Fundamentals of Interferometric Gravitational Wave Detectors

The following sections are included:

• Contents
• List of Figures
• Preface to the Second Edition
• Preface

The next few years should see the commissioning of several new gravitational wave detectors of unprecedented sensitivity. There is real excitement for the Laser Interferometer Gravitational-Wave Observatory (LIGO) in the U.S., Virgo and GEO in Europe, and all of the other gravitational observatories that may soon be built. While ultimate success is not guaranteed, the prospects are good that the world will soon be equipped with a network of gravitational wave detectors sensitive enough to record numerous signals of astronomical origin, broadband enough to allow waveformanalysis thatmay reveal the structure of the sources, and widespread and redundant enough to allow location of the sources on the sky by triangulation. With such a functioning network, it should be possible to experimentally verify the basic physics of gravitational waves, as predicted by the General Theory of Relativity. Furthermore, it is quite likely that we will be able to open a new “gravitational wave window” in astronomy, revealing facets of the Universe previously unseen with electromagnetic waves.

The existence of wave solutions to Einstein’s field equations is, arguably, the most “relativistic” feature of the theory of gravitation known as the General Theory of Relativity. This is true in the sense that it is the instantaneous action-at-a-distance of Newtonian gravity that most offends our notions of causality as understood in the context of the Special Theory of Relativity. Newton’s theory, taken literally, would predict that the gravitational field produced by a mass always has the familiar 1/r² form, with r referring to its present position, no matter how rapidly it might move (or accelerate), and no matter how far away from it we consider its field.

We start this chapter with a discussion of the physics involved in the generation of gravitational waves. The remainder of the chapter is devoted to a review of the possible astrophysical sources of gravitational radiation, and of the characteristics of the signals they would produce.

Gravitational waves, by any practical measure, will cause very weak effects on our measuring devices. Thus, it is natural to make a careful inquiry into the ways in which we will be able to register such small effects, in the presence of a certain unavoidable level of noise. Quantitative discussion of sensitivity is based on the concept of the signal-to-noise ratio. An accurate measurement can be made when the effect one is looking for (the signal) causes an output from your measuring apparatus that is large compared to the random excursions of the output when no signal is present (the noise).

Now we are ready to confront the question, “How well can an interferometer work?” The precision likely to be required is daunting. Metric perturbations h (and thus fractional changes in light travel time) of the order of 10⁻²¹ or smaller are typical of predictions of wave strength. It is convenient to express the wave’s effect as an equivalent motion of the test masses. Recalling the “proper reference frame” description, we have or .

Achieving optimum sensitivity to gravitational waves will require that it take light one half period of the wave to traverse an arm of an interferometer, as we saw in Chapter 2. For a wave with a frequency of about 200 Hz the required time is about 3 msec, in which case the required total (round trip) optical path length in an arm is 1000 km. The impracticality (i.e., the prohibitively high cost, among other problems) of building a device 500 km across has led to the development of schemes for folding a long optical path into a shorter length.

When we discussed the fundamental limits to the sensitivity of an interferometric measuring system, wewere led to consider the inherent discreteness of light, and thus to touch on the quantum mechanical view of the world. An equally fundamental limit exists on the degree to which a test mass can remain at rest. This is the phenomenon usually called thermal noise, a generalization of Brownian motion. In a certain sense, this noise is also due to the discreteness of the world, but here it is the simple graininess of extended objects and their environment (due to the existence of atoms) that is responsible. Thus, as a thermodynamic phenomenon its magnitude depends on Boltzmann’s constant k_B, instead of on Planck’s constant h. (Its proportionality to k_BT is what gives the effect the name “thermal noise”.).

In this chapter, we turn our attention to another important source of displacement noise, the vibration of the terrestrial environment known as seismic noise. Without proper design of the instrument, seismic noise might make gravitational wave detection impossible. (Recall Michelson and Morley’s difficulties caused by vibration of their instrument.) Even with careful instrument design, at some frequencies seismic noise will be the primary limit to an interferometer’s sensitivity.

By now, we have discussed the most important physical mechanisms expected to determine the basic limits to the performance of a gravitational wave interferometer. In this chapter, we will consider two very important practical features of interferometer design. Firstly, now that we understand several important sources of position noise, we will review how one might sensibly go about choosing the overall length of an interferometer’s arms. Secondly, we will examine one other feature that has a strong effect on the sensitivity of such an interferometer as well as on its cost. This is the requirement that the optical beam paths of the interferometer arms be enclosed within a high vacuum system. Finally, we will briefly describe a possible design for an interferometer in space that would make an “end run” around these concerns.

We’ve been discussing the various important component parts of a gravitational wave interferometer. The emphasis has been on how those parts work as parts — how their function can influence the limiting sensitivity to gravitational waves.

In this chapter, we give a sketch of the basics of the theory of feedback control systems. Our purpose is to give the reader enough background to understand the basic design principles of gravitational wave interferometers. To obtain a practical working knowledge of control system design, the reader is encouraged to study specialized texts; a good one at the beginner’s level is Franklin, Powell, and Emami- Naeini’s Feedback Control of Dynamic Systems.

Feedback is essential to the operation of a gravitational wave interferometer. Without it, an interferometer would not work - at the sensitivity required to detect astronomical gravitational waves, terrestrial disturbances would swamp the dynamic range of the instrument. Only by adding a precisely controlled countervailing influence, through a fringe-locking loop, can we hope to utilize the remarkable sensitivity of the interferometer. This is a classic application of the concept of an active null instrument. Once this is included in our picture of a gravitational wave interferometer, we will have sketched all of the basic ideas necessary to understand its successful operation.

Most of the rest of this book is devoted to showing howone might use interferometers to detect the effect of gravitational waves on nearly free masses. In this chapter, we turn our attention to a rather different style of detector, the resonant mass detector, also known colloquially as a “bar”. This is the style of detector first developed by Joseph Weber, and improved substantially in the intervening years. In fact, the best detectors of this type are the most sensitive gravitational wave detectors of any variety yet constructed. This being so, you may wonder, “Why might one prefer interferometric detectors to bars?” We will defer a discussion of this question until the end of the chapter.

For most of this book, we’ve been considering how to make a gravitational wave detector work. Now it is time to consider what we would do with such a detector. This chapter, and the one that follows, can only offer a brief overview of the ways in which we will interpret the outputs of gravitational wave detectors. For additional information, the reader is referred to the chapter by Bernard Schutz in Blair’s The Detection of Gravitational Waves.

Here we finally come to a discussion of the payoff for all of the hard work described in the preceding chapters—interpretation of detected gravitational wave signals. We start the chapter with a discussion of the determination and interpretation of source positions on the sky, because of this topic’s fundamental importance in linking gravitational wave observations with the rest of astronomy. This is followed by a survey of what might be learned from the interpretation of gravitational waveforms themselves. We conclude the chapter with a capsule summary of the results of gravitational wave astronomy to date.

Encouraged by the prospects for astrophysically interesting sensitivity to gravitational waves, research groups around the world have been contructing laboratory scale interferometers since the late 1970’s. Several, including those at MPQ-Garching,⁷⁹ the University of Glasgow²⁴², and Caltech⁶, have achieved rms sensitivities nearly rivalling the best cryogenic bar detectors. The main aim of this work, though, has been the development of techniques that will make possible the successful operation of interferometers with arm lengths of 3 to 4 km.

The following sections are included:

• Introduction
• Physics/Engineering Background (Chapters 4, 10, 11)
• Prehistory of Gravitational Wave Detection (Chapter 1)
• Gravitational Waves and their Interactions with Detectors (Chapter 2)
• Sources of Gravitational Waves (Chapter 3)
• Quantum Measurement Noise (Chapter 5)
• Interferometer Configurations (Chapters 6 and 12)
• Thermal Noise (Chapter 7)
• Seismic Noise (Chapter 8 and Section 11.7.2)
• Resonant Mass Detectors (Chapter 13)
• Large Interferometers (Chapters 9 and 16)
• Data Analysis (Chapter 14)
• Gravitational Wave Astronomy (Chapter 15)

The following section is included:

• References