Detailed ultraviolet asymptotics for AdS scalar field perturbations

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.

A nonrelativistic limit for AdS perturbations

“…In the 'solvable' case d = 4 there is a logarithmic correction to the power law: C λn,λj,λk,λl ∼ ln λ/λ. (Our conclusions on the asymptotic behavior of the interaction coefficients follow from numerical studies, but it should be possible to derive this type of formulas analytically using the methods of [56][57][58].) As follows from (57), in energy supercritical dimensions d ≥ 7 the interaction coefficients grow with the mode number which creates hopes for interesting short wavelength behaviors in these dimensions.…”

“…For even d, the hypergeometric functions in (58) are expressible in terms of elementary functions. In four dimensions in particular, this gives S njkl = 1 (n + 1)(j + 1)(k + 1)(l + 1)…”

Examining instabilities due to driven scalars in AdS

“…This way, stability against a perturbation of order is maintained over time scales of t ∼ −2 . Conventional examinations of perturbative stability using TTF have focused on the reaction of the bulk space to some initial energy perturbation, and have aimed to study the balance between direct and inverse energy cascades [20][21][22][23][24]. Furthermore, numerical examinations of "pumped" scalars and their implications for thermalization of the dual theory have also been examined [25][26][27][28][29].…”

Examining Instabilities Due to Driven Scalars in AdS

“…Conventional examinations of perturbative stability using TTF have focused on the reaction of the bulk space to some initial energy perturbation, and have aimed to study the balance between direct and inverse energy cascades [19][20][21][22][23]. Furthermore, numerical examinations of "pumped" scalars and their implications for thermalization of the dual theory have also been examined [24][25][26][27][28].…”