# Well-posedness of cubic Horndeski theories

## TL;DR

This paper investigates the local well-posedness of cubic Horndeski theories by extending three hyperbolic formulations of Einstein equations, aiming to facilitate numerical simulations in modified gravity models.

## Contribution

It introduces three strongly hyperbolic formulations of the Einstein equations adapted for cubic Horndeski theories, extending existing methods to this class of modified gravity.

## Key findings

- Extended elliptic-hyperbolic system formulation.
- Adapted BSSN formulation with generalized slicing.
- Extended CCZ4 formulation with shift conditions.

## Abstract

We study the local well-posedness of the initial value problem for cubic Horndeski theories. Three different strongly hyperbolic modifications of the ADM formulation of the Einstein equations are extended to cubic Horndeski theories in the weak field regime. In the first one, the equations of motion are rewritten as a coupled elliptic-hyperbolic system of partial differential equations. The second one is based on the BSSN formulation with a generalised Bona-Mass\'o slicing (covering the 1+log slicing) and non-dynamical shift vector. The third one is an extension of the CCZ4 formulation with a generalised Bona-Mass\'o slicing (also covering the 1+log slicing) and a gamma driver shift condition. This final formulation may be particularly suitable for applications in non-linear numerical simulations.