On the Applications of Semiclassical Gravity in Cosmology and Black Hole Physics

The subject of this Ph.D thesis is the study of the propagation of a free quantum scalar matter field over a classical curved spacetime in the realms of Cosmology and Black Hole Physics. It gathers the original research activity of the author in the framework of Semiclassical Gravity and Quantum Field Theory in Curved Spacetimes. In a joint work with N. Pinamonti and D. Siemssen, the author shows the existence and uniqueness of local solutions of the semiclassical Einstein equations in cosmological spacetimes, driven by a free quantum massive scalar field arbitrarily coupled with the background curvature. The remarkable result of the work is to show that the source of regularity issues, which prevents to solve the semiclassical problem directly, is an unbounded operator hidden in the expectation value of the quantum stress-energy. However, this operator admits a more regular inverse, which also respects causality. Thus, the application of its inversion formula to the semiclassical equations allows to formulate a well-posed initial-value problem for local solutions in a small interval of time. In collaboration with N. Pinamonti, S. Roncallo, and N. Zanghì, the semiclassical approach to gravity is applied in the framework of four-dimensional spherically symmetric black holes, which are characterized by dynamical future, outer, trapping horizons. It is shown that the trace anomaly of the quantum stress-energy tensor for a massless, conformally coupled scalar field can be the source of black hole evaporation, after assuming vacuum-like initial conditions in the past and an auxiliary quantum energy condition outside the horizon. As an example, the rate of evaporation induced by the trace anomaly is explicitly evaluated in the Vaidya spacetime. Finally, in a joint paper with N. Pinamonti, the author studies the problem of stability of semiclassical solutions with higher-order derivative terms in a toy-model, consisting of a quantum scalar field in interaction with a classical scalar field. This toy-model mimics also the evolution induced by semiclassical Einstein equations in physically relevant backgrounds, such as cosmological spacetimes. The main result states that, if the quantum field is massive, then the back-reaction can restore stability on the classical background for wide choices of the renormalization constants, because linear perturbations with past compact spatial support decay polynomially in time at large times