A 3D Brans-Dicke Theory Model

Transcription

1 Interntionl Journl of Physics 06 Vol 4 No Avilble online t Science nd Eduction Publishing DOI:069/ijp A 3D Brns-Dicke Theory Model T G do Prdo * E F Reis M C Vergės V Piccirillo J R Cippin Federl University of Technology-Prná Deprtment of Mthemtics Pont Gross PR Brsil *Corresponding uthor: thigogilbertodoprdo@gmilcom Abstrct The evidence of n ccelerted universe nd the gp of 70 percent in the totl energy collected by WMAP re problems tht cnnot be solved by stndrd models bsed on Generl reltivity Therefore the serch for n lterntive theory tht solves the open questions in Generl Reltivity becme n importnt reserch field in the pst few yers A prticulr lterntive is the Brns-Dicke theory which hs been brodly studied s concerned to k- essence type fields in 4D However this theory is lmost unexplored in the context of the dimensionl reduction in 3D In the present work we study the Brns-Dicke Theory in dimensionl reduction context In order to do this we consider the Brns-Dicke theory in the vcuum nd in the presence of mtter fields Keywords: cosmology dimensionl reduction generl reltivity lower dimensionl grvittion Cite This Article: T G do Prdo E F Reis M C Vergės V Piccirillo nd J R Cippin A 3D Brns- Dicke Theory Model Interntionl Journl of Physics vol 4 no 3 (06): doi: 069/ijp Introduction The evidence of n ccelerted expnsion of the universe observed by WMAP [3] opened the discussion of the generl reltivity eventul limits There re mny options for lterntive theories of grvity nd mong them we cn cite sclr-tensor theories like supergrvity Kluz-Klein theories dul string theories M-Theory etc [454] One prticulr kind of sclr-tensor theory to describe n ccelerte expnsion of the universe clled Brns-Dicke theory [67] ws proposed in the erly sixties This theory uses the principle of Mch nd the hypothesis of Dirc [8] considering n eventul vrition in time of the Newton s grvittionl constnt thus ensuring the universlity of free fll (equivlence principle) [0 3] Most of the works which hve been published in this theory up to now consider four flt dimensions nd some of them hve tried to ssocite the sclr field of the Brns-Dicke theory s quintessence field [9] ΛC DM models [9] nd s type of K-essence field [08] Others hve tried to find solution for the observed ccelerted expnsion using dimensionl reduction of the 5D Brns-Dicke theory without mtter [] Concerning three dimensions brod study hs been done in grvittionl theories since the publiction of [3] motivted by the fct tht 3D theories void some complictions found in higher dimensions [56] However there re not s mny results bout 3D sclrtensor theories nd it would be interesting to find some results in this subject more specificlly in 3D Brns-Dicke theory For instnce we cn see some problems like the ssocition of the sclr field of the Brns-Dicke theory to K-essence fields which models the drk energy s done in [0] The gol in this pper is to construct nd then nlyze the behvior of 3D Brns- Dicke model For this we consider Brns-Dicke sclr field which depends only on the cosmic time The 3D Brns-Dicke Theory The ction of Brns-Dicke theory in 3D will be described by [5] 3 ω b S = 6 d x g R gb 3 + d x g M () where R is the sclr curvture ssocited to the 3D metric ( 3 ) g is the sclr FIeld of the Brns-Dicke theory ω b is theory dimensionless prmeter nd M represents the lgrngin of the mtter The mtter FIelds do not depend on the sclr field (this is necessry in order to preserve the wek equivlence principle [5]) Note tht in 3D the field hs dimensions of inverse length The equtions for the grvittionl field derived from () cn be written in the following form 8π ω 3 c Gb = Tb + ( ) b gb c () + ( b gb ) ( ) ( )( ) where = nd T b is the energy momentum tensor ssocited with the mtter fields for 3D From the eqution () it is possible to find tht the field eqution for the sclr field is given by c 8π c = T ω + (3)

2 65 Interntionl Journl of Physics where b T = g T b is the trce of energy momentum tensor In the context of perfect fluid pproximtion it cn be seen from the second term in the eqution () s the energy momentum tensor ssocited with the sclr field nd write ω 8πTb = b b c c ( )( ) g ( )( ) ( b gb ) + (4) We consider perfect fluid pproximtion for the mtter fields Then the energy momentum tensor ssocited with mtter field will be given by ( ρ ) Tb = gb p+ + puu b (5) where p nd ρ represent the pressure nd energy density ssocited with the mtter-fluid nd u is the velocity vector The pressure nd energy density re relted by the stte eqution p = σρ where σ is prmeter of proportionlity Compring this reltion with eqution (3) we see tht the sclr field nd this mtter-fluid re relted by c 8p c = ( ρ p) ω + (6) As the sclr field hs non-miniml coupling with grvity it is impossible to write n expression similr to the eqution (5) relting the ssocited pressure p nd energy density ρ for the field these quntities being identified s the sptil nd temporl components of the energy momentum tensor ssocited with the sclr field given by the eqution (4) However they re relted by the eqution of stte p = σρ 3 3D Brns-Dicke Cosmology in the Absence of Mtter Let the spcetime be described by the 3D FRLW metric tht is given by [5] b dr ds = gbdx dx = dt ( t) + r dϕ kr (7) where ( t ) is dimensionless cosmic scle fctor nd k is prmeter which cn ssume only one of three vlues: (open universe) 0 (t universe) nd + (closed universe) The observtions done by WMAP show tht for 4D stndrd model the universe is bout t Then following this tendency we will consider only the t cse for the 3D model Replcing (7) nd = ( t) in (); we find tht the pressure p nd the energy density ρ ssocited with the sclr field re given by ρ ω = 8π (8) ω p = + 8p (9) If the sclr field is ssumed s perfect fluid then the equtions (8) nd (9) re complemented by the eqution of stte [545] ρ + ( ρ + p ) = 0 (0) Considering the t metric from (7) in the equtions () nd (6) we find = 8 πρ + = 0 () In order to solve this system of coupled equtions we consider the nstz n β ( t) = ( t) ( t) = 0 + t G 0 G 0 () where G 0 denotes the present vlue for the Newton's grvittionl constnt nd β is dimensionless prmeter Using the eqution () in () we get condition tht reltes the α to n prmeters nmely α = n + (3) Now substitution of the nstz in (0) results ω = tht implies in two possibilities for the prmeter n n = 0 or n = The only cceptble possibility for the n prmeter is n = 0 resulting in α = becuse the vlue n = results in divergent prmeter α without physicl mening For this vlue of α the Brns-Dicke sclr field is given by ( t ) G0 = nd the scle fctor hs the form β ( t) = 0 + t G 0 (4) Thus when we consider depending only on time the 3D Brns-Dicke model reduces exctly to the results of the 3D generl reltivity model Therefore for mssless universe the 3D Brns-Dicke theory does not hve ny K-essence behvior 4 3D Brns-Dicke Cosmology in the Presence of Mtter In this section we ddress the cosmology problem derived from the 3D Brns-Dicke theory in the presence of mtter for t bckground As cn be seen in Section II the mtter will be modeled like perfect fluid with

3 Interntionl Journl of Physics 66 ssocited energy density ρ nd pressure p Substituting the FRLW metric (7) in the field equtions () we see tht the resulting Friedmnn equtions in this cse re given by 8π = ρ + 8 πρ ( ) ( p ) ω+ (5) 8p + = ρ- (6) The energy density ρ nd the pressure p re relted by p = σρ Thus the eqution (6) cn be rewritten s ( σ) ( ω+) 8π + = ρ (7) If we require the conservtion of the energy momentum tensor then we see tht the equtions of stte for the mtter fields nd for the sclr field re given by ρ + ( + σ) ρ = 0 (8) ρ + ( ρ + p ) = ρ (9) The non-miniml coupling between the sclr field nd grvity in the eqution () mounts to mixed term in the eqution of stte for involving the mtter density ρ which mkes its solution non-trivil one Also we see tht since the sclr field is mssless nd does not interct with mtter the eqution (8) preserves the wek equivlence principle And by integrting eqution (8) we obtin tht the energy density ρ is relted to the scle fctor ( t ) by mens of ρ ( σ ) + ( t) = ( 0) As in the previous section now we propose the nstz γ β = + = 0 + (0) G0 ( t ) ( t ) ( t ) ( t ) χ χ s solutions for the equtions (5) nd (7) where χ is constnt Replcing this nstz in the coupled field equtions (5) nd (7) gives us the results ( σ ) ( ) ( ) ω + β = γ = ± β () ω+ 4σ + ω+ + ω σ Unlike the sclr-vccum configurtion now it is not possible to determine the prmeter ω ; since the stte eqution (9) is coupled The eqution () shows tht there re two possibilities for γ but for smll vlues of ω the positive vlue of γ llows negtive vlue for the energy density which is forbidden by the strong energy condition [7] Therefore the negtive choice is the only possibility for γ If we replce the nstz (0) in the generl equtions (8) nd (9) we find tht the equtions for the energy density nd pressure ssocited with the sclr field re given by ωγ χ = 8π + ρ γ β ( χt) ωγ χ p = βγ + + γ ( γ ) 8p ( ) + χt () (3) nd the Brns-Dicke energy density nd pressure re relted by p = σρ If the strong energy condition [7] is considered we find tht the decelertion prmeter for the 3D model constructed with the Brns-Dicke theory is given by where ( σ ) ( t) Ω q = +Ω ( σ ) (4) 8 π 8 π Ω= ρ nd Ω = ρ H t ( ) H( t) nd H is the Hubble prmeter We observe tht the first term in the right side is relted to the mtter fields while the second term is relted to the sclr field The behvior of the 3D Brns-Dicke model s Kessence theory depends on the possible vlues for the dimensionless prmeter ω of theory prmeter σ tht reltes the Brns-Dicke energy density nd pressure nd the prmeter σ tht reltes the energy density nd pressure of mtter fields For this lst prmeter there re two possibilities: σ = 0 tht describes Mtter Dominte Er (MDE); σ = Rdition Dominte Er (RDE) For the RDE the 3D Brns Dicke theory recovers the cosmologicl stndrd model results For the MDE the behvior of the prmeter σ is described by Figure Figure σ prmeter in terms of dimensionless prmeter of theory ω

4 67 Interntionl Journl of Physics Anlyzing Figure we found tht σ hs negtive vlues for 3 < ω < 06 However the Brns-Dicke Theory is inconsistent for smll vlues of the dimensionless prmeter ω in solr system level this consistence is restored only for lrger vlues of ω [5] σ does not For lrge vlues of ω( ω ) ssume positive vlues which is inconsistent with the description of n ccelerted expnsion of 3D universe A comprison between the scle fctors of the 3D nd 4D Brns-Dicke models is shown in Figure As expected the models hve different behvior s presented in [0] Figure shows tht there is vlue of time tht the two models converge to the sme result diverging gin shortly therefter Figure Comprison between scle fctor (t) behvior for vcuum of 3D (red) nd 4D (green) models For the MDE the energy density of Brns-Dicke sclr field is non null This is expected becuse the Brns-Dicke sclr field is generted by mtter fields (3) The nlysis of Figure shows tht the Brns-Dicke model cn only σ < 0 for some vlues of describe K-essence field ( ) the dimensionless prmeter ω( < ω < ) 3 06 However for the consistency of the theory only lrger vlues of ω re llowed in solr system level ( ω 50000) [5]; nd for ω > 06 the 3D theory no describe K-essence model ( σ > 0 ) In the RDE the 3D nd 4D models hs the sme behviour The 3D Brns-Dicke model restores the Generl Reltivity results when ω Tble 3D nd 4D Brns-Dicke models Er model 3D 3D vcuum m ( ) ~ t t m> RDE ( t) ~ t ( t) ~ t ω+ ( ω+ ) MDE ( ω+ t ~ t ) t ~ t ω+ 4 t t ( ) ~ ( ) ( ) DNE ( t) ~ t Tble shows comprtion between the scle fctor behvior of 3D nd 4D Brns-Dicke models for the different evolution ers An interesting result is tht there is no difference between vcuum Er nd RDE in the 3D Brns-Dicke model Therefore the 3D model hs only two distinct evolution Ers while the 4D Brns-Dicke model hs four distinct Ers (see Tble ) [0] This comprison shows tht for the sclr tensor theories like Brns-Dicke theory the choice of sptil dimension is very importnt fctor for the theory behvior 5 Conclusions In the bsence of mtter fields (vcuum) the energy density of the Brns-Dicke sclr field vnishes ie ρ = 0 nd the 3D Brns-Dicke theory recovers the 3D Generl Reltivity results This result is expected since the Brns-Dicke sclr field is generted by mtter fields (3) However the 4D Brns-Dicke model does not reduce to the 4D Generl Reltivity model when the mtter fields vnish [0] nd new Er of sclr field emerges in the 4D Brns-Dicke model This new Er in the 4D model is due to K-essence behvior of Brns-Dicke sclr field in 4D model which does not hppen in the 3D model In the presence of mtter fields the energy density of the 3D sclr fields is non-null However even in the presence of mtter fields the 3D Brns-Dick model does not hve K-essence behvior like found in the 4D model [0] Another importnt informtion is tht 3D Brns- Dicke model hs only two distinct evolution Ers the vcuum nd RDE Ers re the sme nd the MDE Er In this sense we conclude tht the difference of scle fctor behvior between 3D nd 4D Brns-Dicke models is minly due to the dimensionl fctor of the field equtions Acknowledgments The uthors thnk CNPq nd CAPES for finncil support nd the Professor Rodrigo Frehse Pereir for the support with the text editor References [] E Berti Astrophysicl Blck Holes s Nturl Lbortories for Fundmentl Physics nd Strong-Field Grvity Brz J Phys 43: [] J C Fbris F T Flcino J Mrto N Pinto-Neto P Vrgs Moniz Dilton Quntum Cosmology with Schrdinger-like Eqution Brz J Phys 4: [3] D N Spergel et l First yer Wilkinson microwve nisotropy probe (WMAP) observtions: determintion of cosmologicl prmeters Astrophys J 48: (rxiv:stro-ph/03009) [4] P Jordn Schwerkrft undweltll Friedrich Vieweg und Sohn Brunschweig 955 [5] Y Fujii K Med The sclr-tensor theory of grvittion Cmbridge University Press Cmbridge 004 [6] C Brns R H Dicke Mch's principle nd reltivistic theory of grvittion Phys Rev 4:95 96 [7] R H Dicke Mch's principle nd invrince under trnsformtion of units Phys Rev 5:63 96 [8] P A M Dirc New bsis for cosmology Proc Roy Soc A65: [9] N Bnerjee D Pvon A Quintessence sclr field in Brns-Dicke theory Clss Qunt Grv 8: [0] H Kim Brns-Dicke sclr field s unique k-essence Phys Lett B606:3 005

5 Interntionl Journl of Physics 68 [] L Qing Y M M Hn D Yu Five-dimensionl Brns-Dicke theory nd cosmic ccelertion Phys Rev D7: [] J Ponce de Leon Lte time cosmic ccelertion from vcuum Dicke theory in 5D Clss Quntum Grv 7: [3] M B~ndos C Teitelboim J Znelli Blck hole in threedimensionl spcetime Phys Rev Lett 69: [4] O Ahrony S S Gubser J Mldcen H Ooguri Y Oz Lrge N Field Theories String Theory nd Grvity Phys Reports 33: [5] A Mloney W Song A Strominger Chirl grvity log grvity nd extreml CFT Phys Rev D8: [6] A A Bytsenko nd M E X Guimrães Truncted het kernel nd one-loop determinnts for the BTZ geometry Eur Phys J C 58:5 008 [7] S W Hwking G F R Ellis The lrge structure of spce-time Cmbridge University Press Cmbridge 973 [8] E Nbulsi A Rmi Effective cosmology l Brns-Dicke with non-minimlly coupling mssive inton field intercting with minimlly coupling mssless field Brz J Phys 40(3):73 00 [9] D A Tretykov A A Shtskiy I D Novikov S Alexeyev Nonsingulr Brns-Dicke cosmology Phys Rev D85: [0] S V Sushkov S M Kozyrev Composite vcuum Brns-Dicke wormholes Phys Rev D84:406 0 [] K H Sidi A Mohmmdi H Sheikhhmdi prmeter nd Solr System constrint in chmeleon-brns-dicke theory Phys Rev D83: [] Feng-Qun Wu nd Xuelei Chen Cosmic microwve bckground with Brns-Dicke grvity II Constrints with the WMAP nd SDSS dt Phys Rev D8: [3] Feng-QunWu Li-e Qing XinWng Xuelei Chen Cosmic microwve bckground with Brns-Dicke grvity I Covrint Formultion Phys Rev D8: [4] S Weinberg Grvittion nd Cosmology Wiley Brbcen 97 [5] C W Misner K S Thorne J A Wheeler Grvittion W H Freemn nd Compny 973