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 ϕ˙/H^p=constant, where ϕ˙ 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=c₂X−c₃X^(p−1)/(2p)□ϕ where c₂,c₃ are constants and X is the kinetic energy of ϕ the DHOST interaction breaks this structure for p≠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=−1−2pH˙/(3H²). 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∝P_ϕ∝H². 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/(μ₁ϕ+μ₂), where μ₁ and μ₂ are constants. For the constant Q, i.e., μ₁=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.