Relation between varying fine structure constant and cosmological components

We propose a simple model based on the assumption that the varying fine structure constant $\alpha$ is an effect of the cosmological expansion to investigate the relation between the varying $\alpha$ and the cosmological components. For a spatially flat, homogeneous, and isotropic universe, the current proportion of cosmological components and age of the universe predicted by the model are consistent with the cosmological observations. Furthermore, the predicted current variation rate of $\alpha$ is also close to the atomic clock measurements. For the early universe, we predict a very strict constraint, which is compatible with the upper limit given by the investigations of cosmic microwave background and big bang nucleosynthesis.

Figure 1

Fitting the binned Keck samples with Eq. (17). The dark blue squares are the weighted mean values of the binned Keck samples and the black bars are the corresponding errors. The red curve is the optimal fitting. The minimal ${\chi }^2$ of the fitting is 4.99, when the parameters ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ and $A$ take the value 0.765 and $8.3\times {10}^{-6}$, respectively. The light blue, yellow, and gray hatched region are the confidence intervals for $1\sigma$, $2\sigma$, and $3\sigma$, respectively.

Figure 2

The ${\chi }^2$ of fitting the binned Keck samples within 0.5 of its minimal value (${\chi }^2=4.99$) in the (${\mathrm{\Omega }}_{\mathrm{\Lambda }}$, $A$) plane.

Figure 3

The panel (a), (b), and (c) show the deviations between Keck and VLT samples measured in the same direction with same redshift, the same direction $\theta =0°$ with the difference of redshift $\mathrm{\Delta }z<0.1$, and the angle of directions $\theta <1°$ with the difference of redshift $\mathrm{\Delta }z<0.1$, respectively. The red lines are the optimal fittings with the proportional function $\mathrm{\Delta }\alpha /{\alpha }_{(\mathrm{VLT}\text{−}\mathrm{Keck})}=kz$. The three slopes given by the optimal fittings are $k=(0.52\pm 0.30)\times {10}^{-5}$, $k=(0.58\pm 0.35)\times {10}^{-5}$, and $k=(0.57\pm 0.31)\times {10}^{-5}$. The light blue, yellow, and gray hatched region are the confidence intervals for $1\sigma$, $2\sigma$, and $3\sigma$, respectively.

Figure 4

Fitting the binned all the samples with Eq. (17). The dark blue squares are the weighted mean values of the binned samples and the black bars are the corresponding errors. The red curve is the optimal fitting. The minimal ${\chi }^2$ of the fitting is 2.50, when the parameters ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ and $A$ take the value 0.713 and $9.8\times {10}^{-6}$, respectively. The light blue, yellow, and gray hatched region are the confidence intervals for $1\sigma$, $2\sigma$, and $3\sigma$, respectively..

Figure 5

The ${\chi }^2$ of fitting the binned all the samples within 0.5 of its minimal value (${\chi }^2=2.50$) in the (${\mathrm{\Omega }}_{\mathrm{\Lambda }}$, $A$) plane.

Figure 6

The curve of $\mathrm{\Delta }\alpha /\alpha$ varying with $H_{0}t$ is predicted with Eq. (13). The blue point denotes the current universe.