Canadian Journal of Physics. Bouncing Cosmologies in Brans-Dicke Theory

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1 Bouncing Cosmologies in Brans-Dicke Theory Journal: Manuscript ID cjp r1 Manuscript Type: Article Date Submitted by the Author: 6-Feb-016 Complete List o Authors: Singh, T.; Banaras Hindu University, Department o Applied Mathematics, Institute o Technology Chaubey, R.; Banaras Hindu University, DST - CIMS Singh, Ashutosh; BHU, DST-CIMS Keyword: FRW model, Brans-Dicke theory, Scalar ield, Scale actor, Bounce

2 Page 1 o 14 Bouncing Cosmologies in Brans-Dicke Theory T. Singh, R. Chaubey and Ashutosh Singh. DST-Centre or Interdisciplinary Mathematical Sciences, Institute o Science, Banaras Hindu University, Varanasi 1005, INDIA. Abstract In this paper it is shown that in Brans-Dicke (BD) theory, i one considers a non-minimal coupling between the matter and scalar ield, it can give rise to a bouncing universe (i.e. an expanding universe preceded by a contracting universe). Two examples o such universes have been considered in a spatially lat FRW universe and their physical properties have been studied. Keywords: FRW Cosmological model; Brans-Dicke theory; Scalar ield; Bounce. PACS No.: jk drtrilokisingh@yahoo.co.in yahoo raghav@redimail.com, rchaubey@bhu.ac.in theprabhu.09@gmail.com 1

3 Page o 14 1 Introduction Gravitation is the dominant orce controlling the large scale behaviour o the universe. Einstein was the irst to give a theory o gravitation in a generally covariant orm. General Relativity is a geometrical theory o space-time. The undamental building block is a metric tensor ield g ij. The Brans- Dicke (BD) theory [1] (whose original motivation was the search or a theory containing Mach s Principle) is the theory where it is considered that there is a scalar partner or the metric tensor or describing geometry o space-time. In this theory the scalar ield is a undamental part, while in other theories the scalar ield is introduced separately in an ad-hoc manner. This theory can pass experimental test rom solar system [, ]. It has been proposed as an alternative to general relativity, other theories are (R) gravity [4, 5], massive theories o gravity [6, 7] to name a ew. The earliest modiication to general relativity was Kaluza-Klein gravity [8] which was intended to uniy gravity with electromagnetic orce. The present universe is undergoing an accelerated phase o expansion as conirmed by the observational data regarding the luminosity red-shit relation o type Ia supernovae [9], the cosmic microwave background radiation [10]. In order to explain the cosmic positive acceleration, an exotic orm o matter dark energy has been proposed within the rame work o general relativity [11, 1]. Positive acceleration can also be explained through modiication o gravity. There have been many attempts to show that BD model can potentially explain the cosmic acceleration. This theory can produce a non-decelerating expansion or low negative value o BD parameter w [1] but this conlicts with lower bound imposed on this parameter by solar system experiments [14]. Some authors propose modiications o Brans-Dicke model by introducing some potential unctions or the scalar ield [15, 16] or considering a ield dependent BD parameter [17]. In a dierent approach within the ramework o general relativity, scalar ield is allowed to interact non-minimally with the matter through an arbitrary unction o scalar ield as a coupling unction. This type o ield is called Chameleon ield [18, 19]. This Chameleon ield can be heavy enough in the environment o the laboratory tests so that the local gravity constraints are satisied. Also, this ield can be light enough in low density cosmological scenario to be considered as a candidate or the dark energy. Chameleon ield can provide a very smooth transition rom a decelerated to accelerated phase o expansion o the universe [0]. In BD theory or its modiication, there is an interaction between scalar ield and geometry. Chameleon ield is non-minimally coupled to normal matter sector. We take a Brans-Dicke ramework in which there is a non-minimal coupling o the scalar ield with

4 Page o 14 matter. The behaviour o an isotropic cosmological model in the early as well as in late time limits in the Brans-Dicke ramework has been investigated by Cliton and Barrow [1]. The non-minimal coupling o a scalar ield with both geometry and matter has been used in literature [,, 4, 5, 6]. An unwanted eature o cosmological theories, the initial singularity problem, is being solved by bouncing cosmology [7, 8, 9, 0, 1,,, 4, 5]. A bouncing universe contracts until a minimal radius is reached, and ater that point it expands. The universe, thereore, never goes to a singular point, thus avoiding the initial singularity. The idea o a bouncing universe in GR ramework has been examined many time since the 190 s [6], loop-quantum gravity [7, 8], matter bounce theories [9, 40, 41], modiied gravity [5, 4] and it oers a consistent description o bouncing cosmologies. The necessary conditions or bounce in dierent theories have been investigated by the several authors [4, 44, 45, 46]. Bouncing solutions in cosmological scenarios resulting rom eective actions in our dimensions which are, under some assumptions, connected with multi-dimensional, super-gravity and string theories has been discussed in the papers [47, 48]. Farajollahi et al. [49] have considered stability analysis and possibility o Phantom crossing with the assumption that potential o the scalar ield and coupling unction are in the power-law orms, and bouncing solutions, some cosmological tests are investigated [50]. Recently, bouncing cosmology with uture singularities in modiied gravity [51], has been discussed with an anstaz or scale actor. Thermodynamics o non-singular bouncing universes governed by GR has been investigated in [5]. The present paper is organised as ollows. Section deals with the basic equations o cosmological model. In this section, it is shown that nonminimally coupled BD scalar models can provide a smooth transition rom the contracting to expanding model. Section contains two classes o exact solutions or a spatially lat FRW universe. The paper ends with a conclusion given in the Section 4. Basic Equations We consider the action unctional: [ gd S = 4 x ()L m + 1 (R w )],i,i (1) where R is the Ricci scalar, = (t) is the BD Scalar ield which is non-minimally coupled to gravity, g is the metric determinant o the metric g ij. Here () is the arbitrary unction o and L m is matter Lagrangian

5 Page 4 o 14 representing the perect luid matter. We take 8πG = c = 1. Varying the action with respect to metric g ij and yields the ield equations: R ij 1 Rg ij = () T ij + 1 (g ij i j) w ( i j 1 g ij( i ) ) () w + = () T ()L m () where T is the trace o energy-momentum tensor T ij = δ( gl m) g δg ij prime denotes the derivation with respect to. The line element o spatially lat FRW model is and ds = dt a (t) [ dr + r (dθ + sin θd ) ] (4) where a(t) is the scale actor. For perect luid, the stress-energy tensor is given by T ij = (ρ + p)u i u j pg ij where, u i is the 4-velocity vector. Here ρ & p are the energy density and pressure respectively. The ield equations or lat FRW universe can be written as: H = () ρ + w H (5) Ḣ + H = () p w H (6) where, H = ȧ is the Hubble parameter. Overhead dot denotes the derivative a with respect to t. The dynamical equation or the scalar ield is: (w + )( + H ) = (ρ p) ρ (7) The matter conservation equation can be written as ρ + H(1 + γ)ρ = (1 + γ)ρ (8) where, γ is the equation o state parameter deined as γ = p. For constant ρ γ, the solution o the above equation can be written as ρ = ρ 0 1+γ a (1+γ) (9) where, ρ 0 is a positive constant. This relation indicates that evolution o matter density depends upon the coupling unction. However i equation o state parameter γ is the unction o time then, rom equations (5) and (6), 4

6 Page 5 o 14 the equation o state parameter γ in terms o geometrical quantities can be written as γ = Ḣ + H + w + H + (10) H + w H We take the scale actor a(t) o the orm: a(t) = a 0 + α t (11) where a 0, α are non-zero positive constants. The above scale actor is the temporal analogue o the toy model traversable wormhole [5]. The Hubble parameter or the above scale actor is given by H(t) = α t a 0 + α t (1) By a bouncing universe, we mean a universe that undergoes a collapse, attains minimum and then subsequently expands. For a successul bounce in FRW model, during contraction phase a(t) is decreasing i.e. (ȧ(t) < 0) and then in the expanding phase, scale actor is increasing i.e. (ȧ(t) > 0). From Fig.1, it is clear that the above scale actor undergoes contracting to expanding phase. At the bounce point i.e. at t = t b, the minimal necessary condition is (i) ȧ(t b ) = 0 and (ii) ä(t) > 0 or t (t b ɛ, t b ) (t b, t b + ɛ) or small ɛ > 0. For a non-singular bounce a(t b ) 0. These conditions may not be suicient or a non-singular bounce. For the above scale actor at t = 0, we get ȧ(t = 0) = 0 and a(0) = a 0 with ä(t) > 0 in small neighbourhood o t = 0, provided a 0 > αt. Thereore, we have a scale actor satisying necessary condition o bounce. In terms o geometrical quantities, we have ä a = Ḣ + H = (αa 0 ) (a 0 + α t ) (1) Thereore, ater the bounce, the universe expands in an accelerated way. The idea o a bounce in lat universe may appear diicult to visualize, but can be understood i we remember that the quantity Ḣb gives a measure o the deviation o the matter world-lines. In this sense, the bounce condition simply means that there exist a phase in which separation between the matter world-lines decreases to a minimum and then increases again. Since, this phenomena is independent o spatial geometry o the space-time, the bounce itsel is independent o it [44]. From equations (5) and (6), it is clear that 5

7 Page 6 o a.5 a,h H t Figure 1: The plot o scale actor (a) and Hubble parameter (H) with t or a 0 = 0.5 and α = 0.5. overall dynamics o the universe depends upon the magnitude and sign o derivatives o. In the next section, we make two anstaz or and discuss the properties o resulting bouncing universe. In terms o energy density and pressure or the perect luid, the energy conditions can be stated as: Null Energy Condition (NEC) is satisied when ρ + p 0. Weak Energy Condition (WEC) is satisied when ρ 0 and ρ + p 0. Strong Energy Condition (SEC) is satisied when ρ + p 0 and ρ + p 0. It is clear that violation o NEC will lead to violation o other energy conditions. Bouncing Universes In this section we discuss two physically viable cosmologies (i.e. = βh = nt), which have physical interest to describe the scalar ield. and.1 Scalar ield as a unction o scale actor Here we make an anstaz: = βh (14) 6

8 Page 7 o 14 where, β is a non-zero constant. From equations (10), (1) and (14), we get γ = (1 β) + (1 + w )β ( β)a (β 1) + w + ( 0 β (β 1) + w β) (15) (αt) From equations (5), (6) and (14), we get (ρ + p) = (β )Ḣ (β + (1 + w)β )H (16) (ρ + p) = (β )Ḣ + ( β (w + )β )H (17) At the bouncing point, or β <, NEC and SEC both are violated or > 0. I β, > 0, NEC and SEC may not be violated at the bouncing point. The deceleration parameter is given by q = d ( ) 1 1 = a 0 (18) dt H (αt) For any a 0, α, q is always negative and ater the bounce, as t increases, q will increase. From equation (1), ä > 0, thereore, ater the bounce, universe a expands in an accelerated way. From equations (1) and (14), we have = 0 (a 0 + α t ) β (19) where, 0 is a constant o integration. For w =, we have w + = 0 and in conormally transormed version o BD theory, it indicates that kinetic energy contribution rom the scalar iled is exactly zero [54]. For this choice o w, rom equation (7), we have () = 1 (0) where, 1 is a constant o integration. Thereore, in this case, we have ρ = ( ) 1 β 0 4 β + 1 α 4 t (a 0 + α t ) β (1) By using p = γρ, we can get pressure or this bouncing universe with w =. From Fig., it is clear that equation o state parameter γ crosses γ = 1. Thereore, this particular phenomenological model o universe is Quintom model [56]. Quintom is a dynamical model o dark energy. It diers rom cosmological constant, Quintessence, Phantom, k-essence and so on, in determination o the cosmological evolution. The behaviour o Brans-Dicke scalar 7

9 Page 8 o 14 γ t Figure : The plot o γ with cosmic time t or a 0 = 0.5, β = 0.5, α = 0.5, w = /..5.5,() () t Figure : The plot o and () with t or a 0 = 0.5, α = 0.5, β = 0.5, 0 = 1.5, 1 =. ield and the unction o scalar ield are shown is Fig.. Here it is observed that the scalar ield attains its minima at the bouncing point o the universe. A thermodynamic description o a perect luid matter system requires the knowledge o particle lux N i = nu i and the entropy lux S i = nσu i, where, n = N, σ = S are respectively the concentration and the speciic entropy a N (per particle) o the created/ annihilated particles. Since the energy density o the matter is given by ρ = nm, where M is mass o each particle, the appearance o the extra term in the continuity equation means that this extra change o ρ can be attributed to a change o n or M. Here, we assume that mass o each matter particle remains constant and the extra term in the continuity equation only leads to a change o number o number density n [5]. In this case, rom continuity equation, we have ṅ + H(1 + γ)n = (1 + γ)n () We take Γ = as the decay rate. We also assume that overall energy transer is adiabatic process in which matter particles are continuously created or 8

10 Page 9 o 14 annihilated while the speciic entropy per particle remains constant during the whole process ( σ = 0) [55]. This means that For w + = 0 case, we have Ṡ S = Ṅ N = Γ () α βt Γ = (4) a 0 + α t Thereore, the second law o thermodynamics, Ṡ 0, can be satisied i β < 0. And, Γ 0 or β < 0. The second law o thermodynamics, is satisied at the bouncing point and rom equation () as Γ 0 in the expanding universe, the ratio Ṡ will ollow the decay rate. S. Scalar ield as an exponential orm Here we make an anstaz: = nt (5) where, n is a non-zero constant. Now, rom equations (10), (1) and (5), we get γ = a ( 0 (α na 0 ) w + + 1) n α 4 t 4 + (α + (w + )na 0 )nα t + (α 4 + (a 0 + w )n a 0 ) t T T (6) where, T = w n α 4 t 4 + (wna 0 α )nα t + ( wna 0 α )na 0 α 4. From equations (5), (6) and (5), we get (ρ + p) = Ḣ + nht (1 + w)n t n (7) (ρ + p) = 6Ḣ 6H nht (w + )n t (8) At the bouncing point, we have H = 0 and Ḣ > 0, thereore or > 0, ρ + p < 0 with n > 0 and ρ + p < 0. Thus in this particular bouncing model, NEC and SEC both are violated or n > 0. As ä > 0, (rom equation (1)), we have an accelerating universe ater a the bounce. For any a 0, α, q is always negative and ater the bounce, as t increases, q will increase. From equations (5), we have ( ) nt = exp (9) 9

11 Page 10 o 14 where, is a constant o integration. For w+ = 0 case, we have () 1. From Fig.4, it is clear that equation o state parameter γ crosses γ = γ t Figure 4: The plot o γ with cosmic time t or n = , α = 0.5, a 0 = 0.5, w = /. 4.5 (),() t Figure 5: The plot o and () with cosmic time t or n = , 1 = 10, =.5. The behaviour o scalar ield is shown is Fig.5. Here it is observed that, at the bouncing point o the universe, the scalar ield and the unction o scalar ield attain its minima and maxima respectively. Now, we have ρ = e nt t ( nα t (a (a 0 + α t ) 0 + α t ) + α w ) n (0) By using p = γρ, we can get pressure or this bouncing universe with w =. In this case, Γ = = nt (1) For n > 0, we clearly have Γ 0. Thereore, rom equation () and (1), the second law o thermodynamics is satisied, ( Ṡ 0) even at and near the bouncing point. 10

12 Page 11 o 14 4 Conclusion For a spatially lat FRW universe, a contracting model ollowed by an expanding model through a bouncing point has been constructed. Two speciic solutions have been considered and their properties have been studied in detail. Acknowledgement The authors wish to place on record their sincere thanks to the reerees or their valuable comments and suggestions. The authors express their sincere thanks to D.S.T., New Delhi or the inancial assistance under the project No. P Reerences [1] C. Brans and R.H. Dicke: Phys. Rev. 14, 95 (1961). [] B. Bertotti et al.: Nature 45, 74 (00). [] S. Sen, A.A. Sen: Phys. Rev. D 6, (001). [4] S. Nojiri and S.D. Odinstov: Phys. Lett. B 599, 17 (004). [5] S. Nojiri and S.D. Odinstov: Int. J. Geom. Methods Mod. Phys. 04, 115 (007). [6] G. Dvali, G. Gabadadze and M. Portti: Phys. Lett. B 485, 08 (000). [7] A. Nicolis, R. Rattazzi, E. Trincherini: Phys. Rev. D 79, (009) [8] M. Allahverdizadeh et al.: Phys. Rev. D 81, (010). [9] A.G. Riess et.al.: Astron. J. 116, 1009 (1998). [10] A. Melchiorri et al.: Astrophys. J. Lett. 56, L6 (000). [11] L. Amendola, S. Tsujikawa: Dark Energy, Cambridge University Press, England (010). [1] M. Li, X.D. Li, S. Wang, W. Wang: Commun. Theor. Phys. 56, 55 (011). 11

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