Generalized theory of varying alpha

In this paper, we formulate a generalization of the simple Bekenstein-Sandvik-Barrow-Magueijo theory of varying alpha by allowing the coupling constant $\omega$ for the corresponding scalar field $\psi$ to depend on $\psi$. We focus on the situation where $\omega$ is exponential in $\psi$ and find the late-time behaviors that occur in matter-dominated and dark-energy dominated cosmologies. We also consider the situation when the background expansion scale factor of the Universe evolves in proportion to an arbitrary power of the cosmic time. We find the conditions under which the fine-structure “constant” increases with time, as in the Bekenstein-Sandvik-Barrow-Magueijo theory, and establish a cosmic no-hair behavior for accelerating universes. We also find the conditions under which the fine-structure constant can decrease with time and compare the whole family of models with astronomical data from quasar absorption spectra.

Figure 1

A plot of the four solutions of Eq. (13) for the various choices $\psi (\ln x=0)\in \{-1,-0.5,0,0.5\}$ and $d\psi /d(\ln x){|}_{\ln x=0}=0$. Here, $x$ is defined as $\ln (t/t_1)$, where $t_1$ is an arbitrary constant, and we have chosen (arbitrarily) the parameters $\mu =2$ and $C=0.1$. Notice that, in this case ($\mu >-2$), all the solutions eventually converge to a common increasing asymptote, independently of the initial conditions.

Figure 2

As in Fig. 1, but with $\mu =6$. All solutions still converge to a common increasing asymptote at late times. However, the slope of the asymptote is more gentle than that in Fig. 1, because the value of $\mu$ has increased. The slope of the asymptote can be made as gentle as desired by taking $\mu$ to be sufficiently large.

Figure 3

A plot of four solutions of Eq. (13) for the various choices $\psi (\ln x=0)\in \{-1,-0.5,0,0.5\}$ and $d\psi /d(\ln x){|}_{\ln x=0}=0$, where, again, $x$ is defined as $\ln (t/t_1)$. The parameter values are now $\mu =-4$ and $C=-0.1$. In this case, $(\mu <-2)$, the solutions eventually converge to a common decreasing asymptote, and this behavior is insensitive to the initial conditions.

Figure 4

As in Fig. 3, but with $\mu =-6$. All solutions still converge to a common decreasing asymptote at late times. However, the slope of the asymptote is more gentle than that in Fig. 3 (because $\mu$ is now more negative). Again, this slope can be made as gentle as desired by taking a sufficiently large negative $\mu$.

Figure 5

As in Fig. 1, but these solutions now occur within an accelerating universe. The parameter values are now $\mu =2$, $C=0.1$, and $M=0.01$. Regardless of the initial conditions, the solutions all tend to constant values after a short initial time. This illustrates how the time variation of $\alpha$ is suppressed in a $\Lambda$-dominated universe.

Figure 6

These contour plots display the value of the ${\chi }^2$ statistic (according to the legend shown in the right-hand color bars) for various choices of $D$, $\mu$, and ${\psi }_0$ in a model that exhibits a smooth transition between the dark-matter and dark-energy dominated eras. Each picture corresponds to a different value of $f(0)\equiv e^{2\psi (0)}$ ranging from ${10}^{-3}$ to ${10}^4$. The vertical and horizontal axes represent the parameters $\mu$ and $D$, respectively. Darker regions in the plots indicate parameter choices that give rise to a lower ${\chi }^2$ statistic, and therefore correspond to a better fit.