Is there any particular reason you made the spatial topology $\mathbf{PR}^3\cong\mathrm{SO}(3)$ rather than $S^3$? That's not going to affect local physics, though, so at least for uniform initial conditions on some $\mathbf{PR}^3$ hypersurface, we can simply scale a restricted round metric
$$d\Omega^2 = d\psi^2 + \sin^2\psi(d\theta^2 + \sin^2\theta d\phi^2),\;\;\;\;0\leq\psi,\theta,\phi\leq\pi.$$
The Einstein equations have the same local form as for $S^3$, so we would get a spacetime that locally looks like the standard $k = +1$ Freidmann-Robertson-Walker solutions of the form $ds^2 = -dt^2 + a^2(t)d\Omega^2$. For example, a lambdavacuum (if energy density due to photons is $\rho\ll|\Lambda|$), radiation-dominated ($\rho\gg|\Lambda|$, $\rho = 3p$, scale factor $\propto t^{1/2}$), etc. These cases are extensively catalogued.
The difference being that the large-scale spatial structure is different, according to the antipodal identification you prescribe. Under the assumption of uniformity, the general answer to how this system evolves is simply: like the corresponding FRW universe. Depending on the conditions, it could even undergo transition from radiation-dominated to matter-dominated, just as our own universe did. I'm unclear as to whether you intended to exclude anything but photons in vacuum at one time or for all time.
