On the Renormalization Group Flow of Gravity

Abstract

Understanding gravity as a fundamental theory implies understanding its behavior as we move across different length scales. In the present thesis, we have investigated this scale-dependent behavior by studying the renormalization group flow of gravity. A central focus of our investigations was the asymptotic safety scenario, which posits the existence of a non-trivial fixed point of the gravitational renormalization group controlling the behavior of the theory in the ultraviolet and providing it with a predictive and well-defined high-energy limit. Our main tool was a continuous Wilsonian renormalization group technique, the functional renormalization group equation. Computations within this framework rely on truncation approximations, whereby the full renormalization group flow is projected onto a subspace of the couplings of the theory. The reliability of the results thus obtained is assessed by verifying their stability under the gradual extension of the truncation subspace. The simplest, non-trivial gravity truncation one may consider is the Einstein-Hilbert truncation, in which evidence was first found for a non-trivial fixed point of gravity in support of the asymptotic safety scenario. The crucial question is then whether or not this fixed point persists under further enlargement of the truncation subspaces. In this thesis, we have tackled this question by following three complementary strategies. First, in Chapter 3, we investigated the renormalization group flow of gravity restricted to the conformal sector. This simplification allowed us to consider truncations containing terms which would be otherwise intractable for technical reasons, and thus obtain a rough picture of the effect of those terms on the behavior of the complete theory. A second strategy, which we followed in Chapters 4 and 5, returning to the case of full gravity, was to construct a functional renormalization group equation that allowed us to study general truncations spanned by arbitrary functions $f(R)$ of the curvature scalar, and hence investigate large sectors of the full renormalization group flow space. Lastly, in Chapter 6, we included in our truncation terms which are known to be particularly problematic from the point of view of perturbative quantization. First, we moved beyond the $f(R)$-case by explicitly including four-derivative tensorial operators in our truncation, and then added a minimally coupled, massless scalar field to our truncation. Remarkably, in all of these extensions a non-trivial ultraviolet fixed point was found. The coherent picture that arises from these studies provides significant evidence in favor of the asymptotic safety scenario, and suggests that quantized gravity remains predictive at arbitrarily small scales.