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Spatially Covariant Theories of a Transverse, Traceless Graviton: Formalism

Justin Khoury University of Pennsylvania, [email protected]

Godfrey E. J. Miller University of Pennsylvania

Andrew J. Tolley Case Western Reserve University

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Recommended Citation Khoury, J., Miller, G. E., & Tolley, A. J. (2012). Spatially Covariant Theories of a Transverse, Traceless Graviton: Formalism. Retrieved from https://repository.upenn.edu/physics_papers/233

Khoury, J., Miller, G. E. J., & Tolley, A. J. (2012). Spatially Covariant Theories of a Transverse, Traceless Graviton: Formalism. Physical Review D, 85(8), 084002. doi: 10.1103/PhysRevD.85.084002 © 2012 American Physical Society

This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/233 For more information, please contact [email protected]. Spatially Covariant Theories of a Transverse, Traceless Graviton: Formalism

Abstract General relativity is a generally covariant, locally Lorentz covariant theory of two transverse, traceless graviton degrees of freedom. According to a theorem of Hojman, Kucharˇ, and Teitelboim, modifications of general relativity must either introduce new degrees of freedom or violate the principle of local Lorentz covariance. In this paper, we explore modifications of general relativity that retain the same graviton degrees of freedom, and therefore explicitly break Lorentz covariance. Motivated by cosmology, the modifications of interest maintain explicit spatial covariance. In spatially covariant theories of the graviton, the physical Hamiltonian density obeys an analogue of the renormalization group equation which encodes invariance under flow through the space of conformally equivalent spatial metrics. This paper is dedicated to setting up the formalism of our approach and applying it to a realistic class of theories. Forthcoming work will apply the formalism more generally.

Disciplines Physical Sciences and Mathematics | Physics

Comments Khoury, J., Miller, G. E. J., & Tolley, A. J. (2012). Spatially Covariant Theories of a Transverse, Traceless Graviton: Formalism. Physical Review D, 85(8), 084002. doi: 10.1103/PhysRevD.85.084002

© 2012 American Physical Society

This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/233 PHYSICAL REVIEW D 85, 084002 (2012) Spatially covariant theories of a transverse, traceless graviton: Formalism

Justin Khoury,1 Godfrey E. J. Miller,1 and Andrew J. Tolley2 1Center for Particle Cosmology, Department of Physics & Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104, USA 2Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106, USA (Received 12 August 2011; published 2 April 2012) General relativity is a generally covariant, locally Lorentz covariant theory of two transverse, traceless graviton degrees of freedom. According to a theorem of Hojman, Kucharˇ, and Teitelboim, modifications of general relativity must either introduce new degrees of freedom or violate the principle of local Lorentz covariance. In this paper, we explore modifications of general relativity that retain the same graviton degrees of freedom, and therefore explicitly break Lorentz covariance. Motivated by cosmology, the modifications of interest maintain explicit spatial covariance. In spatially covariant theories of the graviton, the physical Hamiltonian density obeys an analogue of the renormalization group equation which encodes invariance under flow through the space of conformally equivalent spatial metrics. This paper is dedicated to setting up the formalism of our approach and applying it to a realistic class of theories. Forthcoming work will apply the formalism more generally.

DOI: 10.1103/PhysRevD.85.084002 PACS numbers: 04.50.Kd

I. INTRODUCTION modify the theory while retaining its explanatory power. The two transverse, traceless graviton degrees of freedom For nearly a century, general relativity has been the most are a key feature of general relativity. Though graviton successful paradigm for interpreting and understanding exchange has never been measured and gravitational waves classical gravitational phenomena. To this day, despite have never been detected, there is substantial indirect ongoing experimental efforts, there have been no un- equivocal refutations of general relativity. Nonetheless, evidence for these two graviton degrees of freedom [3]. there are compelling reasons to study alternative gravita- It is therefore natural to ask whether and how we can tional theories. modify the behavior of the known graviton degrees of Perhaps the most obvious reason to consider alternative freedom. gravitational theories is to explain empirical anomalies, In this paper, we construct manifestly consistent mod- most notably the observed magnitude of the cosmic ac- ifications of general relativity that retain the same local celeration. The CDM concordance model achieves a degrees of freedom. Since general relativity is the unique parsimonious fit to cosmological observations by invoking Lorentz covariant theory of a massless spin-2 particle a cosmological constant corresponding to a vacuum en- [4–7], our theories must break Lorentz covariance 4 explicitly. Theories in which Lorentz symmetry is only ergy density ðmeVÞ . However, known quantum 4 120 4 broken spontaneously necessarily rely on additional local corrections to are of order MPl 10 ðmeVÞ ,so CDM suffers from a serious fine-tuning problem [1]. degrees of freedom, which appear in the broken phase as Since we have no handle on the microphysics responsible massless Goldstone modes; an example of such a theory is for the magnitude of vacuum energy, it is an outstanding ghost condensation [8]. theoretical challenge to determine what physical degrees General relativity as formulated by Einstein and of freedom are associated with late-time acceleration. Hilbert is also a generally covariant theory, which means Theories of dynamical dark energy or modified gravity that the equations of motion for the spacetime metric typically introduce new scalar degrees of freedom, but to g take the same form in any coordinate system. date there is no unambiguous evidence for cosmologically Unfortunately, invariance under coordinate transforma- relevant scalars [2]. One motivation for this paper is the tions implies that the theory contains a great deal of possibility that cosmic acceleration might be directly gauge arbitrariness, and the true dynamical degrees of associated with the transverse, traceless graviton degrees freedom of the theory have proven difficult to isolate. of freedom. The inaccessibility of the physical graviton degrees of Apart from any attempt to understand empirical anoma- freedom is a significant obstacle to modifying their be- lies, there remains a compelling theoretical reason to study havior. In fact, the notorious elusiveness of the physical alternatives to general relativity: to determine which of its degrees of freedom is also an obstacle to the canonical features are essential to its experimental success, and quantization of general relativity [9]. which features are merely incidental. To analyze the theory This gauge arbitrariness can be understood most clearly in this manner, we must know what freedom we have to by treating general relativity as a constrained field theory.

1550-7998=2012=85(8)=084002(26) 084002-1 Ó 2012 American Physical Society KHOURY, MILLER, AND TOLLEY PHYSICAL REVIEW D 85, 084002 (2012)

By writing the spacetime metric g in Arnowitt-Deser- We wish our theories to retain the same local degrees of Misner (ADM) form1 and discarding a boundary term, the freedom as general relativity, so in accordance with the Einstein-Hilbert action can be rewritten in canonical form theorem of HKT, our theories cannot be Lorentz covariant. as a theory of a spatial metric hij and a conjugate momen- This aspect of our approach is not necessarily a defect. ij tum tensor subject to four constraints H [10,11]. Since we do not observe exact spacetime symmetry in our ij universe, this property of general relativity is not neces- Though hij and are not themselves generally covariant objects, the general covariance of the theory follows from sarily key to the success of the theory. Simply put, on cosmological scales there is a strong asymmetry between the first class character of the H ’s, which generate gauge transformations corresponding to spacetime diffeomor- the past and the future, and the observable universe has a phisms [12,13]. By representing gauge symmetries as preferred rest frame; these observations are conventionally constraints on phase space, it becomes straightforward to understood as a result of spontaneous symmetry breaking, count degrees of freedom. According to the standard count- but explicit symmetry breaking is another logical ing prescription, the presence of four first class constraints possibility. H in a theory of six canonical coordinates h ensures That being said, on cosmological scales in the cosmo- ij logical rest frame there is substantial evidence for spatial that general relativity contains two local degrees of homogeneity and isotropy. To maximize the verisimilitude freedom; schematically, of our treatment, the theories we consider will retain explicit covariance under spatial diffeomorphisms. To 0 H 0 6 hij s 4 s ¼ 2 degrees of freedom: (1) summarize, we will attempt to modify general relativity while preserving (1) the number of graviton degrees of freedom, and (2) explicit spatial covariance. In this paper, See Sec. II for more detail. In the passage to quantum we develop a general framework within which to explore theory, these transverse, traceless degrees of freedom be- the freedom we have to modify general relativity while come the two polarizations of the graviton. retaining these two desirable properties. To isolate the physical graviton degrees of freedom, one Concretely, we will begin by recasting general relativity H would have to solve the four constraints . One could in spatially covariant form, by solving the Hamiltonian then modify the behavior of the graviton in a straightfor- constraint (which generates local time reparametrizations) ward manner. By taking the configuration space for the while preserving the three momentum constraints (which spatial metric to be Wheeler’s superspace, it is possible to generate spatial diffeomorphisms). We will solve the H solve the three momentum constraints i by fiat, but the Hamiltonian constraint by choosing a cosmologically mo- H Hamiltonian constraint 0 has thus far defied solution in tivated gauge: we will take the determinant of the spatial general. Unless the Hamiltonian constraint can be solved, metric to be the measure of time. This operation destroys the the gauge arbitrariness of general relativity cannot be manifest diffeomorphism covariance and local Lorentz co- eliminated. Fortunately, though no general solution to the variance of the theory. We emphasize that this gauge breaks Hamiltonian constraint has been found, it can be solved in down in the general case when the determinant of the spatial certain circumstances by imposing an appropriate gauge- metric is allowed to evolve nonmonotonically, but it is a fixing condition. natural choice when considering perturbative corrections to We have established that modifying the behavior of the Friedmann-Robertson-Walker (FRW) spacetime. By solv- graviton without new degrees of freedom will force us to ing the Hamiltonian constraint, the determinant of the spa- break Lorentz covariance explicitly, but it is enlightening tial metric and the trace of the momentum tensor drop out of to see how this conclusion arises in the canonical picture. the phase space of the theory. We thereby obtain general H Under the action of the Poisson bracket, the ’s of ~ relativity as a theory of a unit-determinant metric hij and a general relativity obey the Dirac algebra [14,15], which traceless conjugate momentum tensor ~ij subject to three encodes the local Lorentz covariance of a generally cova- first class momentum constraints H~ , which act as the riant theory [16]. In 1974, Hojman, Kucharˇ, and Teitelboim i generators of spatial diffeomorphisms. By the standard (HKT) proved that general relativity is the unique minimal counting prescription, the presence of three first class con- representation of the Dirac algebra [17,18]. It follows H~ ~ immediately that Lorentz covariant modifications of gen- straints i in a theory of five canonical coordinates hij eral relativity introduce additional degrees of freedom guarantees that spatially covariant general relativity con- beyond the two graviton degrees of freedom in general tains two degrees of freedom, as it should; schematically, relativity [19]. To modify general relativity, one must 0 5 h~0 s 3 H~ s ¼ 2 degrees of freedom: (2) either introduce new degrees of freedom or violate the ij i principle of local Lorentz covariance. See Sec. III D for more detail. Our strategy for modifying general relativity relies on 1 0 i.e., in terms of a spatial metric hij, a lapse N N , and a the fact that any theory of five canonical coordinates sub- shift Ni. ject to three first class constraints contains two degrees of

084002-2 SPATIALLY COVARIANT THEORIES OF A TRANSVERSE, ... PHYSICAL REVIEW D 85, 084002 (2012) freedom. To modify general relativity, we will modify the Barbour, Koslowski, and collaborators on the theory of functional form of the physical Hamiltonian density on the Shape Dynamics [51,52]. ~ ij reduced phase space (hij, ~ ), subject to the condition that This paper is organized as follows. In Sec. II, we cover ~ the basic concepts of constrained field theory in the context the momentum constraints H i remain first class; to ensure the consistency of the modification, we will also demand of analyzing the phase space and constraint structure of general relativity. In Sec. III, we show how to impose our that the constraints H~ remain preserved by the equations i cosmological gauge condition and solve the Hamiltonian of motion. Any theory that satisfies these two restrictions constraint to obtain a consistent spatially covariant formu- will retain manifest spatial covariance, and by the counting lation of general relativity. In Sec. IV, we introduce the prescription will necessarily contain two graviton degrees formalism of our approach to modifying gravity in the of freedom. In this paper, we introduce the formalism context of ultralocal theories of the graviton. In Sec. V, necessary to pursue this program of modification and apply we apply our method to derive consistency relations for a the formalism to a class of realistic theories. Forthcoming class of realistic local theories which includes general work will apply the formalism developed here to the goal relativity. of constructing viable alternatives to general relativity [20]. In particular, these alternatives stand a good chance of being consistent with binary pulsar constraints, which II. GENERAL RELATIVITY AS A principally constrain the number of gravitational degrees CONSTRAINED FIELD THEORY of freedom. In this section, we will analyze general relativity by The literature abounds with many and varied approaches treating it as a constrained field theory. In particular, we to the pursuit of modified gravity theories, but generally will examine its phase space and constraint structure, and covariant, locally Lorentz covariant modifications of gen- count its local degrees of freedom. eral relativity that introduce additional degrees of freedom Our starting point is the Einstein-Hilbert action with a have been the most widely explored. The well-known cosmological constant, method for finding such theories is to construct a scalar Z Lagrangian density out of manifestly covariant objects by pffiffiffiffiffiffiffi S ¼ dtd3x gðRð4Þ 2Þ: (3) contracting all free spacetime/Lorentz indices.2 Using this technique, all manner of theories have been explored: From this action, the general covariance of the theory is scalar-tensor theories [21], theories with higher order cur- manifest, but the counting of degrees of freedom is not. vature terms [22–24], theories of massive gravity [25–28], The metric tensor g has ten components, but the theory higher-dimensional gravity theories [29–31], Galileons [32–37], chameleons [38], symmetrons [39,40], cuscutons has only two independent local degrees of freedom. To [41], etc. For a comprehensive review of Lorentz covariant facilitate the counting of degrees of freedom, it is concep- massive gravity theories with detailed references, see [7]. tually simplest to rewrite the action in canonical form, For a comprehensive review of observational tests of modi- which makes the counting manifest. To this end, the space- fied gravity, see [2]. time metric g must first be expressed in ADM form, in i Approaches to gravity which do not assume general terms of a lapse N, a shift N , and a spatial metric hij: covariance and local Lorentz covariance at the outset ds2 ¼ g dxdx have been tried as well. The natural procedure for con- 2 2 i i j j structing such a theory depends on which symmetries it is N dt þ hijðdx þ N dtÞðdx þ N dtÞ: (4) assumed to possess; more often than not, theories without i spacetime symmetry are assumed to maintain explicit spa- The lapse N is a three-scalar, the shift N is a three-vector, tial symmetry. For example, in [15] the action of gravity is and the spatial metric hij is a three tensor. Up to a boundary assumed to be invariant under spatial diffeomorphisms. In term, the Einstein-Hilbert action is equivalent to the ADM [42], Lorentz-violating massive graviton theories were action Z pffiffiffi classified by assuming the graviton mass to be invariant 3 ij 2 under the three-dimensional Euclidean group. A prominent S ¼ dtd x hNðKijK K þ R 2Þ: (5) recent example of a Lorentz-violating theory is Horˇava- Lifshitz gravity [43–50].3 Also of note is the work of In this expression, indices are lowered with hij and raised with its inverse hij, R Rð3Þ is the Ricci scalar of the metric hij, the extrinsic curvature tensor Kij is defined by 1 1 _ Kij N ðhij riNj rjNiÞ; (6) 2In the presence of spinor fields, one must treat spacetime 2 indices and Lorentz indices separately. K hijK , and r rð3Þ is the covariant spatial deriva- 3The original incarnation [43] of Horˇava-Lifshitz gravity ij i i struggled with consistency issues [46–48] which were resolved tive with respect to the metric hij. From Eq. (6), it is clear in [49] by imposing a consistent constraint algebra. that time derivatives in the action (5) act only on hij, not on

084002-3 KHOURY, MILLER, AND TOLLEY PHYSICAL REVIEW D 85, 084002 (2012) i N and N , so the lapse and shift are essentially nondynam- H 0: (16) ical. To obtain the canonical action, one must first define the momentum conjugate to the spatial metric, The symbol denotes weak equality, or equality after the constraints H 0 have been enforced. For example, if L pffiffiffi ij ¼ hðKij KhijÞ X ¼ Y þ H , then X Y. Since the constraints define _ ; (7) hij a surface in phase space, weak equality is also termed the momentum ij is a three-tensor density of unit weight.4 equality on the constraint surface. As an aside, it follows By inverting the relation between ij and Kij and dropping from (11) and (16) that H 0; the vanishing of the a boundary term, one can rewrite the action of general Hamiltonian on the constraint surface is a feature common relativity in canonical form as to generally covariant theories whose coordinates and Z momenta transform as scalars under time reparametriza- 3 ij _ H tions [12]. S ¼ dtd xð hij N Þ; (8) There is no N_ term that would allow us to compute a _ where N0 N and variational expression for N , so the time evolution of N   is unconstrained by the action. The four functions N are pffiffiffi 1 1 H ffiffiffi ij i 2 thus arbitrary until and unless we gauge-fix them. 0 hðR 2Þþp ij ð Þ ; h 2 i jk H i 2hijrk : (9) Constraint properties and degrees of freedom Before examining the constraints more closely, we pause Variation of the action (8) with respect to h and ij yields ij to review some terminology first introduced by Dirac for Hamilton’s equations, describing constrained theories [14]. A quantity whose H H Poisson bracket with each of the constraints vanishes h_ ðxÞ¼ ; _ ijðxÞ¼ ; (10) ij ij (identically or weakly) is termed first class; a quantity ðxÞ hijðxÞ whose Poisson bracket fails to vanish weakly with at least where the Hamiltonian H is one constraint is termed second class.Afirst class con- Z straint has vanishing Poisson bracket with all constraints, 3 H H ¼ d xN : (11) while a second class constraint has nonvanishing Poisson bracket with at least one other constraint. In most cases of To evaluate the above variational derivatives, one must use interest, first class constraints generate gauge symmetries the relations under the action of the Poisson bracket. Second class constraints can usually be solved, either implicitly by using h ðxÞ klðxÞ ij ¼ ¼ kl3ðx yÞ; (12) the ‘‘Dirac bracket’’ [53], or explicitly by expressing some ij ij 5 hklðyÞ ðyÞ phase space variables in terms of others. where By direct calculation—see Appendix A for details—it is possible to prove that the constraints H are first class, kl 1ð k l þ l kÞ ij 2 i j ij : (13) fH ðxÞ; H ðyÞg 0. This means that the symmetry gen- erators close under the action of the Poisson bracket, as Defining the Poisson bracket they must in order to consistently represent a gauge sym- Z   3 A B A B metry. In particular, fA; Bg d z mn mn ; hmnðzÞ ðzÞ ðzÞ hmnðzÞ i 3 fH ðxÞ; H ðyÞg ¼ H ðxÞ@ i ðx yÞ (14) 0 0 x H i 3 ðyÞ@yi ðx yÞ; the equation of motion for any quantity Aðh ;ij;tÞ can be ij H H H 3 written as f 0ðxÞ; iðyÞg ¼ 0ðyÞ@xi ðx yÞ; Z H H H 3 @A @A f iðxÞ; jðyÞg ¼ jðxÞ@xi ðx yÞ A_ ¼ þfA; Hg¼ þ d3yNðyÞfA; H ðyÞg: (15) H 3 @t @t iðyÞ@yj ðx yÞ: (17) If A has no explicit dependence on time, its evolution is This is the Dirac algebra, first discovered by Dirac in the H generated by its Poisson bracket with the ’s. context of parametrized field theories in flat spacetime Variation of the action with respect to N yields the four [14,15]. The gauge symmetry corresponding to this first constraints 5See [12] for a pedagogical treatment of the general theory of 4 According to the standard convention,pffiffiffi the weight of a tensor constrained systems; see [13] for an in-depth analysis of several density is the number of times h multiplies the underlying interesting constrained systems, including electromagnetism and tensor. general relativity.

084002-4 SPATIALLY COVARIANT THEORIES OF A TRANSVERSE, ... PHYSICAL REVIEW D 85, 084002 (2012) ij class algebra is general covariance, and the constraints that any modification of the action for hij and will H generate spacetime diffeomorphisms. destroy this algebra. If we modify the action for the phase ij The geometrical significance of the Dirac algebra was space variables hij and , we must impose an alternative determined by Teitelboim in [16]: it is the algebra of the constraint structure that consistently constrains the phase deformations of a spacelike hypersurface embedded in a space to the same degree as the covariance algebra; this is Lorentzian spacetime manifold. When the H ’s satisfy the approach taken in [15], as well as in [49,51,52]. Since H (17), 0 generates deformations normal to the surface, we take the point of view that full spacetime covariance is a while the H i’s generate deformations parallel to the sur- spurious symmetry, we do not wish our theory to contain a face. The Dirac algebra thus encodes the local Lorentz constraint structure that implies the same degree of redun- covariance of a generally covariant system. In fact, four dancy as the Dirac algebra. first class constraints obeying the Dirac algebra are guar- Though spacetime symmetry is manifestly broken on anteed to arise in any generally covariant field theory cosmological scales (whether spontaneously or explicitly), which satisfies the principle of local Lorentz covariance. there is strong evidence for spatial homogeneity and For the constraints to be consistent with the equations of isotropy, so we will attempt to modify general relativity motion, the constraints must be preserved by the equations while preserving the manifest spatial covariance of the _ of motion, i.e., H 0. Since @H =@t ¼ 0, applying theory. To obtain a spatially covariant formulation of gen- the equations of motion to H yields eral relativity to modify, we will solve the Hamiltonian H Z constraint 0 while leaving the three momentum con- H H_ ðxÞ¼ d3yNðyÞfH ðxÞ; H ðyÞg: (18) straints i intact. The Hamiltonian constraint is famously hard to solve in general, but we are interested in using our From the first class character of the constraints, it follows theories in a cosmological context, so we will solve it using a gauge-fixing constraint which is well-defined on an ex- that H_ 0, as desired. panding FRW background. The Hamiltonian formulation of GR is a theory of a spatial metric h and its conjugate momentum ij, so the ij A. Metric decomposition theory contains 12 canonical (or 6 real) variables. However, these variables are not independent. First, they Before gauge-fixing, we decompose the metric hij into a 1=3 are related by the four constraints H 0. Second, from conformal factor h and a unit-determinant metric ~ Eqs. (10) and (11) it follows that the equations of motion hij, i.e., for h and ij depend on the four arbitrary functions N; ij ¼ ~ to gauge-fix N would require imposing four gauge-fixing hij hij: (20) pffiffiffi constraints [12]. Note that ¼ð hÞ2=3 is a three-scalar density of weight ~ 0 ij0 H 0 0 2=3, while hij is a three-tensor density of weight 2=3. 6 hijs þ 6 s 4 s 4 N s The scalar density we will work withp isffiffiffi not the conformal ¼ 4 canonical DoF: (19) factor , but the volume factor ! h ¼ 3=2, which is The theory therefore has four canonical (or two real) a scalar density of unit weight. We choose ! because its degrees of freedom. conjugate momentum, i L 2i 4 ! ¼ ¼ K; (21) III. SPATIALLY COVARIANT !_ 3! 3 GENERAL RELATIVITY is a three-scalar and hence invariant under spatial confor- mal transformations, which rescale or !; this fact will We would like to depart from general relativity by simplify matters in Secs. IV and V. The momentum con- modifying the equations of motion for the two graviton ~ jugate to hij is degrees of freedom. Ideally, we would like to solve all four     gauge constraints, go down to the physical phase space, ij L ij 1 ij k ij 1 ij and modify the theory at that level. In this way, we would ~ ¼ h k ¼ ! K Kh ; h~_ 3 3 circumvent all the difficulties of consistently modifying a ij constrained field theory. Unfortunately, we do not know (22) how to do this. which is a traceless three-tensor density of weight 5=3; the One possible approach is to modify the equations of quantity motion for the phase space variables h and ij. ij ~ij However, the counting of degrees of freedom in general ~ ij (23) T ! relativity relies on the fact that the four constraints H satisfy a consistent first class algebra, namely, the Dirac is the corresponding traceless three-tensor. By defining the ~kl algebra of Eq. (17), and we know from the HKT theorem traceless projection tensor ij

084002-5 KHOURY, MILLER, AND TOLLEY PHYSICAL REVIEW D 85, 084002 (2012) ~ kl kl 1 ~ ~kl kl 1 kl 0: (29) ij ij 3hijh ; ¼ ij 3hijh ; (24) we can write ~ij more compactly as By direct calculation—see Appendix B for details—one can verify that ij ~ij kl ~ij kl ~ ¼ kl ¼ !klK : (25) fH ðxÞ;ðyÞg ¼ 1iðxÞ3ðx yÞ; (30) ij 0 2 i The phase space variables hij and can thus be written as the constraints H and are thus second class, so we 2=3 ~ ij 2=3 ij 1 ~ij 1=3 0 hij ¼ ! hij;¼ ! ~ þ 2h ! !: (26) expect to be able to solve them. The only wrinkle is that pffiffiffiffiffiffiffiffiffi The decomposition of the spatial metric into a volume 3 fH ðxÞ;ðyÞg ¼ hðxÞ@ i ðx yÞ; (31) factor and a unit-determinant metric is completely general. i x Though the corresponding conjugate momenta were de- so the constraints H i are also second class! By shuffling rived by taking variational derivatives of the Einstein- our constraints slightly, we can obtain a set of two second Hilbert Lagrangian, the decomposition of the momentum class constraints and three first class constraints, and tensor into its trace part and its traceless part is likewise thereby render explicit the spatial covariance of the completely general. Those familiar with the techniques of gauge-fixed action. Indeed, since numerical relativity may be reminded of the York-     pffiffiffiffiffiffiffiffiffi H pffiffiffiffiffiffiffiffiffi Lichnerowicz conformal decomposition or the BSSNOK 0ðxÞ 3 2 hðxÞ@xi k ;ðyÞ hðxÞ@xi ðx yÞ; (32) (Baumgarte, Shapiro, Shibata, Nakamura, Oohara, and kðxÞ Kojima) formalism [54,55]. it follows that the combination ffiffiffi   ffiffiffi   B. Cosmological gauge p H p H H~ H 2 h@ 0 ¼ H 2 hr 0 (33) H i i i k i i k To solve the constraint 0, we must first gauge-fix the k k lapse N with a gauge-fixing constraint for which obeys H = H f 0;g0; this renders 0 second class, and hence H~ H~ H~ H H~ solvable. This process destroys manifest spacetime covari- f iðxÞ; jðyÞg f iðxÞ; 0ðyÞg f iðxÞ;ðyÞg 0: ance. Since we wish to retain explicit spatial covariance, (34) we wish our constraints H i to remain first class. In a cosmological context, it is natural to use the volume The interpretation of this result is simple. The H i’s gen- factor of the spatial metric as a clock, so that t ¼ tð!Þ;we H erate spatial diffeomorphisms, while 0 generates time call this cosmological gauge. As mentioned in the translation. A generic spatial diffeomorphism will alter the introduction, cosmological gauge is only valid when the conformal factor of the spatial metric. If the conformal determinant of the spatial metric evolves monotonically, so factor is taken to be the measure of time, then the H i’s, by this procedure is only valid when considering perturbative altering the conformal factor, will generate time transla- corrections to FRW spacetime. When the evolution of ! is H tion, while 0, by generating time translation, will alter monotonic, tð!Þ is an invertible function, so this gauge is the conformal factor. The H~ ’s generate spatial diffeo- equivalent to taking the volume factor ! to be a function of i morphisms and preserve the conformal factor, so they must time, i.e., ! ¼ !ðtÞ. differ from the H ’s by the gradient of a compensating For cosmological purposes, another good gauge choice i time translation term. would be to take to be a function of time. Since ¼ ! ! H~ 4K=3, this is equivalent to the constant mean curvature From the definition of i, it is apparent that demanding H gauge, in which the trace of the extrinsic curvature tensor 0 and 0 is equivalent to demanding 0, ij H H~ K ¼ hijK is chosen to be a function of time. This gauge 0 0, and i 0. The latter set of constraints has ~ will not be used in the present work, but the constant mean the virtue that the H i are first class, which makes manifest curvature gauge is used in [15,41] and mentioned in [54,55]. the presence of the remaining three gauge symmetries. To impose cosmological gauge, we add to the canonical We therefore take our five constraints to be the two second H action of general relativity a gauge-fixing constraint class constraints and 0 and the threepffiffiffi first class con- H~ H ¼ H~ þ r ðH kÞ ! !ðtÞ; (27) straints i. Using i i 2 h i 0=k , the ~ gauge-fixed action can be rewritten in terms of H i as along with a corresponding Lagrange multiplier . The Z  new gauge-fixed action is 0 3 ij _ 0 i ~ S ¼ dtd x hij N H 0 N H i Z 0 3 ij _    S ¼ dtd xð hij N H Þ: (28) pffiffiffi H 2 hNir 0 : (35) i k Varying the action with respect to then reproduces the k constraint Upon integration by parts, the action becomes

084002-6 SPATIALLY COVARIANT THEORIES OF A TRANSVERSE, ... PHYSICAL REVIEW D 85, 084002 (2012) Z 0 3 ij _ ~H iH~ 0: (43) S ¼ dtd xð hij N 0 N i Þ; (36) ~ Since fðxÞ;ðyÞg ¼ 0, fðxÞ; H iðyÞg 0, and @=@t ¼ where !_ ðtÞ, it follows that pffiffiffi Z h ~ i _ ðxÞ!_ ðtÞþ d3yN~ðyÞfðxÞ; H ðyÞg; N N 2 k riN : (37) 0 k 1 Variation of the action S0 with respect to h and ij yields !_ ðtÞ N~ðxÞi ðxÞ: (44) ij 2 i Hamilton’s equations, Since i = 0, demanding _ 0 allows us to solve for N~, H0 H0 i _ ij h ijðxÞ¼ ij ; _ ðxÞ¼ ; (38) 2_!ðtÞ ðxÞ hijðxÞ ~ N i : (45) i where the new Hamiltonian H0 is ~ Z The functionspffiffiffi N and are thus not arbitrary. Since N ¼ 0 3 ~H iH~ ~ i k H ¼ d xðN 0 þ N i þ Þ: (39) N þ 2 hðriN Þ= k, the lapse N has not been completely gauge-fixed, but its arbitrariness stems solely from its ij i The equation of motion for any quantity Aðhij; ;tÞ is dependence on the three arbitrary functions N . therefore As a check, let us revisit the counting of degrees of freedom in cosmological gauge. For these purposes, the @A A_ ¼ þfA; H0g only effect of gauge-fixing is to replace the first class @t H Z constraint 0 0 and the arbitrary function N with the @A second class constraints H 0 and 0. This modifies ¼ þ d3yðN~ðyÞfA; H ðyÞg þ NiðyÞfA; H~ ðyÞg 0 @t 0 i the left-hand side of Eq. (19), but does not change the final þ ðyÞfA; ðyÞgÞ; (40) tally. 0 ij0 H H~ 0 i0 where the Poisson bracket is defined as in (14). Variation of 6 hij s þ 6 s 1 0 1 3 i s 3 N s 0 ~ i the action S with respect to N, , and N yields the five ¼ 4 canonical DoF: (46) constraints After gauge-fixing, the theory still has four canonical (or H H~ 0 0; 0; i 0: (41) two real) degrees of freedom. The action does not contain time derivatives of the Lagrange multipliers, so at first their evolution appears C. The action of spatially covariant general relativity ~ H unconstrained. Since the H i are first class, the three In this section, we will solve the constraints 0 and functions Ni are indeed arbitrary until and unless we to obtain a spatially covariant formulation of general rela- ~ gauge-fix them. The evolution of N~ and , however, will tivity as a theory of a unit-determinant metric hij and its H ij be determined by demanding the consistency of 0 and conjugate momentum ~ . This will set the stage for with the equations of motion. modifying general relativity in Sec. IV. H For the constraints to be consistent with the equations of Since and 0 are second class, they can be solved motion, they must be preserved by the equations of motion; explicitly to yield expressions for ! and ! in terms of t, H~_ H_ h~ , ~ij, and spatial derivatives. ‘‘Solving’’ for ! is trivial: we therefore demand that i 0, 0 0, and _ 0. ij ~ ~ ! ¼ !ðtÞ. Solving for requires us to take a square root Since the H i are first class and @tH i ¼ 0, it follows at ! H~_ H H H and pick a sign, which amounts to picking either an ex- once that i 0. Since @t 0 ¼ 0, f 0ðxÞ; 0ðyÞg panding or a contracting background. We pause to empha- H H~ 0, and f 0ðxÞ; iðyÞg 0, it follows that size once again that our procedure is only valid in a Z 1 cosmological context, when the conformal factor of the H_ 3 H i 0ðxÞ d yðyÞf 0ðxÞ;ðyÞg ðxÞ ðxÞ: (42) spatial metric can be assumed to be evolving monotoni- 2 i cally. To pick the sign corresponding to an expanding 6 i On a flat FRW background, K ¼ 3a=a_ and hence i ¼ background, first recall that 2 2!K ¼6aa_ . Since we are only considering gravity 4 i = ! ¼K: (47) on an expanding background, we assume that iðxÞ0 3 H_ more generally. The demand 0 0 thus implies On a flat FRW background, K ¼ 3a=a_ and hence ! ¼ 4a=a_ . An expanding FRW background therefore corre- 6 A spatially-flat FRW spacetime corresponds to N ¼ 1, Ni ¼ sponds to ! < 0. Returning to the general case, we 2 0, and hij ¼ a ðtÞij. choose ! < 0 to obtain

084002-7 KHOURY, MILLER, AND TOLLEY PHYSICAL REVIEW D 85, 084002 (2012) sffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Aðh~ ; ij;tÞ 8 ~ ~ij R~ any quantity ij ~ obeys the equation of motion ¼ ij þ 2; (48) ! GR 3 !2 !2=3 @A A_ ¼ þfA; H00g: (57) ~ @t where indices are raised and lowered with hij, and R~ is the ~ i Ricci scalar for hij. Substituting these results for ! and ! Variation of the action with respect to N yields the three back into the action S0 yields the action of general relativity constraints on the reduced phase space (h~ , ~ij), ij H~ Z i 0: (58) S00 ¼ dtd3xð~ijh~_ þ !_ NiH~ Þ; (49) ij ! i As before, the time evolution of Ni is unconstrained by the action; in the absence of a gauge-fixing procedure, the where three functions Ni are arbitrary. ~ ~ ~ jk ~ H i ¼2hijrk~ !ri!; (50) D. Constraint properties and degrees of freedom ~ ~ 7 and ri is the covariant derivative with respect to hij. As ~ By lengthy direct calculation, it is possible to prove discussed in Sec. III D, the constraints H i remain first ~ that the constraints H i are first class, i.e., class and continue to represent spatial covariance, so (49) H~ H~ is the action of spatially covariant general relativity. This f iðxÞ; jðyÞg 0. Furthermore, by applying the equa- ~ ~_ action yields the new Hamiltonian tions of motion to H i, it is possible to show that H i 0, Z so the constraints are preserved by the equations of motion. 00 3 i ~ H ¼ d xð!_ ! þ N H iÞ: (51) We defer demonstrations of these two facts to Sec. V, where we will examine general relativity in the context ij _ ij ~_ of a class of realistic theories. This is an important con- The term hij has split into the term ~ hij and a con- tribution !_ to the physical Hamiltonian density. sistency check, because a priori it is not clear that our ! procedure for solving the Hamiltonian constraint will yield Variation of the action with respect to h~ and ~ij yields ij a consistent action on the reduced phase space. Hamilton’s equations, As a final check, we revisit the counting of degrees H00 H00 of freedom in spatially covariant general relativity. After h~_ ðxÞ¼ ; ~ij ~_ abðxÞ¼ : (52) ij ~ijðxÞ ab h~ ðxÞ imposing cosmological gauge and solving the Hamiltonian ij H constraint 0 0, general relativity is a theory of a unit- ~ To evaluate these variational derivatives, one must use the determinant spatial metric hij and its traceless conjugate relations momentum ~ij, so the theory contains ten canonical (or 5 real) variables. This reduction in the size of the phase space h~ ðxÞ ij ¼ ~kl3ðx yÞ; is compensated by a corresponding reduction in the ~ ij hklðyÞ number of constraints and arbitrary functions: the theory ~ijðxÞ 1 contains three first class constraints H~ 0, and its equa- ¼ h~ij~kl3ðx yÞ; (53) i ~ tions of motion involve three arbitrary functions Ni. hklðyÞ 3 ~ 0 ij0 H~ 0 i0 and 5 hij s þ 5 ~ s 3 i s 3 N s h~ ðxÞ ~ijðxÞ ¼ 4 canonical DoF: (59) ij ¼ 0; ¼ ~ij 3ðx yÞ; (54) ~klðyÞ ~klðyÞ kl Spatially covariant general relativity thus contains four from which follow the operator identities canonical (or two real) degrees of freedom, the same number as fully covariant general relativity. ¼ ~ij ; ¼ ~ab : (55) h~ ab h~ ~ij ij ~ab ij ab E. Modifying spatially covariant general relativity Defining the Poisson bracket appropriate to the reduced We have two criteria in mind for our modified theories of phase space, gravity: two graviton degrees of freedom, and manifest   Z A B A B spatial covariance. Our starting point is the action (49)of fA;Bg d3x ; (56) spatially covariant general relativity, which has both of h~ ðxÞ ~ijðxÞ ~ijðxÞ h~ ðxÞ ij ij these properties. To modify general relativity, we will change the functional form of the scalar quantity !, 7 ~ ¼ The distinction between H i and H i in Eq. (33) vanishes which in general relativity obeys ! GR. This yields H identically after solving 0. the action

084002-8 SPATIALLY COVARIANT THEORIES OF A TRANSVERSE, ... PHYSICAL REVIEW D 85, 084002 (2012) Z motion requires to satisfy an analogue of the renormal- S ¼ dtd3xð~ijh~_ þ !_ NiH~ Þ (60) ! ij ! i ization group equation; scalar momenta satisfying this equation are manifestly invariant under spatial conformal with constraints rescaling of the volume factor !. In the ultralocal case, this H~ ¼ h~ r~ jk !r~ ; is the only consistency condition that arises. In the local i 2 ij k ~ i ! (61) ~ case, demanding that the constraints H i satisfy a first class where ! is an unspecified scalar function of t, the phase algebra is equivalent to demanding that ! obey a rather ~ ij space variables hij and ~ , and spatial derivatives: complicated differential equation.

¼ ðt; h~ ; ~ij;@Þ: (62) ! ! ij i IV. ULTRALOCAL MODIFIED GRAVITY This action leads to the equation of motion The ultralocal limit of a theory is achieved by neglecting @A all terms in the action which are second order or higher in A_ ¼ þfA; Hg; (63) spatial derivatives. Conceptually, this is the limit in which @t each point in space evolves independently of the points where the Hamiltonian H is around it. The ultralocal truncation of a theory is a good Z approximation to the full theory whenever spatial gradients 3 i ~ of fields are small compared to the fields themselves and H ¼ d xð!_ ! þ N H iÞ; (64) their time derivatives. This makes it a natural limit to take in cosmology. and the Poisson bracket is In general relativity, the ultralocal limit simplifies the   Z A B A B form of the Hamiltonian constraint while (1) preserving the fA;Bg d3x : (65) h~ ðxÞ ~ijðxÞ ~ijðxÞ h~ ðxÞ momentum constraints and (2) maintaining a consistent ij ij constraint algebra [56]. This approximation has proven Retaining the manifest spatial covariance of the theory fruitful for analyzing both long wavelength cosmological ~ perturbations [57] and for studying physics near cosmo- amounts to demanding (1) that the modified H i remain first class, i.e., logical singularities [58,59]. The idea of using cosmologi- cal gauge in the ultralocal limit, sometimes referred to as fH~ ðxÞ; H~ ðyÞg 0; (66) the separate universe picture, is treated in [56]. i j In our approach, the ultralocal limit simplifies the form and (2) that the modified constraints be preserved by the of the physical Hamiltonian density ! while preserving ~ modified equations of motion, i.e., the form of the momentum constraints H i. There are two terms in the action (60) which contain !, namely !_ ! ~_ i H i 0: (67) and !N @i!. Though ! appears in the action without a spatial gradient acting on it, vector indices in our theory Any theory satisfying these two points will be manifestly only arise from spatial gradients, so ! cannot contain covariant under spatial diffeomorphisms, with the con- terms linear in spatial gradients. In the ultralocal limit, ! ~ ~ ij straints H i acting as the generators of the gauge symme- is thus a scalar function of t, hij, and ~ that does not try. Moreover, the presence of three first class constraints contain spatial derivatives, i.e., ~ ~ ij H i on the phase space ðhij; ~ Þ guarantees that such a ~ ij theory contains two local degrees of freedom, exactly as ! ¼ !ðt; hij; ~ Þ: (68) desired. In the remainder of the paper, we examine two classes of When ! is of this form, we will say that ! is an ultra- theories. First, for pedagogical purposes, we assume that local function of the phase space variables. We will now does not contain spatial derivatives; this is the ultra- ! show that this form for ! leads to a first class constraint local case. Second, to make contact with general relativity, algebra. we allow ! to depend on spatial derivatives through R~, ~ the Ricci scalar of hij; this is the local case. Forthcoming A. Constraint algebra work will examine more general classes of scalar momenta [20]. In Sec. IV, we use the ultralocal case to introduce the In this section, we will compute the Poisson bracket ~ ~ ~ fH iðxÞ; H aðyÞg assuming that ! is an ultralocal func- formalism needed to determine when the constraints H i remain first class and when the constraints are preserved by tion, and use the result to determine when the constraints H~ the equations of motion. In Sec. V, we apply the formalism i remain first class. To simplify the calculation of ~ ~ ~ to the local case. In both the ultralocal and the local case, fH iðxÞ; H aðyÞg, we split H i into a tensor part J i and a the consistency of the constraints with the equations of scalar part Ki. Concretely, we define the vector densities

084002-9 KHOURY, MILLER, AND TOLLEY PHYSICAL REVIEW D 85, 084002 (2012) Z ~ ~ jk ~ J i 2hijrk~ ; Ki !ri!; (69) 3 jk ~ i ~ ~ ~ i jk FJ ¼ d xf2~ ðrkf Þhij þ 2hijðrkf Þ~

H~ ~ jk ~ i in terms of which i becomes simply þ 2hij~ rkf g: (75)

H~ ¼ J þ K : (70) The first two terms in this integral are in a convenient form i i i ~ for taking variational derivatives with respect to hij and ~ ~ jk The Poisson bracket fH iðxÞ; H aðyÞg can then be written ~ , but the third term requires finessing. To evaluate ~ i ~ i i as the sum of more manageable brackets, rkf , expand the covariant derivative as rkf ¼ @kf þ ~i r ~i ~ krf , where jk is the connection of the metric hij.It H~ H~ J J K K ~ i r ~i f iðxÞ; aðyÞg ¼ f iðxÞ; aðyÞg þ f iðxÞ; aðyÞg follows immediately that rkf ¼ f kr. The identity

þfJ iðxÞ; KaðyÞg þ fKiðxÞ; J aðyÞg: ~i 1 ~im ~ ~ ~ kr ¼ 2h ðrrhkm þrkhrm rmhrkÞ (76) (71) jk ~ ~ i i jk ~ ~ thus implies that 2~ hijrkf ¼ f ~ rihjk,so To simplify the evaluation of these component Poisson Eq. (75) becomes brackets, we will first compute the Poisson brackets of Z 3 jk ~ i ~ ~ ~ i jk the smoothing functionals FJ ¼ d xf2~ ðrkf Þhij þ 2hijðrkf Þ~ Z Z i jk ~ ~ 3 i 3 i þ f ~ rihjkg: (77) FJ d xf J i;FK d xf Ki; Z Z Integrating by parts, this reduces to 3 a 3 a GJ d yg J a;GK d yg Ka; (72) Z 3 jk ~ i ~ ~ i jk ~ FJ ¼ d xf2~ ðrkf Þhij riðf ~ Þhjk where the functions fi and gi are time-independent þ 2h~ ðr~ fiÞ~jkg: (78) smoothing functions. We make the key assumption that ij k the smoothing functions decay so rapidly at infinity that From this expression, it is straightforward to compute when we integrate by parts inside the smoothing func- variational derivatives of FJ, tionals, the boundary term vanishes identically; the F 2 smoothing functions are otherwise arbitrary. With the free- J ¼ 2~mn~jkr~ fi r~ ðfi~mnÞ ~mnr~ fi; ~ ij k i i dom to integrate by parts at will, it is straightforward to hmn 3 compute variational derivatives of the smoothing function- F J ¼ 2~jk h~ r~ fi: (79) als, and thereby to obtain explicit expressions for their ~mn mn ij k Poisson brackets. To obtain the brackets of the vector densities from the brackets of the smoothing functionals, The corresponding results for GJ are we will use the relations G 2 J ¼ 2~mn~bcr~ ga r~ ðga~mnÞ ~mnr~ ga; Z ~ ab c a a hmn 3 3 3 i a J J fFJ;GJg¼ d xd yf ðxÞg ðyÞf iðxÞ; aðyÞg; G J ¼ 2~bc h~ r~ ga: (80) Z mn mn ab c 3 3 i a ~ fFJ;GKgþfFK;GJg¼ d xd yf ðxÞg ðyÞ The variational calculation for the smoothing functional

ðfJ iðxÞ;KaðyÞgþfKiðxÞ;J aðyÞgÞ; FK is less straightforward. After integrating by parts, FK Z becomes 3 3 i a fFK;GKg¼ d xd yf ðxÞg ðyÞfKiðxÞ;KaðyÞg: Z 3 i FK ¼ ! d xð@if Þ!; (81) (73) from which it follows that The fact that Eq. (73) must hold for all sufficiently well- Z behaved functions f and g will allow us to derive explicit 3 i FK ¼ ! d xð@if Þ!: (82) expressions for the Poisson brackets involving J i and Ki. To compute variational derivatives of the smoothing To evaluate ! in full generality would be very difficult, functional FJ, first integrate by parts to obtain so we will make some simplifying assumptions about the Z form of !. In this section, we will assume that ! is an ¼ 3 ~ jkr~ i ~ ij FJ 2 d xhij~ kf ; (74) ultralocal function of t, hij, and ~ . To facilitate calculations, we will enumerate all the ~ from which it follows that scalars that can be built by contracting factors of hij against

084002-10 SPATIALLY COVARIANT THEORIES OF A TRANSVERSE, ... PHYSICAL REVIEW D 85, 084002 (2012) Z factors of ~ij. We begin by recursively defining ijðnÞ, the fF ;G g¼2 d3zfðr~ fiÞðr~ gaÞh~ ~bc linked chain of n factors of ~ij. The chain of zero factors of J J c i ab ij ~ is simply ~ a ~ i ~ jk ðrkg Þðraf Þhij~ ij ~ij ~ i ~ a ~ jk ð0Þh : (83) þðrkf Þraðg hij~ Þ ~ a ~ i ~ bc The process of adding a link to the chain is defined by ðrcg Þriðf hab~ Þg: (90)

ij i kj After integrating by parts, using the definition J i ¼ ðn þ 1Þ~ k ðnÞ: (84) ~ ~ jk ~ ~ 2hijrk~ , and using the identity ðrirj By closing the chain, one obtains scalars, ~ ~ a ~a b rjriÞV ¼ R bijV , this reduces to ðnÞi ðnÞ: (85) Z i 3 i ~ a a ~ i fFJ;GJg¼ d zff J arig g J iraf The ðnÞ are the only scalars that can be built out of i a jk ~ ~ ~ ij þ 2f g ~ ðRjika þ RjaikÞg: (91) connected contractions of hij and ~ . Since ð0Þ¼3 and ð1Þ¼0, ð0Þ¼ð1Þ¼0. For an arbitrary From the symmetries of the Riemann tensor8 and ultralocal function !, it follows that the traceless momentum tensor,9 it follows that ~jkðR~ þ R~ Þ¼0, so the last term in the inte- X1 @ jika jaik ¼ ! ðnÞ: (86) grand vanishes. The connection terms inside the ! @ðnÞ n¼2 remaining covariant derivatives cancel to yield Z For n 2, the variational derivatives of the ðnÞ are fF ;G g¼ d3zðfiJ @ ga gaJ @ fiÞ: (92)   J J a i i a ðnÞðxÞ ~mn ab 1 mn 3 ¼ ab nðnÞ ~ nðn 1Þ ðx yÞ; fJ ðxÞ; J ðyÞg h~ ðyÞ 3 To extract the bracket i a from this re- mn sult, first relabel dummy indices ðnÞðxÞ ~ab 3 Z Z ¼ mnnðn 1Þab ðx yÞ: (87) ~mnðyÞ 3 i a 3 a i fFJ;GJg¼ d xf J a@ig d yg J i@af : (93) The variational derivatives of F are thus K Under the spatial derivatives in this equation, insert   F X1 @ the identities K i ! ~mn jk 1 mn ¼ !ð@if Þ n jk ðnÞ ~ ðn1Þ ; Z h~ @ðnÞ 3 mn n¼2 gaðxÞ¼ d3y3ðx yÞgaðyÞ; X1 FK i @! jk Z ¼ !ð@ f Þ n ~mnðn1Þ : (88) mn i jk i 3 3 i ~ n¼2 @ðnÞ f ðyÞ¼ d x ðx yÞf ðxÞ; (94)

Similarly, the variational derivatives of GK are to obtain X1 Z GK a @! 3 3 i a 3 ¼ !ð@ g Þ m fF ;G g¼ d xd yf ðxÞg ðyÞðJ ðxÞ@ i ðx yÞ ~ a J J a x hmn m¼2 @ðmÞ   3 J iðyÞ@ya ðx yÞÞ: (95) ~mn bc 1 mn bc ðmÞ ~ ðm 1Þ ; 3 Comparing this expression to Eq. (73) yields the G X1 @ identity K ¼ !ð@ gaÞ m ! ~bc ðm 1Þ : (89) ~mn a @ðmÞ mn bc m¼2 J J J 3 f iðxÞ; aðyÞg ¼ aðxÞ@xi ðx yÞ 3 We emphasize that these results for FK and GK rely on the J iðyÞ@ya ðx yÞ: (96) ultralocality assumption, and will be modified in Sec. V. H We are now in a position to compute the Poisson brackets This is the same algebra obeyed by the i in of the smoothing functionals, from which we will extract Eq. (17). This result is completely independent of the Poisson brackets of the vector densities J i and Ki. our choice of !, and will carry over unchanged into (i) fJ iðxÞ; J aðyÞg Sec. V. To obtain the bracket fJ iðxÞ; J aðyÞg, we first com- f g 8 pute FJ;GJ . Combining the FJ and GJ variations R~abcd ¼ R~cdab, R~abcd ¼R~bacd ¼R~abdc. 9 ij ji into the bracket fFJ;GJg yields ~ ¼ ~ .

084002-11 KHOURY, MILLER, AND TOLLEY PHYSICAL REVIEW D 85, 084002 (2012)

(ii) fJ iðxÞ; KaðyÞg þ fKiðxÞ; J aðyÞg This expression depends strongly on the assumed To obtain fJ iðxÞ; KaðyÞg þ fKiðxÞ; J aðyÞg, we first form for !. This result is modified heavily in compute fFJ;GKgþfFK;GJg. Assembling the FJ Sec. VA, where ! is allowed to depend on R~. and GK variations into the Poisson bracket fFJ;GKg (iii) fKiðxÞ; KaðyÞg yields To obtain fKiðxÞ; KaðyÞg, we first compute the Z bracket fFK;GKg. Substituting the FK and GK var- 3 a fFJ;GKg¼! d zð@ag Þ iations into the Poisson bracket fFK;GKg yields X1 X1 X1 @! @! ~ i bc fF ;G g¼!2ð@ fiÞð@ gaÞ mn mðm 1Þbcriðf ~ Þ: K K i a @ðmÞ m¼2 @ðmÞ m¼2 n¼2 @ (97) ! ððnÞbcðm 1Þ @ðnÞ bc By expanding the covariant derivative, simplifying jk ðmÞ ðn 1ÞjkÞ: (104) the ensuing total derivative of !, and recalling that ~ Ki ¼!ri!, this expression reduces to From the definition of the momentum chain ij bc Z Z ðnÞ , it follows that ðnÞ ðm 1Þbc ¼ 3 iK a 3 i a jk fFJ;GKg¼ d zf i@ag ! d zð@if Þð@ag Þ ðmÞ ðn 1Þjk ¼ ðn þ m 1Þ. The terms of the sum thus vanish order by order, so the X1 @ ! mðmÞ: (98) bracket reduces to m¼2@ðmÞ fFK;GKg¼0: (105) Similarly, By comparing this result to Eq. (73), it is apparent Z Z 3 a i 3 i a that fFK;GJg¼ d zg Ka@if þ! d zð@if Þð@ag Þ fKiðxÞ; KaðyÞg ¼ 0: (106) X1 @! mðmÞ; (99) When is an ultralocal function of the @ðmÞ ! m¼2 phase space variables, the Poisson bracket K K so the sum of the two brackets simplifies consider- f iðxÞ; aðyÞg vanishes identically. This will not ~ ably, be the case when ! depends nontrivially on R,as in Sec. VA. fFJ;GKgþfFK;GJg By substituting Eqs. (96), (103), and (106) into Eq. (71), Z H~ J K 3 i a a i and recalling that i ¼ i þ i, we obtain ¼ d zðf Ki@ag g Ka@if Þ: (100) H~ H~ H~ 3 f iðxÞ; jðyÞg ¼ jðxÞ@xi ðx yÞ Integrating by parts and invoking the identity ~ 3 H ðyÞ@ j ðx yÞ: (107) @iKa ¼ @aKi yields i y H fF ;G gþfF ;G g This is the same algebra obeyed by the i in Eq. (17), and J K K J ~ Z by the J i in Eq. (96). Since H i 0, this result implies ¼ d3zðfiK @ ga gaK @ fiÞ: (101) ~ ~ ~ a i i a that fH iðxÞ; H jðyÞg 0, so the constraints H i are first class. To establish this result, we assumed only that was J K ! To extract the quantity f iðxÞ; aðyÞg þ an arbitrary ultralocal function of t, h~ , and ~ij;we fK ðxÞ; J ðyÞg, relabel dummy indices and insert ij i a showed that this was equivalent to making a function the identities in Eq. (94) to obtain ! of t and the scalars ðnÞ defined in Eq. (85). Evidently, ! fFJ;GKgþfFK;GJg can be made any ultralocal function of the phase space Z variables and the momentum constraints will remain first 3 3 i a K 3 ¼ d xd yf ðxÞg ðyÞð aðxÞ@xi ðx yÞ class. K 3 iðyÞ@ya ðx yÞÞ: (102) B. Consistency of constraints with equations of motion Combined with Eq. (73), this result implies that ~_ In this section, we will compute the time derivative H i assuming that ! is an ultralocal function, and use the fJ iðxÞ; KaðyÞg þ fKiðxÞ; J aðyÞg result to determine when the constraints H~ are preserved K 3 K 3 i ¼ aðxÞ@xi ðx yÞ iðyÞ@ya ðx yÞ: ~ by the equations of motion. The time evolution of H i is (103) determined by the equation of motion

084002-12 SPATIALLY COVARIANT THEORIES OF A TRANSVERSE, ... PHYSICAL REVIEW D 85, 084002 (2012) K H~ (ii) f i; !g _ @ i H~ ¼ þfH~ ;Hg; (108) To compute the bracket fK ; g, we first compute i @t i i ! the bracket fFK; !g. By applying the transforma- a 1 where tion @ag ! ! , Eq. (105) becomes Z 3 i ~ fFK; !g¼0: (117) H ¼ d xð!_ ! þ N H iÞ: (109) Along with Eq. (114), this implies that H~ J K J Since i ¼ i þ i and @ i=@t ¼ 0, it follows that K ~ f i; !g¼0: (118) @H i=@t ¼ @Ki=@t. Recalling that Ki ¼!@i!, the first term in Eq. (108) becomes   ~ @ @H i ! By substituting Eqs. (116) and (118) into Eq. (113), we ¼@i !_ ! þ ! : (110) @t @t obtain   To simplify the bracket fH~ ;Hg, we define X1 @ i fH~ ;Hg!@_ mðmÞ ! : (119) Z i i ! @ðmÞ 3 m¼2 ! d x!; (111) Upon inserting Eqs. (119) and (110) into the equation of so that H can be written as motion (108), the !@_ i! terms cancel to yield Z  X1  3 i ~ _ @! @! H ¼!_ ! þ d xN H i: (112) H~ i @i ! þ !_ mðmÞ : (120) @t m¼2 @ðmÞ H~ From the first class character of the constraints i,it H~_ ~ ~ ~ Demanding i 0 implies the consistency condition follows that fH i;Hg!_ fH i; !g. Since H i ¼ X1 J þ K , the second term in Eq. (108) becomes @ @ i i ! ! þ !_ mðmÞ ! fðtÞ; (121) ð Þ ~ @t m¼2 @ m fH i;Hg!_ fJ i; !g!_ fKi; !g: (113) where fðtÞ is an arbitrary function of time. We observe that To compute the brackets fJ ; g and fK ; g, we first i ! i ! the equation of motion (63) is invariant under ! þ compute the smoothing functional brackets ! ! gðtÞ, where gðtÞ is an arbitrary function of time, so we are Z free to apply this transformation to simplify our consis- f g¼ 3 ið ÞfJ ð Þ g FJ; ! d xf x i x ; ! tency condition. If we choose gðtÞ so that !g0ðtÞ¼fðtÞ, Z the consistency condition becomes 3 i fFK; !g¼ d xf ðxÞfKiðxÞ; !g: (114) @ X1 @ ! ! þ !_ mðmÞ ! 0: (122) We have already done all the work needed to evaluate these @t m¼2 @ðmÞ two brackets: since ! can be obtained from GK by the a 1 By assumption, !ðtÞ is an invertible function of time, so substitution @ag ! ! , brackets involving ! can be obtained by applying this substitution to brackets in- @=@t ¼ !@=@!_ . Our consistency condition can thus be written as volving GK. (i) fJ ; g i ! ! 0; (123) To compute the bracket fJ i; !g, we first compute a 1 where we have defined the operator the bracket fFJ; !g. Applying @ag ! ! to Eq. (98) and integrating by parts yields @ X1 @ Z  X1  ! þ mðmÞ : (124) 3 i @! @! m¼2 @ðmÞ fFJ;!g¼ d xf @i ! þ mðmÞ : m¼2 @ðmÞ To rule out the possibility of a ! which satisfies ! 0 ~ (115) while ! Þ 0, we note that the constraints H i contain one power of spatial derivatives, while by assumption the It follows by comparing this result with Eq. (114) scalar momentum is ultralocal. To satisfy 0, that ! ! the quantity would need to depend on the constraints   ! X1 H~ @! i, and would thus need to contain at least one power of fJ i; !g¼@i ! þ mðmÞ : (116) @ðmÞ spatial derivatives. However, applying to ! does not m¼2 increase the number of spatial derivatives. It follows that ! cannot contain any spatial derivatives, and thus

084002-13 KHOURY, MILLER, AND TOLLEY PHYSICAL REVIEW D 85, 084002 (2012) ~ cannot depend on H i. The consistency condition can constraints to be consistent with the equations of motion, therefore be promoted to ! must be invariant under renormalization of the volume factor !. Concretely, ! must obey the renormalization ! ¼ 0: (125) group equation To obtain the most general solution to this equation, we ¼ 0; (131) first note that ð!nðnÞÞ ¼ 0, which motivates us to ! define where ðnÞ X1 ðnÞ : (126) @ @ !nðtÞ ! þ mðmÞ : (132) @! m¼2 @ðmÞ The most general solution to the condition ¼ 0 is ! Satisfying this equation completely fixes the dependence an arbitrary function of the ðnÞ. The explicit time of on !ðtÞ. In the next section, we will generalize our dependence of is thus determined by its dependence ! ! results to a realistic class of scalar momenta. on the phase space variables. To understand this result, we return briefly to the phase ij space ðhij; Þ. To construct three-scalars out of the tensor V. LOCAL MODIFIED GRAVITY ij hij and the traceless tensor ~T , we begin by recursively ij ij The ultralocal ansatz has the virtue of simplifying cal- defining T ðnÞ, a chain of n factors of ~T linked together culations, but the laws of nature are local, not ultralocal. In by factors of hij. In analogy with our construction of the this section, we will apply the formalism developed in the ðnÞ of Eq. (85), we define last section to theories in which ! depends on spatial ~ ij ij 1 ij derivatives of the metric hij through a dependence on the T ð0Þh ¼ ð0Þ; (127) Ricci scalar R~. Since the GR of spatially covariant general and relativity belongs to this class [see Eq. (48)], this is a ij ia bj 1 ia bj realistic class of theories. As we will demonstrate, such T ðn þ 1Þ~T hab ðnÞ¼! ~ fab ðnÞ; ! must obey stringent consistency conditions in order for (128) ~ the H i to generate a consistent first class constraint ij 1 n ij algebra. from which it follows that T ðnÞ¼ ! ðnÞ. The ij contraction hijT ðnÞ yields the desired scalars, A. Constraint algebra ðnÞ ij H H TðnÞhijT ðnÞ¼ n : (129) In this section, we will compute f iðxÞ; aðyÞg assum- ! ~ ing that ! is a function of t, the phase space variables hij The TðnÞ are the only scalars that can be built out of fully ij ~ ij and ~ , and the Ricci scalar R. We will then use the result connected contractions of hij and ~T . In the presence of ~ to determine when the constraints H i remain first class. the constraint ! !ðtÞ, it follows that ~ As before, we decompose H i into a tensor part J i ~ ~ jk ~ TðnÞðnÞ: (130) 2hijrk~ and a scalar part Ki !ri!. Computing H H In other words, the ðnÞ are the scalars on the phase space f iðxÞ; aðyÞg is then a matter of computing the four brackets in Eq. (71). The result for fJ ðxÞ; J ðyÞg carries ðh~ ; ~ijÞ which have the correct conformal weight to have i a ij over unchanged from Eq. (96), but we will have to revisit been derived from three-scalars on the phase space the brackets involving K . To do so, we will first evaluate ðh ;ijÞ. It follows that the ðnÞ are invariant under i ij the smoothing functional brackets fF ;G gþfF ;G g spatial conformal transformations which rescale the J K K J and fFK;GKg. By comparing the ensuing explicit expres- volume factor !, and the condition ! ¼ 0 is thus sions to the formal expressions in Eq. (73), we will derive analogous to a renormalization group equation. explicit expressions for the Poisson brackets involving Ki. Our analysis of the variational derivatives of the smooth- C. Summary ing functional FK defined in Eq. (72) proceeds exactly as in In this section, we developed a formalism for testing the ultralocal case up to Eq. (82), where the quantity ! when our modified theories of gravity lead to a consistent arises. In this section, we assume that ! is a function of t, ~ ij first class constraint algebra, and hence contain two de- hij, ~ , and R~. To simplify calculations, note that this is grees of freedom. To develop the formalism, we made the equivalent to making ! a function of t, R~, and the ðnÞ simplifying assumption that the scalar momentum ! is an defined in Eq. (85). It follows from this assumption that ultralocal function of time t and the phase space variables X1 ~ ij @! @! hij and ~ . This assumption is sufficient to guarantee that ¼ ðnÞþ R:~ (133) ~ ! @ðnÞ @R~ the constraints H i remain first class. However, for the n¼2

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Substituting this result into Eq. (82), using the identity The corresponding results for GK are jk ~ ~ k ~ j ~ R~¼R~ hjkþr r hjk, and integrating by parts X1 GK a @! yields ¼!ð@ag Þ m Z   h~ @ðmÞ X1 @ @ mn m¼2 3 i ! ! ~jk ~   FK ¼ ! d xð@if Þ ðnÞ R hjk 1 @ðnÞ @R~ ~mnðmÞbc ~mnðm1Þ n¼2  bc Z 3   3 j k i @! þ ! d xr~ r~ ð@ f Þ h~ : (134) @! @! i @R~ jk !ð@ gaÞ ~mnR~bc þ!~mnr~ br~ c ð@ gaÞ ; a @R~ bc bc a @R~ Using Eq. (87), it is now straightforward to compute the G X1 @ variational derivatives of F , K ¼!ð@ gaÞ m ! ~bc ðm1Þ : (136) K ~mn a @ðmÞ mn bc F X1 @ m¼2 K ¼ !ð@ fiÞ n ! ~ i We are now in a position to compute the brackets involving hmn n¼2 @ðnÞ   Ki. mn 1 J K K J ~ ðnÞjk ~mnðn1Þ (i) f iðxÞ; aðyÞg þ f iðxÞ; aðyÞg jk 3 J K K J   To compute f iðxÞ; aðyÞg þ f iðxÞ; aðyÞg,we @ @ first compute fF ;G gþfF ;G g. We begin by sub- !ð@ fiÞ ! ~mnR~jk þ!~mnr~ jr~ k ð@ fiÞ ! ; J K K J i @R~ jk jk i @R~ stituting f F X1 @ Eqs. (79) and (136) into the bracket FJ;GKg. After K ¼ !ð@ fiÞ n ! ~jk ðn1Þ : (135) expanding and simplifying a total derivative of ðnÞ, ~mn i @ðnÞ mn jk n¼2 fFJ;GKg turns into

  Z X1 @ Z @ Z @ fF ;G g¼! d3zfið@ gaÞ ! r~ ðmÞþ2! d3zð@ gaÞ ! R~ kr~ fi 2! d3zðr~ fiÞr~ r~ k ð@ gaÞ ! J K a i a ~ i k k i a ~ m¼2@ðmÞ @R @R Z   Z   2 @ 2 @ X1 @ þ ! d3zð@ fiÞr~ r~ c ð@ gaÞ ! ! d3zð@ fiÞð@ gaÞ R~ ! þ mðmÞ ! : (137) i c a ~ i a ~ 3 @R 3 @R m¼2 @ðmÞ ~ ~ i ~ ~ i ~ i ~ ~ j ~ ~ To finesse this expression, integrate by parts, use the identities rirjV ¼ rjriV þ RijV and 2rjRi ¼ riR, simplify a ~ total derivative of !, use the identity Ki ¼!ri!, and expand to obtain Z Z   4 @ fF ;G g¼ d3zfiK @ ga þ ! d3zr~ ð@ fiÞr~ k ð@ gaÞ ! J K i a 3 k i a @R~ Z   2 @ X1 @ ! d3zð@ fiÞð@ gaÞ R~ ! þ mðmÞ ! : (138) i a ~ 3 @R m¼2 @ðmÞ Similarly, Z Z   4 @ fF ;G g¼ d3zgaK @ fi ! d3zr~ ð@ gaÞr~ k ð@ fiÞ ! K J a i 3 k a i @R~ Z   2 @ X1 @ þ ! d3zð@ fiÞð@ gaÞ R~ ! þ mðmÞ ! ; (139) i a ~ 3 @R m¼2 @ðmÞ so the sum of the two brackets reduces to Z Z Z   4 @ fF ;G gþfF ;G g¼ d3zfiK @ ga d3zgaK @ fi þ ! d3zr~ ð@ fiÞr~ k ð@ gaÞ ! J K K J i a a i 3 k i a @R~ Z   4 @ ! d3zr~ ð@ gaÞr~ k ð@ fiÞ ! : (140) 3 k a i @R~

After integrating by parts, expanding, and using the identity @iKa ¼ @aKi, this becomes Z Z Z Z 3 i a 3 a i 3 i a k 3 a i k fFJ;GKgþfFK;GJg¼ d zf Ka@ig d zg Ki@af þ d zð@if Þð@k@ag ÞM d zð@ag Þð@k@if ÞM ; (141) where 4 @ M !r~ ! : (142) k 3 k @R~

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To extract the bracket fJ iðxÞ; KaðyÞg þ fKiðxÞ; J aðyÞg, integrate by parts, relabel dummy indices, and insert the identities in Eq. (94) to yield Z 3 3 i a K 3 K 3 fFJ;GKgþfFK;GJg¼ d xd yf ðxÞg ðyÞð aðxÞ@xi ðx yÞ iðyÞ@ya ðx yÞÞ Z 3 3 i a Mk 3 þ d xd yf ðxÞg ðyÞ@xi ð ðxÞ@xk @xa ðx yÞÞ Z 3 3 i a Mk 3 d xd yf ðxÞg ðyÞ@ya ð ðyÞ@yk @yi ðx yÞÞ: (143)

Z By comparing this expression to Eq. (73), it is clear that 3 3 i a fFK;GKg¼ d xd yf ðxÞg ðyÞ@xi J K K J f iðxÞ; aðyÞg þ f iðxÞ; aðyÞg N k 3 ð ðxÞ@xk @xa ðx yÞÞ 3 3 Z ¼ K ðxÞ@ i ðx yÞK ðyÞ@ a ðx yÞ a x i y 3 3 i a d xd yf ðxÞg ðyÞ@ a Mk 3 y þ @xi ð ðxÞ@xk @xa ðx yÞÞ N k 3 Mk 3 ð ðyÞ@yk @yi ðx yÞÞ: (149) @ya ð ðyÞ@yk @yi ðx yÞÞ: (144) Comparing this expression to Eq. (73), it follows that K K (ii) f iðxÞ; aðyÞg K K N k 3 f iðxÞ; aðyÞg ¼ @xi ð ðxÞ@xk @xa ðx yÞÞ To compute fKiðxÞ; KaðyÞg, we first compute the N k 3 bracket fFK;GKg. Substituting Eqs. (135) and (136) @ya ð ðyÞ@yk @yi ðx yÞÞ: into the bracket fF ;G g yields K K (150) fF ;G g K K   Z @ @ ¼ !2 d3zð@ gaÞ ! r~ jr~ k ð@ fiÞ ! By substituting Eqs. (96), (144), and (150) into Eq. (71), a @~jk i @R~   and recalling that H~ ¼ J þ K , we obtain the identity Z @ @ i i i !2 d3zð@ fiÞ ! r~ jr~ k ð@ gaÞ ! ; i jk a H~ H~ H~ 3 H~ 3 @~ @R~ f iðxÞ; jðyÞg ¼ jðxÞ@xi ðxyÞ iðyÞ@yj ðxyÞ (145) I k 3 þ@xi ð ðxÞ@xk @xj ðxyÞÞ k 3 @ j ðI ðyÞ@ k @ i ðxyÞÞ; (151) where y y y where I k Mk þ N k,or @ X1 @ ! ~bc ! @! @! @! @! ¼ jk n ðn 1Þbc: (146) I 2 ~ j 2 ~ j jk ð Þ k ¼ ! r jk ! jk r @~ n¼2 @ n @R~ @~ @~ @R~ 4 @ !r~ ! : (152) After integrating by parts and expanding, the 3 k @R~ bracket becomes Expanding the derivatives in this expression and using the Z fact that H~ 0 yields 3 i a k i fFK;GKg¼ d zð@if Þð@k@ag ÞN H~ H~ I k I k 3 Z f iðxÞ; jðyÞgð ðxÞþ ðyÞÞ@xi @xj @xk ðxyÞ d3zð@ gaÞð@ @ fiÞN k; (147) I k 3 I k 3 a k i ð@xi ðxÞÞ@xj @xk ðxyÞþð@yj ðyÞÞ@yi @yk ðxyÞ: (153) where The three terms of this equation are algebraically indepen- dent, so the necessary and sufficient condition for the @! @! @! @! N !2 r~ j !2 r~ j : (148) H~ H~ k @R~ @~jk @~jk @R~ Poisson bracket f iðxÞ; jðyÞg to vanish is

I k 0: (154) To extract the bracket fKiðxÞ; KaðyÞg, integrate by parts, relabel dummy indices, and insert the identi- In the ultralocal case the constraints were automatically ties in Eq. (94) to obtain first class, but to generate a first class constraint algebra in

084002-16 SPATIALLY COVARIANT THEORIES OF A TRANSVERSE, ... PHYSICAL REVIEW D 85, 084002 (2012) the local case, the scalar momentum ! must obey the functional brackets fFJ; !g and fFK; !g. As in the ultra- fearsome looking differential equation I k 0. local case, we will obtain brackets involving ! by apply- a 1 As a check, we will now compute the I k arising from the ing the substitution @ag ! ! to brackets involving GK. J GR of spatially covariant general relativity. Recall from (i) f i; !g Eq. (48) that To obtain the bracket fJ i; !g, we first compute the sffiffiffi a 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bracket fFJ; !g. Applying @ag ! ! to 8 ¼ !2ð2Þ!2=3R~ þ 2: (155) Eq. (138) and integrating by parts yields GR 3 Z  2 @ ~ fF ; g¼ d3xfi@ þ R~ ! Since GR is a function only of ð2Þ and R, its partial J ! i ! 3 @R~ derivative with respect to ~ij simplifies, X1  @! 1 ~ k @GR @GR þ mðmÞ ! rkM ; ¼ 2 ~jk: (156) @ðmÞ @~jk @ð2Þ m¼2 (162) After substituting this relation into the definition of I k and ~ ~ jk recalling that J i ¼2hijrk~ , I k becomes where, as before, 4 @ @ @ I ~ GR 2 GR GR J 4 @! kðGRÞ¼ !rk ! k M ¼ !r~ : (163) 3 ~ ~ @ð2Þ k k ~ @R @R  3 @R @ @ @ @ þ2!2~ GR r~ j GR GR r~ j GR : It follows from an application of Eq. (114) that jk @R~ @ð2Þ @ð2Þ @R~  J 2 ~ @! (157) f i; !g¼@i ! þ R 3 @R~  Upon substituting the derivatives X1 @! 1 ~ k þ mðmÞ ! rkM : @GR 4 1 @GR 4 1 @ðmÞ ¼ ; ¼ ; (158) m¼2 @ð2Þ 3!2 @R~ 3!2=3 GR GR (164) I into kðGRÞ, the term in parentheses vanishes. By using the relations K ¼!r~ and H ¼ J þ K ,we i i ! i i i K obtain (ii) f i; !g To compute the bracket fK ; g, we first compute 16 i ! I ð Þ¼ H : (159) the bracket fF ; g. After substituting @ ga ! k GR 2=3 2 k K ! a 9! GR !1 and integrating by parts, Eq. (147) becomes Since H~ 0, the scalar momentum satisfies I 0. Z i GR k 3 i 1 k ~ fFK; !g¼ d xf @ið! rkN Þ; (165) The constraints H i of spatially covariant general relativity thus generate a first class constraint algebra. where, as before, @ @ @ @ B. Consistency of constraints with equations of motion N 2 ! ~ j ! 2 ! ~ j ! k ¼ ! r jk ! jk r : (166) ~_ @R~ @~ @~ @R~ In this section, we will compute the time derivative H i ~ Comparing with Eq. (114) yields for local ! assuming that the constraints H i are first class, and use the result to determine when the constraints fK ; g¼@ ð!1r N kÞ: (167) ~ i ! i k H i are also preserved by the equations of motion. The ~_ analysis of H i proceeds exactly as in the ultralocal case until we arrive at the expression, After substituting Eqs. (164) and (167) into the equation   of motion (160) and recalling that I ¼ M þ N ,we ~_ @! J k k k H i ¼@i !_ ! þ ! !_ f i; !g obtain @t  K _ @ 2 @ !_ f i; !g; (160) H~ ¼@ ! ! þ !_ R~ ! i i @t 3 @R~ where, as before, X1  Z @! 1 ~ k þ !_ mðmÞ !!_ rkI : (168) 3 @ðmÞ ! d x!: (161) m¼2 H~ The point of departure from the ultralocal case is the evalu- Since the i are assumed to be first class, it follows k ~_ ation of the two Poisson brackets fJ i; !g and fKi; !g. necessarily that I 0. Demanding H i 0 thus implies To compute them, we first compute the smoothing the consistency condition

084002-17 KHOURY, MILLER, AND TOLLEY PHYSICAL REVIEW D 85, 084002 (2012) X1 ij @ 2 @ @ on the phase space ðhij; Þ. The scalars R and the ðnÞ ! ! þ !_ R~ ! þ!_ mðmÞ ! fðtÞ; (169) ~ are thus invariant under spatial conformal transformations @t 3 @R m¼2 @ðmÞ which rescale the volume factor !, so once again ! ¼ where fðtÞ is an arbitrary function of time. Recall once 0 is revealed to be analogous to a renormalization group again that the equation of motion (63) is invariant under equation. ! ! ! þ gðtÞ, where gðtÞ is an arbitrary function of As a check, we will now apply the renormalization 0 time. By choosing a function gðtÞ such that !g ðtÞ¼fðtÞ, group equation to the scalar momentum GR of spatially the consistency condition becomes covariant general relativity. Since the constraints of spa- tially covariant general relativity are first class (see @ 2 @ X1 @ ! ! þ !_ R~ ! þ !_ mðmÞ ! 0: (170) Sec. VA), the necessary and sufficient condition for the ~ @t 3 @R m¼2 @ðmÞ constraints to be preserved by the equation of motion is for to satisfy Eq. (173). From Eq. (48), it follows that Since !ðtÞ is assumed to be an invertible function of time, GR sffiffiffi @=@t ¼ !@=@!_ . In analogy with our approach in the ultra- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 local case, we rewrite the consistency condition as ¼ ð2ÞR þ 2; (175) GR 3 ! 0; (171) so the scalar momentum GR depends only on the invariant where we have redefined the operator as quantities ð2Þ and R. It follows at once that GR satisfies @ 2 @ X1 @ the condition ¼ 0, which implies that the con- ! þ R~ þ mðmÞ : (172) GR ~ straints of the theory are preserved by the equations of @! 3 @R m¼2 @ðmÞ motion. It is now clear within the context of our formalism To rule out the possibility of a ! which satisfies ! 0 that the constraints H~ of spatially covariant general Þ H~ i while ! 0, we note that the constraints i contain a relativity generate a consistent first class algebra. This ~ term ri!, making the constraints higher order in spatial result justifies the assertions we made in the first paragraph derivatives than ! itself. However, by examining a series of Sec. III D. expansion of ! in the parameter R~, one can verify that applying to ! does not alter its order in spatial C. Summary derivatives.10 It follows that cannot depend on H~ . ! i In this section, we applied the formalism developed in The condition 0 is therefore equivalent to the ap- ! Sec. IV to determine when scalar momenta built out of parently stronger condition ! ~ ij hij, ~ , and R~ yield a consistent first class constraint ! ¼ 0: (173) algebra. To ensure the first class character of the constraints n 2=3 ~ Since ð! ðnÞÞ ¼ 0 and ð! R~Þ¼0, we are led to H i, it is necessary and sufficient for ! to obey the define the quantities condition ðnÞ R~ I ðnÞ ; R : (174) k 0; (176) !nðtÞ !2=3 where The most general solution to the condition ! ¼ 0 is an arbitrary function of R and the ðnÞ. In this manner, the @ @ @ @ I ¼ !2 ! r~ j ! !2 ! r~ j ! dependence of ! on !ðtÞ is determined by its dependence k @R~ @~jk @~jk @R~ on the phase space variables. 4 ~ @! As before, to understand this result, we return briefly to !rk : (177) ij 3 @R~ the phase space ðhij; Þ. As shown in Sec. IV B, the only scalars that can be built out of the tensor hij and the If @!=@R~ ¼ 0, then I k ¼ 0, so ultralocal scalar momenta ij n traceless tensor ~T are the TðnÞ¼! ðnÞ. If we im- satisfy this condition trivially. The scalar momentum GR pose the gauge-fixing constraint ! !ðtÞ, then TðnÞ of spatially covariant general relativity depends essentially ~ ðnÞ; likewise, the Ricci scalar R of the metric hij obeys on R, and thus satisfies this condition nontrivially. 11 ~ R R. This means that R and the ðnÞ have the correct To guarantee the preservation of the constraints H i by conformal weight to have been derived from three-scalars the equations of motion, the scalar momentum ! must also be invariant under renormalization of the volume 10 factor !. This requires to obey the renormalization Spatial derivatives enter ! solely throughP R~, so the deriva- ! 1 ~k tive expansion of ! can be written ! ¼ k¼0 ckR , where the group equation coefficients ck depend on ! and the ðnÞ. Applying the operator to ! changes the functional form of the ck, but does ! ¼ 0; (178) not generate higher order powers of R~. 11See Eq. (C20) in Appendix C. where

084002-18 SPATIALLY COVARIANT THEORIES OF A TRANSVERSE, ... PHYSICAL REVIEW D 85, 084002 (2012) @ 2 @ X1 @ depend on R~, the Ricci scalar of the metric h~ . In this ! þ R~ þ mðmÞ : (179) ij ~ case, ! must satisfy a corresponding renormalization @! 3 @R ¼ @ðmÞ m 2 group equation, but its form is further restricted by a This generalizes Eq. (131) to include a possible depen- differential equation that relates its dependence on R~ to ~ ij dence of ! on R. The scalar momentum GR satisfies its dependence on the phase space variables h~ and ~ . I ij GR ¼ 0 in addition to kðGRÞ0, so the constraints As a proof of principle, this paper demonstrates the of spatially covariant general relativity generate a consis- possibility of consistently modifying the graviton equa- tent first class constraint algebra. tions of motion, but more remains to be done. In forth- coming work [20], we will apply our formalism to search VI. CONCLUSIONS for viable alternatives to general relativity by attempting to In this paper, we developed a general formalism for modify the ! of general relativity parametrically in the verifying the consistency of spatially covariant modified infrared. If we discover nontrivial modifications of general theories of the transverse, traceless graviton degrees of relativity that contain only two degrees of freedom, it could freedom. It was a long road, so it is worth retracing our open up new lines of theoretical and experimental research. steps to see the logic of our path. A null result, on the other hand, would serve as further In Sec. II, we showed how to express general relativity evidence of the uniqueness of general relativity. It will be interesting to see just how far we can push this program. as a theory of a spatial metric hij and its conjugate mo- mentum ij. In this language, the general covariance and local Lorentz covariance of the theory are encoded by the ACKNOWLEDGMENTS H Dirac algebra obeyed by the four constraints .In We thank N. Afshordi, K. Hinterbichler and M. Trodden Sec. III, we showed how to obtain a spatially covariant for helpful discussions. We also thank the Perimeter version of general relativity. We began in Sec. III A by Institute for Theoretical Physics in Waterloo, Ontario, ij splitting the phase space ðhij; Þ into the phase space where portions of this research were completed. This ð!; !Þ of the spatial volume factor and the phase space work is supported in part by the U.S. Department of ~ ij ðhij; ~ Þ of the unit-determinant metric. In the context of Energy under Grant No. DE-AC02-76-ER-03071 (J. K. cosmology on an FRW background, it is natural to repre- and G. E. J. M.) and the Alfred P. Sloan Foundation (J. K.). sent time diffeomorphism symmetry on the phase space ð!; !Þ and to represent spatial diffeomorphisms on the APPENDIX A: COVARIANT CONSTRAINT ~ ij phase space ðhij; ~ Þ; in Sec. III B, we showed how to ALGEBRA OF GR achieve this splitting using a cosmological gauge condi- Recall the Poisson bracket of GR tion. On an expanding background, ! drops out of the   dynamical phase space of the theory, and its conjugate Z A B A B fA; Bg d3z momentum becomes the scalar part of the physical mn mn ! hmnðzÞ ðzÞ ðzÞ hmnðzÞ Hamiltonian density on the phase space ðh~ ; ~ijÞ;in ij (A1) Sec. III C, we showed how to reduce the phase space by solving the Hamiltonian constraint in cosmological gauge. and the constraints   By successfully projecting the degrees of freedom of gen- pffiffiffi 1 1 H ffiffiffi ij i 2 eral relativity onto the reduced phase space ðh~ ; ~ijÞ,we 0 hðR 2Þþp ij ð iÞ ; ij h 2 have shown how to represent the graviton dynamics of H jk general relativity on the class of conformally equivalent i 2hijrk : (A2) spatial metrics. Our object in this section is to derive the constraint algebra To modify general relativity, we simply modified the H H H i 3 H i 3 f 0ðxÞ; 0ðyÞg¼ ðxÞ@xi ðxyÞ ðyÞ@yi ðxyÞ; functional form of the scalar momentum ! while retain- H H H 3 ing the explicit spatial diffeomorphism symmetry gener- f 0ðxÞ; iðyÞg¼ 0ðyÞ@xi ðxyÞ; H~ ated by the three constraints i. In Sec. IV, we considered H H H 3 H 3 f iðxÞ; jðyÞg¼ jðxÞ@xi ðxyÞ iðyÞ@yj ðxyÞ: the case in which ! is an ultralocal function of the phase ~ ij (A3) space quantities hij and ~ . In this case, the consistency of ~ the constraints H i imposes a single nontrivial condition To evaluate these Poisson brackets, we first define the on the form of !, namely, that it must satisfy a renormal- smoothing functionals ization group equation with flow parameter !. The Z Z 3 0 H 3 i H renormalization group equation encodes the fact that ! FH d xf ðxÞ 0ðxÞ;F d xf ðxÞ iðxÞ; must be invariant under flow through the space of confor- Z Z mally equivalent spatial metrics. In Sec. V, we applied our 3 0 H 3 a H GH d yg ðyÞ 0ðyÞ;G d yg ðyÞ aðyÞ; (A4) formalism to the case in which ! is also allowed to

084002-19 KHOURY, MILLER, AND TOLLEY PHYSICAL REVIEW D 85, 084002 (2012) Z   where the functions f0, fi, g0, and gi are time-independent 1 pffiffiffi F ¼ d3xf0 H hij þ hRij h smoothing functions. We then compute the brackets V 2 V ij Z pffiffiffi Z 3 0 k ij j i 3 3 0 0 H H þ d x hf ðr rkh hij rr hijÞ: (A11) fFH;GHg¼ d xd yf ðxÞg ðyÞf 0ðxÞ; 0ðyÞg; Z Before taking variational derivatives, we exploit our free- fF ;Gg¼ d3xd3yf0ðxÞgaðyÞfH ðxÞ; H ðyÞg; H 0 a dom to integrate by parts to pull the covariant derivatives Z off the metric variation h : 3 3 i a H H ij fF; Gg¼ d xd yf ðxÞg ðyÞf iðxÞ; aðyÞg: (A5)   Z pffiffiffi 3 0 1 ij ij FV ¼ d xf H V h þ hR hij As in Sec. IVA, we assume that the smoothing functions 2 Z ffiffiffi decay so rapidly that they eliminate all boundary terms p þ d3x hðhijr rkf0 rirjf0Þh : (A12) generated by integration by parts, but that they are k ij otherwise arbitrary. This greatly simplifies the explicit evaluation of the brackets of the smoothing functionals. It follows that   By comparing the explicit forms of the brackets to the F 1 pffiffiffi V 0 H mn mn implicit forms in Eq. (A5), we will derive explicit formulae ¼ f Vh þ hR hmn 2 for the brackets of the H ’s. pffiffiffi mn k 0 m n 0 To simplify the calculation of the variational derivatives þ hðh rkr f r r f Þ H of FH, we will split the Hamiltonian constraint 0 into a FV H H ¼ 0: (A13) kinetic piece T and a potential piece V . Explicitly, we mn H H H have 0 ¼ T þ V, where   Similarly, 1 1   H pffiffiffi ij kl G 1 pffiffiffi T hikhjl hijhkl ; V 0 H mn mn h 2 ¼ g Vh þ hR pffiffiffi hmn 2 H pffiffiffi V hðR 2Þ: (A6) mn k 0 m n 0 þ hðh rkr g r r g Þ Similarly, F ¼ F þ F , where G H T V V ¼ 0: (A14) Z Z mn 3 0 H 3 0 H FT d xf ðxÞ TðxÞ;FV d xf ðxÞ V ðxÞ: (A7) Before computing F, we integrate by parts inside F: Z 3 jk i Computing the variation FT is straightforward: F ¼ 2 d xhij rkf : (A15) Z   1 1 3 0 ffiffiffi i kj k ij H ij This simplifies the variational calculation: FT ¼ d xf p ð2 k k Þ Th hij h 2 Z Z Z 3 i jk 3 i jk 3 0 1ffiffiffi k ij F ¼ 2 d xðrkf Þ hij þ 2 d xðrkf Þhij þ d xf p ð2ij hij Þ : (A8) h k Z 3 jk i þ 2 d x hijrkf : (A16) It follows that   i F 1 1 To evaluate rkf , first expand the covariant derivative as T 0 ffiffiffi m kn k mn mn i i i a i a i ¼ f p ð2 Þ H Th r f ¼ @ f þ f . It follows that r f ¼ f . h h k k 2 k k ka k ka mn The identity F 1 T 0 ffiffiffi k mn ¼ f p ð2mn hmn kÞ: (A9) l 1 lm h ki ¼ 2h ðrihkm þrkhim rmhikÞ (A17) jk i i jk Likewise, implies that 2 hijrkf ¼ f rihjk. Substituting   this result into the expression for F and integrating by G 1 1 T 0 ffiffiffi m kn k mn H mn parts yields ¼ g p ð2 k k Þ Th hmn h 2 Z Z 3 i jk 3 i jk G F ¼ 2 d xðrkf Þ hij d xriðf Þhjk T 0 1ffiffiffi k ¼ g p ð2mn hmn Þ: (A10) mn h k Z 3 i jk þ 2 d xðrkf Þhij : (A18) ij j i Keeping in mind that R ¼hijR þrr hij k ij r rkh hij, computing FV is just as straightforward: It follows that

084002-20 SPATIALLY COVARIANT THEORIES OF A TRANSVERSE, ... PHYSICAL REVIEW D 85, 084002 (2012) Z F i jk mn i mn 3 i a ¼ 2ðr f Þ rðf Þ fF; Gg¼ d xf ðxÞH ðxÞ@ i g ðxÞ h k ij i a x mn Z F i jk 3 a H i ¼ 2ðr f Þh : (A19) d yg ðyÞ iðyÞ@ya f ðyÞ; (A25) mn k ij mn

Likewise, then use the identities Z a 3 3 a G a bc mn a mn g ðxÞ¼ d y ðx yÞg ðyÞ; ¼ 2ðrcg Þ ab raðg Þ hmn Z G fiðyÞ¼ d3x3ðx yÞfiðxÞ; (A26) ¼ 2ðr gaÞh bc : (A20) mn c ab mn to write We are now in a position to compute the Poisson brackets Z of interest. 3 3 i a H 3 fF; Gg¼ d xd yf ðxÞg ðyÞð aðxÞ@xi ðx yÞ (i) fH iðxÞ; H jðyÞg H H 3 To calculate f iðxÞ; jðyÞg, we will first calculate H iðyÞ@ya ðx yÞÞ: (A27) fF; Gg. Substituting Eqs. (A19) and (A20) into the Poisson bracket yields By comparing this expression to (A5), we obtain the identity Z 3 bc i a H H H 3 fF; Gg¼2 d zhab ðrcf Þðrig Þ f iðxÞ; jðyÞg ¼ jðxÞ@xi ðx yÞ Z 3 H ðyÞ@ j ðx yÞ: (A28) 3 jk a i i y 2 d zhij ðrkg Þðraf Þ Z 3 a i bc (ii) fH ðxÞ; H ðyÞg 2 d zðrcg Þriðf hab Þ 0 i To calculate fH ðxÞ; H ðyÞg, we will first cal- Z 0 i 3 i a jk culate fFH;Gg¼fFT;GgþfFV ;Gg. Substituting þ 2 d zðrkf Þraðg hij Þ: (A21) Eqs. (A13) and (A20) into the bracket fFT;Gg yields Z 3 0 a After integrating by parts, applying the identity fFT;Gg¼ d zf raðg H TÞ: (A29) a a b ðrirj rjriÞu ¼ R biju , and recalling that H jk i ¼2hijrk , this bracket becomes Assembling Eqs. (A13) and (A20) into the bracket Z fFV ;Gg yields 3 i a a i Z fF; Gg¼ d zðf H arig g H iraf Þ fF ;Gg¼ d3zf0ðr gaÞH Z V a V 3 i a jk Z þ 2 d zf g ðRjika þ RjaikÞ: (A22) pffiffiffi 3 0 a c þ 2 d z hf ðrcg ÞRa It follows from the symmetry (R ¼ R ) and Z pffiffiffi abcd cdab 3 a c 0 þ 2 d z hðrag Þrcr f antisymmetry (Rabcd ¼Rbacd ¼Rabdc) proper- ties of the Riemann tensor that Rjaik ¼Rkija. The Z pffiffiffi ij ji 3 a c 0 symmetry property ( ¼ ) of the momentum 2 d z hðrcg Þrar f : (A30) jk tensor then implies that ðRjika þ RjaikÞ¼0,so Z After integrating the last two terms by parts, the a a 3 i a a i identity ðrarc rcraÞg ¼ Racg implies that fF; Gg¼ d zðf H arig g H iraf Þ: (A23) Z fF ;Gg¼ d3zf0ðr gaÞH Upon expanding the covariant derivatives, the con- V a V Z pffiffiffi nection terms cancel, yielding 3 c 0 a þ 2 d z hRa rcðf g Þ: (A31) Z fF; Gg¼ d3zðfiH @ ga gaH @ fiÞ: (A24) a i i a By integrating the last term by parts, using the c identity 2rcRa ¼rpffiffiffiaR ¼raðR 2Þ, and H H To extract the Poisson brackets f iðxÞ; jðyÞg, recalling that H V ¼ hð2 RÞ, the bracket be- first relabel integration variables, comes

084002-21 KHOURY, MILLER, AND TOLLEY PHYSICAL REVIEW D 85, 084002 (2012) Z Z 3 0 a 3 0 mn 0 fFV ;Gg¼ d zf raðg H V Þ: (A32) fFV ;GTg¼2 d zg rmrnf   Z pffiffiffi 3 0 0 1 1 k mn Combining fFV ;Gg with fFT;Gg and recalling that þ d zf g pffiffiffi H þ 2 hR : H H H 2 V k mn 0 ¼ T þ V yields h Z (A42) 3 0 aH fFH;Gg¼ d zf raðg 0Þ: (A33) The sum of the four brackets reduces to pffiffiffi Z a H Since g is a three-vector and 0= h is a three- fF ;G g¼2 d3zðf0mnr r g0 scalar, H H m n 0 mn 0 aH aH g rmrnf Þ: (A43) raðg 0Þ¼@aðg 0Þ; (A34) After integrating by parts and recalling that H ¼ from which it follows that i r jk Z 2hij k , the bracket becomes 3 0 aH Z fFH;Gg¼ d zf @aðg 0Þ: (A35) 3 0 i 0 0 i 0 fFH;GHg¼ d zðf H rig g H rif Þ: H H To extract the bracket f 0ðxÞ; iðyÞg, first relabel (A44) the variable of integration, Z Upon expanding the covariant derivatives in terms 3 0 a H of partial derivatives and connection terms, the fFH;Gg¼ d xf ðxÞ@xa ðg ðxÞ 0ðxÞÞ; (A36) connection terms cancel to yield Z then use the identity fF ;G g¼ d3zðf0H i@ g0 g0H i@ f0Þ: (A45) Z H H i i a H 3 3 a H g ðxÞ 0ðxÞ¼ d y ðxyÞg ðyÞ 0ðyÞ (A37) H H To extract the bracket f 0ðxÞ; 0ðyÞg, relabel integration variables and use the identities to write Z Z g0ðxÞ¼ d3y3ðx yÞg0ðyÞ; 3 3 0 a H 3 fFT;Gg¼ d xd yf ðxÞg ðyÞ 0ðyÞ@xa ðx yÞ: Z f0ðyÞ¼ d3x3ðx yÞf0ðxÞ (A46) (A38) By comparing this expression to (A5), we obtain the to write Z identity 3 3 0 0 i fFH;GHg¼ d xd yf ðxÞg ðyÞðH ðxÞ 3 fH ðxÞ; H ðyÞg ¼ H ðyÞ@ i ðx yÞ: (A39) 0 i 0 x 3 H i 3 @xi ðx yÞ ðyÞ@yi ðx yÞÞ: H H (A47) (iii) f 0ðxÞ; 0ðyÞg H H To calculate f 0ðxÞ; 0ðyÞg, we will first calcu- By comparing this expression to (A5), we obtain late the identity H H H i 3 fFH;GHg¼fFT;GTgþfFT;GV g f 0ðxÞ; 0ðyÞg ¼ ðxÞ@xi ðx yÞ H i 3 þfFV ;GTgþfFV ;GV g: (A40) ðyÞ@yi ðx yÞ: (A48) It is straightforward to verify that the brackets fFT;GTg and fFV ;GV g vanish identically. To com- pute fFT;GV g, substitute Eqs. (A9) and (A14) into APPENDIX B: CONSTRAINT the Poisson bracket to obtain BRACKETS AFTER IMPOSING 0 Z 3 0 mn 0 We begin with the four constraints H . After introduc- fFT;GV g¼2 d zf rmrng ing the gauge-fixing constraint Z   1 1 pffiffiffi pffiffiffi 3 0 0 ffiffiffi H k mn d zf g p V þ 2 hRmn : h !ðtÞ; (B1) h 2 k (A41) we need to compute the brackets of each of the five con- straints (including ) with . We introduce the smoothing Likewise, functionals

084002-22 SPATIALLY COVARIANT THEORIES OF A TRANSVERSE, ... PHYSICAL REVIEW D 85, 084002 (2012) Z Z H 1 k 3 3 3 f 0ðxÞ;ðyÞg ¼ kðxÞ ðx yÞ: (B13) F d xfðxÞðxÞ;G d ygðyÞðyÞ; (B2) 2 where f and g are arbitrary rapidly-decaying smoothing (ii) fH iðxÞ;ðyÞg functions. We then compute the brackets From Eq. (A19), it follows that Z Z pffiffiffi 3 3 3 i fF;Gg¼ d xd yfðxÞgðyÞfðxÞ;ðyÞg; fF; Gg¼ d zg hrif : (B14) Z 3 3 0 H fFH;Gg¼ d xd yf ðxÞgðyÞf 0ðxÞ;ðyÞg; Integrating by parts and using the fact that g is a Z scalar, this bracket becomes fF; G g¼ d3xd3yfiðxÞg ðyÞfH ðxÞ;ðyÞg: Z pffiffiffi i (B3) 3 i fF; Gg¼ d xf h@ig: (B15) The variation F is To extract the bracket fH ðxÞ;ðyÞg, use the Z i 1 pffiffiffi identity F ¼ d3xf hhijh ; (B4) 2 ij Z 3 3 gðxÞ¼ d ygðyÞ ðx yÞ (B16) so pffiffiffi to write F 1 mn F ¼ f hh ; mn ¼ 0: (B5) Z pffiffiffiffiffiffiffiffiffi hmn 2 3 3 i 3 fF; Gg¼ d xd yf ðxÞgðyÞ hðxÞ@xi ðx yÞ: Likewise, (B17) pffiffiffi G 1 mn G ¼ g hh ; mn ¼ 0: (B6) Comparing to (B3), we obtain the identity hmn 2 pffiffiffiffiffiffiffiffiffi 3 fH ð Þ ð Þg ¼ ð Þ i ð Þ It follows at once that i x ; y h x @x x y : (B18)

fF;Gg¼0: (B7)

Comparing with (B3), we obtain the identity APPENDIX C: CONFORMAL DECOMPOSITION fðxÞ;ðyÞg ¼ 0: (B8) Consider a metric g in a number of dimensions d. Denote the determinant of g by g. Define the positive We now turn to the brackets of with the H . H conformal factor (i) f 0ðxÞ;ðyÞg We split fFH;Gg into fFH;Gg¼fFT;Ggþ jgj1=d > 0 (C1) fFV ;Gg. Assembling Eqs. (A9) and (B5) into the and the metric Poisson bracket fFT;Gg yields Z 1=d g~ jgj g (C2) 3 0 1 k fFT;Gg¼ d zf g k: (B9) 2 so that The bracket fF ;G g vanishes identically, so V g ¼ g~: (C3) Z 1 fF ;G g¼ d3xf0g k : (B10) By construction, the signature of g~ is the same as that of H 2 k g. Denote the determinant of g~ by g~. From the H definition of g~ , it follows that g~ ¼ g=jgj,sog~ ¼1, To extract the bracket f 0ðxÞ;ðyÞg, use the identity Z depending on the signature of g. We therefore call g~ a 3 3 gðxÞ¼ d ygðyÞ ðx yÞ; (B11) unit-determinant metric. The inverse metrics are related by g ¼ g~1.We which yields denote the covariant derivative with respect to g by r, ~ Z and the covariant derivative with respect to g~ by r. The 3 3 0 1 k 3 fF ;G g¼ d xd yf ðxÞg ðyÞ ðxÞ ðx yÞ: connection defined by g is H 2 k 1 (B12) ¼ 2g ð@g þ @g @gÞ; (C4) ~ Comparing to (B3), we obtain the identity while the connection defined by g~ is

084002-23 KHOURY, MILLER, AND TOLLEY PHYSICAL REVIEW D 85, 084002 (2012)

1 ~ ~ ¼ g~ð@ g~ þ @ g~ @ g~ Þ: (C5) We now express R in terms of R and derivatives of ’. 2 Recalling that ¼ d, we find ~ ~ The connection obeys ¼ C, where ~ 1 C ¼ dr’ C ¼ð 2g~ g~Þ@ log: (C6) C C ¼ðd þ 2Þðr~ ’Þðr~ ’Þ2g~ ðr~ ’Þðr~ ’Þ; For convenience, we can write in terms of a scalar field ’ and a constant 0 as (C15) e2’; (C7) 0 so in which case ~ ~ ~ ~ ~ ~ R ¼ R þðd 2Þðr’Þðr’Þ C ¼ r’ þ r’ g~r ’: (C8) ~ ~ ðd 2Þg~ðr’Þðr ’Þ The Riemann tensor of g is ~ ~ ~ ~ ðd 2Þrr’ g~rr ’: (C16) R ¼ @ @ þ ; (C9) The Ricci scalar for g is R ¼ g R; the Ricci scalar while the Riemann tensor of g~ is for g~ is R~ ¼ g~ R~. In terms of covariant derivatives of ~ ~ ~ ~ ~ ~ ’,wehave R~ ¼ @ @ þ : (C10)

~ ~ ~ ~ Using ¼ þ C, the Riemann tensor R can R ¼ R ðd 1Þðd 2Þðr’Þðr ’Þ be rewritten as ~ ~ 2ðd 1Þr r’: (C17) R ¼ R~ þ CC CC þ @C In three dimensions, the Weyl tensor vanishes, so the ~ ~ þ C C @C Riemann tensor is completely determined by the Ricci ~ ~ tensor and the metric via C þ C: (C11) 1 Using R ¼ ðg R g R g R þ g R Þ lkmn d 2 lm kn ln km km ln kn lm ~ ~ rC rC 1 ðglmgkn glngkmÞR: (C18) ~ ~ ðd 1Þðd 2Þ ¼ @C þ C C ~ ~ @C C þ C; (C12) In this case, it suffices to compute the Ricci tensor. When ¼ d 3, our previous formulas reduce to R becomes ~ ~ ~ ~ k Rij ¼ R~ij þðri’Þðrj’Þg~ijðrk’Þðr ’Þ R ¼ R~ þ CC CC ~ ~ ~ ~ ~ k ~ ~ rirj’ hijrkr ’; þ rC rC: (C13) R ¼ R~ ðr~ ’Þðr~ k’Þ r~ kr~ ’: 2 k 4 k (C19) The Ricci tensor of g is R ¼ R ; the Ricci tensor ~ ~ of g~ is R ¼ R . Tracing Eq. (C13) appropriately The condition ! !ðtÞ amounts to ’ ’ðtÞ, so in cos- yields mological gauge we have ~ ~ ~ R ¼ R þ CC CC þ rC rC: Rij R~ij; R R:~ (C20) (C14)

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