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Graviscalar quasinormal modes and asymptotic tails of a thick brane

Qin Tan, Sheng Long, Weike Deng, Jiliang Jing

TL;DR

The paper investigates graviscalar quasinormal modes (QNMs) and late-time tails of a thick brane by analyzing scalar perturbations of the brane metric. It derives the master Schrödinger-like equation for graviscalar KK modes and computes QNMs using WKB, improved AIM, and time-domain evolution, showing a discrete spectrum determined by the brane's internal structure. A key finding is that, unlike the zero mode, the graviscalar sector lacks a localized bound state, yet exhibits QNMs with frequencies related to tensor modes; the late-time tails differ qualitatively between massless ($a=0$) and massive ($a\neq0$) cases, with potential implications for gravitational-wave backgrounds and early-universe cosmology. The work highlights the role of extra dimensions and brane thickness in shaping observable oscillations and tail behavior, and suggests extensions to other thick-brane models.

Abstract

In this work, we investigate the graviscalar quasinormal modes (QNMs) and their asymptotic tail behavior of a thick brane. Considering the scalar perturbations of the thick brane metric, we obtain the main equations of graviscalar Kaluza-Klein modes. Based on these equations, the frequencies of the graviscalar QNMs of the thick brane are obtained by the Wentzel-Kramers-Brillouin, asymptotic iteration, and numerical evolution methods. The results show that the scalar fluctuation of the thick brane has a series of discrete QNMs, similar to the tensor perturbation of the brane. These modes appear as decaying massive scalar particles in four-dimensional spacetime. We also studied in detail the late time tails of these QNMs and found that some modes have slowly decaying oscillatory tails that may be new sources of the gravitational wave backgrounds. Obviously, the QNMs contain the information of the brane and are characteristic modes of the thick brane.

Graviscalar quasinormal modes and asymptotic tails of a thick brane

TL;DR

The paper investigates graviscalar quasinormal modes (QNMs) and late-time tails of a thick brane by analyzing scalar perturbations of the brane metric. It derives the master Schrödinger-like equation for graviscalar KK modes and computes QNMs using WKB, improved AIM, and time-domain evolution, showing a discrete spectrum determined by the brane's internal structure. A key finding is that, unlike the zero mode, the graviscalar sector lacks a localized bound state, yet exhibits QNMs with frequencies related to tensor modes; the late-time tails differ qualitatively between massless () and massive () cases, with potential implications for gravitational-wave backgrounds and early-universe cosmology. The work highlights the role of extra dimensions and brane thickness in shaping observable oscillations and tail behavior, and suggests extensions to other thick-brane models.

Abstract

In this work, we investigate the graviscalar quasinormal modes (QNMs) and their asymptotic tail behavior of a thick brane. Considering the scalar perturbations of the thick brane metric, we obtain the main equations of graviscalar Kaluza-Klein modes. Based on these equations, the frequencies of the graviscalar QNMs of the thick brane are obtained by the Wentzel-Kramers-Brillouin, asymptotic iteration, and numerical evolution methods. The results show that the scalar fluctuation of the thick brane has a series of discrete QNMs, similar to the tensor perturbation of the brane. These modes appear as decaying massive scalar particles in four-dimensional spacetime. We also studied in detail the late time tails of these QNMs and found that some modes have slowly decaying oscillatory tails that may be new sources of the gravitational wave backgrounds. Obviously, the QNMs contain the information of the brane and are characteristic modes of the thick brane.
Paper Structure (4 sections, 49 equations, 4 figures, 2 tables)

This paper contains 4 sections, 49 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Thick brane solution and its scalar perturbation
  3. scalar quasinormal modes of brane
  4. Conclusions

Figures (4)

  • Figure 1: The shape of the effective potential (\ref{['effectivepotentialz']}).
  • Figure 2: The first fifteen graviscalar QNFs of the thick brane solved by the improved AIM.
  • Figure 3: (a): Time evolution of the Gaussian pulse at $kz=10$. (b): Time evolution of the odd wave packet at $kz=10$.
  • Figure 4: Upper panel: Time evolution of the Gaussian pulse with different $a$ at $kz_{\text{ext}}10$. Lower panel: Time evolution of the odd wave packet with different $a$ at $kz_{\text{ext}}=10$.