Tracker and scaling solutions in DHOST theories

Noemi Frusciante; Ryotaro Kase; Kazuya Koyama; Shinji Tsujikawa; Daniele Vernieri

doi:10.1016/j.physletb.2019.01.009

paper

Tracker and scaling solutions in DHOST theories

arXiv:1812.05204 · doi:10.1016/j.physletb.2019.01.009

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 $L = c_{2} X - c_{3} X^{(p - 1) / (2 p)} □ ϕ$ , 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 $w_{ϕ} = - 1 - 2 p H ˙ / (3 H^{2})$ . The scaling solution, which corresponds to $p = 1$ , is the unique case in which all the terms in the field density $ρ_{ϕ}$ 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.

arXiv comment12 pages

Tracker and scaling solutions in DHOST theories

Abstract

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