Nonlinear tails of massive scalar fields around a black hole

TL;DR

This work investigates nonlinear tails in massive scalar perturbations around a Schwarzschild black hole, addressing how nonlinear effects modify late-time decay and whether they leave observable imprints in ringdown signals. Using two setups—a moving-source toy model and a cubic self-interacting scalar field—the authors evolve perturbations with a double-null finite-difference scheme and perform perturbative analyses to second order. They find that, unlike the massless case, the intermediate-time tails for massive fields decay at the same rate as their linear counterparts, $t^{-l-3/2}$, with oscillations set by the mass $\mu$, and largely independent of source details; quadratic QNMs provide a distinct nonlinear signature with frequencies roughly doubling the linear QNMs. The results suggest linear theory suffices for tail modeling in this regime, while nonlinearities reveal themselves primarily through quadratic QNMs, motivating extensions to massive gravitational perturbations of rotating black holes for future gravitational-wave observational tests.

Abstract

Nonlinear effects play a fundamental role in the late-time ringdown of black holes, with direct implications for gravitational-wave observations. For massive fields, these dynamics become richer, yet their nonlinear signatures remain poorly understood. Here, we systematically study nonlinear tails of massive scalar perturbations, from a toy model with ingoing and outgoing sources to a self-interacting scalar model, revealing nonlinear tails and contrasting the results with their linear counterparts. We find that the nonlinear tails of massive scalar fields, opposite to massless ones, decay as the same rate as linear tails in the intermediate time, independent of source parameters or initial conditions. Nevertheless, quadratic quasinormal modes could serve as a probe to the nonlinear effects of massive fields.

Figures (5)

• Figure 1: Time evolution of massless scalar perturbations, comparing outgoing and ingoing cases with a moving source to the source-free case.
• Figure 2: Quasinormal oscillations and tails of massive scalar perturbations for both source-free and source-driven cases, including outgoing and ingoing wavepackets.
• Figure 3: Temporal evolution of massless and massive scalar perturbations for linear and nonlinear cases. The evolved field has been rescaled as ${\Phi _{lm}} = {\lambda ^n}\phi _{lm}^{(n)}$. For the massless case, dashed lines show analytical predictions for both linear and nonlinear decay, in excellent agreement with the numerical data. For the massive case, dashed lines indicate linear analytical predictions for the intermediate-time tails only, yet they accurately capture both linear and nonlinear numerical evolution.
• Figure 4: Temporal evolution of massive scalar perturbations, calculated for the linear $(1,1)$ mode and three nonlinear modes. The evolved field has been rescaled as ${\Phi _{lm}} = {\lambda ^n}\phi _{lm}^{(n)}$. The nonlinear modes $(0,0)$, $(2,2)$, and $(2,0)$ are generated by self-couplings of the $(1,1)$ mode. Linear analytical predictions for the intermediate-time tails (dashed lines) remain in excellent agreement with the numerical evolution, even when nonlinear couplings are present. Changing the initial data type (compact vs. uncompact) affects only the overall amplitude, while the numerical results remain in excellent agreement with the linear analytical predictions.
• Figure 5: Temporal evolution of massive scalar perturbations in nonlinear simulations. The evolved field has been rescaled as ${\Phi _{lm}} = {\lambda ^n}\phi _{lm}^{(n)}$. The $(0,0)$ mode is generated via quadratic self-couplings of the linear $(1,1)$, $(0,0)$, and $(2,0)$ modes. Dashed lines denote linear analytical predictions for the intermediate-time tails, which are in excellent agreement with the corresponding nonlinear numerical evolution.