Singularity avoidance in the hybrid quantization of the Gowdy model

One of the most remarkable phenomena in loop quantum cosmology is that, at least for homogeneous cosmological models, the big bang is replaced with a big bounce that connects our Universe with a previous branch without passing through a cosmological singularity. The goal of this work is to study the existence of singularities in loop quantum cosmology, including inhomogeneities, and check whether the behavior obtained in the purely homogeneous setting continues to be valid. With this aim, we focus our attention on the three-torus Gowdy cosmologies with linearly polarized gravitational waves and use effective dynamics to carry out the analysis. For this model, we prove that all the potential cosmological singularities are avoided, generalizing the results about resolution of singularities to this scenario with inhomogeneities. We also demonstrate that, if a bounce in the (Bianchi background) volume occurs, the inhomogeneities increase the value of this volume at the bounce with respect to its counterpart in the homogeneous case.

Figure 1

(a): The colored area is the region of the parameter space where there is at least one solution to Eq. (35) such that $|x|\le 1$. The left (right) wedge corresponds to the $p$ branch ($m$ branch). Both branches coexist in the darker area. (b): Region compatible with the condition $v^2>0$ for $z=1$. The left (right) wedge corresponds to the $m$ branch ($p$ branch). The lower wedge admits both branches. The line $\beta S=-z$ and the hyperbola $\beta -1+2\beta S(\beta S+z)=0$ are also depicted. (c): Region where both restrictions are satisfied simultaneously.

Figure 2

Evolution of the homogeneous volume $v$ in effective trajectories featuring bounces. The dynamical equations (a2, a3, a4, a5, a6) were integrated numerically with Matlab, choosing the initial values so that the trajectory started in the feasible region: (a) $\beta =100$, $S=-0.994771$, $z=0.977429$; (b) $\beta =1000$, $S=-0.525495$, $z=0.0989956$. For simplicity, the only present inhomogeneous modes were those with $m=1$, $-1$ in both cases.