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 } = { 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 _ { 2 } X - c _ { 3 } X ^ { ( p - 1 ) / ( 2 p ) } \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 - 2 p \dot { H } / ( 3 H ^ { 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 .