Non-genericity of the Nariai solutions: II. Investigations within the Gowdy class

This is the second of two papers in which we study the asymptotics of the generalized Nariai solutions and its relation to the cosmic no-hair conjecture. In the first paper, the author suggested that, according to the cosmic no-hair conjecture, the Nariai solutions are non-generic among general solutions of Einstein's field equations in vacuum with a positive cosmological constant. We explained that this is true within the class of spatially homogeneous solutions. In this current paper, we continue these investigations within the spatially inhomogeneous Gowdy case. On the one hand, we are motivated to understand the fundamental question of cosmic no-hair and its dynamical realization in more general classes than the spatially homogeneous case. On the other hand, the results of the first paper suggest that the instability of the Nariai solutions can be exploited to construct and analyze physically interesting cosmological black hole solutions in the Gowdy class, consistent with certain claims by Bousso in the spherically symmetric case. However, in contrast to that, we find that it is not possible to construct cosmological black hole solutions by means of small Gowdy symmetric perturbations of the Nariai solutions and that the dynamics shows a certain new critical behavior. For our investigations, we use the numerical techniques based on spectral methods which we introduced in a previous publication.