Constraining the variation of $G$ by cosmic microwave background anisotropies

We use the Cosmic Microwave Background Anisotropies (CMBA) power spectra to constrain the cosmological variation of gravitational constant $G$. It is found that the sensitivity of CMBA to the variation of $G$ is enhanced when $G$ is required to converge to its present value. The variations of $G$ from the CMB decoupling epoch $z\sim 1000$ to the present time are modeled by a step function and a linear function of scale factor $a$ respectively, and the corresponding 95% confidence intervals for $G/G_0$ are [0.95, 1.05] and [0.89, 1.13], $G_0$ being the present value. The CMBA constraint is unique in the sense that it entails the range of redshift from $z\approx 1000$ to 0.

Figure 1

The CMBA temperature angular power spectra for three values of the Newtonian gravitational constant with instantaneous stabilization $z_s=0$. The solid, dotted and dashed curves correspond to ${\lambda }_0^2=1$, 0.5 and 3, respectively. The damping effect is only noticeable for $l>500$.

Figure 2

Same as Fig. 1 but with spectra with $z_s=10$ in contrast with those with $z_s=0$. The standard spectrum (solid) nearly coincides with the one with ${\lambda }_0^2=1.2$ and $z_s=0$ (dot-dashed). When instantaneous stabilization occurs at $z_s=10$, marked shifts to larger (smaller) $l$ scales result for ${\lambda }_0^2=1.2$ (dotted) (0.8 (dashed)).

Figure 3

The E-polarization spectra with instantaneous stabilization at $z_s=0$ and 10. The peaks shift to larger (smaller) $l$ scales for ${\lambda }_0^2=2$ (0.6) when instantaneous stabilization takes place at $z_s=10$, in contrast to the cases without stabilization ($z_s=0$).

Figure 4

The effects of linear variation of $G$ on the temperature power spectrum. The solid line shows the spectrum when there is no variation of $G$. When ${\lambda }_0^2=1.2$ (0.8), the peaks shift to larger (smaller) $l$ scales. $a_s$ is set to be 1.

Figure 5

The left (right) panel shows $\delta l_n/n=\pi \delta (d_{*}/r_s)$ vs ${\lambda }_0^2$ for instantaneous (linear) stabilization. For the plot on the left, the thick (red) solid curve corresponds to $a_s=0.1$ and the dashed curve to $a_s=0.5$. The dots correspond to $\delta l_n/n$ averaged over the first three peaks of the spectra in Fig. 2 (${\lambda }_0^2=1.2$, $z_s=10$ and ${\lambda }_0^2=0.8$, $z_s=10$). The two thin solid (black) curves are obtained using the simplified formula Eq. (16) for $a_s=1$ and 0.5, respectively. On the right, the solid and dashed curves correspond to linear stabilization with $a_s=1$ and 0.5, respectively. Similarly we denote by the dots the average $\delta l_n/n$ in Fig. 4 for $a_s=1$, and ${\lambda }_0^2=1.2$ and 0.8, respectively.

Figure 6

The marginal distributions of ${\lambda }_0^2$ and $\ln z_s$ obtained using the three-year WMAP data set in the instantaneous stabilization scenario. Here and thereafter, the maximum of the distribution is normalized arbitrarily to 1.

Figure 7

The contour plot of the joint distribution of ${\lambda }_0^2$ and $\ln z_s$ constrained using the three-year WMAP data set in the instantaneous stabilization scenario. The inner and outer solid lines are the 68% and 95% confidence level contours, respectively.

Figure 8

The marginal distributions of ${\lambda }_0^2$ for the linear stabilization scenario in Eq. (3) with $z_s=0$ and $z_s=0.8$ respectively, constraint using the three-year WMAP data.