Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress tensors

Recently, an interesting scalar field, ‘‘Galileon,’’ theory [1], inspired by the decoupling limit of the DvaliGabadadze-Porrati (DGP) model [2] and its cosmological consequences [3], was introduced. (This model was previously proposed in [6], also in flat space, with a quite different motivation.) Originally formulated in flat spacetime and dimension D 1⁄4 4, its defining property was that, while the action contains both first and second derivatives, the equations of motion uniquely involve the latter. As shown in Ref. [7], the simplest covariantization led to field equations for the scalar and its stress tensor that contained third derivatives; fortunately, [7] also showed how to eliminate these higher derivatives by introducing suitable nonminimal, curvature, couplings. (This cure’s small price was to break an original symmetry of the model, that of shifting the first derivatives of by a constant vector, which is not meaningful in curved space anyhow.) Although the phenomenological relevance of the nonminimal terms has not been studied, [7] furnished a nontrivial example of ‘‘safe,’’ purely second order, class of scalartensor couplings. However, it was restricted to D 1⁄4 4 and involved rather complicated algebra. In the present work, we will provide the transparent and uniform basis in arbitrary D for this, a priori surprising, nonminimal completion. To do so, the Galileon model will first be reformulated in Sec. II; in particular, wewill exhibit its simplest flat-spacetime properties. Section III will incorporate curved backgrounds, in D 1⁄4 4 for concreteness. This will illustrate how the new formulation leads very directly to the original nonminimal couplings of [7]. The final section completes our results by extending them to arbitrary dimensions and backgrounds. Our results are encapsulated in Eqs. (9) for flat, and (35) for general, background. To define our framework more precisely, we will exhibit, starting from a transparent ‘‘canonical’’ flat-space action with purely second-derivative field equation (but still unavoidably higher derivative stress tensor), a ‘‘minimal’’ nonminimal gravitational coupling extension that simultaneously guarantees no higher than second derivatives of either field or metric in both the field equation and stress tensor in any D and background. We do not claim uniqueness for this construction simply because one may add infinitely many (rather trivial because irrelevant) terms, all vanishing in flat space, that also avoid higher derivatives. Examples include Lagrangians such as (any function of) the scalar field times all Gauss-Bonnet-Lovelock or Pontryagin densities, let alone plain scalar curvature. Likewise, starting from a flat ‘‘noncanonical’’ version differing from ours by a total divergence, other nonminimal terms would be generated. Finally, our aim being to avoid higher than second derivatives, we will not discuss, for us trivial, incidental first and zeroth order terms such as Vð Þ.