We show that for Horndeski theories it is possible to derive mathematically compact consistency relations (CR) between physically observable quantities, valid for different classes of theories defined by the behavior of the braiding function ${\alpha }_B$, independent of all other property functions. The CRs establish a parametrization-independent direct relation between the effective gravitational constant, the slip parameter, the gravitational and electromagnetic waves (EMW) luminosity distances, the speed of gravitational waves (GW), and the sound speed. The no-braiding CR is also satisfied by general relativity (GR) and allows one to estimate the gravitational coupling from GWs observations, independently from large scale structure (LSS) observations. A general, less mathematically compact, consistency condition is also derived, valid for any form of the function ${\alpha }_B$ and the other property functions. We apply the CRs to map the large scale structure observational constraints on the effective gravitational constant and the slip parameter to GW-EMW distance ratio constraints, showing that LSS and GWs give independent constraints consistent with no-braiding. Beside allowing one to perform parametrization and model-independent tests of the consistency between different constraints on modified gravity, the CRs allow one to probe the value of the effective gravitational constant with multimessenger observations, independently from LSS observations.

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• Alternative gravity theories
• Dark energy
• Gravitational waves
• Large scale structure of the Universe
• Relativistic aspects of cosmology
• Effective field theory