Exact recession velocity and cosmic redshift based on cosmological principle and Yang-Mills gravity

Based on the cosmological principle and quantum Yang-Mills gravity in the super-macroscopic limit, we obtain an exact recession velocity and cosmic redshift z, as measured in an inertial frame \begin{document}$ F\equiv F(t,x,y,z). $\end{document} For a matter-dominated universe, we have the effective cosmic metric tensor \begin{document}$ G_{\mu\nu}(t) = (B^2(t),-A^2(t), -A^2(t),-A^2(t)),$\end{document} \begin{document}$ \ A\propto B\propto t^{1/2} $\end{document} , where \begin{document}$ t $\end{document} has the operational meaning of time in \begin{document}$ F $\end{document} frame. We assume a cosmic action \begin{document}$ S\equiv S_{\rm cos} $\end{document} involving \begin{document}$ G_{\mu\nu}(t) $\end{document} and derive the ‘Okubo equation’ of motion, \begin{document}$ G^{\mu\nu}(t)\partial_\mu S \partial_\nu S - m^2 = 0 $\end{document} , for a distant galaxy with mass \begin{document}$ m $\end{document} . This cosmic equation predicts an exact recession velocity, \begin{document}$ \dot{r} = rH/[1/2 +\sqrt{1/4+r^2H^2/C_o^2} ] , where \begin{document}$ H = \dot{A}(t)/A(t) $\end{document} and \begin{document}$ C_o = B/A $\end{document} , as observed in the inertial frame \begin{document}$ F $\end{document} . For small velocities, we have the usual Hubble's law \begin{document}$ \dot{r} \approx rH $\end{document} for recession velocities. Following the formulation of the accelerated Wu-Doppler effect, we investigate cosmic redshifts z as measured in \begin{document}$ F $\end{document} . It is natural to assume the massless Okubo equation, \begin{document}$ G^{\mu\nu}(t)\partial_\mu \psi_e \partial_\nu \psi_e = 0 $\end{document} , for light emitted from accelerated distant galaxies. Based on the principle of limiting continuation of physical laws, we obtain a transformation for covariant wave 4-vectors between and inertial and an accelerated frame, and predict a relationship for the exact recession velocity and cosmic redshift, \begin{document}$ z = [(1+V_r)/(1-V_r^2)^{1/2}] - 1 $\end{document} , where \begin{document}$ V_r = \dot{r}/C_o , as observed in the inertial frame \begin{document}$ F $\end{document} . These predictions of the cosmic model are consistent with experiments for small velocities and should be further tested.