Cosmological UV/IR Divergences and de-Sitter Spacetime

TL;DR

This work analyzes one-loop graviton corrections to a massless scalar two-point function in a de Sitter inflationary background, addressing how UV and IR divergences arise in cosmological perturbation theory. By embedding de Sitter space into a UV-complete framework (e.g., string/M-theory), the authors justify a fixed physical UV cutoff and a comoving IR cutoff, and they develop three regularization schemes—brute-force cutoff, dimensional regularization, and Pauli-Villars—showing that all yield the same final result for the scalar propagator. The key finding is that, with physical UV and comoving IR cutoffs, the one-loop corrections produce consistent logarithmic contributions that agree across schemes, clarifying how infrared modes grow with cosmic expansion and how trans-Planckian issues can be addressed within a UV-complete theory. The results have implications for higher-order cosmological perturbations and for understanding the regulation of cosmological loops in inflationary backgrounds. $H$ appears as the Hubble scale, and the regulated correlators depend on $p$, $ $, $$, and physical UV scales via logs such as $\log(p/\mu_{IR})$ and $\log(H/\widetilde{\mu}_{UV})$, with a residual quadratic term in some schemes that is accounted for by the UV completion.$

Abstract

We consider one loop graviton corrections to scalar field Green's functions in the de Sitter phase of an inflationary space-time, a topic relevant to the computation of cosmological observables beyond linear order. By embedding de-Sitter space into an ultraviolet complete theory such as M-theory we argue that the ultraviolet (UV) cutoff of the effective field theory should be taken to be fixed in physical coordinates, whereas the infrared (IR) cutoff is expanding as space expands. In this context, we demonstrate how to implement three different regularization schemes -- the brute force cutoff regularization, dimensional regularization and Pauli-Villars regularization -- obtaining the same result for the scalar propagator if we use any of the three regularization schemes.

Figures (4)

• Figure 1:
• Figure 2: A representation of how modes evolve in de Sitter space. The vertical axis denotes time, the horizontal axis physical space. The grey dashed half wave and the solid grey half wave at time $t_0$ denotes waves of Hubble length. The red slanted solid lines delineate the wavelength of a wave which exits the Hubble radius at time $t_0$ (whose initial value is given by the length between A and B). The solid blue vertical lines indicate the ultraviolet cutoff wavelength (the distance between C and D) which is constant in physical coordinates.
• Figure 3: The two one-loop diagrams which we study here. They are related to the two amplitudes given in the text. The first diagram gives amplitude I and the second one amplitude II.
• Figure 4: The other one-loop interactions in the theory.