Spatially covariant gravity with two degrees of freedom in the presence of an auxiliary scalar field: Hamiltonian analysis

TL;DR

This work analyzes spatially covariant gravity with an auxiliary scalar field and derives, via a strict Hamiltonian constraint analysis, two necessary degeneracy conditions that reduce the propagating degrees of freedom from three to two. The first condition enforces a degeneracy of a Dirac-matrix block, yielding a secondary constraint; the second condition either converts a second-class constraint into a first-class one or generates an additional secondary constraint, selecting the physical branch with only tensorial gravitational waves. A concrete $d=2$ model demonstrates how particular coefficient choices, including a special ansatz for $c_1,c_2,c_3,c_4,d_1,d_2$, realize the two-DOF theory and produce a new first-class constraint, in agreement with perturbative results. The findings provide a nonlinear, Hamiltonian-confirmed pathway to two-DOF gravity theories within spatially covariant frameworks, with potential implications for cosmology and gravitational-wave phenomenology.

Abstract

A class of gravity theories respecting spatial covariance and in the presence of non-dynamical auxiliary scalar fields with only spatial derivatives is investigated. Generally, without higher temporal derivatives in the metric sector, there are 3 degrees of freedom (DOFs) propagating due to the breaking of general covariance. Through a Hamiltonian constraint analysis, we examine the conditions to eliminate the scalar DOF such that only 2 DOFs, which correspond the tensorial gravitational waves in a homogeneous and isotropic background, are propagating. We find that two conditions are needed, each of which can eliminate half degree of freedom. The second condition can be further classified into two cases according to its effect on the Dirac matrix. We also apply the formal conditions to a polynomial-type Lagrangian as a concrete example, in which all the monomials are spatially covariant scalars containing two derivatives. Our results are consistent with the previous analysis based on the perturbative method.