Probing the spacetime fabric: from fundamental discreteness to quantum geometries

This thesis deals primarily with the phenomenology associated to quan- tum aspects of spacetime. In particular, it aims at exploring the phenomeno- logical consequences of a fundamental discreteness of the spacetime fabric, as predicted by several quantum gravity models and strongly hinted by many theoretical insights. The first part of this work considers a toy-model of emergent spacetime in the context of analogue gravity. The way in which a relativistic Bose– Einstein condensate can mimic, under specific configurations, the dynamics of a scalar theory of gravity will be investigated. This constitutes proof-of- concept that a legitimate dynamical Lorentzian spacetime may emerge from non-gravitational (discrete) degrees of freedom. Remarkably, this model will emphasize the fact that in general, even when arising from a relativis- tic system, any emergent spacetime is prone to show deviations from exact Lorentz invariance. This will lead us to consider Lorentz Invariance Viola- tions as first candidate for a discrete spacetime phenomenology. Having reviewed the current constraints on Lorentz Violations and stud- ied in depth viable resolutions of their apparent naturalness problem, the second part of this thesis focusses on models based on Lorentz invariance. In the context of Casual Set theory, the coexistence of Lorentz invariance and discreteness leads to an inherently nonlocal scalar field theory over causal sets well approximating a continuum spacetime. The quantum as- pects of the theory in flat spacetime will be studied and the consequences of its non-locality will be spelled out. Noticeably, these studies will lend support to a possible dimensional reduction at small scales and, in a clas- sical setting, show that the scalar field is characterized by a universal non- minimal coupling when considered in curved spacetimes. Finally, the phenomenological possibilities for detecting this non-locality will be investigated. First, by considering the related spontaneous emission of particle detectors, then by developing a phenomenological model to test nonlocal effects using opto-mechanical, non-relativistic systems. In both cases, one could be able to cast in the near future stringent bounds on the non-locality scale