Linearized modified gravity theories with a cosmological term: advance of perihelion and deflection of light
LLinearized modified gravity theories with a cosmological term: advance of perihelionand deflection of light
Hatice ¨Ozer ∗ Department of Physics, Faculty of Sciences, Istanbul University, Istanbul, Turkey ¨Ozg¨ur Delice † Department of Physics, Faculty of Arts and Sciences, Marmara University, Istanbul, Turkey (Dated: February 7, 2018)Two different ways of generalizing Einstein’s general theory of relativity with a cosmologicalconstant to Brans-Dicke type scalar-tensor theories are investigated in the linearized field approxi-mation. In the first case a cosmological constant term is coupled to a scalar field linearly whereasin the second case an arbitrary potential plays the role of a variable cosmological term. We see thatthe former configuration leads to a massless scalar field whereas the latter leads to a massive scalarfield. General solutions of these linearized field equations for both cases are obtained correspondingto a static point mass. Geodesics of these solutions are also presented and solar system effects suchas the advance of the perihelion, deflection of light rays and gravitational redshift were discussed. Ingeneral relativity cosmological constant has no role on these phenomena. We see that for the Brans-Dicke theory the cosmological constant has also no effect on these phenomena. This is because solarsystem observations require very large values of the Brans-Dicke parameter and the correction termsto these phenomena becomes identical to GR for these large values of this parameter. This result isalso observed for the theory with arbitrary potential if the mass of the scalar field is very light. Fora very heavy scalar field, however, there is no such limit on the value of this parameter and there areranges of this parameter where these contributions may become relevant in these scales. Galacticand intergalactic dynamics is also discussed for these theories at the latter part of the paper withsimilar conclusions.
PACS numbers: 04.25.Nx,04.50.Kd ∗ [email protected] † [email protected] a r X i v : . [ g r- q c ] F e b I. INTRODUCTION
The remarkable observation at the end of 20th century showed that we live in an accelerating universe [1–3]. Ac-cording to well tested theory of gravitation, namely Einstein’s general relativity (GR) theory, this cosmic acceleratingexpansion is caused by a mysterious component of the universe called dark energy. The most clear candidate of darkenergy is Einstein’s cosmological constant, since, this observed behavior of the universe is compatible with a verysmall positive cosmological constant, i.e., Λ ∼ − m − [1–3]. Another candidate is quintessence in the form of aminimally coupled scalar field which varies slowly along its potential [4–6]. For a review on quintessence, see, forexample, see [7]. Considering thus the fact that we live in an asymptotically de Sitter universe, it might be reasonableto investigate the possible effects of a positive cosmological constant into local and global behavior of the universe.Therefore, it is logical to investigate whether such a cosmological constant, despite its smallness, affects the localgravitational phenomena such as bending of light from distant objects or the advance of perihelion of objects inbound orbits.Alternative theories of GR has been a very popular field of research, especially over recent decades. There are severaltheoretical or observational motivations that exist for this active field of research. One of them is to understand themathematical structure, physical predictions and behaviour of GR by studying its alternatives. Another one is thequantization of gravitational interaction, and the fact that it may require some modifications to GR [8, 9]. One morereason is the idea of unification of fundamental interactions, generalizing the Kaluza-Klein idea of unifying gravityand electromagnetism [10, 11] into all interactions, such as string theory [12]. Such attempts require the ideas of theexistence of extra dimensions and compactification. Such a compactification of higher dimensional theories into fourdimensions usually produces a scalar field called dilaton into the four dimensional effective theories [10–12]. Apartfrom these, modified gravity theories, such as f ( R ) theory, are also popular to investigate the possibility that theaccelerating universe may be explained by large scale modifications to GR, without needing a dark energy. We referto the latest reviews for the further motivations and developments of these theories [13–17].Brans-Dicke (BD) theory [18–20] is one of the most simple modifications to GR and usually considered as a suitabletest bed for investigating the effects of possible modifications to GR. After its presentation more than half a decadeago, the properties and outcomes of this theory is investigated in great detail [21–23]. For example, its weak fieldsolution for point particle is obtained and two most interesting weak field phenomena, namely the perihelion precessionof Mercury and light deflection by the Sun is investigated in the original papers of this theory [20, 21]. In its originalform, as we will discuss in the next section, BD theory does not involve neither a cosmological constant or a potentialterm. However, in the later years those extensions were also discussed, mostly in the cosmological scheme.In this paper, we investigate weak field solutions of theories which generalizes Einstein’s general relativity witha cosmological constant to the Brans-Dicke type scalar-tensor theory. Since, as we will discuss in the next section,this generalization can be made, at least, in two different ways, we will consider both cases, separately. There aremany works considering weak field solutions, properties of these solutions and astrophysical implications for differentmodified gravity theories [24–36]. However, in most of the works, except for example [30–32], asymptotic flatness isassumed. In [31] a post-Newtonian extension of BD theory with a potential was presented. Our motivation in thispaper is to investigate the weak field solutions of BD theory in the presence of an asymptotically de Sitter background.This will enable us to shed light into the effects of background curvature on the dynamics of the space time in thepresence of positive cosmological constant in these theories. We transform the linearized field equations in a knownsuitable gauge which makes scalar and tensor equations decouple from each other and makes it easier to obtain thesolutions. We will solve these equations for a static point particle in the coordinates where this gauge is valid forboth cases and transform the obtained solutions into isotropic or Schwarzschild-like coordinates where this gauge isnot valid. In order to obtain physical properties of these solutions, we will discuss the geodesics of both solutionsin Schwarzschild type coordinates. Advance of the perihelion of test particles around this point particle, deflectionof light rays by this point particle in the presence of a curvature background and also the gravitational redshift andgalaxy rotation curves and intergalactic dynamics will be discussed for both solutions. Contribution of the mass of thesource, the cosmological term or the minimum of the potential to these phenomena will be derived using appropriatemethods. The paper is organized as follows. In the next section, we will discuss two different ways of generalizing GRwith a cosmological constant to BD theory and obtain the weak field equations in a chosen gauge for both cases. Insection (III) we will present a static point particle as a source, solve the field equations for both cases in the chosengauge and transform the solutions to the isotropic and Schwarzschild type coordinates. In section (IV) we will obtainradial geodesic equations in Schwarzschild coordinates. We will investigate solar system effects such as the advanceof perihelion in section (V), deflection of light rays in section (VI) and gravitational redshift in section (VII) for bothof the theories. Galactic and intergalactic dynamics is considered in section (VIII). The paper ends with a briefdiscussion. II. WEAK FIELD EQUATIONS
According to Einstein’s general theory of relativity (GR), the gravitational phenomena can be explained by thefollowing action S GR Λ = (cid:90) (cid:112) | g | d x (cid:20) κ ( R − L matter (cid:21) . (1)Here, Einstein’s famous modification of adding a cosmological constant term to the action is already included. We maycall this theory as GRΛ theory. Here κ = 8 πG/c is the gravitational coupling constant, T µν is the energy-momentumtensor, R is the Ricci scalar and Λ is the cosmological constant term and we choose the units where c = G = 1 in thispaper. One of the most studied alternative of GR is the Brans-Dicke (BD) scalar-tensor theory [20], where the Newtongravitational constant κ is replaced by a scalar function as κ → πφ − together with addition of a kinetic term forthis scalar field coupled by a dimensionless constant known as the BD parameter ω . In the original derivation of theBD theory, cosmological constant Λ is set to zero. However, if one wants to extend GR theory with a cosmologicalconstant to scalar-tensor theories, the most straightforward way is to replace κ → πφ − in action (1), similar to theoriginal BD theory. This yields the following action in Jordan frame S BD Λ = (cid:90) (cid:112) | g | d x (cid:26) π (cid:20) φ ( R − − ωφ g µν ∂ µ φ ∂ ν φ (cid:21) + L matter (cid:27) . (2)This action where all curvature related terms R and Λ is coupled with scalar field φ in the same manner, is knownas the Brans-Dicke theory with a cosmological constant [37–39] and its cosmological [40–46], cylindrical [47–49] andother [50] applications were discussed in previous works. Here the value of Λ in BDΛ theory may be different fromits value in GRΛ theory. We call the action (2) as the BDΛ action. Note that as φ becomes a constant, this theoryreduces to GRΛ theory. However, this action is not the only action which reduces to GRΛ when φ is set to a constant.One can replace 2Λ φ term with an arbitrary potential term V ( φ ) to obtain the following action S BDV = (cid:90) (cid:112) | g | d x (cid:26) π (cid:20) φR − ωφ g µν ∂ µ φ∂ ν φ − V ( φ ) (cid:21) + L matter (cid:27) . (3)which can be called as the BD action with a potential, i.e., BDV theory. Here, V ( φ ) plays the role of a variablecosmological term and when φ is set to a constant, the action (3) also reduces to GRΛ theory. There are variousworks considering the addition of such a potential term and its various implications to BD theory [23, 27, 35, 36, 51–55]in various contexts. In general BDΛ and BD V theories and also all possible BD V theories having different potentialsmay have different characteristics and lead to different physics. Hence, there is an arbitrariness in the generalization ofGRΛ theory to scalar-tensor theories. One possible method of identifying differences of different gravitation theoriesis to obtain their weak field solutions and compare with the results of GR. Therefore, here we want to discuss the weakfield solutions of the theories (2) and (3) in the presence of a constant curvature background. Since these differentchoices may have different characteristics, we have to consider these theories separately, though we try to use a unifiedtreatment as much as possible in the text. For example, when we discuss the field equations and their linearizationof the actions (2) and (3) below, to avoid repeated similar equations, we will present those equations for the action(3) but keep in mind that for the case (2) we have to replace V ( φ ) = 2Λ φ in the relevant equations. We will presentthe result of both theories separately whenever their distinction is important.The extended BD actions (2,3) we are considering can also be expressed in other frames [22, 23], such as Ein-stein or string frames, by considering appropriate conformal transformations. For example, the following conformaltransformation, ˜ g µν = φ g µν , (4)and the redefinition of the scalar field ˜ φ = (cid:114) ω + 316 π ln φ, (5)bring the BDV action into the Einstein frame as given by S EV = (cid:90) (cid:40)(cid:112) | ˜ g | d x (cid:34) ˜ R π −
12 ˜ g µν ∂ µ ˜ φ ∂ ν ˜ φ + V [ φ ( ˜ φ )][ φ ( ˜ φ )] (cid:35) + [ φ ( ˜ φ )] − L matter (cid:41) . (6)In this frame, the role of the scalar field is changed from being a part of gravitational interaction to a canonicalscalar-field matter-energy distribution permeating all points of the spacetime. Moreover, it also couples to the matterLagrangian nonminimally. Hence, the manifold is no longer Riemannian and the test particles do not follow geodesics.In Jordan frame, however, the scalar field is a part of gravitational interaction. Therefore, the theory is a metrictheory and test particles follow geodesics in this frame. Actually, there was a debate on which of these frames arephysical or are they equivalent or not. For a review of this debate, see for example [56]. Here we share the originalidea of transformation of units [20] of Brans and Dicke which says that both frames are equivalent and give samephysical results. After some works concerning this debate, there seems to be a concencus on that all these framesare mathematically and physically equivalent [57–62]. Namely, a physical quantity measured on a certain frame doesnot depend on the chosen frame, if the transformations between frames is properly used. Hence, the choice of frameis a matter of convenience since some calculations can be more easily performed in a particular frame. Therefore, inthis work, since we will investigate the motion of test particles in the later stages of this paper, we prefer to work inthe Jordan frame. This is because this frame has a calculational advantage, since in this frame test particles followgeodesics and we do not want to deal with the fifth force arising from the modifications of geodesics equations inEinstein frame.The action of a modified gravity theory ( f ( R ) theory) that is very popular in the recent years [13–17] is S = 12 κ (cid:90) (cid:112) | g | d x ( f ( R ) + L matter ) . (7)It is well known [13–17] that under a Legendre transformation the f ( R ) theories become equivalent to the BDV theory(3) for specific values of ω , namely ω = 0 for metric and ω = − / f ( R ) theories. Thus, BDV theorywith arbitrary ω leads to a more general treatment, includes both f ( R ) theories as special cases. Therefore, we willconsider the theory (3) without any restriction on ω in this paper, except for ω = − / ω = − / f ( R ) theory can be recovered from BDV theory by setting ω = 0 in the resulting expressions.The Jordan frame field equations of the action (3) can be written as 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 ) − V ( φ )2 φ g µν , (8) (cid:3) g φ = 12 ω + 3 (cid:18) π T + φ dV ( φ ) dφ − V ( φ ) (cid:19) , (9)where here T is the trace of the matter energy-momentum tensor T µν and (cid:3) g is the D’Alembertian operator withrespect to the full metric. Now, let us consider the weak field expansion of the above field equations. Hence, the spacetime metric and the BD scalar field can be expanded as g µν = η µν + h µν , g µν = η µν − h µν , (10) φ = φ + ϕ, where η µν = diag( − , , ,
1) is the Minkowski metric, h µν is the tensor representing 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 | (cid:28) ϕ (cid:28) θ µν = h µν − η µν h − η µν ϕφ , (11)together with the gauge θ µν ; ν = 0 , (12)the weak field BD field equations, up to second order , become [27] (cid:3) η θ µν = − πφ ( T µν + τ µν ) + V lin φ g µν , (13) (cid:3) η ϕ = 16 πS. (14)Here τ µν is the energy-momentum pseudo tensor involving quadratic terms and (cid:3) η = η µν ∂ µ ∂ ν is the D’Alembertianof the Minkowski spacetime. The term S is given by S = 14 ω + 6 (cid:20) T (cid:18) − θ − ϕφ (cid:19) + 18 π (cid:18) φ dVdφ − V (cid:19) lin (cid:21) + 116 π (cid:18) θ µν ϕ ,µν + ϕ ,ν ϕ ,ν φ (cid:19) . (15)where in deriving S the relation between Minkowski and curved D’Alembertian operators is used (cid:3) g = (cid:18) θ ϕφ (cid:19) (cid:3) η − θ µν ϕ ,µν − ϕ ,ν ϕ ,ν φ + O (higher terms) . (16)Here subtext lin in equations (13) and (15) means that these terms must also be properly linearized. In our setting,we want to discuss the case where the space time is not asymptotically flat but asymptotically de Sitter, therefore weneed an effective cosmological constant in the linearized field equations. Since this term can be obtained for either ofthe actions (2) and (3) differently, now we have to discuss these cases separately. A. Linearised field equations of Brans-Dicke Theory with a cosmological constant (BD Λ case) For the action (2), the terms involving the potential V can be expanded as V ( φ ) ≈ φ , (cid:18) φ dVdφ − V (cid:19) ≈ − φ , (17)where the terms linear in Λ or ϕ are kept and the terms of the order Λ ϕ or higher are ignored. Using these, we obtainthe following linearised field equations (cid:3) η θ µν = − πφ T µν + 2Λ η µν , (18) (cid:3) η ϕ = 8 π T ω + 3 − ω + 3 Λ φ . (19)In the chosen parametrization (11) and gauge (12), the tensor equations (18) have similar structure to weak field GR Λequations [63]. Thus, all the differences from GR , namely the effects of the BD scalar field will be originated fromscalar field equation (19). The first important observation is that for this case the scalar field is massless . This meansthat BD Λ theory has the similar structure with BD theory where the scalar field has a long range and the existenceof a cosmological constant does not change this behavior. Hence, in BD Λ theory the cosmological constant does notchange the local behavior of the scalar field and acts as a background curvature similar to GR and responsible for theasymptotical nonflatness. Therefore, BD Λ theory is a natural generalization of GR Λ theory to BD theory, mergingthe properties of both theories into a single unified theory.
B. Linearised field equations of Brans-Dicke Theory with an arbitrary potential (BDV case)
Now we consider the action (3). We suppose that the arbitrary potential V is a well behaving function of itsargument and it is Taylor expandable around a constant value of the scalar field, namely φ = φ as: V ( φ ) = V ( φ ) + V (cid:48) ( φ ) ϕ + 12 V (cid:48)(cid:48) ( φ ) ϕ + . . . (20)Here (cid:48) means partial derivative with respect to the scalar field. In the previous works considering this action [27, 35],asymptotic flatness was assumed, which requires vanishing of the first two terms in the expansion. Here we do notimpose such a condition, but it might be reasonable to expect φ to be a minimum of this potential for stability,which require V (cid:48) ( φ ) to be vanishing. Then, the relevant terms in the linearised field equations can be written as V ( φ ) g µν ≈ V ( φ ) η µν , (cid:18) φ dVdφ − V (cid:19) ≈ φ V (cid:48)(cid:48) ( φ ) ϕ − V ( φ ) (21)and the field equations (13,14), in the first order in θ , ϕ and V , become (cid:3) η θ µν = − πφ T µν + V φ η µν , (22)( (cid:3) η − m s ) ϕ = 8 πT ω + 3 − V ω + 3 . (23)Here we used the abbreviations V ≡ V ( φ ) , m s ≡ φ ω + 3 V (cid:48)(cid:48) ( φ ) > . (24)In the parametratization (11) and gauge (12) considered, the tensor equation (22) still has the same structure withthe weak field equations of GRΛ theory [63]. The effect of the scalar fied and arbitrary potential is encoded intothe scalar field equation (23). It is clear that the minimum of potential V plays the role of a constant curvaturebackground or cosmological constant which is responsible for the asymptotic non-flatness. However, there is a slightdifference between Λ in BDΛ theory and V in BDV theory. The terms in the tensor equations of both theories canbe made similar by defining V = 2Λ φ (we put the subscript Λ to distinguish these theories). However, even usingthis redefinition, the coefficients related to cosmological constant Λ and Λ of the scalar field equations (19) and (23)will have different coefficients, 2Λ φ / (2 ω + 3) versus 4Λ φ / (2 ω + 3). This is because the Λ term in BDΛ theory islinear in scalar field whereas V is zeroth order term in the expansion of V ( φ ). The effects of this difference will beseen in the solutions of both theories presented in the next section and also at the physical quantities such as advanceof perihelion due to these terms.Another important difference of these theories is that, this theory leads to a massive scalar field [27, 35] with themass term m s defined as (24) proportional to second derivative of the arbitrary potential V ( φ ). The effect of themass term is to make the scalar field short-ranged. As we will see in the next section, solutions representing isolatedsystems, such as a point particle, contain Yukawa-like terms, which gives a characteristic range l c ∼ /m s where thescalar field related to this source freezes out and the physical properties due to this source becomes indistinguishablefrom GR outside this range. This behavior is in contrast to BD theory where the field has a long range. Hence, theintroduction of an arbitrary potential changes the range of the scalar field. Since metric f ( R ) theory is equivalent to BDV theory for ω = 0, this behaviour persists in this theory too. Note that, the behaviour of scalar field related tominimum of the potential is still long range and persists outside l c . Hence, asymptotically nonflat weak field solutionsmake it possible to open a new window to test these massive BD theories, otherwise they are indistinguishable fromGR outside this range. For example, as we will see in the next section, the advance of perihelion has correctionsdue to minimum of potential, V . This correction is negligable for very light scalar field where l c → ∞ and solarsystem tests require a very large ω for this case. For a heavy scalar case, however, since l c →
0, scalar field due tomass of the isolated source is already frozen and solar system tests have become insensitive to ω . Hence, since ω can take arbitrary values, the correction terms involving ω due to a minimum of potential can be very different fromcorresponding GR Λ solutions.
III. SOLUTIONS TO LINEARIZED FIELD EQUATIONS FOR A POINT MASS
Having obtained linearized metric and scalar field equations in the chosen gauge for both theories, the next stepwould be to obtain a physically relevant solution to these theories. Hence, in the following, we consider a static pointmass solution as a source for both theories. Note that, as far as we know, weak field equations of BDΛ theory withnonzero Λ for a point particle as a source is not discussed before. For BDV theory however, due to its equivalencewith f ( R ) theory, its weak field solutions derived for a nonvanishing V [30–32] using a slightly different method. Thephysical applications we will consider of both theories for nonzero Λ or V , namely, advance of perihelion, deflectionof light, gravitational redshift, and galactic and inter-galactic dynamics were not discussed before. A. A point mass term as a source for BD Λ theory We now consider a point particle located at ¯ r = 0, where ¯ r = ¯ x + ¯ y + ¯ z , described by T µν = m δ (¯ r ) diag(1 , , , . (25)Then, the scalar field equation (19) has the solution ϕ (¯ r ) = 2 m (2 ω + 3) 1¯ r − Λ φ ω + 3) ¯ r . (26)The advantage of using the gauge (11) and (12) is that the resulting tensor equations are decoupled from scalar field.The tensor equation (18) for a diagonal metric ansatze, together with the gauge condition (12) yield the followingnon vanishing components for the solution θ = 4 mφ r − Λ3 ¯ r , θ xx = Λ2 ( y + z ) , (27) θ yy = Λ2 ( x + z ) , θ zz = Λ2 ( x + y ) . Using the trace of θ = θ µµ given by θ = − mφ ¯ r + 4Λ3 ¯ r , (28)and the inverse of (11), the nonzero components of the metric perturbation term becomes h = 2 mφ ¯ r + Λ3 ¯ r + ϕφ ,h ij = (cid:20) mφ ¯ r − Λ6 (¯ r + 3 x i ) − ϕφ (cid:21) δ ij , ( i, j = 1 , , . (29)Clearly, the effects of the nonminimally coupled scalar field reveal themselves as the last terms in the metric pertur-bation tensor. The presence of this nontrivial scalar field may have some physical consequences such as it can modifytest particle trajectories compared to corresponding GR results discussed, for example, in [63].The solution presented above in equation (29) is not in isotropic coordinates. To express this solution in isotropiccoordinates we may consider the following coordinate transformations [63]¯ x i = x (cid:48) i + Λ12 x (cid:48) i . (30)Under these transformations (30) from barred to primed coordinates, the metric perturbation terms, up to linearorder in M and Λ become h (cid:48) = 2 mφ r (cid:48) + Λ3 r (cid:48) + ϕ (cid:48) φ , (31) h (cid:48) ij = (cid:18) mφ r (cid:48) − Λ6 r (cid:48) − ϕ (cid:48) φ (cid:19) δ ij , (32) ϕ (cid:48) = 2 m (2 ω + 3) 1 r (cid:48) − Λ φ ω + 3) r (cid:48) . (33)The explicit expressions of the field variables, with the help of (33), can be expressed as g (cid:48) = − mφ r (cid:48) (cid:18) ω + 3 (cid:19) + Λ r (cid:48) (cid:18) − ω + 3 (cid:19) , (34) g (cid:48) ij = (cid:20) mφ r (cid:48) (cid:18) − ω + 3 (cid:19) − Λ r (cid:48) (cid:18) − ω + 3 (cid:19)(cid:21) δ ij , (35) φ (cid:48) = φ (cid:18) m (2 ω + 3) φ r (cid:48) − Λ r (cid:48) ω + 3) (cid:19) . (36)As well known [20], the mass term in g must be related with weak field GR or Newton potential of a point mass,then φ must be equal to φ = 2 ω + 42 ω + 3 . (37)This implies that, since it is defined for an asymptotically flat space time, when Λ = 0, the post-Newtonian parameter γ is γ BD = h ij | i = j h = ω + 1 ω + 2 . (38)This result, together with the observational result of Cassini mission [64], i.e. γ observed − . ± . − whichsets γ ∼
1, implies that BD parameter must satisfy ω > .
000 for BD theory. For these large value of ω , the abovesolution given by (34) and (35) becomes indistinguishable from the corresponding GR Λ solution [63].Note that, in the case of vanishing Λ, the metric (34) and (35) and scalar field (36) reduce to the linearized BDsolution [20, 21]. These solutions reduce to corresponding linearized GRΛ solutions presented in [63] in the limit( ω → ∞ , φ → ω → ∞ does not always reducethe theory to GR. For a further discusson of the GR limit of BD theory, see, for example [42, 65–69] and referencestherein. B. A point mass term in the BDV theory
For a point mass m given in (25) the corresponding solution of the scalar field equation (23) reads: ϕ (¯ r ) = 2 m (2 ω + 3) e − m s ¯ r ¯ r − V ω + 3) ¯ r . (39)Note that if we set V φ = Λ , (40)then the first order metric equation (22) becomes exactly the same as (18) when Λ is replaced by Λ . Thus, wecan directly use the solutions (27) and also for the metric perturbation terms h µν given in (29) in this case as well.Moreover, with this choice (40), we can use exactly the same transformations (30) to bring the metric solution intoisotropic coordinates. With this notation, the differences between both theories are encoded in ϕ (cid:48) term, which hasthe same form with (39) for this case with ¯ r replaced with r (cid:48) . Then, full metric and scalar field of weak field equationbecomes g (cid:48) = − mφ r (cid:48) (cid:18) e − m s ¯ r ω + 3 (cid:19) + V r (cid:48) φ (cid:18) − ω + 3 (cid:19) , (41) g (cid:48) ij = δ ij (cid:20) mφ r (cid:48) (cid:18) − e − m s ¯ r ω + 3 (cid:19) − V r (cid:48) φ (cid:18) − ω + 3 (cid:19)(cid:21) , (42) φ (cid:48) = φ (cid:18) m e − m s ¯ r (2 ω + 3) φ r (cid:48) − V r (cid:48) φ (2 ω + 3) (cid:19) . (43)Note that this solution is discussed before by using a slightly different approach [30–32]. The vanishing V case isknown as massive BD theory and its weak field solutions were presented before [27, 28, 35]. Also for vanishing V ,the post-Newtonian parameter γ becomes position dependent [30–33]: γ ( r ) = 1 − e − msr ω +3 e − msr ω +3 . (44)The observational result of Cassini mission, namely γ observed − . ± . − , which sets γ ( r ) ∼
1, can beapplied to γ ( r ) in several ways. The first one is to set e − m s r →
0, which requires m s → ∞ , namely the mass of thescalar field must be very heavy. For this case γ = 1 irrespective of the value of ω . If this is not the case, i.e. if themass is not large and if e − m s r is O (1) which requires very light scalar mass as m s →
0, then γ ( r ) reduces to γ BD given in equation (38) and the limit ω > .
000 is again emerged. For intermediate values of m s , however, a numericalinvestigation is required to find the observationally allowed regions of the parameter space ( m s , ω ) and this analysisis done in the work [33].Let us now discuss the effective gravitational constant φ for these theories. In the vanishing of the V term thevalue of φ is fixed by the requirement that the theory must have a correct Newtonian limit, which requires theinvestigation of the term containing the mass of the source in g . The exponential term spoils this expression butfor very light or heavy scalar field mass cases φ can be fixed as discussed below [27, 33, 35]: • For a very massive potential, i.e. m s (cid:29)
1, we can ignore the terms with exponential factor and can set φ = 1 . • For a very light scalar mass case, i.e. m s (cid:28)
1, we can expand e − m s /r term in g in series, keep the firstterm and compare with the Newtonian potential of a point mass. This procedure yields that we must have φ = (2 ω + 4) / (2 ω + 3), as in the original BD theory [20].For the intermediate values of m s , the above prescription does not work to fix φ , but one can define an effectivegravitational coupling term involving the exponential term as G ( r ) = (cid:18) e − m s r ω + 3 (cid:19) φ . (45)A numerical investigation of such case is given in [33] by using the powerful PPN approach, which requires anasymptotically flat spacetime so that V must be vanishing. In our work we will not discuss such an investigationsince we want to discuss the case where the spacetime is not asymptotically flat.As we have discussed in the previous section, another difference between the weak field solutions of BDΛ and BDVtheories is that the factors involving ω in the metric and scalar field expressions of Λ or V have some differences.The result reflects the fact that their couplings with the scalar field are different. Namely V is constant whereas theterm involving Λ is linear in the scalar field. C. Solutions in Schwarzschild Coordinates
Here we will bring both of the solutions given in subsections (III-A) and (III-B) to Schwarzschild type coordinates.This can be done by the following transformations and definitions r (cid:48) = r (cid:18) − mφ r + 112 Λ r + ϕ φ (cid:19) , (46)with the result ds = − (cid:18) − mφ r − Λ r − ϕφ (cid:19) dt + (cid:18) mφ r + Λ r α rφ (cid:19) dr + r d Ω . (47)Here ϕ is given by ϕ = 2 m (2 ω + 3) 1 r − Λ φ ω + 3) r , (48)for BDΛ theory and ϕ = 2 m e − m s r (2 ω + 3) r − V r ω + 3) , (49)for BDV theory. The function α ( r ) in (47) is defined as α = dϕdr . (50)This form of the metric (47) is suitable to represent both solutions with the same metric. Difference from correspondingGRΛ solution is encoded in φ , ϕ and α . IV. MOTION OF TEST PARTICLES
Now we will discuss the effects of the point mass and the presence of the cosmological term on the motions of testparticles and photons. In order to discuss these effects and compare with the previous results exist on the literature,we choose to work in the Schwarzschild like coordinates, which can be written as ds = − A ( r ) dt + B ( r ) dr + r (cid:0) dθ + sin θ d Φ (cid:1) . (51)We also consider equatorial motion by setting θ = π/
2, then the Lagrangian of the test particles or photons can bewritten as 2 L = − A ˙ t + B ˙ r + r ˙Φ . (52)Here overdot means derivative with respect to proper time for time-like particles and an affine parameter for photons.Symmetries of this space time results two first integrals of motion, given by˙ t = − EA , ˙Φ = Lr . (53)Here E and L are related with specific energy and angular momentum of test particles. We can use these results intothe metric itself to obtain a radial equation of motion˙ r = 1 B (cid:18) ε + E A − L r (cid:19) , (54)where ε = 0 for photons and ε = − with the result (cid:18) drd Φ (cid:19) = ˙ r ˙Φ = r (cid:18) εL B + E L AB − Br (cid:19) . (55)0Hereafter we may analyze the equations (54) or (55) for different types of motion for the two different solutions wehave obtained. We should also read the appropriate metric functions A and B (51) from (47) for the solutions we arediscussing and keep the terms in the linear order in mass and cosmological terms in the expressions. It is customaryto use inverse radial coordinate defined by u = 1 r (56)to obtain a modified Binet equation and solve the resulting equation. Then, using the transformation (56), theresulting equation can be written in the linearized order as (cid:18) dud Φ (cid:19) + u = (cid:18) E + εL (cid:19) + Λ3 − ε m uφ L + 2 mu φ − ε Λ3 L u + E ϕ ( u ) L − (cid:0) E + ε − L u (cid:1) α ( u ) L u φ . (57)By differentiating this equation with respect to Φ, one obtains a modified Binet equation as d ud Φ + u = − ε mφ L + 3 mu φ + ε Λ3 L u + 12 ∂∂u (cid:20) E ϕ ( u ) L − (cid:0) E + ε − L u (cid:1) α ( u ) L u φ (cid:21) . (58)Hence, we have obtained a modified Binet equation which is useful for both BDΛ and BDV theories. One can furtheranalyze this equation by appropriately choosing the values of the functions ϕ, α and constant ε . V. PRECESSION OF THE PERIHELION OF THE PLANETS
In order to derive advance of perihelion for these theories, we may try to use the usual perturbative approach tosolve equation (58) for time like particles ε = − , together with appropriate values for ϕ and α . For example, forBDΛ theory, we obtain the following differential equation d ud Φ + u = 2 m ( E + ω + 1)(2 ω + 3) φ L + 6 m ( ω + 1) u (2 ω + 3) φ − ( E + 2 ω + 1)Λ3(2 ω + 3) L u . (59)One can consider the Newtonian elliptical solution as a zeroth order solution of the equation (59). However, thisapproach needs a perturbation extension of geodesic equation (59) second order in mass m . But, this is beyond ourlinearized approximation. Hence, we cannot use the perturbation approach to derive the perihelion advance for bothof the theories. But, there are alternative methods and we will use one of them to derive perihelion shift terms dueto mass of the source and cosmological terms. A. Review of calculation of advance of perihelion by integration of perturbation potential
The calculation is based on the principle that in the weak field of a gravitation theory, we may have a potentialwhich has usual Newtonian term for a central mass distribution as well as other correction terms originated from thetheories modifying Newtonian theory. Using different methods such as directly integrating geodesics equations [70],precession of a cousin of Runge-Lenz vector, i.e., the Hamilton vector in the modified potential background [71], orusing a modification of Landau-Lifshitz method [72] one obtains perihelion shift ∆ as a one dimensional integral ofthe form ∆ = − Lme (cid:90) − z dz √ − z d ˜ V ( z ) dz , (60)where the coordinate transformation r = L/ (1 + e z ) is employed. Here e is eccentricity of elliptic motion. For detailsof the derivation of this integral we refer to the works [70, 73]. This perihelion shift implies that the orbit equationhas the following form u = 1 /r = mL [1 + e cos (1 − (cid:15) )Φ] , ∆ = 2 π(cid:15). (61)In equation (60), ˜ V ( z ) contains the modification terms to the Newtonian potential for central motion which can beread from the full potential of the gravity theory considered of a point mass as given by U ( r ) = − mr + L r + ˜ V ( r ) . (62)1So here we just need to find the ˜ V ( r ) term and replace into the integral (60). To do this we consider the radialgeodesic equation (54), which can be written as ˙ r U ( r ) = ˜ E (63)where ˜ E = E /
2, and U ( r ) = 12 (cid:18) L r (cid:19) (cid:18) − mφ r − Λ r − αrφ (cid:19) + ( αr − ϕ ) E φ . (64)Hence, we need to evaluate U ( r ) for the solutions we have found to read ˜ V ( r ). We will do this in the followingsubsections for both of the theories we consider. B. Advance of perihelion for BD Λ theory Using (47), replacing the value of φ given in (37) and the fact that for a nonrelativistic motion energy per unitmass has the value E = 1 [74], we find that˜ V ( r ) = − (cid:18) ω + 1 ω + 2 (cid:19) mL r − (cid:18) ω + 22 ω + 3 (cid:19) Λ r . (65)Here, some constant terms are discarded because they do not affect the advance of perihelion. Since the result ofintegral (60) is calculated for power-law potentials [70, 73], from their result, the only difference is factors involving ω in ˜ V ( r ), so we see that the perihelion shift can be written as∆ BD Λ = ω + 1 ω + 2 ∆ E + 2 ω + 22 ω + 3 ∆ Λ , (66)where ∆ E is the usual Einstein value of perihelion shift due to mass [75] and ∆ Λ [70, 73, 76, 77] is GRΛ perihelionshift due to cosmological constant. Their expressions are given by∆ E = 6 π ma (1 − e ) , (67)∆ Λ = π Λ m a (cid:112) − e , (68)and ∆ Λ agrees with the one found in [78] only for e → BD Λ theory have similar structures with corresponding GRΛ theory withsame multiplicative factors involving ω . The discussion of some observational consequences of these results will be inthe Section(V D). C. Advance of perihelion for BDV theory
For this case, the potential U given by equation (64) becomes U ( r ) = 12 (cid:18) L r (cid:19) (cid:20) − mφ r − Λ r − m (2 ω + 3) φ (cid:18) m s + 1 r (cid:19) e − m s r − V r φ (cid:21) − E me − m s r (2 ω + 3) φ (cid:18) m s − r (cid:19) + V E r ω + 3) φ . (69)This expression is complicated and resulting ˜ V ( r ) which involve terms containing e − m s r factor cannot be integrated[70] to obtain analytical results. However, for the following special cases it is possible to obtain analytic results forthe advance of perihelion. • For a very heavy scalar field, since as m s → ∞ , e − m s r →
0, the perturbation potential becomes˜ V ( r ) m s − > ∞ = − mL r − (cid:18) ω + 12 ω + 3 (cid:19) Λ r . (70)2From this potential, since φ = 1 for a heavy scalar field, the advance of perihelion is calculated as∆ BDV − heavy = ∆ E + (cid:18) ω + 12 ω + 3 (cid:19) ∆ Λ , (71)where ∆ E is the Einstein value (67) for perihelion shift and ∆ Λ is the GRΛ value (68) with Λ is replaced withΛ . Hence, for a very heavy scalar, when the minimum of the potential is zero, then the perihelion shift isindistinguishable from GR value and independent of ω . This is a well-known result that when the scalar fieldbecomes very short range field, weak field tests yield the same results with GR. However, when the minimumof the potential V is not zero, then the resulting perihelion shift has a term due to the minimum of potentialhaving a factor involving ω . If the value of V or Λ would be fixed by a future observation, then one could puton bounds on ω even for very massive BD theory. For example for metric f ( R ) case ω = 0, the perihelion shiftdue to mass is the same with GR whereas the corresponding term due to the cosmological term is 1 / . • For a very light scalar field, as m s → e − m s r in a series and since m s and m are small we canignore the terms such as m × m s , and using the value of φ given in (37) for this case, then the perturbationterm becomes ˜ V ( r ) m s → = − (cid:18) ω + 1 ω + 2 (cid:19) mL r − (cid:18) ω + 12 ω + 3 (cid:19) Λ r . (72)This implies the perihelion shift as∆ BDV − light = ω + 1 ω + 2 ∆ E + (cid:18) ω + 12 ω + 3 (cid:19) ∆ Λ , (73)Hence, for the case when the mass of the scalar field is very light, the scalar field becomes a long range one,similar to original BD scalar. Thus, the advance of perihelion term of light BD V theory of a point mass becomesexactly the same as the result of BD theory given in [20]. For both heavy or light BDV theories, the effect ofminimum of the potential has same ω dependent factor, which is slightly different from the factor of the resultof BDΛ theory given in equation (66). For f ( R ) theory with very light mass, the perihelion shift due to massof the source becomes one half of corresponding GR value and corresponding term due to the minimum of thepotential is 1 / D. Observability
We have obtained advance of perihelion for both BD Λ theory given in (66) and for heavy or light
BDV theoriesgiven in equations (71) and (73), due to mass of the source and cosmological constant or minimum of potential,respectively. These expressions have a similar structure to the corresponding GR ones given in (67) and (68). Aswe have discussed before, the difference is the different numerical factors involving ω multiplying these terms. Themultiplicative factors due to mass are the PPN parameters γ of these theories. For BD Λ theory and light
BDV theory these parameters are the same as in the original BD theory as given in (38). For these cases, the results agreewith corresponding GR results for large ω since for these cases we have ω > . ω and equal to GR value, 1. Hence for heavy BDV theory, solar systemtests will be satisfied for any value of ω except ω = − / GR Λ theory. The differences are the existence ofmultiplicative terms involving ω . The behaviour of these factors can be seen from figure (1). As it is clear from thisgraph, for positive values of ω , these numerical factors are in the intervals [2 / ,
1) for BD Λ theory and [1 / ,
1) for
BDV theories for 0 ≤ ω < ∞ . Therefore, in these intervals, the correction factors cannot make significant order ofmagnitude changes to these terms. For negative values of ω, however, these factors may have significant effects, asseen from the graph. These factors even vanish at ω = − BD Λ and ω = − / BDV theories or take negativevalues for − / < ω < − BD Λ and − / < ω < − / BDV theories. These factors take positive values for3both theories for ω < − /
2. They blow up as seen from graph as ω → − /
2. Therefore, for small and negative valuesof ω , the deviation from GR can be observed for these theories, in principle.In sumary, the result of the Cassini experiment sets a lower bound for BD parameter ω as ω > .
000 for BD theoryand this behavior is also valid for BD Λ and light
BDV theories. Hence, for both BD Λ and light
BDV theories, themultiplying factors of ω for the terms contributing to the perihelion precession due to the mass of the source andcosmological or minimum potential terms approaches to one as ω approaches to 40 . using the results of [77, 79], which is, Λ ≤ − m − or the same limit for Λ . For heavy BDV theory,however, the local behaviour does not fix ω and in principle this term can take any value. Most significant effects ofthe multiplicative factor is at the negative values of ω . Namely, for negative values of ω there are regions where thefactor (2 ω + 1) / (2 ω + 3) becomes zero, negative, or takes unbounded negative or positive values as ω → − / ω → − / , smaller than current observed valueof Λ, can be compatible with observations. Or conversaly, if ω → /
2, as this factor approaches to zero, a very largeminimum potential compared to observed cosmological constant, may be compatible with observations on perihelionprecession. As a result, there may be an observational window to test heavy
BDV theories with solar system testsif for example the contribution of advance of perihelion due to cosmological constant can be measured with enoughsensitivity in the future observations.
FIG. 1. The behavior of some factors involving ω . The continuous line represents the ratio (2 ω + 2) / (2 ω + 3) whereas thedashed line represents (2 ω + 1) / (2 ω + 3). VI. DEFLECTION OF LIGHT RAYS
In this part we will discuss deflection of light rays for both BDΛ and BDV theories using the geodesic equationsderived in section (IV). The effect of cosmological constant on the deflection angle was a topic with opposing viewswith works confirming [82–97] or denying [98–104] this effect. Therefore, here we first give a short summary of thistopic in the discussion below. Then, we will focus on such effects for the theories we are considering.
1. Calculation of Deflection angle for GR Λ theory using Rindler-Ishak method Here we review deflection of light rays from a compact object in GRΛ theory in the linear approximation. Thisrequires geodesics of photons in the corresponding space-time. We will again use Schwarzschild type coordinates (47)for this discussion as well, hence we can use the orbit equation (58) for photons ε = 0. Note that, whether thecosmological constant affects the light deflection angle has became a source of debate and a lot of work is devoted toclarify this issue using different techniques. The reason for this is the fact that, for GRΛ theory, the geodesic equationfor photons (58) can be reduced to d ud Φ + u = 3 mu . (74)4The fact that this equation is independent of cosmological constant term lead to the conclusion [98] that cosmologicalconstant has no effect in the light deflection. This is because the solution of this equation, given by [78] u (Φ) ≡ u GR (Φ) = 1 r = sin Φ R + 32 mR (cid:18) (cid:19) (75)does not involve cosmological constant explicitly, hence orbit is independent of Λ. Note that the relation betweenintegration constant R and the closest approach distance r , given by setting Φ = π/ r = 1 R + mR . (76)This means that in the orbit equation (75) we can replace R with r in the linearised approximation. Here r is thesolution of the equation dr/d Φ = 0 and from (75) it is given by1 b + Λ3 = 1 r − mr . (77)This means that we can express integration constant R in terms of either r or b and Λ and the latter choice producea Λ dependence in the bending angle expressions. For an asymptotically flat spacetime, the solution (75) implies halfbending angle for Schwarzschild spacetime, as r → ∞ , α E = 2 mR = 2 mb (78)where the last equality is valid in the linearised order only. Note that the asymptote r → ∞ is not valid forSchwarzschild-de Sitter space time because this space time is not asymptotically flat. One might attempt to obtain aΛ dependence by using the relations (76) and (77). However, this was criticized in [90] and argued that despite thisdependence, the orbit is not affected by Λ.In a pionering work proposed in [82], if one considers the measurement of angles which depens on both the local andglobal geometry of the space-time, bending angle can be shown to be affected by the cosmological constant as well.It turns out that, this problem depends both on how to define and measure bending angle and also how to specifyphysical parameters such as impact parameter. However, this is not a generic conclusion and both depends exactlyto the setup used to perform observation to measure, and also the initial conditions as well. Since the spacetimeobtained is not asymptotically flat, the above measurements and definitions will be different from Schwarzschild caseand may lack a universal understanding. We refer the latest works for a more complete review of this topic [83, 90],and in the latter part of the paper we consider bending of light phenomena for BDΛ and BDV theories. There aremany different approaches to this problem but here we only consider Rindler-Ishak method presented in [82, 83] inthis work. Now, let us review their method and results here. FIG. 2. The plane graph corresponding to the orbit equation given in equation (75). The one-sided deflection angle is givenby α = ψ − Φ (The figure is adapted from [82]).
Now, consider the case where both source and observer are static. The cosine of the angle between two coordinatedirections d and δ given in figure (2) is given by cos ψ = g ij d i δ j / [ (cid:112) g ij d i d j (cid:112) g ij δ i δ j ] where g ij is the two dimensionalsubmanifold obtained by setting t = constant , θ = π/ A = B − = 1 − mr − Λ r . (79)5Also, here d = ( dr, d Φ) = ( β, d Φ, with β = dr/d Φ, is the direction of orbit of the photon whereas δ = ( δr,
0) is thedirection of coordinate line Φ = constant. One can obtain β from (75) as β ≡ drd Φ = r R (cid:16) mR sin 2Φ − cos Φ (cid:17) . (80)Using this and GR expressions of metric (51), one finds cos ψ = | β | / (cid:112) β + r /B and from the relation tan ψ = (cid:112) sec ψ − ψ = r | β |√ B . (81)From this expression, one can find that the one-sided bending angle is α = ψ − Φ and one can immediately calculatethe bending angle for small Φ = Φ (cid:28)
1. For the angle Φ to occur, we need from (75)1 r = Φ R + 2 mR , (82)and using this r value, from (80) one finds β = r R (cid:18) m Φ R − (cid:19) ≈ − r R . (83)Moreover, using the weak field GR value for B , whose exact form is given in equation (79), namely B = 1 + mr + Λ r , and from (81), one finds [83] α GR Λ = 2 mR − Λ R r mR − Λ R R Φ + 2 m ) , (84)where r is given by (82) and this expression reduces for Φ = 0 to the result given in [82] as α GR Λ = α E − α Λ , α E = 2 mR , α Λ = Λ R m . (85)Therefore, Λ contributes to the deflection angle and this contribution has the opposite sign compared to the contri-bution due to mass of the source. Here, we have reviewed the deflection angle for a special case where the source andobserver are static. For a more general treatment of calculation of this angle when source or observer may not static,we refer to [90]. Having reviewed Rindler-Ishak method for GRΛ theory, let us now apply this method to both BDΛand BDV theories in the following.
2. Deflection of light rays in BD Λ theory For BDΛ theory, from (58), the orbit equation becomes d ud Φ + u = 2 m (2 ω + 3) φ b + 6 m ( ω + 1) u (2 ω + 3) φ − Λ3(2 ω + 3) b u , (86)where here b = L/E is a constant of motion. Most important observation of this equation is as follows. Unlike inthe GR case [98] where the corresponding equation, i.e., equation (75), is independent of Λ, the last term in equation(86) contains Λ explicitly. Hence Λ clearly affects the path of photons because its solution will directly involve acosmological constant term, even in the linear order. Note also that this term vanishes in the ω → ∞ limit and theequation reduces to the corresponding GR one given in [98]. We will present calculation details in Appendix (A) forclarity and here present only the results.The solution to equation (86) is given in Appendix (A6). By calculating point of closest approach r given in (A7)and its relation with impact parameter b given in (A8) we see that we can use R, r and b interchangeably in the orbitequation for linearized order and this fact enable us to simplify the solution (A6) as u (Φ) = sin(Φ) R + 2 mR (2 ω + 3) φ (cid:26) ω + 12 (3 + cos 2Φ) (cid:27) − Λ R cos Φ6(2 ω + 3) sin Φ . (87)6In comparison with GR solution (75), the solution (A6) involves Λ explicitly. In the GR limit ω → ∞ , this Λ dependentterm vanishes and the solution (87) reduces to (75) in GR limit. Hence, unlike GRΛ case, the orbit of the photons depends on Λ and this dependence vanishes in the GR limit. This implies that, in addition to the Einstein deflectionangle multiplied by a multiplicative factor of ω [20], there should be an extra contribution involving Λ due to orbit ofphotons in BDΛ theory. This extra deflection angle is due to the interaction of cosmological term Λ and the scalarfield and vanishes for the GR limit. If the space time under consideration would be asymptotically flat, then onecould find a deflection angle by measuring from r → ∞ to find α ≈ m/ ( φ R ) − Λ R / [6(2 ω + 3)Φ ]. However, sinceneither the space-time is asymptotically flat nor we can ignore effects of space-time on local measurements, we haveto use an appropriate method to calculate the deflection angle. Hence, using the same method in [82] we will calculatedeflection angle and for clarity we present calculational details in Appendix (A), see equations in appendix(A9-A12).The result, at most in the linear order in m, Φ and Λ, is α BD Λ = 2 mφ R − Λ R ω + 3)Φ − Λ R ω + 3) (cid:20) (2 ω + 1) rR + R Φ r (cid:21) , (88)where r is given in (A10). The first term in (88) is due to mass, the second one is the effect of Λ on orbit and thelast term is due to effect of metric on the measurement of angles. This result reduces to special cases such as GRΛone (84) [82–84] in GR limit ω → ∞ or BD deflection angle [20, 27] for Λ = 0. Also for large ω values where Cassinimission yields the gravitational deflection angle is indistinguishable from corresponding GRΛ expression.
3. Deflection of light rays in for BDV theory
For this case, from (58), corresponding differential equation becomes d ud Φ + u = 3 mu φ − ω + 3) b u + m e − m s /u (2 ω + 3) L φ (cid:2) E − L u + O ( m s , m s ) (cid:3) . (89)Due to the exponential term, this equation is complicated. However, this equation can be analyzed for very massiveor light scalar cases as follows: • For a very massive scalar, m s (cid:29)
1, the exponential term can be ignored and together with φ = 1 for this case,the equation (89) reduces to d ud Φ + u = 3 mu − ω + 3) b u , (90)which is exactly the same with GR case except for the last term in the equation. Hence the linearised solutionwould be a mixture of GR and BDΛ solutions given by u (Φ) = u GR (Φ) − Λ R cos Φ3 b (2 ω + 3) sin Φ , (91)where u GR is solution of GR case given in (75). Thus, the minimum of potential, V = 2Λ , enters in the orbitequation and will affect the light deflection. Repeating similar calculations, one finds that the bending angle inthe linearised order become α BDmassive = α E − Λ R ω + 3)Φ − Λ R ω + 3) (cid:20) (2 ω − r R + R Φ r (cid:21) . (92)Here r is given by (A10) with Λ to be replaced by 2Λ together with φ = 1. Therefore, for a very massive scalarfield, as it is well known, light deflection due to mass is exactly the same with GR, and the effect of minimum ofthe potential acting as a cosmological constant has a slightly different ω dependence compared with the resultof BDΛ theory given in (88). • For a very light scalar, m s (cid:28)
1, we can expand the exponential term in equation (89) and find the followingequation: d ud Φ + u = 2 m (2 ω + 3) φ b + 6 m ( ω + 1) u (2 ω + 3) φ − ω + 3) b u + O ( m × m s ) . (93)7Here we see that the resulting differential equation resembles the same form with the equation (86) of the BDΛcase except Λ is replaced by 2Λ . Therefore its linearised solution (A6) will have same form except Λ to bereplaced by 2Λ . The effect of minimum of the potential acting as a cosmological constant defined by V = 2Λ φ on the total deflection angle can be seen from the total deflection angle expression given by α BDV − lightsc. = 2 mφ R − Λ R ω + 3)Φ − Λ R ω + 3) (cid:20) (2 ω − r R + R Φ r (cid:21) . (94)Here r is given by (A10) with Λ to be replaced by 2Λ . In comparison with the result of BDΛ theory given in(88), there are some slight differences for corresponding results of both light scalar (92) and massive scalar (94)cases. These differences originate from small differences of the metric functions A and B for BDΛ and BDVtheories. Due to observational results, the very light scalar mass case cannot deviate from corresponding GRΛexpression since ω > .
000 limit is also valid for this case. But for BDV theory with very massive scalar field,there is no such limit on ω and the deflection angle due to minimum of potential can be very different fromcorresponding GRΛ expression. However, there is no observational data measuring the deflection angle due tothe cosmological constant, yet. Hence, there is a possibility that can limit the parameters of these theories oreliminate them if such an observation is made in the future. VII. GRAVITATIONAL REDSHIFT
The spacetime described by (51) and (47) is stationary. Hence it admits a timelike Killing vector. In thesecoordinates, the ratio of the measured frequency ν of a light passing through different positions is given by ν ν = (cid:115) A ( r ) A ( r ) (95)Reading metric function A ( r ) from (47) and considering the fact we are working in the linear level, the equation (95)becomes ν ν = 1 + mφ r − mφ r − Λ6 (cid:0) r − r (cid:1) + ϕ ( r ) − ϕ ( r )2 φ . (96)In the GR limit this expression reduces to the one given in [77]. The effects of the scalar field to the gravitationalredshift is given by the last term. Let us evaluate these for BDΛ and BDV theories, separately. A. Gravitational redshift for BD Λ case For this case, reading ϕ from (48), and φ from (37) we find ν ν = 1 + mr − mr − ω + 22 ω + 3 Λ6 (cid:0) r − r (cid:1) , (97)hence the gravitational redshift due to mass is the same as in GR whereas there is a correction term involving BDparameter ω for the gravitational redshift due to the cosmological constant term. Since the result of Cassini missionrequires large ω , this factor approaches to one and the expression becomes identical to GR one for BD Λ theory.
B. Gravitational redshift for BDV case
For this case, by considering (49) we have ν ν = 1 + mφ r (cid:18) e − m s r ω + 3 (cid:19) − mφ r (cid:18) e − m s r ω + 3 (cid:19) − ω + 12 ω + 3 V φ (cid:0) r − r (cid:1) . (98)Hence the gravitational redshift due to mass term is modified since each term is multiplied by a position dependenteffective gravitational term. The term for the minimum of the potential has a similar contribution but the multi-plicative factor involving BD parameter is slightly different than BDΛ case. We can expand the terms involving massterms for a very light or very heavy scalar field mass cases as follows.8 • For a very heavy scalar, m s → ∞ , since e − m s r → φ = 1 we find that ν ν = 1 + mr − mr − ω + 12 ω + 3 V (cid:0) r − r (cid:1) . (99)Similar to the previous case, the gravitational redshift due to mass is the same as in GR and the term dueto minimum of the potential gets a multiplicative factor, same as in advance of perihelion for this theory. For BDV theory with heavy scalar field mass, there is no lower limit for the value of ω . Hence if we regard V asa cosmological constant using the equation V = 2Λ , depending on the value of ω , the redshift term due to V can take any value in the interval ( −∞ , ∞ ) which can be seen from the behavior of multiplicative factor givenin figure (1). Using the argument given in [77] which considers the result of Gravity Probe-A experiment [105]one can put a bound | (2 ω + 1)Λ / (2 ω + 3) | ≤ − m − but this bound is much larger than the current valueof cosmological constant. If future experiments will reach enough sensitivity, then one can use this phenomenato restrict the parameter space ( ω, Λ ) of this theory. • For a very light scalar, m s → e − m s r ∼ − m s r , ignoring multiplication of m with m s and using the value of φ given in equation (37) for this case, we find ν ν = 1 + mr − mr − ω + 12 ω + 3 V φ (cid:0) r − r (cid:1) . (100)Here again the gravitational redshift due to mass is the same with GR case [77], and the gravitational redshiftdue to the minimum of the potential contains a numerical factor involving ω similar to heavy scalar field masscase. However, unlike from heavy case, for a very light scalar, this factor approaches to one since Cassini missionrequires ω ≥ .
000 for light BDV theory. Hence, there is no deviation from GRΛ results for this case.
VIII. GALAXY DYNAMICS
In GR it is well known that the effects of the cosmological constant or dark energy on the solar system scales orgalactic scales are too weak to be observable. However, when the scales comparable or bigger than 1 Mpc, its effectscannot be ignorable anymore. Here, with the help of using the results we have obtained in the previous sections, wewill discuss the effects of cosmological constant or minimum of the potential of BDΛ and BDV theories on the localdynamics of the universe and whether the results agree with GR.
A. Galaxy rotation curves
To discuss these effects in the galactic scale, we can consider galaxy rotation curves. It was observed [106, 107]that the rotation curves of gas at the outer regions of galaxies show a nearly constant velocity up to several galacticluminous radii. To apply our results to this phenomena, now, first let us calculate the rotational velocity of starsaround the center of a static, spherically symmetric galaxy. We can express radial geodesics equation on equatorialplane (54) for timelike particles as ˙ r + U ( r ) = 0 , (101)where U ( r ) = 1 B (cid:18) − E A + L r (cid:19) . (102)The conditions for the existence of stable circular orbits are:˙ r = 0 ( U ( r ) = 0) , U (cid:48) ( r ) = 0 , U (cid:48)(cid:48) ( r ) > . (103)Here (cid:48) denotes derivative with respect to r . From the first two conditions with a little algebra one finds E = 2 A A − rA (cid:48) , L = r A (cid:48) A − rA (cid:48) . (104)Moreover, the second derivative of the potential becomes U (cid:48)(cid:48) = 2 rB (cid:20) rA (cid:48)(cid:48) + A (cid:48) (3 − rA (cid:48) )2 A − rA (cid:48) (cid:21) . (105)9The above conditions were already obtained in the previous works, for example in [108]. A numerical investigationshowed that the last condition in (103) is satisfied in the relevant values of r . From the proper time expression dτ = − ds , considering the definition of four velocity U µ = dx µ /dτ = ( ˙ t, ˙ r, ˙ θ = 0 , ˙Φ) we find that1 = A ( U ) (cid:0) − v (cid:1) , (106)where U is the time component of the four velocity of the particle and v is the spatial velocity defined as v = 1 A (cid:34) B (cid:18) drdt (cid:19) + r (cid:18) d Φ dt (cid:19) (cid:35) = ( v r ) + ( v φ ) , (107)where v r and v φ are the components of the spatial velocity v which is observed in an orthonormal coordinate system.Its Φ component is given by v Φ = r √ A Ω , Ω = d Φ dt , (108)From the first integrals of the geodesics equation and Eq.(104) we can calculate Ω asΩ = d Φ dt = ˙Φ˙ t = Ar LE = (cid:114) A (cid:48) r . (109)Using this value, we find the tangential velocity of a particle in a stable circular motion as follows( v Φ ) = rA (cid:48) A . (110)In the linearized approximation, we find that( v Φ ) = mφ r − Λ r − r ϕ (cid:48) φ . (111)Thus, the effects of the BD scalar field reveals itself in the last term as well as in the constant φ for this phenomena.Let us discuss this term for the theories we consider separately. • For BDΛ theory we have ( v Φ ) BD Λ = mr − ω + 22 ω + 3 Λ r . (112)In order that these expression can have somewhat constant behaviour, the sign of the last term after the minussign must be negative, which is possible for the interval − / < ω < − ω > . BD Λ theory. • For BDV theory we find that( v Φ ) BDV = mφ r (cid:20) e − m s r (2 ω + 3) ( m s r + 1) (cid:21) − ω + 12 ω + 3 V r φ . (113)We can again look for special cases for this expression. For a heavy scalar, exponential term vanishes and the firstterm in tangential velocity becomes similar to BDΛ case since φ = 1. For a very light scalar the term involvingmass becomes again the same as in (112) and for the term containing V we have to take φ as in (37). In all thesecases we see that the numerical factor involving ω does not change the order of magnitude of the term related to Λor V , for positive ω . Hence, these terms cannot explain flat rotation curves of stars in a galaxy for positive ω . Fornegative values of ω , the factors containing ω may be negative and there may be regions where nearly flat rotationcurves possible in principle. However, similar to BD Λ theory,
BDV theory with light scalar mass, solar system testsrequire large positive values of ω and this posibility is ruled out. For a heavy scalar field mass, however, there isno restriction on ω by solar system tests and for − / < ω < − /
2, the coefficient of the last term of (112) afterminus sign becomes negative for positive V , making this term an increasing function of r . Hence, for this range, theminimum of potential can contribute to the flat rotational curves of galaxies. For the values of ω outside this range,0however, the minimum of the potential cannot contribute to flat rotation curves. In GR the flat rotation curves isexplained by the existence of dark matter, usually modeled as a dust or perfect fluid surrounding the galactic corewhich interacts with other particles only via gravity. This behaviour can also be explained by the existence of exoticsources such as a global monopole behaving as a galactic dark matter [109, 110]. It might be interesing to consider adust or perfect fluid source for BD Λ and
BDV theories as a candidate of dark matter. We are currently working onthis problem and we will present our results elsewhere.For intermediate values of m s where these approximations are not valid, the Yukawa type term in (113) mayalso explain the flat rotation curves. This is because Sanders showed in [111] that a Yukawa type phenomenicalgravitational potential can explain the behavior of galaxy rotation curves. In that work the following expression forrotational velocity is obtained ( v Φ ) = G ∞ mr (cid:20) αe − rr (cid:18) rr + 1 (cid:19)(cid:21) . (114)In this expression G ∞ is the gravitational constant measured at infinity, r is a length scale of this potential and α isa coupling constant of this Yukawa type term. Sanders showed that in the presence of a Yukawa type gravitationalpotential, for − . < α < − .
92 there is a region where the general properties of extended galactic rotation curvesare reproduced. Comparing our expression (113) with (114), we see that they are similar if we identify α = (2 ω +3) − , r = 1 /m s , G ∞ = 1 /φ . Then, the above limit on α is equivalent to − . < ω < − .
02. Hence, a generic BDVtheory can explain the observed galaxy rotation curves without needing a dark matter if BD parameter ω is in thisinterval. This result is also discussed in [112] for a generic f ( R, φ ) gravity including BDV theory as a special case. Theproblem here is that the ranges of ω where the observed galaxy rotation curves were reproduced are very restrictednegative and unfavourable values of it. The contribution of minimum of the potential to rotational velocity is in thereducing sense since the multiplicative factor is positive for this value of ω . In summary for a very restricted andnegative value of ω , rotational curves of galaxies can be explained by the mass of the scalar field of BDV theoryleading to a Yukawa type term. B. Inter-Galactic dynamics
Now let us turn our attention to inter-galactic scales. By using the radial geodesics equation ¨ r + Γ rµν ˙ x µ ˙ x ν = 0, andthe first integrals of motion given in (53), we find an equation describing the radial acelerations of galaxies towardseach other as d rdt = − A (cid:48) B = − mφ r + Λ r ϕ (cid:48) φ , (115)where r describes radial separation between two galaxies and m is the total mass of the galaxies. Here inner structuresand relative rotations of galaxies are ignored, merely by treating them as two point particles. Now we evaluate thisequation for the theories we are considering in this paper. • For BDΛ theory, the aceleration equation (115) has the form d rdt = − mr + 2 ω + 22 ω + 3 Λ r , (116)and the only difference with respect to corresponding GRΛ expression [113] is the factor involving ω in front ofthe cosmological constant. In order to better understand the effects of the mass and cosmological constant onthe dynamics of the galaxies let us calculate the ratio of both terms in (118) and denote by q, which is given by q BD Λ = 2 ω + 22 ω + 3 Λ r m = 2 ω + 22 ω + 3 q GR Λ . (117)where at the last step the corresponding GR Λ expression of q , discussed in detail in [114] is identified. Hence,it is clear that the difference between corresponding equation of GRΛ theory is the factor involving ω . Let usnow discuss the behavior of this factor. This factor has the range (2 / ,
1) for ω >
0, hence it does not changethe order of q in this range of ω . The behaviour of this factor is more complicated for negative values of ω ,which can be seen at the graph (1). This factor even vanishes for ω = − − / ≤ ω ≤ − , where in this rangethe effect of positive cosmological constant is attractive rather than repulsive. However, the value of ω is fixed1by solar system tests as ω ≥ .
000 and the dramatic changes of the behaviour of Λ for negative values of itis ruled out. Therefore, we can have similar conclusions given in [114], namely if we take the value of Λ as itsthe recent observed value, then cosmological constant does not affect interplanetary and galactic scales and itseffects becomes significant at the cluster scales for BDΛ theory. This is because for solar system q GR Λ ∼ − ,for galactic scale q GR Λ ∼ − but for cluster scale q GR Λ ∼ O (1) [114]. However, for the theories where thisfactor is not fixed by solar system tests, these extreme behaviors can be still possible. • For BDV theory the acceleration equation (115) becomes d rdt = − mφ r (cid:20) m s r ) e − m s r ω + 3 (cid:21) + 2 ω + 12 ω + 3 V r φ , (118)including a Yukawa like term in the expression. For the case where m s (cid:29) e − m s → φ = 1, wehave d rdt = − mr + 2 ω + 12 ω + 3 V r . (119)Hence for this case the mass term is the same with the GR case and the term related to the minimum of thepotential has the same factor involving ω as in other cases of this theory.For a very light scalar, m s (cid:28) m × m s and we find d rdt = − mr + 2 ω + 12 ω + 3 V r φ . (120)Here again the term containing V has a numerical factor involving ω , different from both BDΛ and massive BDV cases. For both theories the q factor becomes q BDV = 2 ω + 12 ω + 3 V r φ m . (121)For a light scalar, the value of ω is fixed by solar system tests as ω > .
000 and the numerical factor involving ω of (121) has no effect. Hence, similar to BDΛ theory, the effects of the cosmological constant becomes relevantat the cluster scales for BDV theory with very light scalar field mass. For very heavy scalar field mass case,however, solar system tests do not fix the value of ω and the numerical factor may become important for smallor negative values of it. This fact may have two consequences for BD V theory with heavy scalar field mass: • ω is not fixed, significant deviations from the results of GRΛ theory can be possible to observe inprinciple for negative values of ω , since the behaviour of the factor for negative values of ω may be quite largeas seen in the figure (1). •
2) Phenomena at ranges larger then solar system scale may help to limit BD parameter ω for this theorycompared to GRΛ theory if independent measurements determine the value of V and m in (121). IX. CONCLUSIONS
In this paper, we have discussed weak field equations of Brans-Dicke scalar-tensor theory extended by the presenceof either a cosmological constant term coupled linearly to the scalar field or a generic potential in the Jordan frame.The linearized field equations of both cases are obtained in the gauge choice which makes the scalar field termsdecouple from the metric field equations. The most important differences of both theories is that the former leads toa massless scalar field with a source whereas the latter has a massive scalar field where mass term is proportional tosecond derivative of the potential in the Taylor expansion as usual. To our knowledge, the linearized expansion of theformer case is not present in the literature.In the second part of the paper, we have considered the weak field solutions for a massive point particle forboth theories in the linear approximation. The solutions have been first obtained in the gauge employed and thentransformed to some physically relevant coordinates such as isotropic and Schwarzschild type coordinates. As aphysical application, particle motion of test particles has been investigated with the focus on the solar system effectssuch as advance of perihelion, deflection of light rays and gravitational redshift. The effect of mass of the source,2cosmological term or minimum of the potential on these phenomena were derived in the linear order. The effect of themass of the scalar field is also determined for
BDV theory, which contains Yukawa like terms, but analytic solutionswere derived only for very light or very massive scalar field. The effects of the terms responsible for asymptoticalnonflatness, namely Λ or V are similar to cosmological constant in GRΛ theory except some factors involving BDparameter ω , which are different for both theories. This might imply a new observational window in the future, forexample to limit ω for BDΛ or BDV theories. However, the Casini mission limits BD parameter to ω > .
000 [64]for original BD theory, and this limit is also valid for BDΛ theory and BDV theory with very light scalar. Hence, weconclude that the effects of the cosmological constant or the minimum of the potential are indistinguishable for thesetheories. For BDV theory with a very heavy mass, however, since the scalar field has a very short range and freezesout outside this range, the effect of mass of the source to this phenomena becomes identical to corresponding GRone. Hence solar system test are satisfied irrespective of the value of ω . Therefore, the correction to these phenomenadue to the minimum of potential has a factor involving ω , whose value can take much larger and smaller values then O (1) as given in figure (1). Hence, for BDV theory with a very heavy mass, the effect of minimum of potential maybe different from corresponding GR one even if one uses the same observed value of cosmological constant for theminimum of the potential. This fact may even lead to put some bounds on ω for very massive BDV theory if thesephenomena will be measured with enough sensitivity in the future.The latter part of this work is devoted to galactic and intergalactic dynamics of these theories. For the galaxyscale we have calculated rotational velocity of stars in a galaxy and see that the nearly flat region of the galaxyrotation curve cannot be explained by the cosmological constant of BDΛ theory as well as the minimum of potentialfor BDV theory with light scalar field mass. Moreover, the effects of mass and the cosmological constant or minimumof potential becomes indistinguishable from corresponding GR ones since the factors involving ω becomes equal to onefor the observed limit of ω . For a very heavy scalar field mass, however, since there is no limit on ω due to solar systemtests, there is a range of ω where the correction factor becomes negative. Hence the contribution of the minimum ofpotential becomes an increasing function of r , which may contribute to the flat rotation curves for − / < ω < − / ω . This is because the mass of thescalar field introduces a Yukawa like term in rotation velocity expression and this term can produce such a behaviourif − . < ω < − .
02. For the intergalactic scale, we have generalized the GRΛ expression corresponding to theacceleration of two galaxies towards each other where we have treated galaxies as point particles. We have obtained afactor q which can determine the scale where the contribution of the cosmological constant starts to become relevantwhen this factor becomes of the order of unity. Similar to other phenomena we have discussed, this factor becomesindistinguishable for BDΛ or BDV theory with a very light scalar mass from corresponding GRΛ case, due to thelarge value the solar system tests sets on the BD parameter ω . For BDV theory with heavy scalar mass, the scalewhere the factor q becomes at the order of unity can be very different than corresponding GRΛ theory even if we usethe minimum of potential equal to the observed value of the cosmological constant. Hence, these phenomena maylead to test the BDV theory with a very heavy scalar field mass or to limit the range the parameter ω compatible withobservations if in the future there will be observations with enough sensitivity to determine the other parameters ofthe theory. ACKNOWLEDGEMENTS
H.O. is partially supported by TUBITAK 2211/E Programme.
Appendix A: Deflection of light rays in BD Λ theory Here let us find a solution to the orbit equation (86) in the linearized level. In order to find the effect of the pointmass and cosmological constant on a light ray coming from very far region of spacetime, we consider a perturbativeapproach, and consider the following ansatz u (Φ) = u (Φ) + mu (Φ) + Λ u (Φ) . (A1)3Replacing the (A1) into equation (86) we obtain following set of equations in the zeroth order, orders linear on m andΛ as u (cid:48)(cid:48) + u = 0 , (A2) u (cid:48)(cid:48) + u = 2 b (2 ω + 3) φ + 6( ω + 1) u (2 ω + 3) φ , (A3) u (cid:48)(cid:48) + u = − b (2 ω + 3) u . (A4)The solution of the first equation yields a photon following a straight line with u (Φ) = sin(Φ) R . (A5)Replacing this into remaining equations, one obtains the solution u (Φ) = 1 r (Φ) = sin(Φ) R + m (cid:26) b (2 ω + 3) φ + ( ω + 1)[3 + cos (2Φ)](2 ω + 3) φ R (cid:27) − Λ R cos Φ6 b (2 ω + 3) sin Φ . (A6)In comparison with GR case, the solution (A6) involves b and Λ and these parts vanish in the GR limit ω → ∞ . Thissolution implies that, in addition to the Einstein deflection angle multiplied by a multiplicative factor of ω , an extracontribution comes from the cosmological term. This extra deflection angle is due to the interaction of cosmologicalterm Λ and the scalar field and vanishes for the GR limit. Hence, unlike GR case, the orbit of the photons depend on Λ. Note that the relation between integration constant R and closest approach distance r can be found by settingΦ = π/ r = 1 R + 2 m (2 ω + 3) φ (cid:20) b + ( ω + 1) R (cid:21) . (A7)The equation satisfied by closest approach distance dr/d Φ = 0 is given by1 b + (2 ω + 2)Λ3(2 ω + 3) = 1 r − m (2 ω + 4)(2 ω + 3) φ r . (A8)These relations show that in the linearized order in the orbit equation (A6) we can take R = r = b interchangeably.Thus the solution can be expressed in terms of only one of these constants, such as R . Using this fact, the solution(A6) simplifies to (87).In order to calculate the deflection angle, we use the method developed in [82] by Rindler and Ishak. The deflectionangle can be calculated from (81) where here β = drd Φ = mr R ω + 1)(2 ω + 3) φ sin 2Φ − r R cos Φ − Λ Rr ω + 3) cos Φ (cid:18) Φ (cid:19) . (A9)Here, unlike [82], there is singularity in the solutions for Φ = 0, so we measure the deflection angle at Φ = Φ (cid:28) ≈ Φ , cos Φ ≈ . Then,from (A6) we have u = 1 r ≈ Φ R + 2 mφ R − Λ R ω + 3)Φ . (A10)The value of β in (A9) becomes | β | = r R (cid:20) − m (2 ω + 2)(2 ω + 3) φ R Φ + Λ R ω + 3) (cid:18) (cid:19)(cid:21) . (A11)Note that the angle Φ should be at the same order of magnitude as the other parameters in (A6), namely we canchoose Φ ≈ O ( m/R ).Using these results and the metric function B evaluated at the r value (A10), and β given in (A11), the expression(81) yields the following result, at most the linear order of the parameters m, Λ and Φ , as ψ = r | β |√ B = Rr (cid:20) − ω + 2) m Φ (2 ω + 3) φ R + Λ R ω + 3) (cid:18) (cid:19)(cid:21) − (cid:20) − (2 ω + 2) m (2 ω + 3) φ r − (2 ω + 1)Λ r ω + 3) (cid:21) (A12) ≈ Φ + 2 mφ R − Λ R ω + 3)Φ − (2 ω + 1)Λ R ω + 3) (cid:104) Φ R + mφ − Λ R ω +3)Φ (cid:105) − Λ R ω + 3)Φ Φ R + mφ − Λ R ω +3)Φ R . m and Φ . This expression yieldsour result given in equation (88) for half deflection angle for BDΛ theory defined as α = ψ − Φ . [1] A.G. Riess et al. Astron. J. , 1009 (1998).[2] S. Perlmutter et al.,
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