Minimally modified gravity with auxiliary constraints formalism

TL;DR

This paper explores how to reduce degrees of freedom in metric theories of gravity by using auxiliary constraints, aiming to develop minimally modified gravity theories that retain key features of general relativity.

Contribution

It systematically classifies auxiliary constraints needed to construct minimally modified gravity theories with only two tensorial degrees of freedom.

Findings

Scalar-type ACs should be no more than four.

No vector- or tensor-type ACs are suitable.

A concrete model with four ACs is constructed and its gravitational wave dispersion relation is derived.

Abstract

We investigate the possibility of reducing the number of degrees of freedom (d.o.f.) starting from generic metric theories of gravity by introducing multiple auxiliary constraints (ACs), under the restriction of retaining spatial covariance as a gauge symmetry. Arbitrary numbers of scalar-, vector- and tensor-type ACs are considered a priori, yet we find that no vector- and tensor-type constraints should be introduced, and that scalar-type ACs should be no more than four for the purpose of constructing minimally modified gravity (MMG) theories which propagate only two tensorial d.o.f., like general relativity (GR). Through a detailed Hamiltonian analysis, we exhaust all the possible classifications of ACs and find out the corresponding minimalizing and symmetrizing conditions for obtaining the MMG theories. In particular, no condition is required in the case of four ACs, hence in this case the theory can couple with matter consistently and naturally. To illustrate our formalism, we build a concrete model for this specific case by using the Cayley-Hamilton theorem and derive the dispersion relation of the gravitational waves, which is subject to constraints from the observations.