A formal introduction to Horndeski and Galileon theories and their generalizations

Perhaps the simplest way in which to introduce Galileon models is through a question: for a scalar field π in flat spacetime, what is the most general theory which has field equations that are polynomial in second-order derivatives of π, do not contain undifferentiated or only once differentiated π, and do not contain derivatives of order strictly higher than 2? The theories which obey these properties are precisely the flat-spacetime Galileons first presented in the original [1] and later generalized to curved spacetimes [2, 3].

In fact these Galileon theories have a much longer history, having been discovered earlier in different contexts (see [4–7]). They are a subset of Horndeski theories [4], which describe all scalar–tensor theories with second-order field equations in curved four-dimensional spacetime. Flat-spacetime Galileons were also obtained later in a different way by Fairlie et al [5–7] (see also [8]). Here we do not follow these early references, but proceed in the spirit of [1–3, 9].

In D-dimensional flat spacetime, Galileons theories can be defined in several ways. In this section we focus on the simplest Galileon Lagrangian—the reader is referred to section 1.4 for other alternative expressions for the flat-spacetime Galileon Lagrangian.

Consider a Lagrangian of the form

$$\begin{equation} {\mathcal {L}}= {\mathcal {T}}_{(2n)}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n}\nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n} } \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \cdots \pi _{\mu _{\vphantom{()}n} \nu _{\vphantom{()}n}} \end{equation} \tag{ 1 }$$

where here, and in the following, successive derivatives of the scalar π are denoted by¹

$$\begin{equation} \pi _\mu \equiv \partial _\mu \pi , \qquad \pi _{\mu \nu } \equiv \partial _\nu \partial _\mu \pi , \qquad \pi _{\mu \nu \rho } \equiv \partial _\rho \partial _\nu \partial _\mu \pi \qquad {\rm etc}. \end{equation} \tag{ 2 }$$

The integer n counts the number of twice-differentiated π's appearing in the Lagrangian, and the tensor ${\mathcal {T}}_{(2n)}$ appearing in (1) has 2n-contravariant indices. It is a function of π and π_μ only

$$\begin{equation} {\mathcal {T}}_{(2n)} = {\mathcal {T}}_{(2n)}(\pi ,\pi _\mu ) , \end{equation} \tag{ 3 }$$

and is defined to be totally antisymmetric in its first n indices (i.e. in the μ_i, 1 ⩽ i ⩽ n) as well as separately in its last n indices (i.e. in the ν_i, 1 ⩽ i ⩽ n). Thus ${\mathcal {L}}= {\mathcal {L}}(\pi ,\pi _\mu ,\pi _{\mu \nu })$, and the corresponding field equations are $\mathcal{E}=0$ where

$$\begin{equation} {\mathcal {E}} \equiv \left[ \frac{\partial }{\partial \pi } - \partial _\mu \left(\frac{\partial }{\partial \pi _\mu } \right) + \partial _\mu \partial _\nu \left(\frac{\partial }{\partial \pi _{\mu \nu }} \right) \right]\mathcal{L} \end{equation} \tag{ 4 }$$

and the term in square brackets is the Euler–Lagrange operator. The original Galileon model as defined in [1] has Lagrangian $\mathcal{L}^{{\rm Gal},1}$ of the form (1) with

$$\begin{equation} {\mathcal {T}}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n} \nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n} }_{(2n),\rm {Gal},1} \equiv {\mathcal {A}}_{(2n+2)}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n+1}\nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n+1} } \pi _{\mu _{n+1}} \pi _{\nu _{n+1}} , \end{equation} \tag{ 5 }$$

where the 2m-contravariant tensor ${\mathcal {A}}_{(2m)}$ is defined by

$$\begin{equation} {\mathcal {A}}_{(2m)}^{\mu _{\vphantom{()}1} \mu _{\vphantom{()}2} \cdots \mu _{\vphantom{()}m} \nu _{\vphantom{()}1} \nu _{\vphantom{()}2} \cdots \nu _{\vphantom{()}m}} \equiv \frac{1}{(D-m)!} \varepsilon ^{\mu _{\vphantom{()}1} \mu _{\vphantom{()}2} \cdots \mu _{\vphantom{()}m} \sigma _{\vphantom{()}1}\sigma _{\vphantom{()}2}\cdots \sigma _{\vphantom{()}D-m}}_{\vphantom{\mu _{\vphantom{()}1}}} \varepsilon ^{\nu _{\vphantom{()}1} \nu _{\vphantom{()}2} \cdots \nu _{\vphantom{()}m}}_{\hphantom{\nu _{\vphantom{()}1} \nu _{\vphantom{()}2} \ldots \nu _{\vphantom{()}2m}}\sigma _{\vphantom{()}1} \sigma _{\vphantom{()}2}\cdots \sigma _{\vphantom{()}D-m}} . \end{equation} \tag{ 6 }$$

Here the totally antisymmetric Levi-Civita tensor is given by

$$\begin{equation} \varepsilon ^{\mu _{\vphantom{()}1} \mu _{\vphantom{()}2} \cdots \mu _{\vphantom{()}D}} \equiv - \frac{1}{\sqrt{-g}} \delta ^{[\mu _{\vphantom{()}1}}_1 \delta ^{\mu _{\vphantom{()}2}}_2 \cdots \delta ^{\mu _{\vphantom{()}D}]}_D \end{equation} \tag{ 7 }$$

with square brackets denoting unnormalized permutations. (Notice that definitions (6) and (7) (only) are in fact also valid in arbitrary curved spacetimes with metric g_μν and D ⩾ m, and that the metric only enters those definitions via its determinant g and the contraction between σ indices.) Hence the first form of the Galileon Lagrangian we will consider is given by [3]

$$\begin{equation} \mathcal{L}^{\rm {Gal},1}_{N} = \big({\mathcal {A}}_{(2n+2)}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n+1}\nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n+1} }\pi _{\mu _{n+1}} \pi _{\nu _{n+1}} \big) \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \cdots \pi _{\mu _{\vphantom{()}n} \nu _{\vphantom{()}n}} \end{equation} \tag{ 8 }$$

where N indicates the number of times of π occurs;

$$\begin{equation*} N\equiv n+2 \; (\ge 2), \end{equation*}$$

and

$$\begin{equation} N\le D+1 \end{equation} \tag{ 9 }$$

in order for the Lagrangian to be non-zero in D-dimensions. As discussed in [1], the Lagrangian is invariant under the 'Galilean' symmetry π_μ → π_μ + b_μ, π → π + c (where b_μ and c are constants), which is a covariant generalization of the transformation π → π + b_μx^μ + c defined in Minkowski spacetime.

We end this subsection by noting that the Lagrangian $\mathcal{L}^{\rm {Gal},1}_{N}$ can be rewritten in the form

$$\begin{equation} \mathcal{L}^{\rm {Gal},1}_{N}= - \sum _{\sigma \in S_{n+1}} \epsilon (\sigma ) [ \pi _{\vphantom{\mu _1}}^{\mu _{\sigma (1)}} \pi _{\mu _1}] \big [\pi _{\hphantom{\mu _{\sigma (2)}} \mu _2}^{\mu _{\sigma (2)}} \pi _{\hphantom{\mu _{\sigma (3)}} \mu _3}^{\mu _{\sigma (3)}} \cdots \pi _{\hphantom{\mu _{\sigma (n+1)}} \mu _{n+1}}^{\mu _{\sigma (n+1)}}\big ], \end{equation} \tag{ 10 }$$

where σ denotes a permutation of signature (σ) of the permutation group S_{n + 1}. This is the original form presented in [1], and the equality of (8) and (10) can be shown using the identity

$$\begin{equation} \sum _{\sigma \in S_D} \epsilon (\sigma ) g^{\mu _{\sigma (1)}\nu _{\vphantom{()}1}} g^{\mu _{\sigma (2)}\nu _{\vphantom{()}2}} \cdots g^{\mu _{\sigma (D)}\nu _{\vphantom{()}D}} = - \varepsilon ^{\mu _{\vphantom{()}1} \mu _{\vphantom{()}2} \cdots \mu _{\vphantom{()}D}} \varepsilon ^{\nu _{\vphantom{()}1} \nu _{\vphantom{()}2} \cdots \nu _{\vphantom{()}D}}. \end{equation} \tag{ 11 }$$

At first sight, it is not obvious that the field equations obtained from the Lagrangian (8) (or equivalently (1)) are second order—indeed from (4), one might expect up to fourth- order derivatives in the equations of motion. That this is not the case is due to the fact that ${\mathcal {T}}_{(2n)}$ (or ${\mathcal {A}}_{(2n)}$) is totally antisymmetric in its first n indices as well as in its last n indices.

To see this, initially consider the term $\partial _{\mu } \partial _{\nu } ( {\partial \mathcal{L}}/{\partial \pi _{\mu \nu }} )$ in the field equation (4) arising from the variation of a twice-differentiated π. For simplicity we work with the form of the Lagrangian given in (1). On focusing on the 'dangerous terms', namely those containing derivatives of the fields of order 3 or more and indicated by a ∼ below, one finds (in the expressions below we only aim to indicate the form of the terms, and are not rigorous with the index structure):

$$\begin{eqnarray} &&\fl \partial _{\mu } \partial _{\nu } \left( \frac{\partial \mathcal{L}}{\partial \pi _{\mu \nu }} \right) \sim \partial _{\mu } \partial _{\nu } ( {\mathcal {T}}_{(2n)} \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \cdots \pi _{\mu _{\vphantom{()}k} \nu _{\vphantom{()}k}} \cdots ) \nonumber \\ &&\sim \partial _\mu \bigg[ \left(\frac{\partial {\mathcal {T}}_{(2n)}}{\partial \pi _\alpha } \pi _{\alpha \nu } + \frac{\partial {\mathcal {T}}_{(2n)}}{\partial \pi } \pi _{\nu }\right) \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \cdots \pi _{\mu _{\vphantom{()}k} \nu _{\vphantom{()}k}} \cdots \nonumber \\ &&+ \,{\mathcal {T}}_{(2n)} \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \cdots \pi _{\mu _{\vphantom{()}k} \nu _{\vphantom{()}k} \nu _{\vphantom{()}}} \cdots \bigg] + {{\rm etc}} \nonumber \\ &&\sim \frac{\partial {\mathcal {T}}_{(2n)}}{\partial \pi _\alpha } \pi _{\alpha \nu \mu } \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \cdots \pi _{\mu _{\vphantom{()}k} \nu _{\vphantom{()}k}} \cdots + \frac{\partial {\mathcal {T}}_{(2n)}}{\partial \pi _\alpha } \pi _{\alpha \nu } \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \cdots \pi _{\mu _{\vphantom{()}k} \nu _{\vphantom{()}k} \mu } \cdots \nonumber \\ &&+\, \frac{\partial {\mathcal {T}}_{(2n)}}{\partial \pi } \pi _{\nu } \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \cdots \pi _{\mu _{\vphantom{()}k} \nu _{\vphantom{()}k} \mu } \cdots \nonumber \\ &&+\, {\mathcal {T}}_{(2n)} \cdots \pi _{\mu _{\vphantom{()}l} \nu _{\vphantom{()}l} \mu _{\vphantom{()}}}\pi _{\mu _{\vphantom{()}k} \nu _{\vphantom{()}k} \nu _{\vphantom{()}}}\cdots + {\mathcal {T}}_{(2n)} \cdots \pi _{\mu _{\vphantom{()}k} \nu _{\vphantom{()}k} \nu _{\vphantom{()}} \mu _{\vphantom{()}}} \cdots + {{\rm etc.}} \end{eqnarray} \tag{ 12 }$$

Since derivatives commute on flat space, and because of the afore-mentioned antisymmetry properties of the indices of ${\mathcal {T}}_{(2n)}$, the last four terms in (12) vanish. While the first does not, it is, however, straightforward to check that an identical contribution is generated by the second term in the Euler–Lagrange equation (4) (this occurs when the partial derivative ∂π_μ acts on ${\mathcal {T}}_{(2n)}$): since this appears with the opposite sign, these contributions cancel exactly! The second term in (4) also generates a term of the form $\pi _{\mu _k \nu _k \mu }$ which, on contraction with ${\mathcal {T}}_{(2n)}$, vanishes.

Hence a sufficient condition for the field equations derived from Lagrangian (1) to stay of order less or equal to 2 is that the tensor ${\mathcal {T}}_{(2n)}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n}\nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n} }$ is totally antisymmetric in its first n indices as well as separately in its last n indices (see the 'main lemma' of [9]). Furthermore, one easily concludes from the above and equation (5) that the field equations obtained from $\mathcal{L}^{\rm {Gal},1}_{N}$ read $\mathcal{E} = -N \times \mathcal{E}_{N} = 0$ where

$$\begin{eqnarray} \mathcal{E}_{N} &=& -{\mathcal {A}}_{(2n+2)}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n+1}\nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n+1} } \pi _{\mu _1\nu _1} \pi _{\mu _2\nu _2} \cdots \pi _{\nu _{n+1} \mu _{n+1}}, \nonumber \\ &=& \sum _{\sigma \in S_{n+1}} \epsilon (\sigma ) \prod _{i=1}^{n+1} \pi _{\hphantom{\mu _{\sigma (i)}} \mu _i}^{\mu _{\sigma (i)}}. \end{eqnarray} \tag{ 13 }$$

Notice that these equations of motion are of second order only, as advertised, and since they originate from $\mathcal{L}^{\rm {Gal},1}_{N}$ (which contains N factors of π), they contain N − 1 factors of π.

It is interesting to notice from (9) that the largest number of products of fields allowed in D-dimensions is N = D + 1. In that case, n + 1 = D so that it follows from the definition of ${\mathcal {A}}$ in (6) that $\mathcal{E}_{D+1}$ is proportional to the determinant of the matrix of second derivatives π_μν. Then the equation of motion $\mathcal{E}_{D+1} = 0$ is simply the Monge–Ampère equation which has various interesting properties, in particular in relation to integrability (see e.g. [10]). Also, when N = D + 1 the Lagrangian

$$\begin{eqnarray*} \mathcal{L}^{{\rm Gal},1}_{D+1} \propto \det \left(\begin{array}{@{}cc@{}}\pi _{\mu \nu } & \pi _{\nu } \\ \pi _{\mu } & 0\end{array}\right) , \end{eqnarray*}$$

which, when set equal to zero, is the Bateman equation [10, 11].

When D = 4 it follows from (9) that N, the number of times π occurs in the Lagrangian, can take four values, N ∈ {2, 3, 4, 5}. (Reference [1] also includes the tadpole π in the family of Galileon Lagrangians.) Thus there are only four possible non-trivial Galileons Lagrangians of the form (8) in four dimensions, and these are given respectively by²

$$\begin{eqnarray} \mathcal{L}^{\rm {Gal},1}_{2} &=&{\mathcal {A}}_{(2)}^{\mu _{\vphantom{()}1}\nu _{\vphantom{()}1} } \pi _{\mu _{1}} \pi _{\nu _{1}} \nonumber\\ &=& -\pi ^\mu \pi _\mu \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} \mathcal{L}^{\rm {Gal},1}_{3} &=& {\mathcal {A}}_{(4)}^{\mu _{\vphantom{()}1} \mu _{\vphantom{()}2}\nu _{\vphantom{()}1} \nu _{\vphantom{()}2} } \pi _{\mu _{2}} \pi _{\nu _{2}} \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}}\nonumber\\ & =& \pi ^\mu \pi ^\nu \pi _{\mu \nu } - \pi ^\mu \pi _\mu \Box \pi \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray} \nonumber \mathcal{L}^{\rm {Gal},1}_{4} &=& {\mathcal {A}}_{(6)}^{\mu _{\vphantom{()}1} \mu _{\vphantom{()}2} \mu _{\vphantom{()}3}\nu _{\vphantom{()}1}\nu _{\vphantom{()}2} \nu _{\vphantom{()}3} } \pi _{\mu _{3}} \pi _{\nu _{3}} \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \pi _{\mu _{\vphantom{()}2} \nu _{\vphantom{()}2}} \nonumber \\ &=& -(\Box \pi )^2 (\pi _{\mu } \pi ^{\mu }) + 2 (\Box \pi )(\pi _{\mu } \pi ^{\mu \nu } \pi _{\nu }) \nonumber \\ && + (\pi _{\mu \nu } \pi ^{\mu \nu }) (\pi _{\rho } \pi ^{\rho }) -2 (\pi _{\mu }\pi ^{\mu \nu } \pi _{\nu \rho } \pi ^{\rho }) \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} \mathcal{L}^{\rm {Gal},1}_{5} &=& {\mathcal {A}}_{(8)}^{\mu _{\vphantom{()}1}\mu _{\vphantom{()}2}\mu _{\vphantom{()}3}\mu _{\vphantom{()}4} \nu _{\vphantom{()}1}\nu _{\vphantom{()}2}\nu _{\vphantom{()}3}\nu _{\vphantom{()}4}}\pi _{\mu _{4}} \pi _{\nu _{4}} \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \pi _{\mu _{\vphantom{()}2} \nu _{\vphantom{()}2}} \pi _{\mu _{\vphantom{()}3} \nu _{\vphantom{()}3}} \nonumber \\ &=& -(\Box \pi )^3 (\pi _{\mu } \pi ^{\mu }) + 3 (\Box \pi )^2(\pi _{\mu } \pi ^{\mu \nu } \pi _{\nu }) \nonumber \\ && + 3 (\Box \pi ) (\pi _{\mu \nu } \pi ^{\mu \nu }) (\pi _{\rho } \pi ^{\rho }) \nonumber \\ && -6 (\Box \pi )(\pi _{\mu }\pi ^{\mu \nu } \pi _{\nu \rho } \pi ^{\rho }) -2 (\pi _{\mu }^{\hphantom{\mu }\nu } \pi _{\nu }^{\hphantom{\nu }\rho } \pi _{\rho }^{\hphantom{\rho }\mu }) (\pi _{\lambda } \pi ^{\lambda }) \nonumber \\ && -3 (\pi _{\mu \nu } \pi ^{\mu \nu }) (\pi _{\rho } \pi ^{\rho \lambda } \pi _{\lambda }) +6 (\pi _{\mu } \pi ^{\mu \nu } \pi _{\nu \rho } \pi ^{\rho \lambda } \pi _{\lambda }). \end{eqnarray} \tag{ 17 }$$

These Lagrangians (14)–(17) lead respectively to the field equations $\mathcal{E}_N=0$ given by equation (13), and which read

$$\begin{equation} \mathcal{E}_{2} = \Box \pi \end{equation} \tag{ 18 }$$

$$\begin{equation} \mathcal{E}_{3} = (\Box \pi )^2 - \pi _{\mu \nu } \pi ^{\mu \nu } \end{equation} \tag{ 19 }$$

$$\begin{equation} \mathcal{E}_{4} = (\Box \pi )^3 - 3 \Box \pi \pi _{\mu \nu } \pi ^{\mu \nu }+ 2 \pi ^{\mu }_{\hphantom{\mu } \nu } \pi ^{\nu }_{\hphantom{\nu } \rho } \pi ^{\rho }_{\hphantom{\rho } \mu } \end{equation} \tag{ 20 }$$

$$\begin{eqnarray} \mathcal{E}_{5} &=& (\Box \pi )^4 - 6 (\Box \pi )^2 \pi _{\mu \nu } \pi ^{\mu \nu }+ 3 (\pi _{\mu \nu } \pi ^{\mu \nu })^2 \nonumber \\ && + 8 (\Box \pi ) \pi ^{\mu }_{\hphantom{\mu } \nu } \pi ^{\nu }_{\hphantom{\nu } \rho } \pi ^{\rho }_{\hphantom{\rho } \mu } - 6 \pi ^{\mu }_{\hphantom{\mu } \nu } \pi ^{\nu }_{\hphantom{\nu } \rho } \pi ^{\rho }_{\hphantom{\rho } \sigma } \pi ^{\sigma }_{\hphantom{\sigma } \mu } . \end{eqnarray} \tag{ 21 }$$

It is interesting to note that the combination of terms appearing in the equations of motion (18)–(21) are in fact directly related to the elementary symmetric polynomials of the eigenvalue of the matrix $\pi ^\mu _{\;\;\nu }$ see e.g. [12]. (Notice that these also appear in the decoupling limit of massive gravity [13, 14].) In general, for an arbitrary n × n matrix $M^a_{\hphantom{a} b}$ (with a/b a line/column index belonging to {1, ..., n}), the symmetric polynomials e_k with k = 1, 2, ..., n are defined by

$$\begin{equation} e_k(M) = - \frac{1}{k!}{\mathcal {A}}_{(2k)}^{a_1\cdots a_k b_1\cdots b_k} M_{a_1 b_1} M_{a_2 b_2} \cdots M_{a_k b_k} \end{equation} \tag{ 22 }$$

so that, in particular,

$$\begin{equation} e_1 (M) = [M] \end{equation} \tag{ 23 }$$

$$\begin{equation} e_2 (M) = \case{1}{2} ([M]^2 - [M^2]) \end{equation} \tag{ 24 }$$

$$\begin{equation} e_3 (M) = \case{1}{6} ([M]^3 - 3 \,[M][M^2] + 2 \,[M^3]) \end{equation} \tag{ 25 }$$

$$\begin{equation} e_4 (M) = \case{1}{24} ([M]^4 - 6\, [M]^2 \, [M^2] +3\,[M^2]^2 + 8\, [M][M^3]- 6\, [M^4]) \end{equation} \tag{ 26 }$$

where $[M]=M^a_{\hphantom{a} a}$ denotes the trace of M. For an n × n matrix

$$\begin{equation} {\rm det} (M) = e_n(M) . \end{equation} \tag{ 27 }$$

On comparing (22) with (13), it follows that the equations of motion (18)–(21) can be simply rewritten as

$$\begin{equation} \mathcal{E}_{k+1} = k! e_k(\pi ^\mu _{\hphantom{\mu } \nu }). \end{equation} \tag{ 28 }$$

In order to make contact with different formulations of Galileon theories that can be found in the literature, it is useful to note that the Galileon Lagrangian (8) can be written in different, equivalent, ways all of which differ from (8) by an integration by parts. In this section we again work in an arbitrary number of dimensions D.

A first possible alternative Lagrangian for the Galileon with, again, N = n + 2 fields is given by

$$\begin{equation} \mathcal{L}^{\rm {Gal},2}_{N} =\big( {\mathcal {A}}_{(2n)}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n}\nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n} } \pi _{\mu _1} \pi _{\lambda } \pi ^{\lambda }_{\hphantom{\lambda } \nu _1} \big) \pi _{\mu _{\vphantom{()}2} \nu _{\vphantom{()}2}} \cdots \pi _{\mu _{\vphantom{()}n} \nu _{\vphantom{()}n}}, \end{equation} \tag{ 29 }$$

$$\begin{equation} \hphantom{\mathcal{L}^{\rm {Gal},2}_{N}}\equiv {\mathcal {T}}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n} \nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n} }_{(2n),\rm {Gal},2} \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \cdots \pi _{\mu _{\vphantom{()}n} \nu _{\vphantom{()}n}}, \end{equation} \tag{ 30 }$$

where

$$\begin{eqnarray} &&\fl {\mathcal {T}}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n} \nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n} }_{(2n),\rm {Gal},2} = \frac{1}{n} {\mathcal {A}}_{(2n)}^{\alpha _{\vphantom{()}1} \cdots \alpha _{\vphantom{()}n}\nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n} } \big [( \pi ^{\mu _1} \pi _{\alpha _1}) \delta ^{\mu _2}_{\; \; \alpha _2} \cdots \delta ^{\mu _n}_{\; \; \alpha _n} + \delta ^{\mu _1}_{\; \; \alpha _1} ( \pi ^{\mu _2} \pi _{\alpha _2} ) \delta ^{\mu _3}_{\; \; \alpha _3} \cdots \delta ^{\mu _n}_{\; \; \alpha _n} + \cdots \nonumber \\ &&+ \delta ^{\mu _1}_{\; \; \alpha _1} \cdots \delta ^{\mu _{n-1}}_{\; \; \alpha _{n-1}} ( \pi ^{\mu _n} \pi _{\alpha _n} ) \big ]. \end{eqnarray} \tag{ 31 }$$

Similarly, another integration by parts yields

$$\begin{equation} \mathcal{L}^{{\rm Gal},3}_{N} = \big( {\mathcal {A}}_{(2n)}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n}\nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n} } \pi _{\lambda } \pi ^{\lambda }\big) \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \cdots \pi _{\mu _{\vphantom{()}n} \nu _{\vphantom{()}n}} \end{equation} \tag{ 32 }$$

so that

$$\begin{equation} {\mathcal {T}}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n} \nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n} }_{(2n),{\rm Gal},3} = X {\mathcal {A}}_{(2n)}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n} \nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n} } \end{equation} \tag{ 33 }$$

where

$$\begin{equation} X\equiv \pi _\mu \pi ^\mu . \end{equation} \tag{ 34 }$$

The three Lagrangians (8), (29) and (32) are all equal up to a total derivative: from the properties of ${\mathcal {A}}_{(2n)}$, it follows that (see [9] for an explicit proof)

$$\begin{equation} {\mathcal {L}}_{N}^{\rm {Gal},1}=\frac{N}{2}{\mathcal {L}}_{N}^{\rm {Gal},3}-\frac{N-2}{2} \partial _\mu J^\mu _{N}, \end{equation} \tag{ 35 }$$

$$\begin{equation} {\mathcal {L}}_{N}^{\rm {Gal},1}=-N{\mathcal {L}}_{N}^{\rm {Gal},2}+\partial _\mu J^\mu _{N} , \end{equation} \tag{ 36 }$$

where the current $J^\mu _{N}$ is defined by

$$\begin{equation} J^\mu _{N}= X {{\mathcal {A}}}_{(2n)}^{\mu \mu _{2}\cdots \mu _{n}\nu _{1}\nu _{2}\cdots \nu _{n}}\pi _{\nu _{1}}\pi _{\mu _{2}\nu _{2}}\cdots \pi _{\mu _{n}\nu _{n}} . \end{equation} \tag{ 37 }$$

Consequently the equations of motion of all three Galileon Lagrangians are identical, given by (13), and strictly of second order. Finally, observe from (13) and (32) that $\mathcal{L}^{\rm {Gal},3}_{N}$ can be rewritten as

$$\begin{equation} \mathcal{L}^{\rm {Gal},3}_{N} = - X \mathcal{E}_{N-1} \end{equation} \tag{ 38 }$$

where $\mathcal{E}_{N-1}$ are the equations of motion coming from $\mathcal{L}^{\rm {Gal}}_{N-1}$ (where we drop the index 1, 2, 3). In this form, one sees directly that Galileon models containing a given number N of π fields can be obtained from the field equations of the same models with one less field. It is precisely this property which was used by Fairlie et al hierarchical construction of Galileons [5–8].

The property that flat-spacetime Galileons have field equations containing second-order derivatives only is lost in curved spacetime [2] (see also below). However, it is well known that increasing the order of the field equations lead to an increase in the number of propagating degrees of freedom. Hence, when trying to generalize Galileons, it is natural to try to determine the most general scalar theory with field equations containing derivatives of order 2 or less on flat spacetime. This was done in [9] where it was shown that the most general theory in D spacetime dimensions satisfying the three conditions:

• (1)  
  its Lagrangian contains derivatives of order 2 or less of the scalar field π;
• (2)  
  its Lagrangian is polynomial in the second derivatives of π;
• (3)  
  the corresponding field equations are of order 2 or lower in derivatives

is given by

$$\begin{equation} \mathcal{L} = \sum _{n=0}^{D-1} \tilde{\mathcal{L}}_{n}\lbrace f_n \rbrace . \end{equation} \tag{ 39 }$$

Here the f_n are arbitrary functions of π and X (notice that there are D of them);

$$\begin{eqnarray} \tilde{\mathcal{L}}_{n}\lbrace f\rbrace &\equiv & f(\pi ,X) \mathcal{L}^{\rm {Gal},3}_{N= n+ 2}, \nonumber\\ &=& f(\pi ,X) \big( X {\mathcal {A}}_{(2 n)}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n}\nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n} } \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \cdots \pi _{\mu _{\vphantom{()}n} \nu _{\vphantom{()}n}}\big), \end{eqnarray} \tag{ 40 }$$

and the braces indicate that $\tilde{\mathcal{L}}_{n}\lbrace f\rbrace$ is a functional of f. The equations of motion corresponding to each $\tilde{\mathcal{L}}_{n}\lbrace f\rbrace$ are³

$$\begin{eqnarray} &&\fl 0 = 2(f+Xf_{X})\mathcal {E}_{N}+4(2f_{X}+Xf_{XX})\mathcal {L}_{N+1}^{{\rm Gal},2}\nonumber +X[2Xf_{X\pi }-(n-1)f_{\pi }]\mathcal {E}_{N-1} \nonumber \\ &&-n(4Xf_{X\pi }+4f_{\pi })\mathcal {L}_{N}^{{\rm Gal},2} - nXf_{\pi \pi } \mathcal {L}_{N-1}^{{\rm Gal},1} \end{eqnarray} \tag{ 41 }$$

where N = n + 2. When the function f is constant, these equations reduce to $\mathcal {E}_{N}=0$ as in (13). For non-constant f, they depend on π^μν as well as π^μ through f (which also induces a dependence on π), X and the different Galileon Lagrangians which appear in (41). Clearly, therefore, the Lagrangian (39) is no longer invariant under the 'Galilean' symmetry discussed in section 1.1.

Another, different, way in which to generalize flat-spacetime Galileons is to consider theories with several fields (rather than a single scalar π) but imposing that the equations of motion are strictly second order.

One construction of such theories (having some degree of generality) consists in considering sets of p-forms $A^a_p$, where a denotes the type of species. Such forms have field strength $F^a_{p+1} = {\rm d} A^a_p$, such that the exterior derivative dF^a vanishes. Motivated by analysis of section 1.1 for a single-field π, one can consider actions given by the formal expression [15]

$$\begin{equation} \mathcal{L} = \varepsilon ^{\mu _1 \mu _2 \cdots } \varepsilon ^{\nu _1 \nu _2 \cdots \ldots }\, F^a_{\mu _1\mu _2\cdots } F^b_{\nu _1 \nu _2 \dots } \big(\partial _{\mu _k} F^c_{\nu _l \nu _{l+1} \dots }\,\dots \big) \big(\partial _{\nu _j} F^d_{\mu _m \mu _{m+1}\cdots }\,\cdots \big) , \end{equation} \tag{ 42 }$$

where the different species are labelled by (a, b, ...). The number of indices contracted with the first and second Levi-Civita tensors ε must be the same and not greater than D, but the two terms in brackets may now involve different species and therefore a different number of terms. The Bianchi identities (i.e. [d, d] = 0) ensure that only ∂F appears in the field equations, which therefore remain of order 2 in derivatives.

Various examples of this kind are given in [15]. Consider for instance only zero-forms, say two scalar fields π and φ with 'field strengths' given respectively by π_μ ≡ ∂_μπ and φ_μ ≡ ∂_μφ. On using (42) one obtains the following Lagrangian:

$$\begin{eqnarray*} \mathcal{L}(\pi ,\varphi ) &=& {\mathcal {A}}_{(2n+2)}^{\mu _{\vphantom{()}1} \cdots \mu _{\vphantom{()}n+1}\nu _{\vphantom{()}1} \cdots \nu _{\vphantom{()}n+1} } \pi _{\mu _{n+1}} \varphi _{\nu _{n+1}} \pi _{\mu _{\vphantom{()}1} \nu _{\vphantom{()}1}} \varphi _{\mu _{\vphantom{()}2} \nu _{\vphantom{()}2}} \pi _{\mu _{\vphantom{()}3} \nu _{\vphantom{()}3}} \cdots \varphi _{\mu _{\vphantom{()}n-1} \nu _{\vphantom{()}n-1}} \pi _{\mu _{\vphantom{()}n} \nu _{\vphantom{()}n}}, \end{eqnarray*}$$

which has strictly second-order field equations. Similar models with several scalar fields, generally known as 'multi-Galileons', have been constructed in different ways in the literature, for instance by imposing certain internal symmetries (see e.g. [16, 17]), or by using brane-world constructions relying in particular on the analogy between the above described theories and Lovelock actions (see [15]). These latter constructions can then also be used in the curved spacetime extensions of Galileons which we will outline below [18–25].

In the case of a single p-form, we can simply drop the indices a, b etc in (42). But then, it is possible to show that the resulting action always leads to vanishing field equations whenever p is odd! In fact it is not currently known if the equivalent of a Galileon model exists for just a single odd p-form [15]. However, a non-trivial theory for several odd or even (possibly single) p-forms can be obtained easily—once again we follow [15] and quote two simple, mixed 0- and 1-form theories. The first Lagrangian, which is linear in the scalar field and quadratic in the gauge field, is defined for any D ⩾ 3:

$$\begin{eqnarray*} &&\mathcal{L} = \varepsilon ^{\mu \nu \rho } \varepsilon ^{\alpha \beta \gamma } F_{\mu \nu } F_{\alpha \beta } \partial _\rho \partial _\gamma \pi = 4 F^{\mu \rho } F^\nu _{\hphantom{\nu }\rho } \pi _{\mu \nu } -2 F^2 \Box \pi . \end{eqnarray*}$$

Both its π and A_λ field equations are obviously of pure second order, and given by

$$\begin{eqnarray*} &&(F_{\mu \nu ,\rho })^2 - 2 (F^{\mu \nu }_{\hphantom{\mu \nu },\nu })^2 = 0, \\ &&F^{\lambda \mu ,\nu } \pi _{\mu \nu } + F^{\mu \nu }_{\hphantom{\mu \nu },\nu } \pi ^{\lambda }_{\hphantom{\lambda }\mu } - F^{\lambda \mu }_{\hphantom{\lambda \mu },\mu }\, \Box \pi = 0. \end{eqnarray*}$$

Similarly, in D ⩾ 4, the mixed model

$$\begin{eqnarray*} \mathcal{L} &=& \varepsilon ^{\mu \nu \rho \sigma } \varepsilon ^{\alpha \beta \gamma \delta }\, \partial _\mu \pi \partial _\alpha \pi \, \partial _\nu F_{\beta \gamma }\, \partial _\delta F_{\rho \sigma } \\ &=&- 8 (\pi _{\mu } F^{\rho \mu ,\nu } F_{\rho \sigma }^{\hphantom{\rho \sigma },\sigma } \pi _{\nu } ) + 4 (\pi ^{\mu } F_{\mu \nu ,\rho })^2 + 2 (\pi ^{\mu } F_{\nu \rho ,\mu })^2 \\ &&-4 (\pi _{\mu } F^{\mu \nu }_{\hphantom{\mu \nu },\nu })^2 - 2 (\pi _{\mu })^2 (F_{\nu \rho ,\sigma })^2 + 4 (\pi _{\mu })^2 (F^{\nu \rho }_{\hphantom{\nu \rho },\rho })^2 \end{eqnarray*}$$

which is quadratic in both the scalar field and gauge field also yields pure second-order π and A_λ field equations:

$$\begin{eqnarray*} &&\fl 4 (\pi _{\mu \nu } F^{\rho \mu ,\nu } F_{\rho \sigma }^{\hphantom{\mu \sigma },\sigma }) - 2 (F^{\mu \rho ,\sigma } \pi _{\mu \nu } F^\nu _{\hphantom{\nu }\rho ,\sigma }) + 2 (F^{\mu \rho }_{\hphantom{\mu \rho },\rho } \pi _{\mu \nu } F^{\nu \sigma }_{\hphantom{\nu \sigma }, \sigma }) \\ &&-\, (F_{\rho \sigma ,\mu } \pi ^{\mu \nu } F^{\rho \sigma }_{\hphantom{\rho \sigma },\nu }) + (\Box \pi ) (F_{\mu \nu ,\rho })^2 - 2 (\Box \pi ) (F^{\mu \nu }_{\hphantom{\mu \nu },\nu })^2 = 0, \\ &&\fl 2 (\pi _{\mu \rho } F^{\lambda \mu }_{\hphantom{\lambda \mu },\nu } \pi ^{\nu \rho }) + 2 (\pi ^{\lambda \mu }F_{\mu \nu ,\rho } \pi ^{\nu \rho }) + 2 (\pi ^{\lambda \rho } \pi _{\rho \mu } F^{\mu \nu }_{\hphantom{\mu \nu },\nu }) - (\pi _{\mu \nu })^2 (F^{\lambda \rho }_{\hphantom{\lambda \rho },\rho }) \\ &&- \,2 (\Box \pi ) (\pi _{\mu \nu } F^{\lambda \mu ,\nu }) - 2 (\Box \pi ) (\pi ^{\lambda }_{\hphantom{\lambda }\mu } F^{\mu \nu }_{\hphantom{\mu \nu },\nu }) + (\Box \pi )^2 (F^{\lambda \mu }_{\hphantom{\lambda \mu },\mu })= 0. \end{eqnarray*}$$

Again, similar theories can be obtained following different routes, see for instance [24, 26].

These models can also be generalized to non-Abelian gauge bosons $A^a_\mu$ and their field strengths F = dA + A∧A [15]. Indeed, if we denote by $\mathcal{D}$ the gauge covariant derivative, then the Bianchi identities $\mathcal{D}^{\vphantom{a}}_{[\mu }F^a_{\nu \rho ]} = 0$ still hold. Hence Lagrangians of the form $\mathcal{L}= \varepsilon ^{\mu \nu \cdots } \varepsilon ^{\alpha \beta \cdots }\, F^a_{\mu \nu } F^b_{\alpha \beta } (\mathcal{D}_\rho F^c_{\gamma \delta }\, \cdots ) (\mathcal{D}_\epsilon F^d_{\sigma \tau }\, \cdots )$ define nonlinear extensions of Yang–Mills theory, while keeping field equations of second (and lower) order (see also [26, 27]). In this case, indeed, the invariance of the field equations under constant shifts, $A^a_{\mu } \rightarrow A^a_{\mu } + c^a_{\mu }$ and $F^a_{\mu \nu } \rightarrow F^a_{\mu \nu } + k^a_{[\mu \nu ]}$ is lost (just because of the form of the field strength and covariant derivative). This feature is also shared by the generic models introduced in section 2.1, where the original 'Galilean' symmetry is lost, as well as with generalized models also introduced in [15] where undifferentiated F also occur in the action.

Since the expressions we will deal with quickly become complicated, we begin with a specific example. Consider the Lagrangian $\mathcal{L}^{\rm {Gal},1}_4$ of equation (16) in D = 4 dimensions. Written on an arbitrary spacetime (that is, on replacing all partial derivatives by covariant derivatives), the action becomes

$$\begin{equation} S_4=\int {\rm d}^4 x \sqrt{-g}\, \mathcal{L}^{\rm {Gal},1}_4 \end{equation} \tag{ 43 }$$

which, on varying with respect to π, gives the equations of motion $\mathcal{E}_4=0$ (where, as previously, an irrelevant numerical factor has been removed), with

$$\begin{eqnarray} \mathcal{E}_4&\equiv & -\case{1}{2}(\pi _{\mu }\,\pi ^{\mu })(\pi _{\nu \hphantom{\nu }\rho }^{\hphantom{\nu }\nu \hphantom{\rho }\rho } - \pi _{\nu \rho }^{\hphantom{\nu \rho }\nu \rho } ) -\case{1}{2}\, \pi ^{\mu }\,\pi ^{\nu } (2\, \pi _{\mu \rho \nu }^{\hphantom{\mu \rho \nu }\rho } - \pi _{\mu \nu \rho }^{\hphantom{\mu \nu \rho }\rho } - \pi _{\rho \hphantom{\rho }\mu \nu }^{\hphantom{\rho }\rho }) \nonumber \\ &&-\case{5}{2} (\Box \pi ) \pi ^{\mu }(\pi _{\mu \nu }^{\hphantom{\mu \nu }\nu } - \pi _{\nu \hphantom{\nu }\mu }^{\hphantom{\nu }\nu }) -3 \, \pi _{\mu }\,\pi ^{\mu \nu }(\pi _{\rho \hphantom{\rho }\nu }^{\hphantom{\rho }\rho }- \pi _{\nu \rho }^{\hphantom{\nu \rho }\rho }) \nonumber \\ &&- 2\, \pi ^{\mu }\, \pi ^{\nu \rho } ( \pi _{\nu \rho \mu } - \pi _{\mu \nu \rho } )\nonumber \\ && +(\Box \pi )^3 +2 (\pi _{\mu }^{\hphantom{\mu }\nu }\,\pi _{\nu }^{\hphantom{\nu }\rho }\,\pi _{\rho }^{\hphantom{\rho }\mu }) -3 (\Box \pi ) (\pi _{\mu \nu }\, \pi ^{\mu \nu } ). \end{eqnarray} \tag{ 44 }$$

The first two terms contain fourth-order derivatives, the following three terms contain third-order derivatives and the last three terms contain second-order derivatives. Of course the fourth- and third-order derivatives disappear on a flat spacetime. Indeed, commuting the derivatives, $\mathcal{E}_4$ can be rewritten as

$$\begin{eqnarray} \mathcal{E}_4&=& (\Box \pi )^3 +2 (\pi _{\mu }^{\hphantom{\mu }\nu }\,\pi _{\nu }^{\hphantom{\nu }\rho }\,\pi _{\rho }^{\hphantom{\rho }\mu }) -3 (\Box \pi ) (\pi _{\mu \nu }\pi ^{\mu \nu } ) +\case{1}{4}(\pi _{\mu }\,\pi ^{\mu } )(\pi _{\nu } \,R^{;\nu }) \nonumber \\ &&-\case{1}{2}(\pi _{\mu }\,\pi _{\nu }\,\pi _{\rho } R^{\mu \nu ;\rho }) -\case{5}{2} (\Box \pi )(\pi _{\mu }\,R^{\mu \nu }\,\pi _{\nu }) +2 (\pi _{\mu }\,\pi ^{\mu \nu }\,R_{\nu \rho }\,\pi ^{\rho }) \nonumber \\ && + \case{1}{2} (\pi _{\mu }\,\pi ^{\mu }) (\pi _{\nu \rho }\,R^{\nu \rho } ) +2 (\pi _{\mu }\,\pi _{\nu }\,\pi _{\rho \sigma }\,R^{\mu \rho \nu \sigma }). \end{eqnarray} \tag{ 45 }$$

However, one is left over with derivatives of the Ricci tensor and scalar and hence with third-order derivative of the metric.

Similarly the stress–energy tensor $T^{\mu \nu }_4$ (defined in a usual way as $T^{\mu \nu }_4 \equiv (-g)^{-1/2} \delta S_4/\delta g_{\mu \nu }$, and given explicitly in [9]) contains third-order derivatives of π. In fact, these third-order derivatives are still there even if flat spacetime g_μν = η_μν were a solution of Einstein's equations. As a result, one sees that once the metric is dynamical, new degrees of freedom will propagate even on a Minkowski background. Finally, the energy–momentum tensor satisfies the relation

$$\begin{equation} \nabla _\mu T_{4}^{\mu \nu } =2\,\pi ^{\nu }\,\mathcal{E}_4 \end{equation} \tag{ 46 }$$

meaning, in particular, that (on flat spacetime) the third derivatives present in the expression of $T_{4}^{\mu \nu }$ are killed by the application of an extra covariant derivative. From the above discussion, it is clear that a naive covariantization leads to higher order derivatives in the equations of motion.

It turns out that a non-minimal coupling of π to the metric can simultaneously remove all higher derivatives from the field equations of π as well as from the energy–momentum tensor [2]. Indeed, adding to S₄ the action

$$\begin{equation} S^{\rm nonmin}_4 \equiv - \int {\rm d}^4 x \sqrt{-g} (\pi _{\lambda }\,\pi ^{\lambda }) (\pi _{\mu }\,G^{\mu \nu }\,\pi _{\nu }), \end{equation} \tag{ 47 }$$

where G^μν is the Einstein tensor, we obtain the equations of motion for π in the form $\mathcal{E}^{\prime }_4 = 0$, where $\mathcal{E}^{\prime }_4$ is given by

$$\begin{eqnarray} \mathcal{E}^{\prime }_4 &=& (\Box \pi )^3 +2 (\pi _{\mu }^{\hphantom{\mu }\nu }\,\pi _{\nu }^{\hphantom{\nu }\rho }\,\pi _{\rho }^{\hphantom{\rho }\mu }) -3 (\Box \pi ) (\pi _{\mu \nu }\pi ^{\mu \nu } ) -\case{1}{2} (\Box \pi ) (\pi _{\mu }\,\pi ^{\mu }) R\nonumber \\ &&- (\pi _{\mu }\,\pi ^{\mu \nu }\,\pi _{\nu }) R -2(\Box \pi ) (\pi _{\mu } \,R^{\mu \nu }\,\pi _{\nu }) +(\pi _{\lambda }\,\pi ^{\lambda }) (\pi _{\mu \nu }\,R^{\mu \nu })\nonumber \\ &&+4 (\pi _{\mu }\,\pi ^{\mu \nu }\,R_{\nu \rho }\,\pi ^{\rho }) +2 (\pi _{\mu }\,\pi _{\nu }\,\pi _{\rho \sigma }\,R^{\mu \rho \nu \sigma }). \end{eqnarray} \tag{ 48 }$$

This equation does not contain derivatives of order higher than 2, and it obviously reduces to the original form, equation (20), in flat spacetime. However, it involves first-order derivatives of π in curved spacetime meaning that Galileon symmetry is broken. Note also the complex mixing of the field degrees of freedom implied by the presence of second derivatives of both π and g_μν in this equation (this has been dubbed 'kinetic gravity braiding' [28], and can have important effects in a cosmological context). Finally, one can show that the action $S^{\rm nonmin}_4$ is the unique one which eliminates higher derivatives from both the π field equations as well as $T_{4}^{\mu \nu }$ [2, 4].

To summarize, 'covariantization' proceeds as follows: first start with the flat- space Lagrangian and replace partial derivatives by covariant derivatives; then determine the correct 'counterterm(s)' which remove all higher order derivatives (greater than or equal to 3) in the equations of motion.

The results of the previous subsection can be generalized from D = 4 to arbitrary D, and also to the other Galilean Lagrangians (with different n), see equation (8).

The relevant counterterm is now [3]

$$\begin{equation} \mathcal{L}^{{\rm Gal},1}_{(n+1,p)} = {\mathcal {A}}_{(2 n)}^{\mu _{\vphantom{()}1} \ldots \mu _{\vphantom{()}n}\nu _{\vphantom{()}1} \ldots \nu _{\vphantom{()}n} } \mathcal {R}_{(p)} X^p \pi _{\mu _{2p+1}} \pi _{\nu _{2p+1}} \mathcal {S}_{(q)}, \end{equation} \tag{ 49 }$$

where $\mathcal {R}_{(p)}$ and $\mathcal {S}_{(q)}$ are defined by

$$\begin{equation} {\mathcal {R}}_{(p)} \equiv \prod _{i=1}^{p}R_{\mu _{2i-1}\mu _{2i}\nu _{2i-1}\nu _{2i}}, \end{equation} \tag{ 50 }$$

$$\begin{equation} {\mathcal {S}}_{(q)} \equiv \prod _{i=0}^{q-1}\pi _{\mu _{n-i}\nu _{n-i}}, \end{equation} \tag{ 51 }$$

with q = n − 1 − 2p, and we use the convention that ${\mathcal {S}}_{1}=\pi _{\mu _n \nu _n}$ and ${\mathcal {S}}_{q\le 0} = 1$. Notice that when p = 0, there are q = n − 1 terms in second derivatives of π, so that $\mathcal{L}^{{\rm Gal},1}_{(n+1,0)}\equiv \mathcal{L}^{\rm {Gal},1}_{N=n+1}$ given in (8). For arbitrary p, $\mathcal{L}^{{\rm Gal},1}_{(n+1,p)}$ is obtained from $\mathcal{L}^{{\rm Gal},1}_{(n+1,0)}$ by replacing p pairs of twice-differentiated π by a product of Riemann tensors multiplied by X (with suitable indices). Then [3] the action

$$\begin{equation} S = \int {\rm d}^D x \sqrt{-g} \sum _{p=0}^{\lfloor \frac{n-1}{2} \rfloor } \mathcal {C}_{(n+1,p)} \mathcal{L}_{(n+1,p)}, \end{equation} \tag{ 52 }$$

where the coefficients $\mathcal {C}_{(n+1,p)}$ are given by

$$\begin{equation} \mathcal {C}_{(n+1,p)} = \left(-\frac{1}{8}\right)^p \frac{(n-1)!}{(n-1-2p)! (p!)^2} \end{equation} \tag{ 53 }$$

and $\lfloor \frac{n-1}{2} \rfloor$ is the integer part of (n − 1)/2, remarkably leads to field equations both for π and the metric with no more than second derivatives.

A covariantization similar to the one given above also exists for the generalized Galileon of section 2.1. Once again, it is not sufficient to simply take the Lagrangian $\tilde{\mathcal{L}}_{n}\lbrace f\rbrace$ of equation (40) and replace all partial derivatives by covariant derivatives: to cancel 'dangerous' terms in the equation of motion, one must also add the correct finite series of counterterms.

These have been determined in [9], and for a given n (the number of twice-differentiated π's appearing in the Lagrangian), the general term in this series is

$$\begin{equation} \tilde{\mathcal {L}}_{n,p}\lbrace f\rbrace =\mathcal {A}_{(2n)}^{\mu _{1}\cdots \mu _{n}\nu _{1}\cdots \nu _{n}}\mathcal {P}_{(p)} {\mathcal {R}}_{(p)} {\mathcal {S}}_{(q\equiv n-2p)} \end{equation} \tag{ 54 }$$

where we use the convention $\tilde{\mathcal {L}}_{n,0}\lbrace f\rbrace =\tilde{\mathcal {L}}_{n}\lbrace f\rbrace$, the functions $\mathcal {R}_{(p)}$ and $\mathcal {S}_{(q)}$ were defined in equations (50) and (51), and⁴

$$\begin{equation*} \mathcal {P}_{(p)} \equiv \int _{X_{0}}^{X} {\rm d}X_1 \int _{X_{0}}^{X_{1}} {\rm d}X_2 \cdots \int _{X_{0}}^{X_{p-1}} {\rm d}X_p \; f_n(\pi ,X_p)X_p. \end{equation*}$$

In D-dimensions, the suitable combination of $\tilde{\mathcal {L}}_{n,p}\lbrace f\rbrace$ eliminating all higher order derivative is found to be given by [9]

$$\begin{equation} \tilde{\mathcal {L}}^{\rm {cov}}_{n}\lbrace f\rbrace =\sum _{p=0}^{\lfloor \frac{n}{2}\rfloor }\tilde{\mathcal {C}}_{n,p}\tilde{\mathcal {L}}_{n,p}\lbrace f\rbrace , \end{equation} \tag{ 55 }$$

where the coefficients $\tilde{\mathcal {C}}_{n,p}$ are given by⁵

$$\begin{equation} \tilde{\mathcal {C}}_{n,p}=\left(-\frac{1}{8}\right)^{p}\frac{n!}{(n-2p)!p!} = \frac{1}{p!} \mathcal {C}_{(n+1,p)}. \end{equation} \tag{ 56 }$$

To summarize, the full covariantized 'generalized Galileon' Lagrangian in D-dimensions is

$$\begin{equation} \mathcal{L} = \sum _{n=0}^{D-1} \tilde{\mathcal{L}}^{\rm cov}_{n}\lbrace f_n \rbrace \end{equation} \tag{ 57 }$$

which depends on the D functions f_n(π, X), as well as $\sum _{n=0}^{D-1} \lfloor {n/2} \rfloor$ functions of π (the integration 'constants'). However these latter 'constants' can be re-absorbed by a redefinition of the functional coefficients f_n meaning that the Lagrangian depends on D arbitrary functions only.

We end this subsection by comparing the Lagrangian (57) with that constructed by Horndeski in [4]. There the author explicitly constructed the most general scalar–tensor theory in four dimensions which has field equations of second (and lower) order both for the scalar field and the metric. Using the notation of the present paper, the Horndeski Lagrangian reads

$$\begin{equation} \mathcal{L}_{H} = -\mathcal{A}^{\mu _1 \mu _2 \mu _3\nu _1 \nu _2\nu _3}_{(3)} \big(\kappa _1 R_{\mu _1\mu _2\nu _1\nu _2}\pi _{\mu _3\nu _3} - \case{4}{3} \kappa _{1,X} \pi _{\mu _1\nu _1} \pi _{\mu _2\nu _2} \pi _{\mu _3\nu _3}\big) \end{equation} \tag{ 58 }$$

$$\begin{equation} \hphantom{\mathcal{L}_{H} =}-\mathcal{A}^{\mu _1\mu _2\mu _3\nu _1\nu _2\nu _3}_{(3)} ( \kappa _3 R_{\mu _1\mu _2\nu _1\nu _2}\pi _{\mu _3}\pi _{\nu _3} - 4 \kappa _{3,X} \pi _{\mu _1\nu _1} \pi _{\mu _2\nu _2} \pi _{\mu _3}\pi _{\nu _3}) \end{equation} \tag{ 59 }$$

$$\begin{equation} \hphantom{\mathcal{L}_{H} =}-\mathcal{A}^{\mu _1 \mu _2 \nu _1 \nu _2}_{(2)}(F R_{\mu _1\mu _2\nu _1\nu _2} - 4 F_{,X} \pi _{\mu _1\nu _1} \pi _{\mu _2\nu _2}) \end{equation} \tag{ 60 }$$

$$\begin{equation} \hphantom{\mathcal{L}_{H} =}-2 \kappa _8 \mathcal{A}^{\mu _1 \mu _2 \nu _1 \nu _2}_{(2)} \pi _{\mu _1}\pi _{\nu _1} \pi _{\mu _2\nu _2} \end{equation} \tag{ 61 }$$

$$\begin{equation} \hphantom{\mathcal{L}_{H} =}-3(2F_{,\pi } + X \kappa _8) X + \kappa _9, \end{equation} \tag{ 62 }$$

where κ₁, κ₃, κ₈, κ₉ and F are functions of π and X which are related by the constraint

$$\begin{equation} F_{,X} = \kappa _{1,\pi } -\kappa _3 - 2 X \kappa _{3,X}. \end{equation} \tag{ 63 }$$

The relation between Horndeski theories (58)–(62) and theories (57) with D = 4 can be summarized as follows. First, observe that the flat-space restriction of Horndeski theories must be a subset of the most general flat-space theories presented in section 2.1.⁶ Secondly, the theories (57) obtained by covariantizing the theory of section 2.1 must be included in the set of theories discussed by Horndeski. In fact they are exactly equivalent [9, 29], and to see this it is sufficient to rewrite (58)–(62) in the form (57):

$$\begin{equation} \mathcal{L}_{H}=\sum _{n=0}^{3}\tilde{\mathcal{L}}^{{\rm cov}}_{n}\lbrace f_{n}\rbrace \end{equation} \tag{ 64 }$$

and identify

$$\begin{eqnarray*} Xf_{0}(\pi ,X) & = & -\kappa _{9}(\pi ,X)-\frac{X}{2}\int {\rm d}X(2\kappa _{8}-4\kappa _{3,\pi })_{,\pi },\\ Xf_{1}(\pi ,X) & = & X(4\kappa _{3,\pi }+\kappa _{8})-\frac{1}{2}\int {\rm d}X(2\kappa _{8}-4\kappa _{3,\pi })+6F_{,\pi },\\ Xf_{2}(\pi ,X) & = & 4(F+X\kappa _{3})_{,X},\\ Xf_{3}(\pi ,X) & = & \frac{4}{3}\kappa _{1,X}. \end{eqnarray*}$$

We would like to stress that the constructions of [9] and [4] are based on very different starting hypotheses. Reference [9] starts from the most general set of scalar theories in flat D-dimensional spacetime, and then constructs its (possibly non-unique) covariantized generalization (resulting in the Lagrangian (57)). Reference [4] on the other hand determines the unique set of curved spacetime scalar–tensor theories in four dimensions with second-order field equations both for the scalar and the metric. That these two rather different routes end up with identical sets of theories when D = 4 is remarkable! It should be stressed that whether or not this is still true for arbitrary D is still a subject of research, since the extension of Horndeski's construction to D > 4 is currently unknown.

To finish, notice also that the parametrization given by Horndeski can be somewhat misleading because one might conclude that e.g. the terms (58), (60) and (62), which all depend on the function κ₁, are not independent. (The same can be said for (59), (60) and (62) which depend on κ₃.) This is, however, not the case, and one can see easily (see [3, 9]) that each of the terms (58)–(62) lead separately to field equations of second order both for the scalar and the metric. This is transparent in the rewriting (64). Also, Horndeski's parametrization does not elucidate the relation between the two families of Lagrangians $\mathcal{L}^{{\rm Gal},1}$ and $\mathcal{L}^{{\rm Gal},3}$ which appear respectively in equations (59), (61) and equations (58), (60), (62).

As we have discussed above, when D = 4 the covariantized generalized Galileon and Horndeski theories are identical, and their Lagrangian therefore describes all scalar–tensor theories in four dimensions with second-order field equations both for the scalar and the metric. It therefore follows that any other constructions of Galileons in D = 4 curved space must be included in Horndeski theories, as a subcase of the action given in equations (58)–(62) (or equivalently (57)).

This includes in particular the brane constructions of [18–21, 23]. However, the advantage of such (less general) approaches is that they can give insight into different properties of the theories, for example highlighting the origins of some of the metric–scalar field couplings which appear in equations (58)–(62). They can also give a geometric interpretation of the Galileon field π (or its multifield extensions) and of its Galilean symmetry in flat spacetime, as well as allowing for an extension of this symmetry to curved (maximally symmetric) backgrounds. Here we comment briefly on these brane-world constructions, without entering into a detailed description.

The idea is to consider a four-dimensional brane, our universe, evolving in a higher dimensional spacetime (the bulk with metric G_AB). In the simplest case this is five dimensional (so that A = 0, ..., 4), but it can be of higher dimensions—in particular if one is interested in multi-Galileons (see e.g. [19]). Let X^A(x) with denote the brane embedding, with x^μ (μ = 0, 1, 2, 3) the world-volume coordinates. Then the induced metric γ_μν(x) on the brane, and the extrinsic curvature K_μν(x) are defined in the usual way (see for instance [30]) as

$$\begin{equation} \gamma _{\mu \nu }= \frac{\partial X^A}{\partial x^\mu } \frac{\partial X^B}{\partial x^\nu } G_{AB}(X) \qquad K_{\mu \nu }= \frac{\partial X^A}{\partial x^\mu } \frac{\partial X^B}{\partial x^\nu } \nabla _A n_B \end{equation} \tag{ 65 }$$

where n^A is the unit normal vector to the brane. Now work in a gauge in which

$$\begin{equation} X^\mu (x)=x^\mu , \qquad X^5(x)=\pi (x). \end{equation} \tag{ 66 }$$

As a result the bulk is foliated by time-like slices given by the surfaces X⁵(x) = constant, meaning that π can be understood as the transverse position of the brane relative to these time-like slices.

One then considers a 4D probe brane with action consisting of non-trivial Lovelock invariants. These are constructed from the induced metric or 4D boundary terms for bulk Lovelock invariants, and thus contain contributions coming both from the brane induced metric (or 'first fundamental form') and the brane extrinsic curvature (or 'second fundamental form'). The observation of [18] is that these terms are such that the field equations derived from such an action are of second order and fall in the generalized Galileon family. Moreover, whenever the bulk is taken to be simple, e.g. flat, and the brane chosen such that it gives a simple slicing of the bulk (e.g. such that the induced metric is just de Sitter), then the bulk isometries become imprinted on the brane world-volume theory in the form of a symmetry analogous to the flat-spacetime Galileon symmetry [21, 23]. The interesting point though is that π has a physical interpretation as the brane position, whilst its nonlinear symmetries can be identified from the isometries of the bulk. This procedure can be generalized to multi-Galileon theories by considering branes of higher codimension, e.g. [19].

Covariantized versions of the multifield theories introduced in section 2.2, which maintain the second-order nature of the field equations, can also be obtained in a way similar to that introduced for scalar theories: all possible pairs of gradients, ∂F^a∂F^b, must be replaced by suitable contractions of the undifferentiated F^aF^b with the Riemann tensor, and added to the minimally covariantized flat-space action with suitable coefficients [15]. One should, however, pay attention to the fact that, in the p > 0 construction, ∇_μF_αβ... are to be distinguished from ∇_αF_μν..., essentially because of their different ε-index contractions, a distinction irrelevant to the original scalar, π_μα = π_αμ, case. One common feature is that flat-space Galilean invariance is also not restorable by consistent covariantization (nor should it be expected, given the absence of constant vectors or tensors in curved space): the equations now necessarily depend on both second and first derivatives of the fields. It is also interesting to note that actions trivial in flat space can have non-trivial, dynamical, curvature-dependent extensions: consider actions (42) for any single odd p-form (i.e. with just one species of odd p-form), which are vacuous in flat space. Their minimal covariantizations are both nonvanishing and of third order. However, one may also add appropriate counterterms that both remove the offending higher derivatives and remain non-trivial. Indeed, the simplest case is the lowest Galileon D = 5 vector action,

$$\begin{eqnarray} S_{4,{\rm vec}} &=&\int {\rm d}^5 x\, \varepsilon ^{\mu \nu \rho \sigma \tau } \varepsilon ^{\alpha \beta \gamma \delta \epsilon }\, F_{\mu \nu } F_{\alpha \beta }\, \nabla _\rho F_{\gamma \delta }\, \nabla _\epsilon F_{\sigma \tau } \nonumber \\ &=&-\frac{1}{2}\int {\rm d}^5 x\, \varepsilon ^{\mu \nu \rho \sigma \tau } \varepsilon ^{\alpha \beta \gamma \delta \epsilon }\, F_{\mu \nu } F_{\alpha \beta }\, F^\lambda _{\hphantom{\lambda }\rho } F_{\delta \gamma }\, R_{\sigma \tau \lambda \epsilon }. \end{eqnarray} \tag{ 67 }$$

The last equality in (67) exhibits the model's curvature-dependence, and is obtained from the first expression by parts integration. The third derivatives in the resulting field equations can be removed by adding the counterterm

$$\begin{equation} S_{4,{\rm vec}}^{\rm nonmin}=\int {\rm d}^5 x \varepsilon ^{\mu \nu \rho \sigma \tau } \varepsilon ^{\alpha \beta \gamma \delta \epsilon } F_{\mu \nu } F_{\alpha \beta } F^\lambda _{\hphantom{\lambda }\rho } F_{\lambda \gamma } R_{\sigma \tau \delta \epsilon }. \end{equation} \tag{ 68 }$$

It differs from the action (67) itself simply by an overall factor and the index change δ↔λ in the last two terms.