Minimally modified gravity with an auxiliary constraint: a Hamiltonian construction

TL;DR

This paper develops a Hamiltonian framework for minimally modified gravity theories with an auxiliary constraint, reducing degrees of freedom to match those of general relativity and deriving observable gravitational wave properties.

Contribution

It introduces a new Hamiltonian construction for minimally modified gravity with an auxiliary constraint, ensuring only tensorial degrees of freedom propagate.

Findings

Identifies conditions for Hamiltonian and auxiliary constraint for two tensor degrees of freedom.

Defines a new class of type-II minimally modified gravity theories.

Derives gravitational wave dispersion relations constrained by observations.

Abstract

Working directly with a general Hamiltonian for the spacetime metric with the $3+1$ decomposition and keeping only the spatial covariance, we investigate the possibility of reducing the number of degrees of freedom by introducing an auxiliary constraint. The auxiliary constraint is considered as part of the definition of the theory. Through a general Hamiltonian analysis, we find the conditions for the Hamiltonian as well as for the auxiliary constraint, under which the theory propagates two tensorial degrees of freedom only. The class of theories satisfying these conditions can be viewed as a new construction for the type-II minimally modified gravity theories, which propagate the same degrees of freedom of but are not equivalent to general relativity in the vacuum. We also illustrate our formalism by a concrete example, and derive the dispersion relation for the gravitational waves, which can be constrained by observations.