Microscopic quantum mechanics of the $p=\rho$ universe

We present a complete quantum mechanical description of a flat Friedmann-Robertson-Walker universe with equation of state $p=\rho$. We find a detailed correspondence with our heuristic picture of such a universe as a dense black hole fluid. Features of the geometry are derived from purely quantum input.

Figure 1

Nested causal diamonds defining an observer in a time symmetric space-time.

Figure 2

Nested causal diamonds defining a nearest neighbors pair of observers in a time symmetric space-time.

Figure 3

The two causal diamonds ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ ($z$ spatial coordinate suppressed) and the maximal causal diamond ${\mathcal{D}}_M$ that fits in the intersection ${\mathcal{D}}_1\cap {\mathcal{D}}_2$. The picture is valid for Minkowski space-time and more generally conformally flat space-times.

Figure 4

The base spheres $S_{{\mathcal{D}}_1}$ and $S_{{\mathcal{D}}_2}$ of the two causal diamonds ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ (Fig. 3). $\tilde{S}$ is the maximal sphere belonging to the intersection of $S_{{\mathcal{D}}_1}$ and $S_{{\mathcal{D}}_2}$. $\tilde{S}$ coincides with $S_{{\mathcal{D}}_M}$, the base sphere of ${\mathcal{D}}_M$.

Figure 5

The base spheres $S_{{\mathcal{D}}_1}$ and $S_{{\mathcal{D}}_2}$ of the two causal diamonds ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ (Fig. 3), coordinate $z$ suppressed.

Figure 6

The two causal diamonds ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ ($z,y$ spatial coordinates suppressed) and the maximal causal diamond ${\mathcal{D}}_M$ that fits in the intersection ${\mathcal{D}}_1\cap {\mathcal{D}}_2$. ${\eta }_i$, ${\eta }_f$ are the points of ${\mathcal{D}}_1\cap {\mathcal{D}}_2$ with the minimum and maximum values of $\eta$. $\Delta x$ is the separation among the tips of ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ at time ${\eta }_1$.

Figure 7

Detail of the maximal causal diamond ${\mathcal{D}}_M$. $h={\eta }_f-{\eta }_i$ is the height of ${\mathcal{D}}_M$ and $r_{{\mathcal{D}}_M}$ is the radius of the base sphere on ${\mathcal{D}}_M$.

Figure 8

Detail of the maximal causal diamond ${\mathcal{D}}_M$. $r$ is the radius of a generic sphere on ${\mathcal{D}}_M$; $r_M$ is the radius of the sphere of maximal area on ${\mathcal{D}}_M$ i.e. the maximal sphere.