6.4 Speed of gravitational waves6 Gravitational Wave Tests of 6.2 Polarization of gravitational waves

6.3 Gravitational radiation back-reaction 

In the binary pulsar, a test of GR was made possible by measuring at least three relativistic effects that depended upon only two unknown masses. The evolution of the orbital phase under the damping effect of gravitational radiation played a crucial role. Another situation in which measurement of orbital phase can lead to tests of GR is that of the inspiralling compact binary system. The key differences are that here gravitational radiation itself is the detected signal, rather than radio pulses, and the phase evolution alone carries all the information. In the binary pulsar, the first derivative of the binary frequency tex2html_wrap_inline5565, was measured; here the full nonlinear variation of tex2html_wrap_inline5107 as a function of time is measured.

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 tex2html_wrap_inline3921 or better. Since tex2html_wrap_inline5571 during the late inspiral, this means that correction terms in the phasing at the level of tex2html_wrap_inline5573 or higher are needed. More formal analyses confirm this intuition [35, 60, 36, 108].

Because it is a slow-motion system (tex2html_wrap_inline5575), 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 tex2html_wrap_inline5579 to the quadrupole formula were calculated as early as 1976 [135] see TEGP 10.3 [147Jump To The Next Citation Point In The Article]).

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 tex2html_wrap_inline5127 through 2PN order has the form


where tex2html_wrap_inline5583 . The first term is the quadrupole contribution (compare Eq. (55Popup Equation)), the second term is the 1PN contribution, the third term, with the coefficient tex2html_wrap_inline5585, 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 [70].) 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 [28] 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. (72Popup Equation) can be measured, then one again obtains more than two constraints on the two unknowns tex2html_wrap_inline4899 and tex2html_wrap_inline4991, 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 tex2html_wrap_inline5585 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 tex2html_wrap_inline5597, and b is the coefficient of the dipole term, given by tex2html_wrap_inline5601, where tex2html_wrap_inline4585, tex2html_wrap_inline5605, tex2html_wrap_inline5607 are given by Eqs. (69Popup Equation), and tex2html_wrap_inline5609 . 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 tex2html_wrap_inline5463 results in suppression of dipole radiation by symmetry (the sensitivity s turns out to be a relatively weak function of mass near tex2html_wrap_inline5463, 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 tex2html_wrap_inline5463 neutron star and a tex2html_wrap_inline5621 (tex2html_wrap_inline5623) black hole at 200 Mpc, the bound on tex2html_wrap_inline3785 could be 600 (1800) (using advanced LIGO noise curves). The bound increases linearly with signal-to-noise ratio [149]. If one demands that this test be performed annually, thus requiring observation of frequent, and therefore more distant, weaker sources, the bounds on tex2html_wrap_inline3785 will be too weak to compete with existing solar-system bounds (this corresponds to the ``LIGO-VIRGO'' bound on the tex2html_wrap_inline3777 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 tex2html_wrap_inline4191 per year per galaxy, and a black-hole-neutron-star rate tex2html_wrap_inline4353 times that, where tex2html_wrap_inline4353 is highly uncertain. Values of tex2html_wrap_inline5641, 1, and 10 are shown. For the general class of scalar-tensor theories, the corresponding bounds are plotted on the tex2html_wrap_inline3777 - tex2html_wrap_inline3779 plane in Figure  8, under the restrictive assumption of a signal-to-noise ratio (S/N) of 10 [44]. All other factors being equal, the bound achievable on the tex2html_wrap_inline3777 - tex2html_wrap_inline3779 parameters of scalar-tensor gravity is inversely proportional to S/N.


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Figure 10: The bounds on the scalar-tensor coupling constant tex2html_wrap_inline3785 from gravitational wave observations of inspiralling black-hole-neutron-star systems. The solar system bound is around tex2html_wrap_inline3787 .

6.4 Speed of gravitational waves6 Gravitational Wave Tests of 6.2 Polarization of gravitational waves

image The Confrontation between General Relativity and Experiment
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
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