Can a non-local model of gravity reproduce Dark Matter effects in agreement with MOND?
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Can a non-local model of gravity reproduce Dark Matter effectsin agreement with MOND?
Ivan Arraut (1 , Department of Physics, Osaka University,Toyonaka, Osaka 560-0043, Japan and Theory Center, Institute of Particle and Nuclear Studies,KEK Tsukuba, Ibaraki, 305-0801, Japan
Abstract
I analyze the possibility of reproducing MONDian Dark Matter effects by using a non-local modelof gravity. The model was used before in order to recreate screening effects for the CosmologicalConstant (Λ) value. Although the model in the weak-field approximation (in static coordinates)can reproduce the field equations in agreement with the AQUAL Lagrangian, the solutions arescale dependent and cannot reproduce the same dynamics in agreement with MOND.
PACS numbers: 04.20.-q, 04.50.Kd, 95.36.+x, 98.80.-k, 95.35.+d . INTRODUCTION One of the biggest problem in Cosmology is to explain the observed flatness for the galaxyrotation curves and the related observed gravitational lenses effect [1–5]. Many attemptshave been done in order to solve the problem, including Modified Newtonian Dynamics(MOND) [6–10], Modified Gravity (MOG) [11–13], non-localities [14–16], etc. Another bigproblem in Cosmology is to explain the small observed value of the Cosmological ConstantΛ or in other models, to explain the observed accelerated expansion of the universe or Darkenergy [17, 18]. There have been many attempts in order to explain the observed smallvalue of Λ. Among the possibilities, we can find that the introduction of non localities canreproduce the effect of Screening of the Cosmological Constant. Some works around thisissue have been proposed by S. Deser, R. P. Woodard, Odintsov and Sasaki[19–21]. If non-localities can reproduce Dark Matter effects (not necessarily MONDian) and non-localitiescan also screen the Λ value, then it is natural to ask if it is possible to create a non-localmodel of gravity such that both effects could be incorporated.If we observe that the MOND fit parameter is the Λ scale a = r Λ , with r Λ = 10 mt,then it is natural to believe that perhaps Dark Matter and Dark Energy have a commonorigin [6–10]. It is natural then to suspect that the small value of Λ is just related to theexistence of Dark Matter. Under that philosophy, in this paper I take an already suggestednon-local gravity model which introduce two scalar fields (one non-dynamical) in order tocreate an screening effect of the Λ value and I then compare it with the MOND resultsin agreement with the AQUAL-like equation ∇ · (cid:16) µ (cid:16) |∇ Φ | a (cid:17) ∇ Φ (cid:17) = 4 πGρ [22] in order toreproduce the Dark Matter effects at least for the observed Galaxy Rotation curves. TheAQUAL equation provides the appropriate predictions for the extragalactic phenomenologyeven if the AQUAL model itself cannot be realistic since it provides unphysical results [23–25]. It is however already known that an appropriate Relativistic version of MOND mustreproduce the AQUAL equation in the weak field approximation [22].In this paper I find that a non-local model of gravity can reproduce the same AQUALequations in agreement with MOND, but not the same dynamics since the interpolatingfunction parameter µ in this model is a scale-dependent quantity because it depends on thepotential φ rather than on the acceleration ∇ φ . I am not concerned about the origin ofthe non-localities in this paper. The paper is organized as follows: In Section II, I make abrief review of the Lagrangian formulation of MOND. In Section III, I derive the MONDfield equations from the AQUAL Lagrangian. In section IV, I analyze the behavior of theMOND field equations for large distances from the source. In section V, I introduce thenon-local model of gravity originally studied in order to create an screening effect for Λ. Insection VI, I analyze the standard Newtonian limit for the Einstein’s field equations with Λand then I analyze the Newtonian limit for the non-local model of gravity. I then find theexplicit form of the interpolating function µ in terms of the potential. In section VII I findthe explicit solutions for φ and µ as a function of the distance r . I then analyze the differentcases depending on the value taken by the free-parameters of the model. In section VIII Imake a comparison with other non-local models which were able to reproduce MOND.2 I. LAGRANGIAN FORMULATION OF MOND
The modified dynamics assumption in agreement with Milgrom [6–10], can be based inthe following set of minimal assumptions [22–25]: 1). There exist a breakdown of Newtoniandynamics (second law and/or gravity) in the limit of small accelerations.2). In this limit, the acceleration ~a , of a test particle in the gravitating system is givenby ~a (cid:16) ~aa (cid:17) ≈ ~g N , where ~g N is the conventional gravitational field and a is a constant withdimensions of acceleration.3). The transition from the Newtonian regime to the small acceleration asymptotic regionoccurs within a range of order a about a . The value of a is of the same order of magnitudeof cH . The original results obtained by Milgrom [6–10] can be described either of thefollowing ways. A modification of the inertia: mµ (cid:18) aa (cid:19) ~a = ~F (1)Where ~F is an arbitrary static force assumed to depend on its sources in the conventionalway, m is the gravitational mass of the accelerated particle and µ is the interpolating functionwhich will be defined later. In the case of gravity, ~F = m~g N , where ~g N = −∇ φ N and φ N is thegravitational potential deduced in the usual way from the Poisson equation. Alternatively,MOND in agreement with [6–10] can be described as a modification of gravity leaving thelaw of inertia ( m~a = ~F ) intact. Then, ~F = m~g and ~g is a modified gravitational field derivedfrom ~g N using the relation: µ ( g/a ) ~g = ~g N (2)If only gravitational forces were present, both formulations of MOND given by eqns. (1)and (2) would be just equivalent. However, if we consider any force in general, then theprevious formulations are not the same at all. The interpolating function µ ( x ) satisfies thefollowing conditions: µ ( x ) ≈ if x >> µ ( x ) ≈ x if x << µ ( x ),can be defined in different ways. Here I will not be concerned with its definition but on itsasymptotic behavior.In cases of high symmetry (spherical, plane, or cylindrical), the gravitational field ~g as givenby equation (2) is derivable from a scalar potential φ . However, in the most general casesthis is not possible. As has been already explained by Milgrom in his paper [26], MONDcannot be considered as a theory, but only a successful phenomenological scheme for whichan underlying theory can be constructed. One of the reasons is for example, that insideMOND theory there is no momentum conservation. In fact, momentum is only conservedapproximately as far as the mass of the test body is much smaller than the source one [23–25].Milgrom and Bekenstein have already derived a Lagrangian formulation for MOND, where µ and a are introduced by hand. One of the purposes for constructing a fundamentaltheory which can contain MOND as a non-relativistic limit is to obtain the interpolatingfunction µ ( x ) in terms of some fundamental quantities. This is one of the motivations forthis manuscript. A Lagrangian formulation for MOND solves the momentum conservation3roblem associated typically to the original MOND version. A Lagrangian formulation alsoenables to calculate the dynamics of an arbitrary non-relativistic system [23–25].There are two important assumptions inside the MOND theory, they are:1). A composite particle (star or a cluster of stars) moving in an external field, say a galaxy,moves like a test particle according to MOND. Even if within the body, the relativisticaccelerations are large. This assumption is possible as far as the mass of the test particle ismuch smaller than the mass of the galaxy.2). When a system is accelerated as a whole in an external field, the internal dynamics of thesystem is affected by the presence of an external field (even when this field is constant withouttidal forces). In particular, in the limit when the external (center of mass) acceleration ofthe system becomes much larger than the MOND scale a , the internal dynamics approachesto the Newtonian behavior even when the accelerations within the system are much smallerthan a .This second observation due to the Milgrom proposal is quite interesting and one of themotivations for this manuscript since it seems that the internal dynamics of the system canin principle be affected by the presence of some external field. This could in principle bedone due to a non-local connection between the internal and external dynamics.Normally Newtonian gravity is recovered at the non-relativistic regime of General Relativity(GR), and of a number of other relativistic theories of gravity. It is however, necessary toconstruct a new relativistic version of GR such that MOND could be recovered as a naturalnon-relativistic limit [23–25]. This new version could be constructed by considering twopossibilities. 1). Additional degrees of freedom. 2). Non-localities. If we choose to explainthe origin of Dark Matter by introducing non-localities, we must be able to explain the originof such effects. In this manuscript I will not be concerned with the origin of non-localities,I will introduce them in an arbitrary way and then we will write the MOND interpolatingfunction in terms of the field generating the non-local effects.There are two very important reasons to construct a Relativistic version for MOND. 1). Tohelp incorporate principles of MOND into the framework of modern theoretical physics. 2).To provide tools for investigating cosmology in light of MOND. [23–25] III. THE MOND FIELD EQUATIONS
In Newtonian gravity test bodies move with an acceleration equal to ~g N = −∇ φ N , where φ N is the Newtonian gravitational potential. It is determined by the Poisson equation ∇ φ N = 4 πGρ , where ρ is the mass density which produces φ N . The Poisson equation maybe derived from the Lagrangian: L N = − Z d r ( ρφ N + (8 πG ) − ( ∇ φ N ) ) (4)Milgrom and Bekenstein suggested that in searching for a modification of this theory, wewill want to retain the notion of a single potential φ N from which acceleration derives. Wewant φ N to be arbitrary up to an arbitrary additive constant. The most general modificationof L N which yield these features is: L = − Z d r (cid:18) ρφ + (8 πG ) − a f (cid:18) ( ∇ φ ) a (cid:19)(cid:19) (5)4here f ( x ) is an arbitrary function. The scale of acceleration is necessary unless we arein the Newtonian case. If we perform the variation of L with respect to φ , with variation of φ vanishing on the boundaries, we get: ~ ∇ · µ | ~ ∇ φ | a ! ~ ∇ φ ! = 4 πGρ (6)Where µ ( x ) = f ′ ( x ). Eq. (6) is the equation determining the modified potential. Atest particle is assumed to have acceleration ~g = − ~ ∇ φ . We supplement equation (6) by theboundary condition | ~ ∇ φ | → r → ∞ .It is useful to write the field equation in terms of the modified Newtonian field ~g N = − ~ ∇ φ N ,for the same mass distribution, which satisfies the Poisson equation. By eliminating ρ inequation (6), we get: ~ ∇ · µ | ~ ∇ φ | a ! ~ ∇ φ − ~ ∇ φ N ! = 0 (7)The expression in parenthesis, must then be a curl, since its divergence is zero. Then wecan write: µ (cid:18) ga (cid:19) ~g = ~g N + ~ ∇ × ~h (8)It has already been demonstrated by Bekenstein and Milgrom that the present theorysatisfies the basic assumptions of MOND and that the curl term vanishes exactly for thespherically symmetric case and it vanishes at least as fast as r at large distances from thesource in general cases. In the next section I will consider the review of this point. IV. THE FIELD EQUATIONS AT LARGE DISTANCES FROM THE SOURCE
In agreement with [23–25], let’s consider a bound density distribution of total mass Mwith the origin at the center of mass. Following the Bekenstein and Milgrom notation, let’sdefine the vector field ~u as ~u ≡ ~ ∇ φ N − µ (cid:16) | ~ ∇ φ | a (cid:17) ~ ∇ φ . For the reasons explained in the previoussection, the vector ~u , satisfies ~ ∇ · ~u = 0 and it vanishes at infinity. It is possible then towrite ~u in terms of the vector potential ~A : ~u = ~ ∇ × ~A ~A ( ~r ) = (4 π ) − Z ~ ∇ ′ × ~u ( ~r ′ ) | ~r − ~r ′ | d r ′ (9)The only term with an r − behavior at infinity which ~u can have is ~u (2) = ~ ∇ × ( r − ~B ) = − r − ~r × ~B . Where ~B = (4 π ) − R ~ ∇ ′ × ~u ′ d r ′ , which is the lowest order in the multipoleexpansion (9).If we can demonstrate that ~B = 0, then we get that the lowest contributing multiple termto ~u vanishes at least as fast as r − . In the limit of large r, we have: µ | ~ ∇ φ | a ! ~ ∇ φ = ~ ∇ φ N − ~u = r − ( GM~r + ~r × ~B ) + ~O ( r − ) (10)5aking the absolute value, we get: µ | ~ ∇ φ | a ! | ~ ∇ φ | = ( | ~ ∇ φ N | + | ~u | ) / (11)This expression can be translated into: µ | ~ ∇ φ | a ! | ~ ∇ φ | = (cid:18) GMr (cid:19) + B sin r θ ! / (12)As r → ∞ , the full MONDian regime operates and we can assume that µ (cid:16) | ~ ∇ φ | a (cid:17) = | ~ ∇ φ | a .In such a case, the expression (12) becomes: | ~ ∇ φ | = a / r (cid:0) G M + B sin θ (cid:1) / (13)Assuming again the MONDian regime and replacing the previous expression inside of(10), we get: ~ ∇ φ = a / r − ( GM~r + ~r × ~B )( G M + B sin θ ) / + ~O ( r − ) (14)Here θ is the angle between ~r and ~B which we can take along the z-axis without lost ofgenerality. Requiring now that the azimuthal component of ~ ∇ × ( ~ ∇ φ ) vanishes, gives ~B = 0[23–25]. This means that ~u vanishes at large distances from a mass, at least as ~O ( r − ). Withthis result, the equation (8) as r → ∞ , becomes: µ (cid:18) ga (cid:19) ~g = ~g N + ~O ( r − ) (15)Consistent with the MOND predictions explained in eqns. (1) and (2). As r → ∞ , weget: ~g → − ( GM a ) / r ~r + ~O ( r − ) (16)In this limit, the potential becomes: φ → ( GM a ) / ln (cid:18) rr (cid:19) + O ( r − ) (17)Where r is an arbitrary radius. This potential leads to an asymptotically constant cir-cular velocity V ∞ = ( GM a ) / as it is observed in the outskirts of spiral galaxies.The field equation (6) is nonlinear and difficult to solve in general. However, in cases of highsymmetry, the curl term in equation (8) vanishes identically and we have the exact result µ (cid:16) ga (cid:17) ~g = ~g N which is identical to equation (2). For systems for high degree of symmetry,then the solution for φ is straightforward and all the results obtained from the standardNewtonian theory can then be extended to the present formalism.For example, the acceleration field at a distance r from the center in a spherical systemdepends only on the total mass M ( r ), interior to r (in agreement with the Gauss’ theorem),6nd in fact is given by µ (cid:16) ga (cid:17) ~g = − M ( r ) G~rr .The field equation (6) is analogous to the equation for the electrostatic potential in a non-linear isotropic medium in which the dielectric coefficient is a function of the electric fieldstrength [23–25].The field equation (6) is also equivalent to the stationary flow equations of an irrotationalfluid which has a density ˆ ρ = µ (cid:16) | ~ ∇ φ | a (cid:17) , a negative pressure ˆ P = − a f (cid:16) ( ~ ∇ φ ) a (cid:17) , flow velocityˆ ~v = ~ ∇ φ , and a source distribution ˆ S ( ~r ) = 4 πGρ . The fluid satisfies an equation of stateˆ P ( ˆ ρ ) = − a f ([( µ − ( ˆ ρ )] ).An equation of the same form as equation (6) has been studied to describe classical mod-els of quark confinement using a very different form of the function µ at both, large andsmall values of its argument [27]. The conservation laws and other results related to theLagrangian formulation of MOND can be found in [23–25]. In this manuscript I will omitsuch analysis. V. A NON-LOCAL MODEL FOR GRAVITY
The non-local action suggested in [20] is given by: S = Z d x √− g (cid:18) κ ( R (1 + f ( (cid:3) − R )) − l matter ( Q, g ) (cid:19) (18)Where f is some function, (cid:3) is just the D’Alembertian for the scalar field, Λ is theCosmological Constant which is supposed to be screened by the introduced non-localitiesand Q corresponds to the matter fields. We can rewrite the action by introducing two scalarfields ψ and ζ as follows [20]: S = Z d x √− g (cid:18) κ ( R (1 + f ( ψ )) − ζ ( (cid:3) ψ − R ) − l matter (cid:19) = Z d x √− g (cid:18) κ ( R (1 + f ( ψ ) + ζ ) + g µν ∂ µ ζ ∂ ν ψ − l matter (cid:19) (19)If we vary the above action with respect to ζ , then (cid:3) ψ = R or ψ = (cid:3) − R . The variationwith respect to the metric is:0 = 12 g µν (cid:0) R (1 + f ( ψ ) + ζ ) + g αβ ∂ α ζ ∂ β ψ − (cid:1) − R µν (1 + f ( ψ ) + ζ ) −
12 ( ∂ µ ζ ∂ ν ψ + ∂ µ ψ∂ ν ζ ) − ( g µν (cid:3) − ∇ µ ∇ ν )( f ( ψ ) + ζ ) + κ T µν (20)And the variation with respect to ψ gives:0 = (cid:3) ζ − f ′ ( ψ ) R (21)The explicit solutions for the previous equations, can be found if we introduce a metric.In this manuscript, I will focus on spherical symmetric solutions. I will introduce our metricin the next section. 7 . The ghost free condition In [20, 28], it was found that after a conformal transformation to the Einstein frame, weget: ˜ g µν = Ω g µν ˜ R = 1Ω ( R − (cid:3) ln Ω + g µν ∇ µ ln Ω ∇ ν ln Ω)) (22)Ω = 11 + f ( ψ ) + ζ (23)Which gives an action given by [20]: S = Z d x √− g (cid:18) κ ( ˆ R − g µν ∇ µ φ ′ ∇ ν φ ′ + e φ ′ g µν ∇ µ ζ ∇ ν ψ − e φ ′ Λ) + e φ ′ l matter ( Q ; e φ ′ g ) (cid:19) (24)Where: φ ′ ≡ ln Ω = − ln (1 + f ( ψ ) + ζ ) (25)and ˆ R is the resulting Ricci scalar after performing the transformation (22). The conditionfor gravity to have a normal sign is: 1 + f ( ψ ) + ζ > φ ′ and ψ are considered to be the independent fields, then: ζ = e − φ ′ − (1 + f ( ψ )) (27)Then, in terms of the new set of independent variables, the action is: S = Z √− g κ ( R − ∇ µ φ ′ ∇ µ φ ′ − ∇ µ φ ′ ∇ µ ψ − e φ ′ f ′ ( ψ ) ∇ µ ψ ∇ µ ψ − e φ ′ Λ)+ e φ ′ l matter ( Q ; e φ ′ g ) (28)The ghost free condition is simply: f ′ ( ψ ) > f ( ψ ) + ζ > VI. THE NEWTONIAN LIMIT IN THE STANDARD CASE IN S-DS METRIC
In [20], the metric is assumed to be FLRW. In this case, as we are concerned with the DarkMatter effects and we want to compare with the MONDian case, then I will assume thatthe space time metric corresponds to the Newton-Hooke space, which is just the Newtonianlimit for the Schwarzschild-de Sitter space. Explicitly in eqs. (20) and (21), I will assumethe metric to be: 8 s = − (cid:18) − GMr − r r (cid:19) dt + (cid:18) − GMr − r r (cid:19) − dr + r d Ω (30)With d Ω = dθ + sin θdφ . I will work under the condition r s << r << r Λ . Under thatcondition, the weak field approximation is justified. Under the Weak Field approximation,we have to satisfy the standard results: G (1)00 ≈ (cid:3) g ≈ R (1) = 2 R (1)00 (31)Where the weak field approximation for the Ricci tensor is given by: R (1) µν ≡
12 ( (cid:3) h µν − ∂ λ ∂ µ h λν − ∂ λ ∂ ν h λµ + ∂ µ ∂ ν h ) (32)And then, the first order Einstein’s equations become: R (1) µν − η µν R (1) + η µν Λ = − πG N T (1) µν (33)With the metric given by (30), then we get: ∇ g = − πGρ + 2Λ (34)If g µν ≈ η µν + h µν , then ∇ g ≈ ∇ h . Thus: ∇ φ = 4 πG N ρ − Λ (35)With h = − φ = h ij and T ≈ ρ . The spherical symmetry of the metric (30) isimportant since it implies that the curl term in equation (8) can be ignored in agreementwith the analysis performed in the previous section. The results of this section will be usedin the field equations 20 and 21. A. The weak field approximation in non-local gravity and its relation with MOND
In agreement with Bekenstein [22], we have to satisfy at the Newtonian limit an equationsimilar to the AQUAL given already in equation (6). We rewrite it as follows: ∇ φ ≈ µ − ( κ ρ − Λ) − µ − ( ∇ φ ) . ∇ (cid:18) µ (cid:18) |∇ φ | a (cid:19)(cid:19) (36)For the Newtonian limit of the field equations (20), I will make the expansions up tosecond order in the potential φ . Even if the second order terms are most likely negligible,I will keep them in order to get a more accurate result. Take into account that the non-localities, represented by f ( ψ )+ ζ in eq. (20) reproduce an amplification of the non-linearitiesrelated to the space time curvature and it includes the second order contributions. Thisamplification will however depend on a parameter γ which will be defined later. The 0-0component of eq. (20) is then given by:0 ≈ − (cid:0) R (1) (1 + f ( ψ ) + ζ ) + ∇ r ζ ∇ r ψ − (cid:1) − φR (1) (1 + f ( ψ ) + ζ ) − R (1)00 (1 + f ( ψ ) + ζ ) + (1 + 2 φ ) (cid:3) ( f ( ψ ) + ζ ) + ∇ ∇ ( f ( ψ ) + ζ ) − κ ρ (37)9here we have used T = − ρ . In this approach, we neglect the time-dependence ofthe scalar fields. However we take into account the curvature effects through the Christoffelconnections. We can write ζ in terms of f ( ψ ) if we solve the equation (21). For that purpose,we assume an exponential solution like f ( ψ ) = f e γψ as has been suggested in [20]. Thenthe following relations are true: ∇ µ f ( ψ ) = γf ( ψ ) ∇ µ ψ (cid:3) f ( ψ ) = γf ( ψ ) (cid:3) ψ + γ f ( ψ ) ∇ µ ψ ∇ µ ψ (38)We can then prove that the solution for eq. (21) if we expand both sides of the equationand then compare the same order of magnitude terms. The resulting equation is:2 ∇ φ · ∇ ζ + 2 r ∂ζ∂r + ∂ ζ∂r ≈ − γf ( − φ ) (cid:18) ∂φ∂r (cid:19) + 4 r ∂φ∂r + 2 ∂ φ∂r ! (39)The solution for ζ is (ignoring second order contributions): ζ ( ψ ) ≈ f ( ψ ) (40)Where we have used the Lagrange multiplier condition R = (cid:3) ψ . If we replace thesolution for ζ ( ψ ), taking into account eq. (21) and the Lagrange multiplier condition, eq.(37) becomes:0 ≈ − (cid:0) R (1) (1 + 2 f ( ψ )) + γf ( ψ )( ∇ ψ ) − (cid:1) − φR (1) (1 + 2 f ( ψ )) − R (1)00 (1 + 2 f ( ψ ))+2 γ (1 + 2 φ ) f ( ψ ) (cid:3) ψ + 2 ∇ ∇ f ( ψ ) − κ ρ (41)The Christoffel connection component is given by Γ r ≈ ∂ r φ = ∇ r φ . I will take the spatialcomponents of the Einstein’s equations as given by the standard Newtonian approach as itis explained in the standard textbooks. In such a case, I will take the Ricci tensor and thecurvature scalars as: R (1) = 2 R (1)00 ≈ (cid:3) g = (cid:3) h = − (cid:3) φ ≈ − ∇ φ ) · ( ∇ φ ) − ∇ M φ − φ ∇ M φ (42)Note that we are just rewriting the result (32) for the case of a static potential. Note alsothat in the standard Newtonian approach (cid:3) h = ∇ M h , where the subindex M makesreference to the Minkowskian case. However, in this case I consider the expansion up tosecond order and it includes the curvature effects obtained from the Christoffel connections.In principle, the scalar curvature and the Ricci tensor when expanded up to second orderare given by: R ≈ R (1) + R (2) R µν ≈ R (1) µν + R (2) µν (43)Up to first order, then the approximation ψ = − φ is valid in agreement with eq. (42)and the Lagrange multiplier condition. On the other hand, eq. (41) expanded up to secondorder is equivalent to: ∇ M φ ≈ µ − ( κ ρ − Λ) − µ − ∇ φ · ∇ φ (4 ω ) (44)Where µ ( φ ) and ω are defined by: 10 ( φ ) = 2(1 + 2 f ( ψ )(1 + 3 φ − γ − φγ ) + 3 φ ) ω = 1 + 2 f ( ψ ) (cid:18) − γ (cid:19) (45)We can observe that eq. (44) has the same form of eq. (36) which describes the MONDiandynamics. There is however a fundamental difference since in the non-local model, µ definedin eq. (45) is just a function of the potential rather than a function of the acceleration ( |∇ φ | )as it is the case in the MONDian dynamics. For the weak field approximation, the followingapproximations for the eqs. (45) are valid: µ ( φ ) ≈ f (1 + 3 φ − γ − φγ ) + 3 φ − γf φ (1 − γ )) ω ≈ f (cid:18) − γ (cid:19) (46) VII. EXPLICIT SOLUTIONS FOR φ AND µ I will compute the explicit solutions for µ and φ in agreement with the equation (44).Then different regimes will be explored (different values for γ ) and I will identify somespecial values for γ . It is simpler to start by solving µ . For that purpose we have to findthe solutions for the following equations in agreement with the result (45) for the weak fieldapproximation: ∇ µ = − ∇ φ (cid:18) ( − γ ) f −
32 + 2 f γ (1 − γ ) (cid:19) (47)And: ∇ µ = − ∇ φ (cid:18) ( − γ ) f −
32 + 2 f γ (1 − γ ) (cid:19) (48)Then we can write eq. (44) in terms of µ . In vacuum the result is: µ ∇ M µ ≈ C ∇ µ · ∇ µ (49)Where we have defined C as: C = ω (cid:0) ( − γ ) f − + 2 f γ (1 − γ ) (cid:1) (50)The general solution for µ is given by: µ ( r ) = A (cid:18) − Cr + B (cid:19) − C (51)Note that this solution is valid for C = 1. As C = 1, eq. (49) becomes: µ ∇ M µ ≈ ∇ µ · ∇ µ (52)The solution for this equation is: µ ( r ) = De − E/r (53)11 . Solutions for µ and φ for special values of γ There are different possible solutions for φ and µ in agreement with the results obtainedin the previous section. In vacuum and ignoring the Cosmological Constant Λ, we can writethe equation (44) as follows: µ ∇ M φ = C ( ∇ φ ) · ( ∇ µ ) (54)Where we have used the results obtained in eq. (47) and the definition (50). I will analyzesome relevant results summarized in the following table: TABLE I: Relevant values for C as a function of the parameter γ and ω . C γ ω ∞ −∞ −∞ ∞ f −√ f √− f f − f − (cid:0) f − √ f √−
160 + 209 f (cid:1) f + √ f √− f f − f − (cid:0) f + √ f √−
160 + 209 f (cid:1) -1 f + √ f √− f f − f − (cid:0) f + √ f √−
32 + 225 f (cid:1) -1 f −√ f √− f f − f − (cid:0) f − √ f √−
32 + 225 f (cid:1) ∞ f + √ − f +13 f f − f − (cid:16) f + p − f + 13 f (cid:17) −∞ f − √ − f +13 f f − f − (cid:16) f − p − f + 13 f (cid:17) f + f −√ √ f − f f − f − (cid:16) f − √ p
32 + 35 f − f (cid:17) Max f + √ √ f − f f − f − (cid:16) f + √ p
32 + 35 f − f (cid:17)
12 13 f − √ − f +57 f f − f − (cid:16) f − p − f + 57 f (cid:17)
12 13 f + √ − f +57 f f − f − (cid:16) f + p − f + 57 f (cid:17) Where
M in and
M ax correspond to a local minimum and a local maximum respectivelyfor the parameter C as can be easily verified. If we replace the result (51) inside the definitionof µ given in eq. (45), we then obtain the solution for φ consistent with eq. (52). Up to firstorder, the result is: φ = A ( γ ) (cid:18) − Cr + B (cid:19) − C − (cid:18) f (1 − γ )2(3 + 2 f (3 − γ + 4 γ )) (cid:19) (55)Where A ( γ ) is defined as: A ( γ ) = − (cid:18) A (cid:19) Cω (56)The same result for the case C = 1 is: φ = A ( γ ) e − D/r − (cid:18) f (1 − γ )2(3 + 2 f (3 − γ + 4 γ )) (cid:19) (57)12here D is just another integration constant and ω has to be evaluated for the case C = 1. The case C = 1 corresponds to two different values for the parameter γ as can beseen from Table I. The equations (55) and (57) can be rewritten in a compact form as: φ = − (cid:18) A (cid:19) (cid:18) Cω (cid:19) (cid:18) − Cr + B (cid:19) − C + 12 (cid:18) Cω (cid:19) (1 + 2 f (1 − γ )) (58)And: φ = − (cid:18) A (cid:19) (cid:18) ω (cid:19) e − D/r + 12 (cid:18) ω (cid:19) (1 + 2 f (1 − γ )) (59)In both cases, the condition ω = 0 is satisfied. If ω = 0, then C can take 3 differentvalues in agreement with the Table I. The standard Newtonian behavior is recovered for thecase C = 0 = ω . There will be values of γ for which the potential φ will be attractive andother values for which it will be repulsive. B. Special cases for different values of the parameter γ and the ghost-free condi-tion If we calculate the gradient of the potential φ . Without loss of generality, we can set B = 0 for the cases C = − B = 0, we get: ∇ φ = − (cid:18) A (cid:19) (cid:18) Cω (cid:19) (cid:18) − Cr (cid:19) C − C r (60)For C = −
1. This potential can be attractive or repulsive depending of the value of theratio Cω and the relative sign of C with respect to − C = 1 is taken from eq. (59) and it is given by: ∇ φ = − (cid:18) A (cid:19) (cid:18) Dω (cid:19) (cid:18) e − D/r r (cid:19) (61)Which is attractive or repulsive depending on the sign of ω . C. The relevant cases for the potential
If we take into account that dynamically the potential satisfies the condition ~ ∇ φ = v r ,where v is the magnitude of the velocity, then a flat rotation curve for a galaxy can bereproduced only if ∇ φ ∝ r . But it seems that this case the behavior is not reproduced forany value of the parameter C . From eq. (60), it is clear that the Newtonian behavior isreproduced as C = 0. For a well behaved solution, from the table I, we can see that in sucha case, ω = 0. From eq. (50), the relation Cω , then becomes: Cω = 1 f − − f = 00 (62)13here we have introduced the appropriate value for γ taken from the table I. If we replacethis condition inside eq. (60), we then obtain: ∇ φ = − (cid:18) A (cid:19) f − − f ! (cid:18) r (cid:19) = − GMr (63)Where we have imposed the Newtonian limit condition. We have to satisfy the condition: GM = (cid:18) A (cid:19) − f + + f ! (64)Then equation (59), for the full potential becomes: φ = − GMr + 15 f − f (65)Where we have replaced the appropriate values for the constant term in eq. (59). Fromthe previous equations, it is clear that if we want to reproduce the appropriate Newtonianbehavior, then the constant A has to satisfy: A = 4 GM (cid:18) − f + 3518 + 169 f (cid:19) (66)The remaining constant term is not important in order to obtain the Newtonian behavior.It is just a constant quantity which can be ignored for the computations. D. The case C = 1 The case C = 1 is extremely relevant since it looks like a Yukawa-like potential. If wereplace the appropriate value for γ and ω from the Table I, then we can write the equation(61) like: ∇ φ = − (cid:18) A (cid:19) D (cid:0) − f − (23 f − √ f √−
160 + 209 f (cid:1) e − D/r r (67)Where we have used the first value for ω corresponding to C = 1. For the second valueof ω corresponding to C = 1, we can obtain the following result: ∇ φ = − (cid:18) A (cid:19) D (cid:0) − f − (23 f + √ f √−
160 + 209 f (cid:1) e − D/r r (68)The form of this solution just suggest that the behavior of this potential is approximatelyNewtonian as the exponential factor tends to 1. The attractive or repulsive character of thissolution depends on the values taken by f . E. The case C = − The case C = − ∇ φ = −∇ φ · ∇ µ (69)This equation has can be written as: ∇ · ( µ ∇ φ ) = 0 (70)In vacuum, this has the same structure as the equation (7). With the difference that inthe non-local model the interpolating function µ is a function of the potential itself, ratherthan a function of its gradient as MOND suggest. The case C = − B = 0 and B > r in eq. 55, otherwise the potential in such a case becomes complex. VIII. A COMPARISON WITH OTHER MODELS
The non-local model analyzed in this manuscript is able to reproduce the equation (44)with the definitions (45). This equation (after some arrangements) is similar to eq. (6) or(36) which is obtained from the AQUAL Lagrangian (5). However, the present model cannotreproduce the same dynamics due to MOND since the predicted interpolating function µ is a function of the potential ( φ ), rather than a function of the acceleration ( |∇ φ | ) as canbe observed from eq. (45) and the fact that f ( ψ ) = f e γψ with ψ = − φ . This previousrelation is precisely the source of the problem for reproducing MOND appropriately. In [29],it has been demonstrated that in order to reproduce the MONDian dynamics, it is necessaryto add to the Einstein-Hilbert action, a Lagrangian such that it cancels the quadratic partsof the action and also provides some additional terms whose variations are: c a r (( rb ′ ( r )) ) ′ = 8 πGρc (71) c a r ( krb ′ ( r ) − a ( r )) = 0 (72)where a ( r ) and b ( r ) are given by: a ( r ) ≡ A ( r ) − b ( r ) ≡ B ( r ) − ds = − B ( r ) c dt + A ( r ) dr + r d Ω (74)the key point in the work performed in [29] is to change how the potentials depend uponthe source without changing how they depend each other. In fact, the relation between thelinearized potentials is given by: a ( r ) ≈ rb ′ ( r ) (75)This relation is necessary in order to reproduce the appropriate amount of weak lensingconsistent with the data. In the standard formalism of General Relativity, the linearizedpotentials take the form: rb ′ ( r ) ≈ GM ( r ) c r (76)15ut in the MONDian regime, the following relation has to be satisfied: rb ′ ( r ) → p a GM ( r ) c (77)The MOND Lagrangian which cancels the quadratic terms of the Einstein-Hilbert actionand also reproduces the results (71) and (72) is [29]: L MOND → c πG (cid:18) rab ′ ( r ) − a ( r )2 + O ( h ) (cid:19) + c a (cid:18) αa ( r ) r + βa ( r ) b ′ ( r ) + γra ( r ) b ′ ( r ) + δr b ′ ( r ) + O ( h ) (cid:19) (78)the first line of this Lagrangian, just cancels the Einstein-Hilbert action terms given by: L EH = c πG R √− g → c πG (cid:18) − ra ( r ) b ′ ( r ) + a ( r )2 + O ( h ) (cid:19) (79)Where the right-hand side (after the arrow) is obtained after integration by parts andhere we ignore total derivative terms. Note that for the total Lagrangian L = L EH + L MOND ,the Einstein-Hilbert terms vanishes. It has been demonstrated in [29] that no local invariantLagrangian can reproduce the cubic terms of the MONDian action (78). The reason is thatthe curvature tensor and its possible contractions, can only reproduce terms involving twoderivatives acting on one or more weak fields in the following way [29]:(
Curvature ) N ∼ ( h ′′ ) N + O (( h ′ ) ( h ′′ ) N − ) (80)On the other hand, the MOND corrections in eq. (78), involve powers of just one deriva-tive acting on a single weak field like: L MOND ∼ c r πG (cid:18) ( h ′ ) + c a ( h ′ ) + O ( h ) (cid:19) (81)is in this part where the model proposed in this manuscript fails. In this manuscript,the non-localities enter through the function f ( ψ ) with the Lagrange multiplier condition R = (cid:3) ψ (with − φ = ψ ). In such a case, then the non-localities will enter as an algebraicexpansion of the potentials. This can be seen in eq. (45) and the action (19) if we take intoaccount that f ( ψ ) = f e γψ . If we want to reproduce the MONDian dynamics, one possibilityis for example to expand the function f ( ψ ) around the scale defined as the geometric averageof the Gravitational radius and the inverse of the acceleration scale a characteristic ofMOND, the scale is r = q GMa . In such a case, we would get f ( ψ ) ≈ f + f γ ( ∇ ψ ) r = r .But doing this expansion just breaks the nature of the model since in such a case, we areimposing by hand the scale at which the MONDian regime applies rather than obtaining it.The model proposed here cannot reproduce the form for the Lagrangian (78) or (81) in anatural way.In [29], the non-localities are used in order to reduce the number of derivatives for theweak fields and particular components for the curvature are selected by using a time-like4-vector obtained from the gradient of the invariant volume of the past light cone (see [29]for details). The reduction of the number of derivatives is the appropriate such that the16ONDian Lagrangians (78) or (81) with only a single derivative of the weak fields can bereproduced. Remember that the standard Einstein-Hilbert Lagrangian can only reproducepowers of two derivatives acting on weak fields. In the model proposed in this manuscript,it seems that the reduction of derivatives is higher, such that instead of a MONDian actionwith single derivatives on weak fields, we have an action with no derivatives (only algebraicrelations). However the model proposed in this manuscript does not have any problem withthe lenses since the condition (75) is satisfied.Another attempt for reproducing MOND by using non-localities is done in [30]. In sucha case, the model can reproduce galaxy rotation curves but not the observed gravitationallenses. The model in [30] proposes a Lagrangian given by: L = c πG ( R + c − a F ( c a − g µν ǫ, µ ǫ, ν )) √− g (82)Where ǫ is the small potential as it is defined in [30]. The main point here is that aninterpolating function F ( x ) is introduced since the beginning and MOND is embedded insidethis Lagrangian under the assumption that at small x the MONDian dynamics appear. Thefactor inside the interpolating function is a kinetic term for the small potential and it makeseasier to recover the MONDian dynamics. This is the main difference with respect to themodel proposed in this manuscript where, as has been said before, the non-localities areintroduced as algebraic expansion of the potential. If we want to mimic in some sense themodel suggested in [30], we would have to expand the function f ( ψ ) around the neighborhoodof some imposed scale as has been explained before in this section. The model in [30] alsoproposes the same relation (77) but it cannot reproduce the appropriate lenses without theDark Matter assumption. IX. CONCLUSIONS
The non-local model in the present form can reproduce some additional attractive effectsfor some range of the parameter γ . The model cannot reproduce the MONDian dynamicswithout a strong tunning of the parameters. However, it can reproduce the AQUAL fieldequations with a scale-dependent interpolating function µ for some special case given by C = −
1. That case however, requires the additional condition
B > r everywhere. Thereproduction of the AQUAL equations is in agreement with Milgrom and Bekenstein,the first step for getting a Relativistic version of MOND. Every attempt in modifyinggravity in order to reproduce the MONDian dynamics, must reproduce equations like theAQUAL Lagrangian explained before in this manuscript. Further research is needed inorder to see whether or not is viable to reproduce the Dark Matter effects in agreementwith non-localities. Another alternatives for the introduction of non-localities have beenexplored in [29] and [30]. In [29], the non-localities were able to reproduce the MONDiandynamics and the appropriate gravitational lenses. In the case of [30], the non-localitiescould reproduce the dynamics but not the lenses. Acknowledgement
The author would like to thank Jacob Bekenstein for a very useful correspondence as wellas Misao Sasaki for useful discussions and comments. This work is supported by MEXT17The Ministry of Education, Culture, Sports, Science and Technology) in Japan and KEKTheory Center. 18
1] K. Kuijken and G. Gilmore, Mon. Not. R. Astron. Soc. (1989a) 651.[2] H. Babcock,
The rotation of the Andromeda Nebula , Lick Observatory bulletin ; no. ,(1939).[3] L. Volders.
Neutral hydrogen in M 33 and M 101 , Bulletin of the Astronomical Institutes ofthe Netherlands (492): 323.[4] A. Bosma, The distribution and kinematics of neutral hydrogen in spiral galaxies of variousmorphological types , PhD Thesis, Rijksuniversiteit Groningen, 1978, available online at theNasa Extragalactic Database.[5] V. Rubin, N. Thonnard, W. K. Ford, Jr,
Rotational Properties of 21 Sc Galaxies with aLarge Range of Luminosities and Radii from NGC 4605 (R=4kpc) to UGC 2885 (R=122kpc) .Astrophys. J. . Bibcode 1980ApJ...238..471R. doi:10.1086/158003, (1980).[6] K. G. Begeman, A. H. Broeils and R. H. Sanders,
Extended rotation curves of spiral galaxies:dark haloes and modified dynamics , Mon.Not.Roy.Astron.Soc. 249 (1991) 523.[7] M. Milgrom,
A Modification of the Newtonian Dynamics: Implications for Galaxy systems ,Astrophys.
J. 270 (1983) 384.[8] M. Milgrom,
A Modification of the Newtonian dynamics: Implications for galaxies , Astrophys.
J. 270 (1983) 371.[9] M. Milgrom,
A Modification of the Newtonian dynamics as a possible alternative to the hiddenmass hypothesis , Astrophys.
J. 270 (1983) 365.[10] B. Famaey
Modified Newtonian Dynamics (MOND): Observational Phenomenology and Rel-ativistic Extensions , Living Rev.Rel. (2012) 10.[11] J. R. Brownstein and J. W. Moffat, Galaxy rotation curves without non-baryonic dark matter ,Astrophys.
J. 636 (2006) 721.[12] J. W. Moffat,
Scalar-tensor-vector gravity theory , JCAP 0603 (2006) 004.[13] J. R. Brownstein and J. W. Moffat,
Galaxy cluster masses without non-baryonic dark matter ,Mon. Not. Roy. Astron. Soc. (2006) 527.[14] C. Chicone, B. Mashhoon,
Modified Poisson’s equation , J.Math.Phys. (2012) 042501.[15] B. Mashhoon, Nonlocal Gravity , arXiv:1101.3752 [gr-qc].[16] H. J. Blome, C. Chiconeand, F. W. Hehl and B. Mashhoon,
Nonlocal Modification of Newto-nian Gravity , Phys.Rev.
D81 (2010) 065020.[17] V. Sahni,
Dark Matter and Dark Energy , Lect.Notes Phys. (2004) 141.[18] J. Martin,
Everything you always wanted yo know about the Cosmological Constant problem(But were afraid to ask) , Comptes Rendus Physique (2012) 566.[19] S. Deser and R. P. Woodard, Nonlocal Cosmology , Phys. Rev. Lett. , 111301 (2007).[20] S. Nojiri, S. D. Odintsov, M. Sasaki and Y. Zhang, Screening of cosmological constant innon-local gravity , Phys.Lett. B696 (2011) 278.[21] S. Nojiri and S. D. Odintsov,
Modified non-local-F(R) gravity as the key for the inlation anddark energy , Phys. Phys. Lett.
B 659 , (2008), 821.[22] J. D. Bekenstein,
Relativistic gravitation theory for the MOND paradigm , Nucl.Phys.
A827 (2009) 555C.[23] J. D. Bekenstein,
Second Canadian Conference on General Relativity and Relativistic Astro-physics , A. Coley, C. Dyer and T. Tupper, eds. (World Scientific, Singapore 1988), p. 68.[24] J. D. Bekenstein and M. Milgrom,
Does the missing mass problem signal the breakdown of ewtonian gravity? , Astrophys. Journ. , 7 (1984).[25] J. D. Bekenstein in Developments in General Relativity, Astrophysics and Quantum Theory ,F. I. Cooperstock, L.P. Horwitz and J. Rosen, eds. (IOP Publishing, Bristol 1990), p. 156.[26] M. Milgrom,
A Modification of the Newtonian dynamics as a possible alternative to the hiddenmass hypothesis , Astrophys.
J 270 (1983), 365.[27] S. L. Adler and T. Piran,
Relaxation methods for gauge field equilibrium equations , Rev. Mod.Phys. , 140 (1984).[28] Y. Zhang and M. Sasaki, Screening of Cosmological Constant in nonlocal Cosmology , Int. J Mod. Phys.
D21 (2012), 1250006.[29] C. Deffayet, G. E. Far` e se and R. P. Woodard, Nonlocal metric formulations of modified New-tonian dynamics with suffiecient lensing , Phys. Rev.
D84 (2011), 124054.[30] M. E. Soussa and R. P. Woodard,
A nonlocal metric formulation of MOND , Class. Quant.Grav. , (2003), 2737., (2003), 2737.