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 $\dot{\varphi }/H^p=\mathrm{constant}$, where $\dot{\varphi }$ is the time derivative of a scalar field $\varphi$, $H$ is the Hubble expansion rate, and $p$ is a constant. While the tracker is present up to the cubic-order Horndeski Lagrangian $L=c_{2}X-c_3{X}^{(p-1)/(2p)}\square \varphi$, where $c_2,c_3$ are constants and $X$ is the kinetic energy of $\varphi$, the DHOST interaction breaks this structure for $p\ne 1$. 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 $w_{\varphi }=-1-2p\dot{H}/(3H^2)$. 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 $Q(\varphi )=1/({\mu }_1\varphi +{\mu }_2)$, 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.

Comments: 12 pages