Vacuum decay and quadratic gravity

Metastable states are classically stable at zero temperature but can decay due to quantum tunneling. The rate of this process is exponentially small and it may be computed in Euclidean space in the Coleman-de Luccia formalism. The exponential suppression is determined by the Euclidean action computed on a trajectory with definite boundary conditions, known as Coleman-de Luccia instanton, or bounce. In some theories, the bounce may not exist or its on-shell action may be ill-defined or infinite, thus hindering the vacuum decay process. The issue of vacuum stability is, in fact, not just speculation: the Standard Model vacuum state is itself metastable. The Higgs field may tunnel outside its potential well, with catastrophic consequences for all observers. Luckily, the typical lifetime of such a state is predicted to be very long. Still, unknown high energy physics can change it by several orders of magnitude, and particle physics theories as well as cosmological models that predict large decay rates are ruled out thanks to the anthropic principle. Moreover, gravitational effects play an important role in this process, especially in the early Universe. It is thus important to examine in detail vacuum decay phenomena in gravitational settings and to keep the underlying field theory as general as possible. This thesis aims at exploring existence conditions for the Coleman-de Luccia instanton in gravitational settings. The first two chapters are dedicated to outlining the basic formalism and describing preexisting results about vacuum decay in cosmology. The Euclidean path integral approach for decay rate calculations, which was first discussed by Callan and Coleman, is introduced in Chapter 1. A quantum mechanical description of the problem is formulated and then extended to field theory. A detailed analysis of bounce calculations and their physical interpretation as bubbles of true vacuum follows. The Higgs field stability within the Standard Model is also addressed. Gravitational effects on the vacuum decay process are considered in Chapter 2, by focusing on the decay from Minkowski and de Sitter space, as they have important cosmological consequences respectively in the current Universe (due to the smallness of the cosmological constant) and at early times. The implications on Higgs decay are discussed in both settings. The last two chapters are dedicated to new results. Vacuum decay in field theories with a scalar field and quadratic gravity is investigated. An Einstein-Hilbert term, a non-minimal coupling, and a quadratic Ricci scalar are considered while keeping the scalar field potential general. The focus is on decay from Minkowski and de Sitter space, due to their importance in cosmology. Scalar fields with Einstein-Hilbert gravity are discussed in Chapter 3, by showing that the bounce at large Euclidean radii has an analytical form that is almost entirely independent of the potential, which is called the "asymptotic bounce". Bounds on the Hubble parameter in the de Sitter case are also explored, by giving an analytical explanation to numerical evidence present in the literature. These properties are used, in Chapter 4, to test for stabilization of the false vacuum state in quadratic gravity. Conclusions follow.