Coupled dark energy model inspired from general conformal transformation

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