**N. Frusciante**, R. Kase, K. Koyama, S. Tsujikawa, D. Vernieri

**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 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 =c2 X -c3^{X(p-1)/(2p)} □ ϕ, where c_{2} ,c_{3} 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 wϕ = - 1-2 pH ̇/(3H^{2}). The scaling solution, which corresponds to p = 1, is the unique case in which all the terms in the field density ρϕ and the pressure Pϕ obey the scaling relation ρ_{ϕ} ∝P_{ϕ} ∝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.

**Physics Letters B**

Volume 790, Page 167

2019 March