Abstract
We consider theories describing the dynamics of a four-dimensional metric, whose Lagrangian is diffeomorphism invariant and depends at most on second derivatives of the metric. Imposing degeneracy conditions we find a set of Lagrangians that, apart form the Einstein-Hilbert one, are either trivial or contain more than 2 degrees of freedom. Among the partially degenerate theories, we recover Chern-Simons gravity, endowed with constraints whose structure suggests the presence of instabilities. Then, we enlarge the class of parity violating theories of gravity by introducing new ``chiral scalar-tensor theories.'' Although they all raise the same concern as Chern-Simons gravity, they can nevertheless make sense as low energy effective field theories or, by restricting them to the unitary gauge (where the scalar field is uniform), as Lorentz breaking theories with a parity violating sector.
Highlights
It is well known that the Einstein-Hilbert plus cosmological constant action is the unique diffeomorphism invariant action for a four-dimensional metric, whose equations of motion (EOM) are at most of second order [1]
We have studied fully and partially degenerate metric theories in four dimensions whose action is invariant under diffeomorphisms and that contain at most second derivatives of the metric
Apart from the EinsteinHilbert action which propagates 2 physical degrees of freedom, fully degenerate theories are either trivial with no degree of freedom or contain Ostrogradsky modes
Summary
It is well known that the Einstein-Hilbert plus cosmological constant action is the unique diffeomorphism (diff) invariant action for a four-dimensional metric, whose equations of motion (EOM) are at most of second order [1]. A higher order scalar-tensor theory still propagates 3 degrees of freedom (DOF) if, in addition to the usual Hamiltonian and momentum constraints associated with diff invariance, it admits another primary constraint.. A higher order scalar-tensor theory still propagates 3 degrees of freedom (DOF) if, in addition to the usual Hamiltonian and momentum constraints associated with diff invariance, it admits another primary constraint.1 These theories, denoted as degenerate higher order scalartensor (DHOST) theories [or extended scalar-tensor (EST) theories], were introduced in [10] and further analyzed in [11,14,15,16]. A complete classification up to cubic order in second derivatives of the scalar field is given in [17] Their cosmological perturbations, in the framework of the effective theory of dark energy (see e.g., [18]), are studied in [19].
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