Asymptotically Safe Gravity: from spacetime foliation to cosmology

General Relativity provides a remarkably successful description of gravity in terms of the geometric properties of spacetime. Nevertheless, the quantum nature of matter and the existence of regimes in which the classical description of gravity breaks down suggest that the short-distance behavior of spacetime may require a more fundamental quantum theory for the gravitational interaction. However, as it is well known, the quantization of General Relativity leads to a (perturbatively) non-renormalizable theory. The Asymptotic Safety scenario for Quantum Gravity provides a natural mechanism for constructing a fundamental quantum theory for the gravitational force within the framework of Quantum Field Theory. In this scenario the short-distance behavior of gravity is governed by a Non-Gaussian Fixed Point (NGFP) of the underlying renormalization group flow. The high-energy modifications of gravity resulting from the scaling of couplings around the NGFP may have profound implications in astrophysics and cosmology. The Functional Renormalization Group (FRG) constitutes an ideal tool to explore both the fundamental aspects and phenomenological implications of this scenario. The aim of this Ph.D. thesis is twofold. Firstly, we discuss the formulation of a Functional Renormalization Equation tailored to the Arnowitt-Deser-Misner formalism. The construction also includes an arbitrary number of matter fields minimally coupled to gravity. This allows us to analyze the effect of matter on the fixed point structure of the gravitational renormalization group flow. Within a certain class of approximations, it will be shown that most of the commonly studied matter models, including the standard model of particle physics, give rise to a NGFP with real critical exponents. This result is important for the second part of this thesis, the phenomenological implications of Asymptotic Safety. Specifically, using a renormalization group improvement procedure, we study the quantum gravitational corrections arising in two different situations: the inflationary phase in the very early universe and the formation of black holes in the gravitational collapse of massive stars. In the context of cosmology, it will be shown that the predictions of Asymptotic Safety lead to an inflationary model compatible with the recent Planck data. In addition a comparison between the inflationary models derived from foliated quantum gravity-matter systems with observations can put constraints on the primordial matter content of the universe. Finally, the study of gravitational collapse reveals that the anti-screening behavior of the Newton's coupling in the short-distance limit renders the strength of the gravitational tidal forces weaker: the strong singularity appearing in the classical treatment is turned into a weak singularity once corrections from the renormalization group are included.