Towards the non-perturbative cosmological bootstrap

TL;DR

This work develops a non-perturbative framework for quantum field theory on de Sitter space by organizing states into unitary representations of SO(d+1,1). It derives a Källén–Lehmann spectral decomposition for bulk two-point functions, and an inversion formula from sphere data to extract spectral densities, linking bulk spectra to boundary operator content. The authors extend the analysis to boundary four-point functions, establishing a conformal partial wave decomposition and positivity constraints that mirror, yet differ from, ordinary CFT bootstrap due to de Sitter unitarity. Finally, they propose a non-perturbative de Sitter bootstrap program, including regularized crossing and numerical bootstrap approaches, and illustrate the ideas with a concrete dS_2 example. This work lays the groundwork for non-perturbative constraints on QFTs in de Sitter and bridges late-time boundary observables with bulk spectral data.

Abstract

We study quantum field theory on a de Sitter spacetime dS$_{d+1}$ background. Our main tool is the Hilbert space decomposition in irreducible unitary representations of its isometry group $SO(d+1,1)$. As the first application of the Hilbert space formalism, we recover the Källen-Lehmann spectral decomposition of the scalar bulk two-point function. In the process, we exhibit a relation between poles in the corresponding spectral densities and the boundary CFT data. Moreover, we derive an inversion formula for the spectral density through analytical continuation from the sphere and use it to find the spectral decompisiton for a few examples. Next, we study the conformal partial wave decomposition of the four-point functions of boundary operators. These correlation functions are very similar to the ones of standard conformal field theory, but have different positivity properties that follow from unitarity in de Sitter. We conclude by proposing a non-perturbative conformal bootstrap approach to the study of these late-time four-point functions, and we illustrate our proposal with a concrete example for QFT in dS$_2$.

Figures (3)

• Figure 1: Left: de Sitter spacetime dS${}_{d+1}$ as a hollow Minkowski cylinder, cf. equation (\ref{['eq:conformalMetric']}). Time $\tau$ runs upwards from $-\pi/2$ to $\pi/2$. Every horizonal timeslice corresponds to a copy of $S^d$. The infinite past (resp. future) is shown as a solid red (blue) line. The light blue area is the Poincaré patch $X^0 + X^{d+1} \geq 0$; the boundary between the two patches is shown as a dashed line. Right: Penrose diagram of the same spacetime, specializing to $d=1$. Spatial slices $S^1$ are parametrized by an angle $\phi \sim \phi + 2\pi$. Several timeslices of fixed $\eta < 0$ in the conformal coordinates (\ref{['eq:flatMetric']}) are shown as thin purple lines. The left and right sides of the diagram are identified, owing to the periodicity of $\phi$.
• Figure 2: Illustration of contour integrals of Watson-Sommerfeld transformation. Left: sum over non-negative integers as a set of contour integrals around the integers (\ref{['eq:int over gtilde']}). Right: Deforming the contour to a line integral with constant real part.
• Figure 3: Analytic structure of the spectral density $I_{\Delta,\ell=0}$ in the case of a free and a weakly-coupled theory in dS. The solid circles are the locations of the poles for "single-trace" and "double-trace" operators of the dS mean field theory. The single-trace poles appear for instance in the two-point function of the bulk field. The three families of double-trace poles are visible in different correlators, namely $\langle \mathcal{O} \mathcal{O} \mathcal{O} \mathcal{O} \rangle$, $\langle \mathcal{O} \mathcal{O}^\dagger \mathcal{O} \mathcal{O}^\dagger \rangle$ and $\langle \mathcal{O}^\dagger \mathcal{O}^\dagger \mathcal{O}^\dagger \mathcal{O}^\dagger \rangle$. After turning on interactions, the locations of the poles shifts, indicating that boundary operators pick up anomalous dimensions. These shifted poles are shown as crosses in the figure. Of course, new poles may appear too.