Planck constraints on scalar-tensor cosmology and the variation of the gravitational constant

Cosmological constraints on the scalar-tensor theory of gravity by analyzing the angular power spectrum data of the cosmic microwave background (CMB) obtained from the Planck 2015 results are presented. We consider the harmonic attractor model, in which the scalar field has a harmonic potential with curvature ($\beta$) in the Einstein frame and the theory relaxes toward the Einstein gravity with time. Analyzing the $TT$, $EE$, $TE$ and lensing CMB data from Planck by the Markov chain Monte Carlo method, we find that the present-day deviation from the Einstein gravity (${{\alpha }_0}^2$) is constrained as ${{\alpha }_0}^2<2.5\times {10}^{-4-4.5{\beta }^2}$ (95.45% C.L.) and ${{\alpha }_0}^2<6.3\times {10}^{-4-4.5{\beta }^2}$ (99.99% C.L.) for $0<\beta <0.4$. The time variation of the effective gravitational constant between the recombination and the present epochs is constrained as $G_{\mathrm{rec}}/G_0<1.0056$ (95.45% C.L.) and $G_{\mathrm{rec}}/G_0<1.0115$ (99.99% C.L.). We also find that the constraints are little affected by extending to nonflat cosmological models because the diffusion damping effect revealed by Planck breaks the degeneracy of the projection effect.

Figure 1

Time evolution of $\varphi$ in the scalar-tensor $\mathrm{\Lambda }\mathrm{CDM}$ model, with the parameters as indicated in the figure. The other cosmological parameters are fixed to the standard values.

Figure 2

Time evolution of $G(\varphi )/G_0$ in the scalar-tensor models with the same parameters as in Fig. 1. The effective gravitational constant $G(\varphi )$ is inversely proportional to the scalar field $\varphi$ through Eq. (6).

Figure 3

CMB temperature anisotropy spectra in the scalar-tensor models with the $\mathrm{\Lambda }\mathrm{CDM}$ parameters. The data points with error bars represent the Planck data.

Figure 4

95.45% and 95.45% confidence contours in the ${{\alpha }_0}^2-\beta$ plane for the scalar-tensor $\mathrm{\Lambda }\mathrm{CDM}$ models with the other parameters marginalized. The black dashed line shows the function ${{\alpha }_0}^2=2.5\times {10}^{-4-4.5\beta }$ and the gray solid line shows the bound from the Solar System experiment.

Figure 5

Posterior distribution of $G_{\mathrm{rec}}/G_0$.

Figure 6

Comparison of the CMB temperature anisotropy spectra between Planck and WMAP. CMB temperature anisotropy spectra in the scalar-tensor models are also shown, which are same as in Fig. 3. Planck observational data in higher $\ell$ region ($\ell \gtrsim 1000$) provide stronger results of the parameter constraints.

Figure 7

Typical CMB polarization spectra ($EE$) in the scalar-tensor models with the $\mathrm{\Lambda }\mathrm{CDM}$ parameters fixed to the standard values. The data points with error bars show the Planck data.

Figure 8

Same as Fig. 7, but for the temperature polarization cross spectrum.

Figure 9

95.45% and 99.99% confidence contours in the ${{\alpha }_0}^2-\beta$ plane for the scalar-tensor nonflat $\mathrm{\Lambda }\mathrm{CDM}$ models with the other parameters marginalized (red), comparing with those of the flat universe case (blue). The black dashed line and the gray solid line show the function ${{\alpha }_0}^2=2.5\times {10}^{-4-4.5\beta }$ and the bound from the Solar System experiment, respectively.

Figure 10

Posterior distribution of $G_{\mathrm{rec}}/G_0$ for the nonflat models (red dashed) compared with the flat model (black).