[论文解读] Perturbational Treatment of the Gravitational Potential Effect on Binary Black Hole Evolution

Zhoujian Cao, Jui-Ping Yu+3 more|arXiv (Cornell University)|Mar 28, 2012

Pulsars and Gravitational Waves Research参考文献 1被引用 5

一句话总结

本文采用微扰方法研究第三颗超大质量黑洞产生的背景引力势对双黑洞（BBH）演化的影响，而非完整的数值 relativity 方法。结果表明，背景势导致更高阶引力波模态激发，改变波前到达时间，拉长波长，增加辐射能量，并通过红移与偏心率效应的竞争改变并合时间——这对区分位于星系中心的BBH与孤立BBH至关重要。

ABSTRACT

Binary black hole (BBH) systems are usually located in the gravitational potential well formed by a massive black hole (BH), which is mostly located in the center of a galaxy. In most existing studies, the BBH systems are treated as isolated systems, while the effect of the background is ignored. The validity of the approximation is based on the belief that the background gravitational field from other sources is extremely weak compared with the strong gravitational field produced by the BBH itself during the evolution, and can be neglected in gravitational wave detection. However, it is still interesting to check how valid this approximation is. In this work, instead of simulating the three-BH problem with a fully relativistic treatment, we use a perturbational scheme to investigate the effect of the background gravitational potential on the evolution of a BBH, especially on the waveform of its gravitational radiation. Four scenarios are considered including the head-on collision and the inspiral-to-merger process of a BBH which is either freefalling towards or circularly orbiting around a third large BH. The head-on collision and the circular inspiral are two limits of all possible configurations. The existence of the background gravitational potential changes the arrival time of the gravitational wavefront of a BH, prolongs the wavelength, and increases the gravitational radiation energy. And most interestingly, the background gravitational potential induces the higher-order modes of the gravitational wave of a BBH. These interesting phenomena can be explained by the gravitational redshift effect and the change of eccentricity of a BBH's orbit from the background gravitational potential.

研究动机与目标

• 评估在引力波探测中将双黑洞（BBH）视为孤立系统是否合理，尽管它们通常位于星系势阱中。
• 研究来自第三颗大质量黑洞的背景引力势如何影响BBH的动力学与引力波辐射。
• 确定超大质量黑洞的环境效应是否会导致波形畸变，并在源识别中造成误解。
• 探索使用微扰理论而非完整数值相对论模拟研究三黑洞系统之可行性的研究。

提出的方法

• 采用微扰方案，将第三颗大质量黑洞视为固定背景势，BBH系统在其影响下演化。
• 引力势被建模为弱静态场，允许对其对BBH轨道动力学与辐射的影响进行线性化处理。
• 使用 Newman-Penrose 标量 Ψ₄ 提取引力波波形，以分析振幅、相位与模态结构。
• 分析四种情形：BBH发生头对头碰撞，以及自由下落或绕第三颗BH运行的渐近并合过程。
• 该方法假设BBH对第三颗BH无反作用，将三体问题简化为在外势作用下的两体系统。
• 分析重点在于波形特征的变化，如到达时间、波长、振幅，以及高阶（ℓ > 2）模态的激发。

实验结果

研究问题

• RQ1 来自第三颗大质量黑洞的背景引力势如何改变双黑洞系统的引力波形？
• RQ2 背景势在引力辐射中诱导出高阶多极模态（如 ℓ=3, m=0）的程度有多大？
• RQ3 引力红移与潮汐诱导偏心率之间的竞争对BBH并合 timescale 的净效应是什么？
• RQ4 微扰方法能否在无需完整数值相对论模拟的情况下准确捕捉BBH演化中的环境效应？
• RQ5 孤立BBH与处于背景势中的BBH在波形特征（到达时间、振幅、波长）上有何差异？

主要发现

• 背景引力势增加了BBH系统辐射的总引力波能量。
• 该势在引力波中诱导出显著的高阶模态，尤其当BBH在x–z平面内绕转时，ℓ=3, m=0模态明显激发。
• 引力红移效应拉长波长并延迟引力波前到达时间，而潮汐力则增加偏心率并加速并合过程。
• 对并合时间的净效应取决于红移（延迟）与偏心率（加速）效应之间的竞争，导致并合或提前或延后。
• 对于 ℓ < 3 的模态，所有波形量均与背景势强度 m₃/R 呈线性比例，验证了弱场下微扰方法的有效性。
• BBH系统的取向对诱导效应的定性特征影响极小，表明结果在不同构型下具有鲁棒性。