurvaturecorrectionsinDGP brane cosmology.18,19 A comprehensive review on DGP cosmology is dished up in Ref. 20. ∗[email protected] †[email protected] 1 February 2, 2008 16:1 WSPC/INSTRUCTION FILE power-law 2 Yongli Ping, LixinXu, Hongya Liu and Ying Shao The observable universe is presently undergoing an accelerating expansion bas- ing on the observations of Type Ia supernovae.21,22,23 It is possible that such an accelerated expansion could be the result of a modification to the Einstein-Hilbert 24–29 action. Recently,it hasbeenproposedthatacceleratedexpansionofuniverse 30–37 is driven by a tachyon field. This field is derived from string-brane physics and describes the lowest energy level of an unstable Dp-brane or that of a brane- 38 antibrane system. And the notable characteristic of tachyon field is that the 39–44 tachyonfieldhasanegativesquaredmass. Moreover,thereareseveralpapers 45–49 about the brane-worldmodel with a tachyon field. In Ref. 18, DGP brane cosmology with a brane scalar field is introduced. Now, we use a tachyonfield take the place of the scalarfield on the brane.In this paper, it is found that standard Friedmann equation with a brane tachyon field embeds in 5D bulk; then, by adopting a common potential of tachyon field, we obtain the metric which satisfies a power-law expansion on the brane. At the same time, an exact 5D solution is derived. 2. DGP model with a brane tachyon field 18 The action for the DGP model with a scalar field on the brane is written as M3 M2 1 S = 5 d5x√ g + d4x√ q plR ( φ)2 V(φ) +S [q ,ψ ],(1) m µν m 2 − R − " 2 − 2 ∇ − # Z Z where the first term in (1) describes the Einstein-Hilbert action in 5D bulk for a five-dimensional metric g with Ricci scalar . And the second term is the AB R Einstein-Hilbert action for the induced metric q on the brane with a scalar field µν φ and R is the scalar field of brane. M is the Planck mass in five dimensions and 5 M is the induced 4D Planck mass. The last term S is the matter action on the pl m brane with matter field ψ. The metric q is induced fromthe bulk metric g via µν AB q =δAδBg . (2) µν µ ν AB Nowassumingatachyonfieldinsteadofthescalarfieldonthebrane,theaction is rewritten as M3 M2 S = 5 d5x√ g + d4x√ q plR V(T) 1+∂ T∂µT +S [q ,ψ ].(3) µ m µν m 2 − R − " 2 − # Z Z p From the action (3), the Einstein equations are derived as 1 1 M3 g +M2δµδν R q R δ(y)=δµδν (T + )δ(y),(4) 5 RAB − 2 ABR pl A B µν − 2 µν A B µν Tµν (cid:18) (cid:19) (cid:18) (cid:19) here,T and aretheenergy-momentumtensorinthetachyonfieldandmatter µν µν T 16 field respectively. The corresponding junction conditions become 1 1 lim [K ]+ = T + q qαβ(T + ) ǫ→+0 µν − M53 (cid:18) µν Tµν − 3 µν αβ Tαβ (cid:19)(cid:12)y=0 (cid:12) (cid:12) February 2, 2008 16:1 WSPC/INSTRUCTION FILE power-law Power-law cosmological solution derived from DGPbrane with a brane tachyon field 3 M2 1 pl R q qαβR . (5) − M53 (cid:18) µν − 6 µν αβ(cid:19)(cid:12)y=0 From the Lagrangianof the tachyon field, the density(cid:12)and pressure are given (cid:12) V(T) ρ = , (6) T 1 T˙2 − p = pV(T) 1 T˙2. (7) T − − Upon variation of the action, the equationpof the motion for the tachyon field can be written V(T)T¨ a˙ dV(T) +3 V(T)T˙ + =0. (8) 1 T˙2 a dT − The form of the line element is written as in the brane gravity ds2 = n2(y,t)dt2+a2(y,t)γ dxidxj +b2(y,t)dy2, (9) ij − here, γ is a maximally symmetric 3D metric where k = 0, 1 parameterizes the ij ± spatial curvature. Now, we will follow the work in Ref. 16, adopting the Gaussian normal system gauge b2(y,t)=1, the Einstein tensors in the bulk are a˙2 a2 k a G =3n2 ′ + 3n2 ′′, (10) 00 n2a2 − a2 a2 − a (cid:18) (cid:19) a2 a˙2 k ′ G = g ij a2 − n2a2 − a2 ij (cid:18) (cid:19) a na a¨ n˙a˙ n ′′ ′ ′ ′′ +2 + + g + g , (11) a na − n2a n3a ij n ij (cid:18) (cid:19) na˙ a˙ ′ ′ G =3 , (12) 05 na − a (cid:18) (cid:19) a2 a˙2 k na n˙a˙ a¨ ′ ′ ′ G =3 +3 + . (13) 55 a2 − n2a2 − a2 na n3a − n2a (cid:18) (cid:19) (cid:18) (cid:19) The matching condition (5) for the perfect fluid on the brane =ρ, = = =p (14) 00 11 22 33 T T T T read n lim [n]+ = (2(ρ+ρ )+3(p+p )) ǫ→+0 ′ − 3M53 T T (cid:12)y=0 M2 a¨ a˙2 n˙a˙ (cid:12) k + pl2n (cid:12) , (15) M53 (cid:18)n2a − 2n2a2 − n3a − 2a2(cid:19)(cid:12)y=0 M2 a˙2 k (ρ+ρ )a (cid:12) lim [a]+ = pl + T (cid:12). (16) ǫ→+0 ′ − M53 (cid:18)n2a a(cid:19)(cid:12)y=0− 3M53 (cid:12)y=0 Since T + = 0 from (12), we get n/(cid:12)n = a˙ /a˙. Then,(cid:12)in order to simplify, 05 T05 ′ (cid:12) ′ (cid:12) adopting the gauge n(0,t)=1, it is obtained a˙(y,t) n(y,t)= . (17) a˙(0,t) February 2, 2008 16:1 WSPC/INSTRUCTION FILE power-law 4 Yongli Ping, LixinXu, Hongya Liu and Ying Shao Therefore, from (10), (13) and (16), we have M2 a˙2(0,t) k (ρ+ρ )a(0,t) lim[a]+(t)= pl + T , (18) ǫ→0 ′ − M53 (cid:20) a(0,t) a(0,t)(cid:21)(cid:12)y=0− 3M53 (cid:12)y=0 I+ = a˙2(0,t) a′2(y,t)+k a(cid:12)(cid:12)(cid:12)2(y,t) , (cid:12)(cid:12) (19) − y>0 (cid:12) I = (cid:2)a˙2(0,t) a2(y,t)+k(cid:3)a2(y,t)(cid:12) . (20) − ′ (cid:12) − y<0 (cid:12) By taking I+ = I ,(cid:2)the embedding of sta(cid:3)ndard F(cid:12)riedmann cosmology with a − (cid:12) tachyon field is given as follows: a˙2(0,t)+k ρ+ρ T = , (21) a2(0,t) 3M2 pl I = a˙2(0,t) a2(y,t)+k a2(y,t). (22) ′ − From (17) and (22), we can(cid:2)obtain the components(cid:3)of metric for I =0, that is a(y,t)= a(0,t)+ a˙2(0,t)+ky, (23) a¨(0,t) n(y,t)= 1+ p y. (24) a˙2(0,t)+k Therefore,givena specific a(0,t),the epxactsolutionof 5D metric is determinedby equations (23,24). Then, it is presented the mathematic configurationof 5D brane world.Thismetricissimilartotheoneinspace-time-mattertheory50,51 andbrane 52 53 model . There are some link between them. 3. The power-law cosmological solution for a given V(φ) From (8) and (21) for the spatially flat FRW metric, we have 1 3H2 = (ρ+ρ ), (25) 0 M2 T pl and T¨ 1 dV(T) +3H T˙ + =0 (26) 1 T˙2 0 V(T) dT − where H = a˙(0,t)/a(0,t). Then, we assume the tachyon field dominates the uni- 0 verse,i.e.ρ=0.Consideringthetachyonfield,thepotentialofthisfieldisarbitrary. But most adopt the inverse square potential. We utilize this potential as 2 V =2nM2(1 )1/2T 2 (27) T pl − 3n − withn 2/3,whichisshowninRef.32inordertoobtainanacceleratedexpansion ≥ oftheuniversedrivenbytachyonicmatter.Substitutingthispotentialinto(25)and (26) with M2 =1, in the brane the evolution of the tachyon field is pl 2 1/2 T t, (28) ∝ 3n (cid:18) (cid:19) February 2, 2008 16:1 WSPC/INSTRUCTION FILE power-law Power-law cosmological solution derived from DGPbrane with a brane tachyon field 5 and the evolution of the scale factor is obtained a(0,t) tn. (29) ∝ Therefore, the deceleration parameter q is rewritten as 1 q = 1+ . (30) − n The decelerationparameter q is plotted with n. From the Fig.1, we find that when q 2 1.5 1 0.5 n 0.5 1 1.5 2 2.5 3 -0.5 -1 Fig.1. Evolutionofthedeceleration parameterq vs.n 2/3 n < 1, this predicts deceleration on the brane; when n = 1, there is an ≤ uniform speed expansion on the brane; and when n>1, an accelerated universe is obtained on the brane. Substituting (29) into (23) and (24),for k =0 we derive a(y,t)=C tn+ntn 1y (31) − and (cid:2) (cid:3) y n(y,t)=C 1+(n 1) , (32) − t h i whereC isaconstant.Wheny =0,Eq.(31)and(32)returnto(29)andn(0,t)=1. Thereappearscoordinatesingularitiesonthespace-likehyperconey = t/(n 1). ± − This is presumably a consequence of the fact that the orthogonalgeodesics emerg- ing from the brane (which we used to set up b2 = 1) do not cover the full five- dimensional spacetime. Therefore, the exact solution is y ds2 = [1+(n 1) ]dt2+[tn+ntn 1y]γ dxidxj +dy2. (33) − ij − − t This solution can lead to arbitrarily velocity expansion with n on the brane. For getting the uniform speed expansion universe n=1, this solution is rewritten as ds2 = dt2+[t+y]γ dxidxj +dy2, (34) ij − which is a critical situation. February 2, 2008 16:1 WSPC/INSTRUCTION FILE power-law 6 Yongli Ping, LixinXu, Hongya Liu and Ying Shao To simplify, we choose a more simple form of potential as VT =M2T−2. (35) Substituting this potential into (25) and (26), the evolutionof the tachyonfield on the brane is 1/2 4 T t. (36) ∝ 2+√9M2+4 (cid:18) (cid:19) Therefore, when M2 =1, we obtain the tachyon field is 4 1/2 T t, (37) ∝ 2+√13 (cid:18) (cid:19) and the evolution of the scale factor is a(0,t) t(1/3+√13/6). (38) ∝ Substituting (29) into (23) and (24),for k =0 we derive 1 √13 a(y,t)=C t(1/3+√13/6)+( + )yt(√13/6−2/3) (39) 3 6 " # and √13 2 y n(y,t)=C 1+( ) , (40) " 6 − 3 t# whereC isaconstant.However,whena(0,t) t(1/3+√13/6),(1/3+√13/6) 0.93< ∝ ≈ 1.So,thedecelerationparameterq >0.Therefore,wegetanexact5Dcosmological solution which leadS to decelerated expansion on the brane. 4. Conclusion In this paper, we consider the DGP model with a brane tachyon field. In order to make the standard Friedmann cosmology with the tachyon field embed in the 5D bulk, the metric of 5D spacetime is derived. In this form of the metric, we find if the scalefactora isgiventhe exactsolutionofthe 5D bulk willbeobtained.Then, it is known that if the potential of the tachyon field is given, the scale factor a will be described. For a general form of the inverse square potential, an arbitrarily velocity expanding universe with power-law is obtained on the brane. Meanwhile the correspond 5D solution is derived. The velocity of expansion is related to n. If 2/3 n < 1, the universe is decelerating expansion; if n = 1, the universe ≤ is uniform speed expansion; and if n > 1, the accelerated expansion universe is obtained. For example, giving specific potential V = M2T 2, when M2 = 1, we T − have the decelerated expansion universe and the exact solution of 5D bulk. 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