Noncommutative geometry and cosmology

We study some consequences of noncommutativity to homogeneous cosmologies by introducing a deformation of the commutation relation between the minisuperspace variables. The investigation is carried out for the Kantowski-Sachs model by means of a comparative study of the universe evolution in four different scenarios: the classical commutative, classical noncommutative, quantum commutative, and quantum noncommutative. The comparison is rendered transparent by the use of the Bohmian formalism of quantum trajectories. As a result of our analysis, we found that noncommutativity can modify significantly the universe evolution, but cannot alter its singular behavior in the classical context. Quantum effects, on the other hand, can originate nonsingular periodic universes in both commutative and noncommutative cases. The quantum noncommutative model is shown to present interesting properties, as the capability to give rise to nontrivial dynamics in situations where its commutative counterpart is necessarily static.

Figure 1

The typical behavior of the universe characteristic volume $l^3(t)$, volume expansion $\Theta (t)$, and shear $\sigma (t)$ in the commutative and noncommutative versions of the classical Kantowski-Sachs universe. The initial conditions chosen are ${\beta }_0=-11$, $P_{{\beta }_0}=2/5$, and $t_0=0$. In (a)–(c) $\theta =0$ (thick lines) and $\theta =5$ (thin lines). In (d) $\theta =0$ (thick line) and $\theta =-5$ (thin line).

Figure 2

The normalized field of velocities corresponding to the Bohmian differential equations for the commutative Kantowski-Sachs universe with $\mu =1/10$ and $\nu =-1/5$.

Figure 3

The evolution of the universe characteristic volume, volume expansion, and shear in the commutative quantum Kantowski-Sachs universe with $\mu =1/10$ and $\nu =-1/5$. (a),(c), and (e) $\Omega (0)=2$, $\beta (0)=-24$. (b),(d), and (f) $\Omega (0)=7.7$, $\beta (0)=-21.7$. (g),(h) enlarged plots of parts of (e).

Figure 4

The normalized field of velocities corresponding to the Bohmian differential equations for noncommutative Kantowski-Sachs universe with $\mu =1/10$, $\nu =-1/5$, and $\theta =4$.

Figure 5

The evolution of the characteristic volume of the noncommutative quantum Kantowski-Sachs universe (a) $\mu =1/10$, $\nu =-1/5$, $\theta =4$ and $\Omega (0)=2$, $\beta (0)=-24$. (b) $\mu =1/10$, $\nu =-1/5$, $\theta =-4$ and $\Omega (0)=2$, $\beta (0)=-24$. (c) $\mu =1/10$, $\nu =-1/5$, $\theta =1$ and $\Omega (0)=7.7$, $\beta (0)=-21.7$. (d) $\mu =1/10$, $\nu =-1/5$, $\theta =12$ and $\Omega (0)=7.7$, $\beta (0)=-21.7$.