Covariant scalar-tensor theories beyond second derivatives

TL;DR

This paper develops a covariant, gauge-independent framework for scalar-tensor theories involving higher derivatives, extending beyond existing DHOST theories and analyzing their degrees of freedom.

Contribution

It introduces a new basis of diffeomorphism-invariant operators up to four derivatives, extending U-DHOST theories covariantly without gauge fixing or degeneracy assumptions.

Findings

The theory propagates three physical degrees of freedom.

It includes a parity-odd pseudoscalar invariant at four derivatives.

The framework extends beyond degenerate higher-order scalar-tensor theories.

Abstract

We propose a covariant, gauge-independent construction of foliation-based scalar-tensor theories, yielding diffeomorphism-invariant operators involving only gradients on the hypersurfaces where the scalar field is constant, assumed to be spacelike. This defines a basis of independent invariants up to four derivatives of $\phi$, including the first nontrivial parity-odd pseudoscalar at this order, with a straightforward extension to higher derivatives. Our framework goes beyond degenerate higher-order scalar-tensor (DHOST) theories and provides a nonlinear extension of U-DHOST (where $\nabla_\mu\phi$ is supposed to be timelike) directly in covariant form, without using unitary gauge as a starting point or imposing degeneracy a priori. After minimal coupling to gravity, we analyze the theory through its Hamiltonian constraint structure and linear cosmological perturbations about an FLRW background, and show that it propagates three physical degrees of freedom.