Tracker and scaling solutions in DHOST theories

Abstract

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 $\dotϕ/H^p={\rm constant}$, where $\dotϕ$ is the time derivative of a scalar field $ϕ$, $H$ is the Hubble expansion rate, and $p$ is a constant. While the tracker is present up to the cubic-order Horndeski Lagrangian $L=c_2X-c_3X^{(p-1)/(2p)} \square ϕ$, where $c_2, c_3$ are constants and $X$ is the kinetic energy of $ϕ$, 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 $w_ϕ=-1-2p\dot{H}/(3H^2)$. The scaling solution, which corresponds to $p=1$, is the unique case in which all the terms in the field density $ρ_ϕ$ and the pressure $P_ϕ$ obey the scaling relation $ρ_ϕ \propto P_ϕ \propto H^2$. Extending the analysis to the coupled DHOST theories with the field-dependent coupling $Q(ϕ)$ between the scalar field and matter, we show that the scaling solution exists for $Q(ϕ)=1/(μ_1 ϕ+μ_2)$, where $μ_1$ and $μ_2$ are constants. For the constant $Q$, i.e., $μ_1=0$, 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 $ϕ$-matter-dominated epoch.