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Cosmic microwave background with Brans-Dicke gravity II: constraints with the WMAP and SDSS data

Using the covariant formalism developed in a companion paper (paper I), we derive observational constraint on the Brans-Dicke model in a flat FLRW universe with cosmological constant and cold dark matter. The CMB observations we use include the WMAP five year data, the ACBAR 2007 data, the CBI polarization data, and the BOOMERanG 2003 flight data. For the large scale structure we use the matter power spectrum data measured with the LRG survey of the SDSS DR4. We parameterize the Brans-Dicke parameter $ω$ with a new parameter $ζ=\ln(1/ω+1)$, and use the Markov-Chain Monte Carlo method to explore the parameter space. We find that using CMB data alone, one could place some constraint on positive $ζ$ or $ω$, but negative $ζ$ or $ω$ could not be constrained effectively. However, with additional large scale structure data, one could break the degeneracy at $ζ<0$. The $2σ$ (95.5%) bound on $ζ$ is $-0.00837<ζ<0.01018$ (corresponding to $ω< -120.0$ or $ω> 97.8$). We also obtained constraint on $\dot{G}/G$, the rate of change of $G$ at present, as $-1.75 \times 10^{-12} \yr^{-1}<\dot{G}/G < 1.05 \times 10^{-12}\yr^{-1}$, and $δG/G$, the total variation of $G$ since the epoch of recombination, as $ -0.083 < δ{G}/G < 0.095$ at $2σ$ confidence level.