Dynamic Models for the Beginning, Hubble Law and the Future of the Universe Based on Strong Cosmological Principle and Yang-Mills Gravity

We discuss highly simplified dynamic models for the beginning, expansion and future of the universe based on the strong cosmological principle and Yang-Mills gravity in flat space-time. We derive a relativistic Okubo equation of motion for galaxies with a time-dependent effective metric tensor $G_{\mu \nu }(t)$. The strong cosmological principle states that $G_{\mu \nu }(t)={\text{e}}_{\mu \nu }{A}^2(t)$ for $t\ge 0$. In a model (HHK) with Yang-Mills gravity in the super-macroscopic limit, one has $A(t)=a_o{t}^{1\mathrm{/}2}$, which leads to the initial mass run away velocity $\dot{r}(0)=c$, associated with $r(0)=r_o>0$. Thus, the Okubo equation of motion for galaxies predicts a `detonation’ at the beginning of the universe. The Okubo equation also implies $r(\mathrm{\infty })\to \mathrm{\infty }$, $\dot{r}(\mathrm{\infty })\to 0$ with zero redshift for the future of the universe. In addition, the Okubo equation leads to the usual Hubble’s law $\dot{r}(t)\approx H(t)r(t)$, where $H(t)=\dot{A}(t)\mathrm{/}A(t)$ in non-relativistic approximation. We also discuss a model with a strict Hubble linear relation $\dot{r}(t)\approx const\mathrm{.}\times r(t)$ for all time. This model gives a silent beginning of the universe: $\dot{r}(t)=0,
\ddot{r}(t)\to\infty$as$t \to 0$;andfinalradius$r(t) \to \infty,$ final velocity, $\dot{r}(t)\to c$, $\ddot{r}(t)\to 0$ as $t\to \mathrm{\infty }$. In all models with the strong cosmological principle in flat space-time, Hubble’s recession velocities are predicted to have a maximum, i.e., the speed of light, as measured in an inertial frame.