Matrix model cosmology in two space-time dimensions

We examine solutions to the classical Ishibashi-Kawai-Kitazawa-Tsuchiya matrix model equations in three space-time dimensions. Closed, open and static two-dimensional universes naturally emerge from such models in the commutative limit. We show that tachyonic modes are a generic feature of these cosmological solutions.

Figure 1

Parametric plots of the closed universe solutions in the three-dimensional embedding space (with time along the vertical direction) for three different values of $\upsilon <\frac{1}{2}$. Subfigure (b) is the solution (2.8). Subfigures (a) and (c) are solutions for $\upsilon =-1.5$ and $\upsilon =0.4$, respectively, when a cubic term is present in the action [c.f. sec. 3].

Figure 2

Parametric plots of (a) the static universe solution $\upsilon =\frac{1}{2}$ and open universe solutions for two different values (b) $\upsilon =0.6$ and (c) $\upsilon =1$ of $\upsilon >\frac{1}{2}$ in the three-dimensional embedding space (with time along the vertical direction) (c) is the de Sitter solution. The boundary conditions are $a(0)=1$ and $a^{\prime }(0)=0$.

Figure 3

Plots of the string energy (divided by $2\pi \mathcal{T}$) for the three types of solutions as a function of $\upsilon$. The energy for the de Sitter solution is zero (red solid line). The energy for the cylindrical solution is given by the blue dashed line. (The de Sitter solution and cylinder solution are singular at $\upsilon =0$.) The energy of the family of solutions given by inverse elliptic integrals is given by the black dashed-dotted line. It corresponds to closed universes for $\upsilon <\frac{1}{2}$, the static universe for $\upsilon =\frac{1}{2}$ and open universes for $\upsilon >\frac{1}{2}$. The boundary conditions are $a(0)=1$ and $a^{\prime }(0)=0$.