Non-standard gravitational waves imply gravitational slip: on the difficulty of partially hiding new gravitational degrees of freedom
Ignacy Sawicki, Ippocratis D. Saltas, Mariele Motta, Luca Amendola, Martin Kunz
aa r X i v : . [ a s t r o - ph . C O ] A p r Non-standard gravitational waves imply gravitational slip: on the difficulty ofpartially hiding new gravitational degrees of freedom
Ignacy Sawicki,
1, 2
Ippocratis D. Saltas, Mariele Motta, Luca Amendola, and Martin Kunz Départment de Physique Théorique and Center for Astroparticle Physics,Université de Genève, Quai E. Ansermet 24, 1211 Genève, Switzerland Central European Institute for Cosmology and Fundamental Physics,Fyzikální ustáv Akademie věd ČR, Na Slovance 2, 182 21 Praha 8, Czech Republic Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Campo Grande, PT1749-016 Lisboa, Portugal Intitut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
In many generalized models of gravity, perfect fluids in cosmology give rise to gravitational slip.Simultaneously, in very broad classes of such models, the propagation of gravitational waves isaltered. We investigate the extent to which there is a one-to-one relationship between these twoproperties in three classes of models with one extra degree of freedom: scalar (Horndeski andbeyond), vector (Einstein-Aether) and tensor (bimetric).We prove that in bimetric gravity and Einstein-Aether, it is impossible to dynamically hidethe gravitational slip on all scales whenever the propagation of gravitational waves is modified.Horndeski models are much more flexible, but it is nonetheless only possible to hide gravitationalslip dynamically when the action for perturbations is tuned to evolve in time toward a divergentkinetic term. These results provide an explicit, theoretical argument for the interpretation of futureobservations if they disfavoured the presence of gravitational slip.
I. INTRODUCTION
The nature of the late-time acceleration of the Uni-verse remains elusive. Although observations favor Gen-eral Relativity (GR) with a cosmological constant, it isstill far from clear whether the underlying model of grav-ity at large scales does not involve some other, dynamicaldegree of freedom or a genuine modification of GR.The distinction between the various classes of dark-energy models is one of the biggest challenges of futurelarge-scale-structure surveys such as the Euclid satellitemission [1]. It is well known that many models of dark en-ergy exhibit non-vanishing gravitational slip in the pres-ence of perfect-fluid matter sources. Moreover, it wasshown in Refs [2, 3] that the gravitational slip in thebaryon Jordan frame can be expressed solely in termsof quantities observable on the sky, and therefore is it-self a bona fide observable. Fisher matrix forecasts fora Euclid-like survey predict an accuracy of 10% in themeasurement of the quantity η parameterizing the slip,if it is assumed to be scale independent [4].In Ref. [5], we pointed out that models in whichperfect-fluid matter generates gravitational slip at lin-ear order in perturbations on cosmological backgroundsgenerically also has a modified propagation of gravita-tional waves (GWs). Thus a detection of a non-zero slipat late times would in principle imply a genuine modifica-tion of GR. Understanding the connection between thephysics of tensors and scalars has become all the morepertinent since the detection of gravitational waves bythe LIGO collaboration [6]. Depending on the rate ofprogress in cosmological surveys and GW observations,one of these fields should be able to inform on the other.The remaining question is to what extent this relation-ship is guaranteed i.e. whether the vanishing of gravita-tional slip always implies that gravitational waves prop- agate in a standard manner, or whether it is possible tohide the slip and dynamically maintain this under evolu-tion in time . In this article, we refer to the ability of themodel to arrange its degrees of freedom at the linear levelsuch that no gravitational slip is sourced by perfect-fluidmatter as the dynamical shielding of the gravitationalslip. We investigate three very broad classes of mod-ified gravity models for the existence of this property,(i) Horndeski scalar-tensor theories [7, 8], (ii) Einstein-Aether vector-tensor models [9] and (iii) bimetric theoriesof massive gravity [10].For successful dynamical shielding of slip, we requiretwo features:1. We first ask if it is at all possible to maintain a no-slip configuration in time, while GW propagation ismodified. We prove that, in bimetric theories, lin-ear gravitational slip cannot be shielded at all scalesstably in time. On the other hand, in Horndeskiand Einstein-Aether models, particular parameterchoices do exists for which the shielding of gravita-tional slip is dynamically maintained. These mod-els correspond to models where the sound speed ofthe scalar degree of freedom vanishes and as suchthey are rather singular and peculiar limits of thesetheories, where the new fields do not propagatewave-like modes.2. We then ask whether, for these particular choicesof parameters, a no-slip configuration is actuallyan attractor of the dynamics, i.e. whether a genericinitial conditions for perturbations will evolve toit. We find that in Einstein-Aether any slip presentinitially is maintained for all time. It is only in par-ticular (and singular) Horndeski models that theinitial slip decays away and then this no-slip con-dition can be maintained in time.This means that achieving the dynamical shielding ofslip requires a scalar-tensor theory and a severe tuning ofits parameters to a regime where the parameters of theaction for perturbations diverge. Given this, if futureobservations conclude that gravitational slip is stronglydisfavored, then it would be unlikely that gravity could bemodified, in the sense of a modification of GW dynamics.We also investigate the beyond-Horndeski models in-troduced in Refs [11–13] and show that they are an ex-ception to our original assertion in Ref. [5]: they seemto be the unique class of models where gravitational slipis generated from perfect-fluid matter without any mod-ification in tensor propagation. The conclusion of thispaper are summarized in the table below: η = 1 ⇒ mod. GW or BH Section IIIstan. GW ⇒ η = 1 or BH Section III η = 1 ⇒ stan. GW or tuned Horndeski Section IVmod. GW ⇒ η = 1 or tuned Horndeski Section IVTABLE I. Summary of conclusions of this paper. Stan. —standard, mod. — modified, BH — beyond Horndeski opera-tors active. See relevant sections for details and assumptions. We structure the paper as follows: In section II, wepresent the fundamental assumptions and notation weuse in the rest of the paper. In section III, we introducethe three classes of models on which we focus, Horn-deski, Einstein-Aether and bimetric gravity and discusshow gravitational slip and modified tensor propagationemerge, reviewing the results of Ref. [5]. In section IV, wedescribe how to ensure that the shielding of gravitationalslip be preserved under evolution in time and investigatethe requirements on the model space this imposes. Wedraw our conclusions and discuss the implications of ouranalysis in section V.
II. ASSUMPTIONS AND DEFINITIONS
In this section, we lay down the main physical andmathematical assumptions we use throughout this paper.We assume that the Universe is well described by smalllinear perturbations living on a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric. We take asthe line element for the metric on which matter and lightpropagate in the Newtonian gauge:d s = − (1 + 2Ψ)d t + a ( t )(1 − δ ij + h ij ] d x i d x j , where Φ( t, x ) and Ψ( t, x ) are the scalar gravitational po-tentials in Newtonian gauge and h ij ( t, x ) is the trans-verse traceless spatial metric (tensor) perturbation, i.e.the gravitational wave of the Jordan-frame metric. Therequirement of linearity requires that the scalar and ten-sor perturbations and their gradients all be small, | Φ | , | Ψ | , | h ij | ≪ . (1) From here on, we will express all the linear perturba-tion variables in momentum space with wavenumber k .Overdots and primes denote derivatives with respect tocosmic time and conformal time respectively, unless ex-plicitly stated otherwise. H is the Hubble parameter and H is the conformal Hubble parameter. We assume thatthe matter sector behave as dust, neglecting the contribu-tions from the neutrinos and radiation. This assumptionis good enough for the late universe where the relativisticcomponents are extremely subdominant.The gravitational slip is defined as a difference betweenthe two scalar potentials Φ and Ψ and one typically de-fines η ≡ ΦΨ , (2)where in this equation the fields on the right-hand sideare understood to be the root-mean-square values relatedto observables on the sky and therefore positive defi-nite. In ΛCDM cosmology, there is no gravitational slip, η = 1, with small corrections appearing from neutrinofree-streaming. At second order in perturbations, grav-itational slip also always appears even when the matterconsists completely of dust [14], but in the late universeshould be smaller than | η − | . − [14, 15].The gravitational slip is generated through theanisotropy constraint, i.e. the traceless part of the ( ij )linearized Einstein equations. In modified-gravity mod-els in Newtonian gauge, it takes the schematic form C ≡ Ψ( t, k ) − Φ( t, k ) = σ ( t )Π( t, k ) + π m , (3)with Π( t, k ) a functional of the linear perturbation vari-ables of the model as well as potentially some backgroundquantities; σ ( t ) on the other hand is a background func-tion only; π m is the amplitude of the scalar anisotropicstress of the matter components, which for the purposeof the proof in this paper we neglect.Notice that, to avoid ambiguity, we make a distinctionbetween anisotropic stress and gravitational slip. We re-fer to the former as a property of matter and the latter asa property of the geometry. Thus the energy-momentumtensor of a fluid which exhibits anisotropic stress sourcesgravitational slip on the geometrical part of the linearizedEinstein’s equations. In modifications of gravity, grav-itational slip can appear at linear order even withoutanisotropic stress.In section III, we connect the variables σ and Π to theparticular models we will be interested in. For the mo-ment, let us note that modified-gravity models do featurean O (1) correction to the slip parameter at linear orderin perturbations, on at least some scales.On the other hand, it is well known that the value of η can be modified by a change of frame, e.g. a confor-mal rescaling of the metric, making its value seeminglyambiguous.In Refs [2, 3], it was shown that comparing the evo-lution of redshift-space distortions of the galaxy powerspectrum with weak lensing tomography allows us to re-construct η as a function of time and scale in a model-independent manner. Such an operational definition re-moves the frame ambiguity since the measurement picksout the particular metric on the geodesics of which thegalaxies and light move. It is the gravitational slip inthat metric that is being measured by such cosmologicalprobes. With the ambiguity of frame removed, the grav-itational slip is a bona fide observable, rather than just aphenomenological parameter. Fixing the metric also de-termines what is considered a gravitational wave: we callthese the propagating spin-2 perturbations of the metricon which baryonic matter moves.Dynamical models of late-time acceleration can featureinteractions between the new degree of freedom and thematter metric. As we have shown in Ref. [5], these inter-actions lead to modifications in the propagation of tensormodes (gravitational waves). Depending on the model,on the FLRW background, the speed of tensor modes ( c T )can be altered, the effective Planck mass M ∗ can evolvein time or a graviton mass µ can appear, while the pres-ence of a possible second tensor field sources the evolutionthrough the term Γ γ ij , giving for the tensor equation ofmotion h ′′ ij + (2 + ν ) H h ′ ij + c k h ij + a µ h ij = a Γ γ ij . (4)The deviations away from the standard behavior are con-tained in ν , c T and µ , with all these quantities definedin the Jordan frame of the matter and in principle freeto be functions of time. As we will review in section III,for any of these modifications to be present, σ in Eq. (3)must also be non-zero. Thus gravitational slip sourcedfrom perfect-fluid matter is a sign that gravitational-wavepropagation is modified.The question we aim to answer is whether it is possi-ble to find model parameters where the degrees of free-dom arrange themselves in such a way that the grav-itational slip be hidden. We refer to this as the dy-namical shielding of the slip. We purposefully do notuse the more usual term screening , which we restrict tomean the hiding of modifications of gravity (includingthe slip) through non-linear effects in the dynamics (e.g.chameleon [16] or Vainshtein screening [17]). Screeningimplies that the scalar adopts a non-linear profile andtherefore is a change of the background solution. As such,non-linear screening would also result in suppressing themodification to GW propagation. III. GRAVITATIONAL SLIP IMPLIESMODIFIED TENSORS
In Ref. [5], we showed that for very general classes ofmodified gravity theories featuring one extra degree offreedom, whenever perfect-fluid matter sources gravita-tional slip, the propagation of tensor modes is also mod-ified. In this section, we shall briefly introduce the mainproperties of the three classes of models and review the appearance of gravitational slip and modifications of thepropagation of GWs. We will focus on general classes ofmodels which introduce one new scalar, vector or tensordegree of freedom, respectively.
A. General modifications with an extra scalar: TheHorndeski scalar-tensor model and beyond
The Horndeski class of models encompasses all theo-ries which can be constructed from the metric g µν and asingle scalar field φ and which have equations of motionwith no more than second derivatives. It includes themajority of the popular models of late-time accelerationsuch as quintessence, perfect fluids, Brans-Dicke grav-ity, f ( R ) gravity, f ( G ) gravity, kinetic gravity braidingand galileons. We refer to the scalar as the dark en-ergy (DE). The Horndeski Lagrangian is defined as thesum of four terms L to L that are fully specified by anon-canonical kinetic term K ( φ, X ) and three, in prin-ciple arbitrary, coupling functions G , , ( φ, X ), where X = − g µν φ ,µ φ ,ν / m0 , andfour dimensionless, independent and arbitrary functionsof time only, α K , α B , α M and α T , which mix the four freefunctions K and G i present in the action. As we reviewbelow, the parameters α M and α T both affect the propa-gation of gravitational waves and control the appearanceof gravitational slip. The braiding α B controls whetherdark energy clusters at all, while the kineticity α K in turncontrols at what scales this happens and acts to suppressthe sound speed of the scalar modes.In Ref. [18], it was shown that the anisotropy con-straint in the notation of Eq. (3) is σ = α M − α T , (5)Π = α T σ Φ + Hv X , where v X ≡ − δφ/ ˙ φ is a perturbation of the scalar field.On the other hand, the propagation of GWs is modifiedby terms ν = α M , c = 1 + α T , (6) µ = 0 , Γ = 0 , in the notation of Eq. (4).It is clear from Eq. (5) that when both α M = α T = 0,Φ = Ψ. However, this choice of the parameters alsoswitches off all the modifications in Eq. (4). In the con-text of scalar-tensor models, a detection of gravitationalslip sourced by perfect-fluid matter therefore is direct ev-idence that one or both of the parameters α T and α M aredifferent from their concordance values of zero and thatgravity is modified. Beyond Horndeski
Recently, an extension to Horndeski models was dis-covered [11–13]. It contains higher derivatives in the Ein-stein equations, but which all cancel in the equations ofmotion for the real dynamical modes. Thus the theorypropagates no more degrees of freedom than Horndeski.It is currently understood that theories containing onlythe beyond-Horndeski terms are ghost-free. When otherscalar Lagrangian terms are included, the theory contin-ues to propagate just the one scalar degree of freedom,and thus can be ghost-free, provided that the total La-grangian is related to a Horndeski one through a dis-formal transformation. When this is not the case, thesecond degree of freedom reappears (see Refs [22, 23] fordetails). However, since we have fixed the frame by re-quiring that it be the Jordan frame of these baryons,these models do represent new phenomenology.Linear fluctuations in these beyond-Horndeski modelscan be brought into the Horndeski form through a disfor-mal transformation of the metric, but only at the price ofintroducing a non-minimal coupling in the matter sector.Thus, in the Jordan frame of the matter and light, theydo introduce new physics parameterized by a new vari-able α H . However, α H does not enter the GW Eq. (4),implying that the propagation of tensors is not modified.On the other hand, the anisotropy equation (3) doeshave new contributions, σ = α M − α T , (7)Π = α T σ Φ − α H σ (Ψ + ˙ v X ) + Hv X . The α H parameterizes a new type of mixing betweenthe scalar and the metric. In terms of the effective-field-theory notation of Ref. [13, 21], in beyond-Horndeskimodels a α H δN δR appears in the quadratic effective ac-tion instead of α B δN δK which appears in the case of braiding . δN is the fluctuation of the lapse, δK — of theextrinsic curvature and δR — of the intrinsic curvatureof the spatial slice, all in unitary gauge.Since both these terms are made up of products ofscalars, neither of them can contribute to the gravitonequation of motion. On the other hand, δR contains sec-ond spatial derivatives of a scalar curvature fluctuation, ∂ ζ , and therefore it does contribute to the anisotropyconstraint.Thus beyond-Horndeski models are counter-examplesto our assertion in Ref. [5]: in this class of models, gravi-tational slip does not imply that the propagation of grav-itational waves be modified. Thus for a dust-filled uni-verse, standard GWs together with a detection of slip would imply that a beyond-Horndeski modification bepresent. B. General modifications with an extra vector: theEinstein-Aether vector-tensor model
The Einstein-Aether (EA) model introduces a new,spin-1 propagating degree of freedom, denoted as u µ .The vector field spontaneously breaks Lorentz symmetrydue to its non-trivial vacuum expectation value (v.e.v),introducing a preferred direction. The theory spaceof such models is strongly constrained in the Solar-System [24] and even more stringently by observation ofbinary pulsars [25, 26]. It should be noted that this typeof models does not provide a mechanism for accelerationand therefore requires a cosmological constant. Below,unless otherwise stated, we will use the equations andnotation of Ref. [27].The dynamics of the vector field are specified throughthe following Lagrangian: L v = − β ∇ µ u ν ∇ µ u ν − β ( ∇ µ u µ ) −− β ∇ µ u ν ∇ ν u µ + λ ( u α u α + m ) , (8)where the β i ’s are free, constant parameters, λ is a La-grange multiplier which enforces the non-trivial v.e.v of u µ in the action, and m is the value of the vector’s v.e.v.Using a kinematic decomposition of the velocity gradientsin (8) one can see that the coefficient of the shear of thevector field u µ is − ( β + β ). The extra dynamical degree of freedom at the linearlevel in perturbations is the fluctuation of the spatialcomponents v i of the vector u µ , assuming a decomposi-tion of the latter as u µ = ¯ u µ + v µ , where ¯ u µ denotes thebackground field. Conveniently defining the divergenceof the vector’s spatial perturbation as Θ ≡ (cid:16) am (cid:17) ∂ i v i , (9)in Ref. [27], it was shown that the anisotropy constraintEq. (3) becomes Π = (cid:0) a Θ (cid:1) ′ a k , σ = 16 πGm ( β + β ) , (10)with the prime denoting a derivative with respect to con-formal time. On the other hand, the parameters of thetensor equation (4) are ν = 0 , c = (1 − β − β ) − (11) µ = 0 , Γ = 0 . Note that, in full generality, in the Lagrangian (8) one shouldalso include the term ∼ ∇ α u β ∇ β u α , which in a kinematicaldecomposition will give rise to the twist. However, this termvanishes on hypersurface-orthogonal settings [28], such as thecosmological background. It would thus not contribute in linearperturbation theory in cosmology and we neglect it. Notice that, the modification of c T appears through thecoefficient β + β . This corresponds to the link betweengravitational slip and the propagation of tensors for anEA theory, in the presence of perfect-fluid matter. C. General modifications with an extra tensor:bimetric massive gravity model
The Hassan-Rosen bimetric theory [10] is a naturalextension of de Rham-Gabadadze-Tolley massive grav-ity [29, 30] where the fixed reference metric is promotedto a dynamical one. Despite the kinetic term for thereference metric, the theory remains ghost-free and hasthe advantage of allowing consistent flat FLRW solutions.For a recent review on the subject we refer the reader toRefs [31, 32].The model contains two dynamical metric tensors, g µν , f µν , and matter must be minimally coupled to onlyone of the metrics in order to guarantee the absence ofa ghost degree of freedom. We choose g µν to be thismetric. The kinetic terms for each of the metrics arestandard Einstein-Hilbert and the action is augmentedby an interaction potential U of the very specific form, S U = M Z d x √− g X n =0 β n e n (12)where e n are symmetric polynomials built out of thequantity X µν ≡ √ g µα f αν . The five β n are the only freeparameters of this model, defined here to be dimension-ful, absorbing the mass scale of the potential. Since mat-ter is coupled minimally to g , the geodesics are going tobe those of g and therefore the observable potentials andgravitational waves are going to be those of g .In Ref. [34] it was shown that when both the back-ground metrics are flat FLRW, the anisotropy constraintassociated with the metric g , takes the form Π = ∆
E , σ = a r ¯ Z (13)in the notation of Eq. (3). ∆ E is a gauge invariant vari-able, defined as ∆ E ≡ E g − E f , where E g,f is the scalarcoming from the transverse spatial perturbation of themetric; r, ¯ Z are background functions of time only, de-fined in section IV C .On the other hand, the equation of motion for gravi-tational waves of the g metric in notation of (4) is: ν = 0 , c = 1 , (14) µ = r ¯ Z ,
Γ = − r ¯ Z . It is in principle possible to couple to one other particular com-bination of metrics which will cause a ghost to appear above thestrong-coupling scale of the theory [33]. For a similar analysis see also Ref. [35]. In the notation of Ref. [34], r ¯ Z = f . Thus, in these models, if perfect-fluid matter sourcesgravitational slip, r ¯ Z = 0 and the propagation of GWsis also modified. Note that the source γ ij in this modelare the tensor fluctuations of the metric f µν around itsFLRW background. Thus the two tensors are coupled,which turns out to lead to an instability in the radiation-dominated universe [36]. IV. MODIFIED TENSORS IMPLYGRAVITATIONAL SLIP
Having reviewed our previous results, we now turn tothe converse question, which will be the main focus ofthe current work:
If the propagation of GWs is modified in cosmology, doesthis mean that the scalar sector must produce gravita-tional slip for perfect-fluid sources? Or do there existchoices of model parameters which allow for configura-tions of the degrees of freedom where the gravitationalslip is dynamically shielded at all scales and where thisshielding is stable under evolution in time?
Let us start by introducing the mathematical methodwe will be using. We study the scalar perturbations onan FLRW background in a universe containing dust anda modification of gravity. We assume that we are notat some future de-Sitter attractor of the expanding Uni-verse, where the matter has already diluted away. Thisis relevant, since such de Sitter configurations asymptoti-cally lose the matter degrees of freedom and the evolutionof parameters ceases. The dynamics become much sim-pler since there no matter to collapse and source slip.No-slip conditions can then be satisfied, [37]. They donot represent the observable universe, however.First, we eliminate spurious degrees of freedom andauxiliary variables, i.e. fix any gauge freedom and solveall the constraints in the Einstein equations so that onlythe real and independent dynamical variables remain. Inall of the cases considered here, there are two independentscalar degrees of freedom, described by two second-orderequations of motion: one corresponding to the dust, theother — to the new scalar in the dark-energy model,either introduced explicitly, or arising from the helicity-0polarization of the new massive vector or tensor.We can of course choose an arbitrary basis for thephase space variables, which together with their corre-sponding velocities form the phase-space vector, X ( t ; k ) = { ψ , ˙ ψ , ψ , ˙ ψ } , ψ i = ψ i ( t ; k ) . (15) X ( t ; k ) fully characterizes the configuration of the scalarperturbations of the universe at any one time t . Given theanisotropy equation (3), we now require that the initialconfiguration at time t be chosen in such a way thatthere be no gravitational slip, i.e. that the independentdegrees of freedom are arranged to satisfy C [ X ] = A ( t ; k ) · X ( t ; k ) = 0 , (16)where the A i are the compact notation for the coeffi-cients of the coordinates ψ i , ˙ ψ i . The A i are functions ofthe model parameters, and in principle they are all func-tions of time and scale. Their exact form is determinedby the model. The product A ( t ; k ) · X ( t ; k ) should beunderstood as a usual matrix product between two statevectors: one in model space ( A i ), the other in phasespace ( X i ). For the concrete cases described here, C isthe no-slip constraint (3).In view of Eq. (16), the condition that the gravitationalslip vanishes is therefore a 3-dimensional subspace C = 0 with orthogonal vector A and passing though the origin X = 0 in the 4-dimensional space of the possible config-urations. The origin is a point in the phase space whereno perturbations are present, i.e. the exact backgroundcosmology.It is of course always possible to tune the initial condi-tions to satisfy C = 0 . However, the equations of motionfor the system generate an evolution of this configurationin the full phase space, and the question is whether it ispossible to find a model where that evolution would berestricted to the constrained space C = 0 . For this it isnecessary that all time derivatives of C , when evaluatedon the equations of motion, also vanish (i.e. time deriva-tives of constraints do not generate new constraints).Taking time derivatives of C and evaluating them onequations of motion generates new constraints d n C d t n = A n · X ≈ , (17)where we use ≈ to signify equality on the equations ofmotion. Each of the constraints is a three-dimensionalhyperplane passing through the origin with the associ-ated orthogonal vector A n . Since the phase space is four-dimensional, there can at most be four such constraints, C, ˙ C, ¨ C, ... C with their four associated vectors which spanthe phase space.If the four vectors A n are all linearly independent, thenthe only intersection of all the hyperplanes is the origin,i.e. the only configuration for which the vanishing slipis maintained under time evolution is the configurationwith no fluctuations. This is not of interest, but is thegeneric case.Nonetheless, one might be able to tune the parame-ters of the models in such a way that some of the A n become linearly dependent and thus no longer span thewhole phase space. The space of configurations for whichthe no-slip condition is maintained is then the comple-ment of the space spanned by the A n . The fewer linearlyindependent A n there are, the less tuned must be theconfiguration.We will limit our discussion to such cases where atleast ¨ C is linearly dependent on C, ˙ C . The remainingcase, where only ... C is linearly dependent, has only onefield-coordinate free, i.e. would require initial conditionswhich are too tuned to be of interest.The discussion above can be trivially extended to mod-els with more degrees of freedom. Whether the same con- clusions would hold, however, is an open question that wewill not address here. A. Horndeski Scalar-Tensor Model
In this section, we use coordinate time t with its deriva-tive denoted by an overdot and ln a with the derivativedenoted by a prime.For the Horndeski scalar-tensor/dust system, we usethe formulation of the perturbation dynamics describedin Ref. [18] where it was shown that linear perturba-tion theory in an arbitrary Horndeski model can be fullydescribed by specifying a background expansion history H ( t ) , four functions of time defining the perturbations, α K,B,M,T ( t ) and the initial matter density, which thenevolves according to the matter conservation equation.At least one of α M,T must be non-zero in order for thepropagation of gravitational waves to be modified, seeEq. (5). All these functions are in principle arbitrary,corresponding to different model choices and backgroundinitial conditions, and are complemented by algebraicconstraints arising from the requirement of backgroundstability. In particular, we have the requirement thatthe scalar be dynamical and not be a ghost, D ≡ α K + 32 α > , α B = 2 , (18)and that there be no gradient instabilities in the scalarsector (positive sound speed for scalar perturbations) Dc =(2 − α B ) (cid:18) α B α T ) + ( α M − α T ) − ˙ HH (cid:19) −− ˙ α B H + ˜ ρ m H > . (19)Since the effective Planck mass M ∗ evolves in Horndeskimodels, it is helpful to define a non-conserved matterdensity ˜ ρ m ≡ ρ m /M ∗ , which evolves according to, ˙˜ ρ m + 3 H ˜ ρ m = − α M H ˜ ρ m . (20)It is most transparent to pick as the basis for the phasespace the vector, X = { Φ , ˙Φ , Ψ , ˙Ψ } , (21)which is possible provided α M = α T (we return to excep-tions at the end of this section). The no-slip condition(3) is then trivial, C = Ψ − Φ = 0 . (22) It is possible to construct tuned functions
K, G i to give arbitrary α i , Whether such models are reasonable or typical is a separatediscussion, see Ref. [38]. Since this constraint does not involve any derivatives, ˙ C is an independent hypersurface, ˙ C = ˙Ψ − ˙Φ = 0 , (23)and immediately we have that the space of initial condi-tions satisfying C = ˙ C = 0 is at most two dimensional.This case is realized when ¨ C is a linear combination of C and ˙ C on shell, i.e. ¨ C ≈ γ C + γ ˙ C , where γ , aresome arbitrary functions of time. Let us try to find sucha model. ¨ C = ¨Φ − ¨Ψ and therefore both the trace Einstein equa-tion and the equation of motion for the scalar must beused to eliminate the second derivatives. On shell, wehave ¨ C − γ ˙ C − γ C ≈ A ˙Φ + A Φ + A k a Φ = 0 , (24)where the A i are functions of the background propertiesand therefore of time, but not of scale.Rather than using as a basis the four α i functions, itis helpful to redefine them to the set ( R, S, D, α T ) , with R ≡ α M α T , S ≡ α B R − . (25)and D defined in Eq. (18). This is allowed provided α T =0 . The three equations A i = 0 can then be written fullyas S ′ = R ′ − R S (1 + α T ) + 5 + 3Ω P ) + (26) + R + 32 ( S (1 + Ω P ) + Ω m ) ,D ′ = 2 DR ′ S + 6( R − RS − D ( R (1 + α T ) − P )) −− α T ) RS + D (3Ω m − R (5 + 3Ω P ) + 2 R ) S ,R ′′ = R ′ (cid:18) P − (3 + α T ) R −− RS ( R − (1 + α T ) S − D (cid:19) ++ 3 R S (5 − R + 3Ω P )( R − − (1 + α T ) S ) D ++ 12 R (cid:16) ( R (1 + α T ) − − P )(5 − R + 3Ω P )+ 3Ω ′P (cid:17) . where poles at D = 0 and S = 0 motivate the choiceof the basis (25). We have redefined the DE pressure P to the dimensionless variable Ω P ≡ P / H . In derivingthis system, we have already eliminated the DE energydensity using the first Friedmann equation, exchangingit for the matter density fraction Ω m ≡ ˜ ρ m / H , whilewe solve for the derivatives of H using the accelerationequation, H ′ + (3 + Ω P ) H = 0 . (27) The only remaining independent equation is the matter(non-)conservation equation (20), which in terms of thematter density fraction is Ω ′ m = Ω m (3Ω P − Rα T ) . (28)Finally, choosing ln a as the time variable (and ′ to de-note the derivative), allowed us to eliminate the explicitdependence on H .Absent any statements regarding the naturalness ofmodels, the dynamics of Horndeski models must satisfythe matter conservation equation for Ω m , Eq. (28), whilethe only condition on the other variables is that the sta-bility conditions for the scalar and tensor perturbationsbe satisfied. There always exists some, possibly unnatu-ral, Lagrangian and initial conditions for the scalar fieldwhich would give any desired evolution for the other vari-ables, Ω P , α T , D, R, S . Enforcing dynamical screeningin Horndeski requires that the dynamics of the modelparameters be restricted to those satisfying three moreequations, the system (26). We thus have four equationsfor six variables, and a solution could always in principlebe found. Without placing any other requirements (suchas no tuning or a choice of a particular background),we can find a Horndeski model which would screen thegravitational slip, while at the same time modifying thepropagation of gravitational waves. Nonetheless, the question remains whether these mod-els can in some sense be generic or reasonable. For atuning that is not too stringent, model parameters shouldnot evolve with timescales too distinct from cosmologi-cal, i.e. they should be constant or possibly power lawsin the scale factor a , at least over a short-enough timeperiod such as the matter domination era.On the other hand, the equations (26), representingtrajectories which the model parameters should follow toscreen slip, are highly non-linear in the variables D, R, S and therefore the complete phase space contains manytimescales for their evolution. So if the evolution of amodel were to follow a generic trajectory of the system(26), its parameters would have to evolve at a much dif-ferent rate than H . We would then say that this modelis very tuned. Equivalently, if we start a slowly-evolving“natural” model with a generic initial choice of D, R, S ,the values of these parameters required to continue toscreen slip would rapidly move away from those possessedby the model and the screening would cease.Only in the vicinity of the fixed points (FPs) of thephase space of (26) can the evolution of the trajectories Remember that Rα T = α M , so that matter conservation is re-lated to the running of the effective Planck mass. More usually, the freedom of Horndeski models is described as five functions (background and four α functions) and the matterfraction today Ω m0 (e.g. [39]). This statement makes the implicitassumption of the matter conservation equation (28), which wehave chosen to enumerate here explicitly. be slow. Exactly at the FPs, the parameters are con-stant, while close enough to the FPs, the approximateequations are linear and the solutions can be polynomialsof a . Thus, close to the FPs, for some appropriate linearcombination of the model parameters v , the screeningtrajectory requires, v ( a ) = v ∗ + v a n , (29)which is slow enough by our definition and an appro-priate untuned model can be found. Ideally, we wouldfind attractor FPs, with n < , since this would allowfor an indefinite duration of screening for this choice ofmodel parameters. However, even unstable FPs allow fora slow evolution of the model parameters for some shortenough period of time, with the duration dependent onhow closely the initial conditions would lie to the FP. Ouraim is to find and categorize these fixed points, identify-ing the reasonable model parameters required to screenslip.The system (26) contains two external parameterswhich are arbitrary: Ω P and α T . We wish to concentrateon the time scales internal to the D, R, S system (26), sowe choose to remove those to do with the external pa-rameters. For the analysis, we assume a constant α T anda pressure which is such that the matter fraction is con-served, compensating for the evolving Planck mass witha modification of the background, Ω P = Rα T / . The lat-ter choice makes Ω m a constant external parameter. Onthe other hand, this particular choice for Ω P does meanthat the expansion history (27) does not precisely corre-spond to matter domination. Choosing a small enough α T would bring us back close enough to satisfy any ob-servable constraints. The coordinates and nature of thefixed points we find will be functions of the external pa-rameters α T and Ω m . We could then reintroduce a slowevolution of these parameters and would find that thefixed points would move adiabatically without changingtheir nature, until an FP stability criterion were violated.In fact, we choose to present the case Ω m = 1 , sincechanging this value does not introduce any qualitativelynew information.Thus the system we actually study is the three equa-tions (26) with the constant external parameter α T , Ω m = 1 and the pressure Ω P = 3 Rα T . We will referto it as the MD system .The MD system has three fixed points with coordinates ( D, S, R, R ′ ) :FP1: ( D, − , , , (30)FP2: (cid:18) − − α T ) α T (4 + 3 α T )8(1 + α T )(1 + 2 α T ) , − α T )4(1 + 2 α T ) , − α T , (cid:19) , FP3: (cid:18) α T (4 + 3 α T ))2(1 + 2 α T ) , − α T − α T α T , − α T , (cid:19) . Note that it will turn out that R = 0 will be the coordinate ofone of the fixed points, FP1, and therefore the exact matter-domination case is included in our analysis. FP1 is in fact a line independent of the value of D andis also the only fixed point of a system with true mat-ter domination, Ω P = 0 ; the sound speed is positive forsubluminal GWs, − < α T < . Allowing for an evolv-ing Planck mass with a scaling solution has added FP2and FP3. However, FP3 is always either a ghost or hasnegative sound speed, while FP2 is healthy only in therange . < α T < . , which is outside of that impliedby the LIGO measurements [40]. Allowing for Ω m < increases the upper bound for stability at FP1 to − Ω m ,so that GWs are allowed to be superluminal when a non-vanishing density fraction of DE is present. It also in-creases the allowed range of α T for FP2.Linearizing the system around the fixed points showsthat none of them are fully stable: for α T > − , FP1 hasno stable directions, FP2 has two unstable directions for α T < and 3 otherwise, and FP3 — one for − < α T < and two otherwise.In summary, only FP1 is of potential interest, rep-resenting a model with α M = α B = 0 , α K > and amodification of tensor speed. It is however an unstableFP, thus it will offer an approximate solution only for alimited time. Given sufficient tuning of the model’s ac-tion so that the initial model parameters are sufficientlyclose to FP1, this period could of course be made as longas is required, to cover all of matter domination and pos-sibly could be connect to an acceleration era, but we setout to avoid such a tuning and will not consider it further.A general trajectory will rapidly move away from anyof these fixed points toward infinity. As we detail be-low, we have studied the full non-linear dynamics nu-merically for some choices of parameters, confirming thatnone of these fixed points are end points of evolution forthe required model-space trajectories. We find that for − < α T < , essentially all the trajectories flow toward D → ±∞ while S and R converge to fixed values givenby S ∗ ≡ − α T )4(1 + 2 α T ) , R ∗ ≡ − α T , (31)In the limit | D | → ∞ , when linearized around S = S ∗ , R = R ∗ , R ′ = 0 , the MD system reduces to D ′ = 5(1 + α T )2 − α T D (32) δS ′ = − α T )2 − α T δS + 8 + α T (40 + 3 α T )8(1 + 2 α T ) δR + δR ′ δR ′′ = − (16 − α T )2(2 − α T ) δR ′ + 5 + 3 α T − α T ) δR where δS ≡ S − S ∗ and δR ≡ R − R ∗ . In the ( S, R, R ′ ) subspace, the linearized system has eigenvalues − − α T , − , − − α T (33)which are all negative if − < α T < . This confirmsthe behavior we have observed in the numerical study.At S ∗ , R ∗ and | D | → ∞ , the product Dc > providedthat α T > . Thus on trajectories with D > (which ispossible, depending on initial conditions), the model pro-vides a background which is stable to perturbations whengravitational waves are superluminal. Note, however,that this asymptotic configuration does have not a precisematter dominated background, since Ω P = R ∗ α T / . As R ∗ is finite, the background could nonetheless be chosento be close enough to observational constraints by pickinga sufficiently small α T .Thus we have in principle found a limiting “attrac-tor” Horndeski model parameters which would have tobe approached during matter domination if the no-slipcondition were to be preserved at all scales: α K → ∞ , < α T < , (34) α B = 2 − − α T + 3(2 − α T )2(1 + 2 α T ) , α M = 5 α T − α T . Moreover, when this limiting model is taken, the evolu-tion equation for the gravitational slip becomes ¨Σ + 2(4 + α T )2 − α T H ˙Σ + 5(3 + 4 α T + α )(2 − α T ) H Σ = 0 , (35)with Σ ≡ Ψ − Φ , while Ψ → const and therefore, for < α T < , it does decay away over time irrespective ofthe initial conditions. We can thus satisfy both our re-quirements for dynamical shielding during matter domi-nation.Given this understanding, we could now take step backand take the D → ∞ limit of the original system (26) andmodify the pressure to include a cosmological constant-like component, Ω P = Rα T / − (1 − Ω m ) . When thisis done, trajectories evolve from the matter-dominationFP to a future acceleration attractor with some new butfinite values of R and S .However, we need to stress that D → ∞ is a very pecu-liar limit, in which the kinetic energy of the fluctuationsbecomes infinite and therefore the scalar field essentiallyceases to be dynamical. If the action is considered fora rescaled field absorbing this divergence, π ≡ √ Dδφ/ ˙ φ ,one can see that the sound speed vanishes and all theinteractions vanish. This is a limit in which the quantumfluctuations diverge (since they are normalized as / √ c s )and thus the classical theory should not be taken at facevalue. Our opinion is that this is a very pathological sit-uation and it is not worth studying further, despite thefact that the naive linear perturbation theory behaves inthe way we asked. Numerical analysis
To understand the non-linear dynamics of the system,we have carried out a numerical study of the MD sys-tem of equations. Although a fully analytical statementis always desirable, our numerical study confirms thedynamics as we have described them above. For the initial conditions (at t = t min ) we constructed a gridspanned by D = I + , S = I + ∩ I − , R = I + ∩ I − ,with I − = {− − , − − δn , . . . , − n max } , I + = { − , − δn , . . . , n max } with the exponent’s stepset to δn = 0 . and n max = 1 . R ′ was fixed at the value R ′ = 10 − . The numerical simulation was carried forthree values of α T : α T = {− . , . , } , (36)while the integration time was set between t min = log (10 − ) and t max = 0 . For each value of α T , we cal-culated N = 25992 trajectories by employing an implicitEuler method with a fixed step of / .We focus on the case α T = 0 . , it since lies in therange for which the fixed points (30) have the most at-tractive directions. In . of trajectories calculated, D diverged (with half towards + ∞ ), while R → . ( R ′ → ) and S → . as t → t max . These are the limitspredicted in Eqs (IV A). The remaining . of trajecto-ries evolved toward D → , but a closer inspection of theevolution shows that the numerics suffer from instabilityin the vicinity of the D = 0 pole of the MD system andwe conclude that these are not proper solutions.For the other two values of α T , lying either side ofthe interval with maximum stable directions around fixedpoints, we find that of trajectories evolve towarda diverging D and S , with R → R ∗ as given by re-sults (IV A). The remaining of trajectories appearto evolve toward D = 0 , but again these solutions aremarred by numerical instability and we do not considerthem trustworthy.As we pointed out earlier, the choice of basis for thefields (21) and model parameters (25) is inappropriate forcertain α i . We have investigated these cases separatelyand find that they do not provide viable models: • α M = α T : For this choice, the no-slip configuration C = 0 implies that Φ = 0 and thus the gravitationalslip is shielded only when there is no gravitationalfield at all, making this case uninteresting. • α T = 0 : Preserving the no-slip configuration undertime evolution requires that α B = 2 α M . We thenhave D = α K + 6 α and Dc = − D . It is thusimpossible to choose the remaining independent α i to give a model which is simultaneously not a ghostand has a positive sound speed squared.To summarize, we have shown that scalar-tensor the-ories are in principle flexible enough to allow for modelswhich achieve dynamical shielding of gravitational slip.For generic cases, the model parameters need to evolve ontimescales much faster than cosmological which signifies avery special solution. Then, there exists one class of mod-els which can screen for a limited time, but the parametervalues need to be tuned very precisely to make this long0enough for a realistic cosmology. The only model classwhich naturally screens for an extended time requires adivergent kinetic term for the scalar field and thereforevanishing sound speed. This transforms the scalar into akind of dust. This should have been expected of course,since we are essentially requiring that the scalar field fol-low the evolution of dark-matter dust in a very precisemanner. Quasi-Static Horndeski
Frequently, the quasi-static (QS) approximation isused to model the evolution in scalar-tensor modifica-tions of gravity. One neglects the homogeneous part ofthe solution for the scalar field, i.e. one assumes thatthe scalar is not dynamical but rather follows the matterdistribution. This can be justified when the scales underconsideration are both inside the sound horizon of thescalar and the cosmological horizon [41].In Horndeski models, C = 0 does not contain timederivatives or different weights in k and thus is unaf-fected by the QS approximation. The QS approximationin our case is equivalent to enforcing only A = 0 inEq. (24) on the model space while neglecting the tuningrequired by A = A = 0 , i.e. we require α T = H α M ( α B + 2 α M )(2 − α B ) H α M + ˙ H (2 − α B ) − H ˙ α B + ˜ ρ m . (37)This is the same as the QS linear shielding conditions pro-posed in Ref. [42], although there the condition A = 0 is also enforced to cancel the subdominant sources out-side the sound horizon. Provided that the sound speedof the scalar is close to that of light, this QS solutionis approximately valid inside the cosmological horizon.However, in the vicinity of the sound horizon, the QSapproximation breaks down [41] and the deviations fromthe no-slip configuration become large. Indeed when amode cross the sound horizon, it takes time before itdecays to its QS approximation. Thus even inside thehorizon, the slip would vanish only up to corrections oforder max ( aH/c s k, aH/k ) .In particular, in Ref. [37], the covariant Galileon modelwas studied in the QS limit, demonstrating that for ap-propriate choices of parameters, the slip does vanish atparticular moments in time. This vanishing is exactlya QS screening of the sort described above and not acounter-example of the arguments presented in this pa-per: there would be suppressed corrections to the slipeverywhere and it would reappear at scales close the hori-zon. Nonetheless, given the power of any conceivable in-struments at large scales and cosmic variance, this maywell be never detectable for many models. On the otherhand, in Ref. [43], an even stronger QS shielding con-dition is chosen, where also the effective Newton’s con-stant for perturbations takes the standard value and theauthors find that when the full dynamics is solved theoscillations in the scalar would be unobservably small. B. Einstein-Aether theories
Now we ask the same question for the case of EA theo-ries: Does a modified propagation of gravitational wavesalways imply the existence of a gravitational slip? As weshow, within EA theories, there exists a one-parameterfamily of models which preserves a no-slip configurationif one is set up initially. However, the gravitational slip isconserved in these models, so if the initial configurationhas it, it will remain. Thus, EA models cannot provide adynamical shielding mechanism sufficient for our require-ments.Let us start by briefly reviewing the dynamics of thetheory at the linear level of scalar perturbations aroundFLRW, following the equations of Ref. [27]. In the presence of dust, the model describes two prop-agating scalar degrees of freedom: one for the dust, theother – for the helicity-zero component of the vector. Theconservation equation for the energy-momentum tensorfor the vector yields two equations: a constraint equa-tion for the lagrange multiplier λ , and a dynamical equa-tion for the vector respectively. The former allows oneto eliminate λ from the rest of the equations, and usingEinstein equations one can arrive to a set of two second-order equations for two dynamical fields, which can beconveniently chosen to be the spatial divergence of thevector perturbation Θ , defined in Eq. (9) and the New-tonian potential Φ . The equation of motion for Θ takesthe following schematic form Θ ′′ + E Θ ′ ( β i ) H Θ ′ + (cid:18) E Θ(1) ( τ ; β i ) + E Θ(2) ( β i ) k H (cid:19) H Θ= k H (cid:18) β + 3 β + β β ω (cid:19) (cid:0) H Φ + H Φ ′ (cid:1) , (38)with the explicit form of the dimensionless coefficients E i to be read from Ref. [27] and ω ≡ (1 − πGm ( β + β )) .A similar equation follows for Φ from the trace of the ij part of the Einstein equations. Note that the perturba-tion of the dust density does not appear in the equationof motion for Θ if one assumes that the correspondingenergy-momentum tensors are separately conserved. Theconfiguration of this theory is thus specified by the set X = { Θ , Θ ′ , Φ , Φ ′ } . (39)The anisotropy constraint is k (Ψ − Φ) = 16 πGm ( β + β ) (2 H Θ + Θ ′ ) . (40)and thus the requirement of zero anisotropic stress inthis case is equivalent to requiring at least one of thefollowing: i ) m = 0 , ii ) β + β = 0 , iii ) ( a Θ) ′ = 0 . (41) Notice the difference in our convention for the gravitational po-tentials Φ and Ψ compared to [27]. Θ . In particular, the first condition essentiallyrestores the Lorentz symmetry of the theory (setting thev.e.v of u µ to zero) and if imposed, all the non-trivial con-tributions from the vector vanish in the linearized equa-tions, reverting the theory to GR. The second choice willeliminate the shear term in the action and therefore anymodification to GW propagation. The third condition iswhat interests us, since it is a condition on the configu-ration variables , which, if preserved, would imply thatgravitational slip vanishes. Let us investigate it furtherbelow.We want to show whether the no-slip condition C [ X ] ≡ ( a Θ) ′ = 0 , (42)is preserved under evolution in time, i.e. whether C ′ + γ C ≈ , (43)with γ some function of time and the β i .Condition (43) defines a second-order differential equa-tion in time for Θ , which has to be satisfied on-shell. Weuse the equation of motion (38) in (43) to eliminate Θ ′′ .This leads to a new constraint between Θ , Φ and theirfirst time derivatives, which has to be satisfied at all timesand scales in the cosmological evolution: C ′ + γ C ≈ a β ω " (cid:18) H ′ H − (cid:19) (2 β + 3 β + β ) (44) − k H ( β + β + β ) H Θ++ a k H (2 β + 3 β + β ) β ω (cid:0) H Φ + H Φ ′ (cid:1) . This constraint can be satisfied for all no-slip configura-tions provided that β + β + β = 0 , (45) β + 3 β + β = 0 . (46)Conditions (45) and (46) are satisfied simultaneouslywhen β = − β , β = β . (47)The choice (47) defines a family of models parameter-ized by β , for which the no-slip condition is preservedunder time evolution. This family of models then has avanishing sound speed for the scalar and a modified GWspeed: c = β + β + β β + 16 πGm ( β + β )( β + 3 β + β ) = 0 , (48) c = (1 − β − β ) − = (1 + β ) − . We have thus shown that, in models (47), configura-tions with vanishing gravitational slip are maintained un-der time evolution. The second question is whether such no-slip conditions are typical in these models. In models(47), the equation of motion (38) reduces to C ′ = 2 H C , (49)and therefore that C ∝ a . Eq. (40) then implies that Ψ − Φ = const . (50)Contrary to the Horndeski case Eq. (35), in this EAmodel, if there were any slip in the initial configuration,it would be conserved for all time. Thus EA models donot generically evolve to a no-slip configuration and willgenerically exhibit gravitational slip.Moreover, just as is the case with such Horndeski theo-ries, the models (47) have zero sound speed. This leads toproblems with normalizing quantum fluctuations, whichis exacerbated here since the parameters β i are constant,implying that the sound speed for the helicity-zero modewas zero also during inflation.We thus conclude that there are no Einstein-Aethermodels in which the dynamical shielding mechanismwould generically produce a universe with no gravita-tional slip but with modified GW speed. C. Bimetric theories
1. Background
We now study the massive bigravity theories with mat-ter minimally coupled to the metric g . We assume bothmetrics to be spatially flat FLRW at the background level ¯ g µν dx µ dx ν = a ( − dτ + δ ij dx i dx j ) , (51) ¯ f µν dx µ dx ν = b ( − c dτ + δ ij dx i dx j ) , (52)where a and b are the two scale factors. We also definetheir ratio r ≡ b/a . Since the two metrics have in princi-ple independent time coordinates, the lapse c ( τ ) remainsafter choosing conformal time τ of the metric g as thetime coordinate. We also have two a priori independentconformal Hubble parameters, H ≡ a ′ /a and H f ≡ b ′ /cb .In this model, a part of the diffeomorphism group isbroken, so matter conservation must be imposed exter-nally. In such a case, satisfying the background Bianchiidentities leads to two branches of solutions: (i) the al-gebraic branch where r = const and the dynamics onFLRW are strongly coupled (helicity-0 and helicity-1 ofthe massive graviton do not propagate); and (ii) the dy-namical branch, where the Bianchi identities constrainthe lapse c to obey the relation c = 1 + r ′ H r , (53)which implies H f = H , see Ref. [34, 44] for details. Sinceit is the only possibly healthy configuration, we only con-sider the dynamical branch henceforth.2We can define four polynomials of r with coefficientslinear in the constants β n introduced in the action (12), ρ g ≡ β + 3 β r + 3 β r + β r , (54) ρ f ≡ β + 3 β r − + 3 β r − + β r − , (55) Z ≡ β + β r + 2 β r , (56) ¯ Z ≡ β + β cr + β ( c + 1) r , (57)The functions ρ g and ρ f play the role of the backgroundenergy densities arising from the interaction potential(12) in the g and f Friedmann equations. The pressurefrom this potential depends in addition on the combina-tion Z as well as the lapse c . The two pairs of Friedmannequations can be combined to find the expressions H ′ = a cρ m c + 2) − a ( c − Z (cid:0) cr + 2 (cid:1) c + 2) r , (58) H = a r ρ f (59) ρ m = ρ f r − ρ g , (60) c = 3 (cid:0) − r (cid:1) Z + r (cid:0) r ρ f + 3 ρ g (cid:1) − r ) Z − r ρ f , (61)where we have restricted the matter content to be dust.Thus all the instantaneous background dynamics can berecast in terms of the variables { a, r, ρ g , ρ f , Z } . More-over, the evolution of a is given by Eq. (59), r by Eq. (53), ρ f,g by their conservation equations. Thus given that Z ′ = 2 H ( ¯ Z − Z ) , (62)and that ¯ Z ′ can be similarly recast, the model basis set B = { a, r, ρ g , ρ f , Z, ¯ Z } is closed under time evolution andcompletely specifies the evolution of the background ofthe model. In fact, no background functions beyond B enter the coefficients in linear perturbation equations.We should note that we have simply traded the fiveconstants β n for five other variables with a fixed timeevolution. The advantage of this new formulation isthat the expressions are more compact and are explic-itly related to physical parameters such as the gravi-ton mass µ = r ¯ Z . A non-vanishing pressure of thematter p m would provide an additional external vari-able which needs to be specified, but does not changethe discussion otherwise. In what follows we will infact continue to use expressions contaning the extendedset (cid:8) a, c, r, ρ m , H , ¯ Z, Z (cid:9) , since this is somewhat simpler,with the understanding that c, H , ρ m are really given byEqs (59–61).We also point out that in Refs [36, 45] it was shownthat r ′ = 0 , c = 1 in the dynamical branch correspondsto the future de Sitter attractor and therefore will not beconsidered as a relevant background for our proof. The massive and massless fluctuations at linear level correspondto linear combinations of h ij and γ ij [10].
2. Perturbations
Small fluctuations propagate on the background met-rics, g µν = ¯ g µν + h µν , f µν = ¯ f µν + γ µν , the scalar partof which we can described using eight scalars when thegauge is not fixed, h = − a Ψ g ,h i = − a ∂ i B g ,h ij = 2 a ( − Φ g δ ij + ∂ i ∂ j E g ) , (63) γ = − c b Ψ f ,γ i = − cb ∂ i B f ,γ ij = 2 b ( − Φ f δ ij + ∂ i ∂ j E f ) . (64)As was shown in [34], the gravitational slip in the mat-ter g metric is C = a r ¯ Z ∆ E. (65)where ∆ E ≡ E f − E g is a gauge-invariant variable. Inorder to fulfill the no-slip condition C = 0 one of the twofollowing conditions must be satisfied ( i ) r ¯ Z = 0 , (66) ( ii ) ∆ E = 0 . (67)and preserved on equations of motion. Case (i).
Since according to Eq. (IV C), r ¯ Z is thegraviton mass, in this case the propagation of gravita-tional waves is not modified, confirming our hypothesis.Nonetheless, it is interesting to ask when the mass of thegraviton can vanish in cosmology.Since r = 0 , we use Eq. (57) to solve for ¯ Z = 0 obtain-ing c = − ( β + β r )( β + β r ) r . (68)Combining this with the expression for c , Eq. (61), givesa polynomial in r and β n , implying that r must be aconstant. The Bianchi constraint (53) then implies that c = 1 and the Universe is de Sitter. Thus, in bigravitymodels on cosmological solutions, the graviton can onlybe tuned to be massless on exact de Sitter. Case (ii).
The second case is a condition on the con-figuration of the perturbation variables. Requiring thatit be preserved under time evolution places a constrainton the model parameters.The full perturbation equations were first studied inRef. [34] in the gauge-invariant formalism. For our pur-pose, however, the formulation of Ref. [46] is simpler.There it was explicitly shown that fixing the gauge suchthat the matter fluctuations and Φ f are set to zero al-lows one to eliminate all the auxiliary variables, obtainingthe full dynamics in terms of just the fields E g and E f .3The full equations are presented in Ref. [46] and we willnot quote them here. Thus the set of independent linearvariables which fully characterizes scalar perturbationsin this theory is therefore X = { E g , E ′ g , E f , E ′ f } . (69)The evolution of the full matter-bigravity scalar sectoris described by two second-order differential equations,confirming explicitly that in the dynamical branch thereare two degrees of freedom: one from the matter, theother – the propagating helicity-0 mode of the massivegraviton.Since ∆ E ≡ E f − E g is gauge invariant, this expres-sion is also valid in the gauge choice of Ref. [46]. Thusrequiring there be no gravitational slip, C = 0 , in thisgauge is simply the requirement that E g = E f . The slipis not generated when C ′ = a r ¯ Z ( E ′ f − E ′ g ) + γ C = 0 (70)with γ some function of time. This is a new condition,since r ¯ Z = 0 . Thus the best-case scenario has C ′′ onshell be a linear combination of the constraints C, C ′ Taking the ∆ E = ∆ E ′ = 0 initial configuration, it isenough for us to require that ∆ E ′′ = 0 be identicallysatisfied on equations of motion. This is only possible ifthe equations of motion for E g,f are identical for theseinitial conditions, i.e. E ′′ g + H E ′ g − a ρ m E g = 0 , (71) E ′′ f + H E ′ g + k A k E g − a A l E g = 0 , (72)where it will be enough to consider just the small-scalelimit, k → ∞ , and where A k = ( c − Z (cid:0) Z − Z (cid:1) , (73) A l = (2 ¯ Z − Z ) ρ m r + Z (1 − c )( r + 1)(2 ¯ Z − Z ( c + 1)) rZ . (74)Eq. (71) does not contain a k term reflecting the factthat we are using dust for the matter, the sound speedof which is zero. As we will see, already in this limit theconditions we require cannot be satisfied and thereforewe do not need to consider the subleading corrections.Equations (71) and (72) are identical when ( I ) A k = 0 , (75) ( II ) A l = ρ m . (76)Thus ( I ) is satisfied for c = 1 , which is an empty de Sitterand not relevant, or Z = Z . Taking the latter solution, ( II ) gives ( c − (cid:0) r + 1 (cid:1) cZr = ρ m . (77) Using Eqs (60) and (61), we see this constraint is againjust an algebraic equation for r , meaning that r is con-stant. This again implies that c = 1 and therefore canonly be satisfied on de Sitter.We thus have proven that if matter is coupled to oneof the metrics, no model of bimetric gravity in which thegravitational slip is shielded and yet the graviton has amass for gravitational waves exists. V. CONCLUSIONS AND IMPLICATIONS
In this article, we revisited the question of the relation-ship between the propagation of the gravitational wavesand presence of gravitational slip in the scalar sector intheories of modified gravity. In three broad classes oftheories, Horndeski scalar-tensor, Einstein-Aether andbimetric gravity, the appearance of gravitational slipfrom perfect-fluid matter is only possible if the propa-gation of gravitational waves is modified. On the otherhand, beyond-Horndeski models are the first example ofmodels where this relationship is violated: gravitationalslip can be sourced without any corrections to gravita-tional waves. In this respect, this modification of grav-ity behaves more like a correction to the matter energy-momentum tensor.The majority of this paper, however, was devoted toinvestigating to what extent it is possible to pick model parameters so that GW propagation be modified imply-ing that there could be gravitational slip, but it nonethe-less be dynamically shielded at all scales. We have shownthat both the bimetric and Einstein-Aether models aretoo rigid for such a requirement. However, the scalar-tensor models can be tuned to permit such shielding.In the case of Einstein-Aether, there exists a familyof models, parameterized by the speed of GW, where ano-slip initial configuration is maintained under evolu-tion in time. However, if there is any slip in the initialconfiguration, it is preserved. Moreover, the sound speedof the helicity-zero mode is exactly zero, so the initialconditions that one might set during inflation is actuallycompletely out of control.Horndeski models are much more flexible, since themodel parameters are in principle allowed to be func-tions of time. Indeed, we have shown that for any chosencosmological background, it is possible to tune the evo-lution history of the parameters in such a way that ano-slip condition be maintained under evolution in time.However, shielding with generic choices of model param-eters requires that their evolution be much faster thancosmological, signifying the existence of fine tuning orcancellations in the action. As such, this general solu-tion does not point toward a physically interesting model.We attempted to find a more generic model, one whichdid not have to be precisely tuned to maintain screeningfor a long time. We found that such a setup requires adivergent kinetic term for the scalar, implying that onecannot write down a proper action within the Horndeski4class for such a model.Moreover, this setup again produced a model which hasa zero propagation speed for the scalar mode, and there-fore which suffers from the same issues as the Einstein-Aether case. This is not altogether surprising, since weare requiring that the scalar perturbations mimic the dy-namics of the dust exactly, following the collapse of struc-tures in the universe. Notwithstanding all these patholo-gies, in this model, any gravitational slip present in theinitial conditions during matter domination does actu-ally decay away, so the dynamics would shield the gravi-tational slip as we have required.If only an approximate statement is required, it ispossible to choose the Horndeski model parameters insuch a way that, in the quasi-static regime, this shield-ing of slip is effective. The tuning required for this ismuch less onerous. However, this approximation relieson neglecting time derivatives. Taking this approach,one misses out the fact that this tuning nonetheless can-not suppress corrections to the gravitational slip of ordermin ( H /k, H /c s k ) which become very significant at scalesclose to the cosmological/sound horizon. Thus even if themodel of gravity happens to have such parameters thatthe gravitational slip is suppressed at smaller scales, itsmeasurement near horizon scales would still be informa-tive. The question of course is to what precision Euclid orthe Square-Kilometer Array will be capable of measuringthis property, especially if it depends on scale [4, 47, 48].Indeed, the cosmic microwave background is very mucha near-horizon probe and therefore such effects should beunsuppressed if they were present during recombination.Vice-versa, tensor modes do affect the polarization of theCMB and therefore the combination of a gravitationalslip and propagation speed of GWs can be significantlymore informative on the properties of gravity at thatearly time. Indeed, Ref. [49, 50] show that the speed of tensors at recombination can be constrained within about of light. The complementarity of probes of back-ground, scalar and tensor sectors has also been exploredin Ref. [43]. Other probes of gravitational waves of as-trophysical sources have been proposed for constrainingmodifications of gravity, e.g. the decrease of the orbitalperiod of pulsars [51], timing of binary white dwarf sys-tems [52] and the arrival time of GW’s compared to thatof neutrinos and radiation from supernovae and gamma-ray bursts, respectively [53]. Putting all these data to-gether should allow us to answer better what the mech-anism for acceleration of the expansion of the universeis.It is thus our conclusion that dynamical shieldingplaces so onerous a tuning requirement and so strong adependence on cosmological parameters, that it is im-possible to achieve in practice in any natural setting.Thus if future observations were to lead to strong evi-dence against gravitational slip, they would simultane-ously provide a signal that gravitational wave dynam-ics are standard and thus modifications of gravity thatchange graviton propagation are unlikely. ACKNOWLEDGMENTS
We would like to to thank Emilio Bellini andMiguel Zumalacárregui for helpful comments and dis-cussions. The work of L.A. is supported by theDFG through TRR33 “The Dark Universe”. M.K. andM.M. acknowledge funding by the Swiss National Sci-ence Foundation. IDS is supported by FCT underthe grant SFRH/BPD/95204/2013, and further acknowl-edges UID/FIS/04434/2013 and the project FCT-DAAD6818/2016-17. I.S. is supported by the European Re-gional Development Fund and the Czech Ministry of Ed-ucation, Youth and Sports (MŠMT) (Project CoGraDS– CZ.02.1.01/0.0/0.0/15 _003/0000437). [1] L. Amendola et al. , “Cosmology and FundamentalPhysics with the Euclid Satellite,” arXiv:1606.00180 [astro-ph.CO] .[2] L. Amendola, M. Kunz, M. Motta, I. D. Saltas, andI. Sawicki, “Observables and unobservables in darkenergy cosmologies,”
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