Effective field theory of degenerate higher-order inflation
YYITP-20-09
Effective field theory of degenerate higher-order inflation
Hayato Motohashi and Wayne Hu Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Kavli Institute for Cosmological Physics, Department of Astronomy and Astrophysics,Enrico Fermi Institute, The University of Chicago, Chicago, Illinois 60637, USA (Dated: April 28, 2020)We extend the effective field theory of inflation to a general Lagrangian constructed from Arnowitt-Deser-Misner variables that encompasses the most general interactions with up to second derivativesof the scalar field whose background breaks temporal diffeomorphism invariance. Degeneracy con-ditions, corresponding to 8 distinct types – only one of which corresponds to known degeneratehigher-order scalar-tensor models – provide necessary conditions for eliminating the Ostrogradskyghost in a covariant theory at the level of the quadratic action in unitary gauge. Novel implicationsof the degenerate higher-order system for the Cauchy problem are illustrated with the phase spaceportrait of an explicit inflationary example: not all field configurations lead to physical solutionsfor the metric even for positive potentials; solutions are unique for a given configuration only upto a branch choice; solutions on one branch can apparently end at nonsingular points of the metricand their continuation on alternate branches lead to nonsingular bouncing solutions; unitary gaugeperturbations can go unstable even when degenerate terms in the Lagrangian are infinitesimal. Theattractor solution leads to an inflationary scenario where slow-roll parameters vary and running ofthe tilt can be large even with no explicit features in the potential far from the end of inflation,requiring the optimized slow-roll approach for predicting observables.
I. INTRODUCTION
Single-field scalar-tensor theories as inflationary models can be studied in a unified way in the framework of theeffective field theory (EFT) of inflation, where the timelike scalar field is treated as a clock that breaks the timediffeomorphism invariance leaving spatial diffeomorphism invariance unbroken [1, 2]. In general, the EFT of inflationwith higher-derivative operators contains extra ghost degrees of freedom, which may or may not propagate in theregime of validity of the EFT. To consider a regime where higher-derivative interactions also produce interestingobservable phenomenology, one needs to rely on the framework where the ghost degrees of freedom are appropriatelyeliminated. Therefore, an EFT Lagrangian motivated by general ghost-free theories serve as a useful framework. Inthis context, the original EFT framework has been extended in subsequent works [3–7] to include derivative operatorsappearing in the Horndeski [8–13], Gleyzes-Langlois-Piazza-Vernizzi (GLPV) [14, 15] and Horava-Lifshitz [16–18]theories.More general theories involving additional derivative of fields typically propagate ghost degrees of freedom. Theghosts associated with higher-order derivatives are known as the Ostrogradsky ghosts [19, 20], which makes the Hamil-tonian unbounded due to its linear dependence on canonical momenta. Unlike the classically unbounded Hamiltonianof the hydrogen atom, the Ostrogradsky Hamiltonian remains unbounded quantum mechanically as well [21–23].To eliminate the ghost degrees of freedom, one needs to evade the condition of the Ostrogradsky theorem that theLagrangian is nondegenerate with respect to the highest-order derivatives. However, degeneracy with respect to thehighest-order derivative is necessary but not sufficient to evade the unbounded Hamiltonian [24], which is the reasonwhy one needs to impose a certain set of degeneracy conditions to eliminate all aspects of the Ostrogradsky ghosts [25–28]. This argument can be also understood in a broader context in the language of constraints as a generalization ofthe Ostrogradsky theorem [29].The degeneracy conditions were applied to a construction of degenerate higher-order scalar-tensor (DHOST) theorieswith quadratic [25] and cubic interactions [30] of second derivatives of a scalar field, which include the derivative ofthe lapse function. The EFT description of the quadratic and cubic DHOST theories was developed in [31], where thequadratic Lagrangian around the cosmological background was investigated. Cosmological evolution and the linearstability analysis were also investigated [32], focusing on the de Sitter attractor in a shift symmetric quadratic DHOSTmodel. In [31, 32], two assumptions on background dynamics were adopted: that the lapse remains unity and thatthe scalar field is proportional to time coordinate. The first can be imposed as a gauge condition, and the secondshould be satisfied dynamically. For general timelike scalar field evolution on a given phase space trajectory, oneneeds to perform a redefinition of the scalar field to make it proportional to time, which changes all of the DHOST a Present address: Division of Liberal Arts, Kogakuin University, 2665-1 Nakano-machi, Hachioji, Tokyo, 192-0015, Japan a r X i v : . [ h e p - t h ] A p r coefficients (cf. [32] v2). A dynamical lapse is also taken into account in [33, 34] in the context of spatially covariantgravity [35], where the degeneracy conditions were also studied. However, the general EFT framework of degeneratetheories including but not limited to quadratic and cubic DHOST and its application to cosmology have not beenfully investigated yet.In this paper, generalizing our previous work [7], we develop the EFT framework of general degenerate theories, andexplore its peculiar phenomenology. This framework includes quadratic and cubic DHOST as a special subclass as wellas theories where the lapse is nondynamical, e.g. those with second-order equations of motion for the scalar field, as inthe Horndeski case, or the spatial metric in unitary gauge, as in the GLPV case. In § II, we consider the EFT actioncomposed of Arnowitt-Deser-Misner (ADM) geometric quantities including the acceleration and lapse derivative andtheir couplings to intrinsic and extrinsic curvatures. This action includes operators appearing in covariant theoriesinvolving the most general combination of second-order derivatives of scalar field. It also includes Lorentz-violatingtheories such as Horava-Lifshitz gravity [16–18], as well as the scordatura degenerate theory [36] weakly violatingthe degeneracy condition. We derive the background and quadratic actions for various degeneracy classes, whichinclude known DHOST cases, summarized in Appendix A. In § III, we investigate dynamics in degenerate higher-order inflation, which we dub “D-inflation”, and clarify several novel features of degenerate models for both thebackground and perturbations. We provide a detailed study of a specific model, for which the optimized slow-roll(OSR) formalism [37] serves as a powerful tool as the EFT coefficients can exhibit variation on the several efold timescale. In § IV, we discuss conclusions.
II. EFT OF INFLATION
In this section we adopt ADM decomposition and consider the general EFT Lagrangian allowing the most generalcombination of second-order derivatives of scalar field. In § II A, we construct the EFT Lagrangian from geometricquantities including the acceleration and lapse derivative and their arbitrary couplings to intrinsic and extrinsiccurvatures. In § II B, we write down the background and quadratic Lagrangians around cosmological background. Sincevector and tensor perturbations are the same as the previous work [7], in § II C we focus on the scalar perturbations, andreduce the quadratic Lagrangian for a specific degeneracy class. We provide the complete analysis of the constructionof degeneracy conditions in Appendix A.
A. ADM EFT Lagrangian
We work in the 3 + 1 ADM decomposition of the metric ds = − N dt + h ij ( dx i + N i dt )( dx j + N j dt ) , (1)where N , N i , h ij are the lapse, shift, spatial metric, respectively. We define a timelike unit vector n µ ≡ − N t ,µ orthogonal to constant t surfaces, the acceleration a µ ≡ n ν n µ ; ν , and the extrinsic curvature K µν = n ν ; µ + n µ a ν , wheresemicolons on indices here and throughout denote covariant derivatives with respect to g µν .For general scalar-tensor theories, we can choose so-called unitary gauge, where φ = φ ( t ) as long as the gradient ofthe scalar field is always timelike. In unitary gauge, any Lagrangian with up to second derivatives in the field can beexpressed in terms of ADM quantities through φ ,µ = −√− Xn µ ,φ ; µν = √− X ( − K νµ + n ν a µ + n µ a ν − βn µ n ν ) . (2)Here, we define β by β = − n µ (ln X ) ,µ = − ¨ φN ˙ φ + ˙ N − N i ∂ i NN , (3)where X ≡ g µν φ ,µ φ ,ν = − ˙ φ /N is the kinetic term for the scalar. In particular, if we take φ = t , X = − /N andthen β measures the fractional change in the lapse along the normal β = n λ (ln N ) ,λ = ˙ N − N i ∂ i NN , (4)and more generally it determines the fractional change in the elapsed proper time in field coordinates and so (3)involves ¨ φ . In the gauge where φ ∝ t , ¨ φ = 0 and this has been used widely for the purpose of counting the numberof degrees of freedom through the Hamiltonian analysis. However, to keep a normal perturbation analysis wherethe background lapse ¯ N = 1, we use φ = φ ( t ), and retain the ¨ φ term. After solving for a given trajectory φ ( t ), wecan always make a field redefinition ϕ ∝ t ( φ ), which maintains ¯ N = 1 at the expense of redefining the scalar fieldLagrangian, but adopting this at the outset prevents a phase space analysis for background trajectories.We seek to construct a general spatial diffeomorphism invariant EFT Lagrangian involving no more than second-order derivatives of the scalar field which a priori contains N, K µν , a µ , β . So long as we consider theories involvingup to φ ; µν , a i and a j are the only quantities that have one spatial sub/superscript. Hence, a i and a j always appeartogether through α ij ≡ a i a j = h ik (ln N ) ,j (ln N ) ,k , α ≡ α ii . (5)We therefore consider the Lagrangian to be a spatially diffeomorphism invariant function of these quantities S = (cid:90) d xN √ hL ( N, K ij , R ij , α ij , β ; t ) , (6)generalizing [7] to allow it to depend on α ij and β . Here R ij is the Ricci 3-tensor on the spatial slice. Higher-derivativeLagrangians typically contain Ostrogradsky ghosts, but we shall see in the next section that for special combinationsof N, K ij , R ij , α ij , β the Lagrangian only propagates one scalar and the usual two tensor degrees of freedom. Withoutthe α ij dependence and allowing higher-order spatial derivatives, the Lagrangian (6) reduces to the one explored in[33, 34].To explicitly relate this EFT Lagrangian to known ghost-free scalar-tensor theories, we begin with the most generalcovariant Lagrangian that is at most cubic in second derivatives of the field and coupled to the metric as S = (cid:90) d x √− g (cid:34) F + F (cid:3) φ + F R + F G µν φ ; µν + (cid:88) i =1 A i L (2) i + (cid:88) i =1 B i L (3) i (cid:35) , (7)where F i , A i , B i are general functions of φ, X and (4) R is the four-dimensional Ricci scalar. The terms that arequadratic in second derivatives are L (2)1 = φ ; µν φ ; µν = ˙ φ N ( K ij K ji + β − α ) ,L (2)2 = ( (cid:3) φ ) = ˙ φ N ( − K + β ) ,L (2)3 = ( (cid:3) φ ) φ ; µ φ ; µν φ ; ν = ˙ φ N β ( K − β ) ,L (2)4 = φ ; µ φ ; µν φ ; νρ φ ; ρ = ˙ φ N ( α − β ) ,L (2)5 = ( φ ; µ φ ; µν φ ; ν ) = ˙ φ N β , (8)and those that are cubic are L (3)1 = ( (cid:3) φ ) = ˙ φ N ( − K + β ) ,L (3)2 = ( (cid:3) φ ) φ ; µν φ ; µν = ˙ φ N ( − K + β )( K ij K ji + β − α ) ,L (3)3 = φ ; µν φ ; νρ φ ; µ ; ρ = ˙ φ N ( − K ij K jk K ki + 3 α ij K ji + β − αβ ) ,L (3)4 = ( (cid:3) φ ) φ ; µ φ ; µν φ ; ν = − ˙ φ N β ( − K + β ) ,L (3)5 = ( (cid:3) φ ) φ ; µ φ ; µν φ ; νρ φ ; ρ = ˙ φ N ( − K + β )( α − β ) ,L (3)6 = φ ; µν φ ; µν φ ; ρ φ ; ρσ φ ; σ = ˙ φ N β ( − K ij K ji − β + 2 α ) , L (3)7 = φ ; µ φ ; µν φ ; νρ φ ; ρσ φ ; σ = ˙ φ N ( − α ij K ji − β + 2 αβ ) ,L (3)8 = φ ; µ φ ; µν φ ; νρ φ ; ρ φ ; σ φ ; σξ φ ; ξ = ˙ φ N β ( β − α ) ,L (3)9 = (cid:3) φ ( φ ; µ φ ; µν φ ; ν ) = ˙ φ N β ( − K + β ) ,L (3)10 = ( φ ; µ φ ; µν φ ; ν ) = − ˙ φ N β , (9)where we have used (2) to establish the correspondence with the ADM variables. Similarly, we can relate the couplingto the metric using the Gauss-Codazzi relation and integration by parts to rewrite up to boundary terms (see e.g.[3, 38]) (cid:90) d x √− gF R = (cid:90) d x √− g (cid:34) F (cid:0) R + K ij K ji − K (cid:1) − (cid:32) F φ ˙ φN K + 2 F X ˙ φ N ( βK − α ) (cid:33)(cid:35) , (10) (cid:90) d x √− gF G µν φ ; µν = (cid:90) d x √− g (cid:34) F ˙ φN (cid:18) KR − K ij R ji (cid:19) + F φ − F φ φ N R + F φ φ N ( K ij K ji − K )+ F X ˙ φ N (cid:16) ( K ij K ji − K ) β + 2 αK − α ij K ji ) (cid:17) (cid:35) , (11)where F X = F X + F X . (12)While in general, the appearance of α ij and β in the Lagrangian signals an extra degree of freedom since thelapse and shift no longer obey constraint equations, this general Lagrangian (7) contains classes that propagate only 3degrees of freedom and avoids Ostrogradsky ghosts. First, there is the GLPV class which defines a special relationshipbetween the A i , B i , F i coefficients A = − A = 2 F X + XF , A = − A = 2 F ,B = − B B F X XF , − B = B = 2 B = − B = 6 F ,A = B = B = B = 0 . (13)Note that the F and F terms are also arbitrary functions of φ, X . In the Horndeski subclass of GLPV, where thescalar field equations themselves are explicitly second order, F = F = 0 and the remaining functions are moretypically labeled ( G , G , G , G ) = ( F , F , F , F ). It is easy to verify through Eqs. (8,9,10) that this relationshipeliminates the dependence of α ij and β in the GLPV and Horndeski Lagrangians leaving the EFT Lagrangian of theform L ( N, K ij , R ij ; t ). More generally, α ij and β can appear in a Lagrangian which still only propagates 3 degreesof freedom if the functions A i , B i , F i satisfy a certain set of degeneracy conditions (see (40), (41) and [25, 30]). Thisis the DHOST class of models. The Lagrangian (6) also includes the scordatura degenerate theory [36] with a weakviolation of the degeneracy condition.In § II C we generalize these degeneracy conditions to the full EFT Lagrangian (6). Since there the dependence on α ij , β is arbitrary it encompasses theories with further higher-order products of φ ; µν beyond (7). The Lagrangian(6) thus can represent any fully covariant or Lorentz-violating theory involving up to second derivatives of metricand scalar field in the unitary gauge. Furthermore in comparison to EFTs that are explicitly built to encompassDHOST, it allows terms like βR that would only appear with different couplings between the field and the metricthan represented in (7) (cf. [25, 30, 31]). B. Background and quadratic Lagrangian
We next consider the expansion of the ADM EFT Lagrangian (6) to quadratic order in metric perturbations arounda spatially flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) background¯ N = 1 , ¯ N i = 0 , ¯ h ij = a δ ij , (14)for which ¯ K ij = Hδ ij , ¯ R ij = 0 , ¯ α ij = 0 , ¯ β = − ¨ φ ˙ φ , (15)where H ≡ d ln a/dt is the Hubble parameter. Following [7] we define the Taylor coefficients as L (cid:12)(cid:12)(cid:12) b = C ,∂L∂Y ij (cid:12)(cid:12)(cid:12) b = C Y δ ji ,∂ L∂Y ij ∂Z k(cid:96) (cid:12)(cid:12)(cid:12) b = C Y Z δ ji δ (cid:96)k + ˜ C Y Z δ (cid:96)i δ jk + δ ik δ j(cid:96) ) , (16)where “b” denotes that the quantities are evaluated on the background, Y, Z ∈ {
N, K, R, α, β } and the index structureis determined by the symmetry of the background. For notational simplicity we treat scalars and traces with thesame notation; thus implicitly N ii ≡ N , β ii ≡ β and ˜ C NZ = ˜ C βZ = 0.We can further eliminate terms linear in δK = K − H through the identity (cid:90) d x √− gF ( t ) K = − (cid:90) d x √− gn µ F ; µ = − (cid:90) d x √− g ˙ FN , (17)which follows from K = n µ ; µ ignoring boundary terms. The Lagrangian L = N √ hL up to quadratic order in metricperturbations becomes L = N √ h ( C − H C K ) − √ h ˙ C K + N √ h ( C N δN + C R δR + C α δα + C β δβ ) + a (cid:88) Y,Z ( C Y Z δY δZ + ˜ C Y Z δY ij δZ ji ) . (18)In this form δK only shows up in the quadratic-order terms, and we need only its first-order perturbation. In contrast,we expand δR ij , δα ij , δβ up to quadratic order δY ij = δ Y ij + δ Y ij + · · · . (19)We note that δ R ij , δ R ij do not involve δN, δN i , whereas δ α ij = 0 ,δ α ij = 1 a δ ik δN ,j δN ,k ,δ β = − ¯ βδN + ˙ δN ,δ β = ¯ βδN − δN ˙ δN − δN i δN ,i , (20)since n µ ≈ (1 − δN + ( δN ) , − δN i (1 − δN )) , (ln N ) ,µ ≈ δN ,µ (1 − δN ) , (21)up to quadratic order. Note that for the perturbed FLRW metric, α ij is a quadratic-order quantity. The backgroundequations are given by varying the action with respect to N and √ h C − H C K + C N − (3 H + ¯ β ) C β − ˙ C β = 0 , C − H C K − ˙ C K = 0 . (22)With the background equation and integration by parts, the quadratic Lagrangian becomes L = a (cid:34) C R (cid:32) δ R δ √ ha + δ R (cid:33) − C β δN (cid:32) δ √ ha (cid:33) (cid:5) − C β δN i δN ,i + 12 C ββ ˙ δN + C α a δ ij δN ,i δN ,j + 12 (cid:32) C N + C NN − β C βN + ¯ β C ββ + (cid:0) a C β (cid:1) (cid:5) a − (cid:0) a C βN (cid:1) (cid:5) a + (cid:0) a ¯ β C ββ (cid:1) (cid:5) a (cid:33) δN + ( C βK δ K + C βR δ R ) ˙ δN + [( C NK − ¯ β C βK ) δ K + ( C NR + C R − ¯ β C βR ) δ R ] δN + 12 (cid:88) Y = K,R (cid:88) Z = K,R ( C Y Z δ Y δ Z + ˜ C Y Z δ Y ij δ Z ji ) . (23)This expansion can be continued to higher order for the computation of non-Gaussianity (see e.g. [39]). C. Scalar perturbations
From the quadratic Lagrangian (23), we note that the new terms C α , C β , C βY are always accompanied by δN , whichis a natural consequence of (20). Therefore, α ij and β dependencies of the Lagrangian change the dynamics of scalarbut not vector or tensor perturbations which are given explicitly in [7].For scalar perturbations N = 1 + δN, N i = ∂ i ψ, h ij = a e ζ δ ij . (24)Following [7], we use δ √ h = 3 a ζ,δK ij = ( ˙ ζ − HδN ) δ ij − a δ ik ∂ k ∂ j ψ,δK = 3( ˙ ζ − HδN ) − ∂ ψa ,δ R ij = − a ( δ ij ∂ ζ + δ ik ∂ k ∂ j ζ ) ,δ R = − a [( ∂ζ ) − ζ∂ ζ ] ∼ − a ( ∂ζ ) , (25)where the last equality for δ R holds up to a total derivative, to obtain the quadratic Lagrangian in Fourier space as L = 12 c ˙ ζ + c ˙ ζ ˙ δN + 12 c ˙ δN + (cid:18) c + c k a (cid:19) ˙ ζδN + 12 (cid:18) c + c k a (cid:19) k a ζ + c k a ζδN + 12 (cid:18) c + c k a (cid:19) δN + 12 c k a ψ + k a ψ (cid:18) c ζ + c δN + c δN + c k a ζ (cid:19) , (26)where c = 3 a (3 C KK + ˜ C KK ) , c = 3 a C βK , c = a C ββ , c = − a Θ ,c = − a C βR , c = 4 a Ψ , c = 2 a (8 C RR + 3 ˜ C RR ) , c = 4 a Ξ ,c = a Φ , c = 2 a C α , c = a ( C KK + ˜ C KK ) , c = 2 a (2 C KR + ˜ C KR ) , (27)and Φ ≡ C N + C NN − β C βN + ¯ β C ββ + ( C β − C βN + ¯ β C ββ ) (cid:5) + 3 H [ C β − C βN + ¯ β C ββ − C NK − ¯ β C βK ) + ˙ C βK ] + 3 C βK ˙ H + 3 H [3( C βK + C KK ) + ˜ C KK ] , Ψ ≡ C R − C KR − ˙˜ C KR − H (3 C KR + ˜ C KR ) , Ξ ≡ C NR + C R − ¯ β C βR − ˙ C βR − H ( C βR + 3 C KR + ˜ C KR ) , Θ ≡ C β − ( C NK − ¯ β C βK ) + H (3 C KK + ˜ C KK ) . (28)We highlight these four combinations as they involve time derivatives or the Hubble parameter and degeneracyconditions involving them would typically need to arise from integration by parts on the Lagrangian [see e.g. (B1)].In Appendix A, generalizing Ref. [31], we provide a complete analysis of the construction of degeneracy conditionsimposed on the various c i coefficients which we briefly summarize here. The result is 8 types of degeneracy conditions,cases 1 a . . . c (with 3a impossible to satisfy), each of which may be realized by the c i or equivalently the C i coefficientsin various ways. The degeneracy conditions we derive apply for any theory involving second-order derivatives in anyform in the Lagrangian for unitary gauge (6). Since the Lagrangian (6) allows any dependence on ( N, K ij , R ij , α ij , β ),or equivalently on second derivatives (2), this degeneracy conditions applies beyond the quadratic and cubic DHOSTtheories. Furthermore, in general it also applies to Lorentz-violating theories.The first condition required for the single scalar propagating degree of freedom is degeneracy in the temporalstructure of (26). Of the three possibilities, we focus on the case 1 type where the condition c = c /c is satisfiedand the combination ˜ ζ = ζ + c c δN, (29)alone carries the temporal derivatives. The other two cases have the lapse δN as the propagating degree of freedom andwould cause difficulties in recovering an observationally viable theory of gravity. More generally our linear degeneracyconditions should be viewed as necessary, but not necessarily sufficient, conditions for a viable nonlinear scalar-tensortheory of gravity.Under this c = c /c condition, the quadratic Lagrangian (26) for scalar perturbation in unitary gauge wouldappear to propagate only 1 degree of freedom. This degeneracy condition applies to the scalar quadratic Lagrangianin any theory involving second-order derivative of any form in Lagrangian in the unitary gauge (6) and includes theDHOST models as well as the Horndeski or GLPV models where c = 0 and ˜ ζ = ζ .However, the temporal degeneracy condition alone is not sufficient to guarantee that there is only a single degreeof freedom. In terms of the Euler-Lagrange equations, it only removes the fourth-order derivatives and third-orderderivatives still need to be removed to avoid unbounded Hamiltonian [24]. Furthermore, if unitary gauge defines afoliation that corresponds to characteristic surfaces of the second degree of freedom then its dynamics are hidden fromthis temporal structure. Since such a degree of freedom propagates instantaneously on this surface, it is not a Cauchysurface upon which initial conditions can be propagated forwards in time. Hence its temporal kinetic terms vanish.However on a noncharacteristic surface, temporal kinetic terms reappear and can possess a well-posed Cauchy problemas discussed in detail in [40, 41]. One should therefore not take the apparent lack of an extra degree of freedom inunitary gauge as a definitive absence (cf. [35]). Of course, the counting of degrees of freedom cannot depend on thegauge or ADM slicing and so we expect additional degeneracy conditions that involve the spatial derivatives of thekinetic matrix in unitary gauge.For a 1 + 1 dimensional system of linear partial differential equations, including the plane parallel Fourier modesconsidered below, one can exploit the algorithm [41] based on the Kronecker form of a matrix pencil which includesall possible linear combinations of temporal and spatial derivatives to count degrees of freedom and find characteristiccurves in the presence of any hidden constraints (see Appendix of [41]). However for the quadratic and cubic DHOSTtheories, the full covariant and nonlinear degeneracy conditions are already known. As shown in the Appendix, we canobtain the remaining conditions for the quadratic action by demanding that the dispersion relation of remaining degreeof freedom take their normal linear form in unitary gauge. This logic also applies to the wider class of degeneratetheories that originate from a covariant action and so we retain terms that are absent in the quadratic and cubicDHOST Lagrangian in Appendix A.We now focus in particular on the degeneracy conditions given in (A14) in case 1a, as other branches may not havephenomenologically viable theories of gravity associated with them. We emphasize though that this same procedurecan be carried out for any of the branches. In this case, the conditions on the c i coefficients are c = c c , c = c = c = c = 0 , c = 2 c x − c x , (30)where x = c /c , and these conditions imply C ββ = 3 C βK C KK + ˜ C KK , ˜ C KK = −C KK , ˜ C KR = − C KR , ˜ C RR = − C RR , C α = 2 C βK C KK Ξ − C βK C KK Ψ , C βR = 0 . (31)This branch includes the N-I/Ia class of quadratic and cubic DHOST, GLPV and Horndeski theories.Under these conditions we can simplify (26) as L = 12 c ˙˜ ζ + ( c − c ˙ x ) ˙˜ ζδN + 12 c k a ˜ ζ + ( c − c x ) k a ˜ ζδN + 12 (cid:0) c + ˙ c x + c ˙ x − c ˙ x (cid:1) δN + k a ψ (cid:18) c ζ + c − c ˙ x δN (cid:19) . (32)The equation of motion for ψ and δN yields the constraints δN = c c ˙ x − c ˙˜ ζ,k a ψ = 3 c ˙ x − c (cid:20) ( c − c ˙ x ) ˙˜ ζ + ( c − c x ) k a ˜ ζ + (cid:0) c + ˙ c x + c ˙ x − c ˙ x (cid:1) δN (cid:21) , (33)where we have assumed 0 < | Ω | < ∞ , (34)with Ω ≡ c ˙ x − c , which generalizes the condition 2 H C KK (cid:54) = C NK employed in Eq. (33) of [7] to cases where theLagrangian depends α ij , β . Violation of this condition makes unitary gauge perturbations ill-defined. For singular Ω,the kinetic term vanishes and hence the system is strongly coupled. On the other hand, for Ω = 0, unitary gauge itselfis ill-defined. To see this, we follow [7] and move to a comoving gauge defined by the condition that for the perturbedEinstein tensor δG i = 0 for a general metric theory of gravity [42]. The gauge transformation from unitary gaugeto comoving gauge is characterized by the time shift T = − ∆ / ˙ H , where ∆ ≡ HδN − ˙ ζ (see Eq. (B14) of [7]). Using(33), we have ∆ = 1Ω (cid:110) [ c ( Hx − ˙ x ) + c ] ˙ ζ + Hc x ( ˙˜ ζ − ˙ ζ ) (cid:111) , (35)so that Ω = 0 makes ∆ diverge, implying that the gauge transformation between the two gauges requires an infinitetime shift and hence is ill-defined. Note that this is not necessarily a problem if the original system of equations in( δN, ˜ ζ, ψ ) possesses only regular singular points and is hence integrable without first imposing the constraint equation(see [43, 44] for a related discussion). Furthermore if ˜ ζ freezes out but ˜ ζ − ζ continues to evolve outside the horizon,then the two gauges will differ. We construct an explicit model where this occurs in § III (see [45] for a discussion ofrelated cases).Note also that if c = c x the k ˜ ζ term vanishes in the Euler-Lagrange equation (33) for δN , and prevents therecovery of Newtonian gravity for nonrelativistic matter, as found in [31] for one of the DHOST subclasses (seeAppendix A for more details).Substituting the constraints (33) into the Lagrangian (32) and integrating by parts give the usual Mukhanov-Sasakiform for the quadratic Lagrangian L = A ζ ˙˜ ζ − B ζ k a ˜ ζ , (36)where A ζ = c c ( c + ˙ c x ) + c ( c ˙ x − c )( c ˙ x − c ) ,B ζ = a (cid:18) c a c − c xc ˙ x − c (cid:19) (cid:5) − c . (37)From these terms, we can define the scalar sound speed c s and the normalization parameter b s as c s = B ζ A ζ , b s = B ζ a (cid:15) H . (38)For a canonical scalar field c s = b s = 1. These expressions are generalizations of Eq. (37) of [7]. III. D-INFLATION WITH TIME VARYING EFT COEFFICIENTS
In this section we consider models of degenerate higher-order inflation (D-inflation) with time varying EFT coef-ficients, specifically in the quadratic DHOST class. In general, EFT coefficients can vary in time and one needs toevaluate carefully the slow-roll hierarchy of all dynamical parameters, for which the generalized slow-roll approxima-tion developed in [7] provides a systematic framework, based on the evolution of H , (cid:15) H , b s and c s in (38) as well asthe analogous quantities for tensors b t = 2 C R , c t = 2 C R ˜ C KK . (39)D-inflation provides an additional motivation for these time-varying considerations in that one might seek to constructmodels where their novel features are only present during inflation and are absent thereafter where they wouldotherwise impact cosmological and astrophysical observables. We construct our model in § III A, and elucidate thenovel features on background dynamics and evolution of perturbations in § III B and § III C, respectively.
A. D-inflation model
As a concrete example, let us require the inflationary model to satisfy c t − ζ − ζ = 0 at the end of inflation.The former is the requirement for tensor sound speed to be the speed of light, imposed at least at the present epochby observation of gravitational waves from binary neutron star merger and its electromagnetic counterpart [46, 47].The latter is imposed as we would like inflation to become fully canonical so that by reheating everything is as usual.We shall see that enforcing this requirement for all perturbation quantities allows us to avoid instabilities caused byderivative couplings [48].We can concretely implement these requirements using the EFT of a quadratic DHOST model starting with thecase 1a degeneracy conditions (31). Since ˜ C KR = C KR = ˜ C RR = C RR = 0 in this case, the third and the fourthconditions in (31) identically hold. From the second condition in (31) we obtain A = − A . (40)Plugging it into the first and fifth conditions in (31) and solving the two equations for A and A , we obtain A = 2( A + 2 F X ) X − (2 A + 4 F X + XA )[8 F + 2 XA (5 F − XF X ) + X A F − XF F X ]8 X ( F + XA ) ,A = (2 A + 4 F X + XA )[4 A F − A (2 A + 4 F X − XA )]8( F + XA ) , (41)which matches Eqs. (5.1) and (5.2) in [25]. In general a degenerate theory in this class is identified by A , A , F andone can choose F , F as free functions without affecting the degeneracy structure. Note that if C βK = 0 then2 A + 4 F X + XA = 0 , (42)so that A and A are no longer independent; this case corresponds to Eq. (5.3) in [25] and reproduces the GLPVrestriction for quadratic terms in (13).Next, from (8), we obtain the tensor sound speed as c t − C R ˜ C KK − XA F − XA = − XA F + XA , (43)where we used (40) and ˜ ζ − ζ = C βK C KK + ˜ C KK = − X (2 A + 4 F X + XA )4 F + 2 X ( A + 3 A ) . (44)For the Horndeski theory, requiring the right-hand side of (43) to vanish implies that F = F ( φ ). Note also that theright-hand side of (44) identically vanishes for Horndeski and GLPV theories.We would like to choose the functions A , A , F to make these two quantities (43), (44) be nonzero during inflationand evolve to zero by the end of inflation. As a simple example, we set F = − X − V ( φ ) , F = 0 , F = 12 , A = 0 , (45)where we work in natural units M Pl ≡ (8 πG ) − / = 1, and for which the degeneracy conditions (40), (41) yield A = − A , A = A (3 + 8 XA )(1 + 2 XA ) , A = − A (1 + 2 XA ) , (46)and (43), (44) read c t − ζ − ζ ) = 11 + 2 XA − ≡ θ ( φ, X ) . (47)Here, we are interested in a function θ such that it starts from finite value and evolves to zero either from the evolutionof X or an appropriate form for A ( φ, X ).0Under these assumptions, the action is given by S = (cid:90) d xN √ h (cid:34)
12 ( R + K ij K ji − K ) + ˙ φ N − V ( φ ) + A ˙ φ N ( K − K ij K ji − βK + 2 α )+ A − ˙ φ N A ˙ φ N (cid:32) − ˙ φ N A − ˙ φ N A α − β (cid:33) (cid:35) . (48)For simplicity, we will illustrate this model with A = const, or at least nearly so during most of the ∼
60 efoldsbefore the end of inflation. We shall see that in models where the field oscillates at reheating, A needs to vanishbefore this point to avoid gradient or ghost instabilities. However, any late-time change does not affect large-scaleobservables which are well outside the horizon at that point. B. Background dynamics
From (48) we can calculate EFT coefficients. For instance, those which are necessary for the background equa-tions (22) are C = ˙ φ − V − H b (1 − φ A ) , C K = − H b (1 − φ A ) , C N = − ˙ φ − H b [ H − H b (1 − ˙ φ A )] , C β = − A ˙ φ H b , (49)and those for the perturbations areΞ = Ψ = C R = 12 , ˜ C KK = −C KK = 1 − φ A , C βK = − φ A , C βN = 12 ˙ φ A [2(1 − ˙ φ A ) H b − H ]1 − φ A , C NK = 4[(1 − φ A ) H b − (1 + ˙ φ A ) H ] , C NN = 3(1 + 12 A H b ) ˙ φ + 36( H − H b ) H b − H − H b ) − φ A , (50)where [32] H b ≡ H − A ¨ φ ˙ φ − A ˙ φ . (51)The background equations (22) are then given by6 ˙ φ A ( ˙ H b − HH b ) + 3(1 + 2 ˙ φ A ) H b −
12 ˙ φ − V = 0 , (52)2(1 − φ A )( ˙ H b − HH b ) + 5 H b −
10 ˙ φ A H b + 12 ˙ φ − V = 0 . (53)Note that A = 0 recovers the Einstein equations in canonical inflation. While the system involves ... φ via ˙ H b , byvirtue of the degeneracy conditions, it is equivalent to a system whose evolution is determined by initial data in asingle degree of freedom e.g. for the background, the position of the field in phase space ( φ, ˙ φ ).The first step in establishing this equivalence is to eliminate ˙ H b from (52) and (53) to obtain:6(1 − φ A + 6 ˙ φ A ) H b − ˙ φ (1 + ˙ φ A ) − − φ A ) V = 0 . (54)Hence, there are two branches for H b . In general, if A (cid:54) = const we would have a term linear in H b but here, we obtainsimple positive and negative roots H b = σ (cid:115) ˙ φ (1 + ˙ φ A ) + 2(1 − φ A ) V − φ A + 6 ˙ φ A ) , (55)1where σ = ±
1. Next, we choose one of the two branches of H b = H b ( φ, ˙ φ ) and take its time derivative ˙ H b = ˙ H b ( φ, ˙ φ, ¨ φ ).Substituting H b = H b ( φ, ˙ φ ) and ˙ H b = ˙ H b ( φ, ˙ φ, ¨ φ ) into (51) and either of (52) or (53), we obtain two equations for H = H ( φ, ˙ φ, ¨ φ ). Finally eliminating H from the two equations, we obtain an equation for ¨ φ = ¨ φ ( φ, ˙ φ ) governing theevolution of the system from a point in phase space. From this evolution we can then define H = H ( φ, ˙ φ ) and otherbackground quantities which define the slow-roll parameters. Note that equations depend only on m A if one rescalestime to mt . Hence, so long as m A is fixed, the relative evolution in mt is the same for various values of m withonly the amplitudes H ∝ m and ˙ φ ∝ m changing. Given the inflationary dynamics, we can check the condition (34)for whether unitary gauge perturbations are well-defined. In our model,Ω = 6 a [ − H b + ˙ φ A (2 H + 3 H b )] , (56)which should be a finite value.Even at the background level, this procedure produces novel behavior in phase space. First, not all phase spacepositions are allowed, even for a positive potential, and allowed positions can evolve into or from disallowed regions.For definiteness consider the quadratic potential V ( φ ) = m φ /
2. With this potential, for both branches of H b in(55), H ( φ, ˙ φ ) is singular at φ = ± (cid:115) φ A − φ A + 3 ˙ φ A m A (1 + 4 ˙ φ A −
15 ˙ φ A ) , or ˙ φm = ± √ m A . (57)For instance, plugging ˙ φm = √ m A − δ with an infinitesimal variable δ into H ( φ, ˙ φ ) and Taylor expanding around δ = 0 yields Hm = 3 − / ( m A ) − / (cid:114) − m A φ σ √ δ + O ( δ / ) , (58)which is indeed singular at δ = 0 for both σ = ± − m A φ (cid:54) = 0. For 1 − m A φ >
0, theHubble parameter is real for the δ > δ < H = O ( δ − ) → ±∞ on alternate sides of the φ values of the first case in (57).Also, there are boundaries across which H ( φ, ˙ φ ) changes from real to complex value while remaining finite: φ = ± ˙ φm (cid:115) φ A − φ A , or ˙ φm = ± √ m A , (59)where H b is zero or singular respectively. For instance, plugging ˙ φm = √ m A + δ into H ( φ, ˙ φ ) and ¨ φ ( φ, ˙ φ ) and Taylorexpanding around δ = 0 yields Hm = √ m A φ − m A φ + 2 / (7 − m A φ )3( m A ) / (cid:112) − m A φ σ √ δ + O ( δ ) , ¨ φm = 2 / ( m A ) / (cid:112) − m A φ σ √ δ + O ( δ ) . (60)Therefore, for A > − m A φ >
0, approaching the boundary ˙ φm = √ m A from the positive δ side, H ( φ, ˙ φ )changes from real to complex value for both branches. Furthermore, as can be seen from the slope ¨ φ/ ˙ φ , the σ = ± H b are not in general related by time reversal as they would be for H in GR beyond the A = const case, where there would be terms linear in H b in (54). This means that for a giveninitial position in the field phase space, evolution is not unique without specifying the branch choice for the metric.This feature is shared by the class of Galileon or G-inflation models as well [13, 49].We take an example parameter set based on the following rough estimation. Proceeding backwards from the end ofinflation on the σ = +1 branch when the field approaches the origin ( φ, ˙ φ/m ) = (0 , φ A , the background field behaves close to the canonical model with (cid:15) H ≈ /φ and ˙ φ/m ≈ (cid:112) / ∼
60 efolds, we require φ ≈
15 to be in the canonical phase. This phase is bounded at some maximum | φ | by encountering the first of2 - -
10 0 10 20 - - ϕ ϕ / m - -
10 0 10 20 - - ϕ ϕ / m FIG. 1. Phase space portrait of the D-inflation model (48) with V = m φ / m A = 0 .
002 and branches σ = +1 (left), σ = − H > H > (cid:15) H > H singular (solidred) or changing from real to complex (dashed red). The attractor trajectory inevitably crosses into a region where (cid:15) H < the boundaries (57) where H is singular. For small ˙ φ/m this occurs at φ ≈ ± ( m A ) − / , and hence we require | m A φ | < m A < / . Thus, as an example, we set m A = 0 . H from (57) (red solid lines) separate the phase space into disconnected regions with regions where H > (cid:15) H > (cid:15) H > H = + ∞ , whereas with (cid:15) H < H → −∞ at the boundary but then bounces without a curvature singularity when H = 0 and becomesan expanding phase H > H . These nonsingular bounces are generally accompanied by a ghost or gradientinstability in the scalar or tensor sector of unitary gauge. Trajectories flowing from the boundary given by the secondof the conditions in (57) (horizontal solid red) originate from H = + ∞ , whereas H is complex on the other side ofthe boundary.From Fig. 1, we see several other novel features of this model. First, physical solutions do not exist for all possibleinitial phase space points: there are regions where no real solution of H exists on either branch. This occurs outsidethe boundaries (59) (dashed red, no trajectories), e.g. φ = ˙ φ/m = 20.Furthermore, some trajectories in the upper and lower disconnected regions of Fig. 1 appear to end at boundariesacross which H becomes complex by satisfying either the first (dashed red curves) or the second condition (horizontaldashed red lines) in (59). Note that at these boundaries H is finite so that they do not represent curvature singularities.In these cases, as mentioned below (60), the trajectories actually sharply turn so as to be tangent to the boundary atintersection. At intersection, the two branches become degenerate and so solutions continue on the opposite branch,forming a parabola around this point. In other words, trajectories staring on one branch rebound off the boundaryinto the opposite branch so as to never enter the phase space region where only complex solutions exist.Additionally, near this rebound of the trajectories the contracting solution H <
H > H = 0, occurs within one branch before or after hitting the dashedboundaries, whereas the branch change occurs with the rebound of the trajectories at the boundaries with finite(positive or negative) H , which is continuous through the rebound. From (60), we see that there exists an exceptionalcase for this boundary, which is φ = 0, since in this case H = 0 at the boundary and hence H = 0 and branch changeoccurs at the same time. Again, this nonsingular bounce is generally accompanied by a ghost or gradient instabilityin the scalar or tensor sector of unitary gauge. On the other hand, the condition (34) for the well-definedness ofunitary gauge perturbations is itself violated around bounce solutions where the field transits a region where Ω = 0or ±∞ . In these cases, a covariant treatment or full numerical solution is required to assess perturbation pathologies3 - - - - ϕ ϕ / m FIG. 2. Phase space regions for the σ = +1 branch where b s (cid:15) H /c s > c s > (see also [43, 44]). For instance, on the second boundary of (59) where H b diverges at finite H , Ω = ±∞ . Note thatat this boundary, H b and the original higher-order equations of motion (52), (53) appear discontinuous between thebranches but when reduced to a second-order system, the two branches of ¨ φ ( φ, ˙ φ ) join. This property is unique todegenerate models. Finally, there is also a novel feature that some trajectories have the field roll up hill, but we shallsee that in general these regions are associated with ghost or gradient instability as well.On the other hand, the trajectories in the central region of Fig. 1 for σ = +1 are similar to the canonical ones. Alsoas in GR, trajectories start or end on singularities (solid red curves), albeit here at finite field values. This regionalso exhibits an attractor solution which is visually apparent from the converging flows in Fig. 2. To isolate thistrajectory we numerically integrate the reduced evolution equation ¨ φ = ¨ φ ( φ, ˙ φ ). For the initial condition, we adopt( φ, ˙ φ/m ) = (20 . , −
6) at t = 0, which is close to the intersection of the singularity and the attractor and rapidlyevolves onto the attractor. The numerical solution φ = φ ( t ) on the attractor (black curve, the left panel of Fig. 1) isshown for the 60 efolds before the end of inflation. Time t is converted to efolds N by plugging the numerical solution φ = φ ( t ) into the equation H = H ( φ, ˙ φ ) and numerically integrating it. We place the zero point of efolds at the end ofinflation, i.e. (cid:15) H = 1 at N = 0. Note that N = 0 at ( φ, ˙ φ/m ) ≈ (1 . , − .
71) and N = −
60 at ( φ, ˙ φ/m ) ≈ (20 , − . N >
0, the attractor trajectory spirals around the origin and inevitably crosses into a regionof noncanonical behavior where (cid:15) H <
0. We shall see next that this region is associated with gradient instabilities.
C. Perturbations
The central region of Fig. 1, with its attractor solution on the σ = +1 branch, provides a potentially viableinflationary regime and we therefore focus on it for the perturbation analysis. From the EFT coefficients (49), (50)and their time derivatives, we can construct b s , c s , b t , c t and their associated slow-roll parameters. First, the tensorsector is simple. From (39), b t = 1 and c t = (1 − φ A ) − and hence the stability condition b t > c t > | ˙ φ/m | < (2 m A ) − / ≈
16 as it is in the central region.The scalar sector, parametrized by b s and c s , is more complicated. Their explicit forms are too cumbersome toprovide here, but straightforward to obtain. In Fig. 2, we show the regions where (cid:15) H b s /c s > c s > φ, ˙ φ ) → ( − φ, − ˙ φ ) so we only display the lower right quadrant. The attractor itself (black line) remains in thestable region from the end of inflation to ∼
60 efolds prior, but approaching it especially from small velocities may4 c s b s ϵ H c s - - - ϕ / m FIG. 3. Behavior of c s (solid black) and b s (cid:15) H /c s (dashed blue) and as a function of ˙ φ/m near the origin at fixed φ = − .
3. At˙ φ = 0, c s → ±∞ when approached from alternate sides. Similar behaviors can be observed for φ = 0 .
3. There exists gradientinstability near the origin, while no ghost instability. require crossing from a region of gradient instability.At reheating, trajectories spiral around the origin and cross ˙ φ = 0. Here the field will inevitably enter into anunstable regime as we can analytically check as follows. For both branches of H b and a general potential, the Taylorexpansions of H b /H and ˙ H around ˙ φ = 0 are given by H = σ (cid:114) V (cid:104) − σa A ˙ φ + O ( ˙ φ ) (cid:105) ,H b = σ (cid:114) V (cid:34) φ V + O ( ˙ φ ) (cid:35) , ˙ H = A V (cid:104) a + 8 σa ˙ φ + O ( ˙ φ ) (cid:105) , ˙ H b = − A V σa ˙ φ − ˙ φ O ( ˙ φ ) , (61)where a ≡ (cid:113) V V (cid:48) − A V . Note that the Taylor expansion of H b does not include odd powers of ˙ φ since in our model H b in (55) is a function of ˙ φ . The leading order behavior of ˙ H is a constant as ˙ φ → A →
0. Inour model, this is a positive constant so unlike a canonical field (cid:15) H < φ → ˙ φ → c s = ( H − H b )(4 H + H b ) − ˙ H b H − H b ) , = − σ a A ˙ φ (cid:104) O ( ˙ φ ) (cid:105) , lim ˙ φ → b s (cid:15) H c s = 6( H − H b ) H b , = 6 a A ˙ φ (cid:104) O ( ˙ φ ) (cid:105) . (62)The normalization b s (cid:15) H /c s remains positive but approaches zero as ˙ φ →
0, while c s diverges in amplitude. Noticethat this divergence occurs even as A →
0, despite the fact that c s = 1 for A = 0, which indicates a discontinuouslimit. For the potential V ( φ ) = m φ / ˙ φ → c s = − (cid:114) m ˙ φ σφ | φ | − A m φ m A (cid:104) O ( ˙ φ ) (cid:105) . (63)5 - - - - - -
10 00.51.01.52.0 c s b s ϵ H - - - - - -
10 0 - - σ s - - - - - ξ s - - - - δ FIG. 4. Variation of (cid:15) H , b s , c s , and corresponding slow-roll parameters δ , ξ s , σ s along the attractor. Therefore, for both branches, near the origin where 1 − A m φ > c s is determined by φ/ ˙ φ . Hence, for σ = +1 branch, c s → −∞ at the first and third quadrants and c s → + ∞ in the second and fourthquadrants, where the attractor originates. In the former, c s < φ = 0as shown in Fig. 3.Furthermore, from (56), lim ˙ φ → Ω6 a = − H b = − σ (cid:114) V (cid:104) O ( ˙ φ ) (cid:105) , (64)and hence exactly at the origin ( φ, ˙ φ ) = (0 ,
0) of our model, the condition (34) is violated with Ω = 0 and the unitarygauge becomes ill-defined.To avoid gradient instability and unitary gauge being ill-defined, we can relax the assumption that A = const andchoose A ( φ, X ) → | φ | , | ˙ φ/m | (cid:46)
1, the dynamics of perturbations duringinflation will not be affected.Finally we can examine the evolution of b s and c s along the attractor during inflation. In Fig. 4 we show variationof (cid:15) H , b s , c s , and corresponding slow-roll parameters [7] δ ≡ d ln (cid:15) H d N − (cid:15) H , ξ s ≡ d ln b s d N , σ s ≡ d ln c s d N , (65)along the attractor. They are defined based on the quadratic action (36) for ˜ ζ , but as we mentioned above in ourmodel ˜ ζ − ζ evolves to zero so ln ∆ ζ = ln ∆ ζ after inflation. Notice that while all remain perturbative, σ s in particularcan become moderately large around N ∼ −
60 and moreover evolves on the several efold time scale.Such cases can be treated in the optimized slow-roll (OSR) formalism [7, 37], where the slow-roll (SR) result forthe curvature power spectrum after inflation when ˜ ζ = ζ ln ∆ ζ (cid:12)(cid:12)(cid:12) SR = ln (cid:18) H π b s c s (cid:15) H (cid:19) , (66)is corrected by the slow-roll parametersln ∆ ζ ≈ ln ∆ ζ (cid:12)(cid:12)(cid:12) SR − (cid:15) H − δ − σ s − ξ s , (67)6
10 1000 10 - - - - - - k / k l n Δ ζ FIG. 5. The curvature power spectrum ln ∆ ζ evaluated under the improved OSR approximation (67) (solid black) comparedwith the SR approximation (dashed blue). k represents the mode that freezes out in the OSR approximation at N = − and evaluated at freeze-out where k (cid:82) N d N c s /aH ≈ e . , contrary to k (cid:82) N d N c s /aH = 1 in the SR approximation.These approximations are compared in Fig. 5 for the same k , where k is the mode that freezes out at N = − m = 10 − to fix the normalization of H in Planck units and hence that of ∆ ζ to be roughly compatible with observations. Notice that there is a significant running of the tilt pivoting around k/k ∼ or N ∼ −
50 despite being far from the end of inflation and containing no features in the potential there.In this region, the OSR and SR results differ in shape and OSR itself breaks down as an approximation for some N < −
60 where the corrections become order unity. The OSR approximation thus extends the regime of validityfor the calculation into the range − (cid:46) N (cid:46) −
50, which is relevant for the CMB, and is useful in observationallyconstraining D-inflation. We leave such a study and the construction of an observationally viable model to a futurework.
IV. CONCLUSION
In this work, we developed the EFT of inflation for a general Lagrangian constructed from ADM variables, whichencompasses the most general interactions with up to second derivatives of a scalar field whose background sponta-neously breaks temporal diffeomorphism symmetry. The Ostrogradsky ghost usually implied by such higher-orderterms is eliminated by degeneracy conditions, leading to degenerate higher-order (or D-)inflation. We identify 8 typesof degeneracy conditions, one of which corresponds to known DHOST models. For the other cases, which includecurvature couplings not considered in DHOST, we provide necessary conditions for a covariant scalar-tensor theorybased on the dispersion relation of the quadratic action and leave a full assessment of their viability to future work.Higher-order theories imply equations of motions that are higher than second order in the scalar field and typicallylead to an ill-posed Cauchy problem. The degeneracy conditions, which involve the metric as well, restores a well-defined forwards or backwards evolution from initial field and field derivative data on a Cauchy surface but with novelfeatures.We illustrate these features with an explicit example of D-inflation. First, not all field configurations lead tophysical solutions for the metric as illustrated by values where all solutions for the Hubble parameter become complexeven for positive potentials and timelike field gradients. Second, evolution is only uniquely defined up to a branchchoice since the same field configurations lead to distinct expansion histories that are not related by time reversal asthey would be in GR. This feature is present in Horndeski theory as well. Third, trajectories can sharply turn toavoid phase space regions where real solutions fail to exist leading to highly complicated phase space portraits wherecontraction can turn to expansion without encountering a curvature singularity. These bouncing solutions generallytraverse regions of ghost or gradient instabilities in unitary gauge but also cross coordinate singularities in defining itsmetric perturbations (see also [43, 44]). Finally, perturbations can go unstable even in the limit that the additionaldegenerate terms in the Lagrangian are infinitesimal. In our example this occurs for curvature perturbations in thesimplest model of constant, but arbitrarily small, higher-order coefficients during reheating when the inflaton oscillatesaround the minimum of its potential.Our D-inflation model also has novel phenomenology. While the model possesses an attractor which leads to nearly7scale invariant fluctuations across a sufficient number of efolds of inflation, it also can produce substantial running ofthe tilt on CMB scales despite having no features in the potential there and being far from the end of inflation. Inthis case, EFT coefficients vary on the several efold timescale and require an approach that goes beyond the usualslow-roll formalism. We show that corrections captured by the optimized slow-roll approach extends the validity ofpredictions into the large running regime of interest and should be useful in observational tests of the D-inflationscenario.
ACKNOWLEDGMENTS
We thank Marco Crisostomi, Jose Maria Ezquiaga, Kazuya Koyama, and Sam Passaglia for useful comments.H.M. was supported by Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research(KAKENHI) No. JP17H06359 and No. JP18K13565. W.H. was supported by U.S. Dept. of Energy Contract No.DE-FG02-13ER41958 and the Simons Foundation.
Appendix A: Degeneracy Conditions
We can determine the necessary conditions for degeneracy by examining the high k or high frequency limit of thequadratic Lagrangian. We can then find the number of propagating modes and their dispersion relation by assumingsolutions of the form u ( x, t ) = u ( k ) e i ( ωt + kx ) where u = ( δN, ζ, ψ ) T [31].In the limit, we can neglect evolution on the Hubble time scale of the background and the EFT coefficients upto corrections of order ( k/aH ) , which as we detail below is sufficient to establish degeneracy conditions for mostsolutions and easily supplemented in the remaining ones. The quadratic Lagrangian (26) for scalars can be thenwritten as L = 12 u † Ku, (A1)with the kinetic matrix K ≡ c + ω c + c k a ω c + iω (cid:16) c + c k a (cid:17) + c k a − iω c k a + c k a ω c − iω (cid:16) c + c k a (cid:17) + c k a ω c + c k a + c k a − iω c k a + c k a iω c k a + c k a iω c k a + c k a c k a . (A2)For nontrivial solutions of the equation of motion Ku = 0 to exist, we require det K = 0, which can be written as f ω + (cid:40) f (cid:18) ka (cid:19) + f (cid:18) ka (cid:19) + f (cid:41) ω + f (cid:18) ka (cid:19) + f (cid:18) ka (cid:19) + f (cid:18) ka (cid:19) = 0 , (A3)where f ≡ − (cid:16) c − c (cid:17) ( c c − c ) ,f ≡ c ( c c − c ) − c c − c c c − c c ,f ≡ c c c − c ) − (cid:16) c − c (cid:17) ( c c − c c + c c − c c ) ,f ≡ (cid:16) c − c (cid:17) ( c − c c ) ,f ≡ c ( c c − c ) ,f ≡ c ( c c − c ) + c c c − c c + 23 c c c − c c ,f ≡ c (cid:18) c c − c (cid:19) . (A4)In general, this is a fourth order system for ω representing two propagating modes. To remove the second propagatingmode in unitary gauge, we demand f = 0, for which there are several possibilities:81. c = c /c or equivalently C ββ = 3 C βK / (3 C KK + ˜ C KK ). The kinetic terms organize into a single term for˜ ζ = ζ + ( c /c ) δN , which is the propagating degree of freedom.2. c = c = 0 or equivalently 3 C KK + ˜ C KK = C βK = 0. The kinetic term for ζ vanishes and δN is the propagatingdegree of freedom.3. c = c / C KK = 0. The constraint equation for ψ eliminates the kinetic term for ζ and δN isagain the propagating degree of freedom.Below we shall consider each case in Appendix A 1, A 2, A 3, respectively.Furthermore, retaining a higher order in spatial derivatives (or k ) compared with temporal derivatives (or ω ) in afully covariant theory corresponds to the reappearance of the second mode when changing the gauge. Therefore tofind covariant degeneracy conditions, we seek solutions of Eq. (A3) that correspond to a normal dispersion relation ω = c s k . The possible cases area. f f (cid:54) = 0, others = 0 ⇒ f ω + f ( k/a ) = 0.b. f f (cid:54) = 0, others = 0 ⇒ f ω + f ( k/a ) = 0.c. f f (cid:54) = 0, others = 0 ⇒ f ω + f ( k/a ) = 0.There are several caveats regarding this technique that need to be borne in mind. Since we neglect Hubble scaleevolution, we work in the limit ω (cid:29) H and since the c i coefficients generically carry a mass dimension M n i i for some n i and can vary on the Hubble time scale we assume ˙ c i /c i ∼ H (cid:28) M i as well. Whereas the former condition correspondsto c s k/aH (cid:29) c s k/aH ) − correctionsspoil the form of the dispersion relation at c s k/aH (cid:29)
1. This can occur in the “a” case through corrections to f and f which can then dominate over the terms from f and f which form the desired linear dispersion relation. Forthe “b” case these corrections can change the coefficients but not the leading order form and for the “c” case, thecorrections from the other terms are entirely negligible for c s k/aH (cid:29)
1. We therefore further check for supplementaldegeneracy conditions in the “a” or f f (cid:54) = 0 case. Note that since the coefficients in the dispersion relation can alsochange in the “a” and “b” cases from those given by this static technique, the full quadratic Lagrangian should beused to check for ghost and gradient instabilities in those cases.We now consider the various kinetic structures 1 , , a, b, c respectively. Wetreat “1a” in more detail as it serves both as an example of the technique and includes the known DHOST models. c = c /c case Plugging c = c /c to (A4) we have f = 0 and reduced forms for f ... which imply there is a single propagatingdegree of freedom ˜ ζ = ζ + c c δN (A5)in unitary gauge. The degeneracy classes a, b, c where this degree of freedom obeys a linear dispersion relation for c s k/aH (cid:29) f coefficients that remain nonzero. Therefore in each case, we have 5 degeneracyconditions between the various EFT coefficients represented by c i in the static limit. Case 1a can have supplementarydegeneracy conditions beyond the static limit as discussed above.a. First, let us consider the case f f (cid:54) = 0 and all other f ’s zero. Requiring first that f = 0 leads to two branches c = c c or c = 0 , (A6)For each case, f = f = f = 0 should be satisfied. While in general f = 0 has two branches, since f (cid:54) = 0implies c (cid:54) = c /
9, only one solution remains c c = 2 c ( c c + c c ) − c c . (A7)Also, f = 0 yields c c + 19 c c + 23 c c c = c c ( c c − c ) , (A8)9and f = 0 yields c ( c c − c ) + c c c − c c − c c + 23 c c c = 0 . (A9)These are the degeneracy conditions for the static, high k limit.More generally, this “1a” case is subject to corrections which require supplementary degeneracy conditions asdescribed above. Using only the first condition c = c /c , the quadratic Lagrangian (26) reduces to L = 12 c ˙˜ ζ + (cid:18) c − c ˙ x + c k a (cid:19) ˙˜ ζδN + 12 (cid:18) c + c k a (cid:19) k a ˜ ζ + (cid:18) c − c x − c x k a (cid:19) k a ˜ ζδN + 12 (cid:18) ˜ c + ˜ c k a + c x k a (cid:19) δN + 12 c k a ψ + k a ψ (cid:20) c ζ + c − c ˙ x δN + c k a (˜ ζ − xδN ) (cid:21) , (A10)where x ≡ c /c and ˜ c ≡ c + ˙ c x + c ˙ x − c ˙ x, ˜ c ≡ c + ( ˙ c − Hc − c ) x + c x − c ˙ x. (A11)Potentially problematic terms are those where the fields have time derivatives and the coefficients carry additionalfactors of k /a . In the static limit, these terms are arranged to cancel, but beyond the static limit thetime variation of the coefficients breaks this degeneracy relation and changes the dispersion relation even for c s k/aH (cid:29)
1. The only term of this form is c ( k/a ) ˙˜ ζδN . Therefore c = 0 is sufficient as a supplementaldegeneracy condition to ensure a linear dispersion relation for the the single propagating degree of freedom ˜ ζ .This condition may be generalized to nonvanishing c but would then involve tuning between c , a and the other c i coefficients. Due to the appearance of the scale factor a in the generalized degeneracy condition this tuningis unlikely to be preserved in a fully covariant scalar-tensor theory. We therefore take c = 0 and the completeset of degeneracy conditions for case 1a has two branches i ) c = c c , c = c c , c c = 2 c c c − c c , c c = 9 c c ( c c − c ) ,c ( c c − c ) + c c c − c c − c c + 23 c c c = 0 , c = 0 , (A12) ii ) c = c c , c = 0 , c c c − c c = 0 , c c = 9 c c ( c c − c ) ,c ( c c − c ) + c c c − c c − c c + 23 c c c = 0 , c = 0 . (A13)With this complete set of degeneracy conditions, one can explicitly verify that the Euler-Lagrange equationsthat result from Eq. (A10) describe a single propagating degree of freedom ˜ ζ with a linear dispersion relationat c s k/aH (cid:29) a - i we can have c = c c , c = c = c = c = 0 , c = 2 c c c − c c c , (A14)or c = c c , c = c = c = 0 , c = c c c , c = c c c , (A15)and for 1 a - iic = c c , c = c = 0 , c = 2 c c c , c = c c c , c = ( c c − c c ) + c c c c c . (A16)0Models where c = 0 on the (A15) branch are also members of the (A14) branch. On the other hand theconditions c = c c /c , ˜ c = 0 ( c = c c /c , c = 0) must be satisfied for any model on the (A15) branch,including those that are part of the (A14) branch. As pointed out by Ref. [31], this presents a problem if onewants to recover Newtonian gravity for nonrelativistic matter. Since these conditions and c = 0 zero out the k ˜ ζδN and k δN terms in (A10), the Euler-Lagrange equation for δN which usually provides a source to thePoisson equation through the matter density is absent on this branch. Instead the k ˜ ζ term in its equation ofmotion comes from its own Euler-Lagrange equation and has a source in matter pressure. For this reason, in themain text we focus on the (A14) branch. This branch also includes the 2N-I/Ia class of DHOST models [31].The case 1 a - i with (A14) or (A15) corresponds to DHOST class I or II, respectively, and the latter was knownto suffer from gradient instability. On the other hand, the case 1 a - ii with (A16) is not included in DHOSTtheories, as it requires c (cid:54) = 0, namely 8 C RR + 3 ˜ C RR (cid:54) = 0, which can originate from the existence of the quadraticcurvature terms in the covariant Lagrangian.b. Next, we consider f f (cid:54) = 0 and all other f ’s zero. By following the same procedure as in case 1a, we obtain thefollowing four sets of degeneracy conditions i ) c = c c , c = c c , c = c = 0 , c c + 6 c c c + c c = 0 ,ii ) c = c c , c = c c , c = 0 , c = c c , c c + 6 c c c + c c = 0 ,iii ) c = c c , c = 0 , c = c = 0 , c c + 6 c c c + c c = 9 c c ( c c − c ) ,iv ) c = c c , c = 0 , c = 0 , c = c c , c c + 6 c c c + c c = 9 c c ( c c − c ) . (A17)In this case, corrections beyond the static approximation can change the coefficients of the dispersion relationat c s k/aH (cid:29) c = c /c , c = c = c = 0, c = (2 c c c − c c ) /c , the above branch satisfies the degeneracy without requiring c , c , c to be vanishing. By definition c and c are nonvanishing in the presence of quadratic curvatureterms, whereas c is nonvanishing if Lagrangian includes terms such as (cid:0) (4) R + (cid:3) φ (cid:1) . Also, i ) and ii ) does notrequire any condition on c , and iii ) and iv ) requires a different condition c = 0 which can be satisfied with2 c c c − c c (cid:54) = 0 as c does not appear in the degeneracy conditions and hence is a free parameter.c. Finally, we consider f f (cid:54) = 0 and all other f ’s zero. This leads to six possible cases i ) c = c c , c = c = 0 , c = c , c c = c ,ii ) c = c c , c = c = 0 , c = c c c , c = c c c ,iii ) c = c c , c = 0 , c = c c , c = c , c = 3 c c c ,iv ) c = c c , c = 0 , c = c c , c = 2( c c + c c ) c , c c = c c ( c c − c ) + c (cid:18) c c − c c (cid:19) ,v ) c = c c , c = 0 , c = c , c c = c (cid:18) c c − c (cid:19) + c (cid:18) c c − c c (cid:19) = 0 ,vi ) c = c c , c = c c , c = c , c c = ( c c − c c ) c . (A18)Again, note that this branch is not included in DHOST theories as the DHOST conditions c = c /c , c = c = c = 0, c = (2 c c c − c c ) /c are not satisfied in general. Clearly, the conditions i ), ii ) do not include c , c , c ; the condition iii ) does not include c , c , c ; the condition v ) does not include c , c ; and thecondition vi ) does not include c , c . Also, the condition iv ) as well as i ), ii ), vi ) require different conditionson c which do not coincide with the DHOST condition in general.1 c = c = 0 case In case 2, c = c = 0 and the kinetic term for ζ vanishes leaving f = 0 and δN as the propagating degree offreedom. Since models where δN and not ζ is propagating are unlikely to recover Newtonian gravity, we include thiscase for completeness and pedagogical interest.a. Let us begin with the case f f (cid:54) = 0 and all other f ’s zero. In this case we must again check for corrections to thedispersion relation beyond the static limit. Using only the condition c = c = 0, the quadratic Lagrangian (26)reduces to L = 12 c ˙ δN + (cid:18) c + c k a (cid:19) ˙ ζδN + 12 (cid:18) c k a + c k a (cid:19) ζ + c k a ζδN + 12 (cid:18) c + c k a (cid:19) δN + 12 c k a ψ + ψ (cid:18) c k a δN + c k a ζ (cid:19) . (A19)Here, again, the problematic term is c ( k/a ) ˙ ζδN , and hence we impose c = 0 as a supplemental degeneracycondition.Requiring f = f = f = f = 0 leads to four possible cases i ) c = c = 0 , c = c = 0 , c = 0 , c ( c − c c ) + 9 c c + c c − c c c = 0 ,ii ) c = c = 0 , c = c = c − c c = 0 , c = 0 , c c + c c − c c c = 0 ,iii ) c = c = 0 , c = c − c c = 0 , c = 0 , c c c − c c − c c + 6 c c c = 0 ,iv ) c = c = 0 , c = c − c c = 0 , c = 0 , c c + c c − c c c = 0 . (A20)b. Next we consider the case f f (cid:54) = 0 and all other f ’s zero. Requiring f = f = 0 under f = c ( c c − c c ) (cid:54) = 0allows c = c = 0 only. By further requiring f = f = 0, we obtain three possible cases i ) c = c = 0 , c = c = 0 , c = c ( c c − c ) − c c = 0 ,ii ) c = c = 0 , c = c = 0 , c = c − c c = 0 ,iii ) c = c = 0 , c = c = 0 , c = c = 0 . (A21)c. Finally, we consider f f (cid:54) = 0 and all other f ’s zero. This leads to five possible cases i ) c = c = 0 , c = c = 0 , c = 0 , c c − c = 0 ,ii ) c = c = 0 , c = c = 0 , c = 0 , c ( c c − c ) = 0 ,iii ) c = c = 0 , c = c = 0 , c = 0 ,iv ) c = c = 0 , c = c = 0 , c ( c c − c ) − c c = 0 ,v ) c = c = 0 , c = c = 0 , c c + c c − c c c = 0 . (A22) c = c / case Finally we consider the case 3 where c = c /
9. Here the Euler-Lagrange equation for ψ provides a contributionthat cancels the kinetic term for ζ , and δN is again the propagating degree of freedom. As in case 2, this case isunlikely to provide viable theories of gravity. While generally we expect 5 static degeneracy conditions, in this casethere are fewer since some of the f terms are identically zero once other degeneracy conditions are applied.a. The f f (cid:54) = 0 branch does not exist since c = c / f = 0.b. Next, for f f (cid:54) = 0, requiring that additionally f = f = f = 0 leads to two possible cases i ) c = c , c = c c , c = c c + 6 c c c + 9 c c c c − c , c = 0 ,ii ) c = c , c = c c , c = c c + 6 c c c + 9 c c c c − c , c c + 3 c c = 0 . (A23)2c. Finally f f (cid:54) = 0 leads to two possible cases i ) c = c , c = c c , c = c c , c ( c c − c ) + 6 c c c − c c = 0 ,ii ) c = c , c = 0 , c = − c c c − c c − c c c c − c . (A24) Appendix B: Relationship to literature
Our approach is most similar to Ref. [31] and in this Appendix we make the explicit connection to that work anddiscuss the differences. First some of the terms in Ref. [31] take a superficially different form that is related to oursthrough integration by parts. Up to a total derivative N √ h ˜ C KR δK ij δR ji ∼ a (cid:34) ( ˙˜ C KR + H ˜ C KR ) (cid:32) δ √ ha δR + δ R (cid:33) + ˜ C KR δRδK + H ˜ C KR δN δR (cid:35) , (B1)and hence we can rewrite our quadratic Lagrangian (23) in the form of Eq. (1.2) of [31] expose the difference betweenthe two δ L ∼ − a C β (cid:34) δN (cid:32) δ √ ha (cid:33) (cid:5) + δN i δN ,i (cid:35) + a (cid:20) C βR δ R ˙ δN + 12 C RR δ R + 12 ˜ C RR δ R ij δ R ji + (cid:18) C KR + 12 ˜ C KR (cid:19) δKδ R (cid:21) . (B2)The quadratic Lagrangian (23) thus contains terms that differ from Eq. (1.2) of [31]. Since the first term in (B2) has δN or δN ,i , it is nonvanishing only for scalar perturbations. For scalar perturbation, it can be expressed up to a totalderivative as − a C β (cid:34) δN (cid:32) δ √ ha (cid:33) (cid:5) + δN i δN ,i (cid:35) ∼ − a C β (cid:18) ζ − ∂ ψa (cid:19) δN = − a C β ( δK + 3 HδN ) δN, (B3)which can be absorbed into the α B and α K terms in Eq. (1.2) of [31]. On the other hand, the third line of (B2) isnot considered in [31] as these terms have derivatives higher than second order in total. If we assume these terms arevanishing by imposing C βR = C RR = ˜ C RR = C KR + 12 ˜ C KR = 0 , (B4)we have c = c = c = 0 in (27). These conditions hold in the 1a degeneracy subclasses defined by Eq. (A14) and(A15). Ref. [31] considered only these cases. They furthermore assume φ ∝ t and so their V = − ˙ φN β (B5)vanishes in the background ¯ V = 0 or ¯ β = 0 in our notation. Generalizing this does not change the functional formof their Lagrangian, just the mapping between the scalar field and ADM representations and so we retain ¯ β (cid:54) = 0in the correspondences below. Note that if a field redefinition ϕ ∝ t ( φ ) is performed instead after solving for thebackground φ ( t ), which alternately reestablishes the generality of their expressions, then the DHOST coefficients mustcorrespondingly be redefined (cf. [32] v2).In summary, in the subclass of (B4), the quadratic Lagrangian (23) for scalar perturbation takes the same functional3form as Eq. (1.2) of [31] with the correspondence M = ˜ C KK ,α K = 1 H ˜ C KK (cid:18) C N + C NN − β C βN + ¯ β C ββ + ( a C β ) (cid:5) a − ( a C βN ) (cid:5) a + ( a ¯ β C ββ ) (cid:5) a − H C β (cid:19) ,α B = C NK − ¯ β C βK − C β H ˜ C KK ,α T = 2 C R + ˙˜ C KR + H ˜ C KR ˜ C KK − ,α H = 2( C NR + C R ) + H ˜ C KR ˜ C KK − ,α L = − (cid:18) C KK ˜ C KK + 1 (cid:19) ,β = C βK C KK ,β = C ββ ˜ C KK ,β = 2 C α ˜ C KK . (B6)Equivalently, the inverse correspondence between notations for the subclass (B4) is given by a − c = − M (1 + α L ) , a − c = 6 M β , a − c = M β , a − c = M β , a − c = − M α L , (B7)and Θ ≡ − a − c = − HM (1 + α B + α L ) , Ψ ≡ a − c = 12 M (1 + α T ) , Ξ ≡ a − c = 12 M (1 + α H ) , Φ ≡ a − c = H M [ α K − α L ) − α B ] + 6 a − (cid:0) a M Hβ (cid:1) (cid:5) . (B8)with c , c , c vanishing in this class.With these relations we can also translate the degeneracy conditions Eqs. (2.15), (2.16) of [31]:C I : α L = 0 , β = − β , β = − β [2(1 + α H ) + β (1 + α T )] , (B9)C II : β = − (1 + α L ) 1 + α H α T , β = − α L ) (1 + α H ) (1 + α T ) , β = 2 (1 + α H ) α T , (B10)into our notation to confirm that their C I and C II correspond to (A14) and (A15), respectively. The Lagrangian for˜ ζ in (36) is equivalent to Eq. (4.8) of [31] for C I . [1] P. Creminelli, M. A. Luty, A. Nicolis, and L. Senatore, JHEP , 080 (2006), arXiv:hep-th/0606090 [hep-th].[2] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan, and L. Senatore, JHEP , 014 (2008), arXiv:0709.0293 [hep-th].[3] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, JCAP , 025 (2013), arXiv:1304.4840 [hep-th].[4] R. Kase and S. Tsujikawa, Int. J. Mod. Phys. D23 , 1443008 (2014), arXiv:1409.1984 [hep-th].[5] J. Gleyzes, D. Langlois, and F. Vernizzi, Int. J. Mod. Phys.
D23 , 1443010 (2014), arXiv:1411.3712 [hep-th].[6] J. Gleyzes, D. Langlois, M. Mancarella, and F. Vernizzi, JCAP , 054 (2015), arXiv:1504.05481 [astro-ph.CO].[7] H. Motohashi and W. Hu, Phys. Rev.
D96 , 023502 (2017), arXiv:1704.01128 [hep-th].[8] G. W. Horndeski, Int. J. Theor. Phys. , 363 (1974). [9] A. Nicolis, R. Rattazzi, and E. Trincherini, Phys. Rev. D79 , 064036 (2009), arXiv:0811.2197 [hep-th].[10] C. Deffayet, G. Esposito-Farese, and A. Vikman, Phys. Rev.
D79 , 084003 (2009), arXiv:0901.1314 [hep-th].[11] C. Deffayet, S. Deser, and G. Esposito-Farese, Phys. Rev.
D80 , 064015 (2009), arXiv:0906.1967 [gr-qc].[12] C. Deffayet, X. Gao, D. A. Steer, and G. Zahariade, Phys. Rev.
D84 , 064039 (2011), arXiv:1103.3260 [hep-th].[13] T. Kobayashi, M. Yamaguchi, and J. Yokoyama, Prog. Theor. Phys. , 511 (2011), arXiv:1105.5723 [hep-th].[14] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, Phys. Rev. Lett. , 211101 (2015), arXiv:1404.6495 [hep-th].[15] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, JCAP , 018 (2015), arXiv:1408.1952 [astro-ph.CO].[16] P. Horava, Phys. Rev.
D79 , 084008 (2009), arXiv:0901.3775 [hep-th].[17] D. Blas, O. Pujolas, and S. Sibiryakov, Phys. Rev. Lett. , 181302 (2010), arXiv:0909.3525 [hep-th].[18] D. Blas, O. Pujolas, and S. Sibiryakov, Phys. Lett.
B688 , 350 (2010), arXiv:0912.0550 [hep-th].[19] M. Ostrogradsky, Mem. Acad. St. Petersbourg , 385 (1850).[20] R. P. Woodard, Scholarpedia , 32243 (2015), arXiv:1506.02210 [hep-th].[21] M. Raidal and H. Veerme, Nucl. Phys. B , 607 (2017), arXiv:1611.03498 [hep-th].[22] A. Smilga, Int. J. Mod. Phys. A , 1730025 (2017), arXiv:1710.11538 [hep-th].[23] H. Motohashi and T. Suyama, (2020), arXiv:2001.02483 [hep-th].[24] H. Motohashi and T. Suyama, Phys. Rev. D91 , 085009 (2015), arXiv:1411.3721 [physics.class-ph].[25] D. Langlois and K. Noui, JCAP , 034 (2016), arXiv:1510.06930 [gr-qc].[26] H. Motohashi, K. Noui, T. Suyama, M. Yamaguchi, and D. Langlois, JCAP , 033 (2016), arXiv:1603.09355 [hep-th].[27] H. Motohashi, T. Suyama, and M. Yamaguchi, J. Phys. Soc. Jap. , 063401 (2018), arXiv:1711.08125 [hep-th].[28] H. Motohashi, T. Suyama, and M. Yamaguchi, JHEP , 133 (2018), arXiv:1804.07990 [hep-th].[29] K. Aoki and H. Motohashi, (2020), arXiv:2001.06756 [hep-th].[30] J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui, and G. Tasinato, JHEP , 100 (2016), arXiv:1608.08135[hep-th].[31] D. Langlois, M. Mancarella, K. Noui, and F. Vernizzi, JCAP , 033 (2017), arXiv:1703.03797 [hep-th].[32] M. Crisostomi, K. Koyama, D. Langlois, K. Noui, and D. A. Steer, JCAP , 030 (2019), arXiv:1810.12070 [hep-th].[33] X. Gao and Z.-B. Yao, JCAP , 024 (2019), arXiv:1806.02811 [gr-qc].[34] X. Gao, C. Kang, and Z.-B. Yao, Phys. Rev. D99 , 104015 (2019), arXiv:1902.07702 [gr-qc].[35] X. Gao, Phys. Rev.
D90 , 081501 (2014), arXiv:1406.0822 [gr-qc].[36] H. Motohashi and S. Mukohyama, JCAP , 030 (2020), arXiv:1912.00378 [gr-qc].[37] H. Motohashi and W. Hu, Phys. Rev.
D92 , 043501 (2015), arXiv:1503.04810 [astro-ph.CO].[38] E. Gourgoulhon, (2007), arXiv:gr-qc/0703035 [GR-QC].[39] S. Passaglia and W. Hu, Phys. Rev.
D98 , 023526 (2018), arXiv:1804.07741 [astro-ph.CO].[40] P. Motloch, W. Hu, A. Joyce, and H. Motohashi, Phys. Rev.
D92 , 044024 (2015), arXiv:1505.03518 [hep-th].[41] P. Motloch, W. Hu, and H. Motohashi, Phys. Rev.
D93 , 104026 (2016), arXiv:1603.03423 [hep-th].[42] W. Hu and A. Joyce, Phys. Rev.
D95 , 043529 (2017), arXiv:1612.02454 [astro-ph.CO].[43] A. Ijjas, JCAP , 007 (2018), arXiv:1710.05990 [gr-qc].[44] D. A. Dobre, A. V. Frolov, J. T. G. Ghersi, S. Ramazanov, and A. Vikman, JCAP , 020 (2018), arXiv:1712.10272[gr-qc].[45] M. Lagos, M.-X. Lin, and W. Hu, Phys. Rev. D100 , 123507 (2019), arXiv:1908.08785 [gr-qc].[46] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. , 161101 (2017), arXiv:1710.05832 [gr-qc].[47] B. P. Abbott et al. (LIGO Scientific, Virgo, Fermi-GBM, INTEGRAL), Astrophys. J. , L13 (2017), arXiv:1710.05834[astro-ph.HE].[48] H. Ram´ırez, S. Passaglia, H. Motohashi, W. Hu, and O. Mena, JCAP , 039 (2018), arXiv:1802.04290 [astro-ph.CO].[49] C. Deffayet, O. Pujolas, I. Sawicki, and A. Vikman, JCAP1010