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 second-order Hamiltonian and 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, $alpha_B$, clearly discriminates the models from those of the Horndeski theory, 3) the parameters for the higher-order theories, $alpha_H$ and $eta_1$, 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, ($alpha_M$, $alpha_B-alpha_M/2$, $eta_1$).