9 Gravitational Waves from Compact Binaries

We pointed out that the 3.5PN equations of motion, Eqs. (203) or (219*) – (220), are merely 1PN as regards the radiative aspects of the problem, because the radiation reaction force starts at the 2.5PN order. A solution would be to extend the precision of the equations of motion so as to include the full relative 3PN or 3.5PN precision into the radiation reaction force, but the equations of motion up to the 5.5PN or 6PN order are beyond the present state-of-the-art. The much better alternative solution is to apply the wave-generation formalism described in Part A, and to determine by its means the work done by the radiation reaction force directly as a total energy flux at future null infinity.⁶² In this approach, we replace the knowledge of the higher-order radiation reaction force by the computation of the total flux , and we apply the energy balance equation Therefore, the result (232) that we found for the 3.5PN binary’s center-of-mass energy constitutes only “half” of the solution of the problem. The second “half” consists of finding the rate of decrease , which by the balance equation is nothing but the total gravitational-wave flux at the relative 3.5PN order beyond the Einstein quadrupole formula (4*).

Because the orbit of inspiralling binaries is circular, the energy balance equation is sufficient, and there is no need to invoke the angular momentum balance equation for computing the evolution of the orbital period and eccentricity , see Eqs. (9) – (13*) in the case of the binary pulsar. Furthermore the time average over one orbital period as in Eqs. (9) is here irrelevant, and the energy and angular momentum fluxes are related by . This all sounds good, but it is important to remind that we shall use the balance equation (295*) at the very high 3.5PN order, and that at such order one is missing a complete proof of it (following from first principles in general relativity). Nevertheless, in addition to its physically obvious character, Eq. (295*) has been verified by radiation-reaction calculations, in the cases of point-particle binaries [258, 259] and extended post-Newtonian fluids [43, 47], at the 1PN order and even at 1.5PN, the latter order being especially important because of the first appearance of wave tails; see Section 5.4. One should also quote here Refs. [260, 336, 278, 322, 254] for the 3.5PN terms in the binary’s equations of motion, fully consistent with the balance equations.

Obtaining the energy flux can be divided into two equally important steps: Computing the source multipole moments and of the compact binary system with due account of a self-field regularization; and controlling the tails and related non-linear effects occurring in the relation between the binary’s source moments and the radiative ones and observed at future null infinity (cf. the general formalism of Part A).

9.1 The binary’s multipole moments

The general expressions of the source multipole moments given by Theorem 6, Eqs. (123), are worked out explicitly for general fluid systems at the 3.5PN order. For this computation one uses the formula (126*), and we insert the 3.5PN metric coefficients (in harmonic coordinates) expressed in Eqs. (144) by means of the retarded-type elementary potentials (146) – (148). Then we specialize each of the (quite numerous) terms to the case of point-particle binaries by inserting, for the matter stress-energy tensor , the standard expression made out of Dirac delta-functions. In Section 11 we shall consider spinning point particle binaries, and in that case the stress-energy tensor is given by Eq. (378*). The infinite self-field of point-particles is removed by means of the Hadamard regularization; and, as we discussed in Section 6.4, dimensional regularization is used to fix the values of a few ambiguity parameters. This computation has been performed in Ref. [86*] at the 1PN order, and in [64] at the 2PN order; we report below the most accurate 3PN results obtained in Refs. [81*, 80, 62, 63] for the flux and [11*, 74*, 197*] for the waveform.

A difficult part of the analysis is to find the closed-form expressions, fully explicit in terms of the particle’s positions and velocities, of many non-linear integrals. Let us give a few examples of the type of technical formulas that are employed in this calculation. Typically we have to compute some integrals like

where and . When and , this integral is perfectly well-defined, since the finite part deals with the IR regularization of the bound at infinity. When or , we cure the UV divergencies by means of the Hadamard partie finie defined by Eq. (162*); so a partie finie prescription is implicit in Eq. (296*). An example of closed-form formula we get is where we pose and , the brackets surrounding indices denoting the STF projection. Another example, in which the regularization is crucial, is (in the quadrupole case ) where the IR scale is defined in Eq. (42*). Still another example, which necessitates both the and a UV partie finie regularization at the point 1, is where is the Hadamard-regularization constant introduced in Eq. (162*).

The most important input for the computation of the waveform and flux is the mass quadrupole moment , since this moment necessitates the full post-Newtonian precision. Here we give the mass quadrupole moment complete to order 3.5PN, for non-spinning compact binaries on circular orbits, as

where and are the orbital separation and relative velocity. The third term with coefficient is a radiation-reaction term, which will affect the waveform at orders 2.5PN and 3.5PN; however it does not contribute to the energy flux for circular orbits. The two conservative coefficients are and . All those coefficients are [81*, 74*, 197*]

These expressions are valid in harmonic coordinates via the post-Newtonian parameter defined in Eq. (225*). As we see, there are two types of logarithms at 3PN order in the quadrupole moment: One type involves the UV length scale related by Eq. (221*) to the two gauge constants and present in the 3PN equations of motion; the other type contains the IR length scale coming from the general formalism of Part A – indeed, recall that there is a operator in front of the source multipole moments in Theorem 6. As we know, the UV scale is specific to the standard harmonic (SH) coordinate system and is pure gauge (see Section 7.3): It will disappear from our physical results at the end. On the other hand, we have proved that the multipole expansion outside a general post-Newtonian source is actually free of , since the ’s present in the two terms of Eq. (105*) cancel out. Indeed we have already found in Eqs. (93*) – (94*) that the constant present in is compensated by the same constant coming from the non-linear wave “tails of tails” in the radiative moment . For a while, the expressions (301) contained the ambiguity parameters , and , which have now been replaced by their correct values (173).

Besides the 3.5PN mass quadrupole (300*) – (301), we need also (for the 3PN waveform) the mass octupole moment and current quadrupole moment , both of them at the 2.5PN order; these are given for circular orbits by [81*, 74*]

The list of required source moments for the 3PN waveform continues with the 2PN mass -pole and current -pole (octupole) moments, and so on. Here we give the most updated moments:⁶³

All the other higher-order moments are required at the Newtonian order, at which they are trivial to compute with result ()

Here we introduce the useful notation , where and are such that , and . More explicit expressions are ():

where is the usual binomial coefficient.

9.2 Gravitational wave energy flux

The results (300*) – (304) permit the control of the instantaneous part of the total energy flux, by which we mean that part of the flux which is generated solely by the source multipole moments, i.e., not counting the hereditary tail and related integrals. The instantaneous flux is defined by the replacement into the general expression of given by Eq. (68a) of all the radiative moments and by the corresponding -th time derivatives of the source moments and . Up to the 3.5PN order we have

in which we insert the explicit expressions (300*) – (304) for the moments. The time derivatives of these source moments are computed by means of the circular-orbit equations of motion given by Eq. (226*) together with (228). The net result is

The Newtonian approximation agrees with the prediction of the Einstein quadrupole formula (4*), as reduced for quasi-circular binary orbits by Landau & Lifshitz [285]. At the 3PN order in Eq. (307), there was some Hadamard regularization ambiguity parameters which have been replaced by their values computed with dimensional regularization.

To the instantaneous part of the flux, we must add the contribution of non-linear multipole interactions contained in the relationship between the source and radiative moments. The needed material has already been provided in Sections 3.3. Similar to the decomposition of the radiative quadrupole moment in Eq. (88*), we can split the energy flux into the different terms

where has just been obtained in Eq. (307); is made of the quadratic multipolar tail integrals in Eqs. (90) and (95); involves the square of the quadrupole tail in Eq. (90) and the quadrupole tail of tail given in Eq. (91).

We shall see that the tails play a crucial role in the predicted signal of compact binaries. It is quite remarkable that so small an effect as a “tail of tail” should be relevant to the data analysis of the current generation of gravitational wave detectors. By contrast, the non-linear memory effects, given by the integrals inside the 2.5PN and 3.5PN terms in Eq. (92), do not contribute to the gravitational-wave energy flux before the 4PN order in the case of circular-orbit binaries. Indeed the memory integrals are actually “instantaneous” in the flux, and a simple general argument based on dimensional analysis shows that instantaneous terms cannot contribute to the energy flux for circular orbits.⁶⁴ Therefore the memory effect has rather poor observational consequences for future detections of inspiralling compact binaries.

Let us also recall that following the general formalism of Part A, the mass which appears in front of the tail integrals of Sections 3.2 and 3.3 represents the binary’s mass monopole or ADM mass. In a realistic model where the binary system has been formed as a close compact binary at a finite instant in the past, this mass is equal to the sum of the rest masses , plus the total binary’s mass-energy given for instance by Eq. (229). At 3.5PN order we need 2PN corrections in the tails and therefore 2PN also in the mass , thus

Notice that 2PN order in corresponds to 1PN order in .

We give the two basic technical formulas needed when carrying out the reduction of the tail and tail-of-tail integrals to circular orbits (see e.g., [230]):

where is a strictly positive frequency (a multiple of the orbital frequency ), where and is the Euler constant.

Notice the important point that the tail (and tail-of-tail) integrals can be evaluated, thanks to these formulas, for a fixed (i.e., non-decaying) circular orbit. Indeed it can be shown [60, 87*] that the “remote-past” contribution to the tail integrals is negligible; the errors due to the fact that the orbit has actually evolved in the past, and spiraled in by emission of gravitational radiation, are of the order of the radiation-reaction scale ,⁶⁵ and do not affect the signal before the 4PN order. We then find, for the quadratic tails stricto sensu, the 1.5PN, 2.5PN and 3.5PN contributions

For the sum of squared tails and cubic tails of tails at 3PN, we get

By comparing Eqs. (307) and (312) we observe that the constants cleanly cancel out. Adding together these contributions we obtain

The gauge constant has not yet disappeared because the post-Newtonian expansion is still parametrized by instead of the frequency-related parameter defined by Eq. (230*) – just as for when it was given by Eq. (229). After substituting the expression given by Eq. (231), we find that does cancel as well. Because the relation is issued from the equations of motion, the latter cancellation represents an interesting test of the consistency of the two computations, in harmonic coordinates, of the 3PN multipole moments and the 3PN equations of motion. At long last we obtain our end result:⁶⁶

In the test-mass limit for one of the bodies, we recover exactly the result following from linear black-hole perturbations obtained by Tagoshi & Sasaki [395] (see also [393, 397]). In particular, the rational fraction comes out exactly the same as in black-hole perturbations. On the other hand, the ambiguity parameters discussed in Section 6.2 were part of the rational fraction , belonging to the coefficient of the term at 3PN order proportional to (hence this coefficient cannot be computed by linear black-hole perturbations).

The effects due to the spins of the two black holes arise at the 1.5PN order for the spin-orbit (SO) coupling, and at the 2PN order for the spin-spin (SS) coupling, for maximally rotating black holes. Spin effects will be discussed in Section 11. On the other hand, the terms due to the radiating energy flowing into the black-hole horizons and absorbed rather than escaping to infinity, have to be added to the standard post-Newtonian calculation based on point particles as presented here; such terms arise at the 4PN order for Schwarzschild black holes [349*] and at 2.5PN order for Kerr black holes [392*].

9.3 Orbital phase evolution

We shall now deduce the laws of variation with time of the orbital frequency and phase of an inspiralling compact binary from the energy balance equation (295*). The center-of-mass energy is given by Eq. (232) and the total flux by Eq. (314). For convenience we adopt the dimensionless time variable⁶⁷

where denotes the instant of coalescence, at which the frequency formally tends to infinity, although evidently, the post-Newtonian method breaks down well before this point. We transform the balance equation into an ordinary differential equation for the parameter , which is immediately integrated with the result

The orbital phase is defined as the angle , oriented in the sense of the motion, between the separation of the two bodies and the direction of the ascending node (called in Section 9.4) within the plane of the sky. We have , which translates, with our notation, into , from which we determine⁶⁸

where is a constant of integration that can be fixed by the initial conditions when the wave frequency enters the detector. Finally we want also to dispose of the important expression of the phase in terms of the frequency . For this we get

where is another constant of integration. With the formula (318) the orbital phase is complete up to the 3.5PN order for non-spinning compact binaries. Note that the contributions of the quadrupole moments of compact objects which are induced by tidal effects, are expected from Eq. (16*) to come into play only at the 5PN order.

As a rough estimate of the relative importance of the various post-Newtonian terms, we give in Table 3 their contributions to the accumulated number of gravitational-wave cycles in the bandwidth of ground-based detectors. Note that such an estimate is only indicative, because a full treatment would require the knowledge of the detector’s power spectral density of noise, and a complete simulation of the parameter estimation using matched filtering techniques [138*, 350, 284]. We define as

The frequency of the signal at the entrance of the bandwidth is the seismic cut-off frequency of ground-based detectors; the terminal frequency is assumed for simplicity to be given by the Schwarzschild innermost stable circular orbit: . Here we denote by the signal frequency of the dominant harmonics at twice the orbital frequency. As we see in Table 3, with the 3PN or 3.5PN approximations we reach an acceptable accuracy level of a few cycles say, that roughly corresponds to the demand made by data-analysists in the case of neutron-star binaries [139, 137, 138*, 346, 105*, 106*]. Indeed, the above estimation suggests that the neglected 4PN terms will yield some systematic errors that are, at most, of the same order of magnitude, i.e., a few cycles, and perhaps much less.

9.4 Polarization waveforms for data analysis

The theoretical templates of the compact binary inspiral follow from insertion of the previous solutions for the 3.5PN-accurate orbital frequency and phase into the binary’s two polarization waveforms and defined with respect to a choice of two polarization vectors and orthogonal to the direction of the observer; see Eqs. (69).

Our convention for the two polarization vectors is that they form with a right-handed triad, and that and lie along the major and minor axis, respectively, of the projection onto the plane of the sky of the circular orbit. This means that is oriented toward the orbit’s ascending node – namely the point at which the orbit intersects the plane of the sky and the bodies are moving toward the observer located in the direction . The ascending node is also chosen for the origin of the orbital phase . We denote by the inclination angle between the direction of the detector as seen from the binary’s center-of-mass, and the normal to the orbital plane (we always suppose that the normal is right-handed with respect to the sense of motion, so that ). We use the shorthands and for the cosine and sine of the inclination angle.

We shall include in and all the harmonics, besides the dominant one at twice the orbital frequency, consistent with the 3PN approximation [82, 11*, 74*]. In Section 9.5 we shall give all the modes in a spherical-harmonic decomposition of the waveform, and shall extend the dominant quadrupole mode at 3.5PN order [197*]. The post-Newtonian terms are ordered by means of the frequency-related variable ; to ease the notation we pose

Note that the post-Newtonian coefficients will involve the logarithm starting at 3PN order; see Eq. (127*). They depend on the binary’s phase , explicitly given at 3.5PN order by Eq. (318), through the very useful auxiliary phase variable that is “distorted by tails” [87*, 11*] and reads Here denotes the binary’s ADM mass and it is very important to include all its relevant post-Newtonian contributions as given by Eq. (309*). The constant frequency can be chosen at will, for instance to be the entry frequency of some detector. For the plus polarization we have⁶⁹

For the cross polarizations we obtain

Notice the non-linear memory zero-frequency (DC) term present in the Newtonian plus polarization ; see Refs. [427, 11, 189*] for the computation of this term. Notice also that there is another DC term in the 2.5PN cross polarization , first term in Eq. (323f).

The practical implementation of the theoretical templates in the data analysis of detectors follows from the standard matched filtering technique. The raw output of the detector consists of the superposition of the real gravitational wave signal and of noise . The noise is assumed to be a stationary Gaussian random variable, with zero expectation value, and with (supposedly known) frequency-dependent power spectral density . The experimenters construct the correlation between and a filter , i.e.,

and divide by the square root of its variance, or correlation noise. The expectation value of this ratio defines the filtered signal-to-noise ratio (SNR). Looking for the useful signal in the detector’s output , the data analysists adopt for the filter where and are the Fourier transforms of and of the theoretically computed template . By the matched filtering theorem, the filter (325*) maximizes the SNR if . The maximum SNR is then the best achievable with a linear filter. In practice, because of systematic errors in the theoretical modelling, the template will not exactly match the real signal ; however if the template is to constitute a realistic representation of nature the errors will be small. This is of course the motivation for computing high order post-Newtonian templates, in order to reduce as much as possible the systematic errors due to the unknown post-Newtonian remainder.

To conclude, the use of theoretical templates based on the preceding 3PN/3.5PN waveforms, and having their frequency evolution built in via the 3.5PN phase evolution (318) [recall also the “tail-distorted” phase variable (321*)], should yield some accurate detection and measurement of the binary signals, whose inspiral phase takes place in the detector’s bandwidth [105, 106, 159, 156, 3, 18, 111]. Interestingly, it should also permit some new tests of general relativity, because we have the possibility of checking that the observed signals do obey each of the terms of the phasing formula (318) – particularly interesting are those terms associated with non-linear tails – exactly as they are predicted by Einstein’s theory [84, 85, 15, 14]. Indeed, we don’t know of any other physical systems for which it would be possible to perform such tests.

9.5 Spherical harmonic modes for numerical relativity

The spin-weighted spherical harmonic modes of the polarization waveforms have been defined in Eq. (71*). They can be evaluated either from applying the angular integration formula (72*), or alternatively from using the relations (73*) – (74) giving the individual modes directly in terms of separate contributions of the radiative moments and . The latter route is actually more interesting [272*] if some of the radiative moments are known to higher PN order than others. In this case the comparison with the numerical calculation for these particular modes can be made with higher post-Newtonian accuracy.

A useful fact to remember is that for non-spinning binaries, the mode is entirely given by the mass multipole moment when is even, and by the current one when is odd. This is valid in general for non-spinning binaries, regardless of the orbit being quasi-circular or elliptical. The important point is only that the motion of the two particles must be planar, i.e., takes place in a fixed plane. This is the case if the particles are non-spinning, but this will also be the case if, more generally, the spins are aligned or anti-aligned with the orbital angular momentum, since there is no orbital precession in this case. Thus, for any “planar” binaries, Eq. (73*) splits to (see Ref. [197*] for a proof)

Let us factorize out in all the modes an overall coefficient including the appropriate phase factor , where we recall that denotes the tail-distorted phase introduced in Eq. (321*), and such that the dominant mode with conventionally starts with one at the Newtonian order. We thus pose

We now list all the known results for . We assume ; the modes having are easily deduced using . The dominant mode , which is primarily important for numerical relativity comparisons, is known at 3.5PN order and reads [74*, 197]

The other modes are known with a precision consistent with 3PN order in the full waveform [74]:

Notice that the modes with are zero except for the DC (zero-frequency) non-linear memory contributions. We already know that this effect arises at Newtonian order [see Eq. (322a)], hence the non zero values of the modes and . See Ref. [189] for the DC memory contributions in the higher modes having .

With the 3PN approximation all the modes with can be considered as merely Newtonian. We give here the general Newtonian leading order expressions of any mode with arbitrary and non-zero (see the derivation in [272]):

in which we employ the function , also given by Eqs. (305).