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 (spacelike) singularity. Within the cosmological paradigm of a Friedmann-Lemaitre spacetime, where a description of physics in terms of quantum field theories 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 the existence of a pre-inflationary phase as valid, the aim of this thesis has been to develop a customized theoretical framework for interacting scalar quantum field theories (QFTs) on generic Friedmann-Lemaitre backgrounds. Importantly, such a framework cannot be in Euclidean signature as the expanding spacetime generically prohibits 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 the first part of this thesis we present the Anti-Newtonian expansion in a spatially discretized setting, where the flat spatial sections of the  Friedmann-Lemaitre background are replaced with a hypercubical lattice.  In this framework, the solution of the QFT decouples into two sub-problems: (1) The solution of the cosmological quantum mechanics, conceptually associated with the decoupled wordlines in the Anti-Newtonian limit; and (2) The solution of the combinatorial problem that allows one to analytically control the terms of the linked cluster expansion, which is conceptually associated with restoring the spatial interaction between the neighboring world lines. With the goal of making contact to the Functional Renormalization Group, we focus on developing a linked cluster expansion for the Legendre effective action. We show that this expansion may be efficiently recast in terms of graph theoretic methods, with the resulting graph rules largely model independent. Next, we present a ``proof-of-principle'' study showing that the Functional Renormalization Group can be applied to efficiently calculate the critical parameters of the hopping expansion of Euclidean lattice quartic scalar theory.

Motivated by this application of the Functional Renormalization Group, as well as its central role in the asymptotic safety approach to quantum gravity, in the subsequent parts of the thesis we proceed to develop the elements needed for a manifestly Lorentzian formulation of the Functional Renormalization Group method. One of the key differences of this formulation compared to the Euclidean setting is that it necessitates the incorporation of state-dependent data directly into the flow equation formalism. As a consequence, choices analogous to the selection of a state for perturbative QFTs on curved backgrounds need to be made. The Hadamard property of these states entails a universal ultraviolet (UV) behavior of the flow, while the infrared (IR) regime of the flow will be inevitably state dependent. An investigation of the Functional Renormalization Group on all scales requires a Hadamard state with controllable UV and IR properties, which we show to be true for a well-motivated class of vacuum states known as ``States of Low Energy'' (SLE), which are defined on generic Friedmann-Lemaitre backgrounds. In addition to being Hadamard states, we prove
that they admit a convergent infrared expansion  on  generic Friedmann-Lemaitre backgrounds. In the context of the Functional Renormalization Group, this mathematically defines the flow at all scales, in sharp contrast to the commonly used (pseudo-) heat kernel methodology, which inevitably reorganizes only ultraviolet information.
One application is a novel resolution of the infrared divergences that plague   massless modes on  many Friedmann-Lemaitre backgrounds. In this setting,  SLE have the remarkable property of a universal Minkowski-like infrared behavior, yielding infrared finite two-point functions. This feature impacts the priomordial power spectrum, computed for modes based on a SLE, modifying the low angular momentum parts in a way compatible with current Cosmic Microwave Background data. Finally, we  present a simple generalization of the SLE construction to a one-parameter family of Hadamard states, which we envision as being fruitful for the Functional Renormalization Group.

In the final part of this thesis, we present the Lorentzian spatial Functional Renormalization Group. As a first application, we use it in place of the usual trace-log computation to extract the divergent parts of the one-loop effective action. Technically, this is achieved through a generalized resolvent expansion (presented earlier in the thesis), thereby the avoiding the ill-defined pseudo-heat kernel techniques. Despite the non-covariant spatial regulator, the UV divergent parts come out as spacetime covariant (although slightly different from the formal use of a covariant regulator). These UV divergent parts were subsequently cancelled by counterterms in the bare action, as usual. Among the infinitely many Wilsonian couplings, only six are power counting non-irrelevant, and their renormalization flow is been presented. Of particular note is the additional contribution to the renormalization of the Newton constant, induced by the time dependence of the regulator through the cosmological scale factor. In a full quantum gravity computation, this would quantitatively affect the interplay between the matter and gravity sectors. Within the context of the asymptotic safety scenario, it is believed that the interplay between the gravity and matter couplings resolves the triviality of scalar field theories. This has found phenomenological applications, which are however as yet provisionary, because the infrared regime of the flow equation is neither well-posed or controlled. This thesis  prepares the  tools to address this situation.

View the Poster Here