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

Abstract

Within the general framework of spatially covariant theories of gravity, we study the conditions for having only the two tensorial degrees of freedom. Generally, there are three degrees of freedom propagating in the theory, of which two are tensorial and one is of the scalar type. Through a detailed Hamiltonian analysis, we find two necessary and sufficient conditions to evade the scalar type degree of freedom. The first condition implies that the lapse-extrinsic curvature sector must be degenerate. The second condition ensures that the dimension of the phase space at each spacetime point is even, so that the scalar type degree of freedom is eliminated completely. We also compare our results with the previous studies, and apply our formalism to a simple example, in which the Lagrangian is quadratic in the extrinsic curvature. *

Symmetries for scalarless scalar theories

“…In fact, the limit κ 0 → 0 can be taken consistently [when including the appropriate powers of κ 0 as overall coefficients, as in Eq. (38)] and leads to the system (10)- (13). As shown in [14], these correspond to a particular case of DBI-Galileon actions, equipped with the symmetry (3) [or equivalently the κ 0 → 0 limit of (42)].…”

“…A consequence is that these systems spontaneously break Lorentz symmetry: although their Lagrangians are Lorentz invariant, the theories require vacua with a nonvanishing gradient for the scalar, withπ ;μ ≠ 0 (a well-defined square root ffiffiffi ffi X p further requiresπ ;μπ ;μ < 0). Scalar backgrounds that depend only on time,π ¼πðtÞ, are special: making this choice of homogeneous background, the Lagrangians (12) and (13) vanish identically, while (10) and (11) become linear functions of the time derivatives _ π. Any linear combination of them with constant coefficients then reduces to a total derivative, ensuring that the homogeneous scalar configurationπðtÞ automatically satisfies the (trivial) equations of motion in Minkowski space and represents a consistent vacuum for the theory.…”

“…While we focused so far on the specific square root cuscuton potential (24), these very same considerations apply also to the vector versions of all the covariant scalar Lagrangians (11)- (13), which shall contain derivatives of the vector fields. It is sufficient to substitute the scalar first derivatives with the vector A μ in those Lagrangians to determine covariant vector potentials.…”

“…(ii) In Sec. IV B we turn on dynamical gravity and consistently couple our scalar Lagrangians (11)- (13) with tensor modes in curved space. This process generally breaks the SL symmetry that only holds in flat space; nevertheless we show that the scalar-tensor system obeys a novel continuous symmetry, inherited from broken diffeomorphism invariance in the fivedimensional brane-world model.…”

“…The structure of the corresponding theories can lead to new perspectives for the dark energy problem, and for explaining the smallness of the present-day cosmological acceleration rate [8]. Examples of scalar-tensor or massive gravity theories where scalar fluctuations do not propagate have already been discussed in the literature (see, e.g., [9][10][11][12][13]), usually performing a Hamiltonian analysis aimed at determining the constraints that avoid the propagation of scalar modes. In many of the existing examples, scalar fluctuations do not propagate in a unitary gauge, where an homogeneous, timedependent-only configuration for the background scalar field is selected.…”