Expanding ($3+1$)-Dimensional Universe from a Lorentzian Matrix Model for Superstring Theory in ($9+1$) Dimensions

We reconsider the matrix model formulation of type IIB superstring theory in ($9+1$)-dimensional space-time. Unlike the previous works in which the Wick rotation was used to make the model well defined, we regularize the Lorentzian model by introducing infrared cutoffs in both the spatial and temporal directions. Monte Carlo studies reveal that the two cutoffs can be removed in the large-$N$ limit and that the theory thus obtained has no parameters other than one scale parameter. Moreover, we find that three out of nine spatial directions start to expand at some “critical time,” after which the space has SO(3) symmetry instead of SO(9).

Figure 1

The extent of space $R(t{)}^2$ with $N=16$ and $n=4$ is plotted as a function of $t$ for five values of $\kappa$. The peak at $t=0$ starts to grow at some critical $\kappa$.

Figure 2

The nine eigenvalues of $T_{ij}(t)$ with $N=16$ and $n=4$ are plotted as a function of $t$ for $\kappa =4.0$. After the critical time $t_c$, three eigenvalues become larger, suggesting that the SO(9) symmetry is spontaneously broken down to SO(3).

Figure 3

Our data for $R(t{)}^2$ shown in Fig. 1 with $\kappa$ larger than the critical value are replotted against the shifted time $t-t_c$ in units of the size of the Universe $R(t_c)$ at the critical time.