Cosmological test of an extended quintessence model

We investigate the cosmological observational test of extended quintessence model, i.e. a scalar-tensor gravity model with scalar field potential serving as dark energy, by using Planck 2018 cosmic microwave background (CMB) data, together baryon acoustic oscillations (BAO) and redshift-space distortion (RSD) data. As an example, we consider Brans-Dicke kinetic term $\frac{\omega(\phi)}{\phi} \phi_{;\mu} \phi^{;\mu} $ quadratic $V (\phi) = A + B (\phi - \phi_0) \frac{C}{2} \phi_0)^2$, which reduces to general relativity (GR) in limit $\omega(\phi) \to \infty$, constant $B=C=0$. In such typically rolls down oscillates around minimum (\phi)$. find that parameter estimate for CMB+BAO+RSD data set is given $\lg \alpha -3.6 _{-0.54}^{+0.66}~ (68\%)$, corresponding 3.8 \times 10^5 < \omega_0 9.5 10^7~ C 4.9 \pm 1.4~ (68\%) $. However, GR $\Lambda$CDM can fit almost good this favored Akaike information criterion (AIC). The variation gravitational since epoch Recombination constrained be $0.97 G_{\rm rec}/G_0 1.03~ (1 \sigma)$. light recent report CMB favors closed universe, case non-flat geometry our fit, mean value $\Omega_k$ shifts little bit from $-0.049$ $-0.036$, parameters are not degenerate $\Omega_k$.