General relativity limit of Horava-Lifshitz gravity with a scalar field in gradient expansion
aa r X i v : . [ h e p - t h ] M a r IPMU11-0150
General relativity limit of Hoˇrava-Lifshitz gravity with a scalar field in gradientexpansion
A. Emir G¨umr¨uk¸c¨uo˘glu ∗ and Shinji Mukohyama † IPMU, The University of Tokyo, Kashiwa, Chiba 277-8582, Japan
Anzhong Wang ‡ GCAP-CASPER, Physics Department, Baylor University, Waco, TX 76798-7316, USA (Dated: November 8, 2018)We present a fully nonlinear study of long wavelength cosmological perturbations within theframework of the projectable Hoˇrava-Lifshitz gravity, coupled to a single scalar field. Adopting thegradient expansion technique, we explicitly integrate the dynamical equations up to any order ofthe expansion, then restrict the integration constants by imposing the momentum constraint. Whilethe gradient expansion relies on the long wavelength approximation, amplitudes of perturbationsdo not have to be small. When the λ → | λ − | is smaller than the order of perturbations. In the limit λ →
1, thisnew branch allows the theory to be continuously connected to general relativity, with an effectivecomponent which acts like pressureless fluid.
I. INTRODUCTION
Recently, Hoˇrava [1] proposed a new theory of quantum gravity in the framework of quantum field theory. Oneof the essential ingredients of the theory is inclusion of higher-dimensional operators, so that they dominate theultraviolet (UV) behavior and render the theory power counting renormalizable. Improvement of the UV behavior byhigher-dimensional operators has been known for some time [2] but in those previous attempts, higher time derivativeterms led to ghost degrees of freedom. The major modification put forward by Hoˇrava’s theory is that the power-counting renormalizability is achieved without inclusion of higher time derivative terms. This is realized by invokingthe anisotropic scaling between time and space, t → b − z t, ~x → b − ~x , (1)so that higher-dimensional operators include spatial derivatives only. This is reminiscent of Lifshitz scalars [3] incondensed matter physics, hence the theory is often referred to as the Hoˇrava-Lifshitz (HL) gravity. For the 3 + 1dimensional theory to be power-counting renormalizable, the dynamical critical exponent z has to be larger than orequal to 3 [1] (see also [4]). Because of the anisotropic scaling, the theory cannot be invariant under the spacetimediffeomorphism, x µ → x ′ µ ( x ν ) , ( µ, ν = 0 , , , t → t ′ ( t ) , ~x → ~x ′ ( t, ~x ) , (2)denoted usually by Diff( M, F ). The basic variables of the theory are the lapse function N , the shift vector N i , and the3-dimensional spatial metric g ij [5]. Since the lapse function N corresponds to a gauge degree of freedom associatedwith the space-independent time reparametrization, it is natural to restrict the lapse function to be independent ofthe spatial coordinates: N = N ( t ) . (3) ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: Anzhong˙[email protected]
This condition, imposed in the original formulation of the theory, is called the projectability condition .Since its introduction, there has been many cosmological applications of the HL gravity and various remarkablefeatures have been found (see [6, 7] for reviews). In particular, the higher-order spatial curvature terms can give riseto a bouncing universe [8], may ameliorate the flatness problem [9] and lead to caustic avoidance [10]; the anisotropicscaling provides a solution to the horizon problem and generation of scale-invariant perturbations without inflation[11], a new mechanism for generation of primordial magnetic seed field [12], and also a modification of the spectrum ofgravitational wave background via a peculiar scaling of radiation energy density [13]; with the projectability condition,the lack of a local Hamiltonian constraint leads to “dark matter as an integration constant” [14]; in the parity-violatingversion of the theory, circularly polarized gravitational waves can also be generated in the early universe [15].Despite all of its remarkable features, the theory has been challenged by significant questions. In particular, theDiff( M, F ) symmetry allows the existence of an additional spin-0 degree of freedom, often called scalar graviton , andits fate is one of important open issues. Actually, the scalar graviton is known to be unstable either in the UV dueto ghost instability or in the infrared (IR) due to gradient instability [16–18], depending on the value of a couplingconstant λ . In order to avoid the ghost instability, λ must satisfy either λ < / λ >
1. Precisely in these tworanges, the scalar graviton exhibits gradient instability at long distances. We then have to tame this IR instability byexpansion of the universe [19, 20] or have to hide it by the standard Jeans instability. One can formulate a conditionunder which one of these happens [6]. Essentially, the condition says that λ must be sufficiently close to 1 in the IR.However, in the limit λ →
1, the scalar graviton appears to be strongly coupled [20–22]. That is, the “standard”(and naive) perturbative expansion breaks down in the sense that nonlinear terms dominate linear terms in the λ → λ → This may be considered as an analogue of theVainshtein effect [24, 25]. A similar consideration for cosmology was given in [26], where a fully nonlinear analysis ofsuperhorizon cosmological perturbations was carried out.One of limitations of the analysis in [26] is that it is for a purely gravitational system in the absence of ordinarymatter (but with “dark matter as an integration constant”). Since the naive perturbative expansion is known tobreak down not only in the gravity sector but also in the matter sector [20, 21], it is rather important to extend theanalysis of [26] to the system with ordinary matter. Technically speaking, however, this kind of extention is indeed anontrivial challenge since the system now has multi components (ordinary matter and “dark matter as an integrationconstant”) and the gradient expansion technique has not been developed for multi-component systems even in thestandard cosmology in GR.Thus, one of the main objectives of the present paper is to extend the analysis of [26] to the case where HL gravityis coupled to a single scalar field, and provide yet another example indicating that general relativity (plus “darkmatter as an integration constant”) is restored in the λ → | λ − | is smaller than the order of perturbations.The paper is organized as follows. In Sec. II, we briefly review the basic equations in the HL gravity with theprojectability condition (3), while in Sec. III, we analyze the inhomogeneous cosmology in HL gravity using the gradientexpansion method [27], and present the solutions to the equations of motion. In Sec.IV, we present a discussion on thesource of the divergences in the naive perturbative expansion and show that the momentum constraint is dominatedby nonlinear terms in the λ → Specifically, the solutions are continuously connected to the λ = 1 theory, whose action has the exact same form as the Einstein-Hilbertterm (up to high curvature terms negligible at low energies). However, due to the different symmetries, the resulting theory is notexactly GR, but GR with an effective component which acts like dark-matter [14]. This is what we mean by “continuity with GR”throughout the present paper. In the case considered in [6], however, the “dark matter” component is automatically set to zero by theassumed staticity. II. BASIC EQUATIONS
In this section, we review the basic equations of the HL gravity coupled with a scalar field [9, 28], following thenotation in [6], and reformulate them in a way suitable for gradient expansion [26]. In order to make the presentpaper self-contained, some repetition of the material in [26] is inevitable in Secs.II and III, although we shall try ourbest to limit them to a minimum.With Diff( M, F ) and the projectability, the building blocks of the theory are g ij , K ij , D i and R ij , where K ij denotes the extrinsic curvature of constant time hypersurfaces, D i is the covariant derivative compatible with the the3-dimensional spatial metric g ij , and R ij is the three-dimensional Ricci tensor built out of g ij . (This is in contrast toGR or any other theory with general covariance whose building blocks are the 4-dimensional metric and its Riemanntensor.) For the critical exponent z = 3, their momentum dimensions are, respectively, [ K ij ] = [ k ] and [ R ij ] = [ k ] .Throughout the present paper, we shall impose the projectability condition as well as invariance under the spatialparity ( x i → − x i ) and the time reflection ( t → − t ). The number of independent coupling constants in this setup is 11for z = 3 [9, 16]. In fact, with the foliation-preserving diffeomorphisms (2), the projectability condition (3), and theadditional requirements of parity and time reflection symmetry, the most general gravitational action can be specifiedas I g = M P l Z N dt √ g d ~x (cid:0) K ij K ij − λK −
2Λ + R + L z> (cid:1) , (4)where g is the determinant of g ij , and the extrinsic curvature K ij is defined as K ij = 12 N ( ∂ t g ij − D i N j − D j N i ) , (5) K (= g ij K ij ) is the trace of K ij , and R is the Ricci scalar constructed from g ij . To lower and raise an index, g ij andits inverse g ij are used. For the sake of simplicity and clarity, in the remainder of this paper, we choose our unitssuch that M P l = 1.In contrast to GR, the less restricting symmetry allows both kinetic terms K ij K ij and K to be invariant inde-pendently, giving rise to the extra parameter λ , which assumes the value 1 in GR, as mentioned above. Furthermore,in order to realize the power-counting renormalizability, the higher curvature Lagrangian L z> should include up tosixth spatial derivatives. For the analysis in the present paper, the concrete form of L z> is not needed. Addingthe scalar field action I φ that is invariant under spatial parity and time reflection, as well as the foliation preservingdiffeomorphism, the total action is I = I g + I φ ,I φ = Z N dt √ g d ~x (cid:20)
12 ( ∂ ⊥ φ ) − V ( φ, D i , g ij ) (cid:21) , (6)where we define the derivative along vector normal to the hypersurface ∂ ⊥ ≡ N ( ∂ t − N k ∂ k ) , (7)and we decompose the scalar field potential as V ( φ, D i , g ij ) = V ( φ ) + V z ≥ ( φ, D i , g ij ) . (8)Here, V z ≥ summarizes terms with two or more spatial derivatives and like L z> above, its concrete form is not neededfor the purposes of the present paper.Variation of the total action with respect to the 3-dimensional metric g ij leads to the dynamical equation E gij + E φij = 0 , (9)where E gij ≡ g ik g jl N √ g δI g δg kl = − N ( ∂ t − N k D k ) p ij + 1 N ( p ik D j N k + p jk D i N k ) − Kp ij + 2 K ki p kj + 12 g ij K kl p kl − Λ g ij − G ij + E g,z> ,ij , (10) E φij ≡ g ik g jl N √ g δI φ δg kl = g ij (cid:20)
12 ( ∂ ⊥ φ ) − V ( φ ) (cid:21) + E φ,z ≥ ,ij . (11)Here, E g,z> ,ij and E φ,z ≥ ,ij are contributions from L z> and − V z ≥ , respectively, p ij ≡ K ij − λKg ij , and G ij isEinstein tensor of g ij . The trace part and traceless part of Eq.(9) are, respectively,(3 λ − (cid:18) ∂ ⊥ K + 12 K (cid:19) + 32 A ij A ji + 32 ( ∂ ⊥ φ ) + Z = 0 , (12)and ∂ ⊥ A ij + KA ij + 1 N ( A kj ∂ k N i − A ik ∂ j N k ) − (cid:18) Z ij − Zδ ij (cid:19) = 0 , (13)where A ij ≡ K ij − Kδ ij , (14)is the traceless part of K ij and we defined Z ij ≡ Z ig,j + Z iφ,j , Z ≡ Z ii ,Z ig,j ≡ − Λ δ ij − G ij + g ik E z> ,g,kj ,Z iφ,j ≡ − V ( φ ) δ ij + g ik E z ≥ ,φ,kj . (15)Here, Z ig ,j is the variation of the potential part of the gravitational action with respect to the spatial metric; it isa generalization of (minus) the Einstein tensor of g ij to include higher curvature terms, as well as the cosmologicalconstant. The quantity Z iφ ,j is obtained similarly from the potential part of the scalar field action.The variation of the total action with respect to φ yields the remaining dynamical equation0 = − N √ g δI φ δφ = 1 N √ g ∂ t ( √ g ∂ ⊥ φ ) − N √ g D i ( √ g N i ∂ ⊥ φ ) + E φ , (16)where E φ ≡ √ g δδφ Z √ g dt d ~x V ( φ, D i , g ij ) = V ′ ( φ ) + E φ,z ≥ , (17)and E φ,z ≥ is the contribution from V z ≥ .Since the 3-dimensional spatial diffeomorphism is a subgroup of the foliation preserving diffeomorphism, Z ig,j and Z iφ,j satisfy the generalized Bianchi identity and matter conservation, D j Z jg,i = 0 , D j Z jφ,i + E φ ∂ i φ = 0 . (18)For convenience, we decompose the spatial metric and the extrinsic curvature as g ij = a ( t ) e ζ ( t,~x ) γ ij ( t, ~x ) , (19) K ij = 13 K ( t, ~x ) δ ij + A ij ( t, ~x ) , (20)where we have defined ζ ( t, ~x ) so that det γ = 1, and a ( t ) (up to an overall normalization) is defined later in Eq. (37).The trace part and the traceless part of the definition of the extrinsic curvature lead, respectively, to ∂ ⊥ ζ + ∂ t aN a = 13 (cid:18) K + 1 N ∂ i N i (cid:19) , (21)and ∂ ⊥ γ ij = 2 γ ik A kj + 1 N (cid:18) γ jk ∂ i N k + γ ik ∂ j N k − γ ij ∂ k N k (cid:19) . (22)The momentum constraint is obtained by varying the action with respect to N i : D j K ji − λ ∂ i K = ∂ ⊥ φ ∂ i φ . (23)According to the decomposition (20), the momentum constraint is rewritten as ∂ j A ji + 3 A ji ∂ j ζ − A jl ( γ − ) lk ∂ i γ jk −
13 (3 λ − ∂ i K = ∂ ⊥ φ ∂ i φ . (24)It can be shown that the evolution equations we have derived are consistent with vanishing A ii , ln det γ , γ ij − γ ji and γ ik A kj − γ jk A ki [26]. III. GRADIENT EXPANSION
In this section, we analyze the dynamics of nonlinear superhorizon perturbations in the spatial gradient expansionapproach. This approach is valid as long as the characteristic length scale L of the perturbations is much larger thanthe Hubble length H − . By the introduction of small parameter ǫ ∼ / ( H L ), we perform a series expansion on allrelevant quantities and equations. For instance, a spatial derivative acting on a quantity at order ǫ p raises the orderto ǫ p +1 and thus is counted as O ( ǫ ). We then solve the equations order by order in gradient expansion, extending thecalculations of [26] in a spatially flat Friedmann-Robertson-Walker background, to include a single scalar field as thesource. A. Gauge fixing
The foliation preserving diffeomorphism invariance, like all other gauge symmetries, reflects a redundancy in thedescriptions of the theory. By an appropriate choice of gauge conditions, these degrees can be eliminated and physicalquantities can be extracted. In the present paper we adopt the synchronous gauge, or the Gaussian normal coordinatesystem, by setting the lapse function to unity and the shift vector to zero: N = 1 , N i = 0 . (25)This choice fixes the time coordinate but in the spatial coordinates, there remains a gauge freedom of time-independentspatial diffeomorphism, corresponding to the change of coordinates on the initial constant-time hypersurface. Thisresidual gauge degree of freedom will be discussed later in Subsection III F.After the gauge fixing, our basic equations (12), (13), (16), (21) and (22) are simplified to(3 λ − ∂ t K = −
12 (3 λ − K − A ij A ji −
32 ( ∂ t φ ) − Z , (26) ∂ t A ij = − KA ij + Z ij − Z δ ij , (27)0 = ∂ t φ + K∂ t φ + E φ , (28) ∂ t ζ = − ∂ t aa + 13 K , (29) ∂ t γ ij = 2 γ ik A kj , (30)while the momentum constraint (24) has the form ∂ j A ji + 3 A ji ∂ j ζ − A jl ( γ − ) lk ∂ i γ jk −
13 (3 λ − ∂ i K = ∂ t φ ∂ i φ . (31)Hereafter, we assume that λ = 1 /
3; this is consistent with the regime of physical interest λ ≥
1, discussed in theIntroduction section.
B. Basic assumptions and order analysis
We begin by determining the order of all relevant variables. In the limit ǫ →
0, we expect a universe that lookslocally like a Friedmann universe, leading to our starting assumption ∂ t γ ij = O ( ǫ ) . (32)For the scalar field, a similar assumption leads to ∂ i φ = O ( ǫ ). However, in order to simplify the analysis, we imposethe stronger condition ∂ i φ = O ( ǫ ) . (33)That is, we assume that φ (0) , which is the leading order term of φ , is only time dependent: φ (0) = φ (0) ( t ) . (34)The first assumption (32) then implies, from Eq. (30), A ij = O ( ǫ ) , (35)leading, using the constraint equation (31), to ∂ i K = O ( ǫ ) . (36)In other words, the zero-th order part K (0) of K depends on t only. This fact enables us to define a ( t ) by3 ∂ t a ( t ) a ( t ) = K (0) ( ≡ H ( t )) . (37)With this definition of a ( t ), Eq. (29) leads to ∂ t ζ = O ( ǫ ) . (38)To summarize, the relevant quantities in the analysis are expanded as follows: ζ = ζ (0) ( ~x ) + ǫ ζ (1) ( t, ~x ) + ǫ ζ (2) ( t, ~x ) + O ( ǫ ) , (39) γ ij = f ij ( ~x ) + ǫ γ (1) ij ( t, ~x ) + ǫ γ (2) ij ( t, ~x ) + O ( ǫ ) , (40) K = 3 H ( t ) + ǫ K (1) ( t, ~x ) + ǫ K (2) ( t, ~x ) + O ( ǫ ) , (41) A ij = ǫ A (1) ij ( t, ~x ) + ǫ A (2) ij ( t, ~x ) + O ( ǫ ) , (42) φ = φ (0) ( t ) + ǫ φ (1) ( t, ~x ) + ǫ φ (2) ( t, ~x ) + O ( ǫ ) , (43)where a quantity with the upper index ( n ) corresponds to the n -th order term in the gradient expansion. C. Equations in each order
After determining the orders of all physical quantities, we now use this information in the evolution equations(26)–(30) to obtain the evolution equations at each order. In the zero-th order of gradient expansion we have(3 λ − (cid:18) ∂ t H + 32 H (cid:19) = −
12 ( ∂ t φ (0) ) + V ( φ (0) ) + Λ ,∂ t φ (0) + 3 H ∂ t φ (0) + V ′ ( φ (0) ) = 0 , (44)where a prime denotes the ordinary derivative with respect to the indicated argument. By using the second of theabove, the first equation can be integrated to give3 H = 23 λ − (cid:20)
12 ( ∂ t φ (0) ) + V ( φ (0) ) + Λ (cid:21) + e Ca , (45)where e C is an integration constant. The last term in the right hand side of this equation is the “dark matter as anintegration constant” [14], a direct consequence of the projectability condition.The dynamical equations at order O ( ǫ n ) with n ≥
1, are written as a − ∂ t (cid:20) a (cid:18) K ( n ) + 3 φ ( n ) ∂ t φ (0) λ − (cid:19)(cid:21) = − n − X p =1 K ( p ) K ( n − p ) −
32 (3 λ − n − X p =1 h A ( p ) ij A ( n − p ) ji + ∂ t φ ( p ) ∂ t φ ( n − p ) i − ¯ Z ( n ) λ − , (46) a − ∂ t (cid:16) a A ( n ) ij (cid:17) = − n − X p =1 K ( p ) A ( n − p ) ij + ¯ Z ( n ) ij −
13 ¯ Z ( n ) δ ij , (47) a − ∂ t (cid:16) a ∂ t φ ( n ) (cid:17) + (cid:20) V ′′ ( φ (0) ) − ∂ t φ (0) ) λ − (cid:21) φ ( n ) = − (cid:18) K ( n ) + 3 φ ( n ) ∂ t φ (0) λ − (cid:19) ∂ t φ (0) − n − X p =1 K ( p ) ∂ t φ ( n − p ) − ¯ E ( n ) φ , (48) ∂ t ζ ( n ) = 13 (cid:18) K ( n ) + 3 φ ( n ) ∂ t φ (0) λ − (cid:19) − φ ( n ) ∂ t φ (0) λ − , (49) ∂ t γ ( n ) ij = 2 n − X p =0 γ ( p ) ik A ( n − p ) kj , (50)where for later convenience, we introduced new (barred) quantities¯ Z ( n ) ij ≡ Z ( n ) ij + V ′ ( φ (0) ) φ ( n ) δ ij , ¯ Z ( n ) ≡ Z ( n ) + 3 V ′ ( φ (0) ) φ ( n ) , ¯ E ( n ) φ ≡ E ( n ) φ − V ′′ ( φ (0) ) φ ( n ) , (51)to subtract the terms depending on φ ( n ) from (unbarred) Z ( n ) ij and E ( n ) ij , defined in Eqs.(15) and (17). Here, Z ( n ) ij , Z ( n ) and E ( n ) φ are the n -th order terms of Z ij , Z , E φ , respectively. With this definition, ¯ Z ( n ) ij , ¯ Z ( n ) and ¯ E ( n ) φ donot depend on ζ ( n ) , γ ( n ) ij , K ( n ) , A ( n ) ij , nor on φ ( n ) .Similarly, from Eq. (31) we obtain the order O ( ǫ n +1 ) ( n ≥
1) momentum constraint as ∂ j A ( n ) ji + 3 n X p =1 A ( p ) j i ∂ j ζ ( n − p ) − n X p =1 n − p X q =0 A ( p ) j l ( γ − ) ( q ) lk ∂ i γ ( n − p − q ) jk −
13 (3 λ − ∂ i (cid:18) K ( n ) + 3 φ ( n ) ∂ t φ (0) λ − (cid:19) − n − X p =1 ∂ t φ ( p ) ∂ i φ ( n − p ) = 0 , (52)where ( γ − ) ( n ) ij is the n -th order term of the inverse of γ ij , i.e. the inverse ( γ − ) ij is expanded as( γ − ) ij = f ij + ǫ ( γ − ) (1) ij + ǫ ( γ − ) (2) ij + . . . , (53)where f ij = ( γ − ) (0) ij is the inverse of f ij . It is straightforward to show that ( γ − ) ( n ) ij ( n ≥
1) satisfies the followingdifferential equation: ∂ t ( γ − ) ( n ) ij = − n X p =1 A ( p ) i k ( γ − ) ( n − p ) kj . (54)In addition to the dynamical equations and momentum constraint, there are also some useful identities. First, weexpand the generalized Bianchi identity (18) to obtain ∂ j ¯ Z ( n ) ji + 3 n X p =1 (cid:18) ¯ Z ( p ) ji −
13 ¯ Z ( p ) δ ji (cid:19) ∂ j ζ ( n − p ) − n X p =1 n − p X q =0 ¯ Z ( p ) jl ( γ − ) ( q ) lk ∂ i γ ( n − p − q ) jk + n − X p =1 h ¯ E ( n − p ) φ + V ′′ ( φ (0) ) φ ( n − p ) i ∂ i φ ( p ) = 0 , (55)for n ≥
1. Next, expanding the conditions A ii = 0, ∂ i ln det γ = 0, γ ij − γ ji = 0, γ ik A kj − γ jk A ki = 0 and A ij − γ jk A kl ( γ − ) li = 0 leads to the following identities: A ( n ) i i = 0 , n X p =0 ( γ − ) ( p ) jk ∂ i γ ( n − p ) jk = 0 , γ ( n ) ij − γ ( n ) ji = 0 , n − X p =0 (cid:16) γ ( p ) ik A ( n − p ) kj − γ ( p ) jk A ( n − p ) ki (cid:17) = 0 , A ( n ) i j − n − X p =0 n − p − X q =0 γ ( p ) jk A ( n − p − q ) k l ( γ − ) ( q ) li = 0 . (56) D. O ( ǫ ) solution For O ( ǫ ), Eqs.(46)–(50) reduce to ∂ t (cid:20) a (cid:18) K (1) + 3 φ (1) ∂ t φ (0) λ − (cid:19)(cid:21) = 0 , (57) ∂ t (cid:16) a A (1) ij (cid:17) = 0 , (58) a − ∂ t (cid:16) a ∂ t φ (1) (cid:17) + (cid:20) V ′′ ( φ (0) ) − ∂ t φ (0) ) λ − (cid:21) φ (1) = − (cid:18) K (1) + 3 φ (1) ∂ t φ (0) λ − (cid:19) ∂ t φ (0) , (59) ∂ t ζ (1) = 13 (cid:18) K (1) + 3 φ (1) ∂ t φ (0) λ − (cid:19) − φ (1) ∂ t φ (0) λ − , (60) ∂ t γ (1) ij = 2 f ik A (1) kj , (61)where from equations (15), (17) and (51), we have at first order, ¯ Z (1) ij = ¯ Z (1) = ¯ E (1) φ = 0. Integrating the aboveequations, we obtain K (1) + 3 φ (1) ∂ t φ (0) λ − C (1) ( ~x ) a ( t ) , (62) A (1) ij = C (1) ij ( ~x ) a ( t ) , (63) φ (1) = (cid:20) C (1) ( ~x ) Z tt in dt ′ f ( t ′ ) ∂ t ′ φ (0) ( t ′ ) a ( t ′ ) W ( t ′ ) + φ (1)in ( ~x ) (cid:21) f ( t )+ (cid:20) − C (1) ( ~x ) Z tt in dt ′ f ( t ′ ) ∂ t ′ φ (0) ( t ′ ) a ( t ′ ) W ( t ′ ) + ˙ φ (1)in ( ~x ) (cid:21) f ( t ) , (64) ζ (1) = C (1) ( ~x )3 Z tt in dt ′ a ( t ′ ) − λ − Z tt in dt ′ φ (1) ( t ′ ) ∂ t ′ φ (0) ( t ′ ) + ζ (1)in ( ~x ) , (65) γ (1) ij = 2 f ik ( ~x ) C (1) kj ( ~x ) Z tt in dt ′ a ( t ′ ) + γ (1)in ij ( ~x ) , (66)where the integration “constants” C (1) , C (1) ij , φ (1)in , ˙ φ (1)in , ζ (1)in and γ (1)in ij depend only on the spatial coordinates ~x i and satisfy C (1) ii = 0 , f ik C (1) kj = f jk C (1) ki . (67)The functions f i ( t ) ( i = 1 ,
2) are two independent solutions of the homogeneous equation a − ∂ t ( a ∂ t f i ) + (cid:20) V ′′ ( φ (0) ) − ∂ t φ (0) ) λ − (cid:21) f i = 0 ; f ( t in ) = 1 , f ′ ( t in ) = 0 ; f ( t in ) = 0 , f ′ ( t in ) = 1 , (68)and W ( t ) ≡ f ( t ) ∂ t f ( t ) − f ( t ) ∂ t f ( t ) . (69)The two first order integration “constants”, ζ (1)in and γ (1)in ij , can be absorbed into their zero-th order counterparts, ζ (0)in and γ (0)in ij . Thus, without loss of generality, we can set ζ (1)in = 0 , γ (1)in ij = 0 . (70)Finally, the momentum constraint equation (52) with n = 1 leads to the following relation among the remainingintegration constants, C (1) , C (1) ij , ζ (0) and f ij , ∂ j C (1) ji + 3 C (1) ji ∂ j ζ (0) − C (1) jl f lk ∂ i f jk −
13 (3 λ − ∂ i C (1) = 0 . (71)Note that φ (1)in ( ~x ) and ˙ φ (1)in ( ~x ) do not appear in this equation. The physical meaning of φ (1)in ( ~x ) and ˙ φ (1)in ( ~x ) are obvious: φ (1) (cid:12)(cid:12)(cid:12) t = t in = φ (1)in ( ~x ) , ∂ t φ (1) (cid:12)(cid:12)(cid:12) t = t in = ˙ φ (1)in ( ~x ) . (72) E. O ( ǫ n ) solution ( n ≥ ) Equipped with the zero-th and first order solution, we can now determine the general solutions at arbitrary orderin gradient expansion. For any n ≥
1, the solution to Eqs.(46)–(50) is K ( n ) + 3 φ ( n ) ∂ t φ (0) λ − a ( t ) Z tt in dt ′ a ( t ′ ) ( − ¯ Z ( n ) ( t ′ , ~x )3 λ − − n − X p =1 K ( p ) ( t ′ , ~x ) K ( n − p ) ( t ′ , ~x ) −
32 (3 λ − n − X p =1 h A ( p ) ij ( t ′ , ~x ) A ( n − p ) ji ( t ′ , ~x ) + ∂ t ′ φ ( p ) ( t ′ , ~x ) ∂ t ′ φ ( n − p ) ( t ′ , ~x ) i) , (73) A ( n ) ij = 1 a ( t ) Z tt in dt ′ a ( t ′ ) " − n − X p =1 K ( p ) ( t ′ , ~x ) A ( n − p ) ij ( t ′ , ~x ) + ¯ Z ( n ) ij ( t ′ , ~x ) −
13 ¯ Z ( n ) ( t ′ , ~x ) δ ij , (74) φ ( n ) = f ( t ) Z tt in dt ′ f ( t ′ ) r ( n ) ( t ′ , ~x ) W ( t ′ ) − f ( t ) Z tt in dt ′ f ( t ′ ) r ( n ) ( t ′ , ~x ) W ( t ′ ) , (75) ζ ( n ) = Z tt in dt ′ (cid:20) (cid:18) K ( n ) ( t ′ , ~x ) + 3 φ ( n ) ( t ′ , ~x ) ∂ t ′ φ (0) ( t ′ )3 λ − (cid:19) − φ ( n ) ( t ′ , ~x ) ∂ t ′ φ (0) ( t ′ )3 λ − (cid:21) , (76) γ ( n ) ij = 2 Z tt in dt ′ n − X p =0 γ ( p ) ik ( t ′ , ~x ) A ( n − p ) kj ( t ′ , ~x ) , (77)where r ( n ) ( t, ~x ) ≡ (cid:18) K ( n ) ( t, ~x ) + 3 φ ( n ) ( t, ~x ) ∂ t φ (0) ( t )3 λ − (cid:19) ∂ t φ (0) ( t ) + n − X p =1 K ( p ) ( t, ~x ) ∂ t φ ( n − p ) ( t, ~x ) + ¯ E ( n ) φ ( t, ~x ) , (78)and by redefining C (1) , C (1) ij , φ (1)in , ˙ φ (1)in , ζ (0) and f ij , we have set, respectively, K ( n ) (cid:12)(cid:12)(cid:12) t = t in = A ( n ) ij (cid:12)(cid:12)(cid:12) t = t in = φ ( n ) (cid:12)(cid:12)(cid:12) t = t in = ∂ t φ ( n ) (cid:12)(cid:12)(cid:12) t = t in = ζ ( n ) (cid:12)(cid:12)(cid:12) t = t in = γ ( n ) ij (cid:12)(cid:12)(cid:12) t = t in = 0 . (79)We remind that the first order constants have already been fixed in Eq.(70) by redefinition of ζ (0) and f ij , respectively.The initial condition for γ ( n ) ij ( n ≥
1) implies that γ ij | t = t in = f ij , ( γ − ) ij (cid:12)(cid:12) t = t in = f ij and ( γ − ) ( n ) ij (cid:12)(cid:12) t = t in = 0( n ≥ n ≥
1, the solution to Eq.(54) is( γ − ) ( n ) ij = − Z tt in dt ′ n X p =1 A ( p ) i k ( γ − ) ( n − p ) kj . (80)As shown in Appendix A, the solution (73)-(77) automatically satisfies the ( n + 1)-th order momentum constraintequation (52), provided that the redefined integration constants ( C (1) , C (1) ij , ζ (0) , f ij ) satisfy (71) up to O ( ǫ n +1 ).0 F. Number of physical degrees of freedom
The solution we obtained in the previous subsection involves a number of functions depending only on spatialcoordinates, ζ (0) ( ~x ), f ij ( ~x ), C (1) ( ~x ), C (1) ij ( ~x ), φ (1)in ( ~x ) and ˙ φ (1)in ( ~x ) which emerged as integration “constants”.However, not all of the components are independent nor physical. Firstly, they are subject to the constraint (71).Secondly, as stated just after Eq. (25), our gauge condition (25) leaves time-independent spatial diffeomorphism as aresidual gauge freedom. Therefore, the number of physical degrees of freedom included in each integration “constant”is ζ (0) ( ~x ) . . . ,f ij ( ~x ) . . . − ,C (1) ( ~x ) . . . ,C (1) ij ( ~x ) . . . − ,φ (1)in ( ~x ) , ˙ φ (1)in ( ~x ) . . . . (81)This is consistent with the fact that the HL gravity includes not only a tensor graviton (2 propagating degrees offreedom) but also a scalar graviton (1 propagating degree of freedom) and that our system includes a scalar field (1propagating degree of freedom) as well. IV. PERTURBATIVE VS NONPERTURBATIVE APPROACHES
In the previous section, we have derived solutions for nonlinear perturbations in any order of gradient expansion.While gradient expansion relies on the long wavelength approximation, amplitudes of perturbations do not haveto be small. Thus, our analysis in the previous section is totally nonperturbative with respect to amplitudes ofperturbations. The dynamical equations and their solutions do not suffer from any divergences in the λ → λ → λ → | λ − | , the linear terms become less important than nonlinear terms. Hence,neglecting nonlinear terms, blindly solving the linearized momentum constraint and then taking the λ → λ → A. Breakdown of standard perturbative expansion in the λ → limit In this subsection let us briefly review the standard perturbative approach and see that, contrary to the nonper-turbative approach based on the gradient expansion in the previous section, it breaks down in the λ → N = 1 , N i = ∂ i B + n i , g ij = a e ζ T (cid:0) e h (cid:1) ij , (82)where n i is transverse and h ij is transverse and traceless: ∂ i n i = 0, ∂ i h ij = 0 and h ii = 0. Throughout thissubsection, indices are raised and lowered by δ ij and δ ij . We introduce a small parameter ¯ ǫ , consider ζ T , B , n i and h ij as quantities of O (¯ ǫ ), and perform perturbative expansion with respect to ¯ ǫ .In the regime of validity of the standard perturbative expansion, in order to calculate the action up to cubic order,it suffices to solve the momentum constraint up to the first order, which can be written in the form, ∂ i (cid:2) a (3 λ − ∂ t ζ T − ( λ − △ B (cid:3) + 12 △ n i = 0 , (83)1leading to a − △ B = 3 λ − λ − ∂ t ζ T , n i = 0 , (84)where △ ≡ ∂ i ∂ i .It is straightforward to calculate the kinetic action up to the third order. The quadratic part I (2) kin and the cubicpart I (3) kin are [29] I (2) kin = Z dtd ~xa (cid:18) a − ∂ t ζ T △ B + 18 ∂ t h ij ∂ t h ij (cid:19) ,I (3) kin = Z dtd ~xa (cid:20) ζ T (cid:18) a − ∂ t ζ T △ B + 18 ∂ t h ij ∂ t h ij (cid:19) + 12 a − ζ T ∂ i ( ∂ i B △ B + 3 ∂ j B∂ i ∂ j B )+ 12 ( a − ∂ k h ij ∂ k B − ∂ t h ij ζ T ) a − ∂ i ∂ j B − a − ∂ t h ij ∂ k h ij ∂ k B (cid:21) . (85)When B is eliminated by using (84), one can easily see that the quadratic part I (2) kin written in terms of ˜ ζ T = q λ − λ − ζ T is regular. On the other hand, the cubic part I (3) kin written in terms of ˜ ζ T is divergent in the limit λ →
1. Thus, theperturbative expansion breaks down in this limit. More precisely, the regime of validity of the standard perturbativeexpansion is | ζ T | ≪ min( | λ − | , , (86)and disappears in the λ → λ → λ − B. Transformation from transverse to synchronous gauge
In the standard perturbative approach summarized in the previous subsection, we have adopted the transversegauge (82). Instead, in the nonperturbative approach based on the gradient expansion presented in Sec.III, we haveadopted the synchronous gauge (25). In this subsection, we shall investigate the spatial coordinate transformationbetween the two gauges. (Note that in both gauges the space-independent time reparametrization is already fixed bythe condition N = 1.) The transformation is nonlinear but we treat it perturbatively. As we shall see below, thisprovides an alternative way to see the breakdown of the standard perturbative expansion.As described in Appendix B, we start with the transverse gauge, carry out the spatial gauge transformation to thesynchronous gauge, and use the momentum constraint (in the transverse gauge) to eliminate the nondynamical degreeof freedom. In this way, we can express the perturbation in the synchronous gauge in terms of that in the transversegauge. Up to the second order, the result is ζ = −
23 1 λ − ( ζ T − (3 λ − λ −
1) ( ∂ i ζ T ) ( ∂ i △ − ζ T ) + (3 λ − λ − Z t dt ′ ( ∂ i ζ T ) (cid:0) ∂ i △ − ∂ t ′ ζ T (cid:1) − (3 λ − λ − Z t dt ′ △ − (cid:20) ∂ i △ ζ T )( ∂ i △ − ∂ t ′ ζ T ) + (cid:18) ∂ i ∂ j ζ T + 12 △ h ij (cid:19) ( ∂ i ∂ j △ − ∂ t ′ ζ T ) + ( △ ζ T )( ∂ t ′ ζ T ) (cid:21) + 14 Z t dt ′ △ − (cid:20)
12 ( ∂ i ∂ t ′ h jk )( ∂ i h jk ) + 12 ( ∂ t ′ h ij )( △ h ij ) − ∂ i ∂ j ζ T )( ∂ t ′ h ij ) (cid:21) + O (¯ ǫ ) ) , (87)where ζ is the perturbation in the synchronous gauge defined in (19), ζ T and h ij are the metric perturbations in thetransverse gauge defined in (82). One can easily see that the terms quadratic in ζ T are suppressed with respect tothe linear term under the condition (86).Conversely, for a fixed amplitude of ζ T , the expansion with respect to ¯ ǫ in (87) breaks down in the λ → λ → λ −
1) are introduced by the solution of the momentum constraint.2
C. Linear vs nonlinear terms in the momentum constraint
Having understood that the origin of the breakdown of the standard perturbative expansion is the treatment of themomentum constraint, we now discuss the regime of validity of the standard perturbative expansion in the momentumconstraint. Importantly, we shall see that a new branch of solution emerges at the edge of the regime of validity ofthe standard perturbative expansion.For this purpose, we adopt the transverse gauge (82) and expand the momentum constraint with respect to ζ T and h ij , considering them as small quantities but keeping B and n i as nonlinear quantities : ζ T = O ( q ) , h ij = O ( q ) , B = O ( q ) , n i = O ( q ) , (88)where we have introduced a small parameter q to count the order of perturbations ζ T and h ij . In the absence of thescalar field, the momentum constraint is [26]0 = H j ≡ ∂ j A ji + 3 A ji ∂ j ζ T − A jl ( γ − ) lk ∂ i γ jk −
13 (3 λ − ∂ i K, (89)where γ ij = ( e h ) ij , ( γ − ) ij = ( e − h ) ij , while the trace of the extrinsic curvature (21) becomes K = 3 (cid:18) ∂ ⊥ ζ T + ∂ t aN a (cid:19) − N ∂ i (cid:2) ( g − ) ij N j (cid:3) = 3 H − a − △ B + 3 ∂ t ζ T + a − (cid:2) − ( ∂ k ζ T )( ∂ k B ) + 2 ζ T △ B + h kl ∂ k ∂ l B (cid:3) + O ( q ) × n k + O ( q ) , (90)and the traceless part (22) is A ji = 12 ( γ − ) jk ∂ ⊥ γ ki − N (cid:26) ( γ − ) jk γ il ∂ k (cid:2) ( g − ) lm N m (cid:3) + ∂ i (cid:2) ( g − ) jk N k (cid:3) − δ ji ∂ k (cid:2) ( g − ) kl N l (cid:3)(cid:27) = 12 ∂ t h ji − a (cid:26)
12 ( ∂ j n i + ∂ i n j ) + (cid:18) ∂ j ∂ i B − δ ji △ B (cid:19)(cid:27) + 1 a (cid:26) ( ∂ j ζ T )( ∂ i B ) + ( ∂ i ζ T )( ∂ j B ) + 2 ζ T ( ∂ j ∂ i B ) − δ ji (cid:2) ( ∂ k ζ T )( ∂ k B ) + ζ T △ B (cid:3)(cid:27) + 1 a (cid:26) (cid:16) ∂ i h jk + ∂ j h ki − ∂ k h ji (cid:17) ( ∂ k B ) + h jk ( ∂ i ∂ k B ) − δ ji h kl ( ∂ k ∂ l B ) (cid:27) + O ( q ) × n k + O ( q ) . (91)Here, it is understood that ( g − ) ij is the inverse of g ij , that derivatives do not act beyond parentheses and that indicesare raised and lowered by δ ij and δ ij . A straightforward calculation results in the following expansion of H i , H i = − (3 λ − ∂ i ∂ t ζ T + O ( q ) − a [ △ + O ( q )] n i + 1 a (cid:26) ( λ − h δ ji △ + O ( q ) i + (cid:18) △ h ji + ∂ j ∂ i ζ T + δ ji △ ζ T (cid:19) + O ( q ) (cid:27) ∂ j B . (92)Notice that in the above, no assumption is made for B and n i , which are still considered to be nonlinear quantities.It is now clear that the leading term in the coefficient of B relies not only on the order of perturbations, but also onthe value of λ − q ≪ min (1 , λ − a − △ B = 3 λ − λ − ∂ t ζ T + O ( q ) , n i = O ( q ) , for q ≪ min (1 , λ − , (93) Although the discussion in this subsection employs the transverse gauge, the general argument holds in any gauge. In particular, inthe synchronous gauge, where B = n i = 0, the issue arising in B gets transferred to the longitudinal part of h ij . However, for the sakeof clarity, we chose to keep find solutions for the nondynamical fields, gauging away the longitudinal mode instead. λ → λ is sufficiently close to 1 and the condition λ − ≪ q ≪ B in Eq.(92) is dominated by the O ( q ) terms instead of the O ( λ −
1) term. Note thatthis is a nonlinear regime but is still consistent with the assumed smallness of the metric perturbations ζ T and h ij .In this regime, the constraint can be written as H j = − ∂ j ∂ t ζ T + O ( q ) − a ( △ + O ( q )) n j + 1 a (cid:2) M ij + O ( λ −
1) + O ( q ) (cid:3) ∂ i B , (95)where we have defined M ij ≡ △ h ij + ∂ i ∂ j ζ T + δ ij △ ζ T = O ( q ) . (96)The transverse part of the Eq.(95) can be found by evaluating ∂ [ k H j ] , ∂ [ k △ n j ] = ( △ h i [ j )( ∂ k ] ∂ i B ) + ( ∂ [ k △ h i j ] )( ∂ i B )+2( ∂ i ∂ [ j ζ T )( ∂ k ] ∂ i B ) + 2( ∂ [ k △ ζ T )( ∂ j ] B ) + O ( q ) . (97)On the other hand, the longitudinal part can be computed from ∂ j H j as − △ ∂ t ζ T + 1 a ¯ M B = 0 , (98)where we define the operator ¯ M as, ¯ M ≡ M ij ∂ i ∂ j + 2 ( ∂ i △ ζ T ) ∂ i . (99)If this operator is invertible, then we obtain B = 2 a ¯ M − △ ∂ t ζ T + O (cid:18) λ − q (cid:19) + O ( q ) . (100)We note that either the ( λ − /q or q term can provide the largest correction to B , depending on the value of λ − λ −
1. One is (93) in the linear regime, and the other is (100) in the nonlinear regime. The standardperturbative expansion in the previous subsections corresponds to the solution (93). On the other hand, what isrelevant for the nonperturbative recovery of GR (plus “dark matter”) in the λ → D. Yet another consideration
In the previous subsection we have seen that there are two mutually exclusive branches of solution to the momentumconstraint. This explains the reason why the standard perturbative approach breaks down in the λ → λ away from 1, the standard perturbative expansion is valid in the transverse gauge and we have the expansionof the kinetic action as given in subsection IV A. On the other hand, for λ sufficiently close to 1, i.e. in the regime(94), it is the nonlinear solution (100) that should be substituted to the kinetic action.Unlike the kinetic action, the potential part of the action does not depend on λ , when written in terms of ζ T and h ij . This is because B does not appear in the potential part of the action. Therefore, if we could somehow definea field ζ c in such a way that the series of terms in the kinetic action for ζ T sums up to form a standard canonicalkinetic term for ζ c , then the whole action written in terms of ζ c should remain finite in the λ → B ) includes exactly two time derivatives, such a field redefinition shouldbe possible in principle. In practice, however, the field redefinition is not easy to perform since it would be nonlinearand highly nonlocal in space. Nonetheless, this consideration already suggests that the λ → V. SUMMARY AND DISCUSSION
In this paper, we have performed a fully nonlinear analysis of superhorizon perturbations in the HL gravity coupledto a scalar field, by using the gradient expansion technique [27]. After applying the long wavelength expansion to theset of field equations, we integrated these explicitly to the second order. We then showed that the solutions can beextended to any order in gradient expansion, while satisfying the momentum constraint at each order. These solutionsare continuous in the GR limit λ → λ → | λ − | is largerthan the order of perturbations. In other words, the range of validity (86) of these solutions has zero measure in thelimit λ →
1. We found that the divergences are originating from the momentum constraint, where the coefficientsof the terms linear in perturbations vanish in the λ → | λ − | ,the linear terms become less important in comparison to the nonlinear ones. Neglecting nonlinear terms and naivelysolving the linearized momentum constraint, then taking the λ → | ζ | ≪ min( | λ − | , | λ − | ≪ | ζ | ≪ λ → λ → ∞ limit was found to be weakly coupled under a certain condition, and the spectrum of perturbations that weregenerated from quantum fluctuations was calculated in this limit. However, the presence of regular behavior both in λ → ∞ and in λ → λ , one needs to know how theflow of λ is realized and how such a flow affects the evolution of cosmological perturbations. With these considerations,we refrain from exploring the cosmological implications of our results for now.On the other hand, a quantum mechanical extension of our analysis may have a chance to address such issues.One of the major concerns with a proper renormalization analysis in HL gravity is the strong coupling problem inthe λ → λ → B which becomes nonlinear. The solution to the momentum constraint in the two regimes, (93)and (100), gives the nondynamical mode as B ≃ ( λ − λ − a △ − ∂ t ζ = O ( ζ ) , for | ζ | ≪ min( λ − , a ¯ M − △ ∂ t ζ = O (1) , for λ − ≪ | ζ | ≪ . (101)Note that both cases are compatible with small ζ . Thus, substituting this nonlinear solution for B in the action, thenapplying the perturbative expansion for ζ may provide a healthy perturbative action. (However, the reduced actionis nonlocal in space while it is local in time).We also note that the breakdown of the naive perturbative expansion does not necessarily result in loss of renor-malizability. We know that in the regime | ζ | ≪ min( λ − , z = 3, provided that the scalar graviton is assigned a vanishing scaling dimension, i.e. ζ → ζ . This fact is nothing but the power-counting renormalizability of the theory, and is seen after replacing B in the action with the linear solution (93), or the first line of (101). Note also that coefficients of all possible termsin the perturbative expansion are expressed in terms of 11 coupling constants in the action (4). On the other hand,for the regime λ − ≪ | ζ | ≪
1, we need to replace B in the action with the nonlinear solution (100), or the secondline of (101). What is important here is that the scaling dimension of B from the nonlinear solution and that from5the linear solution are exactly the same: B → b B under the scaling (1) with z = 3 in both cases. Therefore, aftersubstituting the nonlinear solution to B in the action, we still conclude that the leading UV contributions in theaction are invariant under the scaling (1) with z = 3, provided that the scalar graviton is assigned a vanishing scalingdimension. In other words, the conditions for power-counting renormalizability of the theory continue to hold in thenonlinear regime.In the present paper, we have considered the projectable version of the HL theory and showed that the generalrelativity (plus dark matter) is safely recovered in the λ → g ij , K ij , D i and R ij but also a i ≡ ∂ i ln( N ) (with [ a i ] = [ k ] for z = 3) [17]. This gives rise to a proliferationof independent coupling constants. For example, for the minimal value of the dynamical critical exponent z = 3, thenumber of independent terms in the gravitational action of the non-projectable extension turns out to be more than70 [22]. In some regime of parameters, the non-projectable extension is claimed to be free from the breakdown of thestandard perturbative expansion method in the λ → U (1) symmetry, U (1) ⋉ Diff( M, F ) [30].It has been shown that the standard perturbative expansion does not break down in the gravitational sector, but doesbreak down in the matter sector, at least apparently [31] (see also [32–36] for more on this extension). It is intriguingto see if a nonperturbative analysis similar to those presented in the present paper can resolve this problem.Finally, the biggest obstacle in front of the HL gravity, and in general, any Lorentz symmetry breaking theory, isthe restoration of the Lorentz invariance in the matter sector at low energies [37, 38]. Even if the Lorentz violationis restricted only to the gravity sector, the radiative corrections from graviton loops will generate Lorentz violationin the matter sector. Such terms can be under control provided that the Lorentz breaking scale is much lower thanthe Planck scale [39]. Another approach is to introduce a mechanism, or symmetry to suppress the Lorentz violatingoperators at low energies, such as supersymmetry [40]. Such an approach was adopted in [41] where a SUSY theorywith anisotropic scaling was constructed. On the other hand, this seems to be a highly nontrivial task for the case ofinteracting models [42, 43]. Acknowledgments
Part of this work was done during AW’s visit at IPMU, the University of Tokyo, Kashiwa. He would like to expresshis gratitude to the Institute for stimulating atmosphere and warm hospitality. The work of A.E.G. and S.M. wassupported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. S.M.also acknowledges the support by Grant-in-Aid for Scientific Research 17740134, 19GS0219, 21111006, 21540278, byJapan-Russia Research Cooperative Program. AW is supported in part by DOE Grant, DE-FG02-10ER41692.
Appendix A: Order O ( ǫ n +1 ) momentum constraint for n ≥ In this Appendix, we prove by induction that the order O ( ǫ n ) solution (73)–(77) satisfies the order O ( ǫ n +1 ) mo-mentum constraint equation (52) for n ≥ n + 1)-th order constraint (52) as a linear combination of lower order constraints by using the explicit solution(73)–(77). To achieve this, we make use of the generalized Bianchi identity (55) as well as the identities in Eq.(56).We also use the following identity for functions f ( t ) and g ( t ) satisfying a ( t in ) f ( t in ) g ( t in ) = 0, f ( t ) g ( t ) = 1 a ( t ) Z tt in dt ′ a ( t ′ ) (cid:2) a ( t ′ ) − ∂ t ′ ( a ( t ′ ) f ( t ′ )) · g ( t ′ ) + f ( t ′ ) · ∂ t ′ g ( t ′ ) (cid:3) . (A1)By applying the identity (A1) to ( f ( t ) , g ( t )) = ( A ( p ) j i , ∂ j ζ ( n − p ) ), ( f ( t ) , g ( t )) = ( A ( p ) j l , ( γ − ) ( q ) lk ∂ i γ ( n − p − q ) jk ) and( f ( t ) , g ( t )) = ( ∂ t φ ( p ) , ∂ i φ ( n − p ) ), the left hand side of the ( n + 1)-th order momentum constraint equation (52) is6rewritten as C ( n +1) i ≡ ∂ j A ( n ) ji + 3 n X p =1 A ( p ) j i ∂ j ζ ( n − p ) − n X p =1 n − p X q =0 A ( p ) j l ( γ − ) ( q ) lk ∂ i γ ( n − p − q ) jk −
13 (3 λ − ∂ i (cid:18) K ( n ) + 3 φ ( n ) ∂ t φ (0) λ − (cid:19) − n − X p =1 ∂ t φ ( p ) ∂ i φ ( n − p ) = ∂ j A ( n ) ji + 1 a ( t ) Z tt in dt ′ a ( t ′ ) ( n X p =1 h a − ∂ t ′ (cid:16) a A ( p ) j i (cid:17) ∂ j ζ ( n − p ) + A ( p ) j i ∂ j (cid:16) ∂ t ′ ζ ( n − p ) (cid:17)i − n X p =1 n − p X q =0 h a − ∂ t ′ (cid:16) a A ( p ) j l (cid:17) ( γ − ) ( q ) lk ∂ i γ ( n − p − q ) jk + A ( p ) j l ∂ t ′ (cid:16) ( γ − ) ( q ) lk (cid:17) ∂ i γ ( n − p − q ) jk + A ( p ) j l ( γ − ) ( q ) lk ∂ i (cid:16) ∂ t ′ γ ( n − p − q ) jk (cid:17)i − n − X p =1 h a − ∂ t ′ ( a ∂ t ′ φ ( p ) ) ∂ i φ ( n − p ) + ∂ t ′ φ ( p ) ∂ t ′ ∂ i φ ( n − p ) i) −
13 (3 λ − ∂ i (cid:18) K ( n ) + 3 φ ( n ) ∂ t φ (0) λ − (cid:19) . (A2)Using Eqs.(73)–(74), (48)–(50) and (54), this is further rewritten as C ( n +1) i = 1 a ( t ) Z tt in dt ′ a ( t ′ ) ( ∂ j − n − X p =1 K ( p ) A ( n − p ) ji ! +3 " n X p =2 − p − X q =1 K ( q ) A ( p − q ) j i ! ∂ j ζ ( n − p ) + n − X p =1 A ( p ) j i ∂ j (cid:18) K ( n − p ) (cid:19) − n X p =1 n − p X q =0 " − p − X r =1 K ( r ) A ( p − r ) j l ! ( γ − ) ( q ) lk ∂ i γ ( n − p − q ) jk + A ( p ) j l − q X r =1 A ( r ) l m ( γ − ) ( q − r ) mk ! ∂ i γ ( n − p − q ) jk + A ( p ) j l ( γ − ) ( q ) lk ∂ i n − p − q − X r =0 γ ( r ) jm A ( n − p − q − r ) mk ! −
16 (3 λ − ∂ i − n − X p =1 K ( p ) K ( n − p ) ! − ∂ i − n − X p =1 A ( p ) jk A ( n − p ) kj ! + n − X p =1 p X q =1 K ( q ) ∂ t ′ φ ( p − q ) ∂ i φ ( n − p ) + 16 n X p =1 ¯ Z ( p ) n − p X q =0 ( γ − ) ( q ) jk ∂ i γ ( n − p − q ) jk ) , (A3)where we have used the generalized Bianchi identity (55). By using the identities (56) we finally obtain C ( n +1) i = − a ( t ) Z tt in dt ′ a ( t ′ ) n − X p =1 K ( n − p ) C ( p +1) i . (A4)Since the O ( ǫ ) constraint in Eq. (71) is already satisfied, i.e. C (2) i = 0, the above relation implies that C ( n +1) = 0 for n ≥ Appendix B: Expansion of the nonlinear perturbations
Here, we present the detail of the calculations to obtain the expression of ζ in terms of ζ T and h ij , given in Eq.(87).While the former ζ is defined in the synchronous gauge N i = 0, the latter ζ T and h ij are defined in in the transverse7gauge δ ik ∂ k h ij = 0. In both gauges, the freedom in the time coordinate is fixed by the choice N = 1. For theperturbative expansion of the spatial metric, we use g ij = a e ζ (cid:0) e h (cid:1) ij = a δ ij + a (2 ζδ ij + h ij ) ! + " a (cid:0) ζ δ ij + 4 ζh ij + h il h lj (cid:1) + O (¯ ǫ ) , (B1)where ¯ ǫ denotes the order of perturbations and the indices of h ij are raised and lowered with Kronecker delta.Throughout this Appendix, when the expansion of a quantity is shown, the terms outside parentheses, in parenthesesand in square brackets are of order ¯ ǫ , ¯ ǫ and ¯ ǫ , respectively.
1. Expansion of the momentum constraint
We first concentrate on the linear perturbation in the transverse gauge. To remove the nondynamical degrees inthe shift vector, we solve the constraint equation order by order. We expand the shift vector while separating thecontributions from each order, as N i = ∂ i B (1) + N T (1) i ! + " (cid:16) ∂ i B (2) + N T (2) i (cid:17) + O (¯ ǫ ) (B2)With these decompositions, we expand the momentum constraint in vacuum H j ≡ D i K ij − λ ∂ j K = 0 , (B3)as a series in perturbations. At first order, we get H (1) j = − (3 λ − ∂ j ∂ t ζ T + λ − a ∂ j △ B (1) − a △ N T (1) j , (B4)which can be solved by △ B (1) = 3 λ − λ − a ∂ t ζ T , N T (1) i = 0 . (B5)Using the second of these results, the next order constraint yields, H (2) j = ( λ − ∂ j (cid:20) △ B (2) + ( ∂ k ζ T ) ( ∂ k B (1) ) − ζ T △ B (1) − h kl ∂ k ∂ l B (1) (cid:21) + a (cid:2) ( ∂ t h kl )( ∂ k h lj ) − h kl ∂ k ∂ t h lj − ( ∂ t h kl )( ∂ j h kl ) (cid:3) + 3 a ∂ k ζ T ) ( ∂ t h kj )+( ∂ k ∂ j ζ T ) ( ∂ k B (1) ) + ( △ ζ T ) ( ∂ j B (1) ) + 12 ( △ h kj )( ∂ k B (1) ) − △ N T (2) j = 0 . (B6)For the following, only the longitudinal part of this relation is needed; by taking its divergence, then using the firstequation of (B5), we end up with △ B (2) = 2 a (cid:18) λ − λ − (cid:19) (cid:2) ζ T ∂ t ζ T + h ij ∂ i ∂ j △ − ∂ t ζ T − ( ∂ i ζ T ) ( ∂ i △ − ∂ t ζ T ) (cid:3) − a (3 λ − λ − △ − (cid:20) ∂ i △ ζ T )( ∂ i △ − ∂ t ζ T ) + (cid:18) ∂ i ∂ j ζ T + 12 △ h ij (cid:19) ( ∂ i ∂ j △ − ∂ t ζ T ) + ( △ ζ T ) ( ∂ t ζ T ) (cid:21) + a λ − △ − (cid:20)
12 ( ∂ i ∂ t h jk ) ( ∂ i h jk ) + 12 ( ∂ t h ij ) ( △ h ij ) − ∂ i ∂ j ζ T ) ( ∂ t h ij ) (cid:21) . (B7)
2. Coordinate transformations
Next, we determine the transformation between the transverse and synchronous gauges. We parametrize thecoordinate transformation as ˜ x µ = x µ + (cid:16) ξ (1) µ (cid:17) + (cid:20)
12 ( ξ (1) ν ∂ ν ξ (1) µ + ξ (2) µ ) (cid:21) + O (¯ ǫ ) , (B8)8where over-tilde denotes quantities in the synchronous gauge. The parameters ξ ( n ) µ are decomposed as ξ ( n ) µ = (cid:16) , ξ ( n ) i + ∂ i ξ ( n ) (cid:17) , (B9)with ∂ i ξ ( n ) i = 0, while the indices of ξ ( n ) i are raised and lowered by δ ij and δ ij . For any tensor field expanded as T = T (0) + (cid:16) δT (cid:17) + (cid:20) δ T (cid:21) + O (¯ ǫ ) , (B10)the transformation at linear and quadratic order proceeds through [44] f δT = δT + £ ξ (1) T (0) , g δ T = δ T + 2 £ ξ (1) δT + £ ξ (1) T (0) + £ ξ (2) T (0) . (B11)For the metric tensor, the transformations become g δg µν = δg µν + g (0) µσ ∂ ν ξ (1) σ + g (0) νσ ∂ µ ξ (1) σ , ^ δ g µν = δ g µν + 2 (cid:16) ξ (1) σ ∂ σ δg µν + δg µσ ∂ ν ξ (1) σ + δg νσ ∂ µ ξ (1) σ (cid:17) + g (0) µσ ∂ ν (cid:16) ξ (1) ρ ∂ ρ ξ (1) σ (cid:17) + g (0) νσ ∂ µ (cid:16) ξ (1) ρ ∂ ρ ξ (1) σ (cid:17) + 2 g (0) ρσ ( ∂ µ ξ (1) σ )( ∂ ν ξ (1) ρ )+ g (0) µσ ∂ ν ξ (2) σ + g (0) νσ ∂ µ ξ (2) σ . (B12)We now determine the transformation ξ µ needed to go from the transverse gauge to the synchronous gauge. For this,we evaluate the 0 i components of (B12) and set g δg i = 0. At first order, we obtain, ∂ i B (1) + N T (1) i + a (cid:16) ∂ i ∂ t ξ (1) + ∂ t ξ (1) i (cid:17) = 0 , (B13)where the transverse and longitudinal parts can be easily separated to give, ξ (1) = − Z t dt ′ B (1) ( t ′ ) a , ξ (1) i = − Z t dt ′ N T (1) i ( t ′ ) a . (B14)Using the solutions (B5) to the linear momentum constraint, the transformation parameters become ξ (1) = − λ − λ − △ − ζ T , ξ (1) i = 0 . (B15)Similarly, the 0 i component of the second order transformation (B12) gives, ∂ i B (2) + N T (2) i + a (cid:16) ∂ i ∂ t ξ (2) + ∂ t ξ (2) i (cid:17) +( ∂ j ξ (1) ) ( ∂ i ∂ j B (1) ) − ( ∂ i ∂ j ξ (1) ) ( ∂ j B (1) ) − ζ T ∂ i B (1) − h ij ∂ j B (1) = 0 , (B16)where we used the second equation of (B15). Using also the first equation of (B15) as well as the first order constraint(B5), the longitudinal part of the second order transformation can be obtained as △ ξ (2) = − Z t dt ′ △ B (2) a + (cid:18) λ − λ − (cid:19) (cid:0) ∂ i ζ T (cid:1) (cid:0) ∂ i △ − ζ T (cid:1) + 2 (cid:18) λ − λ − (cid:19) ζ T +2 (cid:18) λ − λ − (cid:19) Z t dt ′ (cid:20) h ij ∂ i ∂ j △ − ∂ t ′ ζ T − (cid:18) λ + 1 λ − (cid:19) ( ∂ i ζ T ) (cid:0) ∂ i △ − ∂ t ′ ζ T (cid:1)(cid:21) . (B17)Inserting the expression of B (2) from (B7) into the above expression, we finally obtain △ ξ (2) = (cid:18) λ − λ − (cid:19) (cid:0) ∂ i ζ T (cid:1) (cid:0) ∂ i △ − ζ T (cid:1) − λ − λ − Z t dt ′ ( ∂ i ζ T ) ( ∂ i △ − ∂ t ′ ζ T )+ 2 (3 λ − λ − Z t dt ′ △ − (cid:20) ∂ i △ ζ T )( ∂ i △ − ∂ t ′ ζ T ) + (cid:18) ∂ i ∂ j ζ T + 12 △ h ij (cid:19) ( ∂ i ∂ j △ − ∂ t ′ ζ T ) + ( △ ζ T )( ∂ t ′ ζ T ) (cid:21) − λ − Z t dt ′ △ − (cid:20)
12 ( ∂ i ∂ t ′ h jk )( ∂ i h jk ) + 12 ( ∂ t ′ h ij )( △ h ij ) − ∂ i ∂ j ζ T )( ∂ t ′ h ij ) (cid:21) . (B18)9We note that the second order gauge transformation is more divergent than the first order one (B15), in the limit λ → ζ in the synchronous gauge. Since we adopted a nonperturbative decomposition forthe spatial metric, it is useful to express this quantity as, ζ = 16 log (cid:18) det ˜ ga (cid:19) . (B19)Applying the perturbative expansion to the right hand side, we obtain ζ = (cid:18) a g δg ii (cid:19) + (cid:20) a (cid:18) ] δ g ii − a g δg ij g δg ij (cid:19)(cid:21) + O (¯ ǫ ) . (B20)Using the transformed metric from (B12), the above expression becomes ζ = (cid:18) ζ T + 13 △ ξ (1) (cid:19) + (cid:20) ( ∂ i ζ T ) ( ∂ i ξ (1) ) + 16 ( ∂ i ξ (1) ) ( ∂ i △ ξ (1) ) + 16 △ ξ (2) (cid:21) + O (¯ ǫ ) . 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