Minimal exponential measure model in the post-Newtonian limit
YYITP-20-152, IPMU20-0122
MEMe model in the Post-Newtonian limit
Justin C. Feng, Shinji Mukohyama,
2, 3 and Sante Carloni CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico – IST,Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal Center for Gravitational Physics, Yukawa Institute for Theoretical Physics,Kyoto University, Kyoto 606-8502, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of Tokyo Institutes for Advanced Study,The University of Tokyo, Kashiwa, Chiba 277-8583, Japan DIME Sez. Metodi e Modelli Matematici, Universit`a di Genova,Via All’Opera Pia 15, 16145 - Genoa, (Italy).
We examine the post-Newtonian limit of the MEMe model presented in [J. C. Feng, S. Carloni,Phys. Rev. D 101, 064002 (2020)] using an extension of the PPN formalism which is also suitable forother type-I Minimally Modified Gravity theories. The new PPN expansion is then used to calculatethe monopole term of the Post-Newtonian gravitational potential and to perform an analysis ofcircular orbits within spherically symmetric matter distributions. The latter shows that the behaviordoes not differ significantly from that of GR for realistic values of the MEMe model parameter q .Instead the former shows that one can use precision measurements of Newton’s constant G toimprove the constraint on q by up to 10 orders of magnitude. I. INTRODUCTION
A recent article [1] introduced a class of General-ized Coupling Theories (GCTs), the simplest of whichwas termed the Minimal Exponential Measure (MEMe)model. These are modified theories of gravity that donot introduce new dynamical degrees of freedom; rather,they modify the interaction between spacetime and mat-ter in a manner that preserves the Einstein equivalenceprinciple (all matter is minimally coupled to an effec-tive spacetime geometry). According to the classificationscheme of [2–4], GCTs and the MEMe model are Type IMinimally Modified Gravity (MMG) theories, since theyonly have two dynamical degrees of freedom and admitan Einstein frame (in the sense that the theories maybe rewritten as General Relativity [GR] with a modifiedsource). While it was shown in [1] that the dynamical be-havior of the MEMe model differs significantly from GRunder the conditions present in the early universe andwithin a matter distribution, the MEMe model reducesto GR in a vacuum—in this respect, the MEMe modelis qualitatively similar to the Eddington-Inspired BornInfeld (EiBI) theory [5]. However, the predictions of theMEMe model differ from those of GR within a matterdistribution and in its coupling to matter. The purposeof the present article is to determine the degree to whichthese differences can be measured in the post-Newtonianlimit.Modified gravity theories, i.e. those that attempt togo beyond GR, have been extensively studied for atleast three motivations: (i) to understand/solve mys-teries in cosmology such as the origins of dark energy,dark matter, and inflation; (ii) to help develop the the-ory of quantum gravity and (iii) to understand GR itself.Regarding (iii), even if GR is the genuine descriptionof gravity in our universe for a certain range of scales,the only way to prove it experimentally/observationally is to constrain possible deviations from GR by experi-ments/observations. In this regard, it is useful to havea universal parameterization of possible deviations fromGR. For solar system scales, the so-called parameterizedpost-Newtonian (PPN) formalism proved to be partic-ularly useful. The standard PPN formalism includes 10parameters to parameterize deviations from GR and cov-ers a wide range of gravitational theories beyond GR [6].However, there is no guarantee that the standard PPNformalism can be applied to all modified gravity theo-ries. For example, in gravitational theories without thefull diffeomorphism invariance, one cannot, in general,adopt the standard PPN gauge and thus may have to in-troduce additional PPN potentials/parameters (see e.g.[7]). The MEMe model we consider here also requires anextension of the standard PPN formalism for a differentreason: the non-trivial matter coupling inevitably gen-erates potentials that are not included in the standardPPN formalism. These potentials are not only relevantin themselves, but they are also necessary to compute thestandard 10 PPN parameters.In this article, we shall construct an extension of thePPN formalism appropriate for a subclass of Type IMMG and GCTs based on additional PPN potentials.We will then focus on the MEMe model, finding thatone must add to the PPN metric a single new potential,which we denote Ψ, and some additional counterterms.All of the counterterms are proportional to the pressure,mass density, and energy density for a fluid, so they van-ish outside matter sources. This is, however, not the casefor Ψ. Hence the PPN parameters for the MEMe modelin the case of a test particle in an external field agree withthose of GR except for the coefficient associated with Ψ.In the context of the MEMe model, we find that theeffects of the potential Ψ can be absorbed into the New-tonian potential outside a matter distribution. This re-sult suggests that the modification to the matter cou- a r X i v : . [ g r- q c ] N ov plings can in the post-Newtonian limit be reinterpretedas a density-dependent modification of the gravitationalconstant G . Comparing with [8], we argue that currentlaboratory methods can improve the constraint on the(single) parameter q in the MEMe model by 10 orders ofmagnitude over the speed of light constraint discussed in[1].We also study circular orbits in the presence of spher-ically symmetric matter distributions and compare thepredictions of the MEMe model with GR. Our findingssuggest that in most astrophysical systems, the presenceof a dilute matter distribution does not significantly af-fect the motion of matter in the MEMe model.This paper is organized as follows. First, we review theMEMe model and GCTs in Sec. II. We then discuss theNewtonian limit in Sec. III and develop the PPN formal-ism for the MEMe model in Sec. IV. Afterward, in Sec. Vwe consider to post-Newtonian order the monopole termfor the MEMe model and discuss how constraints on thevariation of the effective gravitational constant may beused to constrain the parameter q in the MEMe model.Finally, in Sec. VI we compare the behavior of circularorbits within a spherically symmetric matter distributionin the MEMe model to that of GR. Sec. VII is then de-voted to a summary of the paper and some discussions. II. GENERALIZED COUPLING THEORIESAND THE MEME MODEL
Generalized coupling theories are defined by an actionof the form [1]: S GC = (cid:90) d x (cid:20) κ ( R − − λ (1 − F )]) √− g + L m [ φ, g ·· ] √− g (cid:21) , (1)where the metric g µν is assumed to have the form: g µν = Ξ( A ·· ) A µα A νβ g αβ , (2)and the function F = F ( A ·· ) is chosen so that in a vac-uum, A µα = δ µα is an extremum of the action. Uponvarying the action with respect to the metric and re-membering that A µα is independent of g µν , one obtainsfield equations of the form: G µν + [Λ − λ (1 − F )] g µν = κ Ξ | A ·· | ¯ A αµ ¯ A βν T αβ , (3)( δ µα − A µα ) f να = Ξ | A ·· | (cid:20) T αβ g µ ( α ¯ A β ) ν + T
12 Ξ ∂ Ξ ∂A µν (cid:21) , (4)where ¯ A αµ is the inverse of A µα , and f να = f να ( A ·· ).The MEMe model, discussed at length in [1], is a sim-ple example of a generalized coupling theory. The MEMe model is defined by the following action: S [ φ, g ·· , A ·· ] = (cid:90) d x (cid:26) κ (cid:104) R − (cid:105) √− g + (cid:18) L m [ φ, g ·· ] − λκ (cid:19) √− g (cid:27) , (5)where κ := 8 πG and the Jordan-frame metric g µν is de-fined (with A := A σσ ): g µν = e (4 − A ) / A µα A νβ g αβ , (6)and ˜Λ = Λ − λ , with Λ being the observed value of the cos-mological constant. Unless stated otherwise, indices areraised and lowered using the metric g µν and g µν . Defin-ing the parameter q := κλ , (7)the equation of “motion” for A µα takes the following form A βα − δ βα = q [(1 / T A βα − T βν g αν ] , (8)where T µν is the energy-momentum tensor defined by thefunctional derivative of (cid:82) L m [ φ, g ·· ] √− g d x , and T := g µν T µν . Here, we assume q T (cid:54) = 4. Since Eq. (8) isan algebraic equation for A µα , the tensor A µα does notintroduce additional dynamical degrees of freedom. Thetrace of Eq. (8) implies A = A σσ = 4. The gravitationalequations are (setting A = 4): G µν +[Λ − λ (1 − | A ·· | )] g µν = κ | A ·· | ¯ A αµ ¯ A βν T αβ , (9)where ¯ A αµ is the inverse of A µα as already explained, and | A ·· | = det( A ·· ). One may see from the form of Eq. (9)that the MEMe model admits an Einstein frame in thesense of [2], making this a Type I MMG. Here, the oper-ating definition for an Einstein frame is a choice of vari-ables in which a theory is recast as GR with a modifiedsource, which may involve additional degrees of freedom.We define the
Jordan frame as a choice of variables inwhich matter is minimally coupled to the metric tensor.In the MEMe model, it is the frame in which matter iscoupled to the metric tensor g µν . We should stress how-ever that, despite some similarities, these frames are notrelated to the well-known conformal transformations inmodified gravity. The choice of frame is important alsobecause it specifies the worldlines of free-falling test par-ticles: since matter is minimally coupled to the Jordanframe metric, one expects that small clumps of matter tofollow the worldlines of test particles as defined by theJordan frame metric. For this reason, the Jordan frameis the most physically relevant choice. One should keep in mind that since A βα = δ βα in a vacuum,the Einstein and Jordan frame metrics coincide in the absenceof matter. Equation (8) can be solved exactly for a single perfectfluid. The dual (lowered-index) fluid four-velocity u µ isconstructed from the gradients of the potentials, so it isappropriate to regard u µ to be the metric-independentfluid variables. The energy-momentum tensor for thefluid takes the form: T µν = (cid:0) ρ + p (cid:1) u µ u ν + p g µν , (10)the Jordan frame trace of which is T = 3 p − ρ . Note thatwhile g µν u µ u ν = − g µν u µ u ν (cid:54) = −
1. It is useful alsoto define a dual four-velocity vector which is normalizedwith respect to the Einstein-frame metric g µν . Defining ε := g µν u µ u ν , one can obtain such a four-velocity (defin-ing u µ := g µν u ν ): U µ := u µ / √− ε, (11)where u µ = g µν u ν , and it follows that u µ u ν = − εU µ U ν .Since the MEMe model admits two metric tensors g µν and g µν , one should be careful when raising and loweringthe indices of the four-velocity—while the (dual) vector u µ is the lowered index Jordan frame four-velocity, theraised index Jordan frame four-velocity u µ is defined asthe following: u µ := g µν u ν , (12)which is in general not equal to u µ . One may ob-tain a simple relationship between the respective raisedand lowered components of the Jordan frame fluid four-velocity u µ and u ν by first noting that A µα u α ∝ u µ and ¯ A αµ u α ∝ u µ ; it follows that u µ ∝ u µ (where u µ = g µν u ν ). One may then write u µ = a u µ where a is some factor. Now recall that u µ u µ = ε , and since u µ u µ = u µ u µ g µν = −
1, one can show that a = − /ε and obtain the result: u µ = − ε u µ , (13) U µ = √− ε u µ . (14)It follows that u µ u µ = 1 /ε , and U µ U ν = − ε u µ u ν .Given the following ansatz for A µα : A µα = Y δ µα − ε Z U µ U α , (15)one can easily solve Eq. (8), with the result: Y = 4(1 − p q )4 − q (3 p − ρ ) Z = − q ( p + ρ )[4 − q (3 p − ρ )]4 ( q ρ + 1) ε = −
16 ( q ρ + 1) [4 − q (3 p − ρ )] . (16)The inverse ¯ A αµ = [ δ µα + εZ ( Y + εZ ) − U µ U α ] /Y hasa similar form. The gravitational equation (9) takes theform G µν = κ T µν , (17) where T µν is the effective energy-momentum tensor inthe Einstein frame defined by T µν = ( τ + τ ) U µ U ν + τ g µν , (18)and τ = | A ·· | ( p + ρ ) − τ ,τ = | A ·· | ( p q −
1) + 1 q − Λ κ , (19)with the following expression for the determinant: | A ·· | = det( A ·· ) = 256 (1 − p q ) ( q ρ + 1)[4 − q (3 p − ρ )] . (20)So far, the gravitational field equations (3), (9), and(17) are written as dynamical equations for the metrictensor g µν . One can in principle attempt to reexpressthe field equations in terms of the metric g µν . This canbe done by solving Eq. (6) for g µν and inserting theresulting expression into the Einstein tensor to obtainan expression for the gravitational field equations in theJordan frame. In this case the resulting field equation willcontain derivatives up the second order of the tensor A µα .We do not report here the form of such an equation whichis rather long. However, we wish to highlight this featureof the Jordan frame field equations as it is relevant forthe following discussion on the distinction between theMEMe model and other modified gravity theories, andalso the extension of the PPN formalism that we willpresent in the next section.The reader may note that the MEMe model superfi-cially resembles other modified gravity theories that canbe interpreted as a modification of the gravitational cou-pling, such as scalar-tensor theory or disformal theories[9–13]. Indeed, as pointed out in [1], the Jordan metric g µν may be viewed as a type of vector disformal trans-formation [11]. However, the difference here is that theMEMe model, being an MMG, does not introduce addi-tional dynamical degrees of freedom; the tensor A µα is anauxiliary field, the components of which can be expresseddirectly in terms of the fluid quantities ρ , p , u µ and themetric. As discussed in [14], the addition of an auxiliaryfield in a gravitational theory will generically produceterms involving derivatives of the energy-momentum ten-sor in the field equations. While the MEMe model evadesthis problem in the Einstein frame, the derivatives of A µα present in the Jordan frame equations discussed inthe preceding paragraph will by way of Eq. (8) generateterms containing up to second-order derivatives of T µν .The standard PPN formalism is not equipped to handlesuch terms, and in the following sections, we propose anddevelop methods for dealing with this obstacle. III. NEWTONIAN LIMIT OF THE MEMEMODEL
It is helpful to first consider the Newtonian limit of theMEMe model. In doing so, we will assume that q is atmost of order one. Such a choice is motivated by the val-ues that we have found for the modulus of q in [1]. Thisassumption, combined with the smallness assumption on ρ that is made in the Newtonian and Post-Newtoniananalysis, implies that in our calculation we have at most, qρ ∼ O ( (cid:15) ), where (cid:15) = 1 /c .Our primary aim in this section is to identify and studythe Newtonian potential in the MEMe model. We beginby expressing Eq. (17) in the form R µν = 8 πG (cid:18) T µν − g µν T (cid:19) , (21)where T = g αβ T αβ .In an appropriately chosen coordinate system (see alsoCh. 4 of [6, 15] for further discussion), the (0,0) compo-nent becomes R ≈ πG T , (22)where we have used the fact that in the Newtonian limit T ij T (cid:28) . (23)From Eq. (19), and taking into account the fact that inour approximation det( A ) ≈
1, we obtain T ≈ ρ, (24)so that, defining R = ∆Φ E (with ∆ being the Lapla-cian) ∆Φ E = 4 πGρ. (25)However, from an operational point of view, an ac-celerometer would measure the Newtonian limit of theJordan frame metric g µν . Such a potential would be re-lated to Φ E by the relationΦ J = Φ E + C ∆Φ E . (26)In the case of MEMe, the coefficient C is given by C = 3 q πG . (27)In order to preserve the traditional notation we will fromthis point on work in terms of a potential U satisfyingan equation of the same form as Eq. (25). While it isconvenient to work in terms of a potential satisfying Eq.(25), one should keep in mind that the physically relevantpotential is Φ J , which we will relate to U as we developthe extended PPN formalism in the next section.The expression for Φ J in Eq. (26) brings up a poten-tial conceptual difficulty. If ρ has a sharp discontinuity,as one might expect at the boundary of a star, the gra-dient of Φ J can be large—a similar difficulty has beenidentified in the qualitatively similar EiBI theory [16].However, a large gradient in Φ J implies a strong grav-itational force, which would lead to a rearrangement ofmatter. One would expect this gravitational backreac-tion on the matter distribution to drive the system awayfrom large gradients in Φ J (similar arguments [17] havebeen made for the corresponding difficulty in EiBI—seealso [18]). IV. EXTENDED PPN FORMALISM
Naively, one might expect that the PPN formalismapplied to generalized coupling theories in the Einsteinframe yields a set of PPN parameters which are the sameas those of General Relativity. In the MEMe model, forinstance, the theory is identical to GR if the energy-momentum tensor T µν as defined in Eq. (18) has theperfect fluid form. However, as established in [1], theJordan frame metric is the physically relevant metric,since it is the metric which couples directly to matter.Moreover, the microscopic description of matter is spec-ified by the action of matter fields minimally coupled tothe metric in the Jordan frame and thus gives the equa-tion of state of the matter fluid in the Jordan frame. Itis therefore appropriate to introduce the PPN potentialsand parameters in the Jordan frame. On the other hand,it is more convenient to perform most of the computa-tions in the Einstein frame. Notice that the distinctionbetween the two frames concerns only physical systemsin which matter sources play important roles, and there-fore it does not concern the correction to e.g. celestialmechanics on solar system scales.It may be helpful to provide a brief overview of our pro-cedure, which we first develop for a more general class ofmodified gravity theories and generalized coupling theo-ries, and then apply to the MEMe model. We first at-tempt to apply the PPN formalism to the Jordan framemetric, but we find that to avoid higher-order derivativesof the PPN potentials in the field equations, countertermsmust be added to the Jordan frame metric. We then ex-press the Einstein frame metric in terms of Jordan framevariables so that we can use the simpler field equation(17) in the PPN analysis. A. Standard PPN formalism
We follow the conventions of [6] with the post-Newtonian bookkeeping (with the mass density being de-fined as ρ := ρ (1 + Π)): U ∼ v ∼ p/ρ ∼ Π ∼ O ( (cid:15) ) , (28)so that the velocity components v i are of order O ( (cid:15) / ).It should be mentioned that v i , which are raised compo-nents of the three-velocity in the Jordan frame, do notcorrespond directly to the components of u µ , but to theraised index four-velocity u µ in the Jordan frame. Recallthat the distinction between u µ and u µ is necessary be-cause there are two metric tensors in generalized couplingmodels. The components of u µ have the explicit form: u = (cid:0) u , u (cid:126)v (cid:1) . (29)where (cid:126)v is the coordinate 3-velocity of the fluid in theJordan frame with components v i . In terms of Jordanframe fluid quantities, one may use Eqs. (14) and (18) towrite the source of the gravitational field equation (17)as follows: T µν = − ε ( τ + τ ) u µ u ν + τ g µν . (30)Following [6] (and the coordinate conventions therein),we introduce the conserved rest mass density ρ ∗ which isdefined according to the following formula: ρ ∗ := √− g u ρ = | A ·· |√− g u ρ. (31)Given ρ ∗ , one may then define the following PPN poten-tials by the differential relations: ∆ U = − πG ρ ∗ (32)∆ V i = − πG ρ ∗ v i (33)∆ W i = − πG ρ ∗ v i + 2 ∂ i ∂ t U (34)∆Φ = − πG ρ ∗ v (35)∆Φ = − πG ρ ∗ U (36)∆Φ = − πG ρ ∗ Π (37)∆Φ = − πG p, (38)and the following potentials by integral relations:Φ = G (cid:90) ρ ∗(cid:48) [ (cid:126)v · ( (cid:126)x − (cid:126)x (cid:48) )] | (cid:126)x − (cid:126)x (cid:48) | d x (cid:48) (39)Φ W = G (cid:90) (cid:90) ρ ∗(cid:48) ρ ∗(cid:48)(cid:48) (cid:126)x − (cid:126)x (cid:48) | (cid:126)x − (cid:126)x (cid:48) | · (cid:20) (cid:126)x (cid:48) − x (cid:48)(cid:48) | (cid:126)x (cid:48) − x (cid:48)(cid:48) | (cid:21) d x (cid:48) d x (cid:48)(cid:48) (40) − (cid:90) (cid:90) ρ ∗(cid:48) ρ ∗(cid:48)(cid:48) (cid:126)x − (cid:126)x (cid:48) | (cid:126)x − (cid:126)x (cid:48) | · (cid:20) (cid:126)x − x (cid:48)(cid:48) | (cid:126)x (cid:48) − x (cid:48)(cid:48) | (cid:21) d x (cid:48) d x (cid:48)(cid:48) . In the standard PPN formalism, the metric tensor is ex-panded as follows: g = − U − βU + (2 γ + 1 + α + ζ − ξ )Φ + 2(1 − β + ζ + ξ )Φ + 2(1 + ζ )Φ + 2(3 γ + 3 ζ − ξ )Φ − ( ζ − ξ )Φ − ξ Φ W (41) g j = − (4 γ + 3 + α − α + ζ − ξ ) V j − (1 + α − ζ + 2 ξ ) W j , (42) g ij =(1 + 2 γU ) δ ij . (43)The metric is inserted into the field equations, and ex-panded to PPN order O ( (cid:15) ); one then matches terms pro-portional to each of the potentials in Eqs. (32-40) toobtain the PPN coefficients. We point out to the reader that while the PPN formalism in [6]is equivalent to that of [15], the definitions of the PPN potentialshave changed (though the PPN parameters are the same); wherethe PPN potentials in [15] are defined with respect to ρ , thePPN potentials in [6] are defined with respect to ρ ∗ . This changeresults in a change in the coefficients in front of the potentialsΦ and Φ in Eq. (46) for the metric component ˜ g . B. Extended PPN formalism
The procedure outlined in the preceding section doesnot suffice for certain classes of modified gravity theories.For instance, one might imagine in four dimensions arather general theory of the form (use of the Cayley-Hamilton theorem has been employed on the RHS): R µν + e µν = A ( T ·· ) T µν + A ( T ·· ) T µα T αν + A ( T ·· ) T µα T αβ T βν + B ( T ·· ) δ µν , (44)where e µν contains additional geometric or gravitationalterms, T µν is the energy momentum tensor, and A i ( T ·· )and B ( T ·· ) are scalar functions that are polynomials inscalar invariants of T µν up to third order. Examplesof such a theory include the EiBI [5] or the braneworldmodel of [19]. We also note that Eq. (44) is also a subcaseof the gravitational field equation given in [14].We consider a class of Type-I MMG theories in whichthe source terms in the Einstein frame can be writtenexclusively in terms of the energy-momentum tensor sothat e µν = 0. Expanding the RHS of Eq. (44) to post-Newtonian order, one has a term proportional to ρ , how-ever the PPN expression for the Ricci tensor does notcontain any term that can absorb such a term. One mayremedy this by adding a term to g (41) of the form ν Ψ, where Ψ is a O ( (cid:15) ) potential defined by the follow-ing: ∆Ψ := − πG ρ ∗ ρ = − πG ρ ∗ + O ( (cid:15) ) . (45)We note here that unlike the standard PPN potentials,this additional potential Ψ is dimensionful—since themetric components must be dimensionless, it follows thatthe associated parameter ν must also be dimensionful.We attribute this to the fact that the coefficient for the ρ term which appears in the PPN expansion of (44) in-troduces an additional scale into the theory. Later, weshall see this explicitly when applying this extended PPNformalism to the MEMe model.We now turn to the case of generalized coupling the-ories as described by Eqs. (1–2). For an appropriatechoice of reference frame, the extended PPN metric forthe Jordan-frame metric would take the form (note theaddition of the term 2 ν Ψ in ˜ g ):˜ g = − U − βU + (2 γ + 1 + α + ζ − ξ )Φ + 2(1 − β + ζ + ξ )Φ + 2(1 + ζ )Φ + 2(3 γ + 3 ζ − ξ )Φ − ( ζ − ξ )Φ − ξ Φ W + 2 ν Ψ , (46)˜ g j = − (4 γ + 3 + α − α + ζ − ξ ) V j − (1 + α − ζ + ξ ) W j , (47)˜ g ij =(1 + 2 γU ) δ ij . (48) Here, we follow the conventions of [6]. If one wishes to use thoseof [15], one should instead add a term of the form ν Ψ ◦ where Ψ ◦ is defined similarly to Ψ but with ρ instead of ρ ∗ . However, one still encounters a difficulty when attempt-ing to apply the standard PPN analysis to Eq. (9). Asdiscussed earlier, the gravitational field equations in theJordan frame will contain up to second-order derivativesof T µν . It follows that the direct application of the PPNform to the Jordan frame metric will introduce termsinvolving second derivatives of the fluid potentials andfour-velocity, but the standard PPN formalism and theextended formalism encapsulated in Eqs. (46)-(48) areincapable of absorbing these terms. To see this, considerthe following expression for the Einstein frame metric g µν : g µν = Ξ − ¯ A αµ ¯ A βν g αβ , (49)From Eq. (4), the tensor ¯ A αµ and the factor Ξ = Ξ( A ·· )depend on ρ ∗ , Π and p , and we assume g αβ takes theusual PPN form given in Eqs. (46), (47), and (48). Uponexpanding the Ricci tensor for g µν as given by (49) intoEq. (9), one will obtain terms containing derivatives of ρ ∗ , Π and p , which cannot be absorbed by remainingterms in Eq. (44) if e µν = 0. To eliminate these additional terms, we can add coun-terterms to the metric components ˜ g µν given in Eqs.(46), (47), and (48), and then choose coefficients suchthat Eq. (49) does not contain the quantities ρ ∗ , Π and p . In general, the counterterms take the following form: g =˜ g + c ∆ U + c ∆Φ + c ∆Φ + c ∆Φ + c ∆Φ + c Ψ ∆Ψ + c w ∆Φ W , (50) g j =˜ g j + d V ∆ V j + d W ∆ W j , (51) g ij =˜ g ij + e ∆ U δ ij (52)where we restrict to terms of order g ∼ O ( (cid:15) ), g j ∼O ( (cid:15) / ), and g ij ∼ O ( (cid:15) ). At this stage, one may col-lect terms of order (cid:15) in g which yields the Newtonianpotential in the Jordan frame:Φ J = U + ( c / U, (53)consistently with what was obtained in (26). We thenchoose the coefficients c − , Ψ ,w , d V,W and e so that the One might suppose that e µν contains terms with derivatives of ρ ∗ , Π and p , which can cancel the additional terms introducedby Ξ and ¯ A αµ . Derivatives of ρ ∗ , Π and p correspond to higherorder ( >
2) derivatives of the potentials, which correspond tohigher order derivatives of the metric—one then has a higherorder theory of gravity, which (excluding frame-dependent theo-ries like Hoˇrava-Lifshitz gravity [20] and a certain class of type-IIMMG theories [21–24]) generically suffers from Ostrogradskianinstability [25, 26].
Einstein frame metric takes the desired form: g = − U − βU + (2 γ + 1 + α + ζ − ξ )Φ + 2(1 − β + ζ + ξ )Φ + 2(1 + ζ )Φ + 2(3 γ + 3 ζ − ξ )Φ − ( ζ − ξ )Φ − ξ Φ W + 2 ν Ψ , (54) g j = − (4 γ + 3 + α − α + ζ − ξ ) V j − (1 + α − ζ + 2 ξ ) W j , (55) g ij =(1 + 2 γU ) δ ij . (56)where again we restrict to terms of order g ∼ O ( (cid:15) ), g j ∼ O ( (cid:15) / ), and g ij ∼ O ( (cid:15) ). The reader should keepin mind here that all of the potentials in this expres-sion are those appearing in Eqs. (46), (47), (48), whichare defined with respect to Jordan frame fluid quantities.Therefore this expression is not a PPN expansion of theEinstein frame metric—rather, one should think of Eqs.(54), (55), (56) as the Einstein frame metric expressed interms of (PPN expanded) Jordan frame quantities.It is worth mentioning at this point that to post-Newtonian order, the metric g µν retains the form ex-pected for the PPN gauge in the sense that the spatialcomponents g ij do not acquire cross terms. It shouldalso be mentioned that we are in fact working in a PPNgauge since ˜ g ij is diagonal and depends strictly on thepotentials (32–40)—from Ch. 4 of [6], we expect thata non-PPN gauge will introduce an additional potential.To clarify, one first chooses the gauge in which ˜ g µν hasthe form given in Eqs. (46–48); after the gauge is cho-sen, the set of counterterms in Eqs. (50–52) for g µν issufficient to characterize the PPN expansion.The proposed modification to the PPN parameteriza-tion has been motivated by necessity; without these mod-ifications, one cannot apply the PPN formalism to a classof Type-I MMGs and GCTs whose equations of motioncan be written in the form of Eq. (44) (with e µν = 0),including the MEMe model. Though we have providedhere a preliminary discussion regarding the theoreticalinterpretation for the new potential Ψ, it is perhaps ap-propriate to also understand the physical interpretationof Ψ and the counterterms in a phenomenological con-text. We will attempt to address this point in later sec-tions by studying the net effect of these quantities onsome post-Newtonian systems in the MEMe model. C. MEMe model coefficients
We now apply the extended PPN formalism describedabove to the MEMe model. First, we note that in Eq.(49), Ξ = 1 for the MEMe model (compare Eqs. (2) and(6) and recall that A = 4 on shell). We then demand thatthe Einstein frame metric g µν has the form given in Eqs.(54), (55), (56), and upon comparison with Eq. (49) forthe MEMe model, one obtains the following values forthe coefficients of the counterterms: c = 3 q πG , c = 5 q πG , c = − γ + 2) q πG ,c = 3 q πG , c = 3 q πG , c Ψ = 21 q πG , (57) e = q πG , c w = 0 , d V = − q πG , d W = 0 . (58)The expression for the Einstein frame metric g µν inEqs. (54-56) is then substituted into Eq.(17), and wefind that all of the standard PPN parameters are exactlythe same as that of general relativity ( γ = β = 1, allothers zero). However, the new parameter ν , which hasthe value ν = 0 in general relativity, has the followingvalue in the MEMe model: ν = 3 q G . (59)As anticipated by our remarks in the preceding section,the parameter ν corresponds to the scale q = 1 /λ thatappears in the MEMe model. V. MONOPOLE TERM FOR PPN POTENTIALSA. General analysis
We will now investigate the physical effects of the mod-ification of the PPN monopole term associated with Ψ.We begin by assuming that the matter distribution iscompact and static (so that v i = 0), and consider whathappens outside the matter distribution. One may thendefine an effective gravitational potential in the followingmanner: Φ := 12 (cid:0) g + 2 β U (cid:1) . (60)Outside a matter distribution, the counterterms vanish—recall that outside of a matter distribution, the Einsteinand Jordan frame metrics coincide. For a theory withno preferred location effects ( ξ = 0), the effective gravi-tational potential takes the form (we set v i = 0 so thatΦ = Φ = 0):Φ = U + 2 β Φ + β Φ + 3 β Φ + ν Ψ . (61)where (following the reasoning in Ch. 40 of [27]): β := 12 (1 − β + ζ ) β := 1 + ζ β := γ + ζ . (62)Note that up to an overall factor of 2, Φ consists of allterms in g such that ∆Φ can be written as an algebraicfunction of ρ , Π, p , and U up to fourth order in (cid:15) . We consider the case where the gravitational theory is fullyconservative, with the parameter choices α = α = α = ζ = ζ = ζ = ζ = 0 (in addition to ξ = 0), one has β = (1 − β ) / β = 1, and β = γ .We now consider the multipole expansion for the New-tonian potential:Φ( x ) = (cid:90) G ρ e ( x (cid:48) ) | x − x (cid:48) | d x (cid:48) . (63)where ρ e is an effective energy density given by: ρ e = ρ ∗ (cid:20) β U + β Π + 3 β p/ρ ∗ + ν G ρ (cid:21) . (64)The monopole moment is given by:Φ( x ) = G Mr + O ( r − ) , (65)where: M := (cid:90) ρ e ( x (cid:48) ) d x (cid:48) . (66)The definition given in Eqs. (64) and (66) is motivatedby Eq. (40.4) in [27]; it is in fact identical in the limit ν → W i = V i =0, A = Φ = 0. Making use of the fact that ρ e = ρ ∗ + O ( (cid:15) ), and keeping only the monopole terms, the metricto post-Newtonian order is (cf. Eq. (40.3) of [27]): g = − G Mr − β G M r (67) g j = 0 , (68) g ij = (cid:20) γ G Mr (cid:21) δ ij . (69)It follows that for a spherically symmetric matter dis-tribution, the additional PPN potential can be absorbedinto the mass, as one might have expected. This suggeststhat outside of a spherically symmetric matter distribu-tion, the effects of the additional potential Ψ cannot bedisentangled from the other potentials.To distinguish the effects of the potential Ψ and pa-rameter ν , one should consider the internal structure ofthe source. In particular, if one has a detailed model forthe source itself, it may be possible to disentangle theeffects of the parameter ν from the total mass of a spher-ical source. To see how one might distinguish the effectsof an additional potential Ψ, we consider a given matterdistribution, and split the mass M into two parts, onewhich depends on the original PPN parameters, and onewhich depends on the new parameter ν . Defining thepotential, ¯Φ := Φ − ν Ψ (70)and defining ¯ ρ e := ρ e − ν G ρ ∗ ρ , one has the result¯Φ( x ) = G ¯ Mr + O ( r − ) , (71)where the mass defined with respect to the original PPNpotentials takes the form:¯ M := (cid:90) ¯ ρ e ( x (cid:48) ) d x (cid:48) . (72)Now we consider the standard multipole expansion forthe new PPN potential:Ψ( x ) = (cid:90) G ρ ∗ ( x (cid:48) ) ρ ( x (cid:48) ) | x − x (cid:48) | d x (cid:48) . (73)Now ρ ∗ ρ = ¯ ρ e + O ( (cid:15) ). The monopole moment is givenby: Ψ( x ) = G µ r + O ( r − ) , (74)where: µ := (cid:90) ρ ∗ ( x (cid:48) ) ρ ( x (cid:48) ) d x (cid:48) = (cid:90) ¯ ρ e ( x (cid:48) ) d x (cid:48) + O ( (cid:15) ) . (75)The relationship between ¯ M and µ is sensitive to theinternal structure of the source. For instance, if one con-siders the following Gaussian profile for ¯ ρ e :¯ ρ e ( x ) = ¯ M (cid:0) √ π σ (cid:1) exp (cid:20) − r σ (cid:21) , (76)then one has for µ : µ = ¯ M π / σ . (77)Note that µ depends on the size σ for the source. Moti-vated by the Gaussian expression, one can use Eq. (77)as a parameterization for the internal structure of thesource, with σ being a parameter which represents a char-acteristic length scale for the source. It follows that: M = ¯ M + ν G ¯ M ¯ ρ C √ π , (78)where ¯ ρ C := 3 ¯ M / πσ is the compactness of the source.Given some matter distribution, the mass ¯ M is the post-Newtonian mass in the GR limit ν → B. MEMe model analysis
It should be mentioned that this dependence on thecompactness is only apparent when a detailed descriptionof matter is taken into account. Since MEMe coincideswith GR outside matter sources, the inertial mass outsidethe source is equivalent to the gravitating mass M . It fol-lows that one can only compute the difference betweenthe GR value ¯ M and the MEMe value M when comput-ing the gravitating mass directly from the density. To understand this difference, consider lowering a particlewith a small mass m into a matter distribution satisfy-ing the distribution in (76). We consider this process inthe Einstein frame since the gravitating mass M in theMEMe model is defined in this frame. The gravitationalbinding energy between the particle and the matter dis-tribution is given by m (cid:2) Φ − βU (cid:3) = m (cid:2) ¯Φ − βU + ν Ψ (cid:3) ,where Ψ( x ) = G ¯ M ¯ ρ C √ π erf (cid:0) rσ (cid:1) r . (79)As discussed in [1], a stability argument suggests that q <
0, which in turn suggests ν <
0. Since Ψ( x ) > ν by measuring the mass of an object, disas-sembling it into its constituent parts, and measuring themass of the individual components. C. Constraints on the MEMe model
One can in principle place a constraint on the param-eter ν without requiring the equivalence of inertial andgravitating masses. To see this, first note that one caninterpret Eq. (78) as resulting from a dependence in theeffective gravitational constant on the compactness ¯ ρ C of the source. For a source mass ¯ M and the Gaussianprofile one has the following expression for the effectivegravitational constant: G eff = G (cid:20) ν G ¯ ρ C √ π (cid:21) . (80)Recent experiments [8] with spherical stainless steel (SS316) source masses, which have a density of ∼ . × Kg / m , constrain Newton’s constant to a fractionaluncertainty of about 3 × − . While the experiment in[8] alone cannot place a constraint on ν , one might imag-ine a variation of the experiment in which the spheri-cal source masses can be disassembled into thick spheri-cal shells. If the same experiment is performed for eachshell individually, and then again for the fully reassem-bled source mass, one can search for differences in theeffective gravitational constant—such differences are evi-dent of a weakening or strengthening of the gravitationalbinding energy when masses are brought together. As-suming that fractional uncertainties similar to those of [8]can be achieved, one can in principle constrain ν up to avalue on the order of ν ∼ − m / Kg, or 10 − m / J, inunits of inverse energy. This in turn can place a strongconstraint on q : | q | (cid:47) − m / J . (81)which is ten orders of magnitude stronger than the speedof light constraint ( | q | < × − m / J) in [1], thoughstill 12 orders of magnitude weaker than scales corre-sponding to the inverse of the highest energy densities( ∼ / fm ≈ . × J / m ) probed in acceleratorexperiments to date [28, 29], and 26 orders of magnitudeweaker than that from a TeV scale breakdown. VI. LAPLACIAN COUNTERTERMS ANDORBITSA. Circular orbits for conservative theories
We focus now on the effect of the Laplacian countert-erms in the modified PPN metric (50)–(52) on circulargeodesics in the post-Newtonian limit. For simplicity,we assume that matter sources are spherically symmet-ric and stationary, so that V i = 0, W i = 0, Φ = 0, andΦ = 0. We also consider a conservative theory, whichcorresponds to the choice α = α = α = ζ = ζ = ζ = ζ = 0 in the original PPN analysis of [6]. The lineelement then has the form ( d Ω being the line elementon the unit two -sphere): ds = f dt + h (cid:0) dr + r d Ω (cid:1) , (82)To simplify the analysis, we neglect internal energy den-sity and internal pressure. The functions f and h takethe following forms: f = − U − U + c ∆ U − + c ∆Φ + 2 ν Ψ + c Ψ ∆Ψ , (83) h =1 + 2 U − e ∆ U. (84)For a spherically symmetric matter distribution, one canobtain solutions for the potentials by directly integratinga Poisson equation of the form ∆ ψ = − π G ρ s , which inspherical symmetry may be written explicitly:1 r ∂∂r (cid:18) r ∂ψ ( r ) ∂r (cid:19) = − π G ρ s ( r ) (85)where ρ s is a source function. This can be integrated toobtain the solution: ψ ( r ) = C + (cid:90) rr y (cid:20) C − πG (cid:90) yy ρ s ( y (cid:48) ) y (cid:48) dy (cid:48) (cid:21) dy. (86)Given a Jordan-frame geodesic x µ ( τ ) parameterized byproper time τ , one has the following conserved quantities: e = g µ dx µ dτ = f dtdτ ,l = g µ dx µ dτ = r h dφdτ . (87)From the unit norm condition for the four-velocity, onecan show that the specific energy e must have the form: e = − f h (cid:18) drdτ (cid:19) − fr h l − f. (88) The effective potential may be obtained by consideringthe turning point ( dr/dτ = 0) expression for e : V eff = − f (cid:20) l r h + 1 (cid:21) . (89)We now consider circular orbits and assume sphericalsymmetry ( f = f ( r ), h = h ( r )); circular orbits lie at theminima of the effective potential, and are given by thecondition V (cid:48) eff ( r ) = 0. One can solve V (cid:48) eff ( r ) = 0 for thespecific angular momentum l to obtain: l = r h (cid:115) r f (cid:48) f ( r h (cid:48) + 2 h ) − r h f (cid:48) , (90)and a comparison with Eq. (87) yields the proper tan-gential velocity: r dφdτ = lr h ( r ) . (91)From the line element Eq. (82), one has dt/dτ = (cid:112) − f ( r ) − h ( r ) v , which yields the tangential coordinatevelocity: v ( r ) ≡ r dφdt = (cid:115) − rf (cid:48) ( r ) rh (cid:48) ( r ) + 2 h ( r ) . (92) B. Circular orbits in the MEMe model
The MEMe model is a conservative theory in the senseof [6], as the standard PPN parameters are the same asthat of GR. The extra parameters in the extended PPNformalism have the values given in Eqs. (57)–(59), whichdiffer from that of GR, so one expects circular orbits inthe MEMe model to differ from those of GR, given someprofile for the matter distribution. We first consider aGaussian profile: ρ ∗ = ρ e − r / σ , (93)with ρ being the central density, and σ a characteristicscale. Equation (86) may be used to obtain the poten-tials: U = 2 √ π / G ρ σ r erf (cid:20) r √ σ (cid:21) , Φ = − π / G ρ σ r (cid:26) erf (cid:20) r √ σ (cid:21) (cid:18) √ π r erf (cid:20) r √ σ (cid:21) + 2 √ σ e − r σ (cid:19) − σ erf (cid:104) rσ (cid:105)(cid:27) , Ψ = π / G ρ σ r erf (cid:104) rσ (cid:105) , (94)which may be used to compute the tangential velocity v ( r ) as given by Eq. (92). It turns out that a large0modulus for q is required to obtain rotation curves thatdiffer from q = 0 in a discernible way. For the Gaussianmodel, the tangential velocity of a circular orbit as afunction of radius (rotation curve) is plotted in Fig. 1,for the parameter choices ρ = 10 − , and σ = 1 (with G = c = 1); with one curve corresponding to q = 0 andanother corresponding to q = 10. The rotation curve for q = 10 is virtually identical to that of q = 0 at large radii(as illustrated in the plot for the difference ∆ v := v GR − v MEMe ), and has an increased value for relatively smallvalues of r . One might expect this behavior; for instance,one may note that c ∆ U ∝ − qρ > q <
0) andupon comparison, one finds that the slope for c ∆ U ( r ) ∝ ρ ∗ ( r ) (as given by Eq. (93)) matches the slope for thepotential U ( r ); it follows that the counterterms enhancethe force in the radial direction, which in turn increases v ( r ). The convergence to the GR rotation curve at large r is expected, as one expects the MEMe model to convergeto GR at low density. These general features persist inthe other examples we consider. (a) r v (b) r Δν FIG. 1. Plot (a) illustrates tangential velocity v of circularorbits for the Gaussian matter distribution (93). Two casesare compared: q = 0 [in blue] and q = −
10 [in orange], andour parameter choices are ρ = 10 − , and σ = 1 (with G = c = 1). It should be mentioned that for | q | (cid:54) = 0, v ( r ) generallybecomes imaginary for values of r > v = v GR − v MEMe . Another relevant matter profile is the isothermal one: ρ ∗ = M h π a h r . (95)Such a profile is known to yield flat rotation curves inNewtonian gravity, and is of interest (upon regulariza-tion of the singularity at r = 0) for modeling dark mat-ter halos. The curve v ( r ) is plotted in Fig. 2 for theparameter choices M h = 10 − , and a h = 10 . Again, onesees behavior similar to that of the Gaussian case—the q = −
10 curve only differs (and has a lower value) fromthe q = 0 case at small values for r , as expected. Thedivergence in the rotation curves at small r is expected,since ρ ∗ diverges in the limit r →
0. In Fig. 3, we plot v ( r ) for the combined Gaussian and isothermal matterdistributions ρ ∗ = ρ e − r / σ + M h π a h r , (96)with the same parameter values as those of Figs. 1 and3. Again, we note the velocities are increased at small r . r v FIG. 2. This plot illustrates tangential velocity v of circularorbits for the isothermal matter distribution (95). Two casesare compared: q = 0 [in blue] and q = −
10 [in orange], andthe parameter choices here are M h = 10 − , and a h = 10 (with G = c = 1). Again, as in Fig. 1, for q (cid:54) = 0, v ( r )becomes imaginary for values of r > In all cases, we find that while the Laplacian countert-erms have some effect on the behavior of rotation curves,the value of q must be rather large in order to distinguishthe MEMe model and GR, and even then, this occursonly at small values of r , as illustrated in the plots for∆ v . If one expects the MEMe model to break down atthe TeV scale, then 1 / | q | is expected to be 30 orders ofmagnitude larger than the average density of the Earth;for realistic astrophysical systems (galaxies), one mightexpect 1 / | q | and the matter density to differ by an evengreater amount. For the Gaussian example, the centraldensity ρ in Fig. 1 is six orders of magnitude below thedensity scale 1 / | q | = 10 − at which the MEMe modelbreaks down. For the isothermal example, the average1 r v FIG. 3. Rotation curves for the combined Gaussian (93) andisothermal matter distributions (95), using the same param-eter choices as in Figs. 1 and 3. density 3 M h / πa h is eleven orders of magnitude belowthe density scale 1 / | q | = 10 − .While these results suggest that signatures of theMEMe model are unlikely to appear in galactic rota-tion curves and dilute matter distributions, the MEMemodel may still produce measurable differences in theinteriors of neutron stars. The density for a neutronstar is roughly an order of magnitude less than the high-est energy-density ( ∼ / fm ≈ . × J / m )states of matter probed to date in accelerator experi-ments [28, 29]. If the scale for the cutoff density is as-sumed to be an order of magnitude higher than thatof the quark-gluon density, so that it is two orders ofmagnitude higher than the neutron star density, thenupon modeling a neutron star with a Gaussian matterdistribution, the term c ∆ U can become comparable to − deep within the distribution. In particular, onecan choose ρ = M/ √ π / σ , with the normalization M = G = 1 and σ = 6. In this case, the magnitudeof the counterterm c ∆ U is roughly ∼ .
75 of the post-Newtonian correction − when r = σ/
10, though atthe same radius, one finds − c ∆ U/ U ∼ . × − and − c ∆ U/ U ∼ . × − , so the corrections are stillrather small. However, this rough calculation suggeststhat the corrections from the MEMe model may modifythe properties of the Neutron star in a measurable way. VII. SUMMARY AND DISCUSSION
In this article, we have extended the PPN formal-ism to handle a subclass of Type I MMGs and GCTs,and have applied the extended formalism to the MEMemodel. Outside matter sources the Einstein frame andthe Jordan frame coincide with each other and the fieldequations in either frame agree with those in GR. How-ever, in the non-vacuum case, a PPN analysis for GCTsand the MEMe model should be performed with respectto the Jordan frame metric g µν . In fact, matter is mini- mally coupled to the Jordan frame metric g µν , and it isin this sense that the Jordan frame metric is the physi-cal metric. In order to perform a PPN analysis for g µν ,it is necessary to introduce an additional (dimensionful)potential Ψ and counterterms (the latter vanish outsidea matter distribution) constructed from the Laplaciansof the PPN potentials. This can be understood consider-ing the form of the field equations in the Jordan frame,which contains the energy density and its derivative upto the second order. We have found that with the excep-tion of the counterterm parameters and the parameter ν associated with Ψ, the parameters in the extended PPNformalism are the same as those of GR.The new potential Ψ and its associated parameter ν arenot dimensionless. One might ask whether it is possibleto define a dimensionless potential from Ψ. This can bedone by choosing an appropriate length scale, howeversuch a procedure is not necessarily model-independent.For example, to post-Newtonian order, a theory havingthe form of Eq. (44) would necessarily include a ρ ∗ termon the RHS, the coefficient of which would introduce anadditional scale. Indeed, each of the additional coeffi-cients appearing on the RHS will introduce additionalscales, and any of these can provide a reference scale tomake Ψ dimensionless. To avoid the choice of one scalerather than the other, here we have chosen to leave Ψand ν dimensionful.Given some compact, spherical matter distribution, wehave considered the monopole term in a standard multi-pole expansion and have found that to post-Newtonianorder, the MEMe model is indistinguishable from GR invacuum regions outside the matter distribution. Thisis not particularly surprising, as the Einstein and Jor-dan frame metrics coincide in vacuum, and one can for asingle fluid in the Einstein frame absorb the differencesfrom GR by a redefinition of fluid density and pressure.However, the differences between MEMe and GR becomeapparent when the details of the matter distribution aretaken into account. The monopole expansion indicatesthat in MEMe, the effective gravitational constant G de-pends on the internal structure of the source masses, andwe argue that one can use this dependence to place strongconstraints on the free parameter q of MEMe. In partic-ular, we argue that (conceptual issues aside, see the nextparagraph) a modification of the experiment describedin [8] may improve the constraint on q over the speed oflight constraint of [1] by 10 orders of magnitude. In par-ticular, we propose an experiment in which the sphericalsource masses are disassembled into concentric “thick”shells, and the active gravitational masses of the individ-ual shells and the assembled spheres are compared.This proposal might bring up a conceptual issue re-garding the gravitational binding energy between concen-tric thick shells of matter. In GR, this situation can betreated using the standard junction and thin-shell formal-ism of Israel [30]. Since the geometry outside the shells isessentially that of GR, one might ask whether the bind-ing energy is modified at all. This question depends on2the behavior of the theory at the boundaries of spatiallycompact matter distributions, which can be rather sub-tle in certain theories of modified gravity. In the case ofEiBI gravity [5], which shares a structure similar to thatof the MEMe model in the weak-field limit (it falls intothe class of models described by Eq. (44), and has a New-tonian potential resembling Eq. (26)), it was argued in[16] that discontinuities in matter distributions, such asthose at the boundaries of stars, can generate unaccept-able curvature singularities in EiBI gravity. However, wehave argued that in the Newtonian limit of the MEMemodel, such singularities correspond to strong gravita-tional forces acting on matter which lead to a rearrange-ment of matter distributions, so that the gravitationalbackreaction may resolve such singularities—similar ar-guments have been made for EiBI theory [17] (see also[18]). A detailed investigation of this issue beyond theNewtonian limit in the MEMe model will be left for fu-ture work.Finally, we compared the post-Newtonian predictionsof the MEMe model and GR within a matter distribu-tion to understand the effects of the counterterms thatappear in the gravitational potential. In particular, westudied the behavior of circular geodesics in the pres-ence of spherically symmetric Gaussian and isothermalmatter distributions. Plots of the tangential velocity ro-tation curves indicate that the predictions of the MEMemodel only differ significantly from that of GR only forhigh matter densities and large values for the parame-ter q . It follows from this result that the MEMe modelalone cannot describe galactic rotation curves in the ab-sence of dark matter—in fact, the MEMe model (slightly) increases orbital velocities at small radii—and the differ-ences in the behavior of geodesics between the MEMemodel and GR are minimal even within a distributionof dark matter. These results also indicate that in gen-eral, the counterterms do not have a strong effect on thegeodesics unless the parameter q is increased to an un-realistically large value. On the other hand, a rough es- timate suggests that, for a cutoff density 1 /q an orderof magnitude higher than the highest densities probedin accelerator experiments, the MEMe model may yieldmeasurable corrections to the properties of neutron stars.In the present paper, we have considered the MEMemodel as a type-I MMG theory and have focused onits gravitational aspects. Alternatively, in the Einsteinframe, one can consider the MEMe model as a theory ofa modified matter action minimally coupled to GR. In-deed, after integrating out the auxiliary tensor field A µα the matter action in the Einstein frame is modified insuch a way that the fields in the standard model of par-ticle physics acquire additional (renormalizable and non-renormalizable) interactions among themselves. In futurework, it is certainly interesting to study phenomenologi-cal consequences of those extra interactions (that remaineven in the G → ACKNOWLEDGMENTS
We would like to thank Vitor Cardoso and Chin-moy Bhattacharjee for helpful comments. Some ofthe calculations were performed using the xAct pack-age [31] for
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