The most general second-order field equations of bi-scalar-tensor theory in four dimensions
Seiju Ohashi, Norihiro Tanahashi, Tsutomu Kobayashi, Masahide Yamaguchi
aa r X i v : . [ g r- q c ] M a y Prepared for submission to JHEP
KEK-TH-1826, RUP-15-11
The most general second-order field equations ofbi-scalar-tensor theory in four dimensions
Seiju Ohashi, a Norihiro Tanahashi, b Tsutomu Kobayashi c and Masahide Yamaguchi d a Cosmophys Group, IPNS KEK, 1-1 Oho, Tsukuba 305-0801, Japan b DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA,UK c Department of Physics, Rikkyo University, Toshima, Tokyo 175-8501, Japan d Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
E-mail: sohashi"at"post.kek.jp , N.Tanahashi"at"damtp.cam.ac.uk , tsutomu"at"rikkyo.ac.jp , gucci"at"phys.titech.ac.jp Abstract:
The Horndeski theory is known as the most general scalar-tensor theory with second-orderfield equations. In this paper, we explore the bi-scalar extension of the Horndeski theory. FollowingHorndeski’s approach, we determine all the possible terms appearing in the second-order field equations ofthe bi-scalar-tensor theory. We compare the field equations with those of the generalized multi-Galileons,and confirm that our theory contains new terms that are not included in the latter theory. We also discussthe construction of the Lagrangian leading to our most general field equations. ontents ξ tensors 15B Explicit form of Q I
15C Euler-Lagrange Equations 16D Construction of Lagrangian: single scalar-field case 17
The inflationary scenario [1] is now regarded as a necessary ingredient of modern cosmology becausethe recent observations of cosmic microwave background anisotropies [2–5] strongly suggest the epoch ofinflationary expansion in the early Universe. Since inflation is typically driven by scalar field(s), scalar-tensor theories provide a firm framework to study the dynamics of inflation. On the other hand, theunknown energy called dark energy is shown to be dominant in the present Universe [6–8] and might beunderstood as an outcome of infra-red modification of gravity. Scalar-tensor theories provide powerfultools to realize such a modification, and their phenomenological features have been studied intensivelyto confront them against observations. Although a plethora of inflationary models and modified gravitytheories have been proposed thus far, unfortunately, we have not yet succeeded in finding the real theoryand have kept seeking for it. We have two options to address this problem. The first one is to pursue theultimate (real) theory on the basis of theoretical consistencies, which is often called top-down approach.The other is to construct a framework of theories as general as possible, which is called bottom-upapproach. Of course, though both of approaches are complementary, one of the merits to take the latterapproach is to give a unified understanding of various models proposed individually. Another is that onecan easily pin down or constrain models once characteristic observational results would be reported.– 1 –he Horndeski theory [9] provides a typical working example of the latter approach because it isthe most general single scalar-tensor theory with second-order field equations. It is shown [10] that theHorndeski theory is equivalent to the generalized Galileon [11]. In fact, almost all of the inflationarymodels with single inflaton proposed so far can be described by this theory in a unified manner. Variousaspects of single field inflationary models have been studied in this framework [10], which is useful for usto constrain the models from observational results.In this paper, we take the latter approach and try to extend the Horndeski theory, which includesonly one scalar degree of freedom, to the bi-scalar case. Although no observational results suggest thepresence of multiple (light) scalars during inflation, our framework can clarify the essential differencebetween single and multiple scalar cases, and is useful for further constraining or detecting the presenceof multiple scalar degrees of freedom from observations. Such an attempt to extend the Horndeski theoryhas already been discussed. First of all, multi-field Galileon theory was proposed in the flat spacetime[17–20]. Later, the covariantization of this multi-field Galileon theory, called generalized multi-Galileon,was considered [21] and conjectured that the theory would correspond to the multi-field extension of theHorndeski theory, that is, the most general multi-field scalar-tensor theory with second order equationsof motion. Though the multi-field Galileon theory in the flat spacetime is proven to be the most generalmultiple-scalar field theory in the flat space-time with second order scalar equations of motion [22], it wasshown that the generalized multi-Galileon is not the most general theory [23] because this theory does notcontain the multi-DBI inflation models [24–28], in particular the double-dual Riemann term appearing inthese models. Motivated by these considerations, in this paper, we try to construct a multi-field extensionof the Horndeski theory. Especially we focus on the extension to the bi-scalar case as a first step.The organization of this paper is as follows. In Sec. 2, we construct the field equations of the mostgeneral two-scalar tensor theory with second order field equations following Horndeski’s procedure. Thissection is the main part of this paper. In Sec. 3, we compare our theory with the generalized multi-Galileontheory. We show that terms which are missing in generalized multi-Galileon theory are actually containedin our theory. In Sec. 4, we comment on the construction of the Lagrangian corresponding to the fieldequations we obtain. Finally we summarize our paper and discuss the results in Sec. 5.
Notations and conventions
Before closing the introduction, we summarize the notations and conventions used throughout this paper.We consider a four-dimensional spacetime with a metric g ab and two scalar fields φ I with I = 1 , g ab and φ I with respect to the coordinates x a are denoted as g ab,c ≡ ∂g ab ∂x c , φ I,a ≡ ∂φ I ∂x a , (1.1)respectively. We denote the covariant derivative of φ I with respect to g ab and its scalar product respectivelyas φ I | a ≡ ∇ a φ I , X IJ ≡ − φ I | a φ J | a , (1.2) Another direction is to consider a scalar-vector theory instead of a scalar-tensor theory. Motivated by the earlier workof Horndeski [12], construction of the most general vector theory with second order field equations on flat space, called thevector-Galileon theory, was attempted in Ref. [13]. Recently, the scalar-tensor theory with higher order equations of motionwithout introducing the ghost was proposed as well [14, 15]. See also [16] for yet another possible way of extending theHorndeski theory. – 2 –here X IJ is symmetric in I and J . We use a strike “ | ” also as a separator in (anti-)symmetrization.For example, [ I | J K, L | M ] stands for anti-symmetrization of I and M . Partial derivatives of a function A a...b ( g, ∂g, ∂ g, φ I , ∂φ I , ∂ φ I ) are expressed as A a...b ; cd ≡ ∂A a...b ∂g cd , A a...b ; cd,e ≡ ∂A a...b ∂g cd,e , A a...b ; cd,ef ≡ ∂A a...b ∂g cd,ef ,A a...b ; I ≡ ∂A a...b ∂φ I , A a...b ; cI ≡ ∂A a...b ∂φ I,c , A a...b ; cdI ≡ ∂A a...b ∂φ I,cd , (1.3)and partial derivatives of a function A ( φ I , X JK ) are expressed as A ,I ≡ ∂A∂φ I , A ,IJ ≡ (cid:18) ∂A∂X IJ + ∂A∂X JI (cid:19) . (1.4)In the equations of motion and the Lagrangian, we use the generalized Kronecker delta defined by δ i ...i n j ...j n ≡ n ! δ i [ j . . . δ i n j n ] , δ IKJL ≡ δ I [ J δ KL ] . (1.5)Repeated indices are summed over a = 0 , , , I = 1 , The first step of the construction of the most general scalar-tensor theory of Ref. [9] is to work out themost general equations of motion that are of second order in derivatives and compatible with the generalcovariance. In this section, we generalize this construction to the case with two scalar fields.
The assumptions imposed on the theory we are going to construct are summarized as follows.1. The theory has a Lagrangian scalar density, L .2. The Lagrangian scalar density, L , is composed of a metric, two scalar fields, and their derivativesup to arbitrary order: L = L (cid:0) g ab , g ab,c , g ab,cd , . . . ; φ I , φ I,a , φ
I,ab , . . . (cid:1) , (2.1)where I = 1 , .
3. Field equations are composed of the metric, the two scalar fields, and their derivatives up to secondorder: 0 = δ L δg ab = √− g G ab (cid:0) g cd , g cd,e , g cd,ef ; φ J , φ J,c , φ
J,cd (cid:1) , (2.2)0 = δ L δφ I = √− g E I (cid:0) g ab , g ab,c , g ab,cd ; φ J , φ J,a , φ
J,ab (cid:1) , (2.3)where δ L /δA is the variation of L with respect to a field A .– 3 –et us consider the variation of the action under an infinitesimal coordinate transformation, x a → x a + ξ a , which is given by δ Z d x L = 2 Z d x √− g (cid:18) ∇ b G ab − E I ∇ a φ I (cid:19) ξ a . (2.4)It follows from the assumption of L being a scalar density that Eq. (2.4) vanishes identically, implyingthat the integrand itself vanishes since ξ a may be chosen arbitrarily. Thus, the identity ∇ b G ab = 12 E I ∇ a φ I (2.5)holds. In the case of pure Einstein gravity this identity reduces to the well-known contracted Bianchiidentity, ∇ a G ab = 0. In this sense, the identity (2.5) may be regarded as a generalization of the Bianchiidentity. We are trying to construct the most general bi-scalar-tensor theory with second-order field equa-tions using the identity (2.5). Because both G ab and E I are assumed to be of second order in derivatives,the left-hand side of Eq. (2.5) would yield third derivatives in general while the right-hand side containsat most second derivatives. This indicates that the left-hand side of Eq. (2.5) must be free from thirdderivatives.The construction of the most general second-order equations of motion for bi-scalar-tensor theories isdivided into two parts: we first determine the most general second-order rank-2 tensor whose divergenceremains of second order. After that, we impose the identity (2.5) on the rank-2 tensor to constrain itsform. In this subsection we construct the most general second-order rank-2 tensor, e G ab , whose divergence is alsoof second order. The conditions that ∇ a e G ab has no third derivatives can be expressed as ∂ ∇ b e G ab ∂g cd,efg = 0 , (2.6) ∂ ∇ b e G ab ∂φ I,cde = 0 . (2.7)Using the chain rule, ∇ b e G ab is rewritten as ∇ b e G ab = e G ab ; cd,ef g cd,efb + e G ab ; cd,e g cd,eb + e G ab ; cd g cd,b + e G ab ; cdI φ I,cdb + e G ab ; cI φ I,cb + e G ab ; I φ I,b + e G bc Γ abc + e G ab Γ cbc , (2.8)where Γ abc are the Christoffel symbols. With the help of Eq. (2.8), we can show that the conditions (2.6)and (2.7) are equivalent to e G ab ; cd,ef + e G ae ; cd,fb + e G af ; cd,be = 0 , (2.9) e G ab ; cdI + e G ac ; dbI + e G ad ; bcI = 0 , (2.10)respectively. From the “invariance identity” (see Refs. [29, 30]), we have e G ab ; cd,ef + e G ab ; ce,fd + e G ab ; cf,de = 0 . (2.11)– 4 –y repeated use of Eqs. (2.9), (2.10), and (2.11), we obtain e G ab ; cd,ef = e G cd ; ab,ef = e G ef ; cd,ab , (2.12) e G ab ; cdI = e G cd ; abI . (2.13)For convenience, we now introduce the notion of property S following Ref. [9]. A quantity A a a ...a n − a n is said to have property S if it satisfies the following conditions: (i) it is symmetric in ( a ℓ − , a ℓ ) for ℓ = 1 , , . . . , n ; (ii) it is symmetric under the interchange of any two pairs ( a ℓ − , a ℓ ) and ( a m − , a m )for ℓ, m = 1 , , . . . , n ; (iii) it vanishes if any three of four indices, ( a ℓ − , a ℓ ) and ( a m − , a m ) for ℓ, m = 1 , , . . . , n , are symmetrized. It is shown in Corollary 2.1 of Ref. [9] that A a a ...a n − a n van-ishes if A a a ...a n − a n has property S and n > B a a ...a n +2 m +1 a n +2 m +2 defined by B a a ...a n +2 m +1 a n +2 m +2 ≡ Π ni =1 ∂∂g a i − a i ,a i +1 a i +2 ! Π mj =1 ∂∂φ I j ,a n +2 j +1 a n +2 j +2 ! e G a a . (2.14)This is a n -th derivative with respect to g a i a i +1 ,a i +2 a i +3 and m -th derivative with respect to φ I,a i a i +1 of e G ab .Using Eqs. (2.9)–(2.13), one can easily check that B a a ...a n +2 m +1 a n +2 m +2 has property S . Then, Corollary2 . B a a ...a n +2 m +1 a n +2 m +2 vanishes for 2 n + m ≥
4, leading to the following threesets of identities: ∂∂g cd,ef ∂∂g ij,kl e G ab = 0 , (2.15) ∂∂g cd,ef ∂∂φ I,ij ∂∂φ
J,kl e G ab = 0 , (2.16) ∂∂φ I,cd ∂∂φ
J,ef ∂∂φ
K,ij ∂∂φ
L,kl e G ab = 0 . (2.17)By integrating Eqs. (2.15)–(2.17), we can determine the form of the gravitational field equations.First, integrating Eq. (2.15) yields e G ab = e ξ abcdef g cd,ef + e ξ ab = ξ abcdef R cdef + ξ ab , (2.18)where ξ abcdef and ξ ab are functions of g ab , g ab,c , φ I , φ I,a , and φ I,ab . Note that we have used the identity˜ ξ abcdef g cd,ef = 23 ˜ ξ abcdef R ecdf + ¯ ξ ab (2.19)at the second equality of Eq. (2.18), where ¯ ξ ab are functions of g ab , g ab,c , φ I , φ I,a and φ I,ab . It can be seenthat ξ abcdef and ξ ab have property S . Substituting Eq. (2.18) into Eq. (2.16) and integrating it, we obtain e G ab = ξ abcdefghI R cdef φ I | gh + ξ abcdef R cdef + ξ ab , (2.20)where ξ abcdefgh and ξ abcdefghI are functions of g ab , g ab,c , φ I , and φ I,a , while ξ ab are functions of g ab , g ab,c , φ I , φ I,a and φ I,ab . Here again it can be seen that ξ abcdefgh , ξ abcdefghI and ξ ab have property S . Repeating thesame procedure and integrating Eq. (2.17) give e G ab = ξ abcdefghI R cdef φ I | gh + ξ abcdefghIJK φ I | cd φ J | ef φ K | gh + ξ abcdef R cdef + ξ abcdefIJ φ I | cd φ J | ef + ξ abcdI φ I | cd + ξ ab , (2.21)– 5 –here all of the above ξ tensors are composed of g ab , g ab,c , φ I , and φ I,a , and have property S . Althoughour final goal is to determine the most general equations of motion for the bi-scalar-tensor theory, theequations given up to this point hold irrespective of the number of the scalar fields.Our remaining task in this subsection is to construct explicitly all the possible ξ tensors that haveproperty S and are composed of g ab , g ab,c , φ I , and φ I,a . For this purpose, we can use φ I | a , g ab , and thetotally antisymmetric tensor ε abcd as building blocks, from which the ξ tensors are built by taking theirproducts and linear combinations appropriately. There is no elegant way, and what we will do is toexhaust all the possible combinations of those building blocks yielding the ξ tensors. Let us begin withthe simplest one, ξ ab . It is not difficult to find that the following one is the most general symmetric rank-2tensor composed of φ I , φ I | a , g ab , and g ab,c : ξ ab = a ( φ I , X JK ) g ab + b IJ ( φ I , X JK ) φ I | a φ J | b , (2.22)where a ( φ I , X JK ) and b IJ ( φ I , X JK ) are arbitrary functions of φ I and X JK , and b IJ has the sym-metric property b IJ = b JI . Here, we have used, for the first time in this derivation, the assump-tion that the number of the scalar fields is two, which greatly simplifies the expression of ξ tensorsand the following procedure. Without this restriction we would have for example the term such as c IJKL (cid:16) ε acde φ I | c φ J | d φ K | e φ L | b + ε bcde φ I | c φ J | d φ K | e φ L | a (cid:17) in ξ ab , where c IJKL is arbitrary functions of φ I and X IJ ,and totally anti-symmetric in I, J and K . In a similar manner, we work out ξ abcdI : ξ abcdI = a I (cid:16) g ac g bd + g ad g bc − g ab g cd (cid:17) + b IJK h g ac φ J | b φ K | d + g ad φ J | b φ K | c + g bd φ J | a φ K | c + g bc φ J | a φ K | d − (cid:16) g ab φ J | c φ K | d + g cd φ J | a φ K | b (cid:17)i + c IJKLM (cid:16) φ J | a φ K | c φ L | b φ M | d + φ J | a φ K | d φ L | b φ M | c − φ J | a φ K | b φ L | c φ M | d (cid:17) + d IJKLM (cid:16) φ J | a φ K | c ε bdef φ L | e φ M | f + φ J | a φ K | d ε bcef φ L | e φ M | f (cid:17) , (2.23)where a I , b IJK , c IJKLM , and d IJKLM are arbitrary functions of φ I and X JK satisfying b IJK = b IKJ ,c IJKLM = c IKJLM = c IJKML = c ILMJK ,d IJKLM = − d IKJLM = − d IJKML = d ILMJK . (2.24)The explicit forms of ξ abcdef , ξ abcdefIJ , ξ abcdefgh , and ξ abcdefghI for the bi-scalar case are given in Appendix A.Substituting all the ξ tensors into Eq. (2.21) and rearranging the equation, we arrive at the most generalsecond-order rank-2 tensor e G ab whose divergence is also of second order, e G ab = Aδ ab + B IJ φ I | a φ J | b + C I δ acbd φ I | d | c + D IJK δ acebdf φ I | c φ J | d φ K | f | e + E IJKLM δ acegbdfh φ I | c φ J | d φ K | e φ L | f φ M | h | g + F IJKLM δ acegbdfh (cid:16) ε cepq φ I | p φ J | q φ K | d φ L | f + φ I | c φ J | e ε dfpq φ K | p φ L | q (cid:17) φ M | h | g + G IJ δ acebdf φ I | d | c φ J | f | e + H IJKL δ acegbdfh φ I | c φ J | d φ K | f | e φ L | h | g + Iδ acebdf R dfce + J IJ δ acegbdfh φ I | c φ J | d R fheg + K I δ acegbdfh φ I | d | c R fheg + L IJK δ acegbdfh φ I | d | c φ J | f | e φ K | h | g , (2.25) One might consider other rank-2 tensors such as ε abcd φ I | c φ J | d , but this tensor is excluded because it is not symmetric in a, b . – 6 –here A, B IJ , C I , D IJK , E
IJKLM , F
IJKLM , G IJ , H IJKL , I, J IJ , K I , and L IJK are arbitrary functions of φ I and X IJ , and they are subject to B IJ = B JI , D IJK = D JIK , E
IJKLM = − E KJILM = − E ILKJM = E JILKM ,G IJ = G JI , H IJKL = H JIKL = H IJLK , F
IJKLM = − F JIKLM = − F IJLKM = F JILKM ,J IJ = J JI , L IJK = L JIK = L IKJ . (2.26) In the previous subsection we have obtained the most general second-order rank-2 tensor whose divergenceremains of second order, e G ab . As can be seen, e G ab involves many arbitrary functions. It turns out, however,that those functions are not completely independent in order for the equations of motion to be compatiblewith general covariance. In this subsection, we impose the identity (2.5) that arises due to generalcovariance, i.e., we require that the divergence of e G ab is written as a product of φ I | a and some scalarfunction. This procedure will reduce the number of the arbitrary functions. A straightforward calculationshows that ∇ b e G ab = Q I φ I | a + α IJ δ acebdf φ I | d | c φ J | l R bfel + β IJ δ acebdf φ I | l φ J | lb R dfce + γ IJKL δ acebdf φ K | l φ L | lb φ I | c φ J | m R dfem + ǫ IJK δ aceldfhm φ K | d | c φ I | f | e φ J | g R hmgl + µ I δ acbd R bdcl φ I | l + ν IJKL δ acebdf φ K | l φ L | lb φ I | d | c φ J | f | e + ω IJK δ acbd φ J | l φ K | lb φ I | d | c + ξ IJ φ I | l φ J | a | l + ζ I [ JK ] δ acbd φ I | c φ J | l φ K | m R bdlm + ι IJK δ acegbdfh φ J | l φ K | lb φ I | d | c R fheg + 2 η I [ J | K | L ] δ acebfh φ I | c φ J | g φ K | f | e φ L | l R bhgl + ( λ IJKLM − λ ILMJK ) δ acbd φ L | e φ M | eb φ I | c φ J | f φ K | d | f + σ IJKLMN δ acgbfh φ M | l φ N | lb φ I | c φ J | e φ K | f | e φ L | h | g + 32 τ IJKLM δ acegbdfh φ L | l φ M | b | l φ I | d | c φ J | f | e φ K | h | g + 2 F IJKLM (cid:16) ε bdfh φ I | a φ J | g φ K | d φ L | f + ε aceg φ I | c φ J | e φ K | b φ L | h (cid:17) φ M | l R bhgl + 4 ε bdfh (cid:16) F IJKLM φ I | a φ J | g (cid:17) | b φ K | d φ L | f φ M | h | g + 4 ε aceg (cid:16) F IJKLM φ I | c φ J | e φ K | b φ L | h (cid:17) | b φ M | h | g , (2.27)where the coefficients are functions of φ I and X JK and defined as α IJ = G IJ − J IJ + 2 K I,J − H KLIJ X KL ,β IJ = − I ,IJ + J IJ − K J,I + 2 J KL,IJ X KL ,γ IJKL = − J IJ,KL + H IKJL ,ǫ IJK = K ( I,K ) J − L KIJ ,µ I = 12 C I + 2 I ,I − (cid:0) D JKI + 8 J J [ K,I ] (cid:1) X JK + 4 E JKLMI X JK X LM ,ν IJKL = − G IJ,KL + 3 H K ( IJL ) + 2 H MNIJ,KL X MN − L LIJ,K ,ω IJK = − C I,JK + 2 D J ( IK ) − G IK,J + 2 (cid:0) D LMI,JK − H [ J | LIK, | M ] (cid:1) X LM − E ( I | JLM | K ) X LM − E LMNOI,JK X LM X NO ,ξ IJ = − A ,IJ + B IJ − C J,I − D K [ I | J, | L ] X KL − E KLMNJ,I X KL X MN + 16 E KIMNJ,L X KL X MN ,ζ IJK = − D IJK − J IJ,K + 4 E LMIJK X LM , – 7 – IJKL = 12 H IJKL ,λ IJKLM = 12 D IJK,LM + H IJKM,L − E MLIJK − E NOIJK,LM X NO ,σ IJKLMN = H IJKL,MN − H IMNL,JK ,τ IJKLM = − L [ I | JK,L | M ] ,ι IJK = − K [ I,K ] J . (2.28)We present the explicit form of Q I in Appendix B, though it is irrelevant to the following derivations.In order for the right-hand side of Eq. (2.27) to be proportional to φ I | a , all the terms that are notparallel to φ I | a must vanish identically. We now derive the conditions for this following the procedureof Ref. [9]. Let us first focus on the ǫ IJK and ι IJK terms in Eq. (2.27), which are proportional to φ I | a | b φ J | c | d R egfh : ∇ b e G ab ⊃ ǫ IJK δ aceldfhk φ K | d | c φ I | f | e φ J | g R hkgl + ι IJK δ acegbdfh φ J | l φ K | lb φ I | d | c R fheg + · · · . (2.29)The coefficient of φ A,mn φ B,op g qr,st in this quantity can be extracted by taking a derivative with respectto φ A,mn φ B,op g qr,st as (cid:16) ∇ b e G ab (cid:17) ; mnA ; opB ; qr,st = (cid:16) ǫ IJK δ aceldfhk φ K | d | c φ I | e | f φ J | g R hkgl + ι IJK δ acegbdfh φ J | l φ K | lb φ I | d | c R fheg (cid:17) ; mnA ; opB ; qr,st = 2 ǫ ( A | J | B ) δ aceldfhk δ ( mc g n ) d δ ( oe g p ) f (cid:16) φ J | ( q g r ) k δ ( sl g t ) h + g h ( q δ r ) l φ J | ( s g t ) k (cid:17) + 2 ι IJK δ acegbdfh (cid:16) δ KA δ IB φ J | ( m g n ) b δ ( oc g p ) d + δ IA δ KB φ J | ( o g p ) b δ ( mc g n ) d (cid:17) δ ( qe g r ) h δ ( sg g t ) f . (2.30)Equation (2.30) must vanish when contracted with a vector Y a such that Y a φ I | a = 0 because Eq. (2.5)implies that Y a (cid:16) ∇ b e G ab (cid:17) ; mnA ; opB ; qr,st = Y a (cid:16) E I φ I | a (cid:17) ; mnA ; opB ; qr,st = 12 Y a φ I | a E I ; mnA ; opB ; qr,st = 0 . (2.31)Thus, we obtain a constraint equation given by2 ǫ ( A | J | B ) Y a δ aceldfhk δ ( mc g n ) d δ ( oe g p ) f (cid:16) φ J | ( q g r ) k δ ( sl g t ) h + g h ( q δ r ) l φ J | ( s g t ) k (cid:17) + 2 ι IJK Y a δ acegbdfh (cid:16) δ KA δ IB φ J | ( m g n ) b δ ( oc g p ) d + δ IA δ KB φ J | ( o g p ) b δ ( mc g n ) d (cid:17) δ ( qe g r ) h δ ( sg g t ) f = 0 . (2.32)This constraint has eight free indices m, n, o, p, q, r, s, t , and its any component must be fulfilled. We firsttake the trace of Eq. (2.32) by contracting with g mn g op g qr , giving − ǫ ( A | J | B ) φ J ( s Y t ) + 8 ι ( A | J | B ) φ J ( s Y t ) = − ǫ ( A | J | B ) φ J ( s Y t ) = 0 , (2.33) To obtain this result, we use the fact that derivatives of φ I,ab φ J,cd and R abcd are given by ∂ (cid:0) φ I | ab φ J | cd (cid:1) ∂φ A,mn ∂φ B,op = δ IA δ m ( a δ nb ) δ JB δ o ( c δ pd ) + δ JA δ m ( c δ nd ) δ IB δ o ( a δ pb ) = 12 (cid:16) δ IA δ JB D mnopacbd + δ JA δ IB D mnopcadb (cid:17) ,∂R abcd ∂g qr,st = 14 (cid:0) D qrstabcd + D qrstcdab − D qrstabdc − D qrstbacd (cid:1) , where D ijklabcd ≡ δ i ( a δ jd ) δ k ( b δ lc ) . – 8 –here we have used ι ( A | J | B ) = 0 which follows from the definition (2.28). This equation must be satisfiedfor any φ J | a , and therefore it is necessary to impose ǫ ( A | J | B ) (= ǫ AJB ) = 0. Further constraints can bederived from Eq. (2.32) as follows. Let us project Eq. (2.32) to the basis vectors Y a , ˜ Y a , φ I | a ( I = 1 , Y a Y a = ˜ Y a ˜ Y a = 1 , Y a ˜ Y a = 0 , Y a φ I | a = ˜ Y a φ I | a = 0 for I = 1 , . (2.34)By contracting Eq. (2.32) with Z m W n V o φ C | p φ D | q φ E | r φ F | s φ G | t , where Z a , W a , and V a are either Y a or ˜ Y a , wefind0 = − ǫ ( A | J | B ) δ acbd Y a V b (cid:0) Z c W d + W c Z d (cid:1) (cid:16) X ( J | ( D X E )( F X G ) | C ) − X J ( D X E ) C X F G − X DE X C ( F X G ) J (cid:17) + 4 ι AJB δ acbd Y a V b (cid:0) Z c W d + W c Z d (cid:1) X JC (cid:16) X ( D | ( F X G ) | E ) − X DE X F G (cid:17) . (2.35)Using ǫ ( A | J | B ) = 0 obtained in the previous step, we find ι AJB = 0 as another constraint to be imposed.Repeating a similar procedure for any other products of the second derivative terms in Eq. (2.27), wefind the following constraint equations for the coefficient functions: α IJ = − β JI , γ IJKL = − η I [ J | L | K ] , α AI h δ I ( C X − D ) B − δ IB X − CD i = 4 (cid:0) η ( CD ) AB − η ( C | BA | D ) (cid:1) , (2.36) ǫ IJK = ι IJK = ω IJK = λ ( IJ ) KLM − λ ( I | LM | J ) K = µ I = ζ I [ JK ] = ξ IJ = τ IJKLM = F IJKLM = 0 , (2.37) ν ACDB X − EF − ν AB ( E | C X − | F ) D − σ ( EF ) CADB = 0 , ν B [ A | K | C ] = σ EF [ CA ] DB = 0 , (2.38)where X − IJ is the inverse matrix of X IJ .The constraints (2.36) and (2.37) impose the following conditions on the functions appearing inEq. (2.25): B IJ = − F + 2 W ) ,I,J + A ,IJ + 2 D ( I | K | J ) ,L X KL − E K ( I | MN | J ) ,L X KL X MN − (cid:0) J K ( I,J ) ,L − J KL,I,J (cid:1) X KL , (2.39) C I = − F + 2 W ) ,I + 2 (cid:0) D JKI + 8 J J [ K,I ] (cid:1) X JK − E JKLMI X JK X LM , (2.40) F IJKLM = 0 , (2.41) G IJ = 2 J IJ − K ( I,J ) + 4 J K ( I,J ) L X KL , (2.42) H IJKL = 2 J IJ,KL , (2.43) K [ I,J ] = − J K [ I,J ] L X KL , (2.44) K I,JK = K J,IK , (2.45) L IJK = 23 K ( I,JK ) , (2.46) I = 12 F + W , (2.47)where W = W ( φ I ), and F = F ( φ I , X JK ) is a function satisfying F ,IJ = G IJ , which is integrated to give F = Z G IJ dX IJ = Z (cid:0) J IJ − K I,J + 4 J KI,JL X KL (cid:1) dX IJ . (2.48)– 9 –he conditions ζ I [ JK ] = ω IJK = λ ( IJ ) KLM − λ ( I | LM | J ) K = 0 in Eq. (2.37) imply D I [ JK ] = − J I [ J,K ] + 8 E LMI [ JK ] X LM , (2.49) D I ( JK ) = 12 C J,IK + G JK,I + (cid:0) − D LMJ,IK + 4 H [ I | LJK, | M ] (cid:1) X LM + 8 E ( J | ILM | K ) X LM + 4 E LMNOJ,IK X LM X NO , (2.50)0 = 12 D IJK,LM − D ( I | LM, | J ) K + H IJKM,L − H ( I | LMK, | J ) − (cid:0) E ML ( IJ ) K − E K ( IJ ) LM (cid:1) − (cid:0) E NO ( IJ ) K,LM − E NO ( I | LM, | J ) K (cid:1) X NO , (2.51)and Eq. (2.38) implies G I [ J,K ] L = 0 , (2.52) H IJK [ L,M ] N = 0 , (2.53) G ( IJ,KL ) = 3 H L ( IJK ) + 2 H LM ( IJ,KN ) X MN − K ( I,JK ) ,L . (2.54)Equation (2.52) is nothing but the integrability condition which guarantees F ,IJ,KL = F ,KL,IJ , and hencethe integral (2.48) indeed exists. Now we are at the final stage of deriving the most general second-order field equations of the bi-scalar-tensor theory. Substituting Eqs. (2.39)–(2.47) into Eq. (2.25), we at last obtain G ab = Aδ ab + (cid:2) − F ,I − W ,I + 2 (cid:0) D JKI + 8 J J [ K,I ] (cid:1) X JK − E JKLMI X JK X LM (cid:3) δ acbd φ I | d | c + (cid:0) − F ,I,J − W ,I,J + A ,IJ + 2 D IKJ,L X KL − E KIMNJ,L X KL X MN − J K [ I,L ] ,J X KL (cid:1) φ ( I | a φ J ) | b + D IJK δ acebdf φ I | c φ J | d φ K | f | e + E IJKLM δ acegbdfh φ I | c φ J | d φ K | e φ L | f φ M | h | g + (cid:18) F + W (cid:19) δ acebdf R dfce + F ,IJ δ acebdf φ I | d | c φ J | f | e + J IJ δ acegbdfh φ I | c φ J | d R fheg + 2 J IJ,KL δ acegbdfh φ I | c φ J | d φ K | f | e φ L | h | g + K I δ acegbdfh φ I | d | c R fheg + 23 K I,JK δ acegbdfh φ I | d | c φ J | f | e φ K | h | g . (2.55)This is the main result of this paper. The most general field equations for the single-scalar case [9] arereproduced as should be if one restricts the number of the scalar fields in Eq. (2.55) to one. Note thatone can eliminate W ( φ I ) from the above equation by redefining F → ˆ F ( φ I , X JK ) = F + 2 W . We cansee that Eqs. (2.50)–(2.54) do not reduce the number of the arbitrary functions because these are therelations between derivatives of the functions. In other words, these do not affect the structure of thefield equations (2.55). As we will comment in the final section, these may, however, help us to check theintegrability conditions for the field equations.From the relation (2.5), the scalar-field equations of motion are found to be E I = 2 Q I + δ ceglbdhm (cid:18) − γ JIKL φ K | b φ J | c φ L | d | e R hmgl + 23 σ JIKLMN φ J | c φ M | b φ K | d | e φ L | h | g φ N | m | l (cid:19) . (2.56)– 10 – Comparison with the generalized multi-Galileon theory
The covariant version of the multi-Galileon theory in the flat spacetime [21] was conjectured to be themost general multi-scalar-tensor theory with second-order field equations. However, later it was pointedout that this theory is not the most general one [23]. A counter-example is given by the multi-field DBIGalileons [25]. In this section, we compare the most general second-order field equations obtained in theprevious section with the field equations of the generalized multi-Galileons, and identify the terms thatare missing in the latter theory.The action of the generalized multi-Galileons is given by [21]1 √− g L = G − G I φ I | a | a + G R + G ,IJ (cid:16) φ I | a | a φ J | b | b − φ I | ab φ J | ab (cid:17) + G I G ab φ I | ab − G I,JK (cid:16) φ I | a | a φ J | b | b φ K | c | c − φ I | a | a φ J | c | b φ K | b | c + 2 φ I | b | a φ J | c | b φ K | a | c (cid:17) , (3.1)where G , G I , G and G I are arbitrary functions of φ I and X IJ , and G ab is the Einstein tensor. Thefunctions G I ,JK , G ,IJ,KL , G I,JK and G I,JK,LM are totally symmetric with respect to all of their indices,
I, J, K, L and M in order for the field equations to be of second order. A straightforward calculation leadsto the field equations for the generalized multi-Galileons, E ab ( L ) = (cid:18) − G + G I,J ) X IJ − G ,I,J X IJ (cid:19) g ab + (cid:18) − G ,IJ + G I,J ) − G ,I,J (cid:19) φ I | a φ J | b + (cid:0) − X JK G IJK + G ,I + 2 X JK G IJ,K (cid:1) g l ( a δ b ) cld φ I | d | c + (cid:18) − G IJK + 2 G K ( I,J ) − G K,I,J (cid:19) g l ( a δ b ) celdf φ I | c φ J | d φ K | f | e − (cid:0) G − G IJ X IJ + G I,J ) X IJ (cid:1) g l ( a δ b ) celdf R dfce + (cid:18) G IJ + X KL G IJKL − G I,J ) − X KL G IJK,L (cid:19) g l ( a δ b ) celdf φ I | d | c φ J | f | e + 14 (cid:0) G IJ − G I,J ) (cid:1) g l ( a δ b ) cegldfh φ I | c φ J | d R fheg + 12 (cid:0) G IJKL − G KL ( I,J ) (cid:1) g l ( a δ b ) cegldfh φ I | c φ J | d φ K | f | e φ L | h | g − X JK G IJK g l ( a δ b ) cegldfh φ I | d | c R fheg − (cid:0) G IJK + X LM G IJKLM (cid:1) g l ( a δ b ) cegldfh φ I | d | c φ J | f | e φ K | h | g . (3.2)Comparing Eq. (3.2) with Eq. (2.55), it is easy to see the exact correspondence between each term. It isalso found that terms corresponding to E IJKLM are lacking in the generalized multi-Galileon; this is acompletely new term. We would, however, emphasize that, even setting E IJKLM = 0, Eq. (2.55) coversa wider class of theories than the generalized multi-Galileons. This fact is to be illustrated in a concreteexample presented below. Note in passing that the coefficient functions of the above equation satisfy allthe constraints (2.36)–(2.38) found in the previous section.The double-dual Riemann term deduced from the multi-field DBI Galileons, L = √− g δ IJ δ KL δ acegbdfh φ I | a φ J | b φ K | c φ L | d R fheg , (3.3)– 11 –s not included in the Lagrangian of the generalized multi-Galileon theory [23]. One can however checkthat this term is actually contained in our theory. It is straightforward to derive the field equations fromEq. (3.3): E ab ( L ) = 4 g l ( a δ b ) cegldfh X I [ I φ J ] | c φ J | d R fheg + 8 g l ( a δ b ) cegldfh δ I [ J δ K ] L φ I | c φ J | d φ K | f | e φ L | h | g . (3.4)This is reproduced by setting J IJ = 2 ( δ IJ δ KL − δ IK δ JL ) X KL in our field equations. Having thus determined the most general second-order field equations of the bi-scalar-tensor theory, letus now explore the Lagrangian that gives the field equations we have derived. For the construction of theLagrangian, we employ the same strategy as taken in Ref. [9]. In the single scalar-field case, Horndeskifound that the form of the Lagrangian can be guessed from the trace of the gravitational field equation.In the same way as in the single-field theory, we take the trace of the field equations of the bi-scalar-tensortheory (2.55) and arrive at the terms of the following form as a candidate Lagrangian: L = √− g M (1) I φ I | c | c , (4.1) L = √− g (cid:16) M (2) δ cedf R dfce + 2 M (2) ,IJ δ cedf φ I | d | c φ J | f | e (cid:17) , (4.2) L = √− g M (3) IJK δ cedf φ I | c φ J | d φ K | f | e , (4.3) L = √− g (cid:18) M (4) I δ cegdfh φ I | d | c R fheg + 23 M (4) I,JK δ cegdfh φ I | d | c φ J | f | e φ K | h | g (cid:19) , (4.4) L = √− g (cid:16) M (5) IJ δ cegdfh φ I | c φ J | d R fheg + 2 M (5) IJ,KL δ cegdfh φ I | c φ J | d φ K | f | e φ L | h | g (cid:17) , (4.5) L = √− g M (6) , (4.6) L = √− g M (7) IJKLM δ cegdfh φ I | c φ J | d φ K | e φ L | f φ M | h | g , (4.7)where M (1) , M (2) , M (3) IJK , M (4) I , M (5) IJ , M (6) , and M (7) IJKLM are arbitrary functions of φ I and X IJ satisfying M (3) IJK = M (3) JIK , (4.8) M (5) IJ = M (5) JI , (4.9) M (7) IJKLM = − M (7) KJILM = − M (7) ILKJM = M (7) JILKM . (4.10)In order to maintain the second-order equations of motion for the scalar fields, we have to impose extraconditions on these functions. For example, the Euler-Lagrange equation of L for the scalar field φ I isgiven by E I ( L ) = M (1) J,KI φ J | cd | c φ K | d − M (1) I,JK φ J | dc | c φ K | d − M (1) J,KI,L X KL φ J | c | c + 2 M (1) I,JK,L X JK | c φ L | c + M (1) J,KI,LM X LM | d φ K | d φ J | c | c + M (1) I,JK,LM X JK | c X LM | c + 2 M (1)( I,J ) φ J | c | c − M (1) I,J,K X JK + M (1) J,KI φ J | c | c φ K | d | d − M (1) I,JK φ J | d | c φ K | c | d . (4.11)– 12 –he two terms in the first line of Eq. (4.11) are of third order, while the other terms are of second or firstorder. To eliminate the third-order derivatives, we therefore impose M (1)[ I,J ] K = 0 . (4.12)Performing the same analysis, we find that higher-derivative terms are removed by requiring that M (2) ,I [ J,K ] L = 0 , M (3) IJ [ K,L ] M = 0 , M (4)[ I,J ] K = 0 , M (4) I,J [ K,L ] M = 0 ,M (5) IJ,K [ L,M ] N = 0 , M (7) IJKL [ M,N ] O = 0 . (4.13)The Euler-Lagrange equations for the Lagrangian densities (4.1)–(4.7) are listed in Appendix C, fromwhich one can see the relations between the functions appearing in Eq. (2.55) and those in Eqs. (4.1)–(4.7): A = M (1) I,J X IJ + 4 M (2) ,I,J X IJ − M (3) IJK,L X IJ X KL − (cid:16) M (5) IJ,K,L − M (5) IL,J,K (cid:17) X IJ X KL + 12 M (6) + 8 M (7) IJKLM,N X IJ X KL X MN , (4.14) D IJK = − M (1)( I,J ) K − M (2) , ( I,J ) K + 32 M (3)( IJK ) + (cid:16) M (3) L ( IJ ) ,MK + M (3)( I | LM, | J ) K − M (3) IJL,KM (cid:17) X LM − M (4) K,I,J + 2 (cid:16) M (5) K ( I,J ) − M (5) IJ,K (cid:17) + 4 (cid:16) M (5) L ( I,J ) ,MK − M (5) IJ,L,MK (cid:17) X LM + 12 M (7) IJ ( KLM ) X LM + 8 M (7) IJNLM,OK X LM X NO , (4.15) E IJKLM = 18 (cid:16) δ P QIK δ RSJL + δ P QJL δ RSIK (cid:17) (cid:20) − M (3) P RQ,SM − M (5) P R,Q,SM + M (7) P RQSM + M (7) P RMSQ + M (7) P RQMS − (cid:16) M (7) P RQSN,OM − M (7) P RNSQ,OM − M (7) P NQSO,RM (cid:17) X NO (cid:21) , (4.16) F + 2 W = M (2) − M (2) ,IJ X IJ − M (4) I,J X IJ + 2 M (5) IJ X IJ + 4 M (5) IK,JL X IJ X KL , (4.17) J IJ = − M (2) ,IJ − M (4)( I,J ) + M (5) IJ + (cid:16) M (5) K ( I,J ) L − M (5) IJ,KL (cid:17) X KL , (4.18) K I = − M (4) J,KI X JK . (4.19)In addition, comparing the φ ( I | a φ J ) | b and δ acbd φ I | d | c terms, we see that the following two conditions must besatisfied: − F + 2 W ) ,I,J + A ,IJ + 2 D K ( IJ ) ,L X KL + 16 E KMN ( IJ ) ,L X KL X MN − (cid:0) J K ( I,J ) ,L − J KL,I,J (cid:1) X KL = M (1)( I,J ) + 2 M (2) ,I,J − (cid:16) M (3) IJK,L + 2 M (3) KL ( I,J ) − M (3) K ( IJ ) ,L (cid:17) X KL − (cid:16) M (5) IJ,KL + M (5) KL,IJ − M (5) IK,JL (cid:17) X KL + 12 M (6) ,IJ + 8 (cid:16) M (7) MNKL ( I,J ) − M (7) MN ( KI ) J,L + M (7) MN ( IJ ) K,L (cid:17) X KL X MN , (4.20) − F + 2 W ) ,K + 2 (cid:0) D IJK + 8 J I [ J,K ] (cid:1) X IJ − E IJLMK X IJ X LM = − M (1) I,JK X IJ − (cid:16) M (2) ,K + 2 M (2) I,JK X IJ (cid:17) + 3 M (3)( IJK ) X IJ + 2 M (3) IJM,LK X IL X JM . (4.21)– 13 –he Lagrangian is constructed by solving the above equations for M (1) , . . . , M (7) , and unfortunatelywe have not accomplished this step yet. In Appendix D, we review the construction of the Lagrangian forthe single-scalar case. The lesson from the single-field Lagrangian is that we should probably integrateEq. (4.19) first to identify M (4) compatible with Eq. (4.13). We have not yet succeeded even in solvingthese equations. In addition, it should be kept in mind that the terms we considered ( M (1) , . . . , M (7) )might not be enough to construct the true Lagrangian. Those terms are simply inferred from the trace ofthe equations of motion, and in general there is no guarantee that they are enough though they happenedto be so in the single scalar-field case. Albeit these difficulties, we believe that above calculations areuseful to build the Lagrangian for our theory. We hope to report on this final part of the construction ofthe most general second-order bi-scalar-tensor theory in the near future. In this paper, we have reported our attempt to construct the bi-scalar generalization of the Horndeskitheory in four-dimensional spacetime. Following Horndeski’s method, we have succeeded in derivingall the possible terms appearing in the most general second-order field equations for the bi-scalar tensortheory. We compared our field equations with those of the generalized multi-Galileon theory, and identifiedthe terms that are really not included in that theory. In particular, we confirmed that the double-dualRiemann term, which was shown to be missing in that theory [23], can be reproduced from our results bychoosing the arbitrary functions appropriately. We have also discussed the construction of the Lagrangianyielding the field equations we found. For the construction, we have taken Horndeski’s approach basedon the trace of the field equations as a candidate Lagrangian. From the Euler-Lagrange equations wehave obtained several differential equations for the functions in the Lagrangian, though we could not solvethem to give explicit forms of the functions. In fact, it is still unclear whether or not the candidates ofthe Lagrangian we proposed suffice to generate all the terms in our most general field equations.As discussed e.g. in Ref. [13], we have to impose the integrability conditions on the field equations inorder to ensure the existence of a corresponding Lagrangian. The integrability conditions are summarizedas δ √− g G µν ( x ) δg ρλ ( y ) − δ √− g G ρλ ( y ) δg µν ( x ) = 0 , (5.1) δ √− g G µν ( x ) δφ I ( y ) − δ √− g E I ( y ) δg µν ( x ) = 0 , (5.2) δ √− g E I ( x ) δφ J ( y ) − δ √− g E J ( y ) δφ I ( x ) = 0 , (5.3)where δ/δA denotes variation with respect to a field A . Surprisingly, Horndeski found the correspondingLagrangian in a rather heuristic way without using the integrability conditions explicitly. This impliesthat, in the single scalar case, those conditions are automatically satisfied and do not give rise to any extraconstraints on the functions in the field equations. It is unclear that this property persists in theories withmultiple scalar fields, and we have not been able to determine the corresponding Lagrangian completely.The integrability conditions might give us some clues to accomplish this procedure. We hope to reportthe result obtained from such an approach in the near future.– 14 – cknowledgments We would like to thank Xian Gao, Emir G¨umr¨uk¸c¨uo˘glu, Hideo Kodama, Shinji Mukohyama, VishaganSivanesan and Yi Wang for fruitful discussions and useful comments. S.O. was supported by JSPSGrant-in-Aid for Scientific Research No. 25-9997. N.T. was supported by the European Research Councilgrant no. ERC-2011-StG 279363-HiDGR. This work was supported in part by the JSPS Grant-in-Aid forScientific Research Nos. 24740161 (T.K.), 25287054 and 26610062 (M.Y.).
A The ξ tensors In this appendix, we show the expressions for the ξ tensors introduced in Sec. 2.2 to construct Eq. (2.25).Expressions of ξ ab and ξ abcdI are given by Eqs. (2.22) and (2.23), and we need to construct the other ξ tensors appearing in Eq. (2.21), i.e., ξ abcdef , ξ abcdefIJ , ξ abcdefgh , and ξ abcdefghI . As described in Sec. 2.2,the ξ tensors are constructed by taking the products and linear combinations of φ I | a , g ab and ε abcd thatenjoy property S , and we need to find all such ξ tensors to construct the most general second-order tensorwhose divergence remains of second order. In doing so, we find that not all of the ξ tensors give nontrivialcontributions to Eq. (2.21) because there is some degeneracy and the identical terms appear from morethan one ξ tensor. Those ξ tensors that give nontrivial contributions to Eq. (2.21) are summarized as ξ abcdef = ˆ a IJ (cid:16) ǫ I ace ǫ J bdf + ǫ I bce ǫ J adf + ǫ I ade ǫ J bcf + ǫ I bde ǫ J acf (cid:17) + ˆ b g gh (cid:16) ε aceg ε bdfh + ε bceg ε adfh + ε adeg ε bcfh + ε bdeg ε acfh + ε acfg ε bdeh + ε bcfg ε adeh + ε adfg ε bceh + ε bdfg ε aceh (cid:17) , (A.1) ξ abcdefIJ = ˆ a IJKL (cid:16) ǫ K ace ǫ Lbdf + ǫ K bce ǫ Ladf + ǫ K ade ǫ Lbcf + ǫ K bde ǫ Lacf (cid:17) + ˆ b IJ g gh (cid:16) ε aceg ε bdfh + ε bceg ε adfh + ε adeg ε bcfh + ε bdeg ε acfh + ε acfg ε bdeh + ε bcfg ε adeh + ε adfg ε bceh + ε bdfg ε aceh (cid:17) , (A.2) ξ abcdefgh = ˆ a (cid:16) ε aceg ε bdfh + ε bceg ε adfh + ε adeg ε bcfh + ε bdeg ε acfh + ε acfg ε bdeh + ε bcfg ε adeh + ε adfg ε bceh + ε bdfg ε aceh (cid:17) , (A.3)where ˆ a , ˆ a IJ , ˆ a IJKL , ˆ b and ˆ b IJ are arbitrary functions of φ M and X NO satisfying ˆ a IJ = ˆ a JI and ˆ a IJKL =ˆ a IJLK . Here we have defined ǫ I abc as ǫ I abc = ε abcd φ I | d . (A.4) B Explicit form of Q I The explicit form of Q I is given as follows: Q I ≡ Q ( A ) I + Q ( B ) I + Q ( C ) I + Q ( D ) I + Q ( E ) I + Q ( G ) I + Q ( H ) I + Q ( I ) I + Q ( J ) I + Q ( K ) I + Q ( L ) I , (B.1)with Q ( A ) I = A ,I , (B.2)– 15 – ( B ) I = − B IJ,K X JK − B IJ,KL φ K | c φ L | cb φ J | b + B IJ φ J | b | b , (B.3) Q ( C ) I = C J,I φ J | c | c , (B.4) Q ( D ) I = D JKL,I δ cedf φ J | c φ K | d φ L | e | f + 2 D IJK,L X JL φ K | c | c + D IJK,L φ J | c φ K | cd φ L | d + D IJK,LM δ cebf φ L | d φ M | db φ J | c φ K | e | f − D IJK δ cebf φ J | c | b φ K | e | f − D IJK δ cebf φ J | c φ K | l R bfel , (B.5) Q ( E ) I = E JKLMN,I δ cegdfh φ J | c φ K | d φ L | e φ M | f φ N | g | h − E LJKIM,N δ cegdfh φ L | c φ J | d φ K | e φ N | f φ M | g | h − E LJKIM,NO δ cegbfh φ N | l φ O | lb φ L | c φ K | e φ J | f φ M | g | h + 4 E LJKIM δ cegbfh φ L | c | b φ K | e φ J | f φ M | g | h + E LJKIM δ cegbfh φ L | c φ K | e φ J | f φ M | l R bhgl , (B.6) Q ( G ) I = G JK,I δ cedf φ J | c | d φ K | e | f , (B.7) Q ( H ) I = H JKLM,I δ cegdfh φ J | c φ K | d φ L | e | f φ M | g | h − H IJKL δ cegbfh φ J | c | b φ K | e | f φ L | g | h + 2 H IJKL,M X JM δ egfh φ K | e | f φ L | g | h + 2 H IJKL,M δ cgfh φ J | c φ M | e φ K | e | f φ L | g | h − H IJKL δ cegbfh φ J | c φ K | e | f φ L | l R bhgl + H IJKL,MN δ cegbfh φ M | l φ N | lb φ J | c φ K | e | f φ L | g | h , (B.8) Q ( I ) I = I ,I δ cedf R dfce , (B.9) Q ( J ) I = J JK,I δ cegdfh φ J | c φ K | d R fheg + 2 J IJ,K X JK δ egfh R fheg + 2 J IJ,K δ cgfh φ J | c φ K | e R fheg + J IJ,KL δ cegbfh φ K | d φ L | db φ J | c R fheg − J IJ δ cegbfh φ J | c | b R fheg , (B.10) Q ( K ) I = K J,I δ cegdfh φ J | c | d R fheg − K J,IK δ cegldfhm φ K | c | d φ J | e | f R hmgl − K I δ cegldfhm R dfce R hmgl , (B.11) Q ( L ) I = L JKL,I δ cegdfh φ J | c | d φ K | e | f φ L | g | h − L LJK,IM δ cegldfhm φ M | c | d φ L | e | f φ J | g | h φ K | l | m . (B.12) C Euler-Lagrange Equations
The explicit forms of the Euler-Lagrange equations from L – L are given as follows: E ab ( L ) = M (1) I,J (cid:16) φ I | ( a φ J | b ) + g ab X IJ (cid:17) − M (1) I,JK (cid:16) g l ( b δ a ) celdf φ I | c φ J | d φ K | e | f + 2 X IJ g l ( b δ a ) cld φ K | c | d (cid:17) , (C.1) E ab ( L ) = 2 M (2) ,I,J (cid:16) φ I | ( a φ J | b ) + 2 g ab X IJ (cid:17) − (cid:16) M (2) ,I + 2 M (2) ,K,IJ X JK (cid:17) g l ( a δ b ) cld φ I | c | d − M (2) ,IJ g l ( a δ b ) cegldfh φ I | c φ J | d R fheg + (cid:18) M (2) − M (2) ,IJ X IJ (cid:19) g l ( a δ b ) celdf R dfce − (cid:16) M (2) ,IJ + 2 M (2) ,IJ,KL X KL (cid:17) g l ( a δ b ) celdf φ I | c | d φ J | e | f − M (2) ,IJ,KL g l ( a δ b ) cegldfh φ I | c φ J | d φ K | e | f φ L | g | h − M (2) ,I,JK g l ( a δ b ) celdf φ I | c φ J | d φ K | e | f , (C.2) E ab ( L ) = − M (3) IJK,L X IJ X KL g ab − M (3) IJK,L (cid:16) X KL φ I | ( a φ J | b ) + 2 X IJ φ K | ( a φ L | b ) + 2 X L ( I φ J ) | ( a φ K | b ) (cid:17) + 32 M (3)( IJK ) (cid:16) g l ( a δ b ) celdf φ I | c φ J | d φ K | e | f + 2 X IJ g l ( a δ b ) cld φ K | c | d (cid:17) − M (3) IJK,LM g l ( a δ b ) cegldfh φ I | c φ J | d φ K | e φ L | f φ M | g | h + (cid:16) M (3) ILM,JK − M (3) I [ JM ] ,KL (cid:17) X LM g l ( a δ b ) cglfh φ I | c φ J | f φ K | g | h + 2 M (3) IJK,LM X IL X JK g l ( a δ b ) cld φ M | c | d , (C.3) E ab ( L ) = − M (4) I,JK X IJ g n ( a δ b ) eglnfhm φ K | e | f R hmgl − M (4) I,J X IJ g l ( a δ b ) eglfh R fheg − M (4) I,J g l ( a δ b ) cegldfh φ I | c φ J | d R fheg – 16 – (cid:16) M (4) I,JK,LM X IJ + M (4) K,LM (cid:17) g n ( a δ b ) eglnfhm φ K | e | f φ L | g | h φ M | l | m − M (4) I,J,K g l ( a δ b ) celdf φ J | c φ K | d φ I | e | f − (cid:16) M (4) I,J + M (4) K,IJ,L X KL (cid:17) g l ( a δ b ) celdf φ I | c | d φ J | e | f − M (4) I,K ( J,L ) g l ( a δ b ) cegldfh φ J | c φ L | d φ I | e | f φ K | g | h , (C.4) E ab ( L ) = (cid:16) M (5) IJ + 2 M (5) IK,JL X KL (cid:17) X IJ g l ( a δ b ) eglfh R fheg + h M (5) IJ + (cid:16) M (5) IK,JL − M (5) IJ,KL (cid:17) X KL i g l ( a δ b ) cegldfh φ I | c φ J | d R fheg + 2 h M (5) IJ + (cid:16) M (5) KL,IJ + 4 M (5) IK,JL (cid:17) X KL + 2 M (5) MN,KL,IJ X KM X LN i g l ( a δ b ) celdf φ I | c | d φ J | e | f + 2 h M (5) IK,JL + (cid:16) M (5) IM,JN,KL − M (5) IJ,KL,MN (cid:17) X MN i g l ( a δ b ) cegldfh φ I | c φ J | d φ K | e | f φ L | g | h + 2 h M (5) KI,J − M (5) IJ,K + 2 (cid:16) M (5) IL,J,MK − M (5) IJ,L,MK (cid:17) X LM i g l ( a δ b ) celdf φ I | c φ J | d φ K | e | f − M (5) I [ J,L ] ,K X IJ X KL g ab − (cid:16) M (5) IJ,K,L + M (5) KL,I,J − M (5) IK,J,L (cid:17) X KL φ I | ( a φ J | b ) − M (5) IJ,K,LM g l ( a δ b ) cegldfh φ I | c φ J | d φ K | e φ L | f φ M | g | h , (C.5) E ab ( L ) = 12 M (6) g ab + 12 M (6) ,IJ φ I | ( a φ J | b ) , (C.6) E ab ( L ) = (cid:16) M (7) IJKLM + M (7) IJMLK + M (7) IJKML (cid:17) g l ( a δ b ) cegldfh φ I | c φ J | d φ K | e φ L | f φ M | g | h + 4 (cid:16) M (7) IJKLM + M (7) KLMIJ + M (7) KLIMJ (cid:17) X IJ g l ( a δ b ) celdf φ K | c φ L | d φ M | e | f + (cid:16) − M (7) IJKLN,OM + 2 M (7) IJNLK,OM + 2 M (7) INKLO,JM (cid:17) X NO g l ( a δ b ) cegldfh φ I | c φ J | d φ K | e φ L | f φ M | g | h + 8 M (7) NLIJM,KO X LM X NO g l ( a δ b ) celdf φ I | c φ J | d φ K | e | f + 8 M (7) IJKLM,N X IJ X KL X MN g ab + 8 (cid:16) M (7) IJKLM,N − M (7) IJ ( NK ) M,L + M (7) IJMNK,L (cid:17) X IJ X KL φ M | ( a φ N | b ) . (C.7) D Construction of Lagrangian: single scalar-field case
In this appendix we briefly review the construction of the Lagrangian for the most general single scalar-tensor theory with second order equations of motion (see Ref. [9] for more detail). The most generalsecond order field equations in the single scalar-field case are given by G ab = Aδ ab + (cid:16) − F ′′ + ˙ A + 2 D ′ X (cid:17) φ | a φ | b + (cid:0) − F ′ + 2 DX (cid:1) δ acbd φ | d | c + Dδ acebdf φ | c φ | d φ | f | e + 12 F δ acebdf R dfce + ˙ F δ acebdf φ | d | c φ | f | e + J δ acegbdfh φ | c φ | d R fheg + 2 ˙ J δ acegbdfh φ | c φ | d φ | f | e φ | h | g + Kδ acegbdfh φ | d | c R fheg + 23 ˙ Kδ acegbdfh φ | d | c φ | f | e φ | h | g , (D.1)where φ is a scalar field and X = − ( ∂φ ) /
2. A prime “ ′ ” and a dot “ ˙ ” denote derivatives with respectto φ and X , respectively, and A, D, J , and K are arbitrary functions of φ and X , while F is related tothe other functions as F = Z (cid:16) J − K ′ + 4 ˙ J X (cid:17) dX + W , (D.2)where W is an arbitrary function of φ . As explained in the main text, candidates for the Lagrangian canbe guessed from the trace of the field equations as L = √− g M (1) φ | c | c , (D.3)– 17 – = √− g (cid:16) M (2) δ cedf R dfce + 2 ˙ M (2) δ cedf φ | d | c φ | f | e (cid:17) , (D.4) L = √− g M (3) δ cedf φ | c φ | d φ | f | e , (D.5) L = √− g (cid:18) M (4) δ cegdfh φ | d | c R fheg + 23 ˙ M (4) δ cegdfh φ | d | c φ | f | e φ | h | g (cid:19) , (D.6) L = √− g (cid:16) M (5) δ cegdfh φ | c φ | d R fheg + 2 ˙ M (5) δ cegdfh φ | c φ | d φ | f | e φ | h | g (cid:17) , (D.7) L = √− g M (6) . (D.8)By comparing the most general field equations (D.1) with the Euler-Lagrange equations obtained from(D.3)–(D.8), it can be seen that the free functions are related as A = M (1) ′ X + 4 M (2) ′′ X − M (3) ′ X + 12 M (6) , (D.9) D = −
12 ˙ M (1) − M (2) ′ + 32 M (3) + ˙ M (3) X − M (4) ′′ + 2 M (5) ′ + 4 ˙ M (5) ′ X, (D.10) F = M (2) − M (2) X − M (4) ′ X + 2 M (5) X + 4 ˙ M (5) X , (D.11) J = −
12 ˙ M (2) − M (4) ′ + M (5) + ˙ M (5) X, (D.12) K = − ˙ M (4) X, (D.13) − F ′′ + ˙ A + 2 D ′ X = M (1) ′ + 2 M (2) ′′ − M (3) ′ X + 12 ˙ M (6) , (D.14) − F ′ + 2 DX = − ˙ M (1) X − (cid:16) M (2) ′ + 2 ˙ M (2) X (cid:17) + 3 M (3) X + 2 ˙ M (3) X . (D.15)In order to identify the Lagrangian, we need to solve the above equations for M (1) – M (6) . First, byintegrating Eq. (D.13) we obtain M (4) = − Z KX dX. (D.16)Substituting Eq. (D.16) into Eqs. (D.11) and (D.12), we then find M (2) = − Z (cid:16) M (4) ′ − M (5) − X ˙ M (5) (cid:17) dX, (D.17) M (5) = − Z JX dX. (D.18)Finally, we can integrate Eqs. (D.9), (D.10), (D.14), and (D.15) to identify M (1) , M (3) , and M (6) withthe help of Eqs. (D.16), (D.17) and (D.18), leading to M (1) = − (cid:16) M (2) ′ − XM (3) (cid:17) , (D.19) M (3) = − Z DX dX, (D.20)and M (6) = 2 A + 4 XM (2) ′′ − M (3) ′ . (D.21)– 18 – eferences [1] A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without Singularity,” Phys. Lett. B ,99 (1980). K. Sato, “First Order Phase Transition of a Vacuum and Expansion of the Universe,” Mon. Not.Roy. Astron. Soc. , 467 (1981). A. H. Guth, “The Inflationary Universe: A Possible Solution to theHorizon and Flatness Problems,” Phys. Rev. D , 347 (1981).[2] C. L. Bennett et al. [WMAP Collaboration], “Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP)Observations: Final Maps and Results,” Astrophys. J. Suppl. , 20 (2013) [arXiv:1212.5225 [astro-ph.CO]].[3] G. Hinshaw et al. [WMAP Collaboration], “Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP)Observations: Cosmological Parameter Results,” Astrophys. J. Suppl. , 19 (2013) [arXiv:1212.5226[astro-ph.CO]].[4] P. A. R. Ade et al. [Planck Collaboration], “Planck 2013 results. I. Overview of products and scientificresults,” Astron. Astrophys. , A1 (2014) [arXiv:1303.5062 [astro-ph.CO]].[5] P. A. R. Ade et al. [Planck Collaboration], “Planck 2013 results. XXII. Constraints on inflation,” Astron.Astrophys. , A22 (2014) [arXiv:1303.5082 [astro-ph.CO]].[6] B. P. Schmidt et al. [Supernova Search Team Collaboration], Astrophys. J. , 46 (1998) [astro-ph/9805200].[7] A. G. Riess et al. [Supernova Search Team Collaboration], Astron. J. , 1009 (1998) [astro-ph/9805201].[8] S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. , 565 (1999)[astro-ph/9812133].[9] G. W. Horndeski, “Second-order scalar-tensor field equations in a four-dimensional space,” Int. J. Theor.Phys. , 363 (1974).[10] T. Kobayashi, M. Yamaguchi and J. Yokoyama, “Generalized G-inflation: Inflation with the most generalsecond-order field equations,” Prog. Theor. Phys. , 511 (2011) [arXiv:1105.5723 [hep-th]].[11] C. Deffayet, X. Gao, D. A. Steer and G. Zahariade, “From k-essence to generalised Galileons,” Phys. Rev. D , 064039 (2011) [arXiv:1103.3260 [hep-th]].[12] G. W. Horndeski, “Conservation of Charge and the Einstein-Maxwell Field Equations,” J. Math. Phys. ,1980 (1976).[13] C. Deffayet, A. E. G¨umr¨uk¸c¨uo˘glu, S. Mukohyama and Y. Wang, “A no-go theorem for generalized vectorGalileons on flat spacetime,” JHEP , 082 (2014) [arXiv:1312.6690 [hep-th]].[14] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, “Healthy theories beyond Horndeski,” arXiv:1404.6495[hep-th].[15] X. Gao, “Unifying framework for scalar-tensor theories of gravity,” Phys. Rev. D , 081501 (2014)[arXiv:1406.0822 [gr-qc]].[16] M. Zumalac´arregui and J. Garc´ıa-Bellido, “Transforming gravity: from derivative couplings to matter tosecond-order scalar-tensor theories beyond the Horndeski Lagrangian,” Phys. Rev. D , 064046 (2014)[arXiv:1308.4685 [gr-qc]].[17] A. Padilla, P. M. Saffin and S. Y. Zhou, “Bi-galileon theory I: Motivation and formulation,” JHEP , 031(2010) [arXiv:1007.5424 [hep-th]].[18] A. Padilla, P. M. Saffin and S. Y. Zhou, “Bi-galileon theory II: Phenomenology,” JHEP , 099 (2011)[arXiv:1008.3312 [hep-th]].[19] K. Hinterbichler, M. Trodden and D. Wesley, “Multi-field galileons and higher co-dimension branes,” Phys.Rev. D , 124018 (2010) [arXiv:1008.1305 [hep-th]]. – 19 –
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