FRW Cosmology From Five Dimensional Vacuum Brans Dicke Theory

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1 FRW Cosmology From Five Dimesioal Vacuum Bras Dicke Theory Amir F. Bahrehbakhsh, Mehrdad Farhoudi ad Hossei Shojaie Departmet of Physics, Shahid Beheshti Uiversity, G.C., Evi, Tehra 19839, Ira September 17, 010 arxiv: v [gr-qc] 17 Oct 010 Abstract We follow the approach of iduced matter theory for a five dimesioal (5D) vacuum Bras Dicke theory ad itroduce iduced matter ad iduced potetial i four dimesioal (4D) hypersurfaces, ad the employ a geeralized FRW type solutio. We cofie ourselves to the scalar field ad scale factors be fuctios of the cosmic time. This makes the iduced potetial, by its defiitio, vaishes, but the model is capable to expose variety of states for the uiverse. I geeral situatios, i which the scale factor of the fifth dimesio ad scalar field are ot costats, the 5D equatios, for ay kid of geometry, admit a power law relatio betwee the scalar field ad scale factor of the fifth dimesio. Hece, the procedure exhibits that 5D vacuum FRW like equatios are equivalet, i geeral, to the correspodig 4D vacuum oes with the same spatial scale factor but a ew scalar field ad a ew couplig costat, ω. We show that the 5D vacuum FRW like equatios, or its equivalet 4D vacuum oes, admit accelerated solutios. For a costat scalar field, the equatios reduce to the usual FRW equatios with a typical radiatio domiated uiverse. For this situatio, we obtai dyamics of scale factors of the ordiary ad extra dimesios for ay kid of geometry without ay priori assumptio amog them. For o costat scalar fields ad spatially flat geometries, solutios are foud to be i the form of power law ad expoetial oes. We also employ the weak eergy coditio for the iduced matter, that gives two costraits with egative or positive pressures. All types of solutios fulfill the weak eergy coditio i differet rages. The power law solutios with either egative or positive pressures admit both deceleratig ad acceleratig oes. Some solutios accept a shrikig extra dimesio. By cosiderig o ghost scalar fields ad appealig the recet observatioal measuremets, the solutios are more restricted. We illustrate that the acceleratig power law solutios, which satisfy the weak eergy coditio ad have o ghost scalar fields, are compatible with the recet observatios i rages 4/3 < ω for the couplig costat ad < for depedece of the fifth dimesio scale factor with the usual scale factor. These rages also fulfill the coditio ω > 3/ which prevets ghost scalar fields i the equivalet 4D vacuum Bras Dicke equatios. The results are preseted i a few tables ad figures. PACS umber: h ; Kd ; 04.0.Cv ; e ; Jk Keywords: Bras Dicke Theory; Iduced Matter Theory; FRW Cosmology. 1 Itroductio Attempts to geometrical uificatio of gravity with other iteractios, usig higher dimesios other tha our covetioal 4D space time, bega shortly after ivetio of the special relativity (SR). Nordstrøm was the first who built a uified theory o the base of extra dimesios [1]. Tight af-bahrehbakhsh@sbu.ac.ir m-farhoudi@sbu.ac.ir h-shojaie@sbu.ac.ir 1

2 coectio betwee SR ad electrodyamics, amely the Loretz trasformatio, led Kaluza [] ad Klei [3] to establish 5D versios of geeral relativity (GR) i which electrodyamics rises from the extra fifth dimesio. Sice the, cosiderable amout of works have bee focused o this idea either usig differet mechaism for compactificatio of extra dimesio or geeralizig it to o compact scearios (see e.g. Ref. [4]) such as Brae World theories [5], space time matter or iduced matter (IM) theories [6] ad refereces therei. The latter theories are based o the Campbell Magaard theorem which asserts that ay aalytical N dimesioal Riemaia maifold ca locally be embedded i a (N +1) dimesioal Ricci flat Riemaia maifold [7]. This theorem is of great importace for establishig 4D field equatios with matter sources locally to be embedded i 5D field equatios without priori itroducig matter sources. Ideed, the matter sources of 4D space times ca be viewed as a maifestatio of extra dimesios. This is actually the core of IM theory which employs GR as the uderlyig theory. O the other had, Jorda [8] attempted to embed a curved 4D space time i a flat 5D space time ad itroduced a ew kid of gravitatioal theory, kow as the scalar tesor theory. Followig his idea, Bras ad Dicke [9] iveted a attractive versio of the scalar tesor theory, a alterative to GR, i which the weak equivalece priciple is saved ad a o miimally scalar field couples to curvature. The advatage of this theory is that it is more Machia tha GR, though mismatchig with the solar system observatios is claimed as its weakess [10]. However, the solar system costrait is a geeric difficulty i the cotext of the scalar tesor theories [11], ad it does ot ecessarily deote that the evolutio of the uiverse, at all scales, should be close to GR, i which there are some debates o its tests o cosmic scales [1]. Although it is sometimes desirable to have a higher dimesioal eergy mometum tesor or a scalar field, for example i compactificatio of extra curved dimesios [13], but the most preferece of higher dimesioal theories is to obtai macroscopic 4D matter from pure geometry. I this approach, some features of a 5D vacuum Bras Dicke (BD) theory based o the idea of IM theory have recetly bee demostrated [14], i where the role of GR as fudametal uderlyig theory has bee replaced by the BD theory of gravitatio. Actually, it has bee show that 5D vacuum BD equatios, whe reduced to four dimesios, lead to a modified versio of the 4D Bras Dicke theory which icludes a iduced potetial. Whereas i the literature, i order to obtai acceleratig uiverses, iclusio of such potetials has bee cosidered i priori by had. A few applicatios ad a D dimesioal versio of this approach have bee performed [15, 16]. Though, i Refs. [15], it has also bee claimed that their procedure provides explicit defiitios for the effective matter ad iduced potetial. Besides, some misleadig statemets ad equatios have bee asserted i Ref. [14], ad hece we have re derived the procedure i Sectio. Actually, the reductio procedure of a 5D aalogue of the BD theory, with matter cotet, o every hypersurface orthogoal to a extra cyclic dimesio (recoverig a modified BD theory described by a 4 metric coupled to two scalar fields) has previously bee performed i the literature [17]. However, the key poit of IM theories are based o ot itroducig matter sources i 5D space times. I additio, recet measuremets of aisotropies i the microwave backgroud suggest that our ordiary 4D uiverse should be spatially flat[18], ad the observatios of Type Ia superovas idicate that the uiverse is i a acceleratig expasio phase [19]. Hece, the uiverse should maily be filled with a dark eergy or a quitessece which makes it to expad with acceleratio [0]. The after a itesive amout of work has bee performed i the literature to explai the acceleratio of the uiverse. I this work, we explore the Friedma Robertso Walker (FRW) type cosmology of a 5D vacuum BD theory ad obtai solutios ad related coditios. This model has extra terms, such as a scalar field ad scale factor of fifth dimesio, which make it capable to preset accelerated uiverses beside decelerated oes. I the ext sectio, we give a brief review of the iduced modified BD theory from a 5D vacuum space time to rederive the iduced eergy mometum tesor, as has bee itroduced i Ref. [14], for our purpose to employ the eergy desity ad pressure. I Sectio 3, we cosider a geeralized FRW metric i the 5D space time ad specify FRW cosmological equatios

3 ad employ the weak eergy coditio (WEC) to obtai the eergy desity ad pressure coditios. The, we probetwo special cases of a costat scale factor of the fifth dimesio ad a costat scalar field. I Sectio 4, we proceed to exhibit that 5D vacuum BD equatios, employig the geeralized FRW metric, are equivalet, i geeral, to the correspodig vacuum 4D oes. This equivalecy ca be viewed as the mai poit withi this work which distiguishes it from Refs. [14, 15]. I Sectio 5, we fid exact solutios for flat geometries ad proceed to get solutios fulfillig the WEC while beig compatible with the recet observatioal measuremets. We also provide a few tables ad figures for a better view of acceptable rage of parameters. Fially, coclusios are preseted i the last sectio. Modified Bras Dicke Theory From Five Dimesioal Vacuum Followig the idea of IM theories [6], oe ca replace GR by the BD theory of gravitatio as the uderlyig theory [14, 15, 17]. For this purpose, the actio of 5D Bras Dicke theory ca aalogously be writte i the Jorda frame as S [g AB,φ] = (5) g (φ (5) R ωφ gab φ,a φ,b +16πL m ) d 5 x, (1) where c = 1, the capital Lati idices ru from zero to four, φ is a positive scalar field that describes gravitatioal couplig i five dimesios, (5) R is 5D Ricci scalar, (5) g is the determiat of 5D metric g AB, L m represets the matter Lagragia ad ω is a dimesioless couplig costat. The field equatios obtaied from actio (1) are (5) G AB = 8π φ (5) T AB + ω ( φ φ,a φ,b 1 ) g AB φ, C φ,c + 1 ) (5) (φ ;AB g φ AB φ () ad (5) φ = 8π 4+3ω (5) T, (3) where (5) ;A A, (5) G AB is 5D Eistei tesor, (5) T AB is 5D eergy mometum tesor, (5) T (5) T C C. Also, i order to have a o ghost scalar field i the coformally related Eistei frame, i.e. a field with a positive kietic eergy term i that frame, the BD couplig costat must be ω > 4/3 [1, ]. (5) As explaied i the itroductio, we propose to cosider a 5D vacuum state, i.e. T AB = 0 = (5) T, where equatios () ad (3) read (5) G AB = ω ( φ φ,a φ,b 1 ) g AB φ, C φ,c + 1 ) (5) (φ ;AB g φ AB φ (4) ad 1 (5) φ = 0. (5) For cosmological purposes oe usually restricts attetio to 5D metrics of the form, i local coordiates x A = (x µ,y), ds = g AB (x C )dx A dx B = (5) g µν (x C )dx µ dx ν +g 44 (x C )dy (5) g µν (x C )dx µ dx ν +ǫb (x C )dy, (6) where y represets the fifth coordiate, the Greek idices ru from zero to three ad ǫ = 1. It should be oted that this asatz is restrictive, but oe limits oeself to it for reasos of simplicity. Assumig the 5D space time is foliated by a family of hypersurfaces, Σ, defied by fixed values of the fifth coordiate, the the metric itrisic to every geeric hypersurface, e.g. Σ o (y = y o ), ca be obtaied 1 We have purposely kept the ull term i equatio (4) for later o coveiet. 3

4 whe restrictig the lie elemet (6) to displacemets cofied to it. Thus, the iduced metric o the hypersurface Σ o ca have the form ds = (5) g µν (x α,y o )dx µ dx ν g µν dx µ dx ν, (7) i such a way that the usual 4D space time metric, g µν, ca be recovered. Hece, equatio (4) o the hypersurface Σ o ca be writte as G αβ = 8π φ T (BD) αβ + ω ( φ φ,α φ,β 1 ) g αβφ,σ φ,σ + 1 ( [φ ;αβ g αβ φ 1 ) ] φ V(φ), (8) where T (BD) αβ is a iduced eergy mometum tesor of the effective 4D modified BD theory, which is defied as T (BD) αβ T (IM) αβ +T (φ) αβ, (9) with ad T (IM) αβ φ { b;αβ 8π b b b g αβ ǫ [ b b b g αβ g αβ +gµν g αµ g βν 1 gµν g µν g αβ ( b g αβ b gµν g µν g µν g µν 1 4 gµν g ρσ g µνg ρσ 3 4 g µν g µν) ]} (10) T (φ) αβ ǫ [ {g 8πb αβ φ + ] ( 1 gµν g µν b ) φ +ǫbb,µ φ,µ 1 b g αβφ }. (11) Also, the iduced potetial has bee defied i the formal idetificatio as [14] V[φ] ǫ ω φ b, (1) φ Σ 0 where the prime deotes derivative with respect to the fifth coordiate. Such a idetificatio has bee claimed [3] to be valid depedig o metric backgroud ad cosiderig separable scalar fields. However, this defiitio is differet from what has bee used i Ref. [15]. Reductio of equatio (5) o the hypersurface Σ o gives φ = ǫ [ ( g αβ b φ +φ g αβ )] b b,µ b b φ,µ, (13) which after maipulatio resembles the other field equatio of a modified BD theory i four dimesios with iduced potetial. The defiitio T (BD) αβ ad equatio (13) are all we eed for our purpose i this work ad a iterested reader ca cosult Refs. [14, 15] for further details. I the ext sectio we assume a geeralized FRW metric i a vacuum 5D uiverse to fid its cosmological implicatios. 3 Geeralized FRW Cosmology For a 5D uiverse with a extra space like dimesio i additio to the three usual spatially homogeous ad isotropic oes, metric (6) ca be writte as ds = dt +a (t,y) [ dr 1 kr +r (dθ +si θdϕ ) We have corrected miscalculatios metioed i the Itroductio. ] +b (t,y)dy, (14) 4

5 that ca be cosidered as a geeralized FRW solutio. The scalar field φ ad the scale factors a ad b, i geeral, are fuctios of t ad y. However, for simplicity ad physical plausibility, we assume the extra dimesio is cyclic, i.e. the hypersurface orthogoal space like is a Killig vector field i the uderlyig 5D space time [17]. Hece, all fields are fuctios of the cosmic time oly, ad defiitio (1) makes the iduced potetial vaishes. I this case, we will show that such a uiverse ca have acceleratig ad deceleratig solutios. Note that, the fuctioality of the scale factor b o y, either ca be elimiated by trasformig to a ew extra coordiate if b is a separable fuctio, ad or makes o chages i the followig equatios if b is the oly field that depeds o y. Besides, i the compactified extra dimesio scearios, all fields are Fourier expaded aroudy o, ad heceforth oe ca have terms idepedet of y to be observable, i.e. physics would thus be effectively idepedet of compactified fifth dimesio [4]. Cosiderig metric (14), equatios (4) ad (5) result i cosmological equatios H ω 6 F +HF + k a = ( HB BF ), (15) ( ) Ḣ + F ω +3H + +1 F +HF + k ) (Ḃ a = +B +HB +BF, (16) ad Ḣ +4H + ω 3 F + k a = BF (17) 3 F +F +3HF = BF, (18) which are ot idepedet equatios ad where H ȧ/a, B ḃ/b ad F φ/φ. By employig relatio (9), oe ca iterpret the right had side of equatios (15) ad (16) as eergy desity ad pressure of the iduced effective perfect fluid, i.e. ρ BD T (BD) t t = φ (3HB +BF) (19) 8π ad p BD T (BD) i i = φ 8π (Ḃ +B +HB +BF) = φ HB, (0) 8π where i = 1 or or 3 without summatio o it. The latter equality i (0) comes from equatio (31) which will be derived i the ext sectio. Therefor, the equatio of state is p BD = w eff ρ BD with w eff = 1 F/H +3. (1) The usual matter i our uiverse has a positive eergy desity, this basically has bee demaded by the WEC, i which time like observers must obtai positive eergy desities. Actually, the complete WEC is [4] { ρbd 0 ρ BD +p BD 0. () Now, let us cosider that the scale factor of the fifth dimesio ad the scalar field are ot costat values, i.e. B 0 ad F 0. The, by applyig coditios () ito relatios (19) ad (0), oe gets { B > 0 F 4H (3) or { B < 0 F 3H, (4) 5

6 where we also have assumed expadig uiverses, i.e. H > 0. Usig coditios (3) ad (4) i relatio (1) gives 1 w eff 0 (5) or w eff 0, (6) i where the effective dust matter ca be achieved whe F/H goes to egative or positive ifiity, respectively. I Sectio 5, we explore characteristic of the correspodig uiverses for the above results. Meawhile, i the followig, we cosider two special cases of a costat scale factor of the fifth dimesio ad a costat scalar field. Costat Scale Factor of Fifth Dimesio Whe b is a costat, equatios (15) (18) reduce to H ω 6 F +HF + k a = 0, Ḣ +4H + ω 3 F + k a = 0 ad F +F +3HF = 0. (7) These are exactly the ordiary vacuum BD equatios i 4D space time, with ρ BD = 0 = p BD, as expected. Costat Scalar Field Whe φ is a costat, actio (1) reduces to a 5D Eistei gravitatioal theory that has bee cosidered i Ref. [5] i geeral situatio (i.e. the extra dimesio is ot cyclic). I this case, equatios (15) (18) become H + k a = HB, Ḣ +H + k a = 0 ad Ḃ +B +3HB = 0. (8) Ad, the usual FRW equatios are equipped with p BD = ρ BD /3 HB/8πG, which refers to a radiatio like domiated uiverse for ay kid of geometry without a priori assumptio that the scale factor of the fifth dimesio is proportioal to the iverse of the usual scale factor, i.e. b a 1. Actually, the radiatio like result is expected. For where there is o depedecy o the extra dimesio, the usual four dimesioal part of metric (14) ad the third equatio (8) give a wave equatio for the scale factor of fifth dimesio. Hece, defiitios (10) ad (11) yield a traceless iduced eergy mometum tesor, as metioed i Ref.[5]. Exact solutio of the secod equatio of (8) is a = kt +αt. (9) Substitutig solutio (9) ito the first or third equatio of (8) gives b = βȧ = β kt+α α kt +αt = β 4ka a, (30) where α ad β are costats of itegratio, ad we have assumed that 4D space time has origiated from a big bag. For a closed geometry, solutio (9) admits α > 0 ad predicts a big cruch at t = α for the usual spatial coordiates while the fifth dimesio teds to ifiite size ad is always real, for the maximum value of the usual scale factor is α/. But, a flat geometry expads for ever ad accepts α > 0. A ope geometry also expads for ever ad admits α 0. I this case, α = 0 results i a = t ad b = β. Time evolutio of scale factors correspod to closed, flat ad ope geometries have bee illustrated i Fig. 0 with costat values of α = 1 ad β = 1 as a example. I the ext two sectios, we agai cosider a more geeral situatio i which the scale factor of the fifth dimesio ad the scalar field are ot costats. 6

7 a t (i) b t (ii) Figure 0: Time evolutio of scale factors (i) a ad (ii) b for the special case of a costat scalar field. The dashed, solid ad dotted lies correspod to closed, flat ad ope uiverses, respectively. 4 Correspodece Betwee 5D Equatios ad 4D Oes Let us explore a equatio (if ay) similar to equatio (18) which is a itegrable equatio for whe the rules of F ad B are replaced. For this purpose, addig equatios (15) ad (16), the subtractig equatios (17) ad (18) from it, yields Ḃ +B +3HB +BF = 0. (31) Comparig equatios (18) ad (31) shows that they are equivalet to each other if oe replaces B by F. Ideed, itegratig equatios (18) ad (31) gives φa 3 b = m 1 ad ḃa 3 φ = m, (3) where m 1 0 ad m 0 are costats of itegratio i geeral situatios whe φ ad b are ot costats. Actually, vaishig m 1 or m gives φ or b to be a costat value, respectively, which have bee discussed i the previous sectio. Dividig equatios (3) by each other leads to where m m /m 1. Relatio (33) obviously gives B = m F, (33) ( ) φ m b = b o, (34) φ o where b o ad φ o are iitial values. Now, cosiderig relatio (33), equatios (15) (18) lead to three idepedet equatios H ω 6 F +H F + k a = 0, Ḣ +4H + ω 3 F + k a = 0 ad F + F +3H F = 0, (35) with φ φ m +1 ad ω ω m (m +1), (36) where m 1 ad F = (m +1)F. Equatios (35) are exactly the FRW equatios of 4D vacuum BD theory. However, oe also eeds to check if this ew scalar field is a wave fuctio i 4D vacuum as well. For this purpose, it is easy to show that ) φ = (m +1) (φ m,µ φ,µ +φ m φ = 0, (37) 7

8 where equatios (13) ad (14) for a cyclic extra dimesio have bee employed to get the secod equality. Hece, this procedure exhibits that 5D vacuum FRW like equatios, equatios (15) (18), are equivalet to the correspodig 4D vacuum oes, equatios (35), with the same spatial scale factor but a ew (or modified) scalar field ad a ew couplig costat, φ ad ω, i which to have a o ghost scalar field oe must have ω > 3/ [1, ]. For the special case of m = 1, i.e. whe b φ 1, equatios (15) (18) reduce to H ( ω ) F + k a = 0, Ḣ+3H + From the third equatio of (38) oe gets ( ω +1 Usig relatio (39) ito the first ad secod equatios of (38) yields ) F + k a = 0 ad F +3HF = 0. (38) ( ) 3 ao F = F o. (39) a äa 5 = A ad (ȧ +k)a 4 = A, (40) for ȧ 0 ad ω ad where A (ω +)Fo a6 o /6. These equatios, or actually their divisio i.e. äa +(ȧ +k) = 0, ca be solved by o algebraic procedures, ad their solutios iclude the iverse elliptic fuctios, although we do ot perform it further. For a static uiverse, i.e. ȧ = 0, equatios (38) lead to a flat uiverse with ω =. O the other had, if ω =, the equatios (38) give ä = 0 ad ȧ = k which restrict the geometry either to be flat or ope. For k = 0, oe agai gets a static uiverse with b = b o exp( F o t) ad φ = φ o exp(f o t). I the case k = 1, it leads to a uiform expadig uiverse with a = t, ad the evolutio of scale factor of the fifth dimesio is b = b o exp(f o a 3 o/t ). I the ext sectio we cotiue our ivestigatios for cosmological implicatios of equatios (15) (18) for a flat uiverse compatible with the recet observatios. 5 Exact Solutios for Flat Uiverse Compatible with Observatios Measuremets of aisotropies i the cosmic microwave backgroud radiatio idicate that the uiverse is spatially flat [18], so we cocetrate o solutios with flat 3 spaces. Therefor, equatios (15) (17) yield Ḣ +3H +(B +F)H = 0, (41) that gives ȧa bφ = m 3, (4) where m 3 is a itegratio costat. The case of vaishig m 3 gives a static uiverse which is ot compatible with observatios. I geeral, relatios (3) ad (4) lead to b = b o ( a a o ) ad φ = φ o ( a a o ) m, (43) for m 3 0 ad where m m 1 /m 3 ad m /m 3 = mm, also for geeral situatios B 0 ad F 0, we have 0 ad m 0. Ideed, if i priori, oe had assumed b a (or φ a m ), the equatios (15) (18) would restrict the geometry to be spatially flat, ad automatically would give φ a m (or b a ). Therefor, the power law relatio betwee the scale factor of the fifth dimesio ad the scalar field with the usual scale factor is a characteristic of the spatially flat uiverse. Substitutig solutios (43) ito equatio (41) gives ä ȧ +(m++) ȧ = 0. (44) a 8

9 For m+ 3, equatio (44) has a power law solutio ( ) t s a(t) = a o with H = s t o t, (45) where s (m++3) 1, ad assumptio expadig uiverses makes s > 0. Hece, solutios (43) lead to ( ) t s b(t) = b o with B = s (46) t o t ad ( ) t ms φ(t) = φ o with F = ms. (47) t o t There is also a costrait relatio amog the iitial values, amely a 3 o b oφ o = (m 1 + m + 3)t o. Icidetally, the effective eergy desity ad pressure, equatios (19) ad (0) become ad ρ BD = φ os 8πt ms (m+3)t ms ad p BD = φ os t ms. (48) o 8πt ms o I the case m+ = 3, equatios (43) ad (44) give expoetial solutios a(t) = a o e λ(t to) with H = λ, (49) b(t) = b o e λ(t to) with B = λ (50) φ(t) = φ o e mλ(t to) with F = mλ, (51) where λ is a costat ad its positive values give expadig uiverses, thus we assume λ > 0. Icidetally, the costrait relatio amog the iitial values is a 3 ob o φ o = m 3 /λ. I this case, the eergy desity ad pressure are ρ BD = φ oλ 8πe mλto(m+3)emλt ad p BD = φ oλ 8πe mλtoemλt. (5) Note that, for both groups of solutios, the power law ad expoetial oes, oe has w eff = 1/(3+m). We should emphasis that all solutios of this sectio have bee obtaied without a priori asatz for fuctioality of the scale factor ad the scalar field. I the ext two subsectios, we discuss properties of these solutios. We should also remid that our vaishig iduced potetial case is ot cosistet with zero potetial case of Ref. [15] (where there, it requires ω = 1 oly). 5.1 Power Law Solutios Solutios are geerally cofied withi some costraits that are origiated from mathematical or physical reasos. First of all, due to equatios(15) (17), the parameters ad m are ot idepedet. Substitutig solutios (45) (47) ito either of equatios (15) (17) gives ad hece s ± = m ± = +3± (+3) +6ω(+1) ω (53) ω (ω +1)(+3)± (+3) +6ω(+1). (54) Besides, our costraits are as follows. We have assumed s > 0, m ± 0, 0 ad m ± + 3 for power law solutios. Real solutios of relatio (53) dictate that ( + 3) + 6ω( + 1) 0. By substitutig solutios (45) (47) i the WEC (3) or (4), we get { > 0 (55) m 4 9

10 ω for s + values ω for s values 3 No solutio ( ++3) (+) < ω < 0 3 < 0 < ω (+3) 6(+1) ( ++3) (+) < ω < 0 or 0 < ω (+3) 6(+1) < < 1 ω < ( ++3) (+) or 0 < ω (+3) 6(+1) ω < 0 or 0 < ω (+3) 6(+1) = 1 ω < 4 or ω > 0 No solutio 1 < < 0 (+3) 6(+1) ω < ++3) ( or ω > 0 (+) (+3) 6(+1) ω < 0 or ω > 0 0 < 1 ω > 0 ( ++3) (+) < ω < 0 or ω > 0 > 1 (+3) 6(+1) ω < ++3) ( (+) or ω > 0 (+3) 6(+1) ω < 0 or ω > 0 Table 1: Rages of ad ω for deceleratig power law solutios (i) -10 (ii) Figure 1: Domais of ad ω correspod to Table 1, (i) s + ad (ii) s values. Note that, the lie = 1 is excluded i part (ii). or { < 0 m 3, (56) respectively. Note that, coditios (55) ad (56) are compatible with coditios (5) ad (6), as expected. I the followig, we employ these costraits for whe they lead to cases of decelerated ad especially accelerated uiverses. Meawhile, we should also remid that the deceleratio parameter, q = äa/ȧ, i our model for the power law solutios is q = 1 s. Case Ia: Decelerated Uiverse It is supposed that the uiverse for a log time, whe it was i the radiatio or dust domiated phases, was i a deceleratig regime. I our model, deceleratig solutios ca be obtaied whe 0 < s ± < 1. Acceptable domais of ad ω for such a rage, without cosiderig the WEC, is give i Table 1 ad Fig. 1. Note that, i Fig. 1, the part (ii) completely covers the part (i). Also, adapted values of ad ω with the WECs (55) ad (56) are show i Table with Fig. ad Table 3 with Fig

11 ω for s + values ω for s values < < ω < ( ++3) (+) 3 (+3) 6(+1) ω < ++3) ( (+) No solutio (+3) 6(+1) ω +9 8 Table : Rages of ad ω for deceleratig power law solutios which adapt the WEC (55) (i) (ii) Figure : Domais of ad ω correspod to Table, (i) s + ad (ii) s values. ω for s + values ω for s values 3 No solutio ( ++3) < ω < 0 (+) 3 < 0 < ω (+3) 6(+1) ( ++3) (+) < ω < 0 or 0 < ω (+3) 6(+1) < < 1 ω < ( ++3) (+) or 0 < ω (+3) 6(+1) ω < 0 or 0 < ω (+3) 6(+1) = 1 ω < 4 or ω > 0 No solutio 1 < < 0 (+3) 6(+1) ω < ++3) ( or ω > 0 (+3) (+) 6(+1) ω < 0 or ω > 0 Table 3: Rages of ad ω for deceleratig power law solutios which adapt the WEC (56) (i) -10 (ii) Figure 3: Domais of ad ω correspod to Table 3, (i) s + ad (ii) s values. Note that, the lie = 1 is excluded i part (ii). 11

12 ω for s + values ω for s values 3 No solutio ( +3+6) (+3) < ω < ( ++3) (+) 3 < < ω < ( +3+6) (+3) ω < ( ++3) (+) = ω < 8 No solutio < < 0 ( ++3) (+) < ω < ( +3+6) (+3) No solutio (+3) 6(+1) ω < ++3) ( (+) 0 < < 1 (+3) 6(+1) ω < +3+6) ( (+3) 1 ( ++3) (+) < ω < ( +3+6) (+3) No solutio Table 4: Rages of ad ω for acceleratig power law solutios (i) -10 (ii) Figure 4: Domais of ad ω correspod to Table 4, (i) s + ad (ii) s values. Case IIa: Accelerated Uiverse Recet observatios show that the uiverse is i a acceleratig regime at the preset epoch[19]. This makes s ± > 1, ad acceptable values of ad ω correspodig to this coditio, without cosiderig the WEC, are give i Table 4 ad Fig. 4. I Fig. 4, the maximum value of ω for s + values teds to 5/4 whe = 1, ad for s values teds to 4/3 whe 1. Correspodig cases with the WECs (55) ad (56) are illustrated i Table 5 with Fig. 5 ad Table 6 with Fig. 6, respectively. It should be emphasized that though astroomical tests i the solar system requires a positive large value for ω, but still i the large cosmological scale, oe caot defiitely rule out small or eve egative values of the BD couplig costat. Ideed, these values of ω have achieved cosiderable iterests i the literature. By cosiderig the WEC (55) i relatios (48), oe gets positive eergy desities, as expected, but with egative pressures, where both ρ BD ad p BD decrease with the time. I this case, eve ω for s + values ω for s values 1 < < +9 8 ω < ( +3+6) (+3) ( ++3) (+) < ω < ( +3+6) (+3) No solutio No solutio Table 5: Rages of ad ω for acceleratig power law solutios which adapt the WEC (55). 1

13 Figure 5: Domais of ad ω correspod to Table 5 for s + values. ω for s + values ω for s values 3 No solutio ( +3+6) (+3) < ω < ( ++3) (+) 3 < < ω < ( +3+6) (+3) ω < ( ++3) (+) = ω < 8 No solutio < < 0 ( ++3) (+) < ω < ( +3+6) (+3) No solutio Table 6: Rages of ad ω for acceleratig power law solutios which adapt the WEC (56) (i) (ii) Figure 6: Domais of ad ω correspod to Table 6, (i) s + ad (ii) s values. 13

14 ω for s + values ω for s values 3 No solutio 4 3 < ω < 0 3 < < 1 0 < ω (+3) 6(+1) 4 3 < ω < 0 or 0 < ω (+3) 6(+1) = 1 ω > 0 No solutio 1 < < 0 ω > < ω < 0 or ω > 0 Table 7: Rages of ad ω for deceleratig power law solutios with o ghost scalar fields which adapt the WEC (56) (i) (ii) Figure 7: Domais of ad ω correspod to Table 7, (i) s + ad (ii) s values. Note that, the lie = 1 is excluded i part (ii). though the pressure is egative, but Figs. ad 5 illustrate that oe has deceleratig ad acceleratig solutios. O the other had, usig coditio (56) ito relatios (48) gives positive eergy desities ad pressures. Although, i this situatio the pressure is positive, but still Figs. 3 ad 6 idicate that oe agai has deceleratig ad acceleratig solutios. I this situatio, for decreasig eergy desity ad pressure with the time, oe has to restrict ms <, which most of the solutios fulfill it. Yet we have oe more coditio, amely o ghost scalar fields with ω > 4/3, to be imposed. With this situatio, acceptable solutios are as follows. Case Ib: Decelerated Uiverse Acceptable values of ad ω for the rage ω > 4/3 restrict Table 3 ad Fig. 3, ad the results are show i Table 7 ad Fig. 7. Hece, this model admits a typical decelerated uiverse with o ghost scalar fields, positive iduced eergy desity ad pressure, fulfillig the WEC (56), where the scale factor of fifth dimesio shriks with the time. Icidetally, Fig. illustrates that there is ot ay decelerated solutio with o ghost scalar fields which complies with the WEC (55). Case IIb: Accelerated Uiverse Table 8 ad Fig. 8, which are the reductios of Table 5 ad Fig. 5, illustrate the correspodig domais of ad ω for ω > 4/3. Therefore, the model also admits a typical accelerated uiverse with o ghost scalar fields, positive iduced eergy desity ad egative pressure, fulfillig the WEC (55), where the scale factor of fifth dimesio grows with the time. This situatio restricts 1 < < 3, cotrary to the assumptio of = 1 i Ref. [14]. Also, Fig. 6 idicates that accelerated solutios do ot exist for o ghost scalar fields which fulfill the WEC (56). By employig the recet observatioal measuremets of q, amely 0.9 q o 0.4 [6], we obtai 1.4 s 1.9. This rage of s is oly compatible with the acceleratig Case II, but 14

15 ω for s + values 1 < < ω < ( +3+6) (+3) 5 3 < < ω < +3+6) ( (+3) Table 8: Rages of ad ω for acceleratig power law solutios with o ghost scalar fields which fulfill the WEC (55) Figure 8: Domais of ad ω correspod to Table 8. The area below the dashed lie correspods to Table 9, ad the upper border curve correspods to Table 10 with o ghost scalar fields. imposes more restrictios o domais of ad ω. Ideed, Table 8 ad Fig. 8 of Case IIb with these values of s lead to Table 9 ad the area below the dashed lie i Fig. 8. Thus, oe obtais rages < ad 4/3 < ω These are the best values of ad ω, i this model, that are compatible with the recet observatios. These values also fulfill the coditio ω > 3/, which is required, i Sectio 4, for havig o ghost scalar fields i the equivalet 4D vacuum BD equatios. It is iterestig to ote that, oe ca ifer from relatios (53) ad (54) that m + ad s + do ot allow ω = 0. However, the other costraits does ot obviously show that the zero value of ω is preveted, but the mathematical procedure for all tables ad figures idicates that ω Expoetial Solutios Relatio (49) represets a expoetial growth of the scale factor, i.e. a iflatioary uiverse. However, o such a rapid expasio has bee idicated at preset or throughout almost the whole history of the uiverse except at the very early uiverse stage. Nevertheless, let us probe some properties of these solutios. ω for s + values = ω = < ω 3( ) 5 3 < < ω 3( ) Table 9: Rages of ad ω for acceleratig power law solutios with o ghost scalar fields which fulfill the WEC (55) for s + values, ad are compatible with the recet observatios. 15

16 ω for m + values ω for m values = 1 ω = 5 4 ω = 5 4 > 1 No solutio ω = ( +3+6) (+3) Table 10: Rages of ad ω for expoetial solutios which fulfill the WEC (55). ω for m + values ω for m values < 3 No solutio ω = ( +3+6) (+3) 3 < < 1 ω = ( +3+6) (+3) No solutio 1 < < 0 ω = ( +3+6) (+3) No solutio Table 11: Rages of ad ω for expoetial solutios which fulfill the WEC (56). First of all, we have m + = 3 that, with assumptios m 0 ad 0, restricts m 3, 3 ad m. By employig solutios (49) (51) ito equatios (15) (17), oe gets m ± = 3± (1ω +15) ω + Substitutig coditio m+ = 3 ito relatio (57) gives. (57) ω = ( +3+6) (+3), (58) for 0, 3. Real solutios of relatio (57) impose ω 5/4. Solutios (49) (51) satisfy coditios (55) ad (56) with more restrictios. Usig the WEC (55), i relatios (5) gives positive eergy desities ad egative pressures, where both ρ BD ad p BD decrease rapidly with the time. But, cosiderig coditio (56) i relatios (5), gives positive eergy desities ad pressures. If oe takes m < 0 i coditio (56), the eergy desity ad pressure agai will decrease with the time. Acceptable rages of ad ω for expoetial solutios fulfillig the WECs (55) ad (56) are show i Tables 10 ad 11. These tables idicate that, oly for the rage 1 < 3, solutios do avoid ghost scalar fields. I Fig. 8, the upper border curve illustrates the acceptable values of ad ω for a iflatioary uiverse fulfillig the WEC (55) with o ghost scalar fields. 6 Coclusios Aalogous to the approach of IM theories, oe ca cosider the BD gravity as the uderlyig theory. Hece, extra geometrical terms, comig from the fifth dimesio, are regarded as a iduced matter ad iduced potetial. We have followed, with some correctios, the procedure of Ref. [14] for itroducig the iduced potetial ad have employed a geeralized FRW type solutio for a 5D vacuum BD theory. Hece, the scalar field ad scale factors of the 5D metric ca, i geeral, be fuctios of the cosmic time ad the extra dimesio. However, for simplicity, we have assumed the scalar field ad scale factors to be oly fuctios of the cosmic time, where this makes the iduced potetial, by its defiitio, vaishes. We the have revealed that i geeral situatios, i which the scale factor of the fifth dimesio ad scalar field are ot costats, the 5D equatios, for ay kid of geometry, admit a power law relatio betwee the scalar field ad scale factor of the fifth dimesio. Hece, the procedure exhibits that 5D vacuum FRW like equatios are equivalet, i geeral, to the correspodig 4D vacuum oes 16

17 with the same spatial scale factor but a ew (or modified) scalar field ad a ew couplig costat. This equivalecy ca be viewed as the distiguished poit of this work from Refs. [14, 15]. Ideed, through ivestigatig the 5D vacuum FRW like equatios, we have show that its equivalet 4D vacuum equatios admit accelerated scale factors, cotrary to what oe may have expected from a vacuum space time. Coclusios of the complete ivestigatio of the iduced 4D equatios are as follows. Followig our ivestigatios for cosmological implicatios, we have show that for the special case of a costat scale factor of the fifth dimesio, the 5D vacuum FRW like equatios reduce to the correspodig equatios of the usual 4D vacuum BD theory, as expected. I the special case of a costat scalar field, the actio reduces to a 5D Eistei gravitatioal theory ad the equatios reduce to the usual FRW equatios with a typical radiatio domiated uiverse. For this situatio, we also have obtaied dyamics of scale factors of the ordiary ad extra dimesios for ay kid of geometry without ay priori assumptio amog them. Solutios predict a limited life time for closed geometries ad ulimited oe for flat ad ope geometries. A typical time evolutios of scale factors correspod to closed, flat ad ope geometries have bee illustrated i Fig. 0. The, we have focused o spatially flat geometries ad have obtaied exact solutios of scale factors ad scalar field. Solutios are foud to be i the form of power law ad expoetial oes i the cosmic time. We also have employed the WEC for the iduced matter of the 4D modified BD gravity, that gives two coditios (55) ad (56). We the have pursued properties of these solutios ad have idicated mathematically ad physically acceptable rages of them, ad the results have bee preseted i a few tables ad figures. All types of solutios fulfill the WECs i differet rages, where the expoetial solutios are more restricted. The solutios fulfillig the WEC (55) have egative pressures, but the figures illustrate that for the power law results there are deceleratig solutios beside acceleratig oes. For this coditio, both ρ BD ad p BD decrease with the cosmic time, but the extra dimesio grows. O the other had, the solutios satisfyig the WEC (56) have positive pressures, where the power law results accept acceleratig solutios i additio to deceleratig oes. For this coditio, agai decreasig eergy desity ad pressure with the time ca occur for some solutios, however all with shrikig extra dimesio. The homogeeity betwee the extra dimesio ad the usual spatial dimesios, i.e. b a, ca take place i the solutios, but for the power law oes the WECs exclude it. By cosiderig o ghost scalar fields ad appealig the recet observatioal measuremets, the solutios have bee more restricted. Actually, we have illustrated that the acceleratig power law solutios, which satisfy the WEC ad have o ghost scalar fields, are compatible with the recet observatios i rages 4/3 < ω for the BD couplig costat ad < for depedece of the fifth dimesio scale factor with the usual scale factor. These rages also fulfill the coditio ω > 3/ which prevets ghost scalar fields i the equivalet 4D vacuum BD equatios. Icidetally, this rage is more restricted tha the oe obtaied i Ref. [15], i.e. 1.5 < ω < 1, where the differece may have bee caused by the distict defiitio of the iduced potetial i two approaches of Ref. [14] ad Ref. [15]. However, we should remid that it has also bee show [1] that the WEC, for 5D space times, requires 4/3 ω, i which o other experimetal evideces have bee cosidered. Refereces [1] G. Nordstrøm, Phys. Z. 15, 504 (1914). [] T. Kaluza, Sitz. Preuss. Akad. Wiss. 33, 966 (191). [3] O. Klei, Z. Phys. 37, 895 (196). [4] J.M. Overdui ad P.S. Wesso, Phys. Rep. 83, 303 (1997). 17

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