Bouncing Cosmologies in Brans-Dicke Theory Journal: Manuscript ID cjp-016-0081.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
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.: 98.80.jk drtrilokisingh@yahoo.co.in yahoo raghav@redimail.com, rchaubey@bhu.ac.in theprabhu.09@gmail.com 1
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
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
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
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
Page 6 o 14 4.5 a.5 a,h 1.5 1 0.5 H 0 0.5 6 4 0 4 6 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
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
Page 8 o 14 γ 0 1 4 5 6 7 8 9 10 1 0 1 t Figure : The plot o γ with cosmic time t or a 0 = 0.5, β = 0.5, α = 0.5, w = /..5.5,() 1.5 1 () 0.5 10 5 0 5 10 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
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
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 = 1. 0 4 γ 6 8 10 1 0 1 t Figure 4: The plot o γ with cosmic time t or n = 0.0005, α = 0.5, a 0 = 0.5, w = /. 4.5 (),().5 0 0 10 0 10 0 0 t Figure 5: The plot o and () with cosmic time t or n = 0.0005, 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
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