Abstract
Determining the most general, consistent scalar tensor theory of gravity is important for building models of inflation and dark energy. In this work we investigate the number of degrees of freedom present in the theory of beyond Horndeski. We discuss how to construct the theory from the extrinsic curvature of the constant scalar field hypersurface, and find a simple expression for the action which guarantees the existence of the primary constraint necessary to avoid the Ostrogradsky instability. Our analysis is completely gauge-invariant. However we confirm that, mixing together beyond Horndeski with a different order of Horndeski, obstructs the construction of this primary constraint. Instead, when the mixing is between actions of the same order, the theory can be mapped to Horndeski through a generalised disformal transformation. This mapping however is impossible with beyond Horndeski alone, since we find that the theory is invariant under such a transformation. The picture that emerges is that beyond Horndeski is a healthy but isolated theory: combined with Horndeski, it either becomes Horndeski, or likely propagates a ghost.
Highlights
We discuss how to construct the theory from the extrinsic curvature of the constant scalar field hypersurface, and find a simple expression for the action which guarantees the existence of the primary constraint necessary to avoid the Ostrogradsky instability
The equations of motion contain third order time derivatives, it was argued that they do not lead to the Ostrogradsky instability, since constraints remove this additional degree of freedom
We first rewrote beyond Horndeski in terms of the extrinsic curvature of the constant scalar field hypersurface; we derived the relevant kinetic terms for both beyond Horndeski and Horndeski theories
Summary
Φ) − φ φμν φμν + 2 φμν φνρφμρ φμν ≡ ∇μ∇νφ and K, G3, 4, 5 are arbitrary functions of φ and X, defined as. This is the most general scalar tensor theory of gravity, involving a single scalar field, that leads to covariant second order equations of motion. The cubic Horndeski theory is described by the Lagrangian L3. In this case gravity becomes dynamical only through a mixing with the scalar field, a phenomenon dubbed kinetic gravity braiding, see [20]. Horndeski theories, where the tensor spin-2 degrees of freedom have their own kinetic terms.
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