(ed.). Gravitational waves (IOP, 2001)(422s).

144 LISA: A proposed joint ESA–NASA gravitational-wave missionIt has been emphasized by Cutler et al [72] that the strongest test of a theoryis likely to come from the observed phase of the signal, rather than the amplitude.This is because the cross-correlation of a theoretical template with the observedsignal will be reduced substantially if the two get out of phase by even half a cycleor less during the entire data record. For our case of a perhaps 10M ⊙ black holeorbiting around a 10 5 or 10 6 M ⊙ MBH during the last year before coalescence,there will be roughly 10 5 cycles. Even a very small error in the metric will lead toa continuously increasing error in the orbit, and thus in the phase of the calculatedsignal. While some type of metric error could, in principle, affect the orbit inthe same way as slightly different values of parameters in the problem, such asthe spin and mass of the MBH, it seems unlikely that a conceptual breakdown inthe theory would be nearly equivalent to just having different parameter values.Thus, correctly predicting the signal phase over 10 5 cycles would be an extremelystrong test of the theory.For this case, it is important that the orbit started out being nearly radial.The later evolution of the eccentricity from its initial value very close to unitycan be calculated from the rates of loss of energy and angular momentum due togravitational radiation. The results obtained by Tanaka et al for a SchwarzschildMBH [73] show for a typical case that the orbit never becomes circular, butinstead remains substantially eccentric (0.5 ?) up until the final plunge begins.However, rigorous calculations will be required in order to fit the data well [74].During the last year, the periapsis distance changes only very slowly, but theapoapsis distance decreases substantially. At periapsis, the speed is about halfthat of light, so the dynamics are highly non-Newtonian. In fact, the precessionof periapsis during one radial motion period can be about a whole cycle. Withrelativistic beaming of the gravitational radiation and variation of the strengthof the radiation with time during a radial period, the amplitude observed in agiven direction can vary in a quite complex way. For an orbit plane that is notperpendicular to the spin axis for a rapidly rotating MBH, rapid Lense–Thirringprecession also will be present. In view of the complexity of the relativisticmotion and the large number of cycles over which the phase of the signal canbe followed, such a signal would give a nearly ideal test of the predictions ofgeneral relativity.It is of course necessary to be able to detect the signal in order to test thetheory. It was mentioned earlier that a S/N ratio of about ten would be neededin order to detect such a complex signal. But there also is the question of howdifficult the search for such a signal would be. A very crude estimate of thenumber of templates needed for a brute force search with one year of data gives,within a couple of orders of magnitude, something like 10 18 . Even with rapidadvances in computing power, such a search probably would not be possible.Some improvement could be made by a hierarchical search strategy, but it is notclear that this approach would be sufficient. On the other hand, more powerfulsearch algorithms such as ‘genetic algorithms’ and ‘stimulated annealing’ havebeen developed and demonstrated for substantially more challenging search