Broad-band laser interferometers are especially sensitive to the phase evolution of the gravitational waves, which carry the information about the orbital phase evolution. The analysis of gravitational wave data from such sources will involve some form of matched filtering of the noisy detector output against an ensemble of theoretical ``template'' waveforms which depend on the intrinsic parameters of the inspiralling binary, such as the component masses, spins, and so on, and on its inspiral evolution. How accurate must a template be in order to ``match'' the waveform from a given source (where by a match we mean maximizing the cross-correlation or the signal-to-noise ratio)? In the total accumulated phase of the wave detected in the sensitive bandwidth, the template must match the signal to a fraction of a cycle. For two inspiralling neutron stars, around 16,000 cycles should be detected during the final few minutes of inspiral; this implies a phasing accuracy of or better. Since during the late inspiral, this means that correction terms in the phasing at the level of or higher are needed. More formal analyses confirm this intuition [35, 60, 36, 108].
Because it is a slow-motion system (), the binary pulsar is sensitive only to the lowest-order effects of gravitational radiation as predicted by the quadrupole formula. Nevertheless, the first correction terms of order v and to the quadrupole formula were calculated as early as 1976  see TEGP 10.3 ).
But for laser interferometric observations of gravitational waves, the bottom line is that, in order to measure the astrophysical parameters of the source and to test the properties of the gravitational waves, it is necessary to derive the gravitational waveform and the resulting radiation back-reaction on the orbit phasing at least to 2PN order beyond the quadrupole approximation, and probably to 3PN order.
For the special case of non-spinning bodies moving on quasi-circular orbits (i.e. circular apart from a slow inspiral), the evolution of the gravitational wave frequency through 2PN order has the form
where . The first term is the quadrupole contribution (compare Eq. (55)), the second term is the 1PN contribution, the third term, with the coefficient , is the ``tail'' contribution, and the fourth term is the 2PN contribution, first reported jointly by Blanchet et al. [24, 23, 152]. Calculation of the higher-order contributions is nearing completion.
Similar expressions can be derived for the loss of angular momentum and linear momentum. (For explicit formulas for non-circular orbits, see .) These losses react back on the orbit to circularize it and cause it to inspiral. The result is that the orbital phase (and consequently the gravitational wave phase) evolves non-linearly with time. It is the sensitivity of the broad-band LIGO and VIRGO-type detectors to phase that makes the higher-order contributions to df / dt so observationally relevant. A ready-to-use set of formulae for the 2PN gravitational waveform template, including the non-linear evolution of the gravitational wave frequency (not including spin effects) have been published  and incorporated into the Gravitational Radiation Analysis and Simulation Package (GRASP), a software toolkit used in LIGO.
If the coefficients of each of the powers of f in Eq. (72) can be measured, then one again obtains more than two constraints on the two unknowns and , leading to the possibility to test GR. For example, Blanchet and Sathyaprakash [30, 29] have shown that, by observing a source with a sufficiently strong signal, an interesting test of the coefficient of the ``tail'' term could be performed.
Another possibility involves gravitational waves from a small mass orbiting and inspiralling into a (possibly supermassive) spinning black hole. A general non-circular, non-equatorial orbit will precess around the hole, both in periastron and in orbital plane, leading to a complex gravitational waveform that carries information about the non-spherical, strong-field spacetime around the hole. According to GR, this spacetime must be the Kerr spacetime of a rotating black hole, uniquely specified by its mass and angular momentum, and consequently, observation of the waves could test this fundamental hypothesis of GR [114, 107].
Thirdly, the dipole gravitational radiation predicted by scalar-tensor theories will result in a modification of the gravitational radiation back-reaction, and thereby of the phase evolution. Including only the leading quadrupole and dipole contributions, one obtains, in Brans-Dicke theory,
where , and b is the coefficient of the dipole term, given by , where , , are given by Eqs. (69), and . The effects are strongest for systems involving a neutron star and a black hole. Double neutron star systems are less promising because the small range of masses available near results in suppression of dipole radiation by symmetry (the sensitivity s turns out to be a relatively weak function of mass near , for typical equations of state). For black holes, s =0.5 identically, consequently double black-hole systems turn out to be observationally identical in the two theories.
But for a neutron star and a () black hole at 200 Mpc, the bound on could be 600 (1800) (using advanced LIGO noise curves). The bound increases linearly with signal-to-noise ratio . If one demands that this test be performed annually, thus requiring observation of frequent, and therefore more distant, weaker sources, the bounds on will be too weak to compete with existing solar-system bounds (this corresponds to the ``LIGO-VIRGO'' bound on the axis in Figure 8, which assumes a signal-to-noise ratio of 10). However, if one is prepared to wait 10 years for the lucky observation of a nearby, strong source, the resulting bound could exceed the current solar-system bound. The bounds are illustrated in Figure 10 by the curves marked N =1 and N =1/10. Figure 10 assumes a double-neutron-star inspiral rate of per year per galaxy, and a black-hole-neutron-star rate times that, where is highly uncertain. Values of , 1, and 10 are shown. For the general class of scalar-tensor theories, the corresponding bounds are plotted on the - plane in Figure 8, under the restrictive assumption of a signal-to-noise ratio (S/N) of 10 . All other factors being equal, the bound achievable on the - parameters of scalar-tensor gravity is inversely proportional to S/N.
|The Confrontation between General Relativity and
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
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