Gravitational waves in Brans-Dicke Theory with a cosmological constant
aa r X i v : . [ g r- q c ] J a n Gravitational waves in Brans-Dicke Theory with a cosmologicalconstant
Hatice ¨Ozer ∗ and ¨Ozg¨ur Delice † Department of Physics, Faculty of Sciences,Istanbul University, 34134 Istanbul, Turkey Department of Physics, Faculty of Arts and Sciences,Marmara University, 34722 Istanbul, Turkey (Dated: January 12, 2021)
Abstract
Weak field gravitational wave solutions are investigated in Brans-Dicke (BD) theory in the pres-ence of a cosmological constant. In this setting the background geometry is not flat but asymptoti-cally de-Sitter. We investigate the linearised field equations, and their gravitational wave solutionsin a certain gauge choice. We will show that this theory leads to massless scalar waves as in origi-nal BD theory and in contrast to massive BD theory. The effects of these waves on free particlesand their polarization properties are studied extensively and effects of the cosmological constantis analyzed in these phenomena in detail. The energy flux of these waves are also discussed in thisbackground. By analyzing this flux, we obtain a critical distance where the waves cannot propa-gate further, which extends Cosmic no Hair Conjecture (CNC) to BD theory with a cosmologicalconstant.
PACS numbers: 04.30.-w,04.30.Nk,04.50.Kd ∗ [email protected] † [email protected] . INTRODUCTION Einstein’s theory of General Relativity (GR) is a very successful theory of gravitationwhich perfectly explains all related phenomena and passes all of the tests in the weak grav-ity regime [1]. Its predictions on strong field phenomena, such as on cosmology and blackholes, opened new windows on understanding the structure of the universe. Despite theseachievements, the research on its alternative theories does not seem to come to an end andthey were getting a lot of attention in last years [2, 3]. The motivations of these alternativetheories have several different reasons. First of all, in order to understand the mathematicalstructure and physical predictions of general relativity, its alternative theories should bestudied. In this context, we can make modifications in GR and can study mathematical andphysical consequences of these modifications. Then, we can compare the predictions of GRand its alternative theories and determine the conditions whether these theories could becompatible with the available observational data. Another motivation comes from the at-tempts to quantize gravity, which requires higher order modifications on the Einstein-Hilbert(EH) action and indicate deviations from GR [4, 5]. Some motivation comes from dark com-ponents, namely the dark matter and the dark energy, of the matter-energy composition ofthe Universe. These components were included to make GR compatible with observationson intergalactic and cosmological scales. Although these dark components might possibly beeffects of yet to be discovered particles which might be observed in the scheme of standardmodel or beyond, they certainly imply that it may be worthwhile to investigate the possi-bility that these dark components are just the effects caused by the modifications on largescales of GR. One last motivation we can list is the unification of gravity with other forces,which requires the modifications on EH action [6], as in the Kaluza-Klein theory [7, 8].After the discovery of GR by Einstein, one interesting prediction was made by Einsteinhimself in 1916 [9] that the fabric of space-time could ripple. Namely, he predicted theexistence of gravitational waves, tiny propagating ripples in the curvature of space-time.This topic becomes one of the most important topics of GR, together with black holes andcosmology. Although an indirect evidence is observed decades ago [10], their interaction witha matter distribution when passing through it is extremely tiny that their direct observationis required a century to be passed after their prediction by Einstein. In order to detectgravitational waves directly, very sensitive devices were built such as laser interferometric2ravitational wave antennas LIGO and Virgo. The first direct observation of gravitationalwaves [11] was made on 14 September 2015 by LIGO antennas. The source of these wavesis the catalystic event that the coalescence and merger of two black holes of 65 M ⊙ and 22 M ⊙ [12]. After this observation, many more gravitational waves were observed by LIGOand VIRGO collaborations corresponding to coalescence of black hole binaries, neutron starbinaries [13] in the first, second [14] and third [15] observing runs. All these observationsare a living proof that the gravitational wave observations are opened a new window tothe universe. The GW observations are compatible with GR predictions [16–18] which mayalso help to rule out or limit corresponding predictions of alternative gravity theories withinthe limits of the detectors. Hence, it is important to understand the predictions of thealternative theories [2, 3] about properties of gravitational waves [19–21] in their frameworkto estimate their viability as alternatives to GR.One of the most simple and most studied alternative theory of gravity is Brans-Dicke(BD) scalar tensor theory. In this theory, the gravitational interaction is mediated by boththe curvature of the spacetime represented by a non flat metric tensor and a also scalar field,which takes the role of Newton’s gravitational constant. In the BD theory in its originalform, the extra scalar field is a long range one. This property, together with observationalresults that sets the free parameter of this theory, the so called BD parameter, ω , to veryhigh values, ω > II. WEAK FIELD EQUATIONS
The Einstein- Hilbert (EH) action with a cosmological constant is given by, S GR Λ = Z p | g | d x (cid:20) κ ( R − L m (cid:21) (1)where κ = πGc is the gravitational coupling constant, R is the Ricci scalar, | g | is the absolutevalue of the determinant of the metric tensor g µν and Λ is the cosmological constant term.We may call the theory implied by the action (1), as General Relativity theory in the presenceof a cosmological constant ( GR Λ) theory. Throughout this study we use the units in which c = G = 1 and we use a metric signature ( − , + , + , +).Scalar tensor theories are some of the most studied alternative theories of gravity and BDtheory of gravity is the simplest one of those theories providing a very suitable test bed of theprediction of theories alternative to general Relativity. This theory includes both a scalarfield φ and metric tensor g µν to describe the gravitational interaction. Hence, gravitationalinteractions are partly due to the curvature of the space-time and are partly due to theeffect of the scalar field. In the original Brans-Dicke theory, cosmological constant was notincluded, hence Λ is equal to zero in original BD action. But, in this paper we will considerBD theory with a cosmological constant. To derive such a theory from GRΛ theory withEH action extended with a cosmological constant, we can replace the gravitational couplingconstant κ , with a scalar field, namely we can set κ → πφ − in the action (1) as donein the original BD theory. It is clear that the BD scalar field is inversely proportional tothe gravitational coupling constant. A very straightforward extension of GRΛ theory toBD scalar-tensor theory is just following the original BD prescription by replacing Newtoncoupling constant G with a scalar field φ and adding a dynamical term coupled by anarbitrary parameter ω which is called as the BD parameter. Hence the action of this theorycan be described by the following action in Jordan frame, in which the matter Lagrangianis not coupled to the scalar field, as: S BD Λ = Z p | g | d x (cid:26) π (cid:20) φ ( R − − ωφ g µν ∂ µ φ ∂ ν φ (cid:21) + L m [ ψ m , g µν ] (cid:27) , (2)where φ is BD scalar field and ω is the dimensionless BD parameter. As we have saidbefore, this action may be called as the BD theory with a cosmological constant ( BD Λ).5hen setting Λ = 0, this theory reduces to the original form of the BD theory [27]. Thecosmological and other applications of BD Λ theory were discussed in the previous studies[23, 28–40]. As φ becomes constant, this theory reduces to GR Λ theory. But this actionis not the only one which reduces to GR Λ when φ becomes constant. We can replace 2Λ φ term with an arbitrary potential term V ( φ ), so we obtain the following action in Jordanframe S BDV = Z √− gd x (cid:26) π (cid:20) φR − V ( φ ) − ωφ g µν ∂ µ φ ∂ ν φ (cid:21) + L m [ ψ m , g µν ] (cid:27) . (3)As we have said in the introduction, this action is called as the BD action with a potential,where V ( φ ) acts as a variable cosmological term [23]. If φ is set to a constant, this action alsoreduces to GR action with a cosmological constant. We may call this latter theory as BDV theory. Clearly, there is an arbitrariness in the generalization of GR with a cosmologicalconstant to BD theories. One may ask why consider the theories as different theories since2Λ φ term in (2) can be included as a special case of V ( φ ) in (3). The answer is yes, because BD Λ and
BDV theories have several important different properties in which some of themis given below: • BDΛ theory has a massless scalar field whereas in BDV theory the scalar field attainsan effective mass. • In BDΛ theory scalar field has a long range similar to the original BD theory, keepingthat property of BD theory while introducing a cosmological constant term. Whereasin BDV theory the mass of the scalar field makes the scalar field a short range one.We believe that due to the differences summarized above, these theories deserve a separateanalysis in order to investigate possible different physical consequences of these theories.However since gravitational waves of BD V theory or massive BD theory, which also includes f ( R ) theory as a special case, is already investigated in great detail in the flat backgroundcase, we only consider gravitational wave solutions of BDΛ theory and refer the works [24–26]for corresponding solutions in BDV theory.The field equations of the action (2) can be expressed as G µν + Λ g µν = 8 πφ T µν + ωφ (cid:18) ∇ µ φ ∇ ν φ − g µν ∇ α φ ∇ α φ (cid:19) + 1 φ ( ∇ µ ∇ ν φ − g µν (cid:3) g φ ) , (4) (cid:3) g φ = 12 ω + 3 (8 π T − φ ) , (5)6here T = T µµ is the trace of the energy-momentum tensor T µν and (cid:3) g is the D’Alembertoperator with respect to the metric g µν . In order to obtain the weak field expansion of theabove field equations, we can expand the space time metric and the BD scalar field as g µν = η µν + h µν , g µν = η µν − h µν , (6) φ = φ + ϕ, where η µν = diag( − , , ,
1) is the Minkowski metric, h µν is the metric perturbation tensorrepresenting small deviation from flatness, φ is a constant value of the scalar field and ϕ is a small perturbation to the scalar field i.e., | h µν | ≪ ϕ ≪
1. The linearised fieldequations can be expresses in a very economical way using the above expansion, by defininga new tensor [24] θ µν = h µν − η µν h − η µν ϕφ , (7)and considering the gauge θ µν ; ν = 0 . (8)Finally, the weak BD field equations can be expressed, up to second order , as (cid:3) η θ µν = − πφ ( T µν + τ µν ) + 2Λ η µν + 2Λ ϕφ η µν , (9) (cid:3) η ϕ = 16 πS. (10)Here τ µν is the energy-momentum pseudo tensor involving quadratic terms and (cid:3) η = η µν ∂ µ ∂ ν is the D’Alembert operator of the Minkowski spacetime. The corresponding expressions inthe presence of an arbitrary potential can be found in [24]. The term S is given by S = 14 ω + 6 (cid:20) T (cid:18) − θ − ϕφ (cid:19) − Λ4 π ( φ + ϕ ) (cid:21) + 116 π (cid:18) θ µν ϕ ,µν + ϕ ,ν ϕ ,ν φ (cid:19) . (11)To obtain the expression of S , the relation between Minkowski and curved D’Alembertoperators is used [24]: (cid:3) g = (cid:18) θ ϕφ (cid:19) (cid:3) η − θ µν ϕ ,µν − ϕ ,ν ϕ ,ν φ + O (higher order terms) . (12)In this study one of the our aim is to obtain the linearised field equations of BD the-ory in the presence of a constant background curvature coupled to the scalar field in astraightforward way. Now we will discuss these cases respectively.7 . Linearised Field Equations of Brans-Dicke Theory with a Cosmological Con-stant We will consider the weak field expanded equations of the action (2), given in equations(9,10), in the linear order in the parameters Λ , ϕ, h µν (or θ µν ) by ignoring Λ ϕ terms or othersecond and higher order terms. Then the linearised field equations become, (cid:3) η θ µν = − πφ T µν + 2Λ η µν , (13) (cid:3) η ϕ = 8 πT (2 ω +3) − φ (2 ω +3) . (14)As seen from the above equations, tensor equation has similar structure to GRΛ theory [41]but there is also a scalar field equation unlike GRΛ theory and the scalar field is masslessas in BD theory. Here the scalar field has a long range, the cosmological constant playsthe role of a background curvature as in GRΛ theory and the existence of the cosmologicalconstant does not change this property. B. Point Mass Solutions of linearised Field Equations
We have presented the linearised field equations under weak field expansion and in thecertain gauge for BDΛ theory. Before delving into gravitational wave solutions, let us reviewlocalized point mass solution of this theory. The point mass solution, in the presence ofcosmological constant or for non vanishing minimum potential, was presented and theirphysical properties were discussed comparatively in detail in [23]. Hence in the following wejust give the result in the isotropic spherical coordinates for BDΛ theory.For a point particle at the origin having the energy momentum tensor as T µν = m δ ( r ) diag(1 , , , g = − mφ r ′ (cid:18) ω + 3 (cid:19) + Λ r ′ (cid:18) − ω + 3 (cid:19) , (15) g ij = δ ij (cid:20) mφ r ′ (cid:18) − ω + 3 (cid:19) − Λ r ′ (cid:18) − ω + 3 (cid:19)(cid:21) , (16)8 = φ (cid:18) m (2 ω + 3) φ r ′ − Λ r ′ ω + 3) (cid:19) . (17)The mass term in g must be identical to weak field GR or Newton potential of a pointmass. For this reason φ must be equal to, φ = 2 ω + 42 ω + 3 (18)Also in the absence of Λ, all of the metric and field components reduce to the point masssolution of linearised BD theory [27, 42]. Moreover these solutions also reduce to linearisedGRΛ solutions in the limit of ( ω → ∞ , φ → BD Λ theory the scalar field haslong range, unlike BD V theory where mass term involves a Yukawa type term making thefield a short range one [24, 25]. III. GRAVITATIONAL WAVES IN BRANS-DICKE THEORY WITH A COSMO-LOGICAL CONSTANT
In the previous section, we have investigated BD theories whose weak field equations areof the form (13) and (14). Now we want to obtain the wave solutions of these theories,respectively. Let us expand the space time metric as, g µν = η µν + h Λ µν + h W µν , (19) h Λ ,W µν ≪ , (20)where h Λ µν is the background perturbation due to cosmological term and h Wµν is the gravita-tional wave perturbation. In the same approach the tensor θ µν and the scalar field ϕ can beexpanded as, θ µν = θ Λ µν + θ W µν , (21)and ϕ = ϕ Λ + ϕ W . (22)As it is clear from Eq. (19), (21) and (22), both background modification and gravitationalwave perturbation affects the field equations. Hence we must consider both of these contri-butions. Firstly, we calculate the effects of the background perturbation on the linearisedfield equations by ignoring possible ripples in the spacetime.9 . Background solutions For background solutions, linearised field equations (13), (14) and the gauge condition(8) take the following form, (cid:3) θ Λ µν = 2Λ η µν , (23) (cid:3) ϕ Λ = − φ ω + 3 , (24) θ Λ µν,µ = 0 . (25)The solutions of the field equations, after imposing the Lorentz gauge, become, θ Λ µν = − Λ9 x µ x ν + 5Λ18 η µν x , (26) ϕ Λ = − Λ φ ω + 3) r . (27)We have chosen the background scalar solution ϕ Λ as proportional to r in (27) rather than x = η µν x µ x ν as ϕ Λ = − Λ φ ω +3) x . This choice is made due to the fact that in the linearisedtheory we want the background solution to be static as in line with the Newtonian theory.The tensor part can be a function of time, since one can remove that dependence by asuitable coordinate transformation as in the GR case [41] but we feel that for the scalar fieldwe need to implement this staticity condition by hand. Note that unlike GR there is noBirkhoff theory for BD. Hence there is a possibility that weak field time dependent solutionsare also possible in spherical Schwarzschild coordinates in dS backgrounds. But we will notpursue this case in this paper.By reverting the equation (7), we can obtain the metric perturbation tensor as follows h Λ µν = − Λ9 x µ x ν − η µν x − η µν ϕ Λ φ . (28)In the case of vanishing scalar field, this result reduces exactly to the result of [41].Using these solutions, the space-time metric becomes, ds = − (cid:20) t − r ) − ϕ Λ φ (cid:21) dt + (cid:20) − Λ9 ( − t + 2 r + x i ) − ϕ Λ φ (cid:21) dx i + 2Λ9 t x i dt dx i − x i x j dx i dx j , (29)where i = 1 , , i = j . This line element, although it respects the gauge condition we areconsidering, is neither homogeneous nor isotropic. In order to compare this metric with the10bservations, it might be useful to transform this solution in a better known forms such as ahomogeneous and isotropic form, with the help of appropriate coordinate transformations.Firstly we apply a similar coordinate transformations to convert the solution into the staticsolution as done in GR Λ case [43]: x = ˜ x + Λ9 (cid:16) − ˜ t − ˜ x + ( ˜ y +˜ z ) (cid:17) ˜ xy = ˜ y + Λ9 (cid:16) − ˜ t − ˜ y + ( ˜ x +˜ z ) (cid:17) ˜ yz = ˜ z + Λ9 (cid:16) − ˜ t − ˜ z + ( x ′ +˜ y ) (cid:17) ˜ zt = ˜ t − Λ18 (˜ t + ˜ r )˜ t (30)where ˜ r = ˜ x + ˜ y + ˜ z . When we keep only first order terms, the resulting metric becomes, ds = − (cid:20) − Λ3 ˜ r − ˜ ϕ Λ φ (cid:21) d ˜ t + (cid:20) − Λ6 (˜ r + 3˜ x i ) − ˜ ϕ Λ φ (cid:21) d ˜ x i (31)˜ ϕ Λ = − Λ φ ω + 3) ˜ r . (32)We can obtain a spherically symmetric solution, when we apply the same change of coordi-nates presented in [41], ˜ x = x ′ + Λ12 x ′ , ˜ y = y ′ + Λ12 y ′ , ˜ z = z ′ + Λ12 z ′ , ˜ t = t ′ . (33)Under these transformations, the metric and the scalar field takes the form, ds = − (cid:20) − Λ3 r ′ − ϕ ′ Λ φ (cid:21) dt ′ + (cid:20) − Λ6 r ′ − ϕ ′ Λ φ (cid:21) ( dr ′ + r ′ d Ω ) , (34) ϕ ′ Λ = − Λ φ ω + 3) r ′ . (35)We have converted the solution into the homogeneous and isotropic form. This back-ground solution has the same result with the point mass solution of BD Λ theory presentedin [23] given in equations (15,16,17) if the mass is equal to zero.Let us apply another coordinate transformation to make this metric similar to theSchwarzschild-de Sitter (SdS ) metric, r ′ = r (1 + Λ12 r + ξ Λ φ ) ,t ′ = t. (36)11his transformation brings the line element (34) to the following expression: ds = − (cid:20) − Λ3 r − ϕ Λ φ (cid:21) dt + (cid:20) r + αrφ (cid:21) dr + r d Ω , (37)where we have made a new definition, α = dϕdr = − φ r ω + 3) . (38)The scalar field has the following form ϕ Λ = − Λ φ ω + 3) r . (39)This Schwarzschild type form of the linearised background solution may be useful for futureapplications. B. Wave solutions of BD Λ theory Now let us consider the second case where a gravitational wave solution exists by takinginto account the gravitational wave perturbations that may occur under the presence of asource which may cause fluctuations in the space-time geometry and the scalar field. Forthis case the gauge condition can be written as, θ W µν,µ = 0 . (40)Then the total wave equation is given by, (cid:3) (cid:0) θ Λ µν + θ W µν (cid:1) = 2Λ η µν . (41)The homogeneous part of the field equation (41) is sufficient to represent the gravitationalwaves. Hence we can write the wave equation as, (cid:3) θ W µν = 0 (42)Similarly for the scalar field, (cid:3) ϕ = (cid:3) ( ϕ Λ + ϕ W ) = − φ ω + 3 (43)the homogeneous part (cid:3) ϕ W = 0 , (44)12ill give the wave solution.The solution of the field Eq. (42) can be written as, θ W µν = A µν sin kx + B µν cos kx (45)where A µν and B µν are amplitude tensors and k = ( k , ~k ) is a wave four-vector, i.e., kx = k µ x µ = k t + ~k.~x . If we plug the solution (45) into the Eq.(40) and (42), we find, k µ A µν = k µ B µν = 0 , (46) k = k µ k µ = 0 , (47) A W µµ = B W µµ = 0 . (48)Eq. (46) shows that the amplitude tensors A µν and B µν are orthogonal to the directionof the propagation of the wave and Eq. (47) shows the fact that the gravitational wavespropagates at the speed of light. Hence, considering also (48) we can conclude that thetensorial part of the gravitational waves in BD Λ theory are transverse and traceless wavesmoving with the speed of light.The solution (45) describes a wave with the angular frequency, ν = k = ( k x + k y + k z ) . (49)The solution of the scalar field equation (44) is given by, ϕ W = C sin kx + D cos kx, (50)where C and D are integration constants. Using Eq. (45) and (7), the total metric pertur-bation induced by some source of GW can be written as h Wµν = A µν sin kx + B µν cos kx − η µν φ ( C sin kx + D cos kx ) . (51)The total solution of linearised field equations of BD theory involving a cosmologicalconstant can be written for the tensor θ µν as θ µν = θ Λ µν + θ W µν = − Λ9 x µ x ν + 5Λ18 η µν x + A µν sin kx + B µν cos kx (52)and finally the metric perturbation tensor becomes h µν = h Λ µν + h W µν = − Λ9 x µ x ν − (cid:18) x ϕ Λ φ (cid:19) η µν + A µν sin kx + B µν cos kx − η µν φ ( C sin kx + D cos kx ) (53)13here the scalar field ϕ Λ is given in (27).The total scalar field solution in a constant curvature background is given by ϕ = ϕ Λ + ϕ W = − Λ φ ω + 3) r + C sin kx + D cos kx. (54) C. The Effects of GW on Free Particles in BD Λ theory Here, we want to understand the effect of the gravitational waves in BD Λ theory on freeparticles or detectors. Since the weak equivalence principle holds for BD theory a singleparticle cannot feel the metric perturbations. The simplest way to understand the physicaleffects of GW on matter is to consider the relative motion of two nearby test particles infree fall. Hence, let us consider two nearby freely falling particles of equal mass and specifythe spacetime coordinates of these particles with ( t, x, y, z ) [26, 44]. Then, we calculate thegeodesic deviations caused by the GW. The geodesic deviation equation is defined as [45] d ζ α dτ = R αβνχ U β U ν ζ χ , (55)where ζ µ is the vector connecting the particles , U µ is the four velocity of the two particlesand in the rest frame of the observer [47], U µ = (1 , , , , (56)and we assume a wave moving along the z direction, then the wave vector becomes, k µ = ( ν, , , ν ) . (57)Inserting Eq. (56) into the Eq. (55), we obtain, d ζ α dτ = − R α i ζ i . (58)This result shows that the Riemann tensor is locally measurable by calculating the separationbetween nearby geodesics.For the calculational simplicity, here we consider a gravitational wave moving along z direction. We transform the trigonometric functions such that A µν sin( kx ) + B µν cos kx =˜ A µν cos( kx − δ ) where δ is phase of the wave and we also set δ = 0 for simplicity. We also14se that for waves moving on z direction kx = ν ( t − z ). Then the general solution can besimply written as h µν = h Λ µν + h Wµν = − Λ9 x µ x ν − η µν x − η µν ϕ Λ φ + (cid:16) ˜ A µν − η µν ˜ D (cid:17) cos ν ( t − z ) (59) ϕ = ϕ Λ + ϕ W = − Λ φ ω + 3) r + D cos ν ( t − z ) . (60)where ˜ A µν is the amplitude tensor adapted to the problem by taking real part of the fullsolution with an appropriately chosen phase and ˜ D = D/φ is the real part of the scalaramplitude constructed similarly.Since Riemann tensor is gauge invariant, we can use the linear form of Riemann tensorin TT gauge [46] to calculate the nonvanishing components of this tensor given by2 R abcd = h ad,bc − h bd,ac + h bc,ad − h ac,bd Using this linearised form of Riemann tensor and the solution (59), the necessary com-ponents are found as R x x = 12 (cid:20)(cid:16) ˜ A xx − ˜ D (cid:17) ν − (cid:18) − ω + 3 (cid:19)(cid:21) , (61) R y x = R x y = 12 ν ˜ A xy , (62) R y y = 12 (cid:20)(cid:16) − ˜ A xx − ˜ D (cid:17) ν − (cid:18) − ω + 3 (cid:19)(cid:21) , (63) R z z = 12 (cid:18) − (cid:18) − ω + 3 (cid:19)(cid:19) (64) R z x = R z y = 0 . (65)Now, suppose that the first particle is at the origin and second one is at the point ζ i = ( ζ , , ζ and these particles areinitially at rest. We will analyze how this seperation changes in the presence of an incidentgravitational wave propagating in the z direction. In this case, the relative acceleration offreely falling test particles become, d ζ x dτ = − ζ R x x = ζ (cid:20) − (cid:16) ˜ A xx + ˜ D (cid:17) ν + 2Λ3 (cid:18) − ω + 3 (cid:19)(cid:21) , (66) d ζ y dτ = − ζ R y x = − ζ ν ˜ A xy , (67) d ζ z dτ = − ζ R z x = 0 . (68)15imilarly, the relative acceleration of two nearby freely falling particles seperated by ζ inthe y direction become, ¨ ζ x = − ζ ν ˜ A xy (69)¨ ζ y = ζ (cid:20) ( ˜ A xx − ˜ D ) ν + 2Λ3 (cid:18) − ω + 3 (cid:19)(cid:21) (70)¨ ζ z = d ζ z dτ = 0 . (71)For the third case, lets consider two nearby particles initially seperated by ζ in the z direction.The relative acceleration obey, ¨ ζ x = 0 , (72)¨ ζ y = 0 , (73)¨ ζ z = ζ (cid:20) (cid:18) − ω + 3 (cid:19)(cid:21) . (74)The terms proportional to Λ mimicks usual homogeneous expansion of the universe due tothe cosmological term. The terms proportional to ˜ A xx and ˜ A xy denotes the effects of theusual plus and cross polarizations of the gravitational wave and the terms proportional to˜ D corresponds to the scalar breathing mode of the Brans-Dicke theory. So a gravitationalwave detector feels extra stretching between arms due to the cosmological constant which ispractically zero for a detector in the surface of earth due to the smallness of the cosmologicalconstant. The effect of the BD field is to reduce the amount of this stretching compared toGR and for the critical value ω c Λ = − ω < − ω → ∞ . This effect gets smaller by decreasing ω which vanishesfor ω = ω c Λ and becomes a tension with values of the BD parameter smaller that ω c Λ .Considering all these, we see that, if the wave is propagating along the z direction, themetric perturbation can be expressed as a sum of three polarization states, h Λ µν ( t − z ) = A + ( t − z ) e + µν + A × ( t − z ) e × µν + Φ( t − z ) η µν (75)where e + µν and e × µν denote the usual plus and cross polarization tensors, respectively of16ravitational waves, and A + , A × are amplitudes of these tensors [47]: e + µν = − , e × µν = , η µν = − . (76)In other words, in BDΛ theory, there are two massless spin 2 modes and one massless scalarmode [24–26, 48]. In the geodesic deviation equations, ˜ A xx term denotes the plus modeand ˜ A xy term denotes the cross mode. The terms involving the cosmological constant Λgives the contribution of the cosmological constant on the polarization states. Also, theterms proportional to ˜ D denote the breathing mode resulting from the massless scalar field.Therefore, we can conclude that the BDΛ theory has three polarization states, two of whichdenote plus and cross polarization states and the other one denotes breathing mode causedby the massless scalar field. This is what we were expecting. This result is in agreement withthat obtained in original BD theory without a cosmological constant or a potential and aswe have seen from the analysis of BDΛ theory in [23], the cosmological term does not attaina mass to the scalar field in local gravitational sectors. As we have shown explicitly, thisproperty persists for gravitational waves as well. Besides, the effect of the existence of thecosmological constant on the geodesic deviation equations is the same amount of cosmologicalacceleration in all directions, mimicking the homogeneous cosmological expansion due tocosmological constant.In order to see these results in an other perspective, we transform the wave solutions intoFRW coordinates. Hence we may also apply the following change of coordinates [43], x i = e T √ Λ/3 X i ,t = q Λ3 R + T , (77)where X,Y,Z are comoving coordinates and R = √ X + Y + Z . As a result of the17alculations, the transformed wave-like solution to order √ Λ becomes, h Wµν = ˜ D A xx − ˜ D ) (cid:16) q Λ3 T (cid:17) ˜ A xy (cid:16) q Λ3 T (cid:17)
00 ˜ A xy (cid:16) q Λ3 T (cid:17) ( − ˜ A xx − ˜ D ) (cid:16) q Λ3 T (cid:17)
00 0 0 − ˜ D (cid:16) q Λ3 T (cid:17) × cos ν ( T − Z ) + ν r Λ3 (cid:18) Z − T Z (cid:19) + O (Λ) ! (78)These results clearly shows the effects of the GR wave on the spacetime. The threepolarization degrees of freedom will have same dispersion relation ν ( T − Z ) + ν r Λ3 (cid:18) Z − T Z (cid:19) = n π, (79)which yields approximately Z max ( n, T ) ≃ T − nπν − r Λ3 (cid:18) T − n π ν (cid:19) (80)In these expressions the Brans-Dicke parameter does not enter in the equations. Hence,these expressions showed that the behaviour of tensor part of gravitational waves behaveexactly the same as in GRΛ theory [41] regarding the frequency of the waves. The scalarwaves which does not exists in GR also shows same behaviour with the tensor ones.In order to obtain the linearised solutions in FRW type coordinates which can representthe universe we live in, we have first solved the linearised equations in the coordinates andthe gauge where the linearised equations are valid. The transformed solutions do not satisfythe linearised BDΛ field equations. The important point is the first order correction in FRWtype coordinates in the cosmological constant is in the order of √ Λ rather than Λ as in thecoordinates respecting linearised equations. Hence, for the observational point of view theeffects of Λ is much more relevant despite the smallness of the observed value of cosmologicalconstant.
IV. ENERGY-MOMENTUM TENSOR OF GRAVITATIONAL WAVES
In this section we calculate the gravitational energy carried by the gravitational waves inBDΛ theory as well as the energy of the background geometry. To calculate the background18nergy due to cosmological constant in BD theory we first expand metric tensor as g µν = g ( b ) µν + h µν where g ( b ) µν is a background metric. The scalar field is also expanded as φ = φ + ϕ as given in (6). Using these, then we express relevant scalars and tensors in terms of themas in the expansion of the Ricci tensor is R µν = R ( b ) µν + R (1) µν + R (2) µν + . . . . Using these we canexpress the field equations in the orders of O ( h, ϕ ). We use the following expansion of theRicci tensor in the first and second order [1], where the bar denotes covariant differentiation,in our calculations: R (1) µν = 12 (cid:0) − h | µν − h αµν | α + h ααµ | ν + + h ααν | µ (cid:1) , (81) R (2) µν = 12 (cid:20) h αβ | µ h αβ | ν + h αβ (cid:0) h αβ | µν + h µν | αβ − h αµ | νβ − h αν | µβ (cid:1) (82)+ h α | βν (cid:0) h αµ | β − h βµ | α (cid:1) − (cid:18) h αβ | β − h | α (cid:19) (cid:0) h αµ | ν + h αν | µ − h µν | α (cid:1)(cid:21) . (83)We also consider the fact that the linearized equations are satisfied for first order termsin the field equations and the second order terms contribute to the energy of the backgroundgeometry. Therefore the energy momentum tensor due to cosmological constant in BD theorycan be calculated by subtracting first order terms from total expressions keeping only thesecond order terms. We choose background geometry as the flat Minkowski spacetime. Theenergy momentum tensor of Λ, hence becomes: t µν = φ π (cid:18) R (2) µν − η µν R (2) + 12 η µν h αβ R (1) αβ − Λ h µν + T µν [ ϕ ] (cid:19) (84)with the contribution of the BD scalar to the energy-momentum tensor of the waves isidentified as T µν ( φ ) = ωφ (cid:18) φ ,µ φ ,ν − g µν φ ,α φ ,α (cid:19) + 1 φ ( φ ,µν − g µν (cid:3) φ ) (85)where T µν [ ϕ ] in (84) represents the second order contributions due to the scalar and metricperturbations ϕ and h µν to T µν ( φ ) given in (85).After a long calculation of all relevant tensorial components of the background solution(27,28) we have found the Poynting vector due to cosmological constant as t Λ0 z = φ π (cid:20) − ω + 3) (cid:21) Λ t z (86)where the effect of the scalar field is in the negative for ω > − / ω < − / L , representsthe typical curvature radius and the second one, λ , represents a typical wavelength of thewaves with the assumption that λ ≪ L . Essence of this method considers again the splittingof the metric of the spacetime into a background metric which is a slowly changing functionof spacetime and also the metric perturbation h µν representing high frequency waves with α being the amplitude of waves. Now the perturbation metric satisfies h µν g ( B ) µν × α .In this approximation the derivatives of the metric components vary as g ( B ) µν,α g µν L , , and h µν,α h µν λ . Using steady coordinates, we find the typical order of magnitudes of Riccitensors as R ( B ) µν ∼ αλ , R (1) µν ∼ αλ , R (2) µν ∼ α λ . For our calculations we choose backgroundas Minkowski spacetime, and also split h µν = h Λ µν + h Wµν , signifying the cosmological andwave contributions to the metric perturbation tensor. We can also split the scalar field as φ = φ ( b ) + ϕ , where ϕ may contain wave part as well as other perturbative corrections tothe chosen background scalar φ ( b ) . In this paper we choose φ b = φ and ϕ = ϕ Λ + ϕ W whereformer denote the cosmological background and latter denote the wave contribution to thescalar field.The short wave approximation requires solving the vacuum BDΛ equations under thecircumstances summarized in the previous paragraph. This implies that the field equa-tions already satisfy the linearized equations. Hence the contribution to the waves to theenergy-momentum tensor shows themselves in the second order. Second order field equa-tions contain parts due to cosmological background as well as due to the fluctuations ofthe spacetime and the scalar field. For example, the Ricci tensor can be written as partscontaining background free of ripples as well as parts containing the wave part as follows:The smooth part has the terms: R µν = R ( b ) µν + < R (2) µν > + error and the fluctuating partcontains: R (1) ( h ) + R (2) µν ( h ) − < R (2) µν ( h ) > + error. Note that in GR, both expressions equalto zero due to the vacuum equations require R µν = 0, separately. However, in BD there isalso scalar contributions to the field equations that we will considered in the following. Theoperation < ... > symbolizes averaging of quantity, which is required since the linearizedtensors that we are considering such as Ricci tensor and also the energy-momentum tensorare not gauge invariant in these linearized forms. By averaging over several wavelengths,we may hope to include enough curvature in a small region to make these quantities gaugeinvariant [1, 49–51]. We can identify, as done in [1, 49–51] in GR, the background Einsteintensor in vacuum proportional to the energy-momentum tensor of the waves, meaning that20he energy-momentum of the waves generates the background curvature. Hence, we obtainthat G ( b ) µν ≡ R ( b ) µν − g ( b ) µν R ( b ) = 8 π T ( W ) µν (87)where the energy momentum tensor of the gravitational waves is given by T Wµν = φ π (cid:26) − < R (2) µν ( h ) > + 12 g ( b ) µν < R (2) ( h ) > + 12 h µν R (1) − Λ < h µν > + < T (2) µν ( ϕ ) > (cid:27) (88)where again T µν ( ϕ ) represents the second order contributions of the BD scalar due to waveparts of the perturbations into energy-momentum tensor of gravitational waves identified in(85) and evaluated at the second order in h µν and ϕ .Using the tensor θ µν defined in (7) along with gauge (8) simplifies the energy-momentumtensor given in (88). Moreover, we use an important advantage of using averaging processthat the averages of first derivatives vanish as < ∂ µ X > = 0, which also implies that = − < ( ∂ µ A ) B > . After a long calculation using all these, we found the result t Wµν = φ π (cid:26)(cid:10) θ αβ | µ θ αβ | ν (cid:11) + (4 ω BD + 6) φ (cid:10) ϕ | µ ϕ | ν (cid:11) − h Λ h µν i (cid:27) , (89)which agrees with previous results [24, 25, 48] derived using other methods. Evaluatingthis formula for the solution that we have derived in (59) corresponding to gravitationalwaves moving in z direction, and calculating the averages gives the following result for thePoynting vector t W = φ π (cid:20) − ν (cid:26) | A + | + | A × | + (2 ω BD + 3) φ | D | (cid:27)(cid:21) (90)where we have only used wave parts of the solutions to avoid double counting since thepointing vector for cosmological constant is already calculated in (86).The total Poynting vector combining contributions of cosmological constant and GRwaves becomes t = φ π (cid:26)(cid:18) − ω + 3) (cid:19) Λ t z − ν (cid:20) | A + | + | A × | + (2 ω BD + 3) φ | D | (cid:21)(cid:27) (91)where ν is the frequency of the wave and A + and A × are plus and cross polarization ten-sors, respectively. The first two terms depend on the cosmological constant and shows thecontribution of the cosmological constant to the energy flux. The third and fourth termsare the energy flux of GR theory [1, 52] and the last term is the contribution of the masslessBD scalar field to the energy flux [24, 25, 48].21ow we can define a critical distance as in [53, 54] as the distance where t vanishes,given by t Λ03 = t W . Since both scalar and gravitational waves move with speed of lightin BDΛ theory, we can identify the time with a distance in (86). In terms of observablefrequency f = ν/ π , and curvature radius of dS spacetime L = p / Λ, the critical distancebecome L c = 2 πf A √ a Λ , (92)where a = 8881 − ω + 3) , A = s | A + | + | A × | + (2 ω BD + 3) φ | D | . (93)The expression we have derived is similar to what obtained in [53, 54] with some differ-ences in numerical factors. The critical distance result is valid in the region ω < − / ω > − /
22 due to √ a term. The effect of the cosmological constant is to hinder the prop-agation of gravitational waves distances beyond this length. The waves cannot propagatebeyond L c , since space asymptotically goes to dS spacetime where the inhomogeneities suchas gravitational waves vanish. As discussed in [54], this is due to cosmic no-hair conjecture(CNC) [55], which requires that the inhomogeneities to be dissipated at large distances andlarge times. Since r Λ = p / Λ and f = 1 /λ ≈ r Λ [1], we see that the critical distance isproportional to dS radius and hence the background scale of the spacetime , L c ∼ r Λ . Thisproves that the CNC is also valid for gravitational waves in BD theory in the presence ofa positive cosmological constant where inhomogeneities cannot be propagated further fromdS radius of the spacetime. We have, in this paper, extended this result to the BD theoryin the presence of the cosmological constant Λ. A confirmation of this result can be doneas a future work by studying the global structure of the Robinson-Trautman type radiativespacetime solutions in BD theory [56, 57], as done in [58]. V. CONCLUSION
In this study weak field gravitational wave solutions of Brans-Dicke theory with a cosmolog-ical constant (BDΛ) theory is obtained in the presence of a positive cosmological constant.It is known that the scalar field is massless for this theory and hence the range of the scalarfield is still a long range one, a behaviour contrary to masssive BD theory where the exis-tence of an arbitrary potential introduces an effective mass and makes the scalar field short22ange one. Hence, in this theory, the effect of the cosmological constant is to behave like aconstant background curvature. Therefore, in this paper, what we have found are the weakfield wave solutions on the spacetime having a constant curvature background in BD theory.After obtaining these solutions we have discussed their physical properties. We haveseen that these waves behave similarly to gravitational waves in the original BD theory,rather than massive BD or f ( R ) theories. The weak field gravitational effects of wavelike perturbations on constant positive background curvature of BD theory consist of threeparts. We have two tensorial waves having usual plus and cross polarizations. These wavesare traceless and transverse waves moving with speed of light. The third component, thescalar wave, is massless and moves also with speed of light.We have studied the effects of these waves on test particles and dedectors by calculatingthe geodesics deviation by the gravitational wave of two test particles initially at rest. Wesee that, we have usual plus and cross polarizations as in GR and in addition there is a scalarbreathing mode as in the BD theory. We see that the combined effect of the cosmologicalconstant and the background scalar field is to exert an homogeneous expanding force on theseparticles to extend the length between them similar to Hubble expansion of two galaxiesdue to the dark energy. Note that the force due to scalar background is attractive and actsin the opposite direction which increases with decreasing BD parameter, ω , such that at ω = − ω < −
1, although we have a positive cosmological constant, the force between thesetest particles become attractive.We have also calculated the energy-momentum tensor of gravitational waves for the BDΛtheory calculated by using the showtwave approximation method. The obtained resultsgive the energy-momentum tensor due to the background curvature and background scalarfield as well as those of gravitational waves of this theory. Although the energy momentumtensor of gravitational waves in the original and massive BD theories is well known, wehave obtained those in the BDΛ theory using the shortwave approximation method whichis a different method than the method used to calculate the previous results in original andmassive BD theories in for example [24]. The energy of the background fields is new inthis theory. Analyzing the energy flux for waves propagating in a particular direction, wesee that gravitational waves can not be propagated distances further than the backgroundscale, namely that L c ≈ r Λ , given in (92). This results extend the Cosmic No hair Theorem23CNC) to BD theory in the presence of a cosmological constant for the region ω < − / ω > − / ACKNOWLEDGMENTS
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