# Minimally Modified Gravity: a Hamiltonian Construction

## TL;DR

This paper develops a Hamiltonian framework for minimally modified gravity theories, revealing new classes of theories with two gravitational degrees of freedom that break 4D diffeomorphism invariance but preserve 3D invariance.

## Contribution

It provides a Hamiltonian construction method for minimally modified gravity, including a new class called $f({
m extbf{H}})$ theories, expanding the landscape of viable gravity models.

## Key findings

- Recovered the cuscuton scalar-tensor theories in the Hamiltonian framework.
- Identified a new class of $f({
m extbf{H}})$ theories with arbitrary functions of the Hamiltonian constraint.
- Established conditions for Hamiltonian to propagate two gravitational degrees of freedom.

## Abstract

Minimally modified gravity theories are modifications of general relativity with two local gravitational degrees of freedom in four dimensions. Their construction relies on the breaking of 4D diffeomorphism invariance keeping however the symmetry under 3D diffeomorphisms. Here, we construct these theories from a Hamiltonian point of view. We start with the phase space of general relativity in the ADM formalism. Then, we find the conditions that the Hamiltonian must satisfy for the theory to propagate (up to) two gravitational degrees of freedom with the assumptions that the lapse and the shift are not dynamical, and the theory remains invariant under 3D diffeomorphisms. This construction enables us to recover the well-known "cuscuton" class of scalar-tensor theories in the unitary gauge. We also exhibit a new class of interesting theories, that we dubb $f({\cal H})$ theories, where the usual Hamiltonian constraint $\cal H$ of general relativity is replaced by $f({\cal H})$ where $f$ is an arbitrary function.