Anti-Newtonian Expansions, Hadamard States, and the Spatial Functional Renormalization Group

The validity of general relativity almost up to the Big Bang entails that the Einstein equations themselves can be used to study the detailed structure of spacetime in the vicinity of the singularity. Within the cosmological paradigm of a Friedmann-Lema\^{i}tre spacetime, where a description of physics in terms of quantum field theories (QFTs) on such a curved background is deemed to be valid, this mandates the existence of a pre-inflationary epoch following the Big Bang. Accepting this physically well-motivated scenario of a pre-inflationary phase as valid, this thesis aims to develop a customized theoretical framework for interacting scalar QFTs on generic Friedmann-Lema\^{i}tre backgrounds. Importantly, such a framework cannot be in Euclidean signature (due to ill-definedness of a Wick rotation), nor should it be tailored towards de Sitter spacetime. Motivated by the subdominance of spatial gradients in the approach to the singularity, the major themes of this thesis are variants of spatial averaging and spatial gradient expansions in relation to the dependence on the underlying vacuum-like state. The first of these themes is the Anti-Newtonian expansion in a spatially discretized setting. In this framework, the solution of a QFT decouples into two sub-problems: (i) the solution of the cosmological quantum mechanics; and (ii) the solution of the combinatorial problem that allows one to analytically control the terms of a “spatial hopping” expansion. The second theme is a novel manifestly Lorentzian formulation of the Functional Renormalization Group. The key differences of this formulation compared to the standard Euclidean setting are (i) it necessitates the incorporation of state-dependent aspects directly into the flow equation formalism; and (ii) the purely {\it spatial} mode modulation leads to an additional contribution to the renormalization of the Newton constant. In a full quantum gravity computation, the latter would quantitatively affect the interplay between the matter and gravity sectors. Within the asymptotic safety scenario, this interplay is believed to resolve the triviality of scalar field theories. This has found phenomenological applications, which are however yet provisionary, because the infrared regime of the flow equation is neither well-posed nor controlled. This thesis prepares the tools to address this situation