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$\dot{\phi}$isthetimederivativeofascalarfield$\phi$,$H$istheHubbleexpansionrate,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$c_2, c_3$areconstantsand$X$isthekineticenergyof$\phi$,theDHOSTinteractionbreaksthisstructurefor$p \neq 1$. 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 $w_{\varphi }=-1-2p\dot{H}\mathrm{/}(3H^2)$. The scaling solution, which corresponds to $p=1$, is the unique case in which all the terms in the field density ${\rho }_{\varphi }$ and the pressure $P_{\varphi }$ obey the scaling relation ${\rho }_{\varphi }\propto P_{\varphi }\propto H^2$. Extending the analysis to the coupled DHOST theories with the field-dependent coupling $Q(\varphi )$ between the scalar field and matter, we show that the scaling solution exists for $Q(\varphi )=1\mathrm{/}({\mu }_1\varphi +{\mu }_2)$, 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 $\varphi$-matter-dominated epoch.