# Black hole solutions in shift-symmetric degenerate higher-order scalar-tensor theories

## TL;DR

This paper investigates black hole solutions in shift-symmetric degenerate higher-order scalar-tensor theories, finding that cubic theories are disfavoured by gravitational wave constraints, while quadratic theories remain promising, and provides exact solutions within certain classes.

## Contribution

It identifies which DHOST theories admit exact black hole solutions and clarifies their compatibility with degeneracy and observational constraints.

## Key findings

- Cubic DHOST theories are strongly disfavoured by gravitational wave speed constraints.
- Exact static black hole solutions are found in quadratic and some cubic theories.
- Conditions for coupling functions to admit solutions are explicitly derived.

## Abstract

We show that most classes of shift-symmetric degenerate higher-order scalar-tensor (DHOST) theories which satisfy certain degeneracy conditions are not compatible with the conditions for the existence of exact black hole solutions with a linearly time-dependent scalar field whose canonical kinetic term takes a constant value. Combined with constraints from the propagation speed of gravitational waves, our results imply that cubic DHOST theories are strongly disfavoured and that pure quadratic theories are likely to be the most viable class of DHOST theories. We find exact static and spherically symmetric (Schwarzschild and Schwarzschild-(anti-)de Sitter) black hole solutions in all shift-symmetric higher-derivative scalar-tensor theories which contain up to cubic order terms of the second-order derivatives of the scalar field, especially the full class of Horndeski and Gleyzes-Langlois-Piazza-Vernizzi theories. After deriving the conditions for the coupling functions in the DHOST Lagrangian that allow the exact solutions, we clarify their compatibility with the degeneracy conditions.