Generalized framework for testing gravity with gravitational-wave
  propagation. II. Constraints on Horndeski theory
  Gravitational waves (GW) are generally affected by modification of a gravity
theory during propagation in cosmological distance. We numerically perform a
quantitative analysis on Horndeski theory at cosmological scale to constrain
the Horndeski theory by GW observations in model-independent way. We formulate
a parameterization for a numerical simulation based on the Monte Carlo method
and obtain the classification of the models that agrees with cosmic
accelerating expansion within observational errors of the Hubble parameter. As
a result, we find that a large group of the models in the Horndeski theory that
mimic cosmic expansion of ${\Lambda}$CDM model can be excluded from the
simultaneous detection of a GW and its electromagnetic transient counterpart.
Based on our result and the latest detection of GW170817 and GRB170817A, we
conclude that the subclass of Horndeski theory including arbitrary functions
$G_4$ and $G_5$ can hardly explain cosmic accelerating expansion without
fine-tuning.
