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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 f˙b was measured; here the full nonlinear variation of fb 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 10–5 or better. Since v ∼1/10 during the late inspiral, this means that correction terms in the phasing at the level of v5 or higher are needed. More formal analyses confirm this intuition [6710568214].

Because it is a slow-motion system (v ∼ 10–3), 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 v2 to the quadrupole formula were calculated as early as 1976 [268] (see TEGP 10.3 [281Jump To The Next Citation Point]).

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 preferably 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 f = 2fb through 2PN order has the form

[ ( ) f˙= 96π-f2(πℳf )5∕3 1 − 743-+ 11-η (πmf )2∕3 + 4π(πmf ) 5 336 4 ( 34103 13661 59 ) ] + ------+ -----η + --η2 (πmf )4∕3 + 𝒪 [(πmf )5∕3] , (97 ) 18144 2016 18
where η = m1m2 ∕m2. The first term is the quadrupole contribution (compare Equation (72View Equation)), the second term is the 1PN contribution, the third term, with the coefficient 4π, is the “tail” contribution, and the fourth term is the 2PN contribution, first reported jointly by Blanchet et al. [4039291]. The 2.5PN, 3PN and 3.5PN contributions have also been calculated (see [34] for a review).

Similar expressions can be derived for the loss of angular momentum and linear momentum. Expressions for non-circular orbits have also been derived [12175]. 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 laser interferometric detectors to phase that makes the higher-order contributions to df∕dt so observationally relevant.

If the coefficients of each of the powers of f in Equation (97View Equation) can be measured, then one again obtains more than two constraints on the two unknowns m1 and m2, leading to the possibility to test GR. For example, Blanchet and Sathyaprakash [4241] have shown that, by observing a source with a sufficiently strong signal, an interesting test of the 4 π 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 [231213].

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,

96π 2 5∕3[ − 2∕3] ˙f = -5--f (πℳf ) 1 + b(πmf ) , (98 )
where 3∕5 −4∕5 3∕5 ℳ = (χ 𝒢 )η m, and b is the coefficient of the dipole term, given by b = (5∕48 )(χ −1𝒢4∕3)ξ𝒮2, where χ, 𝒢, 𝒮 are given by Equations (94View Equation), and ξ = 1∕ (2 + ωBD ). Double neutron star systems are not promising because the small range of masses available near 1.4M ⊙ results in suppression of dipole radiation by symmetry. For black holes, s = 0.5 identically, consequently double black hole systems turn out to be observationally identical in the two theories. Thus mixed systems involving a neutron star and a black hole are preferred. However, a number of analyses of the capabilities of both ground-based and space-based (LISA) observatories have shown that observing waves from neutron-star–black-hole inspirals is not likely to bound scalar-tensor gravity at a level competitive with the Cassini bound or with future solar-system improvements [283161236292Jump To The Next Citation Point27Jump To The Next Citation Point28Jump To The Next Citation Point].
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