基于LIGO引力波探测的Ringdown波形理论分析及其数字信号处理
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摘要
根据广义相对论理论,双星环绕-融合-拖尾过程以及致密星体自身引力坍塌都将产生出新的黑洞,而刚形成的此黑洞并不稳定,它将通过引力辐射过程回复到新的平衡态。在这个回复过程中,引力波的波形是由若干似正规模叠加而成的。本文关注的是该波形的末期,即拖尾(Ringdown)波形。该波形的特点是:频率一定,振幅按时间成指数衰减。目前,LIGO(引力波激光干涉仪)有3个位于不同位置的探测器,它们的数据分析都采用匹配滤波的方法,即:已知信号波形,以此为模板与实际数据进行相关分析,提取信噪比超过规定阈值的数据点,将之视为探测信号的可能数据点。在各自进行了上述处理之后,再将提取出的信号相互做偶合事件分析以及时域平移分析,从而大大降低由于各种噪声带来的影响。作为LIGO分析目标之一,Ringdown信号的探测完全按照之前所提及的流程,本文重点研究的对象是距离地球300 Mpc以内中等质量的黑洞,其质量为100-500 M⊙,因为满足这个条件并处于微扰状态的黑洞释放的引力辐射的强度刚好与LIGO的灵敏度相符。
一、论文主要工作
本文讨论的探测波源-处于微扰的中等质量黑洞,它有两种形成机制:①一种大质量自旋恒星(致密星体大约20-500倍太阳质量,但具体的质量有待确定)引力坍塌而成。②较小质量的环绕双星融合而成。目前,关于第①种形成过程的研究工作比较缺乏,而第②种机制的整体波形又无准确的解析解。针对以上问题,本文详细计算和分析了第①种形成过程的整体波形,通过曲线拟合得到了近似的解析解,并且与现有的Ringdown波形做相关性分析。结果表明:拟合效果很好,能够为将来的LIGO数据处理建立新的模板。针对第②种形成机制下的两种现有波形(有效一体波形EOBNR、唯象波形) ,我们将之作为理想信号输入对LIGO第5轮数据做调试,发现唯象波形与EOBNR相比,曲线更为平滑,而且在同样的有效距离上有着更高的准确率。这样我们有望通过唯象波形进一步找到准确的解析解。
二、论文创新点
1、目前,关于单个致密星体自身引力坍塌成中等质量kerr黑洞还缺乏系统的研究,本论文针对这个不足分析了在此过程中产生的整体引力波解析解。
2、现有的处于微扰阶段的黑洞辐射的拖尾引力波波形只是在阶跃函数的后端延伸出来的,这个形式极不准确,因而,我的工作是找到了一个使坍塌阶段和拖尾阶段波形能光滑连接起来的结合点。从而,使LIGO数据处理的模板更为准确。
3、曲线模拟的整体波形与现阶段LIGO拖尾波形的相关性分析是对理论和实验具有指导性作用的研究。通过相关性分析后的数据说明了辐射引力波的能量会因质量和角动量的不同而分布在不同的阶段。质量、角动量及其相关性的分析是非常有价值的。
4、本文利用LIGO第五轮数据进行分析,它是迄今为止观测时间最长,探测覆盖范围最广的一次数据分析。由此得到了关于偶合尺度窗的最优值。并且还对现有的两种理想输入信号(有效一体波形和唯像波形)做了多方面的比较,结果表明:唯像波形与EOBNR相比,曲线更为平滑,而且在同样的有效距离上有着更高的准确率。
三、本论文主要结论
1、在致密星体自身引力坍塌过程中,与自旋一致的方向上没有引力辐射。在质量相等的情况下,自旋角速度越快的星体偏心率越大,自旋越小的星体坍缩过程相对较长;在自旋角速度一定的情况下,质量越大的星体初始偏心率和焦距相应增大。另外,初始条件告诉我们:焦距与初始密度成反比,因而初始密度越大,相应的引力辐射强度反而越弱。
2、自旋愈快质量愈大的致密星体在坍塌过程中,引力辐射的能量大部分集中在Ringdown部分,对于自旋较为缓慢且质量偏小的星体而言,大量的能量将集中在坍塌阶段。
3、综合唯象波形输入信号和EOBNR输入波形后可以得到0.4为最优偶合窗尺度。但是,背景噪声的数目随偶合窗尺度呈单调上升趋势,因而将进一步绘制ROC曲线(探测概率与虚警率关系图)来确定合理的取值。
4、EOBNR理想波形被恢复的点总是比唯象波形对应的点稍远。这是因为在构建EOBNR波形的时候,其振幅比相应位置的唯象波形小,所以对同一个匹配模板而言,EOBNR就被视为较远波源发出的信号。
本文大部分工作完成于美国德克萨斯大学布朗斯维尔分校引力物理中心以及LIGO合作小组,并得到了美国国家自然科学基金的资助。
According to General Relativity, an unstable black hole, which occurs due to the coalescence of a black hole binary following their inspiral and subsequent merger and a compact stellar, will return to a stable configuration by the emission of gravitational radiation in a superposition of quasi-normal modes. In this paper, we focus on the late time of the waveform, which we refer to as Ringdown waveform. It has typical frequency and the amplitude exponential decaying with time. At present, LIGO (Laser Interferometer Gravitational Wave Observatory) locate on 3 different places, which are two in Hanford named H1 and H2 and, another one in Livingston named L1. In each of them, the data analysis process is well-known the method of matched filtering, which means base on the known waveform, we perform the correlation between data and the waveform, and pass the data with higher Signal-Noise-Ratio (SNR) than threshold. In order to reduce background noise, we apply the coincidence Analysis and timeslides analysis on two or three detectors after matched filtering in each detector. As one aim of LIGO data analysis, the Ringdown detection follows such pipline. Here we concern the intermediate mass black holes with 100 and 500 M⊙to a distance of up to 300 Mpc, because LIGO is sensitive to the dominant mode of perturbed black holes with such masses and region.
1 The main works:
As the source of this paper, the perturbed intermedate black holes will be occurred by two different mechanisms:①the compact stellar with spin (mass is near 20-500 M⊙) gravity collapse.②inspiral binary with smaller mass merge to black holes. At present, it is short for the research of the first mechanism; however, there is no accurate analytic whole waveform for the second one. In order to solve these issues, in this paper we calculate and analyze the whole waveform for the first one, get the approximate analytic result through curve fitting, which we make correlation analysis to current Ringdown waveform. The results show it is reasonable to make new template for LIGO data process since such ideal overlap between them. For the second one, there are two kinds of waveform (i.e., EOBNR, phenomenological waveform), which are the injection simulation for S5 LIGO data analysis. We find phenomenological waveform is much smoother and with higher accurate rate. Then it is hopeful to find accurate analytic result.
2 Innovations of this paper
①At present, it is short for the research of compact stellar with spin gravity collapse. This paper calculates the analytic whole waveform in the collapse to fix this shortage.
②The current Ringdown waveform is connected directly to a step function, which is inaccurate. Then as a part work of this paper, we match the collapse waveform to Ringdown smoothly. As a result, the template of LIGO data analysis is much more accurate.
③It is a significant research of curve fitting and current Ringdown correlation. We find the energy of gravitational radiation is in different phase due to mass and spin difference. This conclusion is valuable.
④So far, the S5 LIGO data is in the longest observation time and the largest detection region. With such data, we get the optimal coincidence test window size and compare EOBNR with phenomenological waveform, as a result, phenomenological waveform is much smoother and with higher accurate rate.
3 Main conclusions
①In the collapse, there is no gravitational radiation on the spin direction. With the same mass, the faster spin results bigger eccentricity and shorter collapse period; with the same spin, the bigger mass results bigger initial eccentricity and foci distance. In addition, the bigger initial density results weaker gravitational radiation.
②For the compact stars with faster spin and bigger mass, most of the energy is in Ringdown phase; for the stars with slower spin and smaller mass, most of the energy is in collapse phase.
③According to phenomenological and EOBNR waveform comparison, the optimal size of coincidence test is 0.4. However, the events of background increase with the size, and then we need to plot ROC (detection possibility and false alarm plot) curve to choose reasonable value.
④The found injections of EOBNR are always in further area than phenomenological ones, because when the EOBNR waveform is made, its amplitude is lower than the phenomenological at the same injection time. Therefore, for the same matched template, EOBNR is the signal from further source.
Most of our work is done in the Center for Gravitational Wave Astronomy (CGWA) of University of Texas at Brownsville (UTB) in U.S.A. and international LIGO science cooperation with the support of Unite States national science fund.
一、论文主要工作
本文讨论的探测波源-处于微扰的中等质量黑洞,它有两种形成机制:①一种大质量自旋恒星(致密星体大约20-500倍太阳质量,但具体的质量有待确定)引力坍塌而成。②较小质量的环绕双星融合而成。目前,关于第①种形成过程的研究工作比较缺乏,而第②种机制的整体波形又无准确的解析解。针对以上问题,本文详细计算和分析了第①种形成过程的整体波形,通过曲线拟合得到了近似的解析解,并且与现有的Ringdown波形做相关性分析。结果表明:拟合效果很好,能够为将来的LIGO数据处理建立新的模板。针对第②种形成机制下的两种现有波形(有效一体波形EOBNR、唯象波形) ,我们将之作为理想信号输入对LIGO第5轮数据做调试,发现唯象波形与EOBNR相比,曲线更为平滑,而且在同样的有效距离上有着更高的准确率。这样我们有望通过唯象波形进一步找到准确的解析解。
二、论文创新点
1、目前,关于单个致密星体自身引力坍塌成中等质量kerr黑洞还缺乏系统的研究,本论文针对这个不足分析了在此过程中产生的整体引力波解析解。
2、现有的处于微扰阶段的黑洞辐射的拖尾引力波波形只是在阶跃函数的后端延伸出来的,这个形式极不准确,因而,我的工作是找到了一个使坍塌阶段和拖尾阶段波形能光滑连接起来的结合点。从而,使LIGO数据处理的模板更为准确。
3、曲线模拟的整体波形与现阶段LIGO拖尾波形的相关性分析是对理论和实验具有指导性作用的研究。通过相关性分析后的数据说明了辐射引力波的能量会因质量和角动量的不同而分布在不同的阶段。质量、角动量及其相关性的分析是非常有价值的。
4、本文利用LIGO第五轮数据进行分析,它是迄今为止观测时间最长,探测覆盖范围最广的一次数据分析。由此得到了关于偶合尺度窗的最优值。并且还对现有的两种理想输入信号(有效一体波形和唯像波形)做了多方面的比较,结果表明:唯像波形与EOBNR相比,曲线更为平滑,而且在同样的有效距离上有着更高的准确率。
三、本论文主要结论
1、在致密星体自身引力坍塌过程中,与自旋一致的方向上没有引力辐射。在质量相等的情况下,自旋角速度越快的星体偏心率越大,自旋越小的星体坍缩过程相对较长;在自旋角速度一定的情况下,质量越大的星体初始偏心率和焦距相应增大。另外,初始条件告诉我们:焦距与初始密度成反比,因而初始密度越大,相应的引力辐射强度反而越弱。
2、自旋愈快质量愈大的致密星体在坍塌过程中,引力辐射的能量大部分集中在Ringdown部分,对于自旋较为缓慢且质量偏小的星体而言,大量的能量将集中在坍塌阶段。
3、综合唯象波形输入信号和EOBNR输入波形后可以得到0.4为最优偶合窗尺度。但是,背景噪声的数目随偶合窗尺度呈单调上升趋势,因而将进一步绘制ROC曲线(探测概率与虚警率关系图)来确定合理的取值。
4、EOBNR理想波形被恢复的点总是比唯象波形对应的点稍远。这是因为在构建EOBNR波形的时候,其振幅比相应位置的唯象波形小,所以对同一个匹配模板而言,EOBNR就被视为较远波源发出的信号。
本文大部分工作完成于美国德克萨斯大学布朗斯维尔分校引力物理中心以及LIGO合作小组,并得到了美国国家自然科学基金的资助。
According to General Relativity, an unstable black hole, which occurs due to the coalescence of a black hole binary following their inspiral and subsequent merger and a compact stellar, will return to a stable configuration by the emission of gravitational radiation in a superposition of quasi-normal modes. In this paper, we focus on the late time of the waveform, which we refer to as Ringdown waveform. It has typical frequency and the amplitude exponential decaying with time. At present, LIGO (Laser Interferometer Gravitational Wave Observatory) locate on 3 different places, which are two in Hanford named H1 and H2 and, another one in Livingston named L1. In each of them, the data analysis process is well-known the method of matched filtering, which means base on the known waveform, we perform the correlation between data and the waveform, and pass the data with higher Signal-Noise-Ratio (SNR) than threshold. In order to reduce background noise, we apply the coincidence Analysis and timeslides analysis on two or three detectors after matched filtering in each detector. As one aim of LIGO data analysis, the Ringdown detection follows such pipline. Here we concern the intermediate mass black holes with 100 and 500 M⊙to a distance of up to 300 Mpc, because LIGO is sensitive to the dominant mode of perturbed black holes with such masses and region.
1 The main works:
As the source of this paper, the perturbed intermedate black holes will be occurred by two different mechanisms:①the compact stellar with spin (mass is near 20-500 M⊙) gravity collapse.②inspiral binary with smaller mass merge to black holes. At present, it is short for the research of the first mechanism; however, there is no accurate analytic whole waveform for the second one. In order to solve these issues, in this paper we calculate and analyze the whole waveform for the first one, get the approximate analytic result through curve fitting, which we make correlation analysis to current Ringdown waveform. The results show it is reasonable to make new template for LIGO data process since such ideal overlap between them. For the second one, there are two kinds of waveform (i.e., EOBNR, phenomenological waveform), which are the injection simulation for S5 LIGO data analysis. We find phenomenological waveform is much smoother and with higher accurate rate. Then it is hopeful to find accurate analytic result.
2 Innovations of this paper
①At present, it is short for the research of compact stellar with spin gravity collapse. This paper calculates the analytic whole waveform in the collapse to fix this shortage.
②The current Ringdown waveform is connected directly to a step function, which is inaccurate. Then as a part work of this paper, we match the collapse waveform to Ringdown smoothly. As a result, the template of LIGO data analysis is much more accurate.
③It is a significant research of curve fitting and current Ringdown correlation. We find the energy of gravitational radiation is in different phase due to mass and spin difference. This conclusion is valuable.
④So far, the S5 LIGO data is in the longest observation time and the largest detection region. With such data, we get the optimal coincidence test window size and compare EOBNR with phenomenological waveform, as a result, phenomenological waveform is much smoother and with higher accurate rate.
3 Main conclusions
①In the collapse, there is no gravitational radiation on the spin direction. With the same mass, the faster spin results bigger eccentricity and shorter collapse period; with the same spin, the bigger mass results bigger initial eccentricity and foci distance. In addition, the bigger initial density results weaker gravitational radiation.
②For the compact stars with faster spin and bigger mass, most of the energy is in Ringdown phase; for the stars with slower spin and smaller mass, most of the energy is in collapse phase.
③According to phenomenological and EOBNR waveform comparison, the optimal size of coincidence test is 0.4. However, the events of background increase with the size, and then we need to plot ROC (detection possibility and false alarm plot) curve to choose reasonable value.
④The found injections of EOBNR are always in further area than phenomenological ones, because when the EOBNR waveform is made, its amplitude is lower than the phenomenological at the same injection time. Therefore, for the same matched template, EOBNR is the signal from further source.
Most of our work is done in the Center for Gravitational Wave Astronomy (CGWA) of University of Texas at Brownsville (UTB) in U.S.A. and international LIGO science cooperation with the support of Unite States national science fund.
引文
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