Well-posedness of cubic Horndeski theories

We study the local well-posedness of the initial value problem for cubic Horndeski theories. Three different strongly hyperbolic modifications of the Arnowitt-Deser-Misner 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 Baumgarte-Shapiro-Shibata-Nakamura formulation with a generalized Bona-Mass\'o slicing (covering the 1+log slicing) and nondynamical shift vector. The third one is an extension of the CCZ4 formulation with a generalized 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 nonlinear numerical simulations.