Confirming the First-Ever Detection of Gravitational Waves by Analyzing Laser Interferometer Data

At about 7 a.m. on September 14, 2015, I received an email from my colleagues in Europe notifying me of an extraordinary event: Two detectors in the Laser Interferometer Gravitational-Wave Observatory (LIGO) had simultaneously identified what appeared to be a transient gravitational-wave signal—a “ripple” in space-time.

As a member of the team that developed the LIGO instrumentation, I was excited but also slightly apprehensive. I was excited because if the signal was authentic, it would mark the first time that a gravitational wave had ever been directly observed, confirming Albert Einstein’s prediction of their existence, made a century ago. It would also mark the first observation of a pair of black holes merging to form a single black hole. My colleagues and I were apprehensive, however, because we did not yet know whether the signal came from a genuine gravitational wave or was merely the result of some error in the LIGO control system and instrumentation.

I immediately downloaded the LIGO data onto my laptop, opened MATLAB^®，并开始分析记录的信号并可视化数据。在随后的几个月中，我和我的Ligo同事确认，实际上我们已经检测到了引力波，并确定了其来源：两个黑洞的灾难性碰撞，质量合并为60倍，比我们的太阳在银河系中高60倍距离十亿光年（图1）。

MATLAB is the language used by virtually every team in the world that designs gravitational wave detectors. It is the first tool I go to when I want to design an instrument or look at the data coming out of an instrument. For LIGO, we used it to analyze the fundamental noises that limit gravitational wave detector performance, calculate the optical response of our interferometers, and verify the entire control chain to ensure the validity of the results we observed.

建立世界上最准确的仪器来测量流离失所

在我们观察到的黑洞碰撞期间，超过10³⁰kilograms of mass was converted into gravitational wave energy, most of it emitted in less than a second. The gravitational wave power radiated by this event was more than 10 times greater than the combined light power of every star and galaxy in the observable universe. To detect it, however, we had to design and build the most sensitive instruments ever created for measuring displacement: two laser interferometers capable of measuring changes in distance on the order of 10^-18meters.

Ligo是世界上最大的引力浪潮天文台。它的两个干涉仪位于路易斯安那州相距3000公里以上，另一个在华盛顿。像Ligo中的干涉仪具有两个臂的彼此成直角。在Ligo站点，每个手臂长4公里。我们通过光束分离器将激光束发光，将两个输出梁引导到两个臂的末端，在那里它们被精确取向的镜子反射回（图2）。在稳定状态下，光束穿过每个臂的相同距离，在端子光电探测器相同的端子相遇。当引力波穿过该结构时，它会扭曲干涉仪的臂，短暂缩短第一个并延长第二个，然后逆转这种变化。两臂之间的长度差异很小 - 大约1/1000^thof the diameter of a proton—but it is enough to affect the time the two beams take to travel through the arms. As a result, they arrive at the photodetector out of phase (at different times from one another), producing an interference pattern that we can measure and observe.

Since gravitational waves travel at the speed of light, if one passed by Earth, it would produce similar interference patterns at the Louisiana and Washington detectors within about 10 milliseconds (the time it takes light to travel between the two locations). That is exactly what our systems detected on the morning of September 14.

Noise Analysis and Optical Modeling Tools for Interferometer Design

Numerous noise sources on Earth are capable of causing tiny length changes in the arms of LIGO. To investigate the fundamental noises associated with a gravitational wave detector design, we use aGravitational Wave Interferometer Noise Calculator(GWINC). Developed entirely in MATLAB, this tool is used by physicists around the world to calculate the seismic, thermal, quantum, and other noises that limit gravitational wave detector performance (Figure 3).

The design and continued enhancement of the LIGO interferometers depended on a thorough understanding of noise effects. It also depended on understanding how the optics that make up the system—including beam splitters, lenses, and mirrors—work together in the complete system. To help researchers with this aspect of the interferometer design, I developed Optickle, a MATLAB based tool for frequency-domain modeling and simulation of interferometers. Optickle generates transfer functions for elements of the optical plant, enabling physicists to calculate and visualize the opto-mechanical frequency response of virtually any interferometer, including those in LIGO (Figure 4).

Updating Filters on the Fly

LIGO has thousands of digital filter modules for noise suppression and signal processing. One advantage of the current LIGO setup over its predecessors is the ability to update these filters on the fly. In the past, we had to power down and then restart the system, a process that took up to an hour. Today, we can change the infinite impulse response (IIR) filter coefficients in the modules in about a minute without shutting the system down.

This capability is made possible by a model of the interferometer real-time control system that we created in MATLAB and Simulink^®. For example, to suppress motion of the mirrors below 5 Hz, caused by microseismic peaks from ocean waves hitting the shore, we needed to change the filters to add more gain in the control loop. In MATLAB, we analyzed the feedback loop of the control system and examined the phase margin of that loop. After plotting the necessary transfer functions and determining how much gain we could add, we generated new filter coefficients and loaded them into the filter modules—all while the interferometer was running.

Verifying the First Detection and Anticipating the Next

After the initial detection, I spent days in MATLAB analyzing all the LIGO data from the event to make sure that the signal wasn’t caused by a bug in the system. To do this I reproduced the entire control chain and gravitational wave signal path in MATLAB, including all the filter parameters in use at the time, along with measurements of the detected signal as it moved through the system. This analysis gave us confidence that the signals captured by both interferometers came from a genuine gravitational wave.

在第一个事件发生大约三个月后，在圣诞节前夕，检测到第二次引力波。这是一个较小的，这是两个黑洞合并的结果，比太阳的质量大8和14倍。第二次检测提供了进一步的证据，表明第一个没有氟，它证实了我们进入了重力波天文学的新时代。

Gravitational waves give us an entirely new perspective on our universe, enabling us to study events previously invisible to us. I look forward to exploring the data from each new detection in MATLAB and using the tools we’ve developed to further improve gravitational wave detector instrumentation.