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) approximate to 3H2 - 21 m2 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 << 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.