Constraints on the Brans-Dicke gravity theory with the Planck data

Based on the new cosmic CMB temperature data from the Planck satellite, the 9 year polarization data from the WMAP, the BAO distance ratio data from the SDSS and 6dF surveys, we place a new constraint on the Brans-Dicke theory. We adopt a parametrization \zeta=\ln(1+1/\omega}), where the general relativity (GR) limit corresponds to $\zeta =0$. We find no evidence of deviation from general relativity. At 95% probability, $-0.00246<\zeta <0.00567$, correspondingly, the region $-407.0<\omega <175.87$ is excluded. If we restrict ourselves to the $\zeta >0$ (i.e. $\omega >0$) case, then the 95% probability interval is $\zeta <0.00549$, corresponding to $\omega >181.65$. We can also translate this result to a constraint on the variation of gravitational constant, and find the variation rate today as $\dot{G}=-1.42_{-2.27}^{+2.48}\times {10}^{-13}$yr${}^{-1}$ ($1\sigma$ error bar), the integrated change since the epoch of recombination is $\delta G\mathrm{/}G=0.0104_{-0.0067}^{+0.0186}$ ($1\sigma$ error bar). These limits on the variation of gravitational constant are comparable with the precision of solar system experiments.