Case Study: Gravitational Wave Detection, LIGO
CASE STUDY: GRAVITATIONAL WAVE DETECTION, LIGO
LIGO is an acronym for Laser Interferometer Gravitational-Wave Observatory. The LIGO observatory is funded by the National Science Foundation and is managed jointly by the California Institute of Technology and the Massachusetts Institute of Technology. Several hundred scientists collaborate on observatory design/construction, on predictions of gravitational waveforms, and on methods of analysis of the gravitational wave signals whose detection is expected in the near future. Two LIGO detectors, consisting of Michelson-type interferometers with arms 4 km long, are nearing completion at Hanford, Washington, and Livingston, Louisiana. A third detector with 2-km-long arms is also under construction at the Livingston site.
LIGO is one of several gravitational wave detectors being constructed around the world. The objective of these efforts is not only to detect gravitational waves directly for the first time but also to use the signals as a tool for observational astronomy. Processes that give rise to gravitational radiation typically involve large masses undergoing rapid motions, as in supernova collapse or in the coalescence of massive black holes. Observations of gravitational signals from such events provide details about the dynamics of motion of huge masses deep inside an evolving system; the gravitational radiation produced is usually low in frequency and easily escapes from the system because of the weakness of gravitational interactions. By contrast, electromagnetic radiation comes from individual atomic-size particles, is of high frequency, and is easily obscured by dust or other material at the surface of stars.
Gravitational waves from distant sources have flat (plane) wave fronts by the time they reach the Earth. These waves are actually ripples in space-time that exert forces on matter in their path. Understanding these forces suggests various ways of detecting such waves. Figure 1 illustrates the force fields that are established in the path of a plane gravitational wave. In Figure 1 the propagation direction occurs at right angles to the page. The waves can be fully described in terms of two so-called polarizations. Consider first the polarization in the left-most part of Figure 1. Relative to a reference mass at the origin, a free particle placed anywhere in the same plane will experience a time-varying force—hence an acceleration—in the direction of the arrows. The force reverses every half-period. The strength of the force is greater the greater the distance of the free particle from the origin; this is indicated in the figure by the increasing density of the lines of force away from the center.
The + wave polarization is illustrated in the rightmost part of Figure 1; it is basically the × figure turned 45°. The force fields of the two polarizations are at right angles to each other. An elastic bar lying along the vertical axis of the + wave would be squeezed, while an elastic bar placed along the horizontal axis would be stretched at the same time.
This alternate squeezing and stretching effect is the fundamental basis for interferometric detectors such as LIGO. An interferometer detector is diagrammed in Figure 2. Consider the + polarization. Imagine a reference mass at the origin upon which a beam splitter is placed. Mirrors are placed at the top and at the right of the figure so that light sent through the beam splitter and out along the axes is reflected by the mirrors back to a photodetector near the origin. As the gravitational wave passes through the interferometer, the distance of one of the mirrors from the beam splitter increases (the stretched arm) while the distance of the other mirror decreases (the squeezed arm). This will change the interference conditions on the two beams when they recombine, and the change in the interference pattern can be measured. The structure of such an interferometer, which was invented by A. A. Michelson in an attempt to observe the Earth's absolute motion, is thus very well suited to detect mirror motions in the presence of gravitational waves.
Gravitational wave effects are proportional to a unitless amplitude h called the wave strain. (A unitless strain in a compressed rod, for example, is the fractional change in length, or the change in length divided by starting length.) The force fields of Figure 1 correspond to a plane wave strain that is the same everywhere in the plane; this means that the movement of a reflecting mirror is proportional to the distance of the mirror from the beam splitter, whereas the wavelength of the light propagating in the interferometer arm does not change. The longer the interferometer arms, the greater the change in the interference pattern and the more sensitive the detector will be. Arm lengths of a few kilometers are a practical limit for interferometers placed on the Earth's surface, in part because the entire apparatus must be placed in high vacuum. Plans are under way for an interferometer in orbit, in the vacuum of space, with arms in a triangular configuration about 5 million km on one side (LISA).
Theoretical calculations of the wave strain amplitude are usually based on the theory of general relativity. Such calculations are mathematically very challenging, but for stellar sources of known configuration—such as stellar binaries—they lead to predictions of the wave strain amplitude in which one can place a great deal of confidence. The strains are predicted to be in the neighborhood of 10-20 or smaller. This means that for an interferometer arm with a length of 4 km, the motion of one of the mirrors is only approximately .4 + 10-15 m, which is smaller than the size of an atomic nucleus. That measurements of such small motions can be seriously contemplated is incredible. Only by venturing beyond the frontiers of technology, inspired by a series of brilliant ideas, have these measurements become possible. The LIGO detector has been designed in such a way that as new ideas are proposed and technology advances, improvements can be incorporated into the instrument so that its sensitivity will improve.
One method of increasing the sensitivity of the interferometer detector is to reflect the light beam
in each arm back upon itself with additional mirrors, so that the light bounces back and forth in the arms many times before escaping. Such mirrors are shown between the beam splitter and the end mirrors in Figure 2. This effectively increases the arm length and transforms each arm into a resonant Fabry-Perot cavity. If the arm length becomes too large, however, this advantage is lost as the forces resulting from the wave would reverse and average out while the light circulates. There is thus a storage time limit for photons in the interferometer arms beyond which the sensitivity does not improve. The storage time limit is approximately half the period of the gravitational wave; during this time, the forces act generally to push the mirrors in a consistent direction. For a wave of frequency 500 Hz the storage time limit is0.001 s, and in an arm with a length of 4 km there should not be more than about 75 bounces. The sensitivity of a 4-km interferometer can be pushed to a lower frequency (corresponding to a larger number of bounces) if the mirrors are extremely smooth and made of ultra-low-loss material.
Another improvement in sensitivity can be achieved within a limited frequency range by placing a partially transmitting signal recycling mirror between the beam splitter and the photodetector. This transforms the interferometer into a cavity that resonates and thus gives improved sensitivity within a frequency range controlled by mirror reflectivity and position. This technique is expected to be useful when searching for continuous signals of definite frequencies, such as those from rotating neutron stars.
One of the most serious difficulties to overcome is that of isolating the beam splitter and mirrors from the Earth's seismic vibrations. (Interferometers in space are not subject to seismic noise.) In order to allow the mirrors to respond to a gravitational wave, freedom of motion parallel to the interferometer arm is necessary. This is made possible, while also providing some isolation from seismic vibrations, by suspending the beam splitter and mirrors like pendulums, from supports which are further isolated from vibrations with carefully designed stacks of absorbing material or heavy masses. A pendulum that is free to swing parallel to the arm can be set into motion by a seismic vibration of the support in this direction. However, if the seismic vibration frequency f is large compared to the pendulum frequency f0, the mirror motion will be smaller than the support motion by a factor of (f0/f )2. Several stages of such isolation can be considered, but the pendulum support limits the low-frequency sensitivity of the detector. In future detectors, external equipment may sense seismic motions and actively push the detectors to compensate for seismic disturbance.
Other kinds of vibrations must also be reduced. Vertical motions of the support can couple to horizontal motions, for example, because the mirrors at the ends of very long interferometer arms are actually not parallel—the suspension wires tend to point toward the Earth's center. Also, there may be cross-couplings of vibrations resulting from mechanical misalignments. The suspending wires can vibrate like strings on a violin; all such mechanical oscillations must be severely suppressed. Even if all such vibrations could be completely eliminated, there would still be gravity gradient noise—time-varying gravitational forces on the mirrors arising from people and vehicles moving in the vicinity—that cannot be shielded from the apparatus and that will limit the sensitivity at low frequencies.
There are many other sources of random noise that can cause spurious mirror motions. One of the most important of these is due to random fluctuations in the number of photons that are circulating in the interferometer arms, called shot noise. If the number of circulating photons can be significantly increased, however, the sensitivity of the detector can be improved. The apparatus is most sensitive to mirror motion when there is destructive interference of the recombined light beams at the photodetector; in other words, the photodetector is at the dark port. Then most of the light energy goes back through the beam splitter toward the light source. Another mirror, called a power-recycling mirror, can be placed in the path of this light to reflect it back into the interferometer. This can significantly increase the power circulating in the arms; fluctuations resulting from shot noise then play less of a role. There is a tradeoff, however, because if the power in the arms is too great, there can be undesirable thermal heating and distortion of the mirrors, causing losses of light, as well as fluctuations in radiation pressure on the mirrors. The placement of the power-recycling mirror is shown in front of the laser in Figure 2.
A disadvantage of having high power circulating in the arms is that radiation pressure on the mirrors fluctuates, causing unwanted mirror motions. The effort to reduce shot noise by increasing power in the arms is thus thwarted by radiation pressure. The two effects depend on frequency differently with shot noise tending to increase at frequencies of 200 Hz and higher, and radiation pressure increasing at lower frequencies. Interferometer designers must find a compromise between these two effects—see the discussion of Figure 3 below.
Heat can excite vibration modes of the mirror masses or of the suspension systems, and these can couple into resonances of the system such as pendulum modes and cause unwanted mirror motions. Reducing energy losses in the suspension materials can reduce such effects. Thermal motions have amplitudes that depend on the temperature, and making the interferometer arms long can reduce the harmful effects of thermal excitations.
Other significant sources of noise are instabilities in frequency and power coming from the laser. Usually, the interferometer arms cannot be made exactly equal in length. Then if the frequency of the laser light fluctuates, the interference conditions at the dark port will change, and a spurious signal will be detected. To overcome this, a mode cleaner is inserted between the laser output and the power-recycling mirror. (See Figure 2.) A mode cleaner is an optical assembly that acts like a narrow band filter so that light of unwanted frequency cannot pass through. Power fluctuations can also cause signals at the dark port; partly for this reason, lasers such as Nd:YAG that are easier to control will be used in LIGO.
There are numerous other sources of noise that must be dealt with in order to reduce unwanted signals at the dark port. These include light scattered out of the main light beams and then scattered back in, for example, from the walls of the vacuum chamber. The phase of this scattered light will not agree with the phase of the main beam and so will contaminate the signals. A system of baffles to absorb scattered light within the vacuum system is therefore required. Residual gas in the vacuum chamber can cause fluctuations in the index of refraction that disturb the beams, and gas particles can also bounce off the mirrors, causing slight displacements. Motions from any source (such as seismic disturbances) in the optical systems (as in the mode cleaner) cause the beam position and direction to fluctuate and hence cause some noise at the dark port. Stray electric and magnetic fields can affect the mirrors. For example, if there is stray parasitic charge on any of the optical surfaces, electric fields can exert forces on these elements. Magnetic fields can interact with actuator magnets bonded to the mirrors (such magnets are used to control mirror orientation) and exert forces on them.
The construction of the first-generation LIGO I detectors has been completed (April 2002) and rigorous testing of the many systems—optical, vacuum, laser, control, data analysis—is under way. Figure 3 shows the design sensitivity of LIGO I (curve marked I) and of the advanced LIGO (curve marked II) that features enhanced sensitivity due to the incorporation of many features made possible by developing technology. (Gravitational wave sensitivities h are usually quoted with units 1/√Hz. The squared strain noise in a narrow range of frequencies Δf is then h2 Δf and is dimensionless.) LIGO I's sensitivity lies in the frequency range of 30 to 1,200 Hz. LIGO II should have much improved sensitivity over a wider frequency range.
Interferometer detectors are most sensitive to waves propagating perpendicular to the interferometer plane. However, as the Earth rotates, this direction sweeps across the sky, sampling waves from different directions. The Hanford and Livingston facilities are sufficiently far apart on the Earth's surface that seismic disturbances are likely to be uncorrelated; also, the detectors are oriented differently. A gravitational signal burst would generally arrive at different times at the two sites. Looking for delayed coincidences between signals arriving at the two sites can determine the direction of the source as well as yield better detection sensitivity and eliminate false alarms. The growing worldwide network of gravitational wave observatories (LIGO in the United States, VIRGO and GEO600 in Europe, and TAMA300 in Japan) is working out agreements whereby different projects may save and exchange data. It will thus be possible to operate all of them together in order to observe common coincidences.
There are a few astrophysical sources whose gravitational wave strain amplitudes are large enough to be detected by LIGO I. Estimates of the radiation emitted by the pulsar remnant from the Crab supernova, which occurred in 1054, show that the signal strength may be large enough to be observed. Astronomers have been surprised many times by observations of previously unknown objects when new instruments came online, so it is possible that sources of as yet unknown types exist, which could be observed. LIGO I serves as a proving ground from which much is being learned about noise sources and the methods of detecting gravitational signals. Because signals are weak and many noise sources are present, detection involves continuous observation over many months, so that the noise can be averaged down. Data are analyzed using the technique of matched filtering—correlation of the data with calculated "templates" of waveforms expected from the source.
There are many additional astrophysical sources, however, that should be detectable with LIGO II. These usually involve violent motions of large masses. A few such sources are indicated with heavy lines in Figure 3. For example, the upper heavy line, sloping downward as the frequency increases, corresponds to theoretical estimates of observable signals from the interaction of two ten-solar-mass black holes, at a distance less than 100 Megaparsecs. (1 Megaparsec [Mpc] = 3.3 million light-years. The Milky Way galaxy is about 0.03 Mpc in diameter; there are about
twenty other galaxies containing many potential sources within approximately 1 Mpc of the Milky Way.) As the two black holes orbit around each other, they lose energy because of the emission of gravitational radiation, and spiral inward toward each other. The radiation frequency increases as they come closer, until finally the black holes collide, merge, and probably vibrate violently until the system settles down to a rotating black hole. Each stage of such a process involves different signal frequencies and amplitudes. If the binary system is closer than 100 Mpc, the line will rise on the figure; LIGO I might then be able to detect such signals.
The lower black line at the left of Figure 3 corresponds to black hole–black hole inspiral at distances less than 400 Mpc. The heavy black line toward the right side of Figure 3 shows the signal amplitudes from fast-spinning pulsars of unknown frequencies but at distances less than 0.01 Mpc with an averaging time of three months, if one assumes the pulsars are slightly nonspherical.
Signal recycling can improve LIGO II's sensitivity significantly beyond what is plotted in Figure 3, within a narrow frequency band near approximately 700 Hz (not shown in Figure 3). In this frequency range there are many low-mass X-ray–emitting binary systems that should be observable by LIGO II in a narrow-frequency-band operation mode.
Other binary systems such as neutron star binaries, or neutron star–black hole binaries, are possible sources. In many such sources, the underlying physics is poorly understood; theoretical estimates of the signal amplitudes of waveforms are, at best, approximate. The actual observation of signals would have a very significant impact on understanding the dynamics of such processes.
Heavy stars that exhaust their stores of nuclear energy after many millions of years may undergo catastrophic collapse and subsequently produce a supernova explosion, emitting strong bursts of gravitational radiation. Chunks of matter ejected from such systems could remain in close orbit and radiate at frequencies equal to twice their orbital periods.
Another potential source, not shown in Figure 3, is random gravitational waves arising perhaps from the Big Bang or from events during early stages of expansion, such as transitions from domination of the universe by one type of matter to another.
Clearly, there are many interesting astrophysical processes that serve as potential candidates for gravitational wave detection. Studies of such sources are actively underway by a large number of researchers. At the same time, improvements in detector sensitivity are being actively pursued on many fronts, such as in development of ultrasmooth mirror surfaces and coatings, better suspensions and attachments, actuation of optical components, data analysis, and many other areas.
The LIGO collaboration consists of hundreds of researchers, from all over the world, organized into working groups that concentrate actively on specialized aspects of the overall effort. These efforts are expected to result in the successful observation of gravitational waves, opening up a new and exciting field of astronomy using a new tool that can observe the inner details of large-scale, fast astrophysical processes that cannot be observed by any other technique.
Robertson, N. "Laser Interferometric Gravitational Wave Detectors." Classical and Quantum Gravity17 (15), R19–R40(2000).
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