In this paper first we study Brans–Dicke equations with the cosmological constant to find an exact solution in the spatially flat Robertson–Walker metric. Then we use Observational Hubble data, the baryon acoustic oscillation distance ratio data as well as cosmic microwave background data from Planck to constrain parameters of the obtained Brans–Dicke model. To compare our results and find out the amount of deviation from general relativity, we also constrain concordance cosmological model using the same data. In our theoretical model the Brans–Dicke coupling constant is replaced by a, say new, parameter namely “scalar field density ${\Omega }_{\varphi }$”. Therefore, as ${\Omega }_{\varphi }\to 0$ which is equivalent to $\omega \to \infty$ general relativity is recovered. In general, we found no significant deviation from general relativity. Our estimations show ${\Omega }_{\varphi }=0.01076_{-0.011}^{-0.00035}$ which is equivalent to $\omega >1627$ at 95% confidence level. We also obtained constraint on the rate of change of gravitational constant, $\dot{G}∕G$, at present time as $1.101\times 10^{-13}y{r}^{-1}<\dot{G}∕G<1.152\times 10^{-13}y{r}^{-1}$ (at $1\sigma$ error). The total variation of gravitational constant, since the epoch of recombination, is also constrained as $0.00550<\delta G∕G<0.00575$ at 68% confidence level.

在本文中，我们首先研究宇宙常数的Brans–Dicke方程，以找到空间平坦的Robertson–Walker度量的精确解。然后，我们使用观测哈勃数据，重子声振荡距离比数据以及来自普朗克的宇宙微波背景数据来约束获得的Brans–Dicke模型的参数。为了比较我们的结果并找出相对广义相对论的偏差量，我们还使用相同的数据来约束一致性宇宙学模型。在我们的理论模型中，将Brans–Dicke耦合常数替换为一个新参数，即“标量场密度”。${\Omega }_{\varphi }$”。因此，作为${\Omega }_{\varphi }\to 0$ 相当于 $\omega \to \infty$广义相对论得以恢复。一般而言，我们发现与广义相对论没有显着偏差。我们的估计表明${\Omega }_{\varphi }=0。01076_{-0。011}^{-0。00035}$ 相当于 $\omega >1627$置信水平为95％。我们还获得了对重力常数变化率的约束，$\dot{G}∕G$，目前为 $\mathrm{1个}。101\times \mathrm{1个}{0}^{-13}ÿ{\mathrm{[R}}^{-\mathrm{1个}}<\dot{G}∕G<\mathrm{1个}。152\times \mathrm{1个}{0}^{-13}ÿ{\mathrm{[R}}^{-\mathrm{1个}}$ （在 $\mathrm{1个}\sigma$错误）。自重合时代以来，重力常数的总变化也被约束为$0。00550<\delta G∕G<0。00575$ 置信水平为68％。