Gravitational Waves

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One of the many remarkable predictions of Einstein's general theory of relativity is the existence of gravitational waves (GWs) [27]. Einstein himself elucidated the theoretical existence of GWs as early as 1918 [28]. Today, however, GWs have still not been directly measured, although the measurements of the binary pulsar PSR 1913+16 [29-31], (discovered by Hulse and Taylor and for which they won the Nobel prize) leave little doubt that GWs do, in fact, exist.

The fundamental factor that has led to our failure to directly measure GWs is the exceptional weakness of the gravitational coupling constant. This causes GWs to be too feeble to detect unless produced under extreme conditions. In particular, gravitational waves are produced by accelerating masses (a more exact formulation of this statement appears in Sect. 2.3). Thus, for the highest GW amplitudes, we seek sources with the highest possible accelerations and masses. As a result, astrophysical objects are the most plausible sources of GWs 3.

Further restrictions on viable sources are imposed by our detectors. For instance, for Earth based instruments, even with the best seismic isolation technologies currently available [32, 33], noise from seismic vibrations limits large-scale precision measurements to frequencies above ^30Hz. Through causality, this time-scale limitation implies a maximum length-scale at the source of 104 km. Given the Chandrasekhar limit on the mass of a white dwarf star [34,35] and the white dwarf mass-radius relationship this is approximately the minimum length-scale for white dwarf stars [36].

If searches are restricted to objects more compact than white dwarfs, then within the bounds of current knowledge, gravitational wave astronomy with ground-based interferometers is limited to black holes and neutron stars as sources. According to our current understanding of astrophysical populations, these objects are relatively rare. Thus, the probability of finding them in our immediate stellar neighborhood are small, and any realistic search must be sensitive to these sources out to extragalactic distances to have a reasonable chance of seeing them in an observation time measured in years. Is this a reasonable prospect? The answer is yes, but to understand why, it behooves us to first understand a bit more about what GWs are and how they might be measured.

Gravitational waves arriving at Earth are perturbations of the geometry of space-time. Heuristically, they can be understood as fluctuations in the distances between points in space. Mathematically, they are modeled as a metric tensor perturbation hap on the flat spacetime background. Linearizing the Einstein field equations of general relativity in hap, we find that this symmetric four-by-four matrix satisfies the the wave equation,

in an appropriate gravitational gauge. Here V2 is the usual Laplacian operator, G is the gravitational constant, and Ta p is the stress energy tensor, another symmetric four-by-four matrix which encodes information about the energy and matter content of the spacetime.

From (1), it is obvious that GWs travel at c, the speed of light. To understand the production of gravitational waves by a source, we solve (1) with the stress-energy (Tap) of that source on the right-hand-side. We will discuss this in more detail in Sect. 2.3. First, however, we wish to consider the propagation of gravitational waves.

3 The exception to this statement is the big-bang itself, which should lead to a sto chastic cosmological background of GWs, as mentioned in Sect. 1. Gravitational wave observations have just begun to bound previously viable theoretical models of this background [10]. We will not be considering this background further in this review.

For the propagation of gravitational waves, we require solutions to the homogeneous (Ta3 = 0) version of (1). As usual, such solutions can be expressed as linear combinations of the complex exponential functions

Here, Aa3 is a matrix of constant amplitudes, kM = ( — u, k) is a four vector which plays the role of a wave vector in four dimensions, and xM = (t, x) are the spacetime coordinates. Above, and in what follows, we use the Einstein summation convention that repeated indices, such as the j in k^x^ = —ut + k x, indicate an implicit summation. It can be shown that for gauges in which (1) holds that

In words, this means that the wave vector is orthogonal to the directions in which the GW distorts spacetime, i.e. the wave is transverse.

Since ha p is a four-by-four matrix, it has 16 components. However, because it is symmetric, only 10 of those components are independent. Further, (3) imposes four constraints on hap, reducing the number of free components to six. One can use remaining gauge freedom to impose four more conditions on hap. There are therefore only two independent components of the matrix hap. Details can be found in any elementary textbook on General Relativity such as [37].

The two independent components of hap are traditionally called h+ and hx. These names are taken from the effect that the components have on a ring of freely moving particles laying in the plane perpendicular to the direction of wave propagation, as illustrated in Fig. 1. The change in distance between particles, S£, is proportional to the original distance between them, I, and the amplitude of the gravitational waves, Aap. For a point source, which all astrophysical sources will effectively be, the amplitude decreases linearly with the distance from the source. For astrophysical source populations from which gravitational wave emission have been estimated, the typical gravitational wave strain, h ~ 25i/i, at a detector at Earth would be expected to be less than or of the order of 10~21 [1]. Through interferometry, it is possible to measure S£ ~ 10~18 m. Thus, interferometers of kilometer scales are required to have any chance of measuring these sources.

It might seem that the challenge of attempting to measure gravitational waves is so daunting as to call into question whether it is worthwhile at all. However, there are several factors which make the measurement of gravitational waves attractive. First, astrophysical gravitational wave sources include systems, such as black hole binaries, which are electromagnetically dark. Gravitational waves may therefore be the best way to study such sources. Second, since gravitation couples weakly to matter, gravitational waves propagate essentially without loss or distortion from their source to the detector. Thus, sources obscured by dust or other electromagnetically opaque media may still haß = Aaß exp(±ikMxM).

Fig. 1. Distortion of a ring of freely falling dust as a gravitational wave passes through. The wave is propagating into the page. From left to right are a series of four snapshots of the distortion of the ring. The top row are distortions due to h+. The bottom row are distortions due to hx. The snapshots are taken at times t = 0, t = T/4, t = T/2 and t = 3T/4 respectively, where T is the period of the gravitational wave. The relative phase between h+ and hx corresponds to a circularly polarized gravitational wave.

Fig. 1. Distortion of a ring of freely falling dust as a gravitational wave passes through. The wave is propagating into the page. From left to right are a series of four snapshots of the distortion of the ring. The top row are distortions due to h+. The bottom row are distortions due to hx. The snapshots are taken at times t = 0, t = T/4, t = T/2 and t = 3T/4 respectively, where T is the period of the gravitational wave. The relative phase between h+ and hx corresponds to a circularly polarized gravitational wave.

be observed with gravitational waves. Also, interferometers behave as amplitude sensing devices for GWs (like antennas), rather than energy gathering devices (like telescopes), leading to a 1/r fall-off with distance, rather than the more usual 1/r2 fall-off [2].

But perhaps the most compelling reason for pursuing the measurement of gravitational waves is that they constitute an entirely new medium for astronomical investigation. History has demonstrated that every time a new band of the electromagnetic spectrum has become available to astronomers, it has revolutionized our understanding of the cosmos. What wonders, then, await us when we start to see the Universe through the lens of gravitational waves, (which will surely begin to happen within the next decade as GW detectors continue their inevitable march toward higher sensitivities)? Only time will tell, but there is every reason to be optimistic.

In this article, we concentrate on one of the many sources of gravitational waves for which searches are ongoing - binary neutron star (BNS) systems. In particular, the sensitive frequency band of ground-based interferometers, which is approximately 40 Hz to 400 Hz, dictates that we should be interested in neutron star binaries within a few minutes of coalescence [38]. These sources hold a privileged place in the menagerie of gravitational waves sources that interferometers are searching for. The Post-Newtonian expansion, a general relativistic approximation method which describes their motion, gives us expected waveforms to high accuracy [39-41]. They are one of the few sources for which such accurate waveforms currently exist, and they are therefore amenable to the most sensitive search algorithms available. Furthermore, while the population of neutron star binaries is not well understood, there are at least observations of this source with which to put some constraints on the population [42]. These two factor give BNS systems one of the best (if not the best) chance of discovery in the near future.

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