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

348 Post-Newtonian computation of binary inspiral waveformsa factor in the quadrupole formula (17.41).) On the other hand, we find that Ereads simplyE =− 1 2 µc2 x. (17.49)Next we replace (17.47) and (17.49) into the balance equation (17.39), and findin this way an ordinary differential equation which is easily integrated for theunknown x. We introduce for later convenience the dimensionless time variableτ = c3 ν5Gm (t c − t), (17.50)where t c is a constant of integration. Then the solution readsx(t) = 1 4 τ −1/4 . (17.51)It is clear that t c represents the instant of coalescence, at which (by definition)the orbital frequency diverges to infinity. Then a further integration yieldsφ = ∫ ω dt =−ν5 ∫ x 2/3 dτ, and we get the looked for resultφ c − φ(t) = 1 ν τ 5/8 , (17.52)where φ c denotes the constant phase at the instant of coalescence. It is oftenuseful to consider the number Æ of gravitational-wave cycles which are left untilthe final coalescence starting from some frequency ω:Æ = φ c − φπ= 132πν x −5/2 . (17.53)As we see the post-Newtonian order of magnitude of Æ is c +5 , that is the inverseof the order c −5 of radiation reaction effects. As a matter of fact, Æ is a largenumber, approximately equal to 1.6×10 4 in the case of two neutron stars between10 and 1000 Hz (roughly the frequency bandwidth of the detector VIRGO). Dataanalysts of detectors have estimated that, in order not to suffer a too severereduction of signal to noise, one should monitor the phase evolution with anaccuracy comparable to one gravitational-wave cycle (i.e. δÆ ∼ 1) or better.Now it is clear, from a post-Newtonian point of view, that since the ‘Newtonian’number of cycles given by (17.53) is formally of order c +5 , any post-Newtoniancorrection therein which is larger than order c −5 is expected to contribute to thephase evolution more than that allowed by the previous estimate. Therefore, oneexpects that in order to construct accurate templates it will be necessary to includeinto the phase the post-Newtonian corrections up to at least the 2.5PN or 1/c 5order. This expectation has been confirmed by various studies [21–24] whichshowed that in advanced detectors the 2.5PN or, better, the 3PN approximation isrequired in the case of inspiralling neutron star binaries. Notice that 3PN heremeans 3PN in the centre-of-mass energy E, which is deduced from the 3PNequations of motion, as well as in the total flux Ä, which is computed from a 3PNwave-generation formalism. For the moment the phase has been completed to the2.5PN order [15, 25–27]; the 3PN order is still incomplete (but, see [13, 28, 29]).