Physics, Astrophysics and Cosmology
with Gravitational Waves
Gravitational wave detectors are already operating at interesting sensitivity levels, and they have an upgrade path that should result in secure detections by 2014. We review the physics of gravitational waves, how they interact with detectors (bars and interferometers), and how these detectors operate. We study the most likely sources of gravitational waves and review the data analysis methods that are used to extract their signals from detector noise. Then we consider the consequences of gravitational wave detections and observations for physics, astrophysics, and cosmology.
- 1 A New Window onto the Universe
2 Gravitational Wave Observables
- 2.1 Gravitational field vs gravitational waves
- 2.2 Gravitational wave polarizations
- 2.3 Direction to a source
- 2.4 Amplitude of gravitational waves – the quadrupole approximation
- 2.5 Frequency of gravitational waves
- 2.6 Luminosity in gravitational waves
3 Sources of Gravitational Waves
- 3.1 Man-made sources
- 3.2 Gravitational wave bursts from gravitational collapse
- 3.3 Gravitational wave pulsars
- 3.4 Radiation from a binary star system
- 3.5 Quasi-normal modes of a black hole
- 3.6 Stochastic background
4 Gravitational Wave Detectors and Their Sensitivity
- 4.1 Principles of the operation of resonant mass detectors
- 4.2 Principles of the operation of beam detectors
- 4.3 Practical issues of ground-based interferometers
- 4.4 Detection from space
- 4.5 Characterizing the sensitivity of a gravitational wave antenna
- 4.6 Source amplitudes vs sensitivity
- 4.7 Network detection
- 4.8 False alarms, detection threshold and coincident observation
5 Data Analysis
- 5.1 Matched filtering and optimal signal-to-noise ratio
- 5.2 Suboptimal filtering methods
- 5.3 Measurement of parameters and source reconstruction
6 Physics with Gravitational Waves
- 6.1 Speed of gravitational waves
- 6.2 Polarization of gravitational waves
- 6.3 Gravitational radiation reaction
- 6.4 Black hole spectroscopy
- 6.5 The two-body problem in general relativity
- 6.6 Tests of general relativity
7 Astrophysics with Gravitational Waves
- 7.1 Interacting compact binaries
- 7.2 Black hole astrophysics
- 7.3 Neutron star astrophysics
- 7.4 Multimessenger gravitational-wave astronomy
8 Cosmology with Gravitational Wave Observations
- 8.1 Detecting a stochastic gravitational wave background
- 8.2 Origin of a random background of gravitational waves
- 8.3 Cosmography: gravitational wave measurements of cosmological parameters
- 9 Conclusions and Future Directions
- 10 Acknowledgements
1 A New Window onto the Universe
The last six decades have witnessed a great revolution in astronomy, driven by improvements in observing capabilities across the electromagnetic spectrum: very large optical telescopes, radio antennas and arrays, a host of satellites to explore the infrared, X-ray, and gamma-ray parts of the spectrum, and the development of key new technologies (CCDs, adaptive optics). Each new window of observation has brought new surprises that have dramatically changed our understanding of the universe. These serendipitous discoveries have included:
the relic cosmic microwave background radiation (Penzias and Wilson ), which has become our primary tool for exploring the Big Bang;
the fact that quasi-stellar objects are at cosmological distances (Maarten Schmidt ), which has developed into the understanding that they are powered by supermassive black holes;
pulsars (Hewish and Bell ), which opened up the study of neutron stars and illuminated one endpoint for stellar evolution;
X-ray binary systems (Giacconi and collaborators ), which now enable us to make detailed studies of black holes and neutron stars;
gamma-ray bursts coming from immense distances (Klebesadel et al. ), which are not fully explained even today;
None of these discoveries was anticipated by the observing team, and in many cases the instruments were built to observe completely different phenomena.
Within a few years another new window on the universe will open up, with the first direct detection of gravitational waves. There is keen interest in observing gravitational waves directly, in order to test Einstein’s theory of general relativity and to observe some of the most exotic objects in nature, particularly black holes. But, in addition, the potential of gravitational wave observations to produce more surprises is very high.
The gravitational wave spectrum is completely distinct from, and complementary to, the electromagnetic spectrum. The primary emitters of electromagnetic radiation are charged elementary particles, mainly electrons; because of overall charge neutrality, electromagnetic radiation is typically emitted in small regions, with short wavelengths, and conveys direct information about the physical conditions of small portions of the astronomical sources. By contrast, gravitational waves are emitted by the cumulative mass and momentum of entire systems, so they have long wavelengths and convey direct information about large-scale regions. Electromagnetic waves couple strongly to charges and so are easy to detect but are also easily scattered or absorbed by material between us and the source; gravitational waves couple extremely weakly to matter, making them very hard to detect but also allowing them to travel to us substantially unaffected by intervening matter, even from the earliest moments of the Big Bang.
These contrasts, and the history of serendipitous discovery in astronomy, all suggest that electromagnetic observations may be poor predictors of the phenomena that gravitational wave detectors will eventually discover. Given that 96% of the mass-energy of the universe carries no charge, gravitational waves provide us with our first opportunity to observe directly a major part of the universe. It might turn out to be as complex and interesting as the charged minor component, the part that we call “normal” matter.
Several long-baseline interferometric gravitational-wave detectors planned over a decade ago [Laser Interferometer Gravitational-Wave Observatory (LIGO) , GEO , VIRGO  and TAMA ] have begun initial operations [3, 245, 19] with unprecedented sensitivity levels and wide bandwidths at acoustic frequencies (10 Hz – 10 kHz) . These large interferometers are superseding a world-wide network of narrow-band resonant bar antennas that operated for several decades at frequencies near 1 kHz. Before 2020 the space-based LISA  gravitational wave detector may begin observations in the low-frequency band from 0.1 mHz to 0.1 Hz. This suite of detectors can be expected to open up the gravitational wave window for astronomical exploration, and at the same time perform stringent tests of general relativity in its strong-field dynamic sector.
Gravitational wave antennas are essentially omni-directional, with linearly polarized quadrupolar antenna patterns that typically have a response better than 50% of its average over 75% of the sky . Their nearly all-sky sensitivity is an important difference from pointed astronomical antennas and telescopes. Gravitational wave antennas operate as a network, with the aim of taking data continuously. Ground-based interferometers can at present (2008) survey a volume of order for inspiraling compact star binaries – among the most promising sources for these detectors – and plan to enhance their range more than tenfold with two major upgrades (to enhanced and then advanced detectors) during the period 2009 – 2014. For the advanced detectors, there is great confidence that the resulting thousandfold volume increase will produce regular detections. It is this second phase of operation that will be more interesting from the astrophysical point of view, bringing us physical and astrophysical insights into populations of neutron star and black hole binaries, supernovae and formation of compact objects, populations of isolated compact objects in our galaxy, and potentially even completely unexpected systems. Following that, LISA’s ability to survey the entire universe for black hole coalescences at milliHertz frequencies will extend gravitational wave astronomy into the cosmological arena.
However, the present initial phase of observation, or observations after the first enhancements, may very well produce the first detections. Potential sources include coalescences of binaries consisting of black holes at a distance of 100 – 200 Mpc and spinning neutron stars in our galaxy with ellipticities greater than about . Observations even at this initial level may, of course, also reveal new sources not observable in any other way. These initial detections, though not expected to be frequent, would be important from the fundamental physics point of view and could enable us to directly test general relativity in the strongly nonlinear regime.
Gravitational wave detectors register gravitational waves coherently by following the phase of the wave and not just measuring its intensity. Since the phase is determined by large-scale motions of matter inside the sources, much of the astrophysical information is extracted from the phase. This leads to different kinds of data analysis methods than one normally encounters in astronomy, based on matched filtering and searches over large parameter spaces of potential signals. This style of data analysis requires the input of pre-calculated template signals, which means that gravitational wave detection depends more strongly than most other branches of astronomy on theoretical input. The better the input, the greater the range of the detectors.
The fact that detectors are omni-directional and detect coherently the phase of the incoming wave makes them in many ways more like microphones for sound than like conventional telescopes. The analogy with sound can be helpful, since microphones can be used to monitor environments for disturbances in any location, and since when we listen to sounds our brains do a form of matched filtering to allow us to interpret the sounds we want to understand against a background of noise. In a very real sense, gravitational wave detectors will be listening to the sounds of a restless universe. The gravitational wave “window” will actually be a listening post, a monitor for the most dramatic events that occur in the universe.
1.1 Birth of gravitational astronomy
Gravity is the dominant interaction in most astronomical systems. The big surprise of the last three decades of the 20th century was that relativistic gravitation is relevant in so many of these systems. Strong gravitational fields are Nature’s most efficient converters of mass into energy. Examples where strong-field relativistic gravity is important include the following:
neutron stars, the residue of supernova explosions, represent up to 0.1% (by number) of the entire stellar population of any galaxy;
stellar-mass black holes power many binary X-ray sources and tend to concentrate near the centers of globular clusters;
massive black holes in the range seem almost ubiquitous in galaxies that have central bulges, and power active galaxies, quasars, and giant radio jets;
and, of course, the Big Bang is the only naked singularity we expect to be able to see.
Most of these systems are either dynamical or were formed in catastrophic events; many are or were, therefore, strong sources of gravitational radiation. As the 21st century opens, we are on the threshold of using this radiation to gain a new perspective on the observable universe.
The theory of gravitational radiation already makes an important contribution to the understanding of a number of astronomical systems, such as neutron star binaries, cataclysmic variables, young neutron stars, low-mass X-ray binaries, and even the anisotropy of the microwave background radiation. As the understanding of relativistic phenomena improves, it can be expected that gravitational radiation will play a crucial role as a theoretical tool in modeling relativistic astrophysical systems.
1.2 What this review is about
The first three-quarters of the 20th century were required to place the mathematical theory of gravitational radiation on a sound footing. Many of the most fundamental constructs in general relativity, such as null infinity and the theory of conserved quantities, were developed at least in part to help solve the technical problems of gravitational radiation. We will not cover this history here, for which there are excellent reviews [259, 132]. There are still many open questions, since it is impossible to construct exact solutions for most interesting situations. For example, we still lack a full understanding of the two-body problem, and we will review the theoretical work on this problem below. But the fundamentals of the theory of gravitational radiation are no longer in doubt. Indeed, the observation of the orbital decay in the binary pulsar PSR B1913+16  has lent irrefutable support to the correctness of the theoretical foundations aimed at computing gravitational wave emission, in particular to the energy and angular momentum carried away by the radiation.
It is, therefore, to be expected that the evolution of astrophysical systems under the influence of strong tidal gravitational fields will be associated with the emission of gravitational waves. Consequently, these systems are of interest both to a physicist, whose aim is to understand fundamental interactions in nature, their inter-relationships and theories describing them, and to an astrophysicist, who wants to dig deeper into the environs of dense or nonlinearly gravitating systems in solving the mysteries associated with relativistic phenomena listed in Sections 6, 7 and 8. Indeed, some of the gravitational wave antennas that are being built are capable of observing systems to cosmological distances, and even to the edge of the universe. The new window, therefore, is also of interest to cosmologists.
This is a living review of the prospects that lie ahead for gravitational antennas to test the predictions of general relativity as a fundamental theory, for using relativistic gravitation as a means to understand highly energetic sources, for interpreting gravitational waves to uncover the (electromagnetically) dark universe, and ultimately for employing networks of gravitational wave detectors to observe the first fraction of a second of the evolution of the universe.
We begin in Section 2 with a brief review of the physical nature of gravitational waves, giving a heuristic derivation of the formulas involved in the calculation of the gravitational wave observables such as the amplitude, frequency and luminosity of gravitational waves. This is followed in Section 3 by a discussion of the astronomical sources of gravitational waves, their expected event rates, amplitudes, waveforms and spectra. In Section 4 we then give a detailed description of the existing and upcoming gravitational wave antennas and their sensitivity. Included in Section 4 are bar and interferometric antennas covering both ground and space-based experiments. Section 4 also compares the sensitivity of the antennas with the strengths of astronomical sources and expected signal-to-noise ratios (SNRs). We then turn in Section 5 to data analysis, which is a central component of gravitational wave astronomy, focusing on those aspects of analysis that are crucial in gleaning physical, astrophysical and cosmological information from gravity wave observations.
Sections 7 – 9 treat in some detail how gravitational wave observations will aid in a better understanding of nonlinear gravity and test some of its fundamental predictions. In Section 6 we review the physics implications of gravitational wave observations, including new tests of general relativity that can be performed via gravitational wave observations, how these observations may help in formulating and gaining insight into the two-body problem in general relativity, and how gravitational wave observations may help to probe the structure of the universe and the nature of dark energy. In Section 7 we look at the astronomical information returned by gravitational wave observations, and how these observations will affect our understanding of black holes, neutron stars, supernovae, and other relativistic phenomena. Section 8 is devoted to the cosmological implications of gravitational wave observations, including placing constraints on inflation, early phase transitions associated with spontaneous symmetry breaking, and the large-scale structure of the universe.
This review is by no means exhaustive. We plan to expand it to include other key topics in gravitational wave astronomy with subsequent revisions.
Unless otherwise specified we shall use a system of units in which , which means , . We shall assume a universe with cold dark-matter density of , dark energy of , and a Hubble constant of .
2 Gravitational Wave Observables
To benefit from gravitational wave observations we must first understand what are the attributes of gravitational waves that we can observe. This section is devoted to a short discussion of the nature of gravitational radiation.
2.1 Gravitational field vs gravitational waves
Gravitational waves are propagating oscillations of the gravitational field, just as light and radio waves are propagating oscillations of the electromagnetic field. Whereas light and radio waves are emitted by accelerated electrically-charged particles, gravitational waves are emitted by accelerated masses. However, since there is only one sign of mass, gravitational waves never exist on their own: they are never more than a small part of the overall external gravitational field of the emitter. One may wonder, therefore, how it is possible to infer the presence of an astronomical body by the gravitational waves that it emits, when it is clearly not possible to sense its much larger stationary (essentially Newtonian) gravitational potential. There are, in fact, two reasons:
In general relativity, the effects of both the stationary field and gravitational radiation are described by the tidal forces they produce on free test masses. In other words, single geodesics alone cannot detect gravity or gravitational radiation; we need at least a pair of geodesics. While the stationary tidal force due to the Newtonian potential of a self-gravitating source at a distance falls off as , the tidal force due to the gravitational wave amplitude that it emits at wavelength decreases as . Therefore, the stationary coulomb gravitational potential is the dominant tidal force close to the gravitating body (in the near zone, where ). However, in the far zone () the tidal effect of the waves is much stronger.
The stationary part of the tidal field is a DC effect, and simply adds to the stationary tidal forces of all other objects in the universe. It is not possible to discriminate one source from another. Gravitational waves carry time-dependent tidal forces, and so they can be discriminated from the stationary field if one knows what kind of time dependence to look for. Interferometers are ideal detectors in this respect because they sense only changes in the position of an interference fringe, which makes them insensitive to the DC part of the tidal field.
Because gravitational waves couple so weakly to our detectors, those astronomical sources that we can detect must be extremely luminous in gravitational radiation. Even at the distance of the Virgo cluster of galaxies, a detectable source could be as luminous as the full Moon, if only for a millisecond! Indeed, while radio astronomers deal with flux levels of Jy, mJy and even Jy, in the case of gravitational wave sources we encounter fluxes that are typically Jy or larger. Gravitational wave astronomy therefore is biased toward looking for highly energetic, even catastrophic, events.
Extracting useful physical, astrophysical and cosmological information from gravitational wave observations is made possible by measuring a number of gravitational wave attributes that are related to the properties of the source. In the rest of this section we discuss those attributes of gravitational radiation that can be measured via gravitational wave observations. In the process we will review the basic formulas used in computing the gravitational wave amplitude and luminosity of a source. These will then be used in Section 3 to make an order-of-magnitude estimate of the strength of astronomical sources of gravitational waves.
2.2 Gravitational wave polarizations
Because of the equivalence principle, single isolated particles cannot be used to measure gravitational waves: they fall freely in any gravitational field and experience no effects from the passage of the wave. Instead, one must look for inhomogeneities in the gravitational field, which are the tidal forces carried by the waves, and which can be measured only by comparing the positions or interactions of two or more particles.
In general relativity, gravitational radiation is represented by a second rank, symmetric trace-free tensor. In a general coordinate system, and in an arbitrary gauge (coordinate choice), this tensor has ten independent components. However, as in the electromagnetic case, gravitational radiation has only two independent states of polarization in Einstein’s theory: the plus polarization and the cross polarization (the names being derived from the shape of the equivalent force fields that they produce). In contrast to electromagnetic waves, the angle between the two polarization states is rather than . This is illustrated in Figure 1, where the response of a ring of free particles in the plane to plus-polarized and cross-polarized gravitational waves traveling in the -direction is shown. The effect of the waves is to cause a tidal deformation of the circular ring into an elliptical ring with the same area. This tidal deformation caused by passing gravitational waves is the basic principle behind the construction of gravitational wave antennas.
The two independent polarizations of gravitational waves are denoted and . These are the two primary time-dependent observables of a gravitational wave. The polarization of gravitational waves from a source, such as a binary system, depends on the orientation of the dynamics inside the source relative to the observer. Therefore, measuring the polarization provides information about, for example, the orientation of the binary system.
2.3 Direction to a source
Gravitational wave antennas are linearly-polarized quadrupolar detectors and do not have good directional sensitivity. As a result we cannot deduce the direction to a source using a single antenna. One normally needs simultaneous observation using three or more detectors so that the source can be triangulated in the sky by measuring the time differences in signal arrival times at various detectors in a network. Ground-based detectors have typical separation baselines of , so that at a wavelength of (a frequency of 1 kHz) the network has a resolution of . If the amplitude SNR is high, then one can localize the source by a factor of better than this.
For long-lived sources, however, a single antenna synthesizes many antennas by observing the source at different points along its orbit around the sun. The baseline for such observations is 2 AU, so that, for a source emitting radiation at 1 kHz, the resolution is as good as , which is smaller than an arcsecond.
For space-based detectors orbiting the sun, like LISA, the baseline is again 2 AU, but the observing frequency is some five or six orders of magnitude lower, so the basic resolution is only of order 1 radian. However, as we shall see later, some of the sources that a space-based detector will observe have huge amplitude SNRs in the range of , which improves the resolution to arcminute accuracies in the best cases.
2.4 Amplitude of gravitational waves – the quadrupole approximation
The Einstein equations are too difficult to solve analytically in the generic case of a strongly gravitating source to compute the luminosity and amplitude of gravitational waves from an astronomical source. We will discuss numerical solutions later; the most powerful available analytic approach is called the post-Newtonian approximation scheme. This approximation is suited to gravitationally-bound systems, which constitute the majority of expected sources. In this scheme [79, 169], solutions are expanded in the small parameter , where is the typical dynamical speed inside the system. Because of the virial theorem, the dimensionless Newtonian gravitational potential is of the same order, so that the expansion scheme links orders in the expanded metric with those in the expanded source terms. The lowest-order post-Newtonian approximation for the emitted radiation is the quadrupole formula, and it depends only on the density () and velocity fields of the Newtonian system. If we define the spatial tensor , the second moment of the mass distribution, by the equation
then the amplitude of the emitted gravitational wave is, at lowest order, the three-tensor
This is to be interpreted as a linearized gravitational wave in the distant almost-flat geometry far from the source, in a coordinate system (gauge) called the Lorentz gauge.
2.4.1 Wave amplitudes and polarization in TT-gauge
A useful specialization of the Lorentz gauge is the TT-gauge, which is a comoving coordinate system: free particles remain at constant coordinate locations, even as their proper separations change. To get the TT-amplitude of a wave traveling outwards from its source, project the tensor in Equation (2) perpendicular to its direction of travel and remove the trace of the projected tensor. The result of doing this to a symmetric tensor is to produce, in the transverse plane, a two-dimensional matrix with only two independent elements:
This is the definition of the wave amplitudes and that are illustrated in Figure 1. These amplitudes are referred to as the coordinates chosen for that plane. If the coordinate unit basis vectors in this plane are and , then we can define the basis tensors
In terms of these, the TT-gravitational wave tensor can be written as
If the coordinates in the transverse plane are rotated by an angle , then one obtains new amplitudes and given by
This shows the quadrupolar nature of the polarizations, and is consistent with our remark in association with Figure 1 that a rotation of changes one polarization into the other.
It should be clear from the TT projection operation that the emitted radiation is not isotropic: it will be stronger in some directions than in others111In the case of an inspiraling binary, the root mean square of the two polarization amplitudes in a direction orthogonal to the orbital plane will be a factor larger than in the plane.. It should also be clear from this that spherically-symmetric motions do not emit any gravitational radiation: when the trace is removed, nothing remains.
2.4.2 Simple estimates
It is interesting to observe that the ratio of the wave amplitude to the Newtonian potential of its source at the observer’s distance is simply bounded by
and this bound is attained if the entire mass of the source is involved in the nonspherical motions, so that . By the virial theorem for self-gravitating bodies
where is the maximum value of the Newtonian gravitational potential inside the system. This provides a convenient bound in practice :
The bound is attained if the system is highly nonspherical. An equal-mass star binary system is a good example of a system that attains this bound.
For a neutron star source, one has . If the star is in the Virgo cluster () and has a mass of , and if it is formed in a highly-nonspherical gravitational collapse, then the upper limit on the amplitude of the radiation from such an event is . This is a simple way to get the number that has been the goal of detector development for decades, to make detectors that can observe waves at or below an amplitude of about .
2.5 Frequency of gravitational waves
The signals for which the best waveform predictions are available have well-defined frequencies. In some cases the frequency is dominated by an existing motion, such as the spin of a pulsar. But in most cases the frequency will be related to the natural frequency for a self-gravitating body, defined as
where is the mean density of mass-energy in the source. This is of the same order as the binary orbital frequency and the fundamental pulsation frequency of the body. Even though this is a Newtonian formula, it provides a remarkably good order-of-magnitude prediction of natural frequencies, even for highly relativistic systems such as black holes.
The frequency of the emitted gravitational waves need not be the natural frequency, of course, even if the mechanism is an oscillation with that frequency. In many cases, such as binary systems, the radiation comes out at twice the oscillation frequency. But since, at this point, we are not trying to be more accurate than a few factors, we will ignore this distinction here. In later sections, with specific source models, we will get the factors right.
The mean density and hence the frequency are determined by the size and mass of the source, taking . For a neutron star of mass and radius 10 km, the natural frequency is . For a black hole of mass and radius , it is . And for a large black hole of mass , such as the one at the center of our galaxy, this goes down in inverse proportion to the mass to . In general, the characteristic frequency of the radiation of a compact object of mass and radius is
Figure 2 shows the mass-radius diagram for likely sources of gravitational waves. Three lines of constant natural frequency are plotted: , , and . These are interesting frequencies from the point of view of observing techniques: gravitational waves between 1 and Hz are in principle accessible to ground-based detectors, while lower frequencies are observable only from space. Also shown is the line marking the black-hole boundary. This has the equation . There are no objects below this line, because they would be smaller than the horizon size for their mass. This line cuts through the ground-based frequency band in such a way as to restrict ground-based instruments to looking at stellar-mass objects. No system with a mass above about can produce quadrupole radiation in the ground-based frequency band.
A number of typical relativistic objects are placed in the diagram: a neutron star, a pair of neutron stars that spiral together as they orbit, some black holes. Two other interesting lines are drawn. The lower (dashed) line is the 1-year coalescence line, where the orbital shrinking timescale due to gravitational radiation backreaction (cf. Equation (28)) is less than one year. The upper (solid) line is the 1-year chirp line: if a binary lies below this line, then its orbit will shrink enough to make its orbital frequency increase by a measurable amount in one year. (In a one-year observation one can, in principle, measure changes in frequency of , or Hz.)
It is clear from the Figure that any binary system that is observed from the ground will coalesce within an observing time of one year. Since pulsar binary statistics suggest that neutron-star–binary coalescences happen less often than once every years in our galaxy, ground-based detectors must be able to register these events in a volume of space containing at least galaxies in order to have a hope of seeing occasional coalescences. That corresponds to a volume of radius roughly 100 Mpc. For comparison, first-generation ground-based interferometric detectors have a reach of around 20 Mpc for such binaries, while advanced interferometers should extend that to about 200 Mpc.
2.6 Luminosity in gravitational waves
where the angle brackets denote averages over regions of the size of a wavelength and times of the length of a period of the wave. The energy flux of a wave in the direction is the component.
The gravitational wave luminosity in the quadrupole approximation is obtained by integrating the energy flux from Equation (14) over a distant sphere. When one correctly takes into account the projection factors mentioned after Equation (2), one obtains 
where is the trace of the matrix . This equation can be used to estimate the backreaction effect on a system that emits gravitational radiation.
Notice that the expression for is dimensionless when . It can be converted to normal luminosity units by multiplying by the scale factor
This is an enormous luminosity. By comparison, the luminosity of the sun is only W, and that of a typical galaxy would be W. All the galaxies in the visible universe emit, in visible light, on the order of W. We will see that gravitational wave systems always emit at a fraction of , but that the gravitational wave luminosity can come close to and can greatly exceed typical electromagnetic luminosities. Close binary systems normally radiate much more energy in gravitational waves than in light. Black hole mergers can, during their peak few cycles, compete in luminosity with the steady luminosity of the entire universe!
The above relation can be used to make an order-of-magnitude estimate of the gravitational wave amplitude from a knowledge of the rate at which energy is emitted by a source in the form of gravitational waves. If a source at a distance radiates away energy in a time , predominantly at a frequency , then writing and noting that , the amplitude of gravitational waves is
When the time development of a signal is known, one can filter the detector output through a copy of the expected signal (see Section 5 on matched filtering). This leads to an enhancement in the SNR, as compared to its narrow-band value, by roughly the square root of the number of cycles the signal spends in the detector band. It is useful, therefore, to define an effective amplitude of a signal, which is a better measure of its detectability than its raw amplitude:
Now, a signal lasting for a time around a frequency would produce cycles. Using this we can eliminate from Equation (18) and get the effective amplitude of the signal in terms of the energy, the emitted frequency and the distance to the source:
Notice that this depends on the energy only through the total fluence, or time-integrated flux of the wave. As in many other branches of astronomy, the detectability of a source is ultimately a function of its apparent luminosity and the observing time. However, one should not ignore the dependence on frequency in this formula. Two sources with the same fluence are not equally easy to detect if they are at different frequencies: higher frequency signals have smaller amplitudes.
3 Sources of Gravitational Waves
3.1 Man-made sources
One source can unfortunately be ruled out as undetectable: man-made gravitational radiation. Imagine creating a wave generator with the following extreme properties. It consists of two masses of kg each (a small car) at opposite ends of a beam 10 m long. At its center the beam pivots about an axis. This centrifuge rotates 10 times per second. All the velocity is nonspherical, so in Equation (9) is about . The frequency of the waves will actually be 20 Hz, since the mass distribution of the system is periodic in time with a period of half the rotation period. The wavelength of the waves will, therefore, be m, similar to the diameter of the earth. In order to detect gravitational waves, not near-zone Newtonian gravity, the detector must be at least one wavelength from the source, say diametrically opposite the centrifuge on the Earth. Then the amplitude can be deduced from Equation (9): . This is far too small to contemplate detecting! The story changes, fortunately, when we consider astrophysical sources of gravitational waves, where nature arranges for masses that are times larger than our centrifuge to move at speeds close to the speed of light!
Until observations of gravitational waves are successfully made, one can only make intelligent guesses about most of the sources that will be seen. There are many that could be strong enough to be seen by the early detectors: star binaries, supernova explosions, neutron stars, the early universe. In this section, we make rough luminosity estimates using the quadrupole formula and other approximations, which are usually accurate to within factors of order two, and, very importantly, they show how key observables scale with the properties of the systems. Where appropriate we also make use of predictions from the much more accurate modelling that is available for some sources, such as binary systems and black hole mergers. The detectability depends, of course, not only on the intrinsic luminosity of the source, but on how far away it is. Often the biggest uncertainties in making predictions are the spatial density and event rate of any particular class of sources. This is not surprising, since our information at present comes from electromagnetic observations, and as our earlier remarks about the differences between the mechanisms of emission of gravitational and electromagnetic radiation make clear, electromagnetic observations may not strongly constrain the source population.
3.2 Gravitational wave bursts from gravitational collapse
Neutron stars and black holes are formed from the gravitational collapse of a highly evolved star or the core collapse of an accreting white dwarf. In either case, if the collapse is nonspherical, perhaps induced by strong rotation, then gravitational waves could carry away some of the binding energy and angular momentum depending on the geometry of the collapse. Collapse events are thought to produce supernovae of various types, and increasingly there is evidence that they also produce most of the observed gamma-ray bursts  in hypernovae and collapsars [397, 249]. Supernovae of Type II are believed to occur at a rate of between 0.1 and 0.01 per year in a milky-way equivalent galaxy (MWEG); thus, within the Virgo supercluster, we might expect an event rate of about 30 per year. Hypernova events are considerably rarer and might only contribute observable gravitational-wave events in current and near-future detectors if they involve so much rotation that strong non-axisymmetric instabilities are triggered.
Simulating gravitational collapse is a very active area of numerical astrophysics, and most simulations also predict the energy and spectral characteristics of the emitted gravitational waves . However, it is still beyond the capabilities of computers to simulate a gravitational collapse event with all the physics that might be necessary to give reliable predictions: three-dimensional hydrodynamics, neutrino transport, realistic nuclear physics, magnetic fields, rotation. In fact, it is still by no means clear why Type II supernovae explode at all: simulations typically have great difficulty reversing the inflow and producing an explosion with the observed light-curves and energetics. It may be that the answer lies in some of the physics that has to be oversimplified in order to be used in current simulations, or in some neutrino physics that we do not yet know, or in some unexplored hydrodynamic mechanism . In a typical supernova, simulations suggest that gravitational waves might extract between about and of the total available mass-energy [264, 147, 148], and the waves could come off in a burst whose frequency might lie in the range of 200 – 1000 Hz.
We can use Equation (18) to make a rough estimate of the amplitude, if the emitted energy and timescale are known. Using representative values for a supernova in our galaxy, lying at 10 kpc, emitting the energy equivalent of at a frequency of 1 kHz, and lasting for 1 ms, the received amplitude would be
The upper bound in Equation (11) would give the same amplitude for a source 60 times further away, which reflects the fact that simulations find it difficult to put significant energy into gravitational waves. This amplitude is large enough for current ground-based detectors to observe with a reasonably high confidence, but of course the event rate within 10 kpc is expected to be far too small to make an early detection likely.
3.3 Gravitational wave pulsars
Some likely gravitational wave sources behave like the centrifuge example we used in the first paragraph of this section, only on a grander scale. Suppose a neutron star of radius and mass spins with a frequency and has an irregularity, a deformation of its otherwise axially symmetric shape. We idealize this as a “bump” of mass on its surface, although of course it will really be a distribution of mass leading to an asymmetrical quadrupole tensor. The moment of inertia of the bump will be , and it is conventional to parameterize the bump in terms of the fractional asymmetry it creates in the moment of inertia tensor itself. If we idealize the star as having uniform density, then the spherical moment of inertia is , and so the bump has fractional asymmetry
The bump will emit gravitational radiation at frequency because the star spins about its net center of mass, so it effectively has mass excesses on both sides of the star. The nonspherical velocity will be just . The radiation amplitude will be, from Equation (9),
and the luminosity, from Equation (15) (assuming that roughly four comparable components of contribute to the sum),
The radiated energy would presumably come from the rotational energy of the star . This would lead to a spindown of the star on a timescale
It is believed that neutron star crusts are not strong enough to support fractional asymmetries larger than about , and realistic asymmetries may be much smaller.
From these considerations one can estimate the likelihood that the observed spindown timescales of pulsars are due to gravitational radiation. In most cases, it seems that gravitational wave losses could account for a substantial amount of the spindown: the required asymmetries are much smaller than , often smaller than . But an interesting exception is the Crab pulsar, PSR J0534+2200, whose young age and consequently short spindown time (measured to be s, about 2500 yr) would require an exceptionally large asymmetry. If we take the neutron star’s radius to be 10 km, so that and the speed of any irregularity is , then Equation (24) would require an asymmetry of . Of course, we have made a lot of approximations to get here, only keeping our estimates of amplitudes and energies correct to within factors of two, but a more careful calculation reduces this only by a factor of two to . What makes this interesting is the fact that an asymmetry this large would produce radiation detectable by first-generation interferometers. Conversely, an upper limit from first-generation interferometers would provide direct observational limits on the asymmetry and on the fraction of energy lost by the Crab pulsar to gravitational waves.
From Equation (23) the Crab pulsar would, if its spindown is dominated by gravitational wave losses, produce an amplitude at the Earth of , if its distance is 2 kpc. Is this detectable when present instruments are only capable of seeing millisecond bursts of radiation at levels of ? The answer is yes, if the observation time is long enough. Indeed, the latest LIGO observations have not detected any gravitational waves from the Crab pulsar, which has been used to set an upper limit on the asymmetry in its mass distribution . The limit depends on the model assumed for the pulsar. If one assumes that gravitational waves are produced at exactly twice the pulsar spin frequency and uses the inferred values of the pulsar orientation and polarization angle, then for a canonical value of the moment-of-inertia , one gets an upper limit on the ellipticity of , assuming the pulsar is at 2 kpc. This is a factor of 4.2 below the spindown limit . If, however, one assumes that gravitational waves are emitted at a frequency close, but not exactly equal, to twice the spin frequency and one uses a uniform prior for the orientation and polarization angle, then one gets , which is 0.8 of the limit derived from the spin-down rate.
Indeed, even signals weaker than the amplitude determined by the Crab spindown rate will be observable by present detectors, and these may be coming from a larger variety of neutron stars, in particular low-mass X-ray binary systems (LMXBs). The neutron stars in them are accreting mass and angular momentum, so they should be spinning up. Observations suggest that most neutron stars are spinning at speeds between about 300 and 600 Hz, far below their maximum, which is greater than 1000 Hz. The absence of faster stars suggests that something stops them from spinning up beyond this range. Bildsten suggested  that the limiting mechanism may be the re-radiation of the accreted angular momentum in gravitational waves, possibly due to a quadrupole moment created by asymmetrical heating induced by the accreted matter. Another possible mechanism  is that a “bump” of the kind we have treated is formed by accreting matter channeled onto the surface by the star’s magnetic field. It is also possible that accretion drives an instability in the star that leads to steady emission [308, 270]. In either case, the stars could turn out to be long-lived sources of gravitational waves. This idea, which is a variant of one proposed long ago by Wagoner , is still speculative, but the numbers make a plausible case. We discuss it in more detail in Section 7.3.5.
3.4 Radiation from a binary star system
3.4.1 Radiation from a binary system and its backreaction
A binary star system can also be treated as a “centrifuge”. Two stars of the same mass in a circular orbit of radius have all their mass in nonspherical motion, so that
where is the orbital angular velocity. The gravitational wave amplitude can then be written
Since the internal radius of the orbit is not an observable, it is sometimes convenient to replace by the orbital angular frequency using the above orbit equation, giving
The gravitational wave luminosity of such a system is, by a calculation analogous to that for bumps on neutron stars (assuming that four components of to be significant),
in units given by the fundamental luminosity in Equation (16). This shows that self-gravitating systems always emit at a fraction of , since is always smaller than 1, but it can approach for highly-relativistic systems where .
The radiation of energy by the orbital motion causes the orbit to shrink. The shrinking will make any observed gravitational waves increase in frequency with time. This is called a chirp. The timescale222In Sections 5.1 we will use parameters called chirp times, instead of the masses, to characterize a binary. The timescale defined here is closely related to the chirp times. for this in a binary system with equal masses is
As the binary evolves, the frequency and amplitude of the wave grow and this drives the binary to evolve even more rapidly. The signal’s frequency, however, will not increase indefinitely; the slow inspiral phase ends either when the stars begin to interact and merge or (if they are very compact) when the distance between the stars is roughly at the last stable orbit (LSO) , which corresponds to a gravitational wave frequency of
where we have normalized this to a binary with . This is the last stable orbit (LSO) frequency.
A compact-object binary that coalesces after passing through the last stable orbit is a powerful source of gravitational waves, with a luminosity that approaches the limiting luminosity . This is called a coalescing binary in gravitational wave searches. Since a typical search might last on the order of one year, a coalescing binary can be defined as a system that has a chirp time smaller than one year. In Figure 2 the coalescence line is shown as a straight line with slope 3/4 (set to a constant in Equation (28)). Binary systems below this line have a chirp time smaller than one year. It is evident from the figure that all binary systems observable by ground-based detectors will coalesce in less than a year.
As mentioned for gravitational wave pulsars, the raw amplitude of the radiation from a long-lived system is not by itself a good guide to its detectability, if the waveform can be predicted. Data analysis techniques like matched filtering are able to eliminate most of the detector noise and allow the recognition of weaker signals. The improvement in amplitude sensitivity is roughly proportional to the square root of the number of cycles of the waveform that one observes. For neutron stars that are observed from a frequency of 10 Hz until they coalesce, there could be on the order of cycles, meaning that the sensitivity of a second-generation interferometric detector would effectively be 100 times better than its broadband (prefiltering) sensitivity. Such detectors could see typical coalescences at 200 Mpc. The event rate for second-generation detectors is estimated at around 40 events per year, with rather large error bars [101, 211, 242].
3.4.2 Chirping binaries as standard sirens
When we consider real binaries we must do the calculation for systems that have unequal masses. Still assuming for the moment that the binary orbit is circular, the quadrupole amplitude turns out to be
where we define the chirp mass as
with the reduced mass, the total mass and the symmetric mass ratio. We have left out of Equation (30) a factor of order one that depends on the angle from which the binary system is viewed. The two polarization amplitudes can be found in Equation (132).
Remarkably, the other observable, namely the shrinking of the orbit as measured by the rate of change of the orbital frequency also depends on the masses just through :
In this case, the chirp time is
This is just the equal-mass chirp time of Equation (28) scaled inversely with the symmetric mass ratio . From this equation it is clear that systems with large mass ratios between the components can spend a long time in highly relativistic orbits, whereas equal-mass binaries can be expected to merge after only a few orbits in the highly relativistic regime.
If one observes and , one can infer from Equation (32). Then, from the observed amplitude in Equation (30), the only remaining unknown is the distance to the source. Gravitational wave observations of orbits that shrink because of gravitational energy losses can therefore directly determine the distance to the source . By analogy with the “standard candles” of electromagnetic astronomy, these systems are now being called “standard sirens”. Although our calculation here assumed an equal-mass circular system, the conclusion is robust: any binary, even with ellipticity and extreme mass ratio, encodes its distance in its gravitational wave signal.
This is another way in which gravitational wave observations are complementary to electromagnetic ones, providing information that is hard to obtain electromagnetically. One consequence is the possibility that observations of coalescing compact object binaries could allow one to measure the Hubble constant  or other cosmological parameters. This will be particularly interesting for the LISA project, whose observations of black hole binaries could contribute an independent measurement of the acceleration of the universe [195, 131, 48].
Because chirping systems are so interesting we have also drawn, in Figure 2, a line where the chirp time can be measured in one year. This means that the change in frequency due to the chirp must be larger than the frequency resolution . A little algebra shows that the condition for the chirp to be resolved in an observation time in a binary with period is
Since , this condition leads to a line of slope 7/11 in the logarithmic plot in Figure 2. The line drawn there corresponds to a resolution time of one year. All binaries below this line will chirp in a short enough time for their distances to be measured.
3.4.3 Binary pulsar tests of gravitational radiation theory
The most famous example of the effects of gravitational radiation on an orbiting system is the Hulse–Taylor Binary Pulsar, PSR B1913+16. In this system, two neutron stars orbit in a close eccentric orbit. The pulsar provides a regular clock that allows one to deduce, from post-Newtonian effects, all the relevant orbital parameters and the masses of the stars. The key to the importance of this binary system is that all of the important parameters of the system can be measured before one takes account of the orbital shrinking due to gravitational radiation reaction. This is because a number of post-Newtonian effects on the arrival time of pulses at the Earth, such as the precession of the position of the periastron and the time-dependent gravitational redshift of the pulsar period as it approaches and recedes from its companion, can be measured accurately, and they fully determine the masses, the semi-major axis and the eccentricity of their orbit [394, 344].
Equation (28) for the chirp time predicts that this system would change its orbital period on the timescale (taking and )
From this one can infer that . But this has to be corrected for our oversimplification of the orbit as circular: an eccentric orbit evolves much faster because, during the phase of closest approach, the velocities are much higher, and the emitted luminosity is a very strong function of the velocity. Using equations first computed by Peters and Mathews , for the actual eccentricity of 0.62, one finds (see Equation (109) below) . Observations [394, 388] currently give . There is a significant discrepancy between these, but it can be removed by realizing that the binary system is accelerating toward the center of our galaxy, which produces a small period change. Taking this into account gives a corrected prediction of , and this agrees with the observation within the uncertainties [394, 355]. This is the most sensitive test that we have of the correctness of Einstein’s equations with respect to gravitational radiation, and it leaves little room for doubt in the validity of the quadrupole formula for other systems that may generate detectable radiation.
A number of other binary systems are now known in which such tests are possible . The most important of the other systems is the “double pulsar” in which both neutron stars are seen as pulsars . This system will soon overtake the Hulse–Taylor binary as the most accurate test of gravitational radiation.
3.4.4 White-dwarf binaries
Binary systems at lower frequencies are much more abundant than coalescing binaries, and they have much longer lifetimes. LISA will look for close white-dwarf binaries in our galaxy, and will probably see thousands of them. White dwarfs are not as compact as black holes or neutron stars. Although their masses can be similar to that of a neutron star their sizes are much larger, typically 3,000 km in radius. As a result, white-dwarf binaries never reach the last stable orbit, which would occur at roughly 1.5 kHz for these masses. We will discuss the implications of multi-messenger astronomy for white-dwarf binaries in Section 7.4.
The maximum amplitude of the radiation from a white-dwarf binary will be several orders of magnitude smaller than that of a neutron star or black hole binary at the same distance but close to coalescence. However, a binary system with a short period is long lived, so the effective amplitude (after matched filtering) improves as the square root of the observing time. Besides that, these sources are nearer: there are many thousands of such systems in our galaxy radiating in the LISA frequency window above about 1 mHz, and LISA should be able to see all of them. Below 1 mHz there are even more sources, so many that LISA will not resolve them individually, but will see them blended together in a stochastic background of radiation, as shown in Figure 5.
3.4.5 Supermassive black hole binaries
Observations indicate that the center of every galaxy probably hosts a black hole whose mass is in the range of , with the black holes mass correlating well with the mass of the galactic bulge. A black hole whose mass is in the above range is called a supermassive black hole (SMBH). There is now abundant observational evidence that galaxies often collide and merge, and there are good reasons to believe that when that happens, friction between the SMBHs and the stars and gas of the irregular merged galaxy will lead the SMBHs to spiral into a common nucleus and (on a timescale of some yr) even get close enough to be driven into complete orbital decay by gravitational radiation reaction. In many systems this should lead to a black hole merger within a Hubble time . For a binary with two nonspinning black holes, the frequency of emitted gravitational waves at the last stable orbit is, from Equation (29), ; during and after the merger the frequency rises from 4 mHz to the quasi-normal-mode frequency of 24 mHz (if the spin of the final black hole is negligible). (Naturally, all these frequencies simply scale inversely with the mass for other mass ranges.) This is in the frequency range of LISA, and observing these mergers is one of the central purposes of the mission.
SMBH mergers are so spectacularly strong that they will be visible in LISA’s data stream even before applying any matched filter, although good models of the inspiral and particularly the merger radiation will be needed to extract source parameters. Because the masses of such black holes are so large, LISA can see essentially any merger that happens in its frequency band anywhere in the universe, even out to extremely high redshifts. It can thereby address astrophysical questions about the origin, growth and population of SMBHs. The recent discovery of an SMBH binary  and the association of X-shaped radio lobes with the merger of SMBH binaries  has further raised the optimism concerning SMBH merger rates, as has the suggestion that an SMBH has been observed to have been expelled from the center of its galaxy, an event that could only have happened as a result of a merger between two SMBHs . The rate at which galaxies merge is about 1 yr out to a red-shift of , and LISA’s detection rate for SMBH mergers might be roughly the same.
Modelling of the merger of two black holes requires numerical relativity, and the accuracy and reliability of numerical simulations is now becoming good enough that they will soon become an integral part of gravitational wave searches.
3.4.6 Extreme and intermediate mass-ratio inspiral sources
The SMBH environment of our own galaxy is known to contain a large number of compact objects and white dwarfs. Near the central SMBH there is a disproportionately large number of stellar-mass black holes, which have sunk there through random gravitational encounters with the normal stellar population (dynamical friction). Three body interaction will occasionally drive one of these compact objects into a capture orbit of the central SMBH. The compact object will sometimes be captured [305, 338, 337] into a highly eccentric trajectory () with the periastron close to the last stable orbit of the SMBH. Since the mass of the captured object will be about , while the SMBH will have a far greater mass, we essentially have a “test mass” falling in the geometry of a Kerr black hole. By Equation (33) we would expect that the small body would spend many orbits in the relativistic regime near the horizon of the large black hole: a black hole falling into a black hole would require on the order of orbits. The emitted gravitational radiation [317, 179, 178, 67, 171, 57] would consist of a very long wave train that carries information about the nearly geodesic trajectory of the test body, thereby providing a very clean probe to survey the spacetime geometry of the central object (which could be a Kerr black hole or some other compact object) and test whether or not this geometry is as predicted by general relativity [318, 198, 177, 176, 68].
This kind of event happens occasionally to every SMBH that lives in the center of a galaxy. Indeed, since the SNR from matched filtering builds up in proportion to the square root of the observation time [cf. Equation (33)] and the inherent amplitude of the radiation is linear in [cf. Equation (30)], the SNR varies with the symmetric mass ratio as and typical SNR will be about ten to a thousand times smaller than an SMBH binary at the same distance. LISA will, therefore, be able to see such sources only to much smaller distances, say between 1 to 10 Gpc depending on the mass ratio. For events at such distances LISA’s SNR after matched filtering could be in the range 10 – 100, but matched filtering will be very difficult because of the complexity of the orbit, especially of its evolution due to radiation effects. However, this volume of space contains a large number of galaxies, and the event rate is expected to be several tens to hundreds per year . There will be a background from more distant sources that might in the end set the limit on how much sensitivity LISA has to these events.
Due to relativistic frame dragging, for each passage of the apastron the test body could execute several nearly circular orbits at its periastron. Therefore, long periods of low-frequency, small-amplitude radiation will be followed by several cycles of high-frequency, large-amplitude radiation [317, 179, 178, 67, 171, 57]. The apastron slowly shrinks, while the periastron remains more or less at the same location, until the final plunge of the compact object before merger. Moreover, if the central black hole has a large spin then spin-orbit coupling leads to precession of the orbital plane thereby changing the polarization of the wave as seen by LISA.
Thus, there is a lot of structure in the waveforms owing to a number of different physical effects: contribution from higher-order multipoles due to an eccentric orbit, precession of the orbital plane, precession of the periastron, etc., and gravitational radiation backreaction plays a pivotal role in the dynamics of these systems. If one looks at the time-frequency map of such a signal one notices that the signal power is greatly smeared across the map , as compared to that of a sharp chirp from a nonspinning black-hole binary. For this reason, this spin modulated chirp is sometimes referred to as a smirch . More commonly, such sources are called extreme mass ratio inspirals (EMRIs) and represent systems whose mass ratio is in the range of . Inspirals of systems with their mass ratio in the range are termed intermediate mass ratio inspirals or IMRIs. These latter systems correspond to the inspiral of intermediate mass black holes of mass and might constitute a prominent source in LISA provided the central SMBH grew in mass as a result of a number of mergers of small black holes [30, 31, 32].
While black hole perturbation theory with a careful treatment of radiation reaction is necessary for the description of EMRIs, IMRIs may be amenable to a description using a hybrid scheme of post-Newtonian approximations and perturbation theory. This is an area that requires more study.
3.5 Quasi-normal modes of a black hole
In 1970, Vishveshwara  discussed a gedanken experiment, similar in philosophy to Rutherford’s (real) experiment with the atom. In Vishveshwara’s experiment, he scattered gravitational radiation off a black hole to explore its properties. With the aid of such a gedanken experiment, he demonstrated for the first time that gravitational waves scattered off a black hole will have a characteristic waveform, when the incident wave has frequencies beyond a certain value, depending on the size of the black hole. It was soon realized that perturbed black holes have quasi-normal modes (QNMs) of vibration and in the process emit gravitational radiation, whose amplitude, frequency and damping time are characteristic of its mass and angular momentum [296, 220]. We will discuss in Section 6.4 how observations of QNMs could be used in testing strong field predictions of general relativity.
We can easily estimate the amplitude of gravitational waves emitted when a black hole forms at a distance from Earth as a result of the coalescence of compact objects in a binary. The effective amplitude is given by Equation (20), which involves the energy put into gravitational waves and the frequency at which the waves come off. By dimensional arguments is proportional to the total mass of the resulting black hole. The efficiency at which the energy is converted into radiation depends on the symmetric mass ratio of the merging objects. One does not know the fraction of the total mass emitted nor the exact dependence on . Flanagan and Hughes  argue that . The frequency is inversely proportional to ; indeed, for Schwarzschild black holes . Thus, the formula for the effective amplitude takes the form
where is a number that depends on the (dimensionless) angular momentum of the black hole and has a value between 0.7 (for , Schwarzschild black hole) and 0.4 (for , maximally spinning Kerr black hole). For stellar mass black holes at a distance of 200 Mpc the amplitude is:
For SMBHs, even at cosmological distances, the amplitude of quasinormal mode signals is pretty large:
In the first case we have a pair of black holes inspiraling and merging to form a single black hole. In this case the waves come off at a frequency of around 500 Hz [cf. Equation (13)]. The initial ground-based network of detectors might be able to pick these waves up by matched filtering, especially when an inspiral event precedes the ringdown signal. A black hole plunging into a black hole at a distance of 6.5 Gpc () gives out radiation at a frequency of about 15 mHz. Note that in the latter case the radiation is redshifted from 30 mHz to 15 mHz. Such an event produces an amplitude just large enough to be detected by LISA. At the same distance, a pair of SMBHs spiral in and merge to produce a fantastic amplitude of , way above the LISA background noise. In this case, the signals would be given off at about 7.5 mHz and will be loud and clear to LISA. It will not only be possible to detect these events, but also to accurately measure the masses and spins of the objects before and after merger and thereby test the black hole no-hair theorem and confirm whether the result of the merger is indeed a black hole or some other exotic object (e.g., a boson star or a naked singularity).
3.6 Stochastic background
In addition to radiation from discrete sources, the universe should have a random gravitational wave field that results from a superposition of countless discrete systems and also from fundamental processes, such as the Big Bang. Observing any of these backgrounds would bring useful information, but the ultimate goal of detector development is the observation of the background radiation from the Big Bang. It is expected to be very weak, but it will come to us unhindered from as early as s, and it could illuminate the nature of the laws of physics at energies far higher than we can hope to reach in the laboratory.
It is usual to characterize the intensity of a random field of gravitational waves by its energy density as a function of frequency. Since the energy density of a plane wave is the same as its flux (when ), we have from Equation (17)
But the wave field in this case is a random variable, so we must replace by a statistical mean square amplitude per unit frequency (Fourier transform power per unit frequency) called , so that the energy density per unit frequency is proportional to . It is then conventional to talk about the energy density per unit logarithm of the frequency, which means multiplying by . The result, after being careful about averaging over all directions of the waves and all independent polarization components, is [27, 359]
Finally, what is of most interest is the energy density as a fraction of the closure or critical cosmological density, given by the Hubble constant as . The resulting ratio is called :
The only tight constraint on from non–gravitational-wave astronomy is that it must be smaller than , in order not to disturb the agreement between the standard Big Bang model of nucleosynthesis (of helium and other light elements) and observation. If the universe contains this much gravitational radiation today, then at the time of nucleosynthesis the (blue-shifted) energy density of this radiation would have been comparable to that of the photons and the three neutrino species. Although the radiation would not have participated in the nuclear reactions, its extra energy density would have required that the expansion rate of the universe at that time be significantly faster, in order to evolve into the universe we see today. In turn, this faster expansion would have provided less time for the nuclear reactions to “freeze out”, altering the abundances from the values that are observed today [281, 346]. First-generation interferometers should be able to set direct limits on the cosmological background at around this level. Radiation in the lower-frequency LISA band, from galactic and extra-galactic binaries, is expected to be much smaller than this bound.
Random radiation is indistinguishable from instrumental noise in a single detector, at least for short observing times. If the random field is produced by an anisotropically-distributed set of astrophysical sources (the binaries in our galaxy, for example) then over a year, as the detector changes its orientation, the noise from this background should rise and fall in a systematic way, allowing it to be identified. But this is a rather crude way of detecting the radiation, and a better way is to perform a cross-correlation between two detectors, if available.
In cross-correlation, which amounts to multiplying the outputs and integrating, the random signal in one detector essentially acts as a template for the signal in the other detector. If they match, then there will be a stronger-than-expected correlation. Notice that they can only match well if the wavelength of the gravitational waves is longer than the separation between the detectors: otherwise time delays for waves reaching one detector before the other degrade the match. The outcome is not like standard matched filtering, however, since the “filter” of the first detector has as much noise superimposed on its template as the other detector. As a result, the amplitude SNR of the correlated field grows only with observing time as , rather than the square root growth that characterizes matched filtering .
4 Gravitational Wave Detectors and Their Sensitivity
Detectors of gravitational waves generally divide into two classes: beam detectors and resonant mass detectors. In beam detectors, gravitational waves interact with a beam of electromagnetic radiation, which is monitored in some way to register the passage of the wave. In resonant mass detectors, the gravitational wave transfers energy to a massive body, from which the resultant oscillations are observed.
Both classes include a variety of systems. The principal beam detectors are the large ground-based laser interferometers currently operating in several locations around the globe, such as the LIGO system in the USA. The ESA–NASA LISA mission aims to put a laser interferometer into space to detect milliHertz gravitational waves. But beam detectors do not need to involve interferometry: the radio beams transponded to interplanetary spacecraft can carry the signature of a passing gravitational wave, and this method has been used to search for low-frequency gravitational waves. And radio astronomers have for many years monitored the radio beams of distant pulsars for evidence of gravitational waves; new radio instrumentation is turning this into a powerful and promising method of looking for stochastic backgrounds and individual sources. And at ultra-low frequencies, gravitational waves in the early universe may have left their imprint on the polarization of the cosmic microwave background.
Resonant mass detectors were the first kind of detector built in the laboratory to detect gravitational waves: Joseph Weber  built two cylindrical aluminum bar detectors and attempted to find correlated disturbances that might have been caused by a passing impulsive gravitational wave. His claimed detections led to the construction of many other bar detectors of comparable or better sensitivity, which never verified his claims. Some of those detectors were not developed further, but others had their sensitivities improved by making them cryogenic, and today there are two ultra-cryogenic detectors in operation (see Section 4.1).
In the following, we will examine the principal detection methods that hold promise today and in the near future.
4.1 Principles of the operation of resonant mass detectors
A typical “bar” detector consists of a cylinder of aluminum with a length , a very narrow resonant frequency between and 1.5 kHz, and a mass . A short gravitational wave burst with will make the bar vibrate with an amplitude
To measure this, one must fight against three main sources of noise.
Thermal noise. The original Weber bar operated at room temperature, but the most advanced detectors today, Nautilus  and Auriga , are ultra-cryogenic, operating at . At this temperature the root mean square (rms) amplitude of vibration is
This is far larger than the gravitational wave amplitude expected from astrophysical sources. But if the material has a high (say, ) in its fundamental mode, then that changes its thermal amplitude of vibration in a random walk with very small steps, taking a time to change by the full amount. However, a gravitational wave burst will cause a change in 1 ms. In 1 ms, thermal noise will have random-walked to an expected amplitude change times smaller, or (for these numbers)
So ultra-cryogenic bars can approach the goal of detection near despite thermal noise.
Sensor noise. A transducer converts the bar’s mechanical energy into electrical energy, and an amplifier increases the electrical signal to record it. If sensing of the vibration could be done perfectly, then the detector would be broadband: both thermal impulses and gravitational wave forces are mechanical forces, and the ratio of their induced vibrations would be the same at all frequencies for a given applied impulsive force.
But sensing is not perfect: amplifiers introduce noise, and this makes small amplitudes harder to measure. The amplitudes of vibration are largest in the resonance band near , so amplifier noise limits the detector sensitivity to gravitational wave frequencies near . But if the noise is small, then the measurement bandwidth about can be much larger than the resonant bandwidth . Typical measurement bandwidths are 10 Hz, about times larger than the resonant bandwidths, and 100 Hz is not out of the question .
Quantum noise. The zero-point vibrations of a bar with a frequency of 1 kHz are
This is comparable to the thermal limit over 1 ms. So, as detectors improve their thermal limits, they run into the quantum limit, which must be breached before a signal at can be seen with such a detector.
It is not impossible to do better than the quantum limit. The uncertainty principle only sets the limit above if a measurement tries to determine the excitation energy of the bar, or equivalently the phonon number. But one is not interested in the phonon number, except in so far as it allows one to determine the original gravitational wave amplitude. It is possible to define other observables that also respond to the gravitational wave and can be measured more accurately by squeezing their uncertainty at the expense of greater errors in their conjugate observable . It is not yet clear whether squeezing will be viable for bar detectors, although squeezing is now an established technique in quantum optics and will soon be implemented in interferometric detectors (see below).
Reliable gravitational wave detection, whether with bars or with other detectors, requires coincidence observations, in which two or more detectors confirm each other’s findings. The principal bar detector projects around the world formed the International Gravitational Event Collaboration (IGEC)  to arrange for long-duration coordinated observations and joint data analysis. A report in 2003 of an analysis of a long period of coincident observing over three years found no evidence of significant events . The ALLEGRO bar  at Louisiana State University made joint data-taking runs with the nearby LIGO interferometer, setting an upper limit on the stochastic gravitational-wave background at around 900 Hz of . More recently, because funding for many of the bar detector projects has become more restricted, only two groups continue to operate bars at present (end of 2008): the Rome  and Auriga  groups. The latest observational results from IGEC may be found in .
It is clear from the above discussion that bars have great difficulty achieving the sensitivity goal of . This limitation was apparent even in the 1970s, and that motivated a number of groups to explore the intrinsically wide-band technique of laser interferometry, leading to the projects described in Section 4.3.1 below. However, the excellent sensitivity of resonant detectors within their narrow bandwidths makes them suitable for specialized, high-frequency searches, including cross-correlation searches for stochastic backgrounds . Therefore, novel and imaginative designs for resonant-mass detectors continue to be proposed. For example, it is possible to construct large spheres of a similar size (1 to 3 m diameter) to existing cylinders. This increases the mass of the detector and also improves its direction-sensing. One can in principle push to below with spheres . A spherical prototype called MiniGRAIL has been operated in the Netherlands. A similar prototype called the Schenberg detector is being built in Brazil . Nested cylinders or spheres, or masses designed to sense multiple modes of vibration may also provide a clever way to improve on bar sensitivities .
While these ideas have interesting potential, funding for them is at present (2008) very restricted, and the two remaining bar detectors are likely to be shut down in the near future, when the interferometers begin operating at sensitivities clearly better than .
4.2 Principles of the operation of beam detectors
Interferometers use laser light to measure changes in the difference between the lengths of two perpendicular (or nearly perpendicular) arms. Typically, the arm lengths respond differently to a given gravitational wave, so an interferometer is a natural instrument to measure gravitational waves. But other detectors also use electromagnetic radiation, for example, ranging to spacecraft in the solar system and even pulsar timing.
The basic equation we need is for the effect of a plane linear gravitational wave on a beam of light. Suppose the angle between the direction of the beam and the direction of the plane wave is . We imagine a very simple experiment in which the light beam originates at a clock, whose proper time is called , and is received by a clock, whose proper time is . The beam and gravitational-wave travel directions determine a plane, and we denote the polarization component of the gravitational wave that acts in this plane by , as measured at the location of the originating clock. The proper distance between the clocks, in the absence of the wave, is . If the originating clock puts timestamps onto the light beam, then the receiving clock can measure the rate of arrival of the timestamps. If there is no gravitational wave, and if the clocks are ideal, then the rate will be constant, which can be normalized to unity. The effect of the gravitational wave is to change the arrival rate as a function of the emission rate by
This is very simple: the beam of light leaves the emitter at the time when the gravitational wave of phase passes the emitter, and it reaches the receiver at the time when the gravitational wave of phase is passing the receiver. So in the plane wave case, only the amplitudes of the wave at the emitting and receiving events affect the time delay.
In order to use such an arrangement to detect gravitational waves, one needs two very stable clocks. The best clocks today are stable to a few parts in , which implies that the minimum amplitude of gravitational waves that could be detected by such a two-clock experiment is . However, this equation is also fundamental to the detection of gravitational waves by pulsar timing, in which the originating ‘clock’ is a pulsar. By correlating many pulsar signals, one can beat down the single-pulsar noise. This is described below in Section 4.4.2.
An arrangement that uses only one clock is one that sends a beam out to a receiver, which then reflects or retransmits (transponds) the beam back to the sender. The sender has the clock, which measures variations in the round-trip time. This method has been used with interplanetary spacecraft, which has the advantage that the only clock is on the ground, which can be made more stable than one carried in a spacecraft (see Section 4.4.1). For the same arrangement as above, the return time varies at the rate
This is known as the three-term relation, the third term being the wave strength at the time the beam returns back to the sender.
But the sensitivity of such a one-path system as a gravitational wave detector is still limited by the stability of the clock. For that reason, interferometers have become the most sensitive beam detectors: effectively one arm of the interferometer becomes the ‘clock’, or at least the time standard, that variations in the other arm are compared to. Of course, if both arms are affected by a gravitational wave in the same way, then the interferometer will not see the wave. But this happens only in very special geometries. For most wave arrival directions and polarizations, the arms are affected differently, and a simple interferometer measures the difference between the round-trip travel time variations in the two arms. For the triangular space array LISA, the measured signal is somewhat more complex (see Section 4.4.3 below), but it still preserves the principle that the time reference for one arm is a combination of the others.
4.2.1 The response of a ground-based interferometer
Ground-based interferometers are the most sensitive detectors operating today, and are likely to make the first direct detections . The largest detectors operating today are the LIGO detectors , two of which have arm lengths of 4 km. This is much smaller than the wavelength of the gravitational wave, so the interaction of one arm with a gravitational wave can be well approximated by the small- approximation to Equation (43), namely
(See  for first corrections to the short-arm approximation.) To analyze the full detector, where the second arm will normally point out of the plane we have been working in up till now, it is helpful to go over to a tensorial expression, independent of special coordinate orientations. The gravitational wave will act in the plane transverse to the propagation direction; let us call that direction and let us set up radiation basis vectors and in the transverse plane, such that lies in the plane formed by the wave propagation direction and the arm of our gravitational wave sensor, which lies along the -axis of the detector plane, whose unit vector is . (For a picture of this geometry, see the left-hand panel of Figure 3, where for the moment we are ignoring the -arm of the detector shown there.)
With these definitions, the wave amplitude is the one that has and as the axes of its ellipse. The full wave amplitude is described, as in Equation (6), by the wave tensor
where and are the polarization tensors associated with these basis vectors (compare Equation (4)):
The unique way of expressing Equation (44) in terms of is
This does not depend on any special orientation of the arm relative to the wave direction, and does not depend on the basis we chose in the transverse plane, so we can use it as well for the second arm of the interferometer, no matter what its orientation. Let us assume it lies along the unit vector by . (We do not, in fact, have to assume that the two arms are perpendicular to each other, but it simplifies the diagram a little.) The return-time derivative along the second arm is then given by
. The interferometer responds to the difference between these times,
. By analogy with the wave tensor, we define the detector tensor by 
(If the arms are not perpendicular this expression would still give the correct tensor if the unit vectors lie along the actual arm directions.) Then we can express the differential return time rate in the simple invariant form
where the notation denotes the Euclidean scalar product of the tensors and . Equation (49) can be integrated over time to give the instantaneous path-length (or time-delay, or phase) difference between the arms, as measured by the central observer’s proper time clock:
This is a robust and compact expression for the response of any interferometer to any wave in the long-wavelength (short-arm) limit. Its dependence on the wave direction is called its antenna pattern.
It is conventional to re-express this measurable in terms of the stretching of the arms of the interferometer. Within our approximation that the arms are shorter than a wavelength, this makes sense: it is possible to define a local inertial coordinate system that covers the entire interferometer, and within this coordinate patch (where light travels at speed 1) time differences measure proper length differences. The differential return time is twice the differential length change of the arms:
For a bar detector of length lying along the director , the detector tensor is
although one must be careful that the change in proper length of a bar is not simply given by Equation (51), because of the restoring forces in the bar.
When dealing with observations by more than one detector, it is not convenient to tie the alignment of the basis vectors in the sky plane with those in the detector frame, as we have done in the left-hand panel of Figure 3, since the detectors will have different orientations. Instead it will usually be more convenient to choose polarization tensors in the sky plane according to some universal reference, e.g., using a convenient astronomical reference frame. The right-hand panel of Figure 3 shows the general situation, where the basis vectors and are rotated by an angle from the basis used in the left-hand panel. The polarization tensors on this new basis,
are found by the following transformation from the previous ones:
Then one can write Equation (51) as
where and are the antenna pattern functions for the two polarizations defined on the sky-plane basis by
Using the geometry in the right-hand panel of Figure 3, one can show that
These are the antenna-pattern response functions of the interferometer to the two polarizations of the wave as defined in the sky plane . If one wants the antenna pattern referred to the detector’s own axes, then one just sets . If the arms of the interferometer are not perpendicular to each other, then one defines the detector-plane coordinates and in such a way that the bisector of the angle between the arms lies along the bisector of the angle between the coordinate axes . Note that the maximum value of either or is 1.
The corresponding antenna-pattern functions of a bar detector whose longitudinal axis is aligned along the direction, are
Any one detector cannot directly measure both independent polarizations of a gravitational wave at the same time, but responds rather to a linear combination of the two that depends on the geometry of the detector and source direction. If the wave lasts only a short time, then the responses of three widely-separated detectors, together with two independent differences in arrival times among them, are, in principle, sufficient to fully reconstruct the source location and gravitational wave polarization. A long-lived wave will change its location in the antenna pattern as the detector moves, and it will also be frequency modulated by the motion of the detector; these effects are in principle sufficient to determine the location of the source and the polarization of the wave.
Since the polarization angle of an incoming gravitational wave would generally be expected to be unrelated to its direction of arrival, depending rather on the internal orientations in the source, it is useful to characterize the directional sensitivity of a detector by averaging over the polarization angle . If the wave has a given amplitude and is linearly polarized, then, if we are interested in a single detector’s response, it is always possible to align the polarization angle in the sky plane with that of the wave, so that the wave has pure -polarization. Then the rms response function of the detector is
The function is often simply called the antenna pattern. For a resonant bar, the antenna pattern is
and for an interferometer, it is given by
The antenna pattern of an interferometric detector is plotted in the left panel of Figure 4, which clearly shows the quadrupolar nature of the detector. Note that the response of an interferometer is the best for waves coming from a direction orthogonal to the plane containing the detector, and it is zero for waves in the plane of an interferometer’s arms (i.e., ) that arrive from a direction bisecting the two arms (i.e., ) or from directions differing from this by a multiple of . What is the response of an antenna to a linearly-polarized source at a random location in the sky? This is given by the rms value of over the sky,
which is smaller than the maximum response by a factor of (52%) for a bar detector and by (63%) for an interferometer.
The polarization amplitudes of the radiation from an inspiraling binary, a rotating neutron star, or a ringing black hole, take a simple form as follows:
where is an overall (possibly time-dependent) amplitude, is the signal’s phase and is the angle made by the characteristic direction in the source (e.g., the orbital or the spin angular momentum) with the line of sight. In this case, the response takes a particularly simple form:
Note that , just as , takes values in the range [0, 1]. In this case the average response has to be worked out by considering all possible sky locations, polarizations (which drops out of the calculation) and source orientations. More precisely, the rms response is
For an interferometer the above integral gives 2/5. Thus, the rms response is still 40% of the peak response.
The right-hand panel of Figure 4 shows the percentage area of the sky over which the antenna pattern of an interferometric detector is larger than a certain fraction of the peak value. The response is better than the rms value over 40% of the sky, implying that gravitational wave detectors are fairly omni-directional. In comparison, the sky coverage of most conventional telescopes (radio, infrared, optical, etc.) is a tiny fraction of the area of the sky.
4.3 Practical issues of ground-based interferometers
A detector with an arm length of 4 km responds to a gravitational wave with an amplitude of with
Light takes only about s to go up and down one arm, much less than the typical period of gravitational waves of interest. Therefore, it is beneficial to arrange for the light to remain in an arm longer than this, say for 100 round trips. This increases its effective path length by 100 and hence the shift in the position of a given phase of the light beam will be of order m. Most interferometers keep the light in the arms for this length of time by setting up optical cavities in the arms with low-transmissivity mirrors; these are called Fabry–Pérot cavities.
The main sources of noise against which a measurement must compete are:
Ground vibration. External mechanical vibrations must be screened out. These are a problem for bar detectors, too, but are more serious for interferometers, not least because interferometers bounce light back and forth between the mirrors, and so each reflection introduces further vibrational noise. Suspension/isolation systems are based on pendulums. A pendulum is a good mechanical filter for frequencies above its natural frequency. By hanging the mirrors on pendulums of perhaps 0.5 m length, one achieves filtering above a few Hertz. Since the spectrum of ground noise falls at higher frequencies, this provides suitable isolation. But these systems can be very sophisticated; the GEO600  detector has a three-stage pendulum and other vibration isolation components . The most ambitious multi-stage isolation system has been developed for the Virgo detector .
Thermal noise. Vibrations of the mirrors and of the suspending pendulum can mask gravitational waves. As with vibrational noise, this is increased by the bouncing of the light between the mirrors. Opposite to bars, interferometers perform measurements at frequencies far from the resonant frequency, where the amplitude of thermal vibration is largest. Thus, the pendulum suspensions have thermal noise at a few Hertz, so measurements will be made above 40 Hz in the first detectors. Internal vibrations of the mirrors have natural frequencies of several kHz, which sets an effective upper limit to the observing band. By ensuring that both kinds of oscillations have very high , one can confine most of the vibration energy to a small bandwidth around the resonant frequency, so that at the measurement frequencies the vibration amplitudes are extremely small. This allows interferometers to operate at room temperature. But mechanical s of or higher are required, and this is technically demanding.
Thermal effects produce other disturbances besides vibration. Some of the mirrors in interferometers are partly transmissive, as is the beam splitter. A small amount of light power is absorbed during transmission, which raises the temperature of the mirror and changes its index of refraction. The resulting “thermal lensing” can ruin the optical properties of the system, and random fluctuations in lensing caused by time-dependent thermal fluctuations (thermo-refractive noise) can appear at measurement frequencies. These effects can limit the amount of laser power that can be used in the detector. Other problems can arise from heating effects in the multiple-layer coatings that are applied to the reflective surfaces of mirrors.
Shot noise. The photons that are used to do interferometry are quantized, and so they arrive at random and make random fluctuations in the light intensity that can look like a gravitational wave signal. The more photons one uses, the smoother the interference signal will be. As a random process, the error improves with the square root of the number of photons. Using infrared light with a wavelength , one can expect to measure to an accuracy of
To measure at a frequency , one has to make at least measurements per second, so one can accumulate photons for a time . With light power , one gets photons. In order that should be below m, one needs high light power, far beyond the output of any continuous laser.
Light-recycling techniques overcome this problem, by using light efficiently. An interferometer actually has two places where light leaves. One is where the interference is measured, the difference port. The other is the sum of the two return beams on the beam splitter, which goes back towards the input laser. Normally one makes sure that no light hits the interference sensor, so that only when a gravitational wave passes does a signal register there. This means that all the light normally returns toward the laser input, apart from small losses at the mirrors. Since mirrors are of good quality, only one part in or less of the light is lost during a 1 ms storage time. By placing a power-recycling mirror in front of the laser, one can reflect this wasted light back in, allowing power to build up in the arms until the laser merely resupplies the mirror losses . This can dramatically reduce the power requirement for the laser. The first interferometers work with laser powers of 5 – 10 W. Upgrades will use ten or more times this input power.
Quantum effects. Shot noise is a quantum noise, and like all quantum noises there is a corresponding conjugate noise. As laser power is increased to reduce shot noise, the position sensing accuracy improves, and one eventually comes up against the Heisenberg uncertainty principle: the momentum transferred to the mirror by the measurement leads to a disturbance that can mask a gravitational wave. To reduce this backaction pressure fluctuation, scientists are experimenting with a variety of interferometer configurations that modify the quantum state of the light, by “squeezing” the Heisenberg uncertainty ellipse, in order to reduce the effect of this uncertainty on the variable being measured, at the expense of its (unmeasured) conjugate. The key point here is that we are using a quantum field (light) to measure an effectively classical quantity (gravitational wave amplitude), so we do not need to know everything about our quantum system: we just need to reduce the uncertainty in that part of the quantum field that responds to the gravitational wave at the readout of our interferometer. The best results on squeezing so far  have been obtained during preparations for the GEO-HF upgrade of the GEO600 detector . These techniques may be needed for the second-generation advanced detectors and will certainly be needed for advances beyond that.
Gravity gradient noise. One noise that cannot be screened out is that due to changes in the local Newtonian gravitational field on the timescale of the measurements. A gravitational wave detector will respond to tidal forces from local sources just as well as to gravitational waves. Environmental noise comes not only from man-made sources, but even more importantly from natural ones: seismic waves are accompanied by changes in the gravitational field, and changes in air pressure are accompanied by changes in air density. The spectrum falls steeply with increasing frequency, so for first-generation interferometers this will not be a problem, but it may limit the performance of more advanced detectors. And it is the primary reason that detecting gravitational waves in the low-frequency band below 1 Hz must be done in space.
4.3.1 Interferometers around the globe
The two largest interferometer projects are LIGO  and VIRGO . LIGO has built three detectors at two sites. At Hanford, Washington, there is a 4 km and a 2 km detector in the same vacuum system. At Livingston, Louisiana, there is a single 4 km detector, oriented to be as nearly parallel to the Hanford detector as possible. After a series of “engineering” runs, which helped to debug the instruments, interspersed with several “science runs”, which helped to test and debug the data acquisition system and various analysis pipelines, LIGO reached its design sensitivity goal in the final months of 2005. In November 2005, LIGO began a two-year data-taking run, called S5, which acquired a year’s worth of triple coincidence data among the three LIGO detectors. S5 ended on 30 September 2007. Although interferometers are pretty stable detectors, environmental disturbances and instrumental malfunctions can cause them to lose lock during which the data quality will be either poor or ill defined. The typical duty cycle at one of the LIGO sites is about 80%, and hence about two years of operation was required to accumulate a year’s worth of triple coincident data. Up to date information on LIGO can be found on the project’s website . A recent review of LIGO’s status is .
VIRGO finished commissioning its single 3-km detector at Cascina, near Pisa, in early 2007 and began taking data in coincidence with LIGO in May 2007, thus joining for the last part of S5. VIRGO is a collaboration among research laboratories in Italy and France, and its umbrella organization EGO looks after the operation of the site and planning for the future. There are websites for both VIRGO  and EGO . A recent review of VIRGO’s status is .
A smaller 600-m detector, GEO600, has been operational near Hanover, Germany, since 2001 . It is a collaboration among research groups principally in Germany and Britain. Although smaller, GEO600 has developed and installed second-generation technology (primarily in its suspensions, mirror materials and interferometer configuration) that help it achieve a higher sensitivity. GEO600 technology is being transferred to LIGO and VIRGO as part of their planned upgrades, described below. Full information about GEO can be found on its website . A recent review of GEO600’s status is .
LIGO and GEO have worked together under the umbrella of the LIGO Scientific Collaboration (LSC) since the beginning of science data runs in 2001. The LSC contains dozens of groups from universities around the world, which contribute to data analysis and technology development. The two detector groups pool their data and analyze it jointly. The LSC has a website containing detailed information, and providing access to the publications and open-source software of the collaboration .
VIRGO has signed an agreement with the LSC to pool data and analyze it jointly, thereby creating a single worldwide network of long-baseline gravitational wave detectors. VIRGO is not, however, a member of the LSC.
The LSC has already published many papers on the analysis of data acquired during its science runs, and many more can be expected. The results from these science runs, which will be discussed later, are already becoming astrophysically interesting. The LSC maintains a public repository of its papers and contributions to conference proceedings .
For instance, although the search for continuous waves from known pulsars has not found any definitive candidates, it has been possible to set stringent upper limits on the magnitude of the ellipticity of some of these systems . In particular, in the case of the Crab pulsar, gravitational wave observations have begun to improve  the upper limit on the strength of radiation obtained by radio observations of the spin-down rate.
A yet smaller detector in Japan, TAMA300 , with 300 m arms, was the first large-scale interferometer to achieve continuous operation, at a sensitivity of about . TAMA is seen as a development prototype, and its sensitivity will be confined to higher frequencies (above 500 Hz). An ambitious follow-on detector called the Large-scale Cryogenic Gravitational-Wave Telescope (LCGT) is being planned in Japan, and, as its name suggests, it will be the first to use cooled mirrors to reduce the effects of thermal noise. TAMA  and the LCGT  have websites where one can get more information. A recent review of TAMA’s status is .
There are plans for a detector in Australia, and a small interferometer is operating in Western Australia . The Australian Interferometric Gravitational Observatory (AIGO)  is a proposal of the Australian Consortium for Interferometric Gravitational Astronomy (ACIGA) . The ACIGA collaboration is a member of the LSC and assists in mirror and interferometry development, but it is not yet clear whether and when a larger detector might be funded. From the point of view of extracting information from observations, it is very desirable to have large-scale detectors in Japan and Australia, because of their very long baselines to the USA and Europe. But the future funding of both LCGT and AIGO is not secure as of this writing (2008).
The initial sensitivity levels achieved by LIGO, VIRGO, and GEO are just the starting point. Detailed plans exist for upgrades for all three projects. In October 2007, both LIGO and VIRGO began upgrading to enhanced detectors, which should improve on LIGO’s S5 sensitivity by a factor of roughly two. These should come online in 2009. After a further observing run, called S6, the detectors will again shut down for a much more ambitious upgrade to advanced detectors, to operate around 2014. This will provide a further factor of five in sensitivity, and hence in range. Altogether the two upgrades will extend the volume of space that can be surveyed for gravitational waves by a factor of 1000, and this will make regular detections a virtual certainty. Advanced LIGO has a website giving the plans for the upgrade in the context of development from the initial sensitivity .
GEO600 will remain in science mode during the upgrade to enhanced detectors, just in case a nearby supernova or equally spectacular event should occur when the larger detectors are down. But, when the enhanced detectors begin operating, GEO will upgrade to GEO-HF , a modification designed to improve its sensitivity in the high-frequency region above 1 kHz, where its short arm length does not prevent it being competitive with the larger instruments. GEO is also a partner in the Advanced LIGO project, contributing high-power lasers and high-Q suspensions for controlling thermal noise.
Beyond that, scientists are now studying the technologies that may be needed for a further large step in sensitivity to third-generation detectors. This may involve cooling mirrors, using ultra-massive substrates of special materials, using purely nontransmissive optics, and even circumventing the quantum limit in interferometers, as has been studied for bars. The goal of third-generation detectors would be to be limited just by gravity-gradient noise and quantum effects. A design study for a concept called the “Einstein Telescope” started in Europe in 2008.
4.3.2 Very-high–frequency detectors
The gravitational wave spectrum above the detection band of conventional interferometers, say above 10 kHz, may not be empty, and stochastic gravitational waves from the Big Bang may be present up to megaHertz frequencies and beyond. It is exceedingly difficult to build sensitive detectors at these high frequencies, but two projects have made prototypes: a microwave-based detector that senses the change in polarization as the electromagnetic waves follow a waveguide circuit as a gravitational wave passes by , and a more conventional light-based interferometer .
4.4 Detection from space
Space offers two important ingredients for beam detectors: long arms and a free vacuum. In this section, we describe the three ways that space has been and will be used for gravitational wave detection: ranging to spacecraft (Section 4.4.1), pulsar timing (Section 4.4.2), and direct detection using space-based interferometers (Section 4.4.3).
4.4.1 Ranging to spacecraft
Both NASA and ESA perform experiments in which they monitor the return time of communication signals with interplanetary spacecraft for the characteristic effect of gravitational waves. For missions to Jupiter and Saturn, for example, the return times are of order s. Any gravitational wave event shorter than this will, by Equation (43), appear three times in the time delay: once when the wave passes the Earth-based transmitter, once when it passes the spacecraft, and once when it passes the Earth-based receiver. Searches use a form of data analysis using pattern matching. Using two transmission frequencies and very stable atomic clocks, it is possible to achieve sensitivities for of order , and even may soon be reached .
4.4.2 Pulsar timing
Many pulsars, particularly the old millisecond pulsars, are extraordinarily regular clocks when averaged over timescales of a few years, with random timing irregularities too small for the best atomic clocks to measure. If one assumes that they emit pulses perfectly regularly, then one can use observations of timing irregularities of single pulsars to set upper limits on the background gravitational-wave field. Here, the one-way formula Equation (42) is appropriate. The transit time of a signal to the Earth from the pulsar may be thousands of years, so we cannot look for correlations between the two terms in a given signal. Instead, the delay is a combination of the effects of waves at the pulsar when the signal was emitted and waves at the Earth when it is received. If one observes a single pulsar, then because not enough is known about the intrinsic irregularity in pulse emission, the most one can do is to set upper limits on a background of gravitational radiation at very low frequencies [242, 344].
If one simultaneously observes two or more pulsars, then the Earth-based part of the delay is correlated, and this offers, in addition, a means of actually detecting strong gravitational waves with periods of several years that pass the Earth (in order to achieve the long-period stability of pulse arrival times). Observations are currently underway at a number of observatories. The most stringent limits to date are from the Parkes Pulsar Timing Array , which sets an upper limit on a stochastic background of . Two further collaborations for timing have been formed: the European Pulsar Timing Array (EPTA)  and NanoGrav . Astrophysical backgrounds in this frequency band are likely (see Section 8.2.2), so these arrays have a good chance of making early detections. Future timing experiments will be even more powerful, using new phased arrays of radio telescopes that can observe many pulsars simultaneously, such as the Low Frequency Array (LOFAR)  and the Square Kilometer Array .
Pulsar timing can also be used to search for individual events, not just a stochastic signal. The first example of an upper limit from such a search was the exclusion of a black-hole–binary model for 3C66B .
4.4.3 Space interferometry
Gravity-gradient noise on the Earth is much larger than the amplitude of any expected waves from astronomical sources at frequencies below about 1 Hz, but this noise falls off rapidly as one moves away from the Earth. A detector in space would not notice the Earth’s noisy environment. The Laser Interferometer Space Antenna (LISA) project, currently being developed in collaboration by ESA and NASA with a view toward launching in 2018, would open up the frequency window between 0.1 mHz and 0.1 Hz for the first time [196, 144]. There are several websites that provide full information about this project [24, 153, 266].
We will see below that there are many exciting sources expected in this wave band, for example the coalescences of giant black holes in the centers of galaxies. LISA would see such events with extraordinary sensitivity, recording typical SNRs of 1000 or more for events at redshift one.
A space-based interferometer can have arm lengths much greater than a wavelength. LISA, for example, would have arms km long, and that would be longer than half a wavelength for any gravitational waves above 30 mHz. In this regime, the response of each arm will follow the three-term formula we encountered earlier. The short-arm approximation we used for ground-based interferometers works for LISA only at the lowest frequencies in its observing band.
LISA will consist of three free-flying spacecraft, arranged in an array that orbits the sun at 1 AU, about 20 degrees behind the Earth in its orbit. They form an approximately equilateral triangle in a plane tilted at to the ecliptic, and their simple Newtonian elliptical orbits around the sun preserve this arrangement, with the array rotating backwards once per year as the spacecraft orbit the sun. By passing light along each of the arms, one can construct three different Michelson-type interferometers, one for each vertex. With this array one can measure the polarization of a gravitational wave directly. The spacecraft are too far apart to use simple mirrors to reflect light back along an arm: the reflected light would be too weak. Instead, LISA will have optical transponders: light from one spacecraft’s on-board laser will be received at another, which will then send back light from its own laser locked exactly to the phase of the incoming signal.
The main environmental disturbances to LISA are forces from the sun: solar radiation pressure and pressure from the solar wind. To minimize these, LISA incorporates drag-free technology. Interferometry is referenced to an internal proof mass that falls freely, unattached to the spacecraft. The job of the spacecraft is to shield this mass from external disturbances. It does this by sensing the position of the mass and firing its own jets to keep itself (the spacecraft) stationary relative to the proof mass. To do this, it needs thrusters of very small thrust that have accurate control. The key technologies that have enabled the LISA mission are the availability of such thrusters, accelerometers needed to sense disturbances to the spacecraft, and lasers capable of continuously emitting 1 W of infrared light for many years. ESA is planning to launch a satellite called LISA Pathfinder to test all of these technologies in 2010 .
LISA is not the only proposal for an interferometer in space for gravitational wave detection. The DECIGO proposal is a more ambitious design, positioned at a higher frequency to fill the gap between LISA and ground-based detectors . Even more ambitious, in the same frequency band, is the Big Bang Observer, a NASA concept study to examine what technology would be needed to reach the ultimate sensitivity of detecting a gravitational wave background from inflation at these frequencies .
4.5 Characterizing the sensitivity of a gravitational wave antenna
The performance of a gravitational wave detector is characterized by the power spectral density (henceforth denoted PSD) of its noise background. One can construct the noise PSD as follows; a gravitational wave detector outputs a dimensionless data train, say , which in the case of an interferometer is the relative strain in the two arms, scaled to represent the value of that would produce that strain if the wave is optimally oriented wit