DDegenerate Hořava gravity
Enrico Barausse, Marco Crisostomi, Stefano Liberati and Lotte ter Haar
SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di TriesteIFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy
Abstract:
Hořava gravity breaks Lorentz symmetry by introducing a dynamical timelikescalar field (the khronon), which can be used as a preferred time coordinate (thus select-ing a preferred space-time foliation). Adopting the khronon as the time coordinate, thetheory is invariant only under time reparametrizations and spatial diffeomorphisms. In theinfrared limit, this theory is sometimes referred to as khronometric theory. Here, we ex-plicitly construct a generalization of khronometric theory, which avoids the propagation ofOstrogradski modes as a result of a suitable degeneracy condition (although stability of thelatter under radiative corrections remains an open question). While this new theory doesnot have a general-relativistic limit and does not yield a Friedmann-Robertson-Walker-likecosmology on large scales, it still passes, for suitable choices of its coupling constants, lo-cal tests on Earth and in the solar system, as well as gravitational-wave tests. We alsocomment on the possible usefulness of this theory as a toy model of quantum gravity, as itcould be completed in the ultraviolet into a “degenerate Hořava gravity” theory that couldbe perturbatively renormalizable without imposing any projectability condition. a r X i v : . [ h e p - t h ] J a n ontents Hořava gravity [1] is a gravitational theory that is power-counting renormalizable in theultraviolet (UV), at the expense of giving up Lorentz symmetry. The theory, written interms of 3+1 Arnowitt-Deser-Misner (ADM) variables [2], is indeed invariant only underfoliation-preserving diffeomorphisms (FDiffs), i.e. (monotonic) time reparametrizations andspatial diffeomorphisms, and not under full-fledged four-dimensional diffeomorphisms. Theaction of the theory involves up to six spatial derivatives of the ADM fields, but is onlyquadratic in the time derivatives (which only appear via the extrinsic curvature, i.e. via thetime derivative of the spatial metric). It is this anisotropic scaling between space and timederivatives that ensures power-counting renormalizability.Performing a Stuckelberg transformation, Hořava gravity can be recast as a Lorentz-violating scalar-tensor theory [3]. The scalar field, sometimes referred to as the khronon, isconstrained by a Lagrange multiplier to be timelike (i.e. its gradient must be timelike), andit thus plays the role of a preferred time by selecting a preferred manifold slicing. Using thiscovariant formulation, as opposed to the original 3+1 one (i.e. the unitary gauge where thekhronon is adopted as time coordinate), it becomes more clear why the 3+1 action is builtwithout any time derivatives of the lapse. In fact, to avoid Ostrogradski instabilities [4] (or“ghosts”), one may naively require that the covariant action be quadratic in the unit-norm– 1 –æther” vector field proportional to the khronon’s gradient. This vector field turns out tobe u = − N dt in the unitary gauge, with N the lapse, and one may naively try to obtaintime derivatives of N by introducing the acceleration a ν = u µ ∇ µ u ν . The latter, however,only includes spatial derivatives of N because of the unit norm condition (which implies a µ u µ = 0 ).The absence of time derivatives of the lapse poses a hurdle to proving perturbativerenormalizability (beyond power counting). Calculations of the latter are technically in-volved and have so far only been performed in the unitary gauge [5, 6], where the lapsesatisfies an elliptic equation as a result of the absence of its time derivatives in the action [7].This leads to the “instantaneous” propagator / ( k i k i ) . To overcome this problem, Ref. [5]proved perturbative renormalizability in Hořava gravity under the “projectability condition”,i.e. the assumption that the lapse is a function of time only. The resulting theory, whileconsidered in the first paper by Hořava [1], is disjoint from general (i.e non-projectable)Hořava gravity [7], and is strongly coupled on flat space [7–9].It should be noticed, however, that including derivatives of the lapse (i.e. second timederivatives of the khronon scalar field) does not lead automatically to Ostrogradski ghosts,if the Lagrangian is degenerate . This fact is very well known in the context of Lorentz-symmetric scalar tensor theories, where it led first to beyond-Horndeski theories [10, 11]and then to Degenerate Higher-Order Scalar-Tensor (DHOST) theories [12–14]. Thesetheories have field equations that are higher than second order in time, but still propagateno ghosts.In the following, we will apply this degeneracy program to the infrared (IR) limit ofnon-projectable (i.e. general) Hořava gravity. In that limit, the theory is sometimes referredto as khronometric theory, and while it still violates Lorentz symmetry (being only invariantunder FDiffs in the unitary gauge), it is quadratic in both time and space derivatives (ofthe spatial metric and khronon) [3, 7]. We will show that khronometric theory can bemodified to include second time derivatives of the khronon (i.e. time derivatives of thelapse), while still propagating no Ostrogradski ghost. Although similar constructions werealready obtained in [15–17], here we additionally show that the resulting theory is invariantunder spatial diffeomorphisms and a special set of (monotonic) time reparametrizations(which will turn out to be given by “hyperbolic” time compactifications). While invarianceunder this special group of transformations is sufficient to determine the form of the kineticterm for the lapse, it does not fix its coefficient unambiguously. Therefore, the radiativestability of the fine-tuning of the coupling constants needed to eliminate the ghost remainsan open issue. We will comment on promising ways forward on this issue in the following.As a result of this construction, the novel “degenerate Hořava gravity” theory that wefind does not have a limit to General Relativity (GR). However, somewhat surprisingly,this does not prevent the theory from having the correct Newtonian limit, nor from re-producing (at least for specific values of the coupling constants) the dynamics of GR atthe first post-Newtonian (1PN) order and thus passing solar system tests. However, thebehavior on cosmological scales is wildly different from GR, at least at the background level(i.e. assuming isotropy and homogeneity). We will discuss this issue, its implications, andpossible solutions below. – 2 –his paper is organized as follows. In Sec. 2 we review Hořava gravity both in the uni-tary gauge and in the covariant (Stuckelberg) formalism. In Sec. 3 we then consider theoriesinvariant under a smaller gauge group, i.e. restricted foliation-preserving diffeomorphisms(RFDiffs), as an intermediate step toward Sec. 4, where we introduce our degenerate gen-eralization of Hořava gravity. We study its phenomenology in Sec. 5, and we discuss ourconclusions in Sec. 6. We utilize units in which c = 1 , except in Appendix A, where wediscuss the PN expansion of degenerate Hořava gravity and we thus reintroduce c as abook-keeping parameter. In this section, we review the construction of Hořava gravity and the subgroup of the four-dimensional diffeomorphisms under which the theory is invariant. The distinction betweenspace and time introduces a preferred frame, and thus a preferred time coordinate, whichcorresponds to endowing the space-time manifold with a preferred foliation by space-likesurfaces. This means that the arbitrary reparameterization of time t → ˜ t ( t, x ) is not asymmetry of the theory anymore.The basic ingredients to describe the space-time geometry are the spatial metric γ ij , theshift N i and the lapse function N entering the decomposition of the four-dimensionalmetric [2] d s = − N d t + γ ij (d x i + N i d t )(d x j + N j d t ) . (2.1)The Hořava action is built from quantities invariant under the following unbroken symmetry,which is commonly referred to as FDiffs: x → ˜ x ( t, x ) , t → ˜ t ( t ) , (2.2)where ˜ t ( t ) is a monotonic function of t . Notice that this is the largest possible unbro-ken gauge group that one can have, once a preferred foliation is introduced. Under thissymmetry the fields in (2.1) transform as N → ˜ N = N d t d˜ t , N i → ˜ N i = (cid:18) N j ∂ ˜ x i ∂x j − ∂ ˜ x i ∂t (cid:19) d t d˜ t , γ ij → ˜ γ ij = γ kl ∂x k ∂ ˜ x i ∂x l ∂ ˜ x j . (2.3)Up to dimension six, the action takes the form [1] S = 116 πG (cid:90) d x d t N √ γ (cid:104) (1 − β ) K ij K ij − (1 + λ ) K + α a i a i + (3) R − V (cid:105) , (2.4)where K ij is the extrinsic curvature of the surfaces of constant time K ij = 12 N ( ˙ γ ij − D i N j − D j N i ) (2.5) Note that our definition of K ij differs by an overall sign from the definition used in some textbooks(e.g. [18, 19]), although it agrees e.g. with [20–22]. – 3 –with D i three-dimensional covariant derivatives compatible with the spatial metric γ ij andwith an overdot denoting partial time derivatives), K ≡ K ij γ ij is the trace of the extrinsiccurvature, (3) R is the three-dimensional Ricci scalar and a i ≡ N − ∂ i N (2.6)is the acceleration vector. Besides the bare gravitational constant G , there are three freedimensionless constants: α, β and λ . The “potential” V depends on the three-dimensionalRicci tensor (3) R ij and the acceleration a i , with all possible operators of dimension four andsix. This potential therefore involves only a finite number of these operators, which werefully classified e.g. in [23] and [20]. While crucial for renormalizability, for our purposes thepotential is completely irrelevant. Therefore, we omit to write explicitly its form here, andwe focus on the IR limit of the theory (obtained by neglecting V ) in the rest of this paper.Also notice that the kinetic part of the action (i.e. the part where time derivativesappear) is fully contained in the first two terms of Eq. (2.4). In addition to the helicity-2modes of the graviton, there is also a propagating scalar field, usually referred to as the“khronon” [7]. This formalism allows one to single out explicitly the extra degree of freedom that appearsbecause of the breaking of diffeomorphism invariance. It amounts to rewriting the action ina generally covariant form, at the expense of introducing a compensator field that transformsnon-homogeneously under the broken part of the four-dimensional diffeomorphisms.In more detail, one encodes the foliation structure of the space-time in a scalar field ϕ , such that the foliation surfaces are identified with those of constant ϕ . The action (2.4)then corresponds to the frame where the coordinate time coincides with ϕ (i.e. ϕ = t ). Thischoice of coordinates is often referred to as “unitary gauge”. The action in a generic frameis then obtained by performing the Stuckelberg transformation and reads S = 116 πG (cid:90) d x √− g (cid:2) R − β ∇ µ u ν ∇ ν u µ − λ ( ∇ µ u µ ) + α a µ a µ (cid:3) , (2.7)where u µ = ∂ µ ϕ √− X , X ≡ ∂ µ ϕ∂ µ ϕ , a µ ≡ u ν ∇ ν u µ , (2.8)and R is the four-dimensional Ricci scalar. Notice that this is the action of Einstein-æthertheory when the æther vector field is hypersurface orthogonal [24]. For later purposes, it isconvenient to write the action (2.7) explicitly in terms of the khronon field S = 116 πG (cid:90) d x √− g (cid:20) R + β ϕ µν ϕ µν X + λ ( (cid:3) ϕ ) X − λ ( (cid:3) ϕ ) ϕ µ ϕ µν ϕ ν X +( α − β ) ϕ µ ϕ µρ ϕ ρν ϕ ν X + ( β + λ − α ) ( ϕ µ ϕ µν ϕ ν ) X (cid:21) , (2.9)where to avoid clutter we have introduced the notation ϕ µ ≡ ∂ µ ϕ and ϕ µν ≡ ∇ ν ∂ µ ϕ . Noticethat the action above is invariant under reparameterizations of ϕ , ϕ → ˜ ϕ = f ( ϕ ) , (2.10)– 4 –here f is a (monotonic) arbitrary function. This reflects the invariance (in the unitarygauge) under the FDiffs (2.2).Naively, the higher-order derivatives in the action (2.9) would suggest the presence ofan Ostrogradski ghost in the theory. However, the counting of the degrees of freedom cannotbe straightforwardly performed from the covariant action (2.9), because of the absence ofa standard kinetic term for the khronon and the non-local /X dependence. However, onecan easily perform the counting in the preferred frame, where ϕ cannot be constant andhas a non-vanishing time profile ϕ = t . In this frame, as can be seen in the unitary gaugeaction (2.4), the ghost mode is absent.From the point of view of the covariant action (2.9), the absence of the Ostrogradskimode is guaranteed by the highly non-trivial tuning among the coefficients of the fiveoperators in the action, which translates in the absence of ˙ N terms in the unitary gaugeaction (2.4). Remarkably, this tuning is protected against radiative corrections by thereparameterization invariance (2.10). A detuning of the action coefficients would necessarybreak the symmetry (2.10), generate ˙ N terms in the unitary gauge action, and genericallyreintroduce the ghost mode.Finally, notice that the action (2.9) does not belong to any of the DHOST classesidentified in [13, 25]. This is because when the full diffeomorphism invariance is broken,the degeneracy of the Hessian matrix of the velocities can be achieved in a less restrictiveway. The full diffeomorphism invariant analog of action (2.9) is the Horndeski Lagrangian[26], which in the unitary gauge does not present ˙ N terms. Noticeably, there exists also a smaller unbroken gauge group according to which we canconstruct our Lagrangian, namely the group of RFDiffs x → ˜ x ( t, x ) , t → ˜ t = t + const , (3.1)which differs from FDiffs because the invariance under general time reparametrizations isreplaced by the invariance under time translations. In the Stuckelberg formulation, thissymmetry of the khronon action reduces to the shift symmetry ϕ → ˜ ϕ = ϕ + const , (3.2)which allows for a general dependence of the action on the derivatives of ϕ .Restricting the symmetry from FDiffs to RFDiffs therefore allows one to include inthe action a kinetic term for the lapse N . Moreover, all dimensionless couplings in theLagrangian may now acquire an arbitrary dependence on N , and we can thus include inthe potential V a generic function of N . The kinetic term for N is fixed by the invarianceunder RFDiffs to be of the form (cid:16) ˙ N − N i ∂ i N (cid:17) . (3.3)However, a general dependence of the action on this term inevitably leads to the propagationof an additional ghost scalar degree of freedom [7]. In the Stuckelberg formulation, this– 5 –s the Ostrogradski mode associated with the higher derivatives of the khronon, whichre-appears because of the detuning of the coefficients of the action (2.9). It is well established that if the kinetic term (3.3) and the trace of the extrinsic curvatureappear in the Lagrangian in such a way as to enforce the existence of a primary constraint(which in turn generates a secondary constraint), then the ghost mode can be safely removed[12, 13, 27, 28]. Therefore, it is possible to realize healthy theories within RFDiff gravity,provided that suitable degeneracy conditions are imposed. These models have been fullyclassified in [15–17], but they may not be very attractive for two reasons. First, the presenceof arbitrary functions of N in the Lagrangian results in an infinite number of couplingconstants. Second, the degeneracy conditions are not protected by the RFDiff symmetry,so that radiative corrections will generically induce a detuning of the action and hencereintroduce the ghost. This is very different from Hořava gravity, where the tuning of theaction (2.9) is required by the FDiff symmetry and hence is protected by it. In this section, we present a new class of gravity theories invariant under a symmetryintermediate between FDiffs and RFDiffs. In more detail, the transformation of time isrestricted to take the form of a specific hyperbolic function ˜ t ( t ) , and the symmetry isrealized up to a total derivative.Our starting point is a generalization of the Hořava action (2.4), which includes thetime derivative of the lapse in the RFDiff invariant way (3.3). By introducing the definition V ≡ − N (cid:16) ˙ N − N i ∂ i N (cid:17) , (4.1)we write the action as S = 116 πG (cid:90) d x d t N √ γ (cid:104) ω V + 2 σ K V + (1 − β ) K ij K ij − (1 + λ ) K + α a i a i + (3) R (cid:105) , (4.2)where ω and σ are two additional dimensionless constants. Clearly, the first two terms in(4.2) break FDiff invariance, although they are RFDiff invariant. At this stage, two scalardegrees of freedom propagate, one of which is a ghost [7].We can then impose the existence of a primary constraint by requiring that the deter-minant of the kinetic matrix of the two scalar modes in (4.2) vanishes [29–31]:det (cid:32) ω σσ − λ − β +23 (cid:33) = 0 . (4.3)In Hořava gravity, this condition is trivially realized since ω = σ = 0 , but a non-trivialsolution is also possible and is given by ω = − σ λ + β + 2 . (4.4)– 6 –o completely remove one degree of freedom, the primary constraint, enforced by the con-dition (4.4), must generate a secondary constraint. The conditions for this to happen werederived in complete generality for field theories in [31], and in the case at hand they areautomatically satisfied because of the absence of the following couplings in the action: V ∂ i N , K∂ i N , (3) R · V , (3) R · K . (4.5)Therefore, condition (4.4) is all that is needed to completely eliminate the ghost mode andremain with a single scalar field, the khronon.Thus far, we have not made any progress with respect to degenerate RFDiff theories,of which the action (4.2) – with the condition (4.4) enforced – is a particular case. In fact,nothing prevents the couplings in the Lagrangian from being functions of N and, moredangerously, quantum corrections from spoiling the condition (4.4). Ideally, our aim is todetermine whether there exists a gauge group, smaller than the FDiffs one but larger thanthe RFDiffs one, that could protect the condition (4.4).For this purpose, after imposing the condition (4.4), we transform the action (4.2)under the FDiffs (2.2) and obtain a new action, which now differs from the original one,because V is not invariant. Indeed, using Eq. (2.3), we find that V → ˜ V = V − N d t d˜ t (cid:18) d t d˜ t (cid:19) − . (4.6)By requiring that the new terms generated by Eq. (4.6) in the action are a total derivative,one imposes that the equations of motion are invariant. In this way, we obtain a third-orderdifferential equation for ˜ t ( t ) , which has a non-trivial solution only if the following conditionis imposed on the coefficients of the Lagrangian σ = − λ − β + 23 . (4.7)In this case, there exists a unique family of solutions for ˜ t ( t ) given by ˜ t ( t ) = c c + t + c , (4.8)where c , , are free integration constants. Being (4.8) a hyperbolic function, we refer to thisfamily of time reparametrizations (together with spatial diffeomorphisms) as “hyperbolic-foliation-preserving Diffs” (HFDiffs). Notice that Eq. (4.8) is a monotonic function onintervals not including its pole, which prevents violations of causality.By inspection of the resulting Lagrangian, it is easy to realize that the conditions (4.4)and (4.7) force the kinetic term to be ( V + K ) , (4.9)which is indeed invariant under HFDiffs up to a total derivative since N √ γ ( V + K ) → N √ γ ( V + K ) − (cid:20) ∂ t (cid:18) √ γN ( c + t ) (cid:19) − ∂ i (cid:18) √ γN i ( c + t ) N (cid:19)(cid:21) . (4.10)– 7 –owever, being N √ γK FDiff invariant, if one starts from the generic Lagrangian (4.2),invariance under HFDiffs (up to a total derivative) only requires the kinetic term to be ofthe form V + 2 KV , with an arbitrary coefficient in front. Therefore, this implies thatradiative corrections may induce a running of the coefficients of the Lagrangian. This canpotentially detune the degeneracy condition (4.4) and thus reintroduce the ghost.A way out of this unappealing situation would be to enlarge the group of HFDiffs to acustodial symmetry sufficient to ensure stability of the degeneracy condition under quantumcorrections. Such a gauge group must necessarily lie in between HFDiffs and full-fledgedfour-dimensional diffeomorphisms (since the latter would simply require the theory to beGR). Therefore, one might consider a specific class of time reparametrizations that areexplicit functions of the spatial coordinate, e.g. ˜ t ( t, x ) = c / ( c + t ) + c + f ( x ) , althoughwe have been unable to identify a suitable function f ( x ) of the spatial coordinates.To summarize, we have found a new unbroken gauge group that: (i) allows for a kineticterm for the lapse; (ii) avoids the presence of arbitrary functions of the lapse in the action;although (iii) it does not yet prevent the propagation of a ghost mode. These are theHFDiffs x → ˜ x ( t, x ) , t → ˜ t = c c + t + c , (4.11)and the corresponding action is given by S = 116 πG (cid:90) d x d t N √ γ (cid:20) − (cid:18) λ + β + 23 (cid:19) (cid:0) V + 2 K V (cid:1) +(1 − β ) K ij K ij − (1 + λ ) K + α a i a i + (3) R (cid:105) . (4.12)Comparing with the Hořava action (2.4) we notice that, although there are two new oper-ators, the number of coupling constants is the same. In the this new action, however, evenwhen all couplings are set to zero, we do not recover the GR limit. It is now instructive to look at the new action (4.12) in the Stuckelberg formalism. As inSection 2.1, we perform the Stuckelberg transformation and obtain S = 116 πG (cid:90) d x √− g (cid:20) R + β ϕ µν ϕ µν X + λ ( (cid:3) ϕ ) X + 2( β + 2)3 ( (cid:3) ϕ ) ϕ µ ϕ µν ϕ ν X +( α − β ) ϕ µ ϕ µρ ϕ ρν ϕ ν X + (2 β − α − ϕ µ ϕ µν ϕ ν ) X (cid:21) . (4.13)Comparing with the khronon action for Hořava gravity, Eq. (2.9), we see that the coefficientsof the third and fifth operators have changed. This new highly non-trivial tuning guaranteesthe absence of the Ostrogradski ghost at least at tree level. Moreover, the Lagrangian (4.13)is invariant under the transformation ϕ → ˜ ϕ = c c + ϕ + c , (4.14)– 8 –p to the total derivative − (cid:18) λ + β + 23 (cid:19) ∇ µ (cid:18) c + ϕ ∂ µ ϕ (cid:19) . (4.15)This reflects the invariance, up to a total derivative and in the unitary gauge, under theHFDiffs (4.11).Finally, notice that also in this case, action (4.13) does not belong to any of the DHOSTclasses [13, 25]. Again, because of the absence of a standard kinetic term for the scalar fieldand the non-local /X terms, the counting of the degrees of freedom is easier to performin the preferred foliation. Looking at the form of the kinetic terms, Eq. (4.9), a natural question is whether the newtheory is related to Hořava gravity by a conformal and/or disformal transformation. Tocheck this, it is convenient to work in the Stuckelberg formalism, where these transforma-tions read [32] ¯ g µν = Ω( X ) g µν + Γ( X ) ∂ µ ϕ ∂ ν ϕ , (4.16)with Ω and Γ free functions of X only. (Notice that a ϕ dependence would break even theshift symmetry and therefore it is not allowed).It is well known that Hořava gravity is invariant under the transformation given by [33] Ω = 1 , Γ = ςX , (4.17)where ς is a constant, provided the following rescaling of the coefficients ¯ λ = λ + ς ( λ + 1) , ¯ β = β + ς ( β − . (4.18)A surprising feature of the new theory (which may point to the possible existence of acustodial gauge group in between HFDiffs and four-dimensional diffeomorphisms) is thatit also enjoys this invariance for the very same choice of functions [Eq. (4.17)] and for thesame rescaling of the coefficients [Eq. (4.18)]. Moreover, any choice different from Eq. (4.17)would change the power of X appearing in each of the operators of the Lagrangian, andtherefore cannot connect the two theories.As a consequence, a generic transformation of the form (4.16) cannot map Hořavagravity into its degenerate HFDiff generalization, and both theories are stable under thesame transformation (4.17). Once again, the non-local form of Γ in (4.17) is signaling thatthe two theories make sense only for X (cid:54) = 0 , i.e. the khronon must always be timelike. To study the phenomenology of “degenerate” HFDiff Hořava gravity, we first derive the fieldequations by varying the action (4.12) with respect to the spatial metric γ ij , the shift N i and the lapse N . We denote by D i the covariant derivative defined with respect to γ ij , and– 9 – t ≡ ∂ t − N k D k . We also define the following quantities in terms of the variation of thematter action [21, 22] E = − √ γ δS m δN = N T , (5.1) J i = 1 √ γ δS m δN i = N (cid:0) T i + N i T (cid:1) , (5.2) T ij = 2 N √ γ δS m δγ ij = T ij − N i N j T , (5.3)where T µν is the four-dimensional matter energy-momentum tensor.The variation with respect to N leads to (3) R − β + λ + 11 − β K − K ij K ij + α ( D i N )( D i N )(1 − β ) N − αD i D i N (1 − β ) N + 2(2 + β + 3 λ )3(1 − β ) (cid:20) KV + V − K − D t (cid:18) K + VN (cid:19)(cid:21) = 16 πG E (1 − β ) c , (5.4)which unlike in GR is not a constraint, but rather an evolution equation for N (c.f. thepresence of both second-order space and time derivatives of N ). Notice that in (non-degenerate) Hořava gravity , this equation is instead an elliptic equation for N , to besolved on each slice [21, 22]. In the covariant formalism, this equation becomes indeed(in both non-degenerate and degenerate Hořava gravity) the khronon evolution equation.Varying with respect to N i one obtains the momentum constraint equation D j (cid:34)(cid:18) K ij − λ + 11 − β γ ij K (cid:19) − (2 + β + 3 λ ) γ ij V − β ) (cid:35) − (2 + β + 3 λ )3(1 − β ) (cid:0) D i N (cid:1) (cid:18) K + VN (cid:19) = − πG J i (1 − β ) c , (5.5)while variation with respect to γ ij yields the evolution equation − β (cid:20) (3) R ij − (3) Rγ ij (cid:21) + 1 N D t (cid:18) K ij − λ + 11 − β Kγ ij − (2 + β + 3 λ )3(1 − β ) V γ ij (cid:19) + 2 N D k (cid:18) N ( i [ K j ) k − λ + 11 − β Kγ j ) k − (2 + β + 3 λ )3(1 − β ) V γ j ) k ] (cid:19) + 2 K ik K jk − λ + 1 + β − β KK ij − γ ij (cid:18) K kl K kl + λ + 11 − β K (cid:19) − − β ) N (cid:104) ( D i D j N ) − ( D k D k N ) γ ij (cid:105) + αN (1 − β ) (cid:20) ( D i N )( D j N ) −
12 ( D k N )( D k N ) γ ij (cid:21) + 2(2 + β + 3 λ )3(1 − β ) (cid:20) V γ ij − K ij V − K + VN N ( i ( D j ) N ) (cid:21) = 8 πG (1 − β ) c T ij . (5.6) We often refer to the original Hořava gravity [Eq. (2.4)] as “non-degenerate”, in order to distinguish itfrom its degenerate extension [Eq. (4.12)]. However, it should be by now well understood that even non-degenerate Hořava gravity is a degenerate theory that satisfies (although trivially) the degeneracy condition(4.3). This is even more evident from the tuning of the parameters in its khronometric formulation (2.9). – 10 – .1 Solar-system tests and gravitational-wave propagation
To check the experimental viability of the theory on Earth and in the solar system, weperform a post-Newtonian expansion over flat space and compare to the parametrized PNmetric (PPN) [34, 35]. This will allow us to extract the values of the PPN parameters indegenerate Hořava gravity and compare them to their experimental bounds. The details ofthe calculation follow [22], which performs the same analysis for (non-degenerate) Hořavagravity, and are presented in Appendix A. Here, we will simply summarize the main results.We find that the only PPN parameters differing from GR are the preferred-frameparameters α and α , which take the form α = 4( α − β ) β − , (5.7) α = − ( α − β + 2)(3 α − β − α − β + λ ) + 2( α − β − − α + 28 β + 12 λ + 383( α − . (5.8)Experimental bounds on these parameters are | α | (cid:46) − and | α | (cid:46) − [35]. Comparingto their expressions in (non-degenerate) Hořava gravity [22, 36], we see that while α isunchanged, α gets modified. In (non-degenerate) Hořava gravity, α and α are bothproportional to α − β , i.e. they are both small for α ≈ β .At this point, let us notice that constraints on the propagation speed of gravitationalwaves from GW170817 require | β | (cid:46) − in (non-degenerate) Hořava gravity [37, 38], aswell as in the degenerate version of the theory that we are considering here. Indeed, thekinetic term of the tensor modes is given by K ij K ij and the spatial gradient is containedin (3) R , which gives [c.f. Eqs. (2.4) and (4.12)] a gravitational-wave propagation speed c GW = (1 − β ) − / , which matches the speed of light only for β = 0 .The solar-system bound on α then gives α ≈ β ≈ in both non-degenerate [39, 40]and degenerate Hořava gravity. For α ≈ β ≈ , Eq. (5.8) then yields α ≈ − (1 + 2 λ )(2 + 3 λ )3 λ . (5.9)The experimental constraint | α | (cid:46) − then selects λ ≈ − / or λ ≈ − / . For thelatter, the coefficient in front of V + 2 KV in the action disappears, i.e. one is left withthe non-degenerate version of the theory. Therefore, there exists only one non-trivial setof parameters, namely α ≈ β ≈ and λ ≈ − / , for which degenerate Hořava gravity cansatisfy solar-system tests and the bound on the propagation speed of gravitational waves.For these values the action reads S = 116 πG (cid:90) d x √− g (cid:20) R −
12 ( (cid:3) ϕ ) X + 43 ( (cid:3) ϕ ) ϕ µ ϕ µν ϕ ν X −
23 ( ϕ µ ϕ µν ϕ ν ) X (cid:21) (5.10)or, in the unitary gauge, S = 116 πG (cid:90) d x d t N √ γ (cid:20) − (cid:0) V + 2 K V (cid:1) + K ij K ij − K + (3) R (cid:21) . (5.11)– 11 – .2 Cosmology To test the behavior of the theory on cosmological scales, we assume a standard homoge-neous and isotropic Robertson-Walker metric d s = − d t + a ( t ) δ ij d x i d x j , (5.12)where a ( t ) is the scale factor, and γ ij = a ( t ) δ ij . Notice that we have assumed flat spatialslices, but our conclusions are unchanged if we allow for curvature.Replacing this ansatz in the field equations, the only non-trivial equations are providedby the khronon equation (5.4) and by the trace of the evolution equation (5.6), which givethe system (2 + β + 3 λ )(3 H + 2 ˙ H ) = − πGρ , (5.13) (2 + β + 3 λ )(3 H + 2 ˙ H ) = − πGP , (5.14)with H = ˙ a/a the Hubble rate, while ρ and P are the energy density and pressure of thecosmic matter.As can be seen, this system is completely different from the Friedmann-Robertson-Walker equations of GR. This is no surprise since the theory does not reduce to GR for anyvalues of the coupling constants. More worrisome is the fact that by taking the differenceof the two equations, one obtains that the cosmic matter must necessarily have ρ = P (stifffluid). In other words, the theory does not allow for the usual radiation and matter eras,nor for an early- or late-time accelerated expansion (even in the presence of a cosmologicalconstant). As a curiosity, however, it is worth mentioning that if we set ρ = P in Eqs. (5.13)–(5.14) and solve for H , we find H ( t ) = 2 / [3( t + C )] , with C an integration constant. For C = 0 this reduces to the Hubble rate of the standard matter-dominated era, and isreminescent of the appearance of Dark Matter as an integration constant in projectableHořava gravity [41]. In this work, we have shown that it is possible to construct a novel khronometric theorywith a dynamical lapse, which (via a degenerate Lagrangian) propagates only a gravitonand a khronon. This theory is invariant under a special subgroup of the FDiff symmetry,Eq. (4.11), which we have referred to as hyperbolic-foliation-preserving Diffs (HFDiffs).This new unbroken gauge group selects a specific kinetic term for the lapse (although itdoes not fix its overall coefficient), and it avoids an arbitrary dependence of the action onthe lapse. HFDiffs are not sufficient by themselves to ensure stability of the degeneracycondition under radiative corrections, thus potentially letting the ghost re-appear beyondtree level. However, an enlarged gauge group lying between HFDiffs and four-dimensionaldiffeomorphisms may protect the fine-tuning of the degeneracy condition. We have com-mented on this possibility above, and we will explore it further in future work.Our construction has a two-fold interest for both phenomenology and theory. Onthe phenomenological side, it is a remarkable example of a theory which, despite being– 12 –orentz breaking and not admitting a GR limit, does pass Earth-based, solar-system andgravitational-wave tests, at least for a suitable choice of its coupling constants. Unfortu-nately, the theory fails to reproduce the standard Friedmann-Robertson-Walker cosmologyand can provide an (effective) matter-dominated era only if the universe contains stiff mat-ter alone ( ρ = P ). However, while clearly the cosmology of the theory does not seem towork out of the box, a couple possibilities are worth mentioning. First, Eqs. (5.13)–(5.14)assume a minimal coupling to matter. If matter is instead conformally coupled to gravity,it may be possible to obtain a matter-dominated era and a late-time accelerated expan-sion, although it would still be impossible to accommodate a radiation era and it mightbe tricky to pass solar-system tests (at least in the absence of a screening mechanism pro-tecting local scales from the conformal coupling). Second, and perhaps more importantly,since dark matter seems to arise naturally as an integration constant in our new theory, itmay be worth trying to explain the observed late-time acceleration of the universe in thecontext of non-standard cosmologies that violate the homogeneity/isotropy assumptions ofthe Robertson-Walker ansatz (see e.g. [42] for a review, and references therein).On the theoretical side, if a custodial symmetry protecting the degeneracy conditionis identified, this theory may provide a version of Hořava gravity that does not require theabsence of time derivatives of the lapse to avoid ghosts, and hence may not present the sametechnical hurdles [5, 6] in proving perturbative renormalizability (beyond power counting)that one encounters in FDiff (i.e. non-degenerate) Hořava gravity (at least in its generalnon-projectable form). Acknowledgments
We thank M. Herrero-Valea for insightful conversations on Hořava gravity and its pertur-bative renormalizability. We also thank S. Sibiryakov for pointing out an error in the firstversion of this manuscript. E.B, M.C. and L.t.H acknowledge financial support providedunder the European Union’s H2020 ERC Consolidator Grant “GRavity from Astrophysicalto Microscopic Scales” grant agreement no. GRAMS-815673. S.L. acknowledge fundingfrom the Italian Ministry of Education and Scientific Research (MIUR) under the grantPRIN MIUR 2017-MB8AEZ. – 13 –
Post-Newtonian Expansion
In order to calculate the PPN parameters, we follow [22] and consider a general perturbedflat metric in Cartesian coordinates ( x = ct, x i ) g = − − c φ − c φ (2) + O (cid:18) c (cid:19) ,g i = w i c + ∂ i ωc + O (cid:18) c (cid:19) ,g ij = (cid:18) − c ψ (cid:19) δ ij + (cid:18) ∂ i ∂ j − δ ij ∇ (cid:19) ζc + 1 c ∂ ( i ζ j ) + ζ ij c + O (cid:18) c (cid:19) , (A.1)where under transformations of the spatial coordinates, ψ, ζ, ω, φ, φ (2) transform as scalars, w i , ζ i behave as transverse vectors (i.e. ∂ i w i = ∂ i ζ i = 0 ), and ζ ij is a transverse and tracelesstensor (i.e. ∂ i ζ ij = ζ ii = 0 ). Since we want to use this ansatz in the field equations in theunitary gauge (Eqs. (5.4)–(5.6)), we are not allowed to perform a transformation of thetime coordinate (which is fixed to coincide with the khronon), but we can perform a gaugetransformation of the spatial coordinates to set ζ = ζ i = 0 [22].We supplement this ansatz with an expression for the energy-momentum tensor, whichwe assume to be given by the perfect fluid form T µν = (cid:18) ρ + Pc (cid:19) u µ u ν + P g µν , (A.2)where ρ is the matter energy density, P the pressure and u µ = dx µ /dτ the four-velocityof the fluid elements (with τ the proper time). In the following, we introduce a parameter η , in the action S = 116 πG (cid:90) d x d t N √ γ (cid:20) − η , (cid:18) λ + β + 23 (cid:19) (cid:0) V + 2 K V (cid:1) +(1 − β ) K ij K ij − (1 + λ ) K + α a i a i + (3) R (cid:105) , (A.3)in order to distinguish between (non-degenerate) Hořava gravity (which corresponds to η , = 0 ) from its degenerate generalization ( η , = 1 ).Expanding the evolution equation (5.6) to lowest order in /c , we find ζ ij = O (1 /c ) from the off-diagonal part, while the trace gives ψ = φ + O (cid:18) c (cid:19) . (A.4)From this we can write ψ = φ + δψc + O (cid:18) c (cid:19) , (A.5)which we can substitute in the other equations. Using this expression and expandingEq. (5.4) to lowest order in /c , we obtain the modified Poisson equation ∇ φ N = 4 πG N ρ + O (cid:18) c (cid:19) , (A.6)– 14 –here we define the rescaled gravitational constant G N = 2 G/ (2 − α ) . Notice that G N isthe gravitational constant measurable by a local experiment, while G is merely the baregravitational constant appearing in the action.We can then expand the momentum constraint (5.5) to lowest order in /c to find the1PN equation for the “frame-dragging” potential w i : ∇ w i + 2 (cid:18) β + λβ − (cid:19) ∂ i ∇ ω = 16 πGρv i − β − η , )3 (cid:18) β + 3 λβ − (cid:19) ∂ i ∂ t φ . (A.7)This is the first place where one can see a modification with respect to (non-degenerate)Hořava gravity.One can then expand both the trace of the evolution equation (5.6) and the khrononequation (5.4) to next-to-leading order in /c , obtaining respectively ∇ δψ = − πGp − πρv − (cid:16) α (cid:17) ∂ i φ∂ i φ − φ ∇ φ + (2 + β + 3 λ ) (cid:0) ∂ t ∇ ω + (3 + η , ) ∂ t φ (cid:1) , (A.8)and (cid:126) ∇ · (cid:34)(cid:18) − α (cid:19) (cid:126) ∇ (cid:18) φ + φ (2) c (cid:19)(cid:35) = 4 πGρ + 1 c (cid:16) πGρv + 12 πGp + (2 − α ) (cid:126) ∇ φ · (cid:126) ∇ φ −
16 (2 + β + 3 λ ) (cid:0) (3 + η , ) ∂ t ∇ ω + (9 + 7 η , ) ∂ t φ (cid:1)(cid:17) . (A.9)Then, we use the same methods and notation described in Appendix A of [22] and definethe potentials X ( (cid:126)x, t ) = G N (cid:90) d x (cid:48) ρ ( (cid:126)x (cid:48) , t ) | (cid:126)x − (cid:126)x (cid:48) | , (A.10) V i = G N (cid:90) d x (cid:48) ρ ( (cid:126)x (cid:48) , t ) v (cid:48) i | (cid:126)x − (cid:126)x (cid:48) | , (A.11) W i = G N (cid:90) d x (cid:48) ρ ( (cid:126)x (cid:48) , t ) (cid:126)v (cid:48) · ( (cid:126)x − (cid:126)x (cid:48) ) ( x − x (cid:48) ) i | (cid:126)x − (cid:126)x (cid:48) | , (A.12) Φ = G N (cid:90) d x (cid:48) ρ ( (cid:126)x (cid:48) , t ) v (cid:48) | (cid:126)x − (cid:126)x (cid:48) | , (A.13) Φ = − G N (cid:90) d x (cid:48) ρ ( (cid:126)x (cid:48) , t ) φ N ( (cid:126)x (cid:48) , t ) | (cid:126)x − (cid:126)x (cid:48) | , (A.14) Φ = G N (cid:90) d x (cid:48) P ( (cid:126)x (cid:48) , t ) | (cid:126)x − (cid:126)x (cid:48) | , (A.15)– 15 –hich obey the following relations ∇ X = − φ N , (A.16) ∇ V i = − πG N ρv i , (A.17) ∇ Φ = − πG N ρv , (A.18) ∇ Φ = 4 πG N ρφ N , (A.19) ∇ Φ = − πG N P, (A.20) ∂ i V i = ∂ t φ N , (A.21) ∂ i V i = − ∂ i W i , (A.22) ∂ t ∂ i X = W i − V i . (A.23)We can then take the divergence of Eq. (A.7) and solve it for ω , obtaining ω = 3 α + 2 η , + ( β + 3 λ )(3 + η , )6( β + λ ) ∂ t X . (A.24)Replacing this solution again into Eq. (A.7), we obtain w i = 2 − αβ − V i + W i ) . (A.25)This allows us to evaluate g i as g i = w i c + ∂ i ωc + O (cid:18) c (cid:19) = − η , (2 + 3 λ − β ) + β ( β + 3 λ )(3 + η , ) − α (1 + β + 2 λ ) + 3 λ + 9 β β − β + λ ) W i c + η , (2 + 3 λ − β ) − β ( β + 3 λ )(3 + η , ) + 3 α (1 − β − λ ) + 21 λ + 15 β β − β + λ ) V i c + O (cid:18) c (cid:19) . (A.26)We can also solve Eq. (A.9) for φ (2) φ (2) = φ N − − − + (3 α − β + η , ( α − β − λ ))(2 + β + 3 λ )6( α − β + λ ) ∂ t X . (A.27)While the solutions that we found completely describe the metric at 1PN order, inorder to read off the PPN parameters one needs to transform the metric from the unitarygauge that we used for the calculation to the standard PN gauge [22, 34, 35]. We do that,following again [22], by performing a gauge transformation t → t + δt , where we choose δt ∝ ∂ t X , with η , appearing in the transformation. This finally yields g = − − φ N c − φ N c + 4 Φ c + 4 Φ c + 6 Φ c + O (cid:18) c (cid:19) , (A.28) g i = − (cid:16) α − α (cid:17) V i c − (cid:16) α (cid:17) W i c + O (cid:18) c (cid:19) , (A.29) g ij = (cid:18) − φ N c (cid:19) δ ij + O (cid:18) c (cid:19) , (A.30)– 16 –rom which we can read off the parameters α and α α = 4( α − β ) β − , (A.31) α = η , − α + 3 β + 2 λ )(2 + β + 3 λ )3( α − β + λ )+ ( α − β )[ − β (3 + β + 3 λ ) − λ + α (1 + β + 2 λ )]( α − β − β + λ ) . (A.32) References [1] P. Horava, “Quantum Gravity at a Lifshitz Point,”
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