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 $p \neq 1$. 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 $w_{\phi}=-1-2p\dot{H}/(3H^2)$. 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.