Effective field theory of degenerate higher-order inflation

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 derivatives of the scalar field whose background breaks temporal diffeomorphism invariance. Degeneracy conditions, corresponding to 8 distinct types -- only one of which corresponds to known degenerate higher-order scalar-tensor models -- provide necessary conditions for eliminating the Ostrogradsky ghost in a covariant theory at the level of the quadratic action in unitary gauge. Novel implications of the degenerate higher-order system for the Cauchy problem are illustrated with the phase space portrait of an explicit inflationary example: not all field configurations lead to physical solutions for the metric even for positive potentials; solutions are unique for a given configuration only up to a branch choice; solutions on one branch can apparently end at nonsingular points of the metric and their continuation on alternate branches lead to nonsingular bouncing solutions; unitary gauge perturbations can go unstable even when degenerate terms in the Lagrangian are infinitesimal. The attractor solution leads to an inflationary scenario where slow-roll parameters vary and running of the 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 the effective field theory (EFT) of inflation, where the timelike scalar field is treated as a clock that breaks the time diffeomorphism invariance leaving spatial diffeomorphism invariance unbroken [1,2]. In general, the EFT of inflation with higher-derivative operators contains extra ghost degrees of freedom, which may or may not propagate in the regime of validity of the EFT. To consider a regime where higher-derivative interactions also produce interesting observable phenomenology, one needs to rely on the framework where the ghost degrees of freedom are appropriately eliminated. Therefore, an EFT Lagrangian motivated by general ghost-free theories serve as a useful framework. In this context, the original EFT framework has been extended in subsequent works [3][4][5][6][7] to include derivative operators appearing in the Horndeski [8][9][10][11][12][13], Gleyzes-Langlois-Piazza-Vernizzi (GLPV) [14,15] and Horava-Lifshitz [16][17][18] theories.
More general theories involving additional derivative of fields typically propagate ghost degrees of freedom. The ghosts associated with higher-order derivatives are known as the Ostrogradsky ghosts [19,20], which makes the Hamiltonian unbounded due to its linear dependence on canonical momenta. Unlike the classically unbounded Hamiltonian of the hydrogen atom, the Ostrogradsky Hamiltonian remains unbounded quantum mechanically as well [21][22][23]. To eliminate the ghost degrees of freedom, one needs to evade the condition of the Ostrogradsky theorem that the Lagrangian is nondegenerate with respect to the highest-order derivatives. However, degeneracy with respect to the highest-order derivative is necessary but not sufficient to evade the unbounded Hamiltonian [24], which is the reason why one needs to impose a certain set of degeneracy conditions to eliminate all aspects of the Ostrogradsky ghosts [25][26][27][28]. This argument can be also understood in a broader context in the language of constraints as a generalization of the Ostrogradsky theorem [29].
The degeneracy conditions were applied to a construction of degenerate higher-order scalar-tensor (DHOST) theories with quadratic [25] and cubic interactions [30] of second derivatives of a scalar field, which include the derivative of the lapse function. The EFT description of the quadratic and cubic DHOST theories was developed in [31], where the quadratic Lagrangian around the cosmological background was investigated. Cosmological evolution and the linear stability analysis were also investigated [32], focusing on the de Sitter attractor in a shift symmetric quadratic DHOST model. In [31,32], two assumptions on background dynamics were adopted: that the lapse remains unity and that the scalar field is proportional to time coordinate. The first can be imposed as a gauge condition, and the second should be satisfied dynamically. For general timelike scalar field evolution on a given phase space trajectory, one needs 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 arXiv:2002.07967v2 [hep-th] 27 Apr 2020 coefficients (cf. [32] v2). A dynamical lapse is also taken into account in [33,34] in the context of spatially covariant gravity [35], where the degeneracy conditions were also studied. However, the general EFT framework of degenerate theories including but not limited to quadratic and cubic DHOST and its application to cosmology have not been fully investigated yet.
In this paper, generalizing our previous work [7], we develop the EFT framework of general degenerate theories, and explore its peculiar phenomenology. This framework includes quadratic and cubic DHOST as a special subclass as well as theories where the lapse is nondynamical, e.g. those with second-order equations of motion for the scalar field, as in the Horndeski case, or the spatial metric in unitary gauge, as in the GLPV case. In §II, we consider the EFT action composed of Arnowitt-Deser-Misner (ADM) geometric quantities including the acceleration and lapse derivative and their couplings to intrinsic and extrinsic curvatures. This action includes operators appearing in covariant theories involving the most general combination of second-order derivatives of scalar field. It also includes Lorentz-violating theories such as Horava-Lifshitz gravity [16][17][18], as well as the scordatura degenerate theory [36] weakly violating the degeneracy condition. We derive the background and quadratic actions for various degeneracy classes, which include known DHOST cases, summarized in Appendix A. In §III, we investigate dynamics in degenerate higherorder inflation, which we dub "D-inflation", and clarify several novel features of degenerate models for both the background 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 time scale. 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 general combination of second-order derivatives of scalar field. In §II A, we construct the EFT Lagrangian from geometric quantities including the acceleration and lapse derivative and their arbitrary couplings to intrinsic and extrinsic curvatures. In §II B, we write down the background and quadratic Lagrangians around cosmological background. Since vector and tensor perturbations are the same as the previous work [7], in §II C we focus on the scalar perturbations, and reduce the quadratic Lagrangian for a specific degeneracy class. We provide the complete analysis of the construction of degeneracy conditions in Appendix A.

A. ADM EFT Lagrangian
We work in the 3 + 1 ADM decomposition of the metric 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 , where semicolons 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 of the scalar field is always timelike. In unitary gauge, any Lagrangian with up to second derivatives in the field can be expressed in terms of ADM quantities through Here, we define β by where X ≡ g µν φ ,µ φ ,ν = −φ 2 /N 2 is the kinetic term for the scalar. In particular, if we take φ = t, X = −1/N 2 and then β measures the fractional change in the lapse along the normal and more generally it determines the fractional change in the elapsed proper time in field coordinates and so (3) involvesφ. In the gauge where φ ∝ t,φ = 0 and this has been used widely for the purpose of counting the number of degrees of freedom through the Hamiltonian analysis. However, to keep a normal perturbation analysis where the background lapseN = 1, we use φ = φ(t), and retain theφ term. After solving for a given trajectory φ(t), we can always make a field redefinition ϕ ∝ t(φ), which maintainsN = 1 at the expense of redefining the scalar field Lagrangian, 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 secondorder derivatives of the scalar field which a priori contains N, K µν , a µ , β. So long as we consider theories involving up 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 appear together through We therefore consider the Lagrangian to be a spatially diffeomorphism invariant function of these quantities generalizing [7] to allow it to depend on α i j and β. Here R i j is the Ricci 3-tensor on the spatial slice. Higher-derivative Lagrangians typically contain Ostrogradsky ghosts, but we shall see in the next section that for special combinations of N, K i j , R i j , α i j , β the Lagrangian only propagates one scalar and the usual two tensor degrees of freedom. Without the α i j 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 general covariant Lagrangian that is at most cubic in second derivatives of the field and coupled to the metric as 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 are quadratic in second derivatives are and those that are cubic are where we have used (2) to establish the correspondence with the ADM variables. Similarly, we can relate the coupling to the metric using the Gauss-Codazzi relation and integration by parts to rewrite up to boundary terms (see e.g. [3,38]) where While in general, the appearance of α i j and β in the Lagrangian signals an extra degree of freedom since the lapse and shift no longer obey constraint equations, this general Lagrangian (7) contains classes that propagate only 3 degrees of freedom and avoids Ostrogradsky ghosts. First, there is the GLPV class which defines a special relationship between the A i , B i , F i coefficients Note that the F 4 and F 5 terms are also arbitrary functions of φ, X. In the Horndeski subclass of GLPV, where the scalar field equations themselves are explicitly second order, F 4 = F 5 = 0 and the remaining functions are more typically labeled (G 2 , G 3 , G 4 , G 5 ) = (F 0 , F 1 , F 2 , F 3 ). It is easy to verify through Eqs. (8,9,10) that this relationship eliminates the dependence of α i j and β in the GLPV and Horndeski Lagrangians leaving the EFT Lagrangian of the form L(N, K i j , R i j ; t). More generally, α i j and β can appear in a Lagrangian which still only propagates 3 degrees of freedom if the functions A i , B i , F i satisfy a certain set of degeneracy conditions (see (40), (41) and [25,30]). This is the DHOST class of models. The Lagrangian (6) also includes the scordatura degenerate theory [36] with a weak violation of the degeneracy condition.
In §II C we generalize these degeneracy conditions to the full EFT Lagrangian (6). Since there the dependence on α i j , 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 metric and scalar field in the unitary gauge. Furthermore in comparison to EFTs that are explicitly built to encompass DHOST, it allows terms like βR that would only appear with different couplings between the field and the metric than 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 around a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) background for whichK where H ≡ d ln a/dt is the Hubble parameter. Following [7] we define the Taylor coefficients as where "b" denotes that the quantities are evaluated on the background, Y, Z ∈ {N, K, R, α, β} and the index structure is determined by the symmetry of the background. For notational simplicity we treat scalars and traces with the same notation; thus implicitly We can further eliminate terms linear in δK = K − 3H through the identity which follows from K = n µ ;µ ignoring boundary terms. The Lagrangian L = N √ hL up to quadratic order in metric perturbations becomes 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 i j , δα i j , δβ up to quadratic order We note that since up to quadratic order. Note that for the perturbed FLRW metric, α i j is a quadratic-order quantity. The background equations are given by varying the action with respect to N and √ h With the background equation and integration by parts, the quadratic Lagrangian becomes 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 , which is a natural consequence of (20). Therefore, α i j and β dependencies of the Lagrangian change the dynamics of scalar but not vector or tensor perturbations which are given explicitly in [7].
For scalar perturbations Following [7], we use where the last equality for δ 2 R holds up to a total derivative, to obtain the quadratic Lagrangian in Fourier space as where and We highlight these four combinations as they involve time derivatives or the Hubble parameter and degeneracy conditions 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 conditions imposed on the various c i coefficients which we briefly summarize here. The result is 8 types of degeneracy conditions, cases 1a . . . 3c (with 3a impossible to satisfy), each of which may be realized by the c i or equivalently the C i coefficients in various ways. The degeneracy conditions we derive apply for any theory involving second-order derivatives in any form in the Lagrangian for unitary gauge (6). Since the Lagrangian (6) allows any dependence on (N, K i j , R i j , α i j , β), or equivalently on second derivatives (2), this degeneracy conditions applies beyond the quadratic and cubic DHOST theories. 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 temporal structure of (26). Of the three possibilities, we focus on the case 1 type where the condition c 3 = c 2 /c 2 1 is satisfied and the combinationζ alone carries the temporal derivatives. The other two cases have the lapse δN as the propagating degree of freedom and would cause difficulties in recovering an observationally viable theory of gravity. More generally our linear degeneracy conditions should be viewed as necessary, but not necessarily sufficient, conditions for a viable nonlinear scalar-tensor theory of gravity. Under this c 3 = c 2 /c 2 1 condition, the quadratic Lagrangian (26) for scalar perturbation in unitary gauge would appear to propagate only 1 degree of freedom. This degeneracy condition applies to the scalar quadratic Lagrangian in any theory involving second-order derivative of any form in Lagrangian in the unitary gauge (6) and includes the DHOST models as well as the Horndeski or GLPV models where c 2 = 0 andζ = ζ.
However, the temporal degeneracy condition alone is not sufficient to guarantee that there is only a single degree of freedom. In terms of the Euler-Lagrange equations, it only removes the fourth-order derivatives and third-order derivatives still need to be removed to avoid unbounded Hamiltonian [24]. Furthermore, if unitary gauge defines a foliation that corresponds to characteristic surfaces of the second degree of freedom then its dynamics are hidden from this temporal structure. Since such a degree of freedom propagates instantaneously on this surface, it is not a Cauchy surface 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 problem as discussed in detail in [40,41]. One should therefore not take the apparent lack of an extra degree of freedom in unitary gauge as a definitive absence (cf. [35]). Of course, the counting of degrees of freedom cannot depend on the gauge or ADM slicing and so we expect additional degeneracy conditions that involve the spatial derivatives of the kinetic matrix in unitary gauge.
For a 1 + 1 dimensional system of linear partial differential equations, including the plane parallel Fourier modes considered below, one can exploit the algorithm [41] based on the Kronecker form of a matrix pencil which includes all possible linear combinations of temporal and spatial derivatives to count degrees of freedom and find characteristic curves in the presence of any hidden constraints (see Appendix of [41]). However for the quadratic and cubic DHOST theories, the full covariant and nonlinear degeneracy conditions are already known. As shown in the Appendix, we can obtain the remaining conditions for the quadratic action by demanding that the dispersion relation of remaining degree of freedom take their normal linear form in unitary gauge. This logic also applies to the wider class of degenerate theories that originate from a covariant action and so we retain terms that are absent in the quadratic and cubic DHOST Lagrangian in Appendix A.
We now focus in particular on the degeneracy conditions given in (A14) in case 1a, as other branches may not have phenomenologically viable theories of gravity associated with them. We emphasize though that this same procedure can be carried out for any of the branches. In this case, the conditions on the c i coefficients are where x = c 2 /c 1 , and these conditions imply This branch includes the 2 N-I/Ia class of quadratic and cubic DHOST, GLPV and Horndeski theories. Under these conditions we can simplify (26) as The equation of motion for ψ and δN yields the constraints where we have assumed with Ω ≡ c 1ẋ − c 4 , which generalizes the condition 2HC KK = C N K employed in Eq. (33) of [7] to cases where the Lagrangian depends α i j , 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 itself is ill-defined. To see this, we follow [7] and move to a comoving gauge defined by the condition that for the perturbed Einstein tensor δG 0 i = 0 for a general metric theory of gravity [42]. The gauge transformation from unitary gauge to comoving gauge is characterized by the time shift T = −∆/Ḣ, where ∆ ≡ HδN −ζ (see Eq. (B14) of [7]). Using (33), we have so that Ω = 0 makes ∆ diverge, implying that the gauge transformation between the two gauges requires an infinite time 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 ifζ freezes out butζ − 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 of related cases). Note also that if c 8 = c 6 x the k 2ζ term vanishes in the Euler-Lagrange equation (33) for δN , and prevents the recovery of Newtonian gravity for nonrelativistic matter, as found in [31] for one of the DHOST subclasses (see Appendix A for more details).
Substituting the constraints (33) into the Lagrangian (32) and integrating by parts give the usual Mukhanov-Sasaki form for the quadratic Lagrangian where From these terms, we can define the scalar sound speed c 2 s and the normalization parameter b s as For a canonical scalar field c 2 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 coefficients, specifically in the quadratic DHOST class. In general, EFT coefficients can vary in time and one needs to evaluate carefully the slow-roll hierarchy of all dynamical parameters, for which the generalized slow-roll approximation developed in [7] provides a systematic framework, based on the evolution of H, H , b s and c s in (38) as well as the analogous quantities for tensors D-inflation provides an additional motivation for these time-varying considerations in that one might seek to construct models where their novel features are only present during inflation and are absent thereafter where they would otherwise impact cosmological and astrophysical observables. We construct our model in §III A, and elucidate the novel 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 2 t −1 = 0 andζ −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 epoch by 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 by derivative couplings [48].
We can concretely implement these requirements using the EFT of a quadratic DHOST model starting with the case 1a degeneracy conditions (31). SinceC KR = C KR =C RR = C RR = 0 in this case, the third and the fourth conditions in (31) identically hold. From the second condition in (31) we obtain Plugging it into the first and fifth conditions in (31) and solving the two equations for A 4 and A 5 , we obtain which matches Eqs. (5.1) and (5.2) in [25]. In general a degenerate theory in this class is identified by A 2 , A 3 , F 2 and one can choose F 0 , F 1 as free functions without affecting the degeneracy structure. Note that if C βK = 0 then so that A 2 and A 3 are no longer independent; this case corresponds to Eq. (5.3) in [25] and reproduces the GLPV restriction for quadratic terms in (13). Next, from (8), we obtain the tensor sound speed as where we used (40) andζ For the Horndeski theory, requiring the right-hand side of (43) to vanish implies that F 2 = F 2 (φ). Note also that the right-hand side of (44) identically vanishes for Horndeski and GLPV theories. We would like to choose the functions A 2 , A 3 , F 2 to make these two quantities (43), (44) be nonzero during inflation and evolve to zero by the end of inflation. As a simple example, we set where we work in natural units M Pl ≡ (8πG) −1/2 = 1, and for which the degeneracy conditions (40), (41) yield and (43), (44) read Here, we are interested in a function θ such that it starts from finite value and evolves to zero either from the evolution of X or an appropriate form for A 2 (φ, X).
Under these assumptions, the action is given by For simplicity, we will illustrate this model with A 2 = const, or at least nearly so during most of the ∼ 60 efolds before the end of inflation. We shall see that in models where the field oscillates at reheating, A 2 needs to vanish before this point to avoid gradient or ghost instabilities. However, any late-time change does not affect large-scale observables 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 equations (22) are and those for the perturbations are where [32] The background equations (22) are then given by Note that A 2 = 0 recovers the Einstein equations in canonical inflation. While the system involves ... φ viaḢ b , by virtue of the degeneracy conditions, it is equivalent to a system whose evolution is determined by initial data in a single 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 eliminateḢ b from (52) and (53) to obtain: Hence, there are two branches for H b . In general, if A 2 = const we would have a term linear in H b but here, we obtain simple positive and negative roots where σ = ±1. Next, we choose one of the two branches of H b = H b (φ,φ) and take its time derivativeḢ b =Ḣ b (φ,φ,φ). (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 forφ =φ(φ,φ) governing the evolution of the system from a point in phase space. From this evolution we can then define H = H(φ,φ) and other background quantities which define the slow-roll parameters. Note that equations depend only on m 2 A 2 if one rescales time to mt. Hence, so long as m 2 A 2 is fixed, the relative evolution in mt is the same for various values of m with only the amplitudes H ∝ m andφ ∝ m changing. Given the inflationary dynamics, we can check the condition (34) for whether unitary gauge perturbations are well-defined. In our model, which should be a finite value. Even at the background level, this procedure produces novel behavior in phase space. First, not all phase space positions 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 2 φ 2 /2. With this potential, for both branches of H b in (55), H(φ,φ) is singular at For instance, pluggingφ m = 1 √ 3m 2 A2 − δ with an infinitesimal variable δ into H(φ,φ) and Taylor expanding around which is indeed singular at δ = 0 for both σ = ±1 branches so long as 1 − 3 2 m 2 A 2 φ 2 = 0. For 1 − 3 2 m 2 A 2 φ 2 > 0, the Hubble parameter is real for the δ > 0 side of the boundary, and imaginary for the δ < 0 side. On the other hand H = O(δ −1 ) → ±∞ 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: where H b is zero or singular respectively. For instance, pluggingφ m = 1 √ 2m 2 A2 + δ into H(φ,φ) andφ(φ,φ) and Taylor expanding around δ = 0 yields Therefore, for A 2 > 0 and 1 − 2m 2 A 2 φ 2 > 0, approaching the boundaryφ m = 1 √ 2m 2 A2 from the positive δ side, H(φ,φ) changes from real to complex value for both branches. Furthermore, as can be seen from the slopeφ/φ, the σ = ±1 trajectories form two halves of a parabola whose minimum intersects the boundary. Taylor expansion around another boundary in (59) also has a similar structure.
Another interesting point is that the branches of H b are not in general related by time reversal as they would be for H in GR beyond the A 2 = const case, where there would be terms linear in H b in (54). This means that for a given initial 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 of inflation on the σ = +1 branch when the field approaches the origin (φ,φ/m) = (0, 0), we see that since D-inflation effects for the background scale asφ 2 A 2 , the background field behaves close to the canonical model with H ≈ 2/φ 2 andφ/m ≈ 2/3 on the slow-roll attractor. Thus for the canonical phase to last at least ∼ 60 efolds, we require φ ≈ 15 to be in the canonical phase. This phase is bounded at some maximum |φ| by encountering the first of the boundaries (57) where H is singular. For smallφ/m this occurs at φ ≈ ±(m 2 A 2 ) −1/2 , and hence we require |m 2 A 2 φ 2 | < 1 or m 2 A 2 < 1/15 2 . Thus, as an example, we set m 2 A 2 = 0.002. The phase space portrait for this set of parameters is depicted in Fig. 1 for both branches. Singularities in H from (57) (red solid lines) separate the phase space into disconnected regions with regions where H > 0 (yellow), H > 0 (blue) or both (green) shaded. Trajectories (blue arrows) flowing from the boundary given by the first of the conditions in (57) (curved solid red) with H > 0 begin at H = +∞, whereas with H < 0 begin in a collapsing phase with H → −∞ at the boundary but then bounces without a curvature singularity when H = 0 and becomes an expanding phase H > 0 too rapidly to be resolved in Fig. 1. The same is true for trajectories flowing into the boundaries but with reversed signs for H. These nonsingular bounces are generally accompanied by a ghost or gradient instability in the scalar or tensor sector of unitary gauge. Trajectories flowing from the boundary given by the second of the conditions in (57) (horizontal solid red) originate from H = +∞, whereas H is complex on the other side of the boundary.
From Fig. 1, we see several other novel features of this model. First, physical solutions do not exist for all possible initial phase space points: there are regions where no real solution of H exists on either branch. This occurs outside the 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 boundaries across which H becomes complex by satisfying either the first (dashed red curves) or the second condition (horizontal dashed 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 at intersection. 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 boundary into 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 < 0 bounces to expansion H > 0 as well. For the most cases, the bounce itself, H = 0, occurs within one branch before or after hitting the dashed boundaries, 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 exceptional case for this boundary, which is φ = 0, since in this case H = 0 at the boundary and hence H = 0 and branch change occurs at the same time. Again, this nonsingular bounce is generally accompanied by a ghost or gradient instability in the scalar or tensor sector of unitary gauge. On the other hand, the condition (34) for the well-definedness of unitary gauge perturbations is itself violated around bounce solutions where the field transits a region where Ω = 0 or ±∞. In these cases, a covariant treatment or full numerical solution is required to assess perturbation pathologies (see also [43,44]). For instance, on the second boundary of (59) where H b diverges at finite H, Ω = ±∞. Note that at this boundary, H b and the original higher-order equations of motion (52), (53) appear discontinuous between the branches but when reduced to a second-order system, the two branches ofφ(φ,φ) join. This property is unique to degenerate models. Finally, there is also a novel feature that some trajectories have the field roll up hill, but we shall see 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. Also as in GR, trajectories start or end on singularities (solid red curves), albeit here at finite field values. This region also exhibits an attractor solution which is visually apparent from the converging flows in Fig. 2. To isolate this trajectory we numerically integrate the reduced evolution equationφ =φ(φ,φ). For the initial condition, we adopt (φ,φ/m) = (20.3, −6) at t = 0, which is close to the intersection of the singularity and the attractor and rapidly evolves onto the attractor. The numerical solution φ = φ(t) on the attractor (black curve, the left panel of Fig. 1) is shown 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 of inflation, i.e. H = 1 at N = 0. Note that N = 0 at (φ,φ/m) ≈ (1.0, −0.71) and N = −60 at (φ,φ/m) ≈ (20, −3.5). During inflation on the attractor unitary gauge perturbations are well-defined since Ω given in (56) is finite and nonzero.
After inflation when N > 0, the attractor trajectory spirals around the origin and inevitably crosses into a region of noncanonical behavior where 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 viable inflationary 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 tensor sector is simple. From (39), b t = 1 and c 2 t = (1 − 2φ 2 A 2 ) −1 and hence the stability condition b t > 0 and c 2 t > 0 for the present case is satisfied if |φ/m| < (2m 2 A 2 ) −1/2 ≈ 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 to provide here, but straightforward to obtain. In Fig. 2, we show the regions where H b s /c 2 s > 0 (yellow), c 2 s > 0 (blue), or both (green, instability free) near the attractor of the central region. Note that in our example these are invariant for (φ,φ) → (−φ, −φ) so we only display the lower right quadrant. The attractor itself (black line) remains in the stable region from the end of inflation to ∼ 60 efolds prior, but approaching it especially from small velocities may require crossing from a region of gradient instability.
At reheating, trajectories spiral around the origin and crossφ = 0. Here the field will inevitably enter into an unstable regime as we can analytically check as follows. For both branches of H b and a general potential, the Taylor expansions of H b /H andḢ aroundφ = 0 are given by where a 1 ≡ 3 V V 1−2A2V . Note that the Taylor expansion of H b does not include odd powers ofφ since in our model H b in (55) is a function ofφ 2 . The leading order behavior ofḢ is a constant asφ → 0 which vanishes if A 2 → 0. In our model, this is a positive constant so unlike a canonical field H < 0 asφ → 0 as shown in Fig. 1.
Using (61), we can expand the sound speed and normalization as liṁ φ→0 The normalization b s H /c 2 s remains positive but approaches zero asφ → 0, while c 2 s diverges in amplitude. Notice that this divergence occurs even as A 2 → 0, despite the fact that c 2 s = 1 for A 2 = 0, which indicates a discontinuous limit. For the potential V (φ) = m 2 φ 2 /2, Therefore, for both branches, near the origin where 1 − A 2 m 2 φ 2 > 0 is satisfied, the sign of c 2 s is determined by φ/φ. Hence, for σ = +1 branch, c 2 s → −∞ at the first and third quadrants and c 2 s → +∞ in the second and fourth quadrants, where the attractor originates. In the former, c 2 s < 0 occurs only in a small neighborhood aroundφ = 0 as shown in Fig. 3.
Furthermore, from (56), and hence exactly at the origin (φ,φ) = (0, 0) of our model, the condition (34) is violated with Ω = 0 and the unitary gauge becomes ill-defined. To avoid gradient instability and unitary gauge being ill-defined, we can relax the assumption that A 2 = const and choose A 2 (φ, X) → 0 at the origin. Provided this occurs only for |φ|, |φ/m| 1, the dynamics of perturbations during inflation will not be affected.
Finally we can examine the evolution of b s and c 2 s along the attractor during inflation. In Fig. 4 we show variation of H , b s , c 2 s , and corresponding slow-roll parameters [7] along the attractor. They are defined based on the quadratic action (36) forζ, but as we mentioned above in our modelζ −ζ evolves to zero so ln ∆ 2 ζ = ln ∆ 2 ζ after inflation. Notice that while all remain perturbative, σ s1 in particular can 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 for the curvature power spectrum after inflation whenζ = ζ is corrected by the slow-roll parameters to be roughly compatible with observations. Notice that there is a significant running of the tilt pivoting around k/k 0 ∼ 10 4 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 validity for the calculation into the range −60 N −50, which is relevant for the CMB, and is useful in observationally constraining D-inflation. We leave such a study and the construction of an observationally viable model to a future work.

IV. CONCLUSION
In this work, we developed the EFT of inflation for a general Lagrangian constructed from ADM variables, which encompasses the most general interactions with up to second derivatives of a scalar field whose background spontaneously breaks temporal diffeomorphism symmetry. The Ostrogradsky ghost usually implied by such higher-order terms is eliminated by degeneracy conditions, leading to degenerate higher-order (or D-)inflation. We identify 8 types of degeneracy conditions, one of which corresponds to known DHOST models. For the other cases, which include curvature couplings not considered in DHOST, we provide necessary conditions for a covariant scalar-tensor theory based 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 typically lead to an ill-posed Cauchy problem. The degeneracy conditions, which involve the metric as well, restores a welldefined forwards or backwards evolution from initial field and field derivative data on a Cauchy surface but with novel features.
We illustrate these features with an explicit example of D-inflation. First, not all field configurations lead to physical solutions for the metric as illustrated by values where all solutions for the Hubble parameter become complex even for positive potentials and timelike field gradients. Second, evolution is only uniquely defined up to a branch choice since the same field configurations lead to distinct expansion histories that are not related by time reversal as they would be in GR. This feature is present in Horndeski theory as well. Third, trajectories can sharply turn to avoid phase space regions where real solutions fail to exist leading to highly complicated phase space portraits where contraction can turn to expansion without encountering a curvature singularity. These bouncing solutions generally traverse regions of ghost or gradient instabilities in unitary gauge but also cross coordinate singularities in defining its metric perturbations (see also [43,44]). Finally, perturbations can go unstable even in the limit that the additional degenerate terms in the Lagrangian are infinitesimal. In our example this occurs for curvature perturbations in the simplest model of constant, but arbitrarily small, higher-order coefficients during reheating when the inflaton oscillates around the minimum of its potential.
Our D-inflation model also has novel phenomenology. While the model possesses an attractor which leads to nearly scale invariant fluctuations across a sufficient number of efolds of inflation, it also can produce substantial running of the tilt on CMB scales despite having no features in the potential there and being far from the end of inflation. In this case, EFT coefficients vary on the several efold timescale and require an approach that goes beyond the usual slow-roll formalism. We show that corrections captured by the optimized slow-roll approach extends the validity of predictions into the large running regime of interest and should be useful in observational tests of the D-inflation scenario. and f 6 = 0 yields 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 as described above. Using only the first condition c 3 = c 2 2 /c 1 , the quadratic Lagrangian (26) reduces to Equivalently, the inverse correspondence between notations for the subclass (B4) is given by 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 .