We consider the recently introduced "galileon" field in a dynamical spacetime. When the galileon is assumed to be minimally coupled to the metric, we underline that both field equations of the galileon and the metric involve up to third-order derivatives. We show that a unique nonminimal coupling of the galileon to curvature eliminates all higher derivatives in all field equations, hence yielding second-order equations, without any extra propagating degree of freedom. The resulting theory breaks the generalized "Galilean" invariance of the original model.Comment: 10 pages, no figure, RevTeX4 format; v2 adds footnote 1, Ref. [12], reformats the link in Ref. [14], and corrects very minor typo

We introduce a large class of scalar-tensor models with interactions containing the second derivatives of the scalar field but not leading to additional degrees of freedom. These models exhibit peculiar features, such as an essential mixing of scalar and tensor kinetic terms, which we have named kinetic braiding. This braiding causes the scalar stress tensor to deviate from the perfect-fluid form. Cosmology in these models possesses a rich phenomenology, even in the limit where the scalar is an exact Goldstone boson. Generically, there are attractor solutions where the scalar monitors the behaviour of external matter. Because of the kinetic braiding, the position of the attractor depends both on the form of the Lagrangian and on the external energy density. The late-time asymptotic of these cosmologies is a de Sitter state. The scalar can exhibit phantom behaviour and is able to cross the phantom divide with neither ghosts nor gradient instabilities. These features provide a new class of models for Dark Energy. As an example, we study in detail a simple one-parameter model. The possible observational signatures of this model include a sizeable Early Dark Energy and a specific equation of state evolving into the final de-Sitter state from a healthy phantom regime.1 Similar models were introduced before in another context [17,18]. Also, recently, such models have been extended to vectors and p-forms [19].2 Generic choices of K, G give equations of motion with explicit dependence on the first derivative ∇µφ.Hence, the Lagrangians (1.1) do not have the Galilean symmetry which characterizes the interactions of [7] and [8].We will therefore study the equation of state of the fluid representing the excess over the attractor, with energy density J = − * . Assuming that one can invert J(X, ρ ext ) to find X(J, ρ ext ), we can simply expand the total energy density (J, ρ ext ) =φ(J, ρ ext )J − K(X(J, ρ ext )) for small J to find, at lowest order in Notice that the factor in front of J is a function of ρ ext , implying that J does not dilute like dust as could have been naively expected and as is seen in the case of ghost condensates.

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We consider minimal extensions of the recently proposed Mimetic Dark Matter and show that by introducing a potential for the mimetic non-dynamical scalar field we can mimic nearly any gravitational properties of the normal matter. In particular, the mimetic matter can provide us with inflaton, quintessence and even can lead to a bouncing nonsingular universe. We also investigate the behaviour of cosmological perturbations due to a mimetic matter. We demonstrate that simple mimetic inflation can produce red-tilted scalar perturbations which are largely enhanced over gravity waves.

The k-essence theories admit in general the superluminal propagation of the perturbations on classical backgrounds. We show that in spite of the superluminal propagation the causal paradoxes do not arise in these theories and in this respect they are not less safe than General Relativity.