Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order

Abstract

Abstract

We present all scalar-tensor Lagrangians that are cubic in second derivatives of a scalar field, and that are degenerate, hence avoiding Ostrogradsky instabilities. Thanks to the existence of constraints, they propagate no more than three degrees of freedom, despite having higher order equations of motion. We also determine the viable combinations of previously identified quadratic degenerate Lagrangians and the newly established cubic ones. Finally, we study whether the new theories are connected to known scalar-tensor theories such as Horndeski and beyond Horndeski, through conformal or disformal transformations.

Published for SISSA by Springer

Received: September 12, 2016 Revised: November 14, 2016 Accepted: December 9, 2016

Published: December 20, 2016

Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order

J. Ben Achour,a M. Crisostomi,b K. Koyama,b D. Langlois,c K. Nouid,c andG. Tasinatoe

aCenter for Field Theory and Particle Physics, Fudan University,

20433 Shanghai, China

bInstitute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, U.K.

cLaboratoire APC Astroparticule et Cosmologie, Universit Paris Diderot Paris 7, 75013 Paris, France

dLaboratoire de Mathmatiques et Physique Thorique, Universit Franois Rabelais, Parc de Grandmont, 37200 Tours, France

eDepartment of Physics, Swansea University,

Swansea, SA2 8PP, U.K.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We present all scalar-tensor Lagrangians that are cubic in second derivatives of a scalar eld, and that are degenerate, hence avoiding Ostrogradsky instabilities. Thanks to the existence of constraints, they propagate no more than three degrees of freedom, despite having higher order equations of motion. We also determine the viable combinations of previously identied quadratic degenerate Lagrangians and the newly established cubic ones. Finally, we study whether the new theories are connected to known scalar-tensor theories such as Horndeski and beyond Horndeski, through conformal or disformal transformations.

Keywords: Classical Theories of Gravity, Cosmology of Theories beyond the SM

ArXiv ePrint: 1608.08135

JHEP12(2016)100

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP12(2016)100

Web End =10.1007/JHEP12(2016)100

Contents

1 Introduction 1

2 Degenerate Scalar-Tensor theories 32.1 Action 42.2 Covariant ADM decomposition 52.3 Horndeski Lagrangians and kinetic structure of the action 72.4 Degeneracy conditions and primary constraints 8

3 Classication of cubic theories 103.1 Minimally coupled theories 103.2 Non-minimally coupled theories 123.3 Minkowski limit 12

4 Merging quadratic with cubic theories 13

5 Conformal and disformal transformation 155.1 Conformal transformation on Beyond Horndeski 155.2 Conformal and disformal transformation on Horndeski 16

6 Conclusions 17

A Curvature dependent Lagrangians 18

B Tensorial structure implied by the degeneracy conditions 20

C Quadratic theories 21C.1 Minimally coupled theories 21C.2 Non-minimally coupled theories 22

D Identifying cubic theories 23D.1 Minimally coupled theories 24D.2 Non-minimally coupled theories 28

1 Introduction

General Relativity (GR) is the unique consistent classical theory for a massless, self-interacting spin two eld in four dimensional spacetime [1]. It describes accurately gravitational phenomena spanning a large interval of scales, from short distances probed by table top experiments, to large distances probed by astronomy and astrophysics [2]. By including

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a positive cosmological constant term to the Einstein-Hilbert action, GR can also describe the current acceleration of the universe, but only if one is willing to accept the enormous ne tuning that observations require on the value of the cosmological constant [3]. Attempts to avoid such ne tuning motivate the study of gravitational theories more general than GR, the simplest option being scalar-tensor theories of gravity (see e.g. [4] for a review). Theories that involve derivative scalar interactions, in the family of Galileons [5], are characterised by interesting screening e ects, as for example the Vainsthein mechanism [6], which are able to reduce the strength of the scalar fth force to a value compatible with present constraints on deviations from GR.

Intriguingly, although the subject has been studied for many decades by now, we still do not know the structure of the most general consistent scalar-tensor theory, i.e. a theory describing a scalar interacting with a spin-2 tensor eld in four dimensions. Horndenski [7] analysed the most general actions for scalar-tensor theories which lead to second order equations of motion (EOMs), and avoid Ostrogradsky instabilities [8, 9]. In four dimensional spacetime, this condition allows one to consider actions which contain at most three powers of second derivatives of the scalar eld. However, as realised only recently, there also exist viable theories beyond Horndeski [1012], which do not su er from the Ostrogradsky instability even though the corresponding Euler-Lagrange equations are higher order. Such theories have interesting consequences for cosmology and astrophysics. In particular, they lead to a breaking of the Vainshtein mechanism inside matter, which can modify the structure of nonrelativistic stars [1318], as well as that of relativistic ones [19].

The aim of the present paper is to determine the maximal generalization of Horndenski theories in four dimensions, by which we mean all scalar-tensor theories that contain at most three powers of second derivatives of the scalar eld, and that propagate at most three degrees of freedom.

As demonstrated in [20], a systematic way to identify scalar-tensor theories that contain at most three degrees of freedom, i.e. without Ostrogradsky ghost, is to consider Lagrangians that are degenerate, i.e. whose Hessian matrix obtained by taking the second derivatives of the Lagrangian with respect to velocities is degenerate. For scalar-tensor theories, such a degeneracy can depend on the specic coupling between the metric and the scalar eld. From the Hamiltonian point of view, the degeneracy of the Lagrangian translates into the existence of constraints on phase space, in addition to the usual Hamiltonian and momentum constraints due to di eomorphism invariance, and explains why one degree of freedom is eliminated, even if the equations of motion are higher order. A detailed Hamiltonian analysis conrms the direct link between this degeneracy and the elimination of the Ostrogradsky ghost [21]. For Lagrangians depending on the accelerations of several variables, the degeneracy of the Lagrangian is not su cient to eliminate the multiple Ostrogradsky ghosts and extra conditions must be imposed, as shown in [22] for classical mechanics systems (see also [23] for a slightly di erent approach, reaching the same conclusion). The singularity of the Hessian matrix (this time obtained by taking the second derivatives of the Lagrangian with respect to the lapse and shift) nds application also in other contexts like massive gravity: indeed, it is this condition that provides the tertiary constraint necessary to remove the Boulware-Deser ghost mode [24].

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The degeneracy criterium, which provides a powerful and simple method to identify viable theories, was used in [20] to identify all scalar tensor theories whose Lagrangian depends quadratically on second order derivatives of a scalar eld. Degenerate higher derivative Lagrangians, later dubbed EST (Extended Scalar Tensor) in [26], or DHOST (Degenerate Higher Order Scalar Tensor) in [27], include Horndeski theories as well as their extensions beyond Horndeski. As stressed in [20] and [25], only specic combinations of Horndeski theories and of their extensions beyond Horndeski are (Ostrogradsky) ghost-free. Quadratic degenerate theories are further studied in [2628], in particular how they change under disformal transformations of the metric.

In the present work, we extend the systematic classication of degenerate theories to include Lagrangians that possess a cubic dependence on second order derivatives, so to nd the most general extension of Horndenski scalar-tensor theory of gravity. We also allow for non-minimal couplings with gravity and show that the only viable Lagrangian, among all possible ones involving the Riemann tensor contracted with the second derivative of the scalar eld, is of the form G . The class of theories we consider thus encompasses

Horndeski Lagrangians and our analysis conrms that all Horndeski theories are degenerate, as expected. We also nd new classes of cubic Lagrangians that are degenerate. In total, we identify seven classes of minimally coupled cubic theories, and two classes of non-minimally coupled cubic theories. We study in which cases it is possible to combine any of these cubic theories with the previously identied quadratic theories to obtain more general Lagrangians. We investigate which cubic theories admit a well-dened Minkowski limit,i.e. when the metric is frozen to its Minkowski value. We also study whether the new cubic theories are related to known Lagrangians through conformal or disformal transformations. Technical appendixes contain details of the calculations leading to the results we present in the main text.

2 Degenerate Scalar-Tensor theories

Scalar-Tensor theories involving second order derivatives of the scalar eld in the action are generally plagued by an Ostrogradsky instability, unless the Lagrangian is degenerate, i.e. there is a primary constraint that leads to the removal of the additional undesired mode.1 In order to study these theories, it is useful to recast the action into ordinary rst order form via the introduction of a suitable auxiliary variable. This can be done by replacing all rst order derivatives by the components of a vector eld A, as rst explained

in [20], and by imposing the relation

A = , (2.1)

1In this paper we do not perform a full Hamiltonian analysis (see refs. [21] and [29] for Hamiltonian formulations of beyond Horndeski theories); however, we expect that, in general, the primary constraint is second-class and leads to a secondary constraint that is also second-class, so that both constraints remove one degree of freedom, as shown explicitly in [21] for the quadratic case. Note that the primary constraint can also be rst-class in some very particular cases.

3

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using a Lagrangian multiplier. Therefore, after introducing the general action we investigate, we will focus on its kinetic structure by identifying the time derivatives of the elds contained in A .

2.1 Action

In this paper we consider the most general action involving quadratic and cubic powers of the second derivative of the scalar eld:

S[g, ] = Z

d4x g

,

(2.2)

where the functions f2 and f3 depend only on and X (we use a mostly

plus convention for the spacetime metric). The tensors C(2) and C(3) are the most general tensors constructed with the metric g and the rst derivative of the scalar eld .

As we will see in detail in the next subsection, when written in terms of the auxiliary variable A, each second derivative of yields terms linear in velocities. By contrast, the curvature depends quadratically on the velocities of the metric and one can introduce terms non-minimally coupled to gravity, such as f2 R and f3 G , leading to second or third powers in velocities respectively. A priori, one could also envisage many more terms of this kind involving the Riemann tensor contracted in various ways. However, as shown in appendix A, the only viable Lagrangians among all the possible ones with appropriate powers in velocities, turn out to be these two (up to integrations by parts).

Note that one could also include in our general action (2.2) terms of the form P (X, ) or terms depending linearly on . We have not included such terms explicitly because they do not modify the degeneracy conditions, but one should keep in mind that they can always be added to the Lagrangians that will be identied in our analysis.

Due to the way the tensors C(2) and C(3) are contracted in the action, one can always impose, without loss of generality, the symmetry relations:

C (2) = C (2) = C (2) and C (3) = C (3) = C (3) = C (3) . (2.3)

As a consequence, they can be expressed as

C (2) = hha1 gg + a2 g g + a3 g + a4g + a5 ii , (2.4) C (3) = hhb1 g gg + b2 g g g + b3 gg g + b4 g g

+b5 g g + b6 gg + b7 g g + b8 g

+b9 g + b10 ii , (2.5)

where the functions as and bs depend only on and X. The notation hh. . . ii means that

these expressions are symmetrised so to satisfy eq (2.3). Explicitly, we have

5

f2 R + C (2) + f3 G + C (3)

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C (2) + C (3) =

4

Xi=1aiL(2)i +

10

Xi=1biL(3)i , (2.6)

where

and

where the tensors C (2) and C (3) are now expressed in terms of A and . Clearly, the two Lagrangians (2.2) and (2.9) are equivalent.

Although we do not perform explicitly a Hamiltonian analysis here,2 let us briey comment about the role of the Lagrangian multipliers and the relations they enforce. Since the action (2.9) does not involve the velocities of , the corresponding conjugate momenta p appear in the total Hamiltonian HT as primary constraints that weakly vanish.

The evolution of pi gives the secondary constraints iAi 0. By contrast, the evolution of

p0 allows one to solve for the multiplier used in HT to impose the other primary constraint 0 0, where is the momentum of . The evolution of 0, on the other hand,

xes the multiplier associated with p0. All these constraints are second class and therefore can be consistently imposed in the Hamiltonian analysis; in particular the constraints i Ai 0 enables us to eliminate the velocity of Ai in favour of the spatial derivative of

A0, as explained in detail in the next section. It is thus clear that the constraints that follow from the in (2.9) do not get mixed up with the (potential) extra primary constraint necessary to eliminate the Ostrogradsky mode, which characterises degenerate theories.

2.2 Covariant ADM decomposition

In order to study the kinetic structure of the action (2.9), we must perform a 3 + 1 decomposition of its building blocks. We now assume the existence of an arbitrary slicing of spacetime with 3-dimensional spacelike hypersurfaces. We introduce the unit vector n normal to the spacelike hypersurfaces, which is time-like and satises the normalization condition nn = 1. This induces a three-dimensional metric, corresponding to the

projection tensor on the spatial hypersurfaces, dened by

h g + nn . (2.10)

2All the details about the complete Hamiltonian analysis of quadratic theories can be found in [21].

5

L(2)1 = , L(2)2 = ()2 , L(2)3 = () ,

L(2)4 = , L(2)5 = ( )2 ; (2.7)

L(3)1 = ()3 , L(3)2 = () , L(3)3 = ,

L(3)4 = ()2 , L(3)5 = , L(3)6 = ,

L(3)7 = , L(3)8 = ,

L(3)9 = ( )2 , L(3)10 = ( )3 . (2.8)

Introducing the auxiliary variable A as in (2.1), the general action (2.2) becomes

S[g, ; A, ] =

Z

d4x g

f2 R + C (2)A A (2.9)

+f3 G A + C (3)A A A + ( A)

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,

Following the construction of [20], we dene the spatial and normal projection of A,

respectively h A , A An . (2.11)

Let us now introduce the time direction vector t = /t associated with a time coordinate t that labels the slicing of spacelike hypersurfaces. One can always decompose t as

t = Nn + N, (2.12)

thus dening the lapse function N and the shift vector N orthogonal to n. We also dene the time derivative of any spatial tensor as the spatial projection of its Lie derivative with respect to t. In particular, we have

A tA ,

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, h h h Lth . (2.13)

Due to the symmetric property of A = A, it is possible to express

h Lt

in terms

of DA and h , therefore the only velocities (time derivative of the elds) involved in

A areA = N V + NDA , h = 2 NK + D(N )

, (2.14)

where V nA , K is the extrinsic curvature tensor and D denotes the 3-dimen

sional covariant derivative associated with the spatial metric h .

Instead of using the velocities h and A , it is convenient to work with the covariant objects K and V and interpret them as covariant velocities associated with the elds h and A . Working with these covariant quantities allows us to avoid dealing with the lapse and the shift vector.

Using these denitions, as well as the property A = A, the 3+1 covariant decomposition of A is given by

A = D A K + 2 n(K ) n(D )A

+ nn

V a

, (2.15)

where a n n is the acceleration vector. One can rewrite (2.15) as

A = V + K + D 2 n(D )A

a, (2.16)

with

nn ,

A h(h ) + 2 n(h( )

) . (2.17)

These two tensors fully characterise the velocity structure of the building block A that appears in the action (2.9) and will play an essential role in deriving the degeneracy conditions.

6

2.3 Horndeski Lagrangians and kinetic structure of the action

As an example of theories of the type (2.2), and as a useful step for the general case, let us rst consider the particular case of the so-called quartic and quintic Horndeski Lagrangians3

LH4 = f2R 2f2X(

2 ) , (2.18)

LH5 = f3G + 13f3X(

3 3

+ 2 ) , (2.19)

which correspond, respectively, to a quadratic and a cubic Lagrangian in our terminology. Indeed, they are of the form (2.2), with

a1 = a2 = 2f2X , a3 = a4 = a5 = 0 , (2.20)

and

3b1 = b2 =

2

A2

and we have introduced the projection tensor (orthogonal to the directions n and)

P h

where A2 AA,

. (2.25)

Notice that the tensors (2.23) and (2.24) are orthogonal to the vector n, therefore the kinetic terms do not contain the velocity V . This is the peculiarity of Horndeski Lagrangians which reects in second order equations of motions.

Let us now turn to the general action (2.2). In order to extract its kinetic part, it is convenient to re-express the curvature terms in the action as Horndeski Lagrangians so that one can use the results above. The action (2.2) is thus rewritten as

S[g, ] =

Z

d4x g

, (2.26)

3We use the subscripts 4 and 5, referring to quartic and quintic, only for the Horndeski Lagrangians themselves. According to our terminology, for all other associated variables we use instead the subscripts(2) and (3) referring to quadratic or cubic types of theory.

4Note that h[ h] denotes the anti-symmetrisation on ( , ) and h[ h||h| | ] denotes the anti-symmetrisation on the second index of each tensor h. The same notation holds for the terms dened with the projector P.

7

32b3 = f3X , bi = 0 (i = 4, . . . , 10) . (2.21)

It is instructive to extract the kinetic part of these two Lagrangians as the result will be useful for the general case. The kinetic structure of the original Lagrangians (2.18) and (2.19) is the same as the following ones

LH4kin = C (2)H and LH5kin = C (3)H , (2.22)

with4

C (2)H =

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hf2 h [h] + 2f2X

A2h [h] 2 P [ P]

A2h[ h||h| | ] 2 P[ P|| P| | ] , (2.24)

i

, (2.23)

C (3)H =

2f3X

A2

2

1 2

LH4 + (2) + LH5 + (3)

where the tensors (2) and (3) are of the form (2.4)(2.5) with the new functions

1 = a1 2f2X ,2 = a2 + 2f2X , (2.27) ~b1 = b1

1 3f3X ,

while all the other functions remain unchanged.

Replacing the Lagrangians LH4 and LH5 in (2.26) with the kinetically equivalent ones (2.22), one nds that the kinetic structure of the total action is described by the tensors

C (2) = C (2)H + (2) and C (3) = C (3)H + (3) . (2.29)

Only these tensors are relevant for the degeneracy conditions, which we derive below.

2.4 Degeneracy conditions and primary constraints

We now introduce the Hessian matrix of the Lagrangian with respect to the velocities V and Kij. This matrix can be written in the form (introducing a factor 1/2 for convenience)

H = A Bij

Bkl Kij,kl !

with

KijKkl . (2.31)

The degeneracy of the theory is associated with the degeneracy of its Hessian matrix, i.e. detH = 0. Equivalently, one can nd a non trivial null eigenvector (v0, Vkl) such thatv0A + BklVkl = 0 , v0Bij + Kij,klVkl = 0 . (2.32)

These conditions translate into the existence of a primary constraint, which takes the form

v0 + Vijij + 0 , (2.33)

where we have introduced the covariant momenta conjugated respectively to A and h ,

~b2 = b2 + f3X , ~b3 = b3

2

3f3X , (2.28)

, (2.30)

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1

2

1

2

1

2

2L

2L

2L

A

V 2 , Bij

V Kij , Kij,kl

LKij , (2.34)

and the dots indicate momentum-independent terms, involving only the elds and their spatial derivatives. Note that we will always assume v0 6= 0 since we are interested in

removing the Ostrogradsky mode: therefore in the following we will x v0 = 1 without loss of generality.

It is important to keep in mind that the primary constraint (2.33) is a scalar constraint involving only the scalar components of ij, i.e. Vijij. It is indeed responsible for removing

the scalar Ostrogradsky mode. However, there could still be extra primary constraints in the vector sector of ij, which can further reduce the number of degrees of freedom (dof) (as

8

LV , ij

pointed out in [27] and further stressed in [28]). Indeed, as we will show in what follows, some classes of theories that possess the constraint (2.33), also enjoy the two following primary constraints:

i Pjkij + 0 , (2.35) where we have used the projector (2.25). These constraints remove the two helicity-2 dof present in the metric sector, leaving the theory with only one dof.

In order to compute the Hessian matrix of (2.26), one needs to keep all terms quadratic and cubic in the velocities. The Hessian matrix decomposes into its quadratic and cubic contributions denoted

H(2) = A(2) Bij(2)

Bkl(2) Kij,kl(2) !

A(2) C (2) , Bij(2) C (2) ij, Kij,kl(2) C (2) ij kl, (2.37)

A(3) 3C (3) , Bij(3) 3C (3) ij , Kij,kl(3) 3C (3) ij kl .

(2.38)

Introducing the tensor

L + ij Vij , (2.39) the conditions (2.32) (with v0 = 1) for purely quadratic theories read

C (2) L = 0 , C (2) ij L = 0 . (2.40)

On the other hand, the cubic Hessian matrix contains velocities. Therefore, the degeneracy conditions must be satised for arbitrary values of . This implies that can be factorised and the conditions (2.32) in the cubic case are analogous to the quadratic ones, namely

C (3) L = 0 , C (3) ij L = 0 . (2.41)

The above equations mean that, in order to get a degenerate Lagrangian, the projections of the tensors C (2)L or C (3)L, respectively via and ij , must vanish. As shown in appendix B, this implies that these tensors are necessarily of the form

C (2)L = 2A( ) , (2.42)

and

C (3)L = 41 A(h )(A) + 42 A( )A() , (2.43)

where , 1 and 2 are arbitrary scalar quantities.

By solving the conditions (2.42), one recovers the quadratic theories identied in [20]; they are summarised in appendix C. Conditions (2.43) are solved in detail in appendix D and in the next section we report the various classes of purely cubic theories. Then, we will consider the possibility to merge quadratic and cubic theories. In this case the additional condition to impose is that L is the same in (2.42) and (2.43), i.e. we have to use the same Vij.

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, H(3) = A(3) Bij(3)

Bkl(3) Kij,kl(3) !

, (2.36)

with

3 Classication of cubic theories

The degeneracy conditions for quadratic theories (i.e. with f3 = bi = 0) have already

been solved and the corresponding theories identied in [20]. These quadratic theories were then examined in more details in [2628]. In this section we thus focus our attention on the purely cubic theories, i.e. characterized by f2 = ai = 0. Solving the degeneracy conditions here is much more involved than in the quadratic case and rewriting them in the tensorial form (2.41) is instrumental to obtain the full classication. Below, we simply present the full classication, indicating for each class the free functions among the bi and the constraints satised by the other functions. All the cubic theories we identify are summarised at the end of the section in table 1. The details of how we have identied these classes are given in appendix D, where the reader can also nd the explicit expression of the null eigenvectors associated with the degeneracy. The latter are indispensable to identify the healthy combinations of quadratic and cubic Lagrangians, which will be given in the next section.

3.1 Minimally coupled theories

We start with the minimally coupled case, corresponding to f3 = 0. There are seven di erent classes of theories.

3M-I: Four free functions b1, b2, b3 and b4 (with 9b1+2b2 6= 0). All the other functions

are determined as follows:

b5 =

2X b2 , b6 =

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9b1b3 + 3b4X(b2 + b3) 2b22 X(9b1 + 2b2) ,

b7 =

3X b3 , b8 =

9b1b3 6b4X(b2 + b3) + 6b2b3 + 4b22 X2(9b1 + 2b2) ,

b9 = 1X2(9b1 + 2b2)2

h3b24X2(9b1 + 3b2 + b3) 2b4X 9b1(b2 b3) + 4b22

+24b1b22 + 54b21b2 + 27b21b3 + 4b32

i

,

b10 = 1X3(9b1 + 2b2)3

h3b34X3(9b1 + 3b2 + b3) 6b2b24X2(9b1 + 3b2 + b3) +2b4X 81b21(b2 + b3) + 18b1b2(3b2 + 2b3) + 2b22(5b2 + 3b3)

2 54b21b2(b2 + 2b3) + 4b1b22(7b2 + 9b3) + 81b31b3 + 4b32(b2 + b3)

i

. (3.1)

This class includes the pure quintic beyond Horndeski Lagrangian:

LbH5 = f(, X)

hX ()3 3

+ 2

(3.2)

3 ()2 2 +2 i ,

which corresponds to the choice of functions

b1X =

b2 3X =

b32X =

b4

3 =

b5

6 =

b6

b7

3 =

6 = f . (3.3)

10

The above combination is special as it leaves the Lagrangian linear in V , therefore in (2.38) A3 = 0.

Notice that in this class 9b1 + 2b2 6= 0. The condition 9b1 + 2b2 = 0 leads to the next

three classes.

3M-II. Three free functions b1, b3, b6 (with 9b1 2b3 6= 0). All the other functions

are given by

b2 =

92b1 , b4 =

3X b1 , b5 =

9X b1 ,

b7 =

3b3 2b6X X2 ,

b9 = 9b1(b3 + 2b6X) 81b21 2b26X23X2(9b1 2b3)

3X b3 , b8 =

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,

b10 =

h18b6X 12b1b3 + 27b21 + 2b23

36b3 8b1b3 + 18b21 + b23

12b3b26X2 + 4b36X3

ih

9X3(9b1 2b3)2i1

,

In this class 9b1 2b3 6= 0. The case 9b1 2b3 = 0 (and 9b1 + 2b2 = 0) is described

by the next two classes.

3M-III. A single free function b1. All the other functions are determined in terms of b1 as follows:

b2 =

92b1 , b3 =

92b1 , b4 =

3b1

X , b5 =

9X b1 , b6 =

9b1 2X ,

b7 =

272X b1 , b8 =

9b12X2 , b9 =

3 b12X2 , b10 =

b1 X3 .

3M-IV. Five free functions b1, b4, b5, b8, b10. The other functions are given by

b2 =

92b1 , b3 =

92b1 , b6 = 3b4

92X b1 ,

3b1 2X(2b4 + b5)

2X2 .

3M-V. Two free functions, b1 and b4, while the other functions are given by

b2 = b3 = b5 = b6 = b7 = b8 = 0 , b9 = b24

3b1 , b10 =

b7 = 3b5 +

272X b1 , b9 =

b34 27b21

. (3.4)

There is only one (scalar) dof that propagates due to the primary constraints (2.35), and their associated secondary constraints.

3M-VI. Six free functions b1, b4, b5, b8, b9 and b10. All the other functions vanish:

b2 = b3 = b6 = b7 = 0 . (3.5)

Again, there is only one (scalar) dof that propagates.

3M-VII. Four free functions b5, b7, b8 and b10. The remaining functions vanish, except b9:

b1 = b2 = b3 = b4 = b6 = 0 , b9 =

b5X . (3.6)

11

3.2 Non-minimally coupled theories

We now consider the purely cubic Lagrangians with f3 6= 0. There are two classes of

theories.

3N-I. In addition to f3, the functions b1 and b4 are free (with the only restriction b1 6= 0). The other functions are determined as follows:b2 = 3 b1 , b3 = 2 b1 , b6 = b4 ,

b5 = 2(f3X 3b1)2 2b4f3XX

3b1X , b7 =

2b4f3XX 2(f3X 3b1)2

3b1X ,

b8 = 2(3b1 + b4X f3X) (f3X 3b1)2 b4f3XX 9 b21X2

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,

b9 = 2b4(3b1 + b4X f3X)

3b1X , b10 =

2b4(3b1 + b4X f3X)2

9b21X2

. (3.7)

Quintic Horndeski (2.19), as well as the combination of quintic Horndeski plus quintic beyond Horndeski (3.2), is included in this class of models.

3N-II. Free functions b5, b8 and b10, in addition to f3. The other functions are given by

b1 = b2 = b3 = 0 , b7 = b5 ,

b4 = b6 =

f3X

X , b9 =

2 f3X + Xb5

X2 .

3.3 Minkowski limit

Here we discuss which ones among the classes of theories described above admit a healthy Minkowski limit, i.e. the limit where the metric is given by g = and the metric

uctuations are ignored. In this limit, only the scalar sector is dynamical and the Hessian matrix reduces to its purely scalar component, i.e. A. For cubic theories, the degeneracy

is thus expressed by the condition A(3) = 0, which imposes the relations

b1 =

b2

3 =

b3

b5

b7

2 , b4 =

2 = b6 =

2 , b8 = b9 = b10 = 0 . (3.8)

The only classes that satisfy these conditions are

3M-I: Beyond Horndeski theory,

3N-I: Beyond Horndeski and Horndeski theory,

3N-II: Imposing also b5 = 2f3X/X and b8 = b10 = 0 .

This shows that there is a new theory, 3N-II, which propagates three degrees of freedom on curved spacetime and has a healthy Minkowski limit. On the other hand, theories that do not satisfy (3.8) could still have a healthy decoupling limit around a non-trivial background.

12

Minimally coupled theories

Classication # dof Free functions Minkowski limit Examples 3M-I 3 i=1,2,3,4 X (bH) bH, bH(1)

3M-II 3 i=1,3, 6 X

3M-III 3 i=1 X

3M-IV 3 i=1,4,5,8,10 X

3M-V 1 i=1,4 X

3M-VI 1 i=1,4,5,8,9,10 X

3M-VII 3 i=5,7,8,10 X

Non-minimally coupled theories

Classication # dof Free functions Minkowski limit Examples3N-I 3 f3, i=1,4 X (H, bH) H, H+bH, H(2), ( , )H(3)

3N-II 3 f3, i=5,8,10 X

Table 1. Summary of all cubic degenerate classes. The subscript i in free functions indicates which functions among bi are free. Examples: (1): theories obtained by the generalised conformal transformation ( ) from beyond Horndeski (bH). (2): theories obtained by the generalised disformal transformation ( ) from Horndeski (H). This is equivalent to a combination of Horndeski and beyond Horndeski [25]. (3): theories obtained by the generalised conformal and disformal transformation from Horndeski. See section 5 for discussions about the generalised conformal and disformal transformation.

4 Merging quadratic with cubic theories

In this section we wish to determine all the theories of the form (2.2), i.e. quadratic plus cubic Lagrangians, that are degenerate. Adding two degenerate Lagrangians does not always yield a degenerate one. This is the case only if the null eigenvectors associated with the two Lagrangians coincide. Therefore, in order to see whether the combination of two Lagrangians is viable, one needs to compare their eigenvectors, which are all listed in appendix C for quadratic theories and in appendix D for cubic ones, and check when they are equal.

We present four tables describing all the di erent possibilities for merging quadratic and cubic theories. We indicate with Xtheories that can be freely combined, with X theories that cannot be combined, and with (n) theories that can be combined imposing the additional condition(s) (n) listed below each table.

Minimally coupled quadratic plus minimally coupled cubic theories.

3M-I 3M-II 3M-III 3M-IV 3M-V 3M-VI 3M-VII

2M-I (1) (2) X X (3) X X

2M-II X X X X X X (4)

2M-III X X X X X X (5)

13

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Conditions:

(1). b4 = 6a1b1+4a2b2+a3X(9b1+2b2)2X(a1+3a2)

(2). b6 = 3(6a1b1+4a2b3+a3X(2b39b1))4X(a1+3a2)

(3). b4 = 3b1(2a13a3X)2X(a1+3a2)

(4). b7 = 3b5

(5). b7 = 0 (1 dof)

Notice that condition (5) eliminates also the 2 tensor dof, leaving the joined classes

2M-III + 3M-VII with only one scalar dof.

The quartic beyond Horndeski theory LbH4 is included in 2M-I, while the quintic beyond Horndeski theory LbH5 (3.2) is included 3M-I. They satisfy the condition (1) thus the combination LbH4 + LbH5 is still viable [20, 25].

Non-minimally coupled quadratic plus minimally coupled cubic theories.

3M-I 3M-II 3M-III 3M-IV 3M-V 3M-VI 3M-VII

2N-I (1) & (3) (1) & (6) (1) X (1) & (4) X X

2N-II X X X X X X (7)

2N-III (3) (6) X X (4) X X

2N-IV (2) & (3) (2) & (6) (2) X (5) X X

Conditions:

(1). a3 = 8a1f2Xf2 + 6a1+4f2XX 4f2X2

(2). a3 = 12a2f2Xf2 8(a2f2X)X 6f2X2

(3). b4 = 2f2X(9b1+2b2)f2 2(6b1+b2)X

(4). b4 = 6b1

(5). b4 = 3b1(X(a3X+4f2X)2f2)2X(a2X+f2)

(6). b6 = 3(6b1f29b1f2XXb3f2+2b3f2XX)f2X

(7). b7 = b5

The quartic Horndeski theory LH4 (2.18) is included in 2N-I. The combination LH4+LbH5 does not satisfy the conditions (1) and (3), thus this combination is not degenerate [20, 25].

14

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3f2Xf22X

Minimally coupled quadratic plus non-minimally coupled cubic theories.

2M-I 2M-II 2M-III

3N-I X X X

3N-II X X X

The quintic Horndeski theory LH5 (2.19) is included in 3N-I. As can be seen from the table, it is not possible to combine 3N-I and 2M-I thus the combination LH5 + LbH4 is not viable [25].

Non-minimally coupled quadratic plus non-minimally coupled cubic theories.

2N-I 2N-II 2N-III 2N-IV

3N-I (1) X X X

3N-II X X X X

Conditions:

(1). b4 = a1f3XX6b1f2+6b1f2XX+2f2f3Xf2X

a3 = 2(

The classes 2N-I and 3N-I contain three free functions each, thus the combination 2N-I + 3N-I contains four free functions due to the conditions (1). In the next section, we show that this theory can be obtained by the generalised conformal and disformal transformation from LH4 + LH5.

5 Conformal and disformal transformation

We now investigate which ones among the cubic theories can be obtained from known Lagrangians through conformal and disformal transformations. The same analysis for quadratic theories can be found in [26, 27]. First we identify the class of theories minimally coupled with gravity (i.e. f3 = 0) that can be obtained from beyond Horndeski (3.2) by a conformal transformation. Then, we study the class of theories that can be obtained from Horndeski theory (2.19) by a conformal together with a disformal transformation.

5.1 Conformal transformation on Beyond Horndeski

It was shown in [25] that under the generalised disformal transformation

g = g + (X) , (5.1)

beyond Horndeski theory is transformed into itself:

LbH5[ f] = LbH5[f], (5.2)

15

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b1(9a1f2X12a1f2XX2+6f2f2XX6f22)+2f3X(f2a1X)2)

3b1f2X2

where f = f/(1 + X )7/2. On the other hand, under the generalised conformal transformation

g = (X)g , (5.3)

it transforms asLbH5[ f] = LbH5[f] +

XibiL(3)i , (5.4)

where

f =

f 2 ,

b4 =

b6 = 3

fX X 3 ,

b8 = 6

f X

3 ,

b9 = 6

f X (X X ) 4 ,

b10 = 6

f 2X (X X ) 5 , (5.5)

and the other bi vanish. In terms of the total bi, this gives

b1 = Xf, b2 = 3Xf, b3 = 2Xf, b4 = b6 = 3f +

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3fX X

, b5 = b7 = 6f,

6f 2X (X X ) 3 . (5.6)

These bs satisfy conditions (3.1), thus this theory is included in class 3M-I.

5.2 Conformal and disformal transformation on Horndeski

The generalised conformal and disformal transformation

g = (X)g + (X) , (5.7)

transforms the Horndenski action as

LH5[ f3] = LH5[f3] + LbH5[f] +

XibiL(3)i , (5.8)

where

f3 =

b8 = 6f X

, b9 =

6f X (X X ) 2 , b10 =

+ X + Z

f3

f3 ( X X) + X X

2 ( + X )3/2 dX , (5.9)

f =

f3 X ( X + X X)3 ( + X )5/2 , (5.10)

b4 = b6 =

f3 X X

( + X )5/2 , (5.11)

b5 = b7 =

2 f3 X X [X ( X + X X) ]

( + X )5/2 , (5.12)

b8 = 2X

f3 X X [ X ( X + X X) + X]

3/2( + X )5/2 , (5.13)

b9 =

2X f3 X X X

( + X )5/2 , (5.14)

2X f3 X 2X X

3/2 ( + X )5/2 . (5.15)

16

b10 =

and the other bi vanish. One can check that this theory satises the conditions (3.7), thus it is included in class 3N-I. Theories in class 3N-I have three free functions. On the other hand, the action (5.8) contains f3, and . Thus there is the same number of free functions. Indeed we can relate f3 X , X and X to f3X, f and b4 as

f3 X = f3X ( + X )5/2

[ X ( X + X X)]

X = b4

3Xf + f3X , (5.17)

X = (3f b4)

X (3Xf + f3X) . (5.18)

Thus, theories in class 3N-I can be mapped to Horndeski if the transformation (5.7) is invertible.

Finally we consider the generalised conformal and disformal transformation from LH4 + LH5. Using the result for the transformation of LH4 obtained in [26, 27], we can show that this theory corresponds to the combination of 2N-I and 3N-I and satises the condition(1). This theory has four free functions, which correspond to f2, f3, and . Thus this theory can be regarded as the Jordan frame version of the Horndenski theory where the gravitational part of the Lagrangian is described by Hordenski with the metric g ,

LH4[g] + LH5[g], while the matter is non-minimally coupled through g . By performing the generalised conformal and disformal transformation, the gravitational action is described by the combination of 2N-I and 3N-I and the metric is minimally coupled to matter.

6 Conclusions

In this paper, we presented all Ostrogradsky ghost-free theories that are at most cubic in the second derivative of the scalar eld, and that propagate at most three degrees of freedom. Extending Horndeskis results, we have found new Lagrangians, which lead to higher order equations of motion but avoid Ostrogradsky instabilities by means of constraints that prevent the propagation of dangerous extra degrees of freedom.

In order to achieve our results, we used the degeneracy criterium introduced in [20], and classied the Lagrangians that are degenerate, i.e. whose Hessian matrix, obtained by taking the second derivatives of the Lagrangian with respect to velocities, is degenerate. In total, we identied seven classes of minimally coupled cubic theories and two classes of non-minimally coupled cubic theories, which contain as subclasses all known scalar-tensor theories which are cubic in second derivatives of the scalar eld. We also investigated which cubic theories admit a well-dened Minkowski limit, i.e. when the metric is frozen to its Minkowski value. Our results are summarised in the table 1.

We then studied in which cases it is possible to combine any of these cubic theories with the previously identied quadratic ones. Note that one can also add arbitrary terms of the form P (X, ) and Q(X, ) without changing the degeneracy of the total Lagrangian. We conrmed the previous nding that the combination of quartic or quintic

17

, (5.16)

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beyond Horndeski with a di erent Horndeski is not viable. Finally, we studied whether our cubic theories are related to known Lagrangians through generalised conformal or disformal transformations. We identied the theory, with four free functions, that is obtained by the generalised conformal and disformal transformation from the combination of quartic and quintic Horndeski Lagrangians.

Various interesting developments are left for the future. First, phenomenological aspects of these new theories should be investigated, in particular studying the existence of stable cosmological FLRW solutions possibly self-accelerating and their properties, by using for instance the e ective description of dark energy (see e.g. [30] for a review and [31] for a recent generalization that includes non-minimal couplings to matter). It would also be worth analysing possible distinctive features of screening mechanisms in these set-ups. Secondly, on the theory side, it would be interesting to analyse further generalizations of scalar-tensor theories containing higher powers of second derivatives of the scalar eld. Such theories do not admit a well-dened Minkowski limit, and some explicit examples have been discussed in [26] and in [32]. A more complete classication using the techniques we presented should be feasible, and left for future investigations.

Acknowledgments

KK is supported by the U.K. Science and Technologies Facilities Council grants ST/K00090X/1 and ST/N000668/1 and the European Research Council through grant 646702 (CosTesGrav). GT is partially supported by STFC grant ST/N001435/1. MC and GT are grateful to CERN Theoretical Physics Department for hospitality and nancial support during the development of this project.

A Curvature dependent Lagrangians

Curvature tensors depend quadratically on the extrinsic curvature so, according to the kinetic structure presented in section 2.2, their combination with the second derivative of the scalar eld yields cubic powers in velocities. All the possible quadratic and cubic terms in velocities involving the curvature are

LR =

where

and

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2

Xi=1

L2[fi] +

9

Xi=3

L3[fi] , (A.1)

L2[f1] = f1 R , L2[f2] = f2 R ; (A.2)

L3[f3] = f3 R , (A.3)

L3[f4] = f4 R , L3[f5] = f5 R , (A.4)

L3[f6] = f6 R , L3[f7] = f7 R , (A.5)

L3[f8] = f8 R , L3[f9] = f9 R , (A.6)

18

where fi are arbitrary functions of and X. Only one of the two quadratic Lagrangians in (A.2) is independent, since it is possible to express one in terms of the other through integrations by parts: we worked with L2[f2]. Also the cubic Lagrangians (A.3)(A.6) are not all independent: we can obtain L3[f9] from L3[f6] and L3[f8] using integrations by parts, and L3[f8] from L3[f4], L3[f5] and L3[f3] using also the Bianchi identity. Therefore, we are left with ve cubic independent Lagrangians (A.3)(A.5). To keep contact with Horndeski theory, without loss of generality it is useful to replace (A.3) with the following expression

L3[f3] = f3 G , (A.7)

that we studied in the main text.

In this appendix we concentrate separately on the four remaining cubic non-minimally coupled Lagrangians (A.4)(A.5). What characterises these Lagrangians in comparison with (A.7) is that they all feature time (and space) derivatives of the extrinsic curvature. This indicates the possible presence of additional Ostrogradsky modes, this time coming from the metric sector of the theory, unless there are suited extra primary constraints that remove them.

The covariant 3+1 decomposition of (A.4)(A.5) shows that the only components of the extrinsic curvature that acquire time derivatives are the scalar ones:

E

Their covariant velocities appear in the form

VE nE , VF nF , (A.9)

in analogy to what we encountered in section 2.2. Therefore, applying the same kind of eld redenition used for the scalar eld (2.1), Lagrangians (A.4)(A.5) generally propagate two more Ostrogradsky modes, E and F , in addition to A . To avoid their propagation, we need two more primary constraints.

Dening the conjugate momenta associated to the new elds

E

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ij

2 Kij , F

PijKij . (A.8)

LVF , (A.10) for the set of Lagrangians (A.4)(A.5) we obtain

= VE + VF + . . . , E = V , F = V , (A.11)

where

= 2f4 Xf6 + A2 (2f5 + Xf7) , = 2f4 + A2 (2f5 + f6) A4 f7 , (A.12) and the dots in represent non relevant terms. From the form of the momenta (A.11), it is clear that a total of three primary constraints can only be obtained in the trivial way, i.e.

0 , E 0 , F 0 . (A.13)

Hence = = 0 and, due to the Lorentz invariance of fi, relations (A.12) give

f4 = f5 = f6 = f7 = 0 . (A.14)

LVE , F

B Tensorial structure implied by the degeneracy conditions

First, we show the equivalence between the relations (2.40) and (2.42) for quadratic theories. It is simple to show that (2.40) is equivalent to

M C(2)(L) = 0 with C(2)(L) C (2)L (B.1)

and

M A g (g ) + 2n(g( )A ) . (B.2)

Indeed, decomposing (B.1) in the directions n n , hi hj and n hi leads to the equations (2.40). As a consequence, C(2)(L) is necessarily in the kernel of M viewed as an operator acting on symmetric 4 dimensional matrices. A matrix V is in the kernel

of M when

M V = 0 A V = 2 nV ( A ) (B.3)

V s.t. V = V ( A ) with Vn = 0 . (B.4) Furthermore, the only available vector V in the theory which is orthogonal to n is in the direction. Hence, there exists a scalar such that

C (2)L = 2A( ) , (B.5)

which is the relation (2.42).

The generalization to cubic theories is rather immediate. Let us show that (2.43) and (2.41) are equivalent. Following the same strategy as previously, we rst show that (2.41) is equivalent to

M C (3)(L) = 0 with C (3)(L) C (3)L , (B.6) with M dened as in the quadratic case by (B.2).

Now, both M and C(3)(L) can be viewed as operators acting on symmetric 4 dimensional matrices. Thus, (B.6) means that C(3)(L) and M are orthogonal, or equivalently that the image of C(3)(L) lies in the kernel of M. To go further, we recall that the kernel of

M is dened by (B.3). The vector space orthogonal to n is three dimensional and a basis is given by h where labels the elements of the basis (only 3 out of the 4 components of h are independent). Thus, if we use the notation V for a basis of Ker(M) where labels the elements of the basis, then V = h( A ) which is clearly of the form (B.3). Hence, due to symmetries, C(3)(L) can we written as

C (3)(L) = m V V (B.7)

where m is a symmetric matrix. Due to the covariance, the symmetric matrix m is necessarily of the form m = 41g + 42A A where 1 and 2 are scalars. Notice that there is no components of the form A( n ) nor of the form n n in m because V n = 0. As a conclusion, (B.6) is true if and only if there exist scalars 1 and 2 such that:

C (3)(L) = 41 A(h )(A) + 42 A( )A() . (B.8)

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C Quadratic theories

We review the quadratic theories proposed in [20] and further classied in [26] and [27].

C.1 Minimally coupled theories

2M-I. Three free functions a1, a2, and a3, together with

a4 = 2

a1

X , a5 =

4a1 (a1 + 2a2) 4a1a3X + 3a23X24 (a1 + 3a2) X2 . (C.1)

We assume a2 6= a1/3. This case includes beyond Horndeski theory. The corre

sponding null eigenvector is given by

v1 =

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X(2a2 + a3X)

A (2a1 (A2 + 2X) + 2a2 (2A2 + 5X) a3X (A2 + X))

, (C.2)

v2 = 2a1 4a2 + a3XA (2a1 (A2 + 2X) + 2a2 (2A2 + 5X) a3X (A2 + X))

. (C.3)

This class was called M-I in [26] and IIIa in [27].

2M-II. Three free functions a1, a4, a5 and

a2 =

a1

2 a1

3 , a3 =

3 X . (C.4)

The corresponding null eigenvector is given by

v1 = XA (A2 + X) , v2 =

1A (A2 + X) . (C.5)

This class was called M-II in [26] and IIIb in [27].

2M-III. Four free functions a2, a3, a4, a5 and the unique condition

a1 = 0 . (C.6)

The eigenvector is given by

v1 =

2A2 + X2A (A2 + X)2 . (C.7)

This class was called M-III in [26] and IIIc in [27].

For minimally coupled quadratic theories, the vector components of ij (i.e.i Pjkij) are proportional to a1, therefore, as noticed in [28], this class propagates only one scalar dof.

21

X2A (A2 + X) , v2 =

C.2 Non-minimally coupled theories 2N-I. Three free functions f2, a1 and a3. The conditions are

a2 = a1 6=

f2X , (C.8)

a4 = 18(f2 a1X)2

4f2 3(a1 2f2X)2 2a3f2

a3X2(16a1f2X + a3f2) +4X 3a1a3f2 + 16a21f2X 16a1f22X 4a31 + 2a3f2f2X

, (C.9)

a5 = 18(f2 a1X)2

(2a1 a3X 4f2X) [a1(2a1+ 3a3X 4f2X) 4a3f2] . (C.10)

The combination of Horndeski and beyond Horndeski theories is included in this class. The corresponding null eigenvector is given by

v1 = D A (a1X f2)(2a1 a3X 4f2X) , (C.11)

v2 = D A (a1(2a1 + a3X 4f2X) 2a3f2) , (C.12)

with

D1 a1 A2 a3X2 + 2f2 + 12f2XX

+ A4 (4f2X a3X) 6f2X + 8f2XX2

2a21 3A2 X + A4

+ f2

A2

+ X

a

3 2A2 + X

4f2X

+ 4f2

.

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This class was called N-I in [26] and Ia in [27].

2N-II. Three free functions f2, a4, a5 and

a2 = a1 =

2 (f2 2Xf2X)X2 (C.13)

The corresponding null eigenvector is given by

v1 = 0 , v2 =

f2X , a3 =

A (A2 + X)2

. (C.14)

This class was called N-II in [26] and Ib in [27].

For non-minimally coupled quadratic theories, the vector components of ij are in

stead proportional to f2 Xa1, hence also here there are not tensor modes [28]. 2N-III. Three free functions f2, a1 and a2. The conditions are

a1 + a2 6= 0 , and a1 6=

f2X , (C.15)

4f2

a3 = 4f2X(a1 + 3a2)f2

2(a1 + 4a2 2f2X)X

X2 , (C.16)

a4 = 2f2

X2 +

8f22Xf2

2(a1 + 2f2X)

X , (C.17)

a5 = 2 f22X3

4f32 + f22X(3a1 + 8a2 12f2X)+8f2 f2XX2(f2X a1 3a2) + 6f22XX3(a1 + 3a2)

. (C.18)

22

The corresponding null eigenvector is given by

v1 = X(f2 2f2XX)A (2A2 (f2 f2XX) + X(3f2 2f2XX))

v2 = 2f2XX 2f2A (2A2 (f2 f2XX) + X(3f2 2f2XX))

This class was called N-III (i) in [26] and IIa in [27].

2N-IV. Three free functions f2, a2 and a3. The conditions are

a1 + a2 6= 0 (C.21)

a1 = f2X , (C.22)

a4 = 8f22Xf2

a5 = 1

4f2X3(f2 + a2X)

The corresponding null eigenvector is given by

v1 = 2EX(a2X + f2)(f2 2f2XX) , (C.25)

v2 = E 4X(a2X +f2)(f22f2XX)A2 f2X(4a2+a3X 4f2X)8a2f2XX2+2f22

2 A + X ,(C.26)

with

E1 A3 f2X(4a2 + a3X 4f2X) 8a2f2XX2 + 2f22

+A X2(2a2(f2 4f2XX) + a3f2X 4f2f2X) . (C.27)

This class was called N-III (ii) in [26] and IIb in [27].

Due to the condition f2 Xa1 = 0, only one scalar dof is present in this class [28].

D Identifying cubic theories

In this appendix we solve in details the conditions (2.43) for purely cubic theories. Let us rst note that, since Vij lies in the hyper-surface orthogonal to n, it can be decomposed

as follows

Vij = v1hij + v2ij , (D.1)

where v1 and v2 are scalar quantities. The tensor L , introduced in (2.39), can thus be written as

L + ij Vij = + v1 ij hij + v2 ij

ij ,

= 1 nn + 2 2 n(A ) A (v1g + v2AA ) , (D.2)

23

, (C.19)

. (C.20)

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4f2X

X , (C.23)

f2a23X4 4f32 8f22X(a2 2f2X)

4f2X2 (4f2X (f2X 2a2) + a3f2) + 8f2XX3(a3f2 4a2f2X)

. (C.24)

with

1 1 + A v1 + A (2X + A2 )v2 and 2 v1 + Xv2 . (D.3)

D.1 Minimally coupled theories

A long but straightforward calculation shows that the tensorial relation (2.43) leads to the following 12 equations:

1b2 = 0 , 1b3 = 0 , 1b6 = 0 , (D.4)

(1A + X2)b5 + 22b2 = 0 , (1A + X2)b7 + 32b3 = 0 , (D.5)

1b7 = 12(m1 + A2 m2) , (1A + X2)b8 + 2(2b6 + b7) = 12A m2 , (D.6)1(b9 + 3A2 b10) + 22A (3Xb10 + b8 + b9)

A [(v1 + Xv2)(2b8 + 3Xb10) + (4v1 + Xv2)b9 + 2v2b6 + v2b7] = 12m2 , (D.7) 1(b4 + A2 b9) + 2A (2b9X + 2b4 + b5)

A [b4(4v1 + Xv2) + (b5 + Xb9)(v1 + Xv2) + (b6v1 + b2v2))] = 0 , (D.8) 1(b5 + b8A2 ) + 22A (b8X + b5 + b7)

A [b5(4v1 + Xv2) + 3b3v2 + b7(3v1 + 2Xv2) + b8X(v1 + Xv2)] = 12m1 , (D.9)1(3b1 + b4A2 ) + 22A (b4X + 3b1)

A [3b1(4v1 + v2X) + 2b2v1 + b4X(v1 + Xv2)] = 0 , (D.10)1(b2 + b6A2 ) + 22A (b6X + b2)

A [b2(4v1 + Xv2) + 3b3v1 + b6X(v1 + Xv2)] = 0 . (D.11)

The two equations in (D.6) enable us to solve for 1 and 2, yielding

1 = 112 [(1 A 2)b7 A (1A + X2)b8 2A 2b6] , (D.12)

2 = 112A [(1A + X2)b8 + 2(2b6 + b7)] . (D.13)

Hence, if we replace these expressions in the previous system, we end up with 10 equations for the 10 unknown bi. These 10 equations can be written in a matrix form as follows:

JHEP12(2016)100

b2 b3 b6 b5 b7 b1 b4 b8 b9 b10

A 0

= 0 , (D.14)

C B

24

where

A

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

22 0 0 T 0 0

0 32 0 0 T 0

S 3A v1 A T 0 0 0

, B

A T 0 0 0

0 2A T 0 0

S 0 A T 0

0 T SA 3A2 T

, (D.15)

C

2A v1 0 0 0 0 3S

0 3A v2 2A 2 S 1 + XA v2 0

A v2 0 A v1 0 0 0 0 0 2(A2 v2 + 2) 0 (A2 v2 + 2) 0

,

and 0 denotes a 64 matrix of zeros. We have also introduced the notation T 1A +X2 and S 1 + 2A 3v1A .

The resolution of the system depends on the rank of the matrices A and B. To solve the system, it is useful to separate the vector in (D.14) into two pieces

b+ = (b2, b3, b6, b5, b7, b1) and b = (b4, b8, b9, b10) . (D.16)

Hence, we solve successively the following two matrix equations

Ab+ = 0 and Cb+ + Bb = 0 . (D.17)

We can distinguish several cases, depending on whether 1 or T vanish.

1 = 0 and T 6= 0In that case v2 is related to v1 by

v2 =

JHEP12(2016)100

1 + A v1

A (A2 + 2X) . (D.18)

The matrix A is highly degenerate with rank=3 and Ab+ = 0 produces 3 conditions. Two of them give

b5 =

2X b2 , b7 =

3X b3 , (D.19)

and the third one is

A A2 (2b2 + 3b3 Xb6) + X(5b2 + 6b3 Xb6)

v1 = X(b2 + Xb6) . (D.20)

Equation (D.20) plus the four remaining equations Cb+ + Bb = 0 are solved into three sectors.

3M-I). 9b1 + 2b2 6= 0:

b6 = 9b1b3 + 3b4X(b2 + b3) 2b22

X(9b1 + 2b2) ,

25

b8 = 9b1b3 6b4X(b2 + b3) + 6b2b3 + 4b22 X2(9b1 + 2b2) ,

b9 = 1X2(9b1 + 2b2)2

h3b24X2(9b1 + 3b2 + b3) 2b4X 9b1(b2 b3) + 4b22

+24b1b22 + 54b21b2 + 27b21b3 + 4b32

i

,

b10 = 1X3(9b1 + 2b2)3

h3b34X3(9b1 + 3b2 + b3) 6b2b24X2(9b1 + 3b2 + b3) +2b4X 81b21(b2 + b3) + 18b1b2(3b2 + 2b3) + 2b22(5b2 + 3b3)

2 54b21b2(b2 + 2b3) + 4b1b22(7b2 + 9b3) + 81b31b3 + 4b32(b2 + b3)

i

,

JHEP12(2016)100

and therefore

v1 =

X(3b1 + b4X)

A (A2 (6b1 + 2b2 b4X) + X(15b1 + 4b2 b4X))

(D.21)

As a conclusion, we end up with four free parameters b1, b2, b3 and b4.

3M-II). 9b1 + 2b2 = 0 and 9b1 2b3 6= 0:

b2 =

92b1 , b4 =

3X b1 , b8 =

3b3 2b6X X2 ,

b9 = 9b1(b3 + 2b6X) 81b21 2b26X23X2(9b1 2b3)

,

b10 =

h18b6X 12b1b3 + 27b21 + 2b23

36b3 8b1b3 + 18b21 + b23

12b3b26X2 + 4b36X3

ih

9X3(9b1 2b3)2i1

,

and

v1 = X(2b6X 9b1)

A [2A2 (9b1 3b3 + b6X) + X(45b1 12b3 + 2b6X)]

(D.22)

The three parameters b1, b3, b6 are free.

3M-III). 9b1 + 2b2 = 0 and 9b1 2b3 = 0:

b2 =

9b1

9b1

2 , b3 =

2 , b4 =

3b1

X ,

b6 = 9b1

2X , b8 =

9b12X2 , b9 =

3 b12X2 , b10 =

b1 X3 .

We obtain that b1 is free whereas there is no constraint on v1.

1 = 0 and T = 0: 3M-IV

This case is characterized by the fact that v1 and v2 are totally xed by

v1 = X

A (X + A2 ) , v2 =

1A (X + A2 ) . (D.23)

26

Furthermore, Ab+ = 0 xes

b3 = b2 . (D.24)

The four remaining equations give

b2 =

92b1 , b6 = 3b4

92X b1 ,

b7 = 3b5 +

272X b1 , b9 =

3b1 2X(2b4 + b5)

2X2 ,

and b1, b4, b5, b8, b10 are free.

1 6= 0 and T 6= 0: 3M-VIn that case, B is invertible and A reaches its maximal rank = 5. Hence, from Ab+ = 0 we get

b2 = b3 = b5 = b6 = b7 = 0 . (D.25)

The four remaining equations Cb+ + Bb = 0 give

b8 = 0 , (D.26)

together with three equations. If b1 = 0 all the other functions must be zero, therefore we assume b1 6= 0 and we obtain

b9 = b243b1 , b10 =

JHEP12(2016)100

b34 27b21

, (D.27)

plus one relation between v1 and v2:

A b4(A + A2 +X

. (D.28)

As a conclusion, only two parameters, b1 and b4, are free. One of the two components v1 or v2 of the eigenvector is also a free parameter.

This class possesses two more primary constraints of the form (2.35), hence there is only one scalar dof.

1 6= 0 and T = 0 v1 is xed by

v1 =

v1 + A2 +X

2 v2) = 3b1 1 + 3A v1 + A A2 +X

v2

A

X + A2 (X + A2 )v2 . (D.29)

Solving Ab+ = 0 leads to

b2 = b3 = b6 = 0 . (D.30)

Furthermore, the equations Cb+ + Bb = 0 give two branches:

27

3M-VI). b7 = 0and the component v2 is xed to

v2 = X 2A2 2A (A2 + X)2

. (D.31)

The components b1, b4, b5, b8, b9 and b10 are free.

Also in this class, tensor modes are eliminated by the primary constraints (2.35), so only one scalar dof is left.

3M-VII). b1 = b4 = 0 , b9 = b5/X and the component v2 is xed by

2A b5 A2 + X

2 v2 = X(b5 + b7) 2A2 b5 . (D.32)

The components b5, b7, b8 and b10 are free.

D.2 Non-minimally coupled theories

The resolution follows the same strategy as in the minimally coupled case. First, we write the generalised conditions in a form analogous to (D.14)

A 0 C B

!

JHEP12(2016)100

~b+

~b

!

= X f3X

A , (D.33)

where is a matrix given by

=

2 22/3

v2 2A v2

2A v2

2 + A2 v2

(2 + A2 v2) 0v2

0

. (D.34)

Hence, the solution for ~b = (~b+, ~b) is the sum of the general solution of the homogeneous equation (with f3 = 0) and a particular solution. Again, we solve them according to whether 1 and T vanish or not.

When 1 = 0 necessarily v1 = v2 = 0, which would imply in turn that 1 = 1. This is an inconsistency, hence there is no solution when 1 = 0.

28

1 6= 0 and T 6= 0: 3N-IWe need to assume b1 6= 0 otherwise T = 0, and we end up in the next class of

theories.

b2 = 3 b1 , b3 = 2 b1 , b6 = b4 ,

b5 = 2(f3X 3b1)2 2b4f3XX

3b1X , b7 =

2b4f3XX 2(f3X 3b1)2

3b1X ,

b8 = 2(3b1 + b4X f3X) (f3X 3b1)2 b4f3XX 9 b21X2

b9 = 2b4(3b1 + b4X f3X)

3b1X , b10 =

and b1, b4 and f3 are free. Furthermore, the eigenvector is given by

v1 = A (3b1 + b4X f3X)

A2 (b4X 3b1 + f3X) + A4 b4 + f3XX

A b4

A2 (b4X 3b1 + f3X) + A4 b4 + f3XX

. (D.36)

This is the particular solution, now we need to nd which homogeneous solution is compatible with it. To ensure the full theory to be degenerate, the eigenvectors of the homogeneous and the particular solutions must coincide. It is easy to show that it cannot be supplemented with any of the minimally coupled theories.

1 6= 0 and T = 0: 3N-IIWe obtain

b1 = b2 = b3 = b5 = b7 = 0 ,

b4 = b6 =

Furthermore, the eigenvector is given by

v1 = 0 , v2 =

and b5, b8, b10 and f3 are free.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

29

,

JHEP12(2016)100

2b4(3b1 + b4X f3X)2

9b21X2

,

, (D.35)

v2 =

f3X

X , b9 =

2 f3X

X2 .

A (X + A2 )2 . (D.37)

Now we study which homogeneous solution can be added to this particular one. It is possible to add only 3M-VII where b5 + b7 = 0. Therefore, the full conditions for this class of theories are

b1 = b2 = b3 = 0 , b7 = b5 , b4 = b6 =

f3X

X , b9 =

2 f3X + Xb5

X2 .

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