An Efficient Higher-Order WKB Code for Quasinormal Modes and Greybody Factors

Abstract

The higher-order WKB Mathematica^® code for computing quasinormal modes, whose accuracy was significantly enhanced through extensions to higher orders and, in particular, through the use of Pad´e resummation, has been widely employed in numerous studies over the past several years. In this work, we present an updated and optimized version of the code. The main improvement consists in expanding the effective potential in a Taylor series around its maximum, rather than evaluating the full analytic expression of the WKB formula for each specific potential. This modification leads to a substantial reduction in computation time. In cases where the effective potential is complicated and involves non-rational functions, the speed gain can reach several orders of magnitude, while preserving the accuracy of the method.

Keywords

WKB approximation / black holes / quasinormal modes / grey-body factors

Author Contributions

R.A.K.: conceptualization, methodology, software, writing; J.M.: conceptualization, methodology, software, writing; A.Z.: conceptualization, methodology, software, writing. All authors have read and agreed to the published version of the manuscript.

Funding

A.Z. was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The Wolfram Mathematica^® package and examples of its applications are publicly available online at https://arxiv.org/src/2603.12466v1/anc.

Conflicts of Interest

Given their editorial roles, Roman A. Konoplya (Editor-in-Chief) and Alexander Zhidenko (Editorial Board Member) had no involvement in the peer review of this paper and had no access to information regarding its peer-review process. Full responsibility for the editorial process of this paper was delegated to another editor of the journal.

Use of AI and AI-Assisted Technologies

During the preparation of this work, the authors used ChatGPT developed by OpenAI for language editing and text refinement. After using this service, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.

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