Coupled dark energy model inspired from general conformal transformation
  We study the coupled dark energy model constructed from the general conformal
transformation in which the coefficient of the conformal transformation depends
on both the scalar field and its kinetic term. Under this conformal
transformation, the action for subclass of Degenerate Higher-Order
Scalar-Tensor (DHOST) theories is related to the Einstein-Hilbert action. The
evolution of the background universe has the scaling fixed point which
corresponds to acceleration of the universe at late time. For the choices of
parameters which make the late-time scaling point stable, the fixed point
corresponding to $\phi$-matter-dominated-era ($\phi$MDE) is a saddle point, and
the universe can evolve from radiation dominated epoch through $\phi$MDE before
reaching the scaling point at late time with the cosmological parameters which
satisfy the observational bound. During the $\phi$MDE, the effective equation
of state parameter is slightly positive, so that one of possible mechanisms for
alleviating the $H_0$ tension can be achieved. In this coupled dark energy
model, the effective gravitational coupling for dark matter perturbations on
small scales can be smaller than that in the $\Lambda$CDM model. Therefore a
growth rate of the dark matter perturbations is suppressed compare with the
$\Lambda$CDM model, which implies that the $\sigma_8$ tension could be
alleviated.
