Tracker and scaling solutions in DHOST theories

In quadratic-order degenerate higher-order scalar-tensor (DHOST) theories compatible with gravitational-wave constraints, we derive the most general Lagrangian allowing for tracker solutions characterized by $dot{phi}/H^p={rm constant}$, where $dot{phi}$ is the time derivative of a scalar field $phi$, $H$ is the Hubble expansion rate, and $p$ is a constant. While the tracker is present up to the cubic-order Horndeski Lagrangian $L=c_2X-c_3X^{(p-1)/(2p)} square phi$, where $c_2, c_3$ are constants and $X$ is the kinetic energy of $phi$, the DHOST interaction breaks this structure for $p eq 1$. Even in the latter case, however, there exists an approximate tracker solution in the early cosmological epoch with the nearly constant field equation of state $w_{phi}=-1-2pdot{H}/(3H^2)$. The scaling solution, which corresponds to $p=1$, is the unique case in which all the terms in the field density $rho_{phi}$ and the pressure $P_{phi}$ obey the scaling relation $rho_{phi} propto P_{phi} propto H^2$. Extending the analysis to the coupled DHOST theories with the field-dependent coupling $Q(phi)$ between the scalar field and matter, we show that the scaling solution exists for $Q(phi)=1/(mu_1 phi+mu_2)$, where $mu_1$ and $mu_2$ are constants. For the constant $Q$, i.e., $mu_1=0$, we derive fixed points of the dynamical system by using the general Lagrangian with scaling solutions. This result can be applied to the model construction of late-time cosmic acceleration preceded by the scaling $phi$-matter-dominated epoch.