Quantum cosmology, eternal inflation, and swampland conjectures

In light of the recent swampland conjectures, we explore quantum cosmology and eternal inflation beyond the slow roll regime. We consider a model of a closed universe with a scalar field $\phi$ in the framework of tunneling approach to quantum cosmology. The scalar field potential is assumed to have a maximum at $\phi=0$ and can be approximated in its vicinity as $V(\phi)\approx 3H^{2}-\frac{1}{2}m^{2}\phi^{2}$. Using the instanton method, we find that for $m<2H$ the dominant nucleation channel for the universe is tunneling to a homogeneous, spherical de Sitter space. For larger values of $m/H$, the most probable tunneling is to an inhomogeneous closed universe with a domain wall wrapped around its equator. We determine the quantum state of the field $\phi$ in the nucleated universe by solving the Wheeler-DeWitt equation with tunneling boundary conditions. Our results agree with earlier work which assumed a slow-roll regime $m\ll H$. We finally show that spherical universes nucleating with $m<2H$ undergo stochastic eternal inflation with inflating regions forming a fractal of dimension $d>2$. For larger values of $m$ the field $\phi$ is unstable with respect to formation of domain walls and cannot be described by a perturbative stochastic approach.