Cosmological evolution of viable models in the generalized scalar-tensor theory

We investigate the parameter distributions of the viable generalized scalar-tensor theory with conventional dust matter after GW170817 in a model-independent way. We numerically construct the models by computing the time evolution of a scalar field, which leads to a positive definite secondorder Hamiltonian, and the models are consistent with the observed Hubble parameter. We show the model parameter distributions in the degenerate higher-order scalar-tensor (DHOST) theory and its popular subclasses, e.g., Horndeski and GLPV theories, etc. We find that (1) the Planck mass run rate, alpha(M), is insensitive to distinguish the theories; (2) the kinetic-braiding parameter, alpha(B), marginally discriminates the models from those of the Horndeski theory in some range; and (3) the parameters for the higher-order theories, alpha(H) and beta(1), are relatively smaller in magnitude (by several factors) than alpha(M) and alpha(B), but can still be used for discriminating the theories except for the GLPV theory. Based on the above three facts, we propose a minimal set of parameters that sensibly distinguishes the subclasses of DHOST theories, (alpha(M), alpha(B) - alpha(M)/2, beta(1)).