Variation of the speed of light with temperature of the expanding universe

From an extended relativistic dynamics for a particle moving in a cosmic background field with temperature $T$, we aim to obtain the speed of light with an explicit dependence on the background temperature of the universe. Although finding the speed of light in the early universe much larger than its current value, our approach does not violate the postulate of special relativity. Moreover, it is shown that the high value of the speed of light in the early universe was drastically decreased before the beginning of the inflationary period. So we are led to conclude that the theory of varying speed of light should be questioned as a possible solution of the horizon problem.

Figure 1

This figure shows two graphics, namely, $R(t)$, which is the size (radius) of the universe as a function of time, and $c(T)$, representing the speed of light with dependence on the temperature of the universe according to Eq. (7). At the beginning of the universe when it was a singularity with a minimum radius of the order of the Planck radius, i.e., $R_P\sim {10}^{-35}\mathrm{m}$, having the Planck energy scale $E_p\sim {10}^{19}\mathrm{GeV}$, which corresponds to the Planck temperature $T_P\sim {10}^{32}\mathrm{K}$ and the Planck time $t_P\sim {10}^{-43}\mathrm{s}$, the speed of light $c^{\prime }$ was infinite since there was no spacetime [see Eq. (7) for $c(T)$]. But immediately after, when $T_1={10}^{31}\mathrm{K}$, the speed of light had already assumed a value close to the current value as shown by Eq. (7) for $c(T)$, and therefore a cone of light (a spacetime) had been formed; i.e., with $c=2.99792458\times {10}^8\mathrm{m}/\mathrm{s}$ for the present time, then, according to the function $c(T)$, we find $c_1=c(T_1)=3.16008998\times {10}^8\mathrm{m}/\mathrm{s}$ (see the figure). Subsequently, for $T_2={10}^{30}\mathrm{K}$, we find $c_2=c(T_2)=3.01302757\times {10}^8\mathrm{m}/\mathrm{s}$. For $T_3={10}^{29}\mathrm{K}\Rightarrow c_3=c(T_3)=2.99942467\times {10}^8\mathrm{m}/\mathrm{s}$. And for $T_4={10}^{28}\mathrm{K}\Rightarrow c_4=c(T_4)=2.99807447\times {10}^8\mathrm{m}/\mathrm{s}$. Finally, for $T_5={10}^{27}\mathrm{K}\Rightarrow c_5=c(T_5)=2.99793955\times {10}^8\mathrm{m}/\mathrm{s}$. From this temperature $T_5={10}^{27}\mathrm{K}$, when $t=t_5\sim {10}^{-35}\mathrm{s}$, corresponding to the energy scale of the grand unified theory (GUT) with ${10}^{14}\mathrm{GeV}$, the universe inflated very quickly, starting with a radius $R_5\sim {10}^{-25}\mathrm{m}$ and reaching $R_6\sim {10}^{25}\mathrm{m}$ at the time $t_6\sim {10}^{-32}\mathrm{s}$; i.e., the size of the universe increased rapidly 50 orders of magnitude. Since the speed of light $c_5\approx c$, VSL should be put in doubt. Hence, perhaps the vacuum energy had played a fundamental role in that epoch.