Expanding universe as a classical solution in the Lorentzian matrix model for nonperturbative superstring theory

Recently we have shown by Monte Carlo simulation that an expanding ($3+1$)-dimensional universe appears dynamically from a Lorentzian matrix model for type IIB superstring theory in ($9+1$) dimensions. Here we study the classical equation of motion as a complementary approach. While the Monte Carlo results represent the behavior at earlier times, the classical equation of motion is expected to be valid at later times. In particular, we find a class of SO(3) symmetric solutions, which exhibits the time-dependence compatible with the expanding universe, while having no spatial noncommutativity. Based on this result, we speculate that the spatial noncommutativity, which plays a crucial role in the spontaneous breaking of rotational symmetry, vanishes at later times due to some dynamical mechanism.