Minisuperspace model of compact phase space gravity

The kinematical phase space of classical gravitational field is flat (affine) and unbounded. Because of this, field variables may tend to infinity, leading to appearance of singularities, which plague Einstein’s theory of gravity. The purpose of this article is to study the idea of generalizing the theory of gravity by compactification of the phase space. We investigate the procedure of compactification of the phase space on a minisuperspace gravitational model with two-dimensional phase space. In the affine limit, the model reduces to the flat de Sitter cosmology. The phase space is generalized to the spherical case, and the case of loop quantum cosmology is recovered in the cylindrical phase space limit. Analysis of the dynamics reveals that the compactness of the phase space leads to both UV and IR effects. In particular, the phase of recollapse appears, preventing the Universe from expanding to infinite volume. Furthermore, the quantum version of the model is investigated and the quantum constraint is solved. As an example, we analyze the case with the spin quantum number $s=2$, for which we determine transition amplitude between initial and final state of the classical trajectory. The probability of the transition is peaked at $\mathrm{\Lambda }=0$.

Figure 1

Momentum and position polymerization (cylindrical) limits of the spherical phase space.

Figure 2

Evolution of the canonical variable $q$.

Figure 3

Phase trajectories for the system under consideration.

Figure 4

Evolution of the vector $\overrightarrow{S}=(S_x,S_y,S_x)$ for (from left to right) $\delta =0.1$ (red thick curve), $\delta =0.5$ (black thick curve), $\delta =0.9$ (blue thick curve). There are symmetric solutions in the three other quadrants of the sphere. The $S_y$ component remains constant during the evolution and is given by $S_y=S\sqrt{\delta }$. The $t\to \pm \infty$ limits are located at the ends of the curves (on the equator). The maximal value of the $S_z$ component corresponds to the maximal value of the $q$ variable that is reached on a trajectory.

Figure 5

Modulus square of the transition amplitude as a function of the parameter $\delta =\mathrm{\Lambda }/{\mathrm{\Lambda }}_{*}\in [0,1]$ and $\delta \ne \frac{7}{18}\approx 0.389$. The $|W(\delta ){|}^2$ decreases from $|W(0){|}^2=(\frac{6}{13}{)}^2\approx 0.213$ to $|W(4/21){|}^2=0$; then the probability of transition increases until reaching $|W(1){|}^2={(\frac{867}{1976})}^2\approx 0.193$. For the special case $\delta =7/18$ we have $|W(7/18){|}^2\approx 0.017$.