Spatially covariant gravity with velocity of the lapse function: the Hamiltonian analysis

Abstract

We investigate a large class of gravity theories that respect spatial covariance, and involve kinetic terms for both the spatial metric and the lapse function. Generally such kind of theories propagate four degrees of freedom, one of which is an unwanted scalar mode. Through a detailed Hamiltonian analysis, we find that the condition requiring the kinetic terms to be degenerate is not sufficient to evade the unwanted scalar mode in general. This is because the primary constraint due to the degeneracy condition does not necessarily induce a secondary constraint, if the mixing terms between temporal and spatial derivatives are present. In this case, the second condition that we dub as the consistency condition must be imposed in order to ensure the existence of the secondary constraint and thus to remove the unwanted mode. We also show how our formalism works through an explicit example, in which the degeneracy condition is not sufficient and thus the consistency condition must be imposed.

Spatially covariant gravity theories with two tensorial degrees of freedom: The formalism

“…In this section, we perform the Hamiltonian analysis to derive the conditions of the theory in order to have two tensorial degrees of freedom. We will follow the formalism developed in [26].…”

“…Generally, N and h ij are independent and thus both acquire "time" derivatives through £ n N , £ n h ij , etc. This case was discussed in [26]. In this paper, as a first step, we concentrate on case with only £ n h ij ≡ 2K ij .…”

“…In Sec. II, we setup our formalism for the Hamiltonian analysis, following the formalism developed in [26]. In Sec.…”

Effective field theory of degenerate higher-order inflation

“…Plugging it into the first and fifth conditions in (31) and solving the two equations for A 4 and A 5 , we obtain…”

“…[30] v2). A dynamical lapse is also taken into account in [31,32] in the context of spatially covariant gravity [33], where the degeneracy conditions were also studied. However, the general EFT framework of degenerate theories including but not limited to quadratic and cubic DHOST and its application to cosmology have not been fully investigated yet.…”