Quantum depletion in gravitational vacuum

The corpuscular theory of gravity proposed by Gia Dvali and Cesar Gomez describes gravitational backgrounds as Bose-Einstein Condensates (BEC) of weakly interacting soft gravitons, offering a new perspective on the possible unification of Einstein's General Relativity (GR) and the quantum nature of matter. Such a quantum description of gravitational backgrounds provides a new interpretation of particle production phenomena, such as Hawking's Radiation(HR) in Black Holes (BHs). In fact these are no longer seen as vacuum processes, but are instead related to the quantum depletion of the condensate. The aim of this work is to try to get more insight on this connection between the quantum depletion of a BEC and the HR. To do so we focus on the description of a BEC confined in a spherical volume and with a long range $1/r$ interaction. Starting from the Bogoliubov approach to the study of weakly interacting BEC, we will first consider a simplified one-dimensional case, to be interpreted as a simplified situation in which we can implement the 1/r interaction between the constituents. We will then proceed to the study of the three-dimensional spherically symmetric case, in which we vary the field expansion to try to account for propagating modes. Our work will show that the approach used in this text turns out to be incapable of efficiently describe particles escaping from the condensate and will give hints on possible ways to solve this issue.