### Gravitational Wave Detection with LIGO

The paper that I’ll be reviewing today is the famous Observation of Gravitational Waves from a Binary Black Hole Merger by B.P. Abott et al. This paper records the detection of gravitational waves for the first time in human history, and was awarded the Nobel Prize in 2017.

## Introduction

What are gravitational waves, and how are they caused? Gravitational waves are caused by the change with time of the quadrupole moment of a system of masses. What does that mean? Let us take an example. Imagine the rotation of the earth. As it is rotating, its mass doesn’t change (first moment), and neither does the relative position of its center of mass (second moment). However, during rotation, the poles get flattened a little bit. When this flattening is happening, the quadrupole moment, or third moment, changes. One way of thinking of it is to say that the “shape” changes. If one was calculating the gravitational potential energy of the earth at a distant point in space, it would change a little because of the flattening of the earth near the poles. It will have reduced. The energy that is lost by the system is radiated out in the form of gravitational waves.

But what do gravitational waves do? Simplistically streaking, they stretch and compress space itself. One may imagine the universe to be a rubber sheet. Then gravitational waves elongate and compress this mat with time. This is exactly the property of waves that the LIGO exploited to detect gravitational waves. A more mathematical description of gravitational waves can be found here. In this mathematical explanation, they “simplify” Einstein’s Field Equations to make them linear, and then use the fact that however we deform the universe, although metrics and other observables might change, the physical laws will always remain the same (the justification for gauge transformations). With different kinds of such deformations, we get different solutions for how the “shape and size” (metric) of the universe changes with time. All these are valid descriptions of the universe. One such description is that gravitational waves are emitted after certain phenomena. Although these are solutions only for “simple” and “approximate” Einstein’s Field Equations, it has been rigorously proven that gravitational waves indeed are solutions of Einstein’s Field Equations in the complicated, non-linear case.

An important caveat is that this is not the first time that gravitational waves have been observed. The detection of the binary pulsar system PSR B1913+16 proved that gravitational waves do indeed exist. However, the presence of gravitational waves was inferred from other observations, and not directly detected. This paper presents proof for the first actual detection of gravitational waves, and also the first observation of the merger of two black holes in a binary black hole system.

## Observation

The merger of two black holes so far away in space and time was observed on September 14, 2015. The initial detection was completed by low latency searches, which means that some preliminary gravitational wave analysis was done quickly (3 minutes) so that scientists could quickly figure out the approximate location of the merger of the black holes, and turn their telescopes to the portion of the sky where this signal was coming from. If astronomical observations co-incided with this wave detection, then they would have proof that this wave was not a technical glitch, but the result of an astronomical event.

Two observatories observed this event: one in Hanford, WA and another in Livingston, LA. The gravitational waves reached Hanford 10 ms before Livingston, and this discrepancy in time of observation helped scientists locate the binary black hole system in the sky. The observed signals had a signal-to-noise ratio of 24.

Some features of the observed signal were these: over 0.2 seconds, both the amplitude and the frequency of the wave increased in 8 cycles from 35 to 150 Hz, where the amplitude reached its maximum. Let us focus on the maxima: where the frequency is 150 Hz. Two bodies orbiting their common center of mass would have to be orbiting 75 times every second to produce this signal. After noting the frequency and the rate of change of frequency, the scientists concluded that two compact masses of a certain total mass had to be orbiting each other at a distance of approximately 350 km for these gravitational waves to have been produced. They concluded that these two point masses could not be neutron stars (too light) or a neutron star black hole binary (would have coalesced at a lower orbital frequency). Hence, these had to be two black holes.

## Detectors

The detection of gravitational waves happened at two observatories. This was so that random noise from the environment could be eliminated (most sources of noise would be local, and hence would be detected by one observatory but not the other), and the location of the astronomical event in the sky could be determined (mainly by the time difference of observation by the two observatories). The instrument that was used was a laser interferometer, and its working principle is explained beautifully here. Simplistically speaking, and interferometer has two arms that are perpendicular to each other, and there are light beams inside both. They are arranged in such a way that these light beams interfere destructively at the detector. However, gravitational waves stretch one arm, and compress the other one (unless the arms are at equal angles to the direction of propagation of the wave, which is extremely unlikely for a randomly generated gravitational wave in the sky). This causes the light to not interfere destructively, and we can hence detect gravitational waves by detecting visible interference patterns of laser beams. The instrument uses a resonant optical cavity to increase the intensity of the laser- we end up getting a 100kW laser light from a 700W light. Hence, even a small change in the interference pattern of light can be detected easily.

The mirrors used in the experiment are also mounted on quadruple-pendulum systems, so that these mirrors remain almost stationary in the event of earthquakes or other sources of external noise. To reduce Rayleigh scattering of photons, ultrahigh vacuum is created, and the pressure inside the arms is reduced to $10^{-6}$ Pa. Other instruments like seismometers, etc were also mounted near both observatories so that if localized causes did create a disturbance in the instrument, they would be detected by those instruments also, and hence such detections could be discarded.

## Methods of searching

The blackhole merger event, codenamed GW150914, was verified by two different techniques. Because these two techniques use completely independent methods of analysis, and the fact that both report the same event boosts the plausibility of this being an actual gravitational wave and not another random disturbance. All events that are observed are assigned a detection-statistic value by both methods, which measures their likelihood of being blackhole merger events. GW150914 was given very high values by both methods. But how can we calibrate the instrument to ensure that we give high detection-statistic value only to actual black hole merger events? We measure the rate at which the instrument gives the same or higher detection-statistic values to events that are not black hole mergers. If this happens very rarely, say once in 5,000 years, then we can be reasonably sure that the event that has been given a high detection-statistic value is an actual black hole merger, and not a freakish once-in-5000-years-event.

The two methods of analysis used are:

• Generic Transient Search– In this technique, there are no pre-constructed gravitational waves with which incoming signals should be compared. After observing a signal lying within a certain frequency and time range, we construct a gravitational wave form having those properties. Then this wave form and the incoming signal is compared. If the comparison is favorable, we can reasonably infer that the incoming signal is a gravitational wave. The detection-statistic value given to GW150914 was 20. Only once in 22,500 years will this method give such a high value to an event that is not the merger of two black holes producing gravitational waves.
• Binary Coalescence Search– We have approximately 225,000 pre-constructed gravitational wave forms. The incoming signal is then compared with these wave forms, and the better the match, the higher the detection statistic value. The detection statistic value that this search gave to GW150914 was 23.6, and this method would erroneously give such a high value only once in 203,000 years.

## Conclusions

Some properties of the black hole merger event could be calculated from the observations:

No deviation from General Relativity was found with regard to mass and spin of black holes, and also the coefficients in the mathematical expression for the phase of the gravitational waves (expressed as a power series in $f^{1/3}$). Moreover, better lower bounds for the Compton wavelength of a graviton were obtained.

The authors conclude this paper with the assertion that there are probably a lot of black hole binary systems out there that are merging much more often than we can observe, the predicting or analyzing the interference patterns of the gravitational waves created by such events could help us decode the history of the universe.