Meudon Cedex, France. Institut des Hautes Etudes Scientiques, Bures sur Yvette, France. St. Louis, Missouri (January 25, 1995) Abstract

Transcription

1 Gravitational-Radiation Daping of Copact Binary Systes to Second Post-Newtonian order Luc Blanchet, Thibault Daour 2;, Bala R. Iyer 3, Cliord M. Will 4, and Alan G. Wisean 5 Departeent d'astrophysique Relativiste et de Cosologie y, Observatoire de Paris, 9295 Meudon Cedex, France 2 Institut des Hautes Etudes Scientiques, 9440 Bures sur Yvette, France 3 Raan Research Institute, Bangalore , India 4 McDonnell Center for the Space Sciences, Departent of Physics, Washington University, St. Louis, Missouri Departent of Physics and Astronoy, Northwestern University, Evanston, Illinois (January 25, 995) Abstract The rate of gravitational-wave energy loss fro inspiralling binary systes of copact objects of arbitrary ass is derived through second post-newtonian (2PN) order O[(G=rc 2 ) 2 ]beyond the quadrupole approxiation. The result has been derived by two independent calculations of the (source) ultipole oents. The 2PN ters, and in particular the nite ass contribution therein (which cannot be obtained in perturbation calculations of black hole GR-QC spaceties), are shown to ake a signicant contribution to the accuulated phase of theoretical teplates to be used in atched ltering of the data fro future gravitational-wave detectors w, Nn, Jd, Lf Typeset using REVTEX

2 One of the ost proising astrophysical sources of gravitational radiation for detection by large-scale laser-interferoeter systes as the US LIGO or the French-Italian VIRGO projects [] is the inspiralling copact binary. This is a binary syste of neutron stars or black holes whose orbit is decaying toward a nal coalescence under the dissipative eect of gravitational radiation reaction. For uch of the evolution of such systes, the gravitational wavefor signal is an accurately calculable \chirp" signal that sweeps in frequency through the detectors' sensitive bandwidth, typically between 0 Hz and 000 Hz. Estiates of the rate of such inspiral events range fro 3 to 00 per year, for signals detectable out to hundreds of Mpc by the advanced version of LIGO [2]. In addition to outright detection of the waves, it will be possible to deterine paraeters of the inspiralling systes, such as the asses and spins of the bodies [3{5], to easure cosological distances [6], to probe the non-linear regie of general relativity [7], and to test alternative gravitational theories [8]. This is ade possible by the technique of atched ltering of theoretical wavefor teplates, which depend on the source paraeters, against the broad-band detector output [9]. Roughly speaking, any eect that causes the teplate to dier fro the actual signal by one cycle over the 500 to 6,000 accuulated cycles in the sensitive bandwidth will result in a substantial reduction in the signal-to-noise ratio. This necessitates knowing the prediction of general relativity for gravitational radiation daping, and its eect on the orbital phase, to substantially higher accuracy than that provided by the lowest-order quadrupole, or Newtonian approxiation. If post-newtonian corrections to the quadrupole forula scale as powers of v 2 =r (G = c = ), then, say, for a double neutron-star inspiral in the LIGO/VIRGO bandwidth, with =r typically around 0 2, corrections at least of order (=r) will be needed in order to be accurate to one cycle out of the 6,000 cycles accuulated for this process. This corresponds to corrections of second post-newtonian (2PN) order. Although nuerous corrections to the quadrupole energy-loss forula have previously been calculated (for a suary, see [0]), the 2PN contributions to the energy loss rate 2

3 for arbitrary asses have not been derived. This paper presents these contributions for the rst tie, discusses their signicance for gravitational-wave data analysis, and outlines the derivation. The central result is this: through 2PN order, the rate of energy loss, de=dt, fro a binary syste of two copact bodies of ass and 2, orbital separation r, and spins S and S 2, in a nearly circular orbit (apart fro the adiabatic inspiral) is given by de dt = r + r + r = r! 2 i 73 2 i ^L i ^L ^L 2 ; () where ^L is a unit vector directed along the orbital angular oentu, = + 2, = 2 = 2, = S = 2, 2 = S 2 = 2 2 and i denotes the su over i =;2. The ters in square brackets in Eq. () are respectively: at lowest order, Newtonian (quadrupole); at order =r, PN []; at order (=r) 3=2, the non-linear eect of \tails" (4 ter) [2,3], and spin-orbit eects [4,5]; and at order (=r) 2, the 2PN ters (new with this paper), and spin-spin eects [4{6]. For the special case of a test ass orbiting a assive black hole, perturbation theory has been used to derive an analogous analytic forula (apart fro spin-spin eects) [2,7], and for non-rotating holes, to extend the expansion through the equivalent of 4PN order [8]. The test-body = 0 liit of Eq. () agrees copletely with these results to the corresponding order. The equations of otion for circular orbits, correct to 2PN order including spin eects, yield for the orbital angular velocity! v=r and the orbital energy E [9,4,0] 3=2! i 2 2 i 2 +3! 2 = r 3 r (3 ) r ^L i + r

4 3 2 E = 2 2r 2 4 r (7 )+ r ^L i r 3^L ^L 2 ; (2a) 3=2i 2 8 ( ) 2 2 i 2+! 3^L ^L2 : (2b) Cobining Eqs. () and (2), one can express the rate of change _! of the angular velocity as a function of!, and get _! = =3! = (!) 2=3 +(4 )(!) (!) 4=3 ; (3) where the spin-orbit () and spin-spin () paraeters are given by = 2 i(3 2 i =2 + 75)^L i, and =(=48)( ^L ^L 2 ). Fro that one calculates the accuulated nuber of gravitational-wave cycles N = R (f= _ f)df, where f =!= is the frequency of the quadrupolar waves, in ters of the frequencies at which the signal enters and leaves the detectors' bandwidth. In order to avoid coplications caused by spininduced precessions of the orbital plane [20,5], we assue that the spins are aligned parallel to the orbital angular oentu (in particular and reain constant). Using 0 Hz as the entering frequency of LIGO/VIRGO-type detectors, set by seisic noise, and, as the exit frequency, the saller of either 000 Hz (set by photon-shot noise) or the frequency corresponding to the innerost stable circular orbit and the onset of plunge [for sall ass ratio, f ax ==(6 3=2 ) [2]], we nd contributions to the total nuber of observed wave cycles fro the various post-newtonian ters listed in Table. Because for black holes, and < 0:63 0:74 for neutron stars (depending on the equation of state, see [22]), and are always less than 9:4 and 2:5, respectively. However, if we consider odels for the past and future evolution of observed binary pulsar systes such as PSR and PSR 93+6, we nd (using a conservative upper liit 4

5 for oents of inertia) that < 5:2 0 3, 93+6 < 6:5 0 3, and we expect 2 < If such values are typical, both the spin-orbit and (a fortiori) the spin-spin ters will ake negligible contributions to the accuulated phase. Table deonstrates that the 2PN ters, and notably the nite-ass (-dependent) contributions therein (which cannot be obtained by test-body approaches), ake a signicant contribution to the accuulated phase, and thus ust be included in theoretical teplates to be used in atched ltering. The additional question of how the presence of these ters will aect the accuracy of estiation of paraeters in the teplates can only be answered reliably using a full atched lter analysis [4,5]. This is currently in progress [23]. The reainder of this paper outlines the derivations leading to this result. Two entirely independent calculations were carried out, using dierent approaches, one by BDI, using their previously developed generation foralis [24,25], the other by WW, using a foral slow-otion expansion originated by Epstein and Wagoner [26]. Details of these calculations will be published elsewhere [27]. Both approaches begin with Einstein's equations written in haronic coordinates (see [25] for denitions and notation). We dene the eld h, easuring the deviation of the \gothic" etric fro the Minkowski etric = diag( ; ; ; ): h = p gg. (Greek indices range fro 0 to 3, while Latin range fro to 3). Iposing the haronic coordinate h = 0 then leads to the eld equations 2h =6( g)t + (h) 6 ; (4) where 2 denotes the at spacetie d'alebertian operator, T is the atter stress-energy tensor, and is an eective gravitational source containing the non-linearities of Einstein's equations. It is a series in powers of h and its derivatives; both quadratic and cubic nonlinearities in play an essential role in our calculations. Post-Minkowski atching approach (BDI). This approach proceeds through several steps. The rst consists of constructing an iterative solution of Eq. (4) in an inner doain (or near zone) that includes the aterial source but whose radius is uch less than a gravitational 5

6 wavelength. Dening source densities = T 00 + T kk, i = T 0i, ij = T ij, and the retarded potentials V = 42 R, V i = 42 R i, and W ij = 42 R [ ij +(4) (@ i V@ j V 2 ij@ k V@ k V)], where 2 R denotes the usual at spacetie retarded integral, one obtains the inner etric h in to soe interediate accuracy O(6; 5; 6): h 00 in = 4V +4(W ii 2V 2 )+O(6), h 0i in = 4V i + O(5), h ij in = 4W ij + O(6), where O(n) eans a ter of order " n in the post- Newtonian paraeter " v=c. Fro this, one constructs the inner etric with the higher accuracy O(8; 7; 8) needed for subsequent atching as h in = 2 R [6 (V;W)]+O(8; 7; 8), where (V;W) denotes the right-hand-side of Eq. (4) when retaining all the quadratic and cubic nonlinearities to the required post-newtonian order in the near zone, and given as explicit cobinations of derivatives of V;V i and W ij. The second step consists of constructing a generic solution of the vacuu Einstein equations (Eq. (4) with T = 0), in the for of a ultipolar-post-minkowskian expansion that is valid in an external doain which overlaps with the near zone and extends into the far wave zone. The construction of h in the external doain is done algorithically as a functional of a set of paraeters, called the \canonical" ultipole oents M i :::il (t), S i :::il(t) which are syetric and trace-free (STF) Cartesian tensors. Scheatically, h ext = F [M L ;S L ] where L i :::i l and where the functional dependence includes a non-local tie dependence on the past \history" of M L (t) and S L (t). The third, \atching" step consists of requiring that the inner and external etrics be equivalent (odulo a coordinate transforation) in the overlap between the inner and the external doains. This requireent deterines the relation between the canonical oents and the inner etric (itself expressed in ters of the source variables). Perforing the atching through 2PN order [25] thus deterines M L (t) =I L [ ]+O(5), S L (t) = J L [ ]+O(4), where the \source" oents I L and J L are given by soe atheatically well-dened (analytically continued) integrals of the quantity (V;W) which appeared as source of h in. When coputing the source oents we neglect all nite size eects, such as spin (which to 2PN accuracy can be added separately) and internal quadrupole eects. The nal result for the 2PN quadrupole oent reads, in the case of a circular binary, 6

7 I ij = STF ij [Ar i r j + Br 2 v i v j ]; (5) with A = 42 (=r)( + 39) 52 (=r)2 ( ), B = 2 ( 3) (=r)( ). The nal step consists of coputing fro the external etric h ext the gravitational radiation eitted at innity. This entails introducing a (non haronic) \radiative" coordinate syste X =(T;X i ) adapted to the fall-o of the etric at future null innity. The transverse-traceless (TT) asyptotic wavefor h TT ij can be uniquely decoposed into two sets of STF \radiative" ultipole oents U L, V L which are then coputed as soe non linear functionals of the canonical oents, and therefore of the source ultipole oents. For instance, up to O(5), Z U ij (T )=I (2) T! ij (T )+2 dt 0 ln T T 0 I (4) ij (T 0 ) ; (6) 2b which contains a non-local \tail" integral (in which b = be =2 where b is a freely speciable paraeter entering the coordinate transforation x! X : T R = t jxj 2ln(jxj=b)). The superscript (n) denotes n tie derivatives. The energy loss is given by integrating the square TT ij =@T over the sphere at innity. At 2PN order this leads to de dt = 5 U () ij U () ij + 89 U () ijk U () + 6 ijk 45 V () ij V () ij U () ijk U () ijk + 84 V () ijk V () ijk : (7) Inserting the 2PN expression (5) of the source quadrupole into the radiative quadrupole (6), and using the previously derived PN expressions for the other ultipole oents, we end up with the energy loss (). Epstein-Wagoner approach (WW). This approach starts by considering h = 2 R (6 ) as a foral solution of Eq. (4) everywhere. One then expands the retarded integral 2 R to leading order in =R in the far zone, while the retardation is expanded in a slow-otion approxiation. Using identities, such as ij = ( 00 x i x j )=@t 2 + spatial divergences, which result = 0 (a consequence of the haronic gauge condition), we express the radiative eld as a sequence of Epstein-Wagoner ultipole oents, 7

8 h ij d 2 TT = 2 R dt 2 X =0 n k :::n k I ijk :::k EW (t R) TT ; (8) where I ijk :::k EW are integrals over space of oents of the source (e.g. I ij EW = R 00 x i x j d 3 x; see [26,28] for forulae). To sucient accuracy for the radiative eld, the 2-index oent ust be calculated to 2PN order, the 3 and 4-index oents to PN order, and the 5 and 6-index oents to Newtonian order. The oents of the copact-support source distribution ( g)t are straightforward. Contrary to what happens in the BDI calculation where the atching leads to atheatically well-dened forulas for the source ultipole oents, the EW oents of the non-copact source are given by forally divergent integrals. To deal with this diculty we dene a sphere of radius Rr=" centered on the center of ass of the syste, and integrate the non-copact oents within the sphere. Many integrations by parts are carried out to siplify the calculations, and the resulting surface ters are evaluated at R and kept. The divergent ters are proportional to R, and signal the failure of the slow-otion expansion procedure extended into the far zone. We discard the divergent ters. In order to copare directly with BDI, we transfor the EW oents into STF oents using the projection integrals given by Thorne [29]. For the quadrupole oent, for instance, that transforation is given by I ij = STF ij [IEW ij + ijkk 2 (IEW 2IEW kijk +4IEW kkij )+ 63 ijkkll (23IEW 32I kijkll EW +0IEW kkijll +2IEW klklij )]. We nd that those STF oents agree exactly with the \source" oents of BDI, e.g. Eq. (5). Note that the foral EW approach isses the tail eects in the wavefor (see (6)). They ust be added separately. ACKNOWLEDGMENTS This work is supported in part by CNRS, the NSF under Grant No (Washington University), and NASA under Grants No. NAGW 3874 (Washington University), NAGW 2936 (Northwestern University) and NAGW 2897 (Caltech). BRI acknowledges the hospitality of IHES. 8

9 REFERENCES Present address: Theoretical Astrophysics 30-33, California Institute of Technology, Pasadena, California 925 y Unite Propre de Recherche no. 76 du Centre National de la Recherche Scientique [] A. Abraovici, et al., Science 256, 325 (992); C. Bradaschia, et al., Nucl. Instru. & Methods A289, 58 (990). [2] R. Narayan, T. Piran, and A. Shei, Astrophys. J. 379 L7 (99); E. S. Phinney, ibid. 380, L7 (99). [3] C. Cutler, et al., Phys. Rev. Lett. 70, 2984 (993). [4] L. S. Finn, Phys. Rev. D 46, 5236 (992); L. S. Finn and D. F. Cherno, ibid. 47, 298 (993). [5] C. Cutler and E. Flanagan, Phys. Rev. D 49, 2658 (994). [6] B. F. Schutz, Nature 323, 30 (986). [7] L. Blanchet and B. S. Sathyaprakash, Phys. Rev. Lett. (in press). [8] C. M. Will, Phys. Rev. D 50, 6058 (994). [9] K. S. Thorne, in 300 Years of Gravitation, edited by S.W.Hawking and W. Israel (Cabridge University Press, Cabridge, 987), p. 330; B. F. Schutz, in The Detection of Gravitational Waves, edited by D. G. Blair (Cabridge University Press, Cabridge, 99), p [0] C. M. Will, in Relativistic Cosology, edited by M. Sasaki (Universal Acadey Press, Tokyo, 994), p. 83. [] R. V. Wagoner and C. M. Will, Astrophys. J. 20, 764 (976); ibid 25, 984 (977). [2] E. Poisson, Phys. Rev. D 47, 497 (993). 9

10 [3] L. Blanchet and T. Daour, Phys. Rev. D 46, 4304 (992); A. G. Wisean, ibid. 48, 4757 (993); L. Blanchet and G. Schafer, Class. Quantu Grav. 0, 2699 (993). [4] L. E. Kidder, C. M. Will, and A. G. Wisean, Phys. Rev. D 47, R483 (993). [5] L. E. Kidder, subitted to Phys. Rev. D. [6] In the presence of spin-dependent interactions, a circular orbit only exists as an approxiation; the forulas presented here are averaged over one orbit. The only other eects that ight contribute are nite-size, quadrupole eects; these would be signicant only for fast-rotating copact bodies, for which they scale as (R 0 =r) 2 (=r) 2, i.e. 2PN relative to Newtonian order, where R 0 is the size of the copact body. However they enter via the equations of otion, not via the radiation eld at 2PN order. [7] E. Poisson, Phys. Rev. D 48, 860 (993). [8] H. Tagoshi and T. Nakaura, Phys. Rev. D 49, 406 (994); H. Tagoshi and M. Sasaki, Prog. Theor. Phys. 92, 745 (994). [9] T. Daour and N. Deruelle, C.R. Acad. Sc. Paris 293, 877 (98); T.Daour, ibid. 294, 355 (982); T.Daour and G. Schafer, Nuovo Ciento 0B, 27 (988). [20] T. A. Apostolatos, et al., Phys. Rev. D 49, 6274 (994). [2] For plunge frequencies for other ass ratios, see L. Kidder, C. M. Will and A. G. Wisean, Phys. Rev. D 47, 328 (993). [22] M. Salgado, et al., Astron. Astrophys. 29, 55 (994); Astron. Astrophys. Suppl. 08, 455 (994); G. B. Cook, et al., Astrophys. J. 424, 823 (994). [23] E. Poisson and C. M. Will, private counication. 0

11 [24] L. Blanchet and T. Daour, Phil. Trans. R. Soc. London A 320, 379 (986); Phys. Rev. D 37, 40 (988); Ann. Inst. H. Poincare (Phys. Theorique) 50, 377 (989); L. Blanchet, Proc. R. Soc. Lond. A 409, 383 (987); T. Daour and B. R. Iyer, Ann. Inst. H. Poincare (Phys. Theorique) 54, 5 (99). [25] L. Blanchet, Phys. Rev. D, in press. [26] R. Epstein and R. V. Wagoner, Astrophys. J. 97, 77 (975). [27] L. Blanchet, T. Daour, and B. R. Iyer, subitted to Phys. Rev. D; C. M. Will and A. G. Wisean, in preparation. [28] A. G. Wisean, Phys. Rev. D 46, 57 (992). [29] K. S. Thorne, Rev. Mod. Phys. 52, 299 (980).

12 TABLES TABLE I. Contributions to the accuulated nuber N of gravitational-wave cycles in a LIGO/VIRGO-type detector. Frequency entering the bandwidth is 0 Hz (seisic liit); frequency leaving the detector is 000 Hz for 2 neutron stars (photon shot noise), and 360 Hz and 90 Hz for the two cases involving black-holes (innerost stable orbit). Spin paraeters and are dened in the text. Nubers in parentheses indicate contribution of nite-ass () eects. 2 :4M 0M +:4M 20M Newtonian 6, First PN 439(04) 22(26) 59(4) Tail Spin-orbit Second PN 9(3) 0(2) 4() Spin-spin 2 3 2