Detailed ultraviolet asymptotics for AdS scalar field perturbations

TL;DR

This paper develops methods to accurately evaluate the asymptotic behavior of integrals involving Jacobi polynomials with large degrees, crucial for understanding short-wavelength mode interactions in AdS spacetime perturbations.

Contribution

It introduces new asymptotic techniques for integrals of Jacobi polynomials and applies them to analyze scalar field perturbations in AdS, revealing detailed leading-order behavior.

Findings

Derived explicit asymptotic formulas for integrals of Jacobi polynomials with large degrees.

Applied asymptotic methods to scalar field perturbations in AdS, identifying key interaction behaviors.

Provided closed-form expressions for coefficients in the asymptotic expansions.

Abstract

We present a range of methods suitable for accurate evaluation of the leading asymptotics for integrals of products of Jacobi polynomials in limits when the degrees of some or all polynomials inside the integral become large. The structures in question have recently emerged in the context of effective descriptions of small amplitude perturbations in anti-de Sitter (AdS) spacetime. The limit of high degree polynomials corresponds in this situation to effective interactions involving extreme short-wavelength modes, whose dynamics is crucial for the turbulent instabilities that determine the ultimate fate of small AdS perturbations. We explicitly apply the relevant asymptotic techniques to the case of a self-interacting probe scalar field in AdS and extract a detailed form of the leading large degree behavior, including closed form analytic expressions for the numerical coefficients appearing in the asymptotics.