Abstract
We review how the (absence of) Ostrogradsky instability manifests itself in theories with multiple fields. It has recently been appreciated that when multiple fields are present, the existence of higher derivatives may not automatically imply the existence of ghosts. We discuss the connection with gravitational theories like massive gravity and beyond Horndeski which manifest higher derivatives in some formulations and yet are free of Ostrogradsky ghost. We also examine an interesting new class of Extended Scalar-Tensor Theories of gravity which has been recently proposed. We show that for a subclass of these theories, the tensor modes are either not dynamical or are infinitely strongly coupled. Among the remaining theories for which the tensor modes are well-defined one counts one new model that is not field-redefinable to Horndeski via a conformal and disformal transformation but that does require the vacuum to break Lorentz invariance. We discuss the implications for the effective field theory of dark energy and the stability of the theory. In particular we find that if we restrict ourselves to the Extended Scalar-Tensor class of theories for which the tensors are well-behaved and the scalar is free from gradient or ghost instabilities on FLRW then we recover Horndeski up to field redefinitions.
Highlights
In particular we find that if we restrict ourselves to the Extended Scalar-Tensor class of theories for which the tensors are well-behaved and the scalar is free from gradient or ghost instabilities on FLRW we recover Horndeski up to field redefinitions
While the existence of higher time derivatives automatically involve an Ostrogradsky instability, there has been a recent revived interest in how the Ostrogradsky instability manifest itself in theories with multiple fields where the notion of higher derivatives can be more subtle as it can change under field redefinitions
In this paper we have reviewed different methods one can diagnose the existence of absence of Ostrogradsky instability in a theory with two scalar fields, and indicate that similar arguments can be applied to more involved theories such as those involving gravity and additional scalar fields, as in massive gravity or scalar-tensor theories of gravity
Summary
For a single field theory, the existence of higher-time derivatives leads to the well-known Ostrogradsky instability [39] which manifests itself as an additional ghost degree of freedom (dof) which suffers from a kinetic instability and leads to an inconsistent theory (see ref. [9] for the different consequences of this instability). In what follows we shall emphasize the distinction between setting a variable to a given value and taking an appropriate decoupling limit This distinction is important in the case of gravity where h may for instance symbolically play the role of the metric (or the metric fluctuation about flat spacetime). First we point out that taking a healthy theory with auxiliary variables and making these auxiliary variables dynamical is not a consistent procedure and can very well change the number of dofs This confusion is at the origin of the results of [46, 47] where the terms proposed manifestly exhibit a ghost as shown explicitly in [45, 48,49,50]. One could argue that by taking this road one could in principle allow the theory to have a preferred frame and allow theories which manifestly evade the Ostrogradsky ghost in a specific frame
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