5.1 Preliminary Estimate

We first note that the quantization can be interpreted as giving a relation among the parameters of the model, in our case the charge $Q$ and the de Sitter cosmological horizon $H$. Let us take for example a total action of the order of the quantum, $l _{\mathrm{P}} ^{2}$, i.e. associated to a state with quantum number $n = 1$, for a de Sitter cosmological horizon $H = \sqrt{2}$. Then $\bar{S} \approx 1 / 2$, i.e. $\Theta = Q/H \approx 0.55$, so that $Q \approx 0.78$. This shows, as a preliminary estimate, that a small gravitational system with quantum properties is conceivable.

We can also see how the action behaves for variations of the parameters $Q$ and $H$ by a numerical plot of the level curves of the action. This is shown in figure 10.

Figure: Graph of the level curves for the action for the level values $1 , 2 , \dots , 50$. The levels are obtained using the numerically evaluated action with the function ContourPlot of Mathematica$^{\circledR}$. The thicker dashed black line displays the limit $H = (4/3) ^{3/2} Q$ of the condition $Q / H > (3/4) ^{3/2}$ for which bounded trajectories exist. The grayed region is shown, blew up, in figure 11.

To get a clearer plot in the region close to the limiting line $H = (4/3) ^{3/2} Q$ for small $Q$, a smaller region of the plot is shown, enlarged, in figure 11.

Figure 11: Blow up of the region of small $Q$, $H$ parameters. This better shows that the quantization condition implies a non-vanishing minimum allowed value for both the charge $Q$ and the de Sitter horizon $H$, i.e. a non-vanishing minimum allowed value for the charge $Q$ together with a maximum allowed value for the cosmological constant $1/H$.