Detecting a Tail Effect in Gravitational-Wave Experiments

Future gravitational-wave experiments looking at inspiralling compact binaries could achieve the detection of a very small effect of phase modulation induced by the tails of gravitational waves. Once a binary signal has been identified, further analysis of data will provide a measure of the total mass-energy M of the binary, which enters as a factor in this tail effect, by means of optimal signal processing. The detection of the effect will then consist in showing the compatibility of the measured values of M and of the other parameters depending on the two masses of the binary. This illustrates the high potentiality of gravitational-wave experiments for testing general relativity. PACS numbers: 04.80.Nn, 04.30.Db, 97.60.Lf, 97.80.— d The first direct detection of gravitational radiation will probably take place in future gravitational-wave experiments such as LIGO and VIRGO. For the moment, the detection of gravitational radiation has only been indirect, thanks to the very precise timing observations of the binary pulsar 1913 + 16 [1]. Among the best candidate sources for a direct detection of gravitational radiation are binary systems of compact objects (neutron stars or black holes) in their late inspiralling stages of evolution [2]. The number of neutron-star coalescences is expected to be a few per year out to a distance of 100 Mpc [3] (with maybe a comparable number of black-hole coalescences), at which distance LIGO and VIRGO might observe the waves with a signal-to-noise ratio (SNR) — 10. Such a premiere will open a totally new field in astronomy, and will permit verification of some fundamental predictions of general relativity. Often quoted is the possibility of verifying that the waves are of pure helicity two, with no admixture of other spin states. The purpose of our work (this Letter and the detailed account [4]) is to show, on the basis of a particular effect related to the so-called gravitational-wave tail effect, that the observations of inspiralling compact binaries will permit also verification of some aspects of the nonlinear structure of general relativity. This verification is made possible by the now recognized fact [5] that a very precise general relativity prediction is needed to reach full potential accuracy on the measurement of the binary's parameters. The tail effect is essentially due to the propagation of gravitational radiation on the curved background spacetime generated by its own source. More specifically, the tail of the radiation results, at lowest order, from the nonlinear interaction between the time-varying quadrupole moment of the source (which generates the linear radiation) and its monopole moment, or total mass-energy M (which generates the background). The tail radiation has the distinctive property ("nonlocality" in time) of depending on the source's dynamics at arbitrary remote instants in the past, anterior to the simply retarded time t — r/c. This reflects the fact that gravity propagates not only on the light cone (direct propagation with the speed of light c), but also within the light cone (averaged propagation with all velocities less than c). (See [6] for references on tails and related nonlinear effects. ) The detection of the tail effect (or of effects immediately related to it) in future gravitational-wave experiments will provide direct evidence that gravity propagates on a curved space-time — that generated by its own source. (Note that indirect evidence from the observations of the binary pulsar is probably out of reach [6].) This will represent an interesting test of the nonlinearity of general relativity in the "gravitodynamics" regime of the theory, involving rapidly varying and strong gravitational fields. This will also provide an independent measurement of the total mass-energy M of the source. The tail effect arises at the so called 1 5 postNewtonian (1.5-PN) approximation in the radiation, i.e., at the relative order c 3 beyond the usual quadrupole radiation. Let us consider the radiation emitted by a general isolated source, at a large distance r from the source (neglecting terms that die out like 1/r~). More precisely, we denote by h(t) that linear combination of the components of the wave which is directly felt by some detector [e.g. , h(t) is the relative variation of the arm's length of a laser interferometric detector]. Then the expression of h(t), including all terms in the post-Newtonian expansion up to the order c, can be written [7] as