Tracker and scaling solutions in DHOST theories. (arXiv:1812.05204v1 [gr-qc])
<a href="http://arxiv.org/find/gr-qc/1/au:+Frusciante_N/0/1/0/all/0/1">Noemi Frusciante</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Kase_R/0/1/0/all/0/1">Ryotaro Kase</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Koyama_K/0/1/0/all/0/1">Kazuya Koyama</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Tsujikawa_S/0/1/0/all/0/1">Shinji Tsujikawa</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Vernieri_D/0/1/0/all/0/1">Daniele Vernieri</a>
In quartic-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 $dot{phi}/H^p={rm
constant}$, where $dot{phi}$ is the time derivative of a scalar field $phi$,
$H$ is the Hubble expansion rate, and $p$ is a constant. While the tracker is
present up to the cubic-order Horndeski Lagrangian $L=c_2X-c_3X^{(p-1)/(2p)}
square phi$, where $c_2, c_3$ are constants and $X$ is the kinetic energy of
$phi$, 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
$w_{phi}=-1-2pdot{H}/(3H^2)$. The scaling solution, which corresponds to
$p=1$, is the unique case in which all the terms in the field density
$rho_{phi}$ and the pressure $P_{phi}$ obey the scaling relation
$rho_{phi} propto P_{phi} propto H^2$. Extending the analysis to the
coupled DHOST theories with the field-dependent coupling $Q(phi)$ between the
scalar field and matter, we show that the scaling solution exists for
$Q(phi)=1/(mu_1 phi+mu_2)$, 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 $phi$-matter-dominated epoch.
In quartic-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 $dot{phi}/H^p={rm
constant}$, where $dot{phi}$ is the time derivative of a scalar field $phi$,
$H$ is the Hubble expansion rate, and $p$ is a constant. While the tracker is
present up to the cubic-order Horndeski Lagrangian $L=c_2X-c_3X^{(p-1)/(2p)}
square phi$, where $c_2, c_3$ are constants and $X$ is the kinetic energy of
$phi$, 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
$w_{phi}=-1-2pdot{H}/(3H^2)$. The scaling solution, which corresponds to
$p=1$, is the unique case in which all the terms in the field density
$rho_{phi}$ and the pressure $P_{phi}$ obey the scaling relation
$rho_{phi} propto P_{phi} propto H^2$. Extending the analysis to the
coupled DHOST theories with the field-dependent coupling $Q(phi)$ between the
scalar field and matter, we show that the scaling solution exists for
$Q(phi)=1/(mu_1 phi+mu_2)$, 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 $phi$-matter-dominated epoch.
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