Cuscuta-Galileon cosmology: Dynamics, gravitational constants, and the Hubble constant

We discuss cosmology based on a Cuscuta-Galileon gravity theory, which preserves just two degrees of freedom. Although there exist no additional freedom, introduction potential scalar field changes the dynamics. The is completely determined by matter fields. Giving an exponential as example, we cosmological gravitational ``constant'' ${G}_{\mathrm{F}}$ appeared in effective Friedmann equation becomes time dependent. also present how to construct when know evolution Hubble parameter. When assume $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ for background evolution, find form. then analyze density perturbations, characterized only change ${G}_{\mathrm{eff}}$, From observational constraints such constraint from big-bang nucleosynthesis and time-variation constant, restrict parameters our models. dependence constant equation, may have chance explain tension problem.