New class of generalized coupling theories

Justin C. Feng1 and Sante Carloni1

1CENTRA, Departamento de Fısica, Instituto Superior Tecnico IST,Universidade de Lisboa UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal

We propose a new class of gravity theories which are characterized by a nontrivial coupling betweenthe gravitational metric and matter mediated by an auxiliary rank-2 tensor. The actions generatingthe field equations are constructed so that these theories are equivalent to general relativity in avacuum, and only differ from general relativity theory within a matter distribution. We analyze indetail one of the simplest realizations of these generalized coupling theories. We show that in thiscase the propagation speed of gravitational radiation in matter is different from its value in vacuumand that this can be used to weakly constrain the (single) additional parameter of the theory. Ananalysis of the evolution of homogeneous and isotropic spacetimes in the same framework showsthat there exist cosmic histories with both an inflationary phase and a dark era characterized by adifferent expansion rate.

I. INTRODUCTION

In recent years, we have witnessed considerable ad-vances in the accuracy and methodology of experimentaland observational investigation of gravitational phenom-ena. The wealth of new data increasingly exacerbatesa puzzle that has been present for several decades, withregard to our current understanding of the gravitationalinteraction. On one hand, the detection of gravitationalwaves (e.g., Refs. [1, 2]) and the observation of the blackhole at the center of M87 [3] have brought extraordi-nary confirmation of the predictions of general relativity(GR) in the strong field regime. On the other hand, ithas become increasingly clear that GR alone is unableto correctly describe the dynamics of objects at galac-tic and extra-galactic scales [4], the current acceleratedexpansion of the Universe [5], and the tension in the es-timation of the present value of the Hubble parameter[6].

As pointed out in Ref. [7], one way to interpolate be-tween these contrasting results is to reevaluate the inter-action between spacetime and matter, rather than assum-ing that gravity behaves differently at different scales.The motivation for such a point of view lies in the re-alization that deviations from GR only appear in space-times in which the role of matter cannot be neglected,like cosmology and the gravitational behavior of galaxiesand clusters of galaxies.

Indeed, the weakest assumption in the construction ofthe celebrated Einstein equations is the way in whichmatter and spacetime are coupled to each other. A keyprinciple that guided Einstein was the local conserva-tion of the energy-momentum tensor (the divergence-freeproperty), which in the modern framework is encoded inthe existence of a natural variational principle able togenerate the field equations [8]. However, there is nocompelling reason not to consider more complicated con-nections between the spacetime geometry (Einstein ten-sor) and the energy-momentum tensor for matter.

If one is willing to consider the possibility that thecoupling between these objects is more complex than a

simple proportionality, one could consider the followingequation [7],

Gµν = χµναβ Tαβ , (1)

where the coupling tensor χµναβ is a generic, nonsingu-

lar, fourth-order tensor which mediates (and generalizes)the response of spacetime to a given matter distribution.

The structure of (1) can be engineered in such a waythat its phenomenology in vacuum is exactly that of GR.Such a generalization avoids the difficulties that normallyafflict modifications of GR. In particular, many modifiedgravity theories have a nontrivial vacuum phenomenol-ogy which is strongly constrained by the measurement ofpost-Newtonian effects and, more recently, gravitationalwave detections and black hole phenomenology. Equa-tion (1) is compatible with all these constraints. Phe-nomenological differences only appear within a matterdistribution, like in the (very different) case of torsion inthe Einstein-Cartan-Sciama-Kibble theory [9].

Equation (1), although interesting, is still rather am-biguous as a theory. In particular, i) to avoid deviationsfrom GR in vacuum, one must provide a mechanism thatdrives the coupling tensor χµν

αβ to the product of twoKronecker deltas δαµδ

βν (up to a factor of 8πG) in vac-

uum, and ii) one should be able to construct a variationalprinciple that generates the gravitational field equations.A first objective of the present work is to construct such atheory. We find that there is, in fact, a common solutionof both of these problems at the cost of a modification ofEq. (1).

In this article, we provide a fundamental motivation forEq. (1) in the framework of semiclassical gravity. Thismotivation is useful because it provides a natural inter-pretation for a key parameter as a vacuum energy in ourfinal theory. We then construct a general class of actionsthat can generate an equation similar to (1) in which thecoupling tensor χµν

αβ is a function of a rank-2 tensorAµ

α. These actions contain no derivatives of Aµα, i.e.,

this field is nondynamical (or auxiliary) and their varia-tion leads to (algebraic) equations which constrain Aµ

α

to be a Kronecker delta in a vacuum.

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We should remark that the strategy of employing aux-iliary fields in modified gravity theories is not new. In theliterature, other theories characterized by a similar set-ting have been explored. In Ref. [10], for example, it wasshown, under some rather general assumptions, that theintroduction of auxiliary fields in GR will generally intro-duce higher derivatives of the energy-momentum tensorin the field equations. One of the assumptions in theirapproach is that the matter fields couple to the metricin the usual way, so that the matter Lagrangian reducesto the usual one in a local frame. In this article, wedemonstrate that one can avoid higher derivatives of theenergy-momentum tensor by relaxing this condition. Indoing so, we obtain an example of a theory which gener-alizes the coupling between matter and gravity withoutintroducing dynamical degrees of freedom or introducinghigher derivatives of the matter fields.

We will study in detail an explicit example of such the-ories [the Minimal Exponential Measure (MEMe) model],and examine its basic features and phenomenology. Re-markably, we will find that for a single perfect fluid, thenondynamical auxiliary fields in the MEMe model inducea vector disformal transformation [11, 12] of the metricwithin a matter distribution. While disformal general-ized matter couplings have been explored in the recentliterature [13, 14], we are not aware of any disformal the-ory [11–16] which avoids introducing degrees of freedomthrough the use of auxiliary fields. A consequence ofthis is that gravitational waves propagate at the speedof light in a vacuum (consistent with the vacuum phe-nomenology of GR), but propagate with a different speedwithin a matter distribution. Additionally, we will showthat MEMe cosmologies possess an unstable (de Sitter)inflationary era and also a (de Sitter) dark energy erain which the expansion rate is different. In fact, when acosmological constant is introduced, the presence of thecoupling tensor is able to alleviate, albeit not completelysolve, the coincidence problem.

The paper is organized in the following way. SectionII concerns a semiclassical gravity interpretation for (1),which serves as a motivation for our work. Section III ex-plores the general features of the rank-2 theory, in partic-ular its derivation from a variational principle, the clas-sification of different subclasses of theories, and the formof the field equations. In Sec. IV, the simplifications tothe theory that follow when the matter model is a perfectfluid are described. Section V presents the MEMe model,its exact solution in the case of a single perfect fluid, andits general features. Section VI shows how data fromgravitational wave signals can constrain the parametersof generalized coupling theories, and presents a param-eter constraint for the MEMe model. Section VII con-tains the analysis of the cosmology of the MEMe modelvia phase space analysis. Section VIII concludes with asummary and discussion of future work.

We adopt the MTW signature “(−,+,+,+)” [17] anduse natural units c = 1, defining κ = 8πG. Since indexplacement is critical in our analysis, the placement of

indices in indexed quantities which appear as argumentsin functions and and functionals will be indicated by dots.

II. SEMICLASSICAL GRAVITY FRAMEWORK

Here, we propose a framework for semiclassical gravitywhich relaxes the coupling between matter and the grav-itational field. The purpose of this section is to providea fundamental motivation for generalized coupling theo-ries; in particular, this discussion will allow us to lateridentify a key parameter in the theory with the vacuumenergy. We first sketch a derivation of the semiclassicalEinstein equations from the effective action. A more de-tailed discussion of these topics may be found in Refs.[18–22]. We then discuss a modification of this deriva-tion and obtain a framework in which the gravitationalfield does not couple directly to matter, but is mediatedby a rank-4 tensor.

A quantum field theory for some field ϕ on curvedspacetime endowed with a classical metric gµν is definedby a generating functional Z[J, g··], which has the formalfunctional integral expression

Z[J, g··] =

∫Dϕei(S[ϕ]+〈Jϕ〉x), (2)

where 〈X〉x :=∫X√|g|d4x, J is an external current,1

and S[ϕ] is the action for matter fields. For the restof this section, we suppress the functional dependence ongµν , and unless stated otherwise, Z[J ] and all functionalsconstructed from it are implicitly functionals of gµν . Onecan construct the following actionlike functional W [J ]:

W [J ] := −i lnZ[J ]. (3)

From W [J ], one may obtain the expression for the formalexpectation value φ = 〈ϕ〉 of the field ϕ,

φ(x) =δW [J ]

δJ(x)

∣∣∣∣J=0

. (4)

The field equations governing φ are obtained from theeffective action Γ[φ], which may be implicitly defined asa functional Legendre transformation of W [j],

Γ[φ] = −〈jφ〉x +W [j], (5)

where now j is an external current defined by

j(x) :=δΓ[φ]

δφ(x). (6)

1 The external current J is typically introduced as a calculationaltool for computing N -point correlation functions in quantumfield theory and is set to zero at the end of the calculation; foradditional details, consult Ref. [23].

3

At this point, one can see that in the absence of theexternal current j, the functional derivative vanishes, andone recovers the principle of stationary action for Γ[φ].To one-loop order, the effective action has the form

Γ[φ] = S[φ] + ~Γ(1)[φ] +O(~2), (7)

where S[φ] is the classical action evaluated on the ex-pectation value φ and Γ(1)[φ] is a functional, the explicitexpression for which may be found in Ref. [18]. One maytherefore interpret the effective action Γ[φ] to be a quan-tum corrected classical action. However, such an actionis divergent, and, as is customary in quantum field the-ory, one typically adds counterterms in the Lagrangianto absorb these divergences; i.e., we perform a renormal-ization.

In curved spacetime, one can show that some of thedivergent terms in Γ[φ] are purely geometrical; our anal-ysis here focuses primarily on these terms. Therefore, anappropriate regularization at one loop level can be ob-tained by adding geometric counterterms to the effectiveaction [20, 21, 24]. In particular, these counterterms willhave the form

Sct[g··] =

∫d4x√−g

[γ0 + γ1 R + γ2,1 R

2

+ γ3,1 C2 +O(R····

3)

],

(8)

where γi are coupling constants, R is the Ricci scalar,C2 := Cαβµν C

αβµν is the square of the Weyl tensor, andthe remaining terms quadratic in curvature have been ab-sorbed into the topological Gauss-Bonnet integral. Thetotal action is therefore

Σsg[φ, g··] = Γr[φ, g

··] + Sct[g··]. (9)

where Γr includes φ-dependent counterterms. At thispoint, one may recover the semiclassical Einstein actionby choosing the constants γi so that Sct[g

··] completelycancels all curvature terms except for the Einstein Hilbertterm and the vacuum energy term. Then, upon applyingthe stationary action principle to Σsg[φ, g

··], one obtains

Gµν + Λ gµν = κTµν [φ], (10)

where Gµν is the Einstein tensor for the metric gµνand the energy-momentum tensor Tµν [φ] depends on therenormalized coupling constants, the expectation valueof the field φ, and gµν .

Up to this point, the derivation we have presented isstandard [21]. We now discuss a similar procedure whichdiffers in that one drops the assumption that the metricthat appears in the effective action is the gravitationalmetric. Instead, we postulate that the metric gµν is re-lated to the gravitational metric gµν in the following way,

gµν := χµναβ gαβ , (11)

where the rank-4 tensor χµναβ is constructed from other

fields, which we shall specify later in this paper. We

then propose a choice of constants γi in the countertermaction Sct[g

··] such that all terms involving the curva-ture Rα

βµν in the effective action are canceled, includingthe Einstein-Hilbert term. In doing so, we effectivelypostulate that there is some mechanism which stronglysuppresses the curvature terms in this model.

Assuming that the dynamics for coupling tensor χµναβ

is provided by an action of the form Sχ[g··, χ····], the ac-

tion for the renormalized one loop theory has the form

Σ[φ, g··, χ····] = Γr[φ, g

··]+Sct[g··]+SG[g··]+Sχ[g··, χ··

··],(12)

where SG[g··] is the Einstein-Hilbert action.The action Σ[φ, g··, χ··

··] now describes a framework inwhich the metric tensor gµν is no longer directly cou-pled to the matter fields; the coupling is mediated by thetensor χµν

αβ . In the remainder of this article, we showthat the tensor χµν

αβ does not necessarily require the in-troduction of additional dynamical degrees of freedom inthe low-energy classical limit and that one can constructχµν

αβ entirely from nondynamical auxiliary fields. Also,since these auxiliary fields do not introduce derivativesof the energy-momentum tensor in the field equations,generalized coupling theories evade the no-go result ofRef. [10]. In fact, such a no-go result assumes that thematter fields couple to gµν in the usual way, which is nolonger the case when the couplings between the matterfields and gµν are mediated by the tensor χµν

αβ . Welater identify and study a theory that is natural in thesense that Sχ is simply the vacuum energy term.

III. COUPLING TENSOR THEORIES:GENERAL CONSIDERATIONS

A. Couplings

Here, we explore a class of theories in which the rank-4 coupling tensor χµν

αβ is constructed from invertiblerank-2 coupling tensors Aµ

α, with inverse Aµα. In par-ticular, we assume that χµν

αβ may be decomposed in thefollowing manner,

χµναβ = Ψ(A·

·)AµαAν

β , (13)

where Ψ(A··) is a scalar function of Aµ

α that has theproperty Ψ(δ·

·) = 1 when Aµα = δµ

α, where δµα is

the Kronecker delta. Note that Aµα = δµ

α is a tenso-rial equation, but only when one index is raised and theother is lowered—this is because δµ

α is a tensor,2 but δµνand δµν are not. For this reason, it is important to payparticular attention to index placement when perform-ing variations—see Appendix A. To simplify the analy-sis, the coupling tensors are assumed to be symmetric so

2 To see this, recall the expression δνµ = gµσ gνσ .

4

that Aµα = Aαµ and Aµα = Aα

µ. From these tensors,one constructs a physical3 metric gµν and its inverse gµν ,

gµν = Ψ(A··)Aµ

αAνβ gαβ , (14)

gαβ = Ψ−1(A··) Aαµ A

βν g

µν . (15)

Unless explicitly stated otherwise, indices are raised andlowered using the metric gµν and its inverse gµν . The

covariant derivative ∇µ is defined with respect to gµν .We shall slightly abuse some terminology for the sake ofconvenience: throughout this article, we shall refer to thephysical metric gµν as the “Jordan frame” metric and gµνas the “Einstein frame” metric.

Of course, since Aµα are square matrices, Eq. (13)

does not describe the most general coupling that onecan construct from Aµ

α; one could alternatively con-struct4 χµν

αβ = Θµα Θν

β from a general power se-ries in Aµ

α, labeled Θµα, with coefficients that are

scalar functions of Aµα. For instance, one may choose

Θµα = exp(δµ

α − Aµα). It is also worth mentioning

that one can also consider generalized couplings con-structed from one-forms. For instance, one could con-sider a generalized vector disformal transformation of theform gµν = ω2 gµν + σ AµAν , where ω and σ are func-tions of A2 = gµνAµAν and auxiliary scalar fields ψ, butone should be aware that unless ω and σ are independentof Aµ, such a coupling introduces an additional depen-dence on gµν , which will generate additional terms in thegravitational equations. For the purposes of the presentarticle, we will not explicitly5 consider these alternativecouplings, restricting only to those which have the formgiven in Eq. (13).

The idea of considering theories of gravitation with twometrics related as in (14) or, more generally, in (11) of-fers an interesting connection with continuum mechanicsand electromagnetism. Such relations have been stud-ied before in various realizations; see Refs. [13, 14, 25].The difference with respect to these works is that in thepresent work, the behavior of the coupling tensor is ex-plicitly given through a variational principle.

B. Classification of theories

We wish to construct theories with the property thatthe coupling tensors satisfy Aµ

α = δµα in the absence of

3 In the sense that matter couples to gµν .4 The reader might observe that this is similar to what is done

with tetrads eµa in the tetrad formalism. The main differencehere is that both indices of the tensor Θµα are in the coordi-nate basis (there are no internal Lorentz indices). However, onecan nonetheless imagine Θµα to be a transformation of the met-ric tensor between one adapted to gravitational dynamics (theEinstein frame) and one adapted to matter (the Jordan frame).

5 One might, however, imagine that the tensor Aµα could in princi-ple be a composite field constructed out of other auxiliary fields.

matter. We do so by way of a variational principle, witha functional of the form

SA[A··, g··] = −λ

κ

∫d4x√−gF (A·

·). (16)

There is no unique functional that yields Aµα = δµ

α asa solution. However, it is straightforward to constructsuch actions. A simple example is

SAq = − λκ

∫d4x√−g

(1

2A− 1

4Aβ

αAαβ

). (17)

It is also straightforward to verify that the variation withrespect to Aα

β yields the algebraic “equation of motion”Aµ

α = δµα, as intended. More generally, one can con-

struct a functional of the form

SAp = −λκ

∫d4x√−gP (A·

·), (18)

where P (A··) is a polynomial function of Aα

β of finite or-der satisfying the property P (δ·

·) = 1. We call this classof theories polynomial class theories. The coefficients forP (A·

·) which yield the solution Aµα = δµ

α may be ob-tained by factoring the derivative of P (A·

·) and demand-ing that at least one of the factors be (Aµ

α − δµα). Inparticular, one can choose coefficients in the polynomialP (A·

·) such that

∂P

∂Aαβ= (Aα

σ − δασ)fβσ , (19)

where fβσ is some quotient polynomial. It is not too diffi-cult to demonstrate that to second order, the form of theaction SAq (17) is the one that uniquely yields the solu-tion Aµ

α = δµα. One may note that higher-order poly-

nomial class theories may yield additional nondegeneratesolutions, but since Aµ

α must satisfy an algebraic equa-tion, it suffices to specify initial conditions that satisfyAµ

α = δµα.

Another class of simple theories have actions of theform

SAe = −λκ

∫d4x√−g |A|nE(A·

·), (20)

where |A| = det(A··), and again, E(A·

·) is a functionsatisfying the property E(δ·

·) = 1. The variation of theabove takes the form

δSAe =− λ

κ

∫d4x√−g |A|n×[(

∂E

∂Aβα+ nE Aβα

)δAβ

α − 1

2E gµν δg

µν

],

(21)and the variation with respect to Aβ

α yields an equa-tion which may (after a straightforward integration) bewritten as

∂ lnE

∂Aβα= −n Aβα. (22)

5

This suggests that E(A··) has the form

E(A··) = exp (k − fp(A··)) , (23)

where k = fp(δ··) and fp(A·

·) is a finite polynomial that,to second order and above, satisfies the following:

∂(nA− fp)∂Aβα

= (Aασ − δασ)fβσ . (24)

Again, fβσ is some quotient polynomial. The simplestcase is the choice fp = nA (in which case k = 4n). SinceEq. (23) is an exponential, theories of this type will betermed exponential class theories.

The theories considered so far are homogeneous, mean-ing that the actions depend explicitly on the tensor Aµ

α

or its inverse, but not both. One can also construct in-homogeneous theories in which the action is an explicitfunctional of both Aα

β and Aαβ . It can be difficult toobtain analytical solutions for a general polynomial orexponential class theory, and it will become increasinglydifficult to obtain analytical solutions for the more com-plicated inhomogeneous theories. For this reason, we willnot study inhomogeneous theories any further, and willfocus on the simplest theory which can be solved exactlyfor a perfect fluid.

C. Gravitational action

The theories described in the previous section are con-structed so that when Aµ

α = δµα, the action SA has the

value

SA[δ··, g··] = −λV

κ, (25)

V :=∫d4x√−g being the four-volume. This is enforced

by the requirement that when Aµα = δµ

α, the integrandof the action SA satisfies the property P (δ·

·) = E(δ··) =

1. Later, we find that the parameter λ must be largein order to maintain consistency with late-time experi-mental and observational constraints, so we must add acounterterm 2λ in the gravitational action. In particular,we assume that the dynamics for the gravitational metricgµν is provided by an action of the form

Sg[g··] =

1

2κ

∫d4x√−g

(R+ 2 Λ

)=

1

2κ

∫d4x√−g [R− 2 (Λ− λ)] ,

(26)

where Λ is a gravitational parameter related to the ob-served value of the cosmological constant Λ according tothe formula λ− Λ = Λ. It follows that in the generalizedcoupling theories we have constructed, gµν satisfies thevacuum Einstein field equations with cosmological con-stant Λ = λ− Λ,

Gµν + Λ gµν = 0, (27)

in the absence of matter. This result implies that generalcoupling theories do not avoid the fine-tuning problem as-sociated with the cosmological constant, since one mustrequire |λ − Λ|/|λ| 1 to fit observational data. How-ever, as we shall argue later, the fine-tuning problem canbe mitigated to some degree in the particular generalizedcoupling theory we study.

D. Generalized coupling in matter action

Consider a matter action of the following form,

Sm = Sm[φ, g··] =

∫d4x√−gLm[φ, g··], (28)

where φ (field indices suppressed) is a tensor field as-sumed to be minimally coupled to the metric gµν . Up toboundary terms, the variation of the matter action Smhas the following form,

δSm =

∫d4x√−g

(E[φ, g··] δφ− 1

2Tαβ δg

αβ

), (29)

where E[φ, g··] is the Euler-Lagrange operator yieldingthe field equations E[φ, g··] = 0, and the Jordan frameenergy-momentum tensor is defined as

Tαβ := − 2√−gδSmδgαβ

. (30)

One may relate Tµν to the Einstein frame energy-momentum tensor τµν by making use of the chain rule

τµν := − 2√−gδSmδgµν

= −2 Ψ2 |A|√−gδSmδgαβ

∂gαβ

∂gµν. (31)

Using Eq. (15), one may obtain the following:

τµν = Ψ |A|Tαβ Aαµ Aβν . (32)

The variation of gαβ , as given by Eq. (15), is

δgαβ =Ψ−1 Aαµ Aβν δg

µν

−(

2 gσ(β Aα)τ + Ψ−1 gαβ

∂Ψ

∂Aστ

)δAσ

τ .(33)

The variation of the action then takes the following form,

δSm =

∫d4x√−g

[Ψ2 |A|E[φ, g··] δφ− 1

2τµν δg

µν

+ Ψ2 |A|(Tαβ g

σ(α Aβ)τ +

T

2 Ψ

∂Ψ

∂Aστ

)δAσ

τ

].

(34)where for convenience we define the “trace” T :=Tαβ g

αβ . Note that, since the variation δSm dependson the variation δAσ

τ , the presence of matter will con-tribute additional terms to the field equation for Aµ

α, aswe shall demonstrate shortly.

6

E. General field equations

We can now join together the previous results and givethe general action for a generalized coupling theory. Wehave

S[φ, g··, A··] =

∫d4x

(R− 2 [Λ− λ(1− F )])

√−g

+ 2κLm[φ, g··]√−g

,

(35)where F = F (A·

·). Upon variation with respect to themetric and remembering that Aµ

α is independent of gµν ,one obtains

Gµν + [Λ− λ (1− F )] gµν = κΨ |A| Aαµ Aβν Tαβ . (36)

The form of this equation allows one to draw some gen-eral conclusions on the physics of these models. We no-tice immediately that the theory will generate a varyingcosmological constant, which is dependent, via Aµ

α, onthe matter distribution. Additionally, the presence of thequantity Ψ |A| Aαµ Aβν contracted with Tαβ “scrambles”the gravitational sources in a nontrivial way.

Notice also the differences between the (36) and (1).In Eq. (1), there is no effective cosmological term andthe energy-momentum tensor is a function of the Einsteinmetric gµν . This might suggest that the two theories arecompletely different. However, as stated in Ref. [7], in(1), χµν

αβ is completely general and thus can be chosento return the structure of (35). In this sense, the twoequations are still related.

The variation with respect to Aµα yields the field equa-

tion for Aµα,

(δµα−Aµα)fνα = Ψ2 |A|

[Tαβ g

µ(α Aβ)ν + T

1

2 Ψ

∂Ψ

∂Aµν

],

(37)where, as before, fµν is some tensor constructed from Aµ

α

such that

δF

δAµν= (Aν

α − δνα)fµα . (38)

As we have anticipated, the matter action Sm contributesadditional terms to the field equation (37) for Aµ

α. Theseadditional terms will generally drive the coupling ten-sor Aµ

α away from the condition Aµα = δµ

α. However,since we are assuming minimal coupling, Eq. (37) con-tains no covariant derivatives of Aµ

α. Thus, the resultingfield equation is an algebraic equation for Aµ

α, and thecondition Aµ

α = δµα is only violated at points where

Tαβ 6= 0. It follows that at points where the energy-momentum tensor Tαβ vanishes, the coupling tensors sat-isfy Aµ

α = δµα. In this sense, the field Aµ

α has the samebehavior as the torsion tensor in the context of Einstein-Cartan theory [9].

Before moving on to a specific example, we note that ingeneral, the energy-momentum tensor Tαβ contains fac-tors of gµν and gαβ , which generally introduce additional

factors of Aµα into the field equation. It follows that in

general, Eq. (37) can be a high-order algebraic equationfor Aµ

α. For example, in the context of a homogeneouspolynomial class theory, a matter action like Eq. (17) willlead to an equation that is formally quadratic in Aµ

α. Itis also possible that Eq. (37) may admit no real solutions.This implies that certain energy-momentum tensors Tαβmay require that Aµ

α be complex valued. For complex-valued Aµ

α, one generally has a complex metric gµν overa real manifold. The resulting complex matter action Smand the action SA should be replaced with the respectivereal actions (Sm+S∗m)/2 and (SA+S∗A)/2 to ensure a con-sistent coupling to the real-valued gravitational degreesof freedom gµν . Fortunately, this problem is not too se-vere; one can show that upon expanding Aµ

α about δµα,

one does not need to consider complex values for Aµα

until fifth order in the expansion parameter. Also, themetric is only complex within a matter distribution; in avacuum, one has Aµ

α = δµα so that gµν = gµν . As we

will show in the next section, it turns out that for a singleperfect fluid, this problem does not arise in the specifictheory we examine later in this article—we will in factobtain an exact real-valued solution for Aµ

α.

IV. PERFECT FLUID ANSATZ

Since we construct generalized coupling theories byway of a variational principle, it is appropriate to ap-peal to the variational principle for relativistic fluids, asdiscussed in Refs. [26–31]. The variational principle fora relativistic perfect fluid may be formulated on an arbi-trary background spacetime, which need not satisfy theEinstein equations, so that the relativistic Euler equa-tions

gαβ∇βTαγ = 0 (39)

are in general independent of the contracted Bianchiidentities. We remind the reader that ∇µ is the covariantderivative compatible with the Jordan frame metric gµν ,and that matter is assumed to be minimally coupled togµν .

Equation (39) leads to another reason for appealing toa variational principle. In generalized coupling theories,the contracted Bianchi identities do not imply that Eq.(39) is divergence free, but only that the source of theEinstein tensor satisfies the divergence-free property onshell; the relativistic Euler equations must be supplied bya variational principle. One can, on the other hand, showfrom the diffeomorphism invariance of the matter actionthat Eq. (39) must hold on shell. For a more detaileddiscussion, we refer the reader to Appendix B.

Variational principles for perfect relativistic fluids usu-ally involve constrained variations, and are formulated interms of gradients of velocity potentials, so the covariant(lowered index) fluid four-velocity uµ does not have alocal dependence on the metric [28, 31]. In terms of uµ,

7

the energy-momentum tensor for a perfect fluid takes theform

Tµν = (ρ+ p)uµuν + p gµν . (40)

It should be stressed that in general, uµ uµ = uµuνg

µν 6=−1; the fluid four-velocity only has unit norm with re-spect to the Jordan frame metric

uµ uν gµν = −1. (41)

Note that, since for a perfect fluid the trace is

T := Tαβ gαβ = 3p− ρ, (42)

only one factor of gαβ appears in the field equation (37).For perfect fluids, it is appropriate to employ the fol-

lowing ansatz for the solution,

Aβα = Y δβ

α + Z uβ uα, (43)

where uµ is the fluid four-velocity. This ansatz will bejustified in the next section for the specific model westudy, but it can be considered nonetheless general forthe case of a single perfect fluid. It is straightforwardto show that the inverse of Aµ

α as given in Eq. (43) is(assuming Y + εZ 6= 0)

Aαβ =1

Y

(δβα − Z

Y + εZuα uβ

), (44)

where

ε := uµ uµ. (45)

Now one can define a timelike unit vector Uµ by rescalinguµ (which is also assumed to be timelike, so that ε < 0),

Uµ := uµ/√−ε, (46)

and it follows that uµ uν = −εUµ Uν . Equation (43) maythen be written in the alternate form

Aµα = Y δµ

α − εZ Uµ Uα. (47)

This form for Aµα is useful for computing the determi-

nant |A| = det(A··), which reads

|A| = det(A··) = Y 3 (Y + εZ) . (48)

V. MINIMAL EXPONENTIAL MEASUREMODEL

Consider a matter action of the following form,

Smλ = Sm −λ

κ

∫d4x√−g, (49)

where Sm is given by Eq. (28) and

√−g =√−g|A|Ψ2. (50)

If the factor Ψ2 is chosen such that the variation of thevolume element

√−g yields the desired field equations forAµ

α, then one may interpret the parameter λ in Eq. (49)to be something akin to a vacuum energy density gener-ated by matter fields. This is in fact the interpretationprovided by the semiclassical procedure outlined in Sec.II. In the framework of effective field theory, the value ofλ then corresponds to the energy scale at which a fieldtheoretical description for matter breaks down.

To construct such a theory, we seek an expression forthe factor Ψ such that the variation of

√−g yields thefield equation Aµ

α = δµα in a vacuum. Assuming Ψ is

an explicit function of Aµα only, Eq. (50) shows that

the volume element in Eq. (49) has the same form ofthe integrand of the action SAe for an exponential classtheory in Eq. (20).

Hence, if we wish to recover the field equation Aµα =

δµα, Ψ should have the form of an exponential of a poly-

nomial in Aµα. The simplest such form for Ψ is

Ψ = exp

(4−A

2 s

). (51)

This form for Ψ introduces an exponential of the sim-plest polynomial of Aµ

α into the Jordan frame mea-sure

∫d4x√−g, and for this reason, we call the resulting

theory the Minimal Exponential Measure model, or theMEMe model. Defining the parameter

q :=κ

λ, (52)

the variation of the volume functional is

−1

qδ

∫d4x√−g =− 1

q

∫d4x√−g|A|Ψ2×[(

Aστ −1

sδτσ

)δAσ

τ − 1

2gµν δg

µν

],

(53)which leads to the parameter choice s = 1. The variationof Sm with respect to Aσ

τ yields the field equations

Aαβ − δβα = q[Tµν g

αν Aµβ − (1/4)T δβα]. (54)

Upon multiplying through by Aµβ , we obtain an expres-

sion that is formally linear in the components of Aµα,

Aβα − δβα = q [(1/4)TAβ

α − Tβν gαν ] . (55)

We note here that the MEMe model is the simplest ho-mogeneous theory that one can construct in the sensethat one obtains an equation that is effectively linear inAµ

α.The trace of Eq. (55) yields an equation linear in the

trace, A = Aαα

A− 4 = q T (A/4− 1) , (56)

which implies that A = 4, which in turn implies Ψ = 1,so that remarkably, all exponential factors disappear on

8

shell. Note that in deriving this result, we have not yetmade any assumption about the matter model; this resultholds for any energy-momentum tensor.

We now consider the case of a perfect fluid. To justifythe ansatz for Aµ

α given in Eq. (43), we consider thecontraction of Eq. (55) with the vector uα. The result is

Aβα uα = −uβ

1 + q ρ

4 [4− q (3 p− ρ)], (57)

and it follows that Aβα uα ∝ uβ , which is consistent with

the ansatz for Aµα given in Eq. (43).

To obtain explicit expressions for Y and Z, we use theansatz (43) to rewrite the field equation (55) in the form

Wαβ = W1 δβ

α +W2 uβ uα = 0, (58)

where

W1 =1

4Y [4− q (3 p− ρ)] + p q − 1,

W2 =q (p+ ρ)

(εZ + Y )2+

1

4Z [4− q (3 p− ρ)];

(59)

i.e., Eq. (55) is reduced to the two scalar equations W1 =0 and W2 = 0. As W1 is completely independent of ε andZ, it can be used to find an explicit expression for Y :

Y =4(1− p q)

4− q (3 p− ρ). (60)

The trace of the ansatz (43) is A = 4Y + εZ, and since(56) implies A = 4, we obtain the following expressionfor εZ,

εZ = 4 (1− Y ) =4 q (p+ ρ)

4− q (3 p− ρ), (61)

where we have used Eq. (60) for Y in the second equal-ity. We then solve the equation W2 = 0 to obtain anexpression for Z,

Z = −q (p+ ρ)[4− q (3 p− ρ)]

4 (q ρ+ 1)2, (62)

and from (61), we obtain the expression for ε,

ε = − 16 (q ρ+ 1)2

[4− q (3 p− ρ)]2. (63)

Using Eq. (49) and Eq. (26) [cf. Eq. (36)], we obtainthe gravitational field equations

Gµν +[Λ− λ

(1− e4−A|A|

)]gµν = κe(4−A)/2|A|AαµAβνTαβ ,

(64)where the determinant |A| is given by the explicit expres-sion

|A| = 256 (1− p q)3(q ρ+ 1)

[4− q (3p− ρ)]4. (65)

It is useful to write Eq. (64) in the form

Gµν = κTµν , (66)

where Tµν is the effective energy-momentum tensor de-fined by

Tµν = T1 Uµ Uν + T2 gµν , (67)

and

T1 = |A| (p+ ρ),

T2 =|A| (p q − 1) + 1

q− Λ

κ.

(68)

Notice that, though q appears in the denominator of thefirst term in T2, |A| is a function of q. Thus, in the limitq → 0,

T1 → p+ ρ,

T2 → p− Λ/κ.(69)

In motivating the MEMe model, we have put forward theinterpretation of λ/κ = 1/q as a form of vacuum energydensity for quantum fields. If the MEMe model is re-garded to be a low energy description for some quantumgravity theory, it is natural to expect the vacuum energy1/q to be the Planck energy density, but whether thisis the case ultimately depends on the specific details ofthe fundamental theory. Barring any specific knowledgeabout the underlying theory, we point out that q can inprinciple be independent of the Planck scale. In fact, itturns out that the parameter q sets the scale at which gµνfails to be a suitable spacetime metric and thus providesa natural regularization scale for quantum fields. To seethis, note that for positive q, the determinant |A| vanisheswhen p q → 1, or when the pressure (assuming w > 0) ison the order 1/q. In this limit, Aµ

α and gµν fail to be in-vertible, so that gµν cannot serve as a spacetime metric.Thus, q sets the scale at which the usual formulation ofquantum field theory on a spacetime background givenby gµν breaks down, and, as consequence, 1/q providesa natural regularization scale for the vacuum energy ofquantum fields. This scale can in principle be indepen-dent of the Planck scale, as it concerns the breakdownin the effective spacetime metric gµν , rather than thegravitational metric gµν . In this sense, the MEMe modelis phenomenologically compatible with a regularizationscale for vacuum energy far below the Planck scale.

It is worth remarking that the breakdown of the de-scription of the MEMe model as a bimetric theory doesnot imply that the model itself breaks down. Indeed, inthe limit p q → 1, the gravitational metric gµν remainswell defined and the gravitational field equations havethe form

Gµν ≈ (λ− Λ) gµν . (70)

A similar mechanism is present in the case of negativeq, which corresponds to a negative vacuum energy den-sity for matter fields. For negative q, the determinant

9

|A| vanishes in the limit ρ q → −1. In this case, one canalso recover Eq. (70), but now λ is negative-valued. Thissuggests that for negative λ, one has a de Sitter phase inthe limit ρ q → −1, or when the density ρ approaches theregularization scale 1/|q|. We will revisit this in greaterdetail when we discuss the evolution of cosmologies ob-tained from the MEMe model.

The early de Sitter phase aside, there are more funda-mental reasons to choose a negative value of q, or equiv-alently, a negative λ. One might suppose, for example,that in the fundamental theory, the fermionic contribu-tion to the vacuum energy dominates, which would alsoimply a negative vacuum energy density and a negativevalue for λ. We remark that a negative vacuum energydensity may be of particular interest in the context ofstring theory,6 in which a negative vacuum energy den-sity is natural [33]. A stronger case can me made if oneimagines the MEMe model to be some limit of a dynam-ical theory; if (λ/κ)

√−g is interpreted to be potentialfor some dynamical theory, then dynamically stable so-lutions should lie at a minima of the potential. Whenevaluated on the solution Aστ = δτ

σ, the second deriva-tive of the potential taken with respect to Aµ

α, becomes

λ

κ

∂2√−g∂Aµα ∂Aνβ

∣∣∣∣Aστ=δτσ

= −λ√−gκ

δµβ δνα. (71)

The solution Aστ = δτσ corresponds to a minimum of the

potential if λ < 0, and the same solution is a maximumof the potential if λ > 0. A negative λ is therefore re-quired to ensure the dynamical stability of the solutions,assuming that MEMe is a limit for some dynamical the-ory.

Earlier, we pointed out that generalized coupling mod-els still suffer from the fine-tuning of the cosmologicalconstant (in particular, Λ/|λ| = |λ−Λ|/|λ| 1), and theMEMe model is no exception in this regard. However,we have argued that the vacuum energy in the MEMemodel is independent of the Planck scale. If the magni-tude of the vacuum energy is far below the Planck scale,the fine-tuning problem becomes less severe than that ofthe standard cosmological constant problem.

VI. PROPAGATION SPEED FORGRAVITATIONAL WAVES

The MEMe model can be thought of as a bimetric the-ory, and unless the two metrics are related by a confor-mal transformation, bimetric theories generally have theproperty that the light cones defined by each metric do

6 An important consideration, which falls beyond the scope of thisarticle, is whether any theory which reduces to the MEMe modelin some limit must lie in the “swampland,” in particular whetherthe MEMe model is incompatible with vacua allowed by stringtheory [32].

not necessarily coincide. To see this, we consider theform of the Jordan frame metric gµν for a perfect fluidgiven the ansatz (43) for the tensor Aµ

α,

gµν = Ψ[Y 2 gµν − εZ (2Y + εZ)Uµ Uν

], (72)

which we note is a type of vector disformal transforma-tion [12].

Since electromagnetic waves propagate according tothe metric gµν , and linearized gravitational waves prop-agate on a background metric gµν , Eq. (72) may be usedto compute the relative propagation speed between lightand gravitational waves. The relationship between prop-agation speeds can be computed explicitly by consideringa vector kµ that is null with respect to the gravitationalmetric gµν . In an orthonormal frame (defined with re-spect to gµν) adapted to uµ, Eq. (72) yields the disper-sion relation

− (1 + ΨZ (Y + εZ)) (k0)2 + (k)2 = 0. (73)

The speed of gravitational waves cg with respect to themetric gµν is then given by the expression

cg = c√

1 + ΨZ (Y + εZ), (74)

where c is the speed of light. It is straightforward to showthat if q > 0, then for sufficiently small energy densities,cg < c. We now define the quantity

∆ := (cg/c)2 − 1 = ΨZ (Y + εZ). (75)

For the MEMe model, ∆ has the expression

∆ = −q (p+ ρ)

1 + q ρ≈ −q (p+ ρ) + q2 ρ (p+ ρ) +O(q3),

(76)i.e., the theory predicts that gravitational waves propa-gating within a matter distribution will travel at a differ-ent speed compared to gravitational waves in a vacuum.It is worth pointing out that the difference in propagationspeed is not necessarily a problem in the MEMe model.A generic argument is given in Ref. [34] and says thatbimetric theories admitting two different speeds of lightdo not lead to causal paradoxes, so long as the system iswell posed.

For positive q or ρ q < −1, gravitational waves inthe MEMe model slow down in the presence of matter(∆ < 0), so that they will refract in a manner similarto that of light in a medium. On the other hand, if qis negative and ρ < 1/|q|, then gravitational waves canexceed the speed of light (∆ > 0). To lowest order inq, the amount by which gravitational waves speed up orslow down in a medium is linear in the energy density.The Earth provides a relatively dense medium throughwhich detectable gravitational waves propagate,7 so one

7 One can use the constraints on the speed of gravitational waves

10

may constrain the parameter q using the uncertainty intime delay for gravitational waves. Recall that for a pairof gravitational wave detectors, the order of the arrivaland time delay for gravitational wave signals can local-ize the source to a circular ring in the sky. However,such a localization assumes that gravitational wave sig-nals propagate at cg = c; cg > c will tend to “widen” thepredicted localization rings (more precisely, the enclosedsolid angle increases), and cg < c will tend to “narrow”the predicted localization rings. If one has an opticalcounterpart, then the location of the optical source inthe sky may be different from the apparent localizationif q 6= 0.

The gravitational detection GW170817 and its opti-cal counterpart GRB170817A may be used to place aconstraint on |q| [35]. The signals were localized in thesouthern sky. The signal first arrived at the Virgo de-tector, then at the LIGO-Livingston 22 ms afterwards,and finally at LIGO-Hanford, 3 ms after its appearanceat LIGO-Livingston [1]. This order of events, combinedwith the fact that the signal was localized in the south-ern sky, indicates that the GW signal passed through theEarth. There is no significant discrepancy between thelocalization of GW170817 and the position of the sig-nal GRB170817A, so we may immediately place a roughconstraint on |q| from the uncertainty in the differenceof arrival times between different gravitational detectors,which we estimate to be roughly 5%. From the uncer-tainty in the time delay, one expects the uncertainty in cgto also be roughly 5%, so that |q| < 0.1/ρE . On average,the energy density of the Earth is ρE ∼ 5 × 1012 J/m3,which places the following upper bound on |q|:

|q| < 2× 10−14 m3/J. (77)

Fundamentally, λ/κ = 1/q in the MEMe model corre-sponds to the vacuum energy density for quantum fields,but phenomenologically, the value of |q| sets the valueof the energy density at which the model breaks down.The highest energy scale probed to date is the TeV scale,which from l ∼ ~c/E, corresponds to a length scalel ∼ 2 × 10−19m. In a similar manner, one may use thisto compute an inverse energy density via the expression|q| = l4/~c, so that if one expects new physics at theTeV scale, then one might expect |q| ∼ 5 × 10−50m3/J,which is about 35 orders of magnitude smaller than theconstraints from GW170817; Eq. (77) is still a ratherweak constraint. Of course, our analysis here is prelim-inary, and a more careful analysis may provide strongerconstraints on λ.

from GW170817 and GRB170817A [35] to place constraints onq (see for instance Ref. [36]), but this actually places a mucha weaker constraint on |q| than the propagation of gravitationalwaves through the Earth because the average energy density ofbaryonic and dark matter (Λ does not significantly affect ourresult here) in the Universe is 30 orders of magnitude smallerthan the average energy density of the Earth.

VII. COSMOLOGY

We now explore the features of cosmological modelsbuilt on the field equations (64) of the MEMe model.We will accomplish this task by employing dynamicalsystems techniques. Dynamical systems tools have nowbeen used for a long time to explore cosmological modelsin GR (see, e.g., Ref. [37] and references therein) andmodified gravity (e.g., Ref. [38]). We will perform herea quick phase space analysis with the aim of evaluatingthe potential of (64) to provide a theoretical frameworkfor an inflationary/dark Universe.

The first step in constructing the cosmological model isto select a class of reference observers. This step, whichis often made tacitly in GR, is crucial in the context oftheories like the MEMe model. A choice which is akinto the selection of fundamental observers in GR is the“Jordan velocity” uµ of the matter fluid. However, sucha choice would pose a serious problem: the field uµ isonly a valid velocity frame if ε is different from zero, i.e.,qρ 6= −1.

We prefer to use frames which have no such limita-tions, so as to avoid spurious singularities. A convenientchoice is a frame Uµ which is parallel to uµ so that itis still orthogonal to the three-surfaces of homogeneitydescribed by hµν = gµν + uµ uν and therefore preventsany “tilting” effect [39]. In this way, we can suppose thatfor a homogenous and isotropic fluid source, the metricof the spacetime in the frame specified by Uµ has theFriedmann-Lemaıtre-Robertson-Walker form

ds2 = −dt2 + S2 (t)

[dr2

1− kr2+ r2dΩ2

], (78)

where k = −1, 0, 1 is the spatial curvature, dΩ2 the in-finitesimal solid angle and S is the scale factor.

In the Uµ frame, the cosmological equations can bewritten as

3q

(H2 +

k

S2

)=

256κ(1− pq)3(qρ+ 1)2

[4 + q(ρ− 3p)]4+ qΛ− κ,

(79)

6q(H +H2

)=− 256κ(pq − 1)3(qρ+ 1)[2− q(ρ+ 3p)]

[4 + q(ρ− 3p)]4

+ 2(qΛ− κ), (80)

where H = S/S and we have assumed a barotropic equa-tion of state p = wρ for the fluid. There is, however, animportant caveat: the equations above only correspondto the equations for the MEMe model if uµ is well de-fined. We must therefore require that |A| 6= 0 (ε 6= 0) in

our analysis. In the same way, either gασ∇σTαβ = 0 or∇αTαβ = 0 give the conservation law

ρ = −3Hρ(w + 1)[q2ρ2w(3w − 1) + ρ(q − 7qw) + 4

]q2ρ2w(3w − 1)− qρ (3w2 + 13w + 2) + 4

.

(81)

11

As in GR, the three equations above are redundant. Thestructure of the above equations shows that in the MEMemodel, the gravitation of a perfect fluid is very differentform the GR equivalent.

Let us now construct the phase space. Assuming H >

0, we define the dimensionless variables

χ =qH2

κ, K =

k

H2S2, (82)

Ω =κρ

3H2, L =

Λ

3H2, (83)

choosing also a dimensionless time variable N =log (S/S0) and indicating with a prime the derivativewith respect to N . The cosmological equations may berewritten as the autonomous system

χ′ =2(L− 1)χ+256(3χΩ + 1)(3wχΩ− 1)3[3χΩ(1 + 3w)− 2]

3[3(1− 3w)χΩ + 4]4− 2

3,

Ω′ =1

3Ω

2

χ+ 6(1− L)− 256(3χΩ + 1)(3wχΩ− 1)3[3χΩ(1 + 3w)− 2]

3[3(1− 3w)χΩ + 4]4

− 9(w + 1)(3wχΩ− 1)[3(3w − 1)χΩ− 4]

4− 3 (3w2 + 13w + 2)χΩ + 9w(3w − 1)χ2Ω2

,

L′ =1

3L

2

χ+ 6(1− L)− 256(3χΩ + 1)(3wχΩ− 1)3[3χΩ(1 + 3w)− 2]

3[3(1− 3w)χΩ + 4]4

,

(84)

together with the constraint

L−K − 256(3χΩ + 1)2(3wχΩ− 1)3

3χ[3(1− 3w)χΩ + 4]4− 1

3χ= 1, (85)

given by Eq. (79).Although not immediately clear from the form of the

(84), the system admits three invariant submanifolds Ω =0, L = 0, and χ = 0. Therefore, a global attractor canonly exist at the origin. Setting to zero the lhs of (84),we find two fixed points (A and B) and two lines of fixedpoints (L1 and L2). Notice that in the case w = 0, L2

becomes asymptotic.The solutions associated to the fixed points can be

found using the modified Raychaudhuri equation (80)written in the form

H

H2=

128 (3χ∗Ω∗ + 1) (3wχ∗Ω∗ − 1)3

[3χ∗Ω∗ (3w + 1)− 2]

3χ∗ [3(1− 3w)χ∗Ω∗ + 4]4

+ χ∗ − 1 +1

3L∗,

(86)where the asterisks indicate the values of the dynamicalvariables at a fixed point.

Using (86), we obtain the result that A and B corre-spond, respectively, to a Milne and Friedmann solution,whereas lines L1 and L2 correspond to de Sitter solutionswith time constants

γ1 =

√Λ

3,

γ2 =

√Λq − κ

3q+

256κ (qρ0 + 1)2

(qρ0w − 1)3

q[4 + 3q4ρ40 (1− 3w)

4]

,

(87)

respectively. The cosmology then presents two differentexponential expansion phases. Notice that the pointsin line L2 are related to the behavior of the system inthe limit pq → 1 mentioned at the end of Sec. V. As wehave written our equations in the frame Uµ, the boundaryρq = −1 does not correspond to any visible feature of thephase space. Nonetheless, one should bear in mind thatphysical orbits only correspond to Ωχ 6= −1 (and indeedΩχ < −1).

Writing the conservation law (81) in the same way asthe modified Raychaudhuri equation, we can deduce thebehavior of the matter energy density at a fixed point,

ρ

ρ=

3H(w + 1) (1− 3wχ∗Ω∗) [3χ∗Ω∗(3w − 1)− 4]

9w(3w − 1)χ2∗Ω

2∗ − 3 (3w2 + 13w + 2)χ∗Ω∗ + 4

,

(88)where H is given by the modified Raychaudhuri equation(86). Apart from L2, the behavior of matter is the usualone. At the fixed point L2, the corrections to GR returna peculiar behavior in the energy density. In particularthe energy density remains constant even if the spacetimeis expanding. This implies that orbits near to this pointhave a very slowly decreasing energy density.

The Hartmann-Grobmann theorem can be used to de-duce the stability of the fixed subspaces. It turns outthat the lines are always attractors and the fixed pointsare saddles. In particular, as point A (the origin) is asaddle, no global attractor is present in the phase space.Therefore, orbits will go toward the lines L1 and L2.

The fixed points with their stability and the associatedsolutions for the scale factor and matter energy densityare given in Table I. We illustrate the phase space in Figs.

12

1 and 2.As the phase space is not compact, one should study

the behavior at infinity; however, for the purpose of thiswork, this analysis is not very relevant. This happensbecause, as we have mentioned, our equations cease tobe valid if |A| = 0, i.e., in the subspaces 3wΩχ = 1and 3Ωχ = −1, respectively. Consequently, all asymp-totic fixed points that lie beyond the line L1 and/or theboundary 3Ωχ = −1 are irrelevant for our purposes.The same conclusion holds for the asymptotic points atΩ → ∞, χ → 0 and Ω → 0, χ → ∞: they do notconstitute physically relevant states for the cosmology.

As we know from Secs. V and VI that q must be small,and if we wish to have a de Sitter phase at high densities,q must also be negative. If we assume high densities atan early time, we expect that physically relevant initialconditions will be close to 3Ωχ = −1 (ρ q = −1) at highΩ. Therefore, one expects initial conditions for physicalmeaningful orbits to be close to the subspace 3Ωχ = −1.Relevant orbits stemming from this subspace “bounce”against the general Friedmann fixed point (B) and then,maintaining a low value for χ, the orbits evolve towardone of the attractors of L1. In Fig. 3 we give examplesof these types of orbits.

These scenarios are interesting from the point of viewof inflationary dark cosmology. In fact, they both ad-mit an early and late accelerated expansion phase withdifferent effective cosmological constants and thus solvenaturally the “graceful exit” problem. The final value ofthe cosmological constant is Λ, while the inflationary ex-pansion is given by a de Sitter solution with time constantvery close to Λ + λ. The differences in the two historiesare related to the expansion rate, the length, and time ofoccurrence of the decelerated expansion phase.

Another observation concerns the case in which thecosmological constant is zero. In this case, because ofthe existence of the invariant submanifold L = 0, thedynamics is represented by the lower planes of Figs. 1and 2. It is clear that we still recover an inflationaryphase, but no dark energy era is dynamically achievableas in the L = 0 subspace, as the only finite attractor ispoint A. In principle there are homoclinic orbits thatstart at Ω → ∞, χ → 0 and bounce against point B,but even neglecting the unphysical nature of the initialpoint, the picture in Fig. 3 suggests that these orbitsnever represent accelerated expansion.

It is worth noticing that the scale factor S we havededuced is not the one “measured” by the observers atrest with respect to matter, and therefore it is not initself a physically meaningful quantity. Fortunately, forour choice of frame and symmetries, the behavior of theJordan frame scale factor SJ is easy to calculate. UsingEq. (72), we can easily show that

SJ = ΨY 2S =16(1− pq)2

[4 + q(ρ− 3p)]2S. (89)

We will use this relation to calculate SJ correspondingto the solutions at the fixed points (see Table I). For

negative q and 0 < w < 1/3, SJ differs from S by a factorof order unity as long as ρ ≤ 1/|q|, so SJ remains welldefined when the determinant of the Jordan frame metricvanishes. On the other hand, for positive q and 0 <w < 1/3, SJ vanishes as the pressure p approaches 1/q.Among the fixed points the biggest difference between Sand SJ is present in the solutions associated to point B,which only differ significantly at small t. These resultssuggest that critical constraints for the MEMe model canbe found by looking closely at the phenomenology of thematter dominated era.

VIII. CONCLUSIONS

In this article, we have proposed a new class of mod-ified gravity theories in which the interaction of matterand spacetime is mediated by a rank-4 tensor χµν

αβ =Ψ(A·

·)AµαAν

β , where Aµα are nondynamical auxiliary

fields. It is tempting to compare the role of this fieldwith the one that the Higgs field has in particle physics:as the Higgs field determines the inertial mass of parti-cles, χµν

αβ in a sense determines the active gravitationalmass of matter. However, unlike the case of the Higgsmechanism, the fundamental mechanism that leads to ageneralized coupling theory is not immediately evident atthe present stage of investigation. Nonetheless, we haveattempted to motivate generalized coupling theories in aframework that is more fundamental than simply speci-fying an action for the classical theory; in particular, weconstructed the gravitational field equations via a pro-cedure analogous to the one used to construct the semi-classical Einstein field equations. While this constructiondoes require some fine-tuning, it provides a natural inter-pretation of λ as the vacuum energy and may be usefulas a starting point for finding a more fundamental the-ory from which a generalized coupling theory emerges asa limit.

It has been pointed out that auxiliary field theories canemerge in a strong coupling limit, as in the case of Ref.[40], and one might imagine that a generalized couplingtheory also emerges in some limit. Alternately, one mayimagine that the MEMe model emerges as the result ofintegrating out degrees of freedom in a more fundamentaltheory. However, even under such scenarios, fine-tuningproblems remain: in order to obtain an effective actionable to generate Eq. (1), we were forced to postulate theexistence of some mechanism which suppresses the termsconstructed from the curvature invariants for the met-ric gµν . The need for such a mechanism may perhapsprovide some guidance for constructing a more funda-mental theory. In particular, a more detailed analysis isneeded to determine whether a generalized coupling the-ory can naturally emerge from some effective field theory,or whether one must go beyond the framework of effec-tive field theory to justify the suppression of curvatureterms.

In light of the semiclassical derivation we have pre-

13

TABLE I. Fixed point, stability and associated solutions for the system (84).

Point χ,Ω, L,K Attractor Repeller Scale factor Energy density Jordan scale factor

A 0, 0, 0,−1 Never Never S = S0(t− t0) ρ = 0 SJ = S

B 0, 1, 0, 0 Never Never S = S0(t− t0)2

3(w+1) ρ = ρ0(t−t0)2

SJ =16[(t−t0)2−qρ0w]2

[qρ0(1−3w)+4(t−t0)2]2S

L1 χ0, 0, 1, 0 0 ≤ w ≤ 1 Never S = S0eγ1(t−t0) ρ = 0 SJ = S

L2 13wΩ0

,Ω0, 1 + wΩ0, 0 0 < w ≤ 1 Never S = S0eγ2(t−t0) ρ = ρ0 SJ = 16(1−qρ0w)2

(qρ0(1−3w)+4)2S

χ

Ω

L

L1

O

B

FIG. 1. A representation of the finite phase space of the system (84) in the case of dust w = 0 and q < 0. The representationis constructed reporting on the χ < 0, Ω > 0, L > 0 faces of the octant the invariant submanifolds χ = 0, Ω = 0, L = 0. Theblack line of point corresponds to L1, whereas the red line represents the intersection of the singular plane 3χΩ + 4 = 0, whichis present for w = 0.

14

χΩ

L

L1

O

B

FIG. 2. A representation of the finite phase space of the system (84) in the case of radiation w = 1/3 and q < 0. Therepresentation is constructed reporting on the χ < 0, Ω > 0, L > 0 faces of the octant the invariant submanifolds χ = 0, Ω = 0,L = 0. The black line corresponds to line L1, and, differently form the dust case, there is no singular plane. This phase spaceis qualitatively very similar to the one in Fig. 1.

15

χΩ

L

L1

O

B

FIG. 3. Three different orbits in the case of dust w = 0 and q < 0. The blue and magenta orbits correspond to cosmic historieswith a matter era close to that of GR. The yellow region corresponds to decelerating expansion, while the red surface delineatesthe 3Ωχ = −1 (qρ = −1) part of the phase space. For the sake of clarity here, we did not show the initial point of these orbits,as it is located far in the Ω direction, close to the surface 3Ωχ = −1.

sented, it is also tempting to construct a fundamentallysemiclassical theory (in the sense that gµν is fundamen-tally classical) from the generalized coupling theories wehave studied. A major obstacle to constructing such atheory concerns the fact that the time evolution for quan-tum fields depends on the background spacetime geome-try, which is in turn dependent on the state of the quan-tum field. This interdependence will generically intro-duce nonlinear time evolution in quantum states [41, 42].One way around this issue might be to employ somesort of measurement-feedback scheme for objective col-lapse models, which has been successfully implementedin Newtonian toy models [43], but a complete relativis-tic implementation of this approach is presently lack-ing. Even if a relativistic implementation can be con-structed, one might expect the renormalized parametersin Σ[φ, g··, χ··

··] to be scale dependent, so that the re-sulting energy-momentum tensor Tµν is not unique; thisnonuniqueness forms another obstacle to constructing afundamental theory with our framework. Indeed, any

attempt to construct a fundamental theory within ourframework must supply a mechanism to regulate the di-vergences of quantum field theory. In the absence of sucha mechanism, we take the conservative view that that ourproposed theory is a low energy/coarse grained descrip-tion for a more fundamental (and fully quantum) theory.

The generalized coupling theories we propose sharefeatures with other modified gravity theories. We haveshown explicitly that a convenient framework of analysisis to interpret them as bimetric theories of gravity. Bi-metric theories also modify the gravitational propertiesof standard matter in a manner akin to nonminimallycoupled theories. Nonetheless, our generalized couplingtheories have an advantage over these other frameworksin that the vacuum phenomenology remains essentiallythat of GR. Of course, whether one is truly in a vacuumdepends on the details of the definition of matter in agravitational theory. In GR, the source of the Einsteinequations is modeled as a continuum, and in theories likethe one we have proposed, one has to ask how well this

16

approximation works in a given framework. Considerfor example the issue of a photon traveling between twogalaxies in a cosmological setting. Should one considerit as traveling in vacuum or in a continuum? In classicalcosmology, these problems become almost irrelevant, butthey might be of crucial importance in the generalizedcoupling case.

On a cosmological level, the field equations (36) showclearly the presence of a dynamical cosmological con-stant and a modification of the gravitational responseto the thermodynamical potentials ρ and p for a fluid.The field equations (36) also share a common structurewith many well-studied cosmological models like LoopQuantum Cosmology [44], the Randall-Sundrum type IIbraneworld model [45], and some effective cosmologicalmodels, e.g., Refs. [46, 47]. Whether our theory can berelated to any of those theories and how this can happenremains to be determined.

We have constructed a simple realization of our gener-alized coupling theories, which we call the Minimal Ex-ponential Measure model, given by the action

S[φ, g··, A··] =

∫d4x

[R− 2 Λ

]√−g+ 2κ

(Lm[φ, g··]− λ

κ

)√−g, (90)

where gµν = e(4−A)/2AµαAν

β gαβ [see (26) and (49)],

and Λ = Λ−λ. This choice of gµν is the simplest that onecan make which does not require the choice of a particularform for the action of Aµ

α.The MEMe model has three appealing features: i) it is

equivalent to GR in a vacuum; ii) its (single) additionalparameter λ corresponds to a regularization scale thatcan in principle be independent of the Planck scale; andiii) for negative q (with ρ |q| < 1), the MEMe modelhas inflationary behavior at early times without requiringadditional dynamical degrees of freedom. Furthermore,a dynamical systems analysis indicates that the MEMemodel can qualitatively describe cosmic histories whichinclude an inflationary era, a graceful exit, and a darkenergy era. There are a number of interesting questionsthat stem from these results. One might be concernedthat the duration of the inflationary era is too dependenton the initial conditions. This is true if one thinks aboutinflation purely in terms of de Sitter expansion. Figure3 shows that there is a wide volume of the phase spacewhich corresponds to accelerated inflation; therefore, awide variety of parameter choices and initial conditionsmight lead to an inflationary phase of the right duration.The exact calculation, however, cannot be immediatelymade on the basis of the phase space analysis we havepresented and will be left for future work. One mightalso ask whether, given the complex form of Eq. (81),the MEMe model is also able to predict the establishedsequence of cosmic eras, i.e., the fact that the Universeat early time is dominated by relativistic particles (w =1/3), successively by nonrelativistic matter (w = 0), and

then by spatial curvature. One can show that this is thecase by recalling that q must be small. Expanding in qto first order, Eq. (81) reads

ρ ≈ −3

4Hρ(w + 1)

(3qρ(w + 1)2 + 4

). (91)

Choosing the integration constant so that when a = a0,ρ = ρ0, one obtains

ρd =4ρd,0a

30

4a3 + 3qρd,0 (a3 − a30)

(92)

ρr =3ρr,0a

40

3a4 + 4qρr,0 (a4 − a40)

(93)

for w = 0, 1/3, respectively. Since in our model we con-sidered only ρq < −1, ρd and ρr must always be finite,and we can neglect the singularity in the above expres-sions. For ρq < −1, the behavior of both ρd and ρrfollows closely the behavior of the standard cosmologicalmodel. This indicates that in MEMe cosmologies, thecosmic eras have the same chronological ordering as inthe standard cosmological model.

It should be stressed, at this point, that the interest-ing cosmic histories we have found are only achievablethrough some (additional) fine-tuning. Indeed, we haveto ensure that Λ |λ|, but, as we have argued, this fine-tuning can be made less severe than that of the cosmolog-ical constant problem because, again, λ can in principlebe independent of the Planck scale. A future research tar-get will also be to determine whether the MEMe modelcan fit the observational data available so far. Such atask will entail a full analysis of the cosmological phe-nomenology to determine, for example, the inflationarypower spectra for primordial fluctuations, the cosmic mi-crowave background spectrum etc.

Since one can view the MEMe model (and indeed anygeneralized coupling theory) as a bimetric theory in thepresence of matter, the MEMe model predicts a differentpropagation speed for gravitational waves within matterdistributions. We were able to obtain a constraint onthe parameter q = κ/λ from the timing uncertainty ofgravitational wave detections; however, this constraint isweak compared to the value of q that one expects if newphysics appears at the TeV scale. One can in principleobtain stronger constraints on q by studying gravitationalwaves propagating through dense matter distributions,for instance the gravitational waves from the ringdownof a NS-BH merger propagating through the cloud ofejecta from the disrupted neutron star. Another inter-esting phenomenon predicted by the MEMe model is therefraction of gravitational waves by matter. These willalso be the focus of future studies.

ACKNOWLEDGMENTS

We thank Vitor Cardoso, David Hilditch, MicheleMaggiore, Jose Mimoso, Masato Minamitsuji, Shinji

17

Mukohyama, Adrian del Rio Vega, and Daniele Vernierifor useful discussions. J.C.F. acknowledges supportfrom FCT Grant No. PTDC/MAT-APL/30043/2017,and S.C. acknowledges support from FCT Grant No.UID/FIS/00099/2019 and Grant No. IF/00250/2013.

Appendix A: Variation of the trace

In this Appendix, we examine the variation of the traceA = Aσ

σ. The purpose of this is to address a potentialobjection to the fact that we assume A to be independentof gµν . For instance, one might argue that, since A =Aµν g

µν , A is dependent on gµν so that the variationswill ultimately depend on gµν .

The key point we wish to make here is that index place-ment matters when choosing the variables we vary, andthat this choice determines whether the variations of Adepend on gµν . To see this, consider the variation ofA = Aµν g

µν ,

δA = gµν δAµν +Aµν δgµν . (A1)

If we regard Aµν and gµν to be independent, the aboveexpression suffices. However, if we instead demand thatAµ

ν and gµν are independent variables, then we mustrewrite δAµν in terms of δAµ

ν and δgµν . In particular,we perform the variation of Aµν = gνσ Aµ

σ :

δAµν = gνσ δAµσ +Aµ

σ δgνσ

= gνσ δAµσ −Aµσ gνα gβσ δgαβ

= gνσ δAµσ −Aµβ gνα δgαβ .

(A2)

The second line makes use of δgνσ = −gνα gβσ δgαβ ,which follows from the condition δ(gµσ gνσ) = 0. Plug-ging this result back into (A1) yields

δA = gµν gνσ δAµσ −Aµβ gµν gνα δgαβ +Aµν δg

µν

= gµν gνσ δAµσ −Aµν δgµν +Aµν δg

µν .(A3)

The last two terms cancel, and one obtains the result

δA = δσµ δAµ

σ. (A4)

This demonstrates that A is independent of gµν if wechoose Aµ

ν and gµν to be independent variables.

Appendix B: Divergence-free property

Here, we show using variational methods that thesource Tµν of the Einstein tensor must satisfy ∇µTµν = 0if the field equations are satisfied. We also demon-strate that the field equations also imply gασ∇σTαβ = 0.Though the general proof is standard, and can be foundin textbooks (Refs. [48], for instance), we present it hereto demonstrate that both Tµν and Tαβ must satisfy the

divergence-free property on shell. Consider first an ac-tion of the form, defined on some domain U ,

S[g··, A··, ϕ] = SEH [g··] + SA[g··, A·

·] + Sm[g··, A··, ϕ],

(B1)where SEH is the Einstein-Hilbert action. The variationhas the form

δS =

∫U

d4x√−g

[1

2κ(Gµν − κTµν) δgµν

+Ψ2 |A|q

Wµν δAµ

ν + Ψ2 |A|E[ϕ, g··] δϕ

],

(B2)

where SA and Sm are assumed to have a volume elementof the form d4x

√−g = d4x√−gΨ2|A|, and

Tµν :=−2√−g

δ (SA + Sm)

δgµν,

Wαβ :=

q√−gδS

δAαβ,

E[ϕ, g··] :=1√−g

δS

δϕ.

(B3)

Now, consider a differomorphism generated by a vectorfield wµ which vanishes on the boundary ∂U ,

wµ|∂U = 0. (B4)

The action S, being constructed from covariant quan-tities, is diffeomorphism invariant; under the boundarycondition (B4), a diffeomorphism generated by wµ can-not change the value of the action, so that for an in-finitesimal diffeomorphism of the form

x→ x+ εwµ, (B5)

the first-order variation of the action resulting from thediffeomorphism must satisfy δS = 0. If the infinitesimalparameter ε is constant, the variation of the metric takesthe form:

δεgµν = ε£wg

µν = 2 ε(∇(µwν)

). (B6)

Upon integrating by parts and making use of the Bianchiidentities∇µGµν = 0 and Eq. (B4), the condition δS = 0yields

ε

∫U

d4x√−g wν ∇µTµν = −

∫U

d4x√−g×[

Ψ2 |A|q

Wµν δεAµ

ν + Ψ2 |A|E[ϕ, g··] δεϕ

].

(B7)If the field equations E[ϕ, g··] = 0 and Wα

β = 0 aresatisfied, one has

ε

∫U

d4x√−g wν ∇µTµν = 0, (B8)

and if one demands that δS = 0 for any infinitesimaldiffeomorphism, Eq. (B8) must hold for all wµ, and

18

it follows that ∇µTµν = 0. This demonstrates that∇µTµν = 0 holds if the field equations E[ϕ, g··] = 0 andWα

β = 0 are satisfied.A similar argument may be used to demonstrate that

Tαβ satisfies the divergence-free property on shell. In

particular, one may show that on shell, gασ∇σTαβ = 0.Recalling that Sm = Sm[ϕ, g··] [cf. (28)], the variation ofSm takes the form (29)

δSm =

∫d4x√−g

(E[ϕ, g··] δϕ− 1

2Tαβ δg

αβ

), (B9)

where Tαβ is defined in Eq. (30). Under the diffeomor-phism (B5), the variation in gαβ has the form

δεgαβ = ε£wg

αβ = 2 ε ∇σ(gσ(αwβ)

), (B10)

and upon demanding δSm = 0, one obtains

ε

∫U

d4x√−gwβgασ∇σTαβ = −

∫d4x√−g [E[ϕ, g··]δεϕ] .

(B11)

On shell, E[ϕ, g··] = 0, and upon demanding that δSm =

0 for all wµ, gασ∇σTαβ = 0. Note that this resultdepends only on the diffeomorphism invariance of Sm,and does not require that gµν satisfy the gravitationalfield equations—the argument is valid for any metric gµν .

This demonstrates that the property gασ∇σTαβ = 0 isindependent of the Bianchi identities. We note that both∇µTµν = 0 and gασ∇σTαβ = 0 require that the fieldequations E[ϕ, g··] = 0 are satisfied, and that ∇µTµν = 0depends also on the equation Wα

β = 0. Furthermore,one can derive∇µTµν = 0 without including the Einstein-Hilbert action. Thus, the view that the divergence-freeproperty of the energy-momentum tensor is enforced bythe gravitational field equations is somewhat inaccuratefrom a fundamental perspective. A more accurate viewis that the divergence-free property and local conserva-tion laws follow from the diffeomorphism invariance ofthe action and the field equations.

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