Ultraviolet asymptotics and singular dynamics of AdS perturbations

TL;DR

This paper investigates the high-frequency behavior of coefficients in a time-averaged theory describing spherically symmetric AdS-scalar field perturbations, revealing implications for singularity formation and turbulence in higher dimensions.

Contribution

It provides asymptotic analysis of the coefficients in the time-averaged equations, highlighting gauge dependence and deriving recursive relations for efficient numerical evaluation.

Findings

Coefficients grow more rapidly at high frequencies in higher dimensions.

Asymptotics inform understanding of finite-time singularities.

Recursive relations facilitate numerical simulations.

Abstract

Important insights into the dynamics of spherically symmetric AdS-scalar field perturbations can be obtained by considering a simplified time-averaged theory accurately describing perturbations of amplitude epsilon on time-scales of order 1/epsilon^2. The coefficients of the time-averaged equations are complicated expressions in terms of the AdS scalar field mode functions, which are in turn related to the Jacobi polynomials. We analyze the behavior of these coefficients for high frequency modes. The resulting asymptotics can be useful for understanding the properties of the finite-time singularity in solutions of the time-averaged theory recently reported in the literature. We highlight, in particular, the gauge dependence of this asymptotics, with respect to the two most commonly used gauges. The harsher growth of the coefficients at large frequencies in higher-dimensional AdS suggests strengthening of turbulent instabilities in higher dimensions. In the course of our derivations, we arrive at recursive relations for the coefficients of the time-averaged theory that are likely to be useful for evaluating them more efficiently in numerical simulations.