Gravitational waves contain tail effects that are due to the backscattering
of linear waves in the curved space-time geometry around the source. The
knowledge as well as the accuracy of the two-body inspiraling post-Newtonian
(PN) dynamics and of the gravitational-wave signal has been recently improved,
notably by computing the spin-orbit (SO) terms induced by tail effects in the
gravitational-wave energy flux at the 3PN order. Here we sketch this
derivation, which yields the phasing formula including SO tail effects through
the same 3PN order. Those results can be employed to improve the accuracy of
analytical templates aimed at describing the whole process of inspiral, merger,
and ringdown.Comment: 6 pages; proceeding of the 9th LISA Symposium, Pari

Blanchet, Luc

Buonanno, Alessandra

Faye, Guillaume

English

arXiv

Gravitational waves contain tail effects that are due to the backscattering of linear waves in the curved space-time geometry around the source. The knowledge as well as the accuracy of the two-body inspiraling post-Newtonian (PN) dynamics and of the gravitational-wave signal has been recently improved, notably by computing the spin-orbit (SO) terms induced by tail effects in the gravitational-wave energy flux at the 3PN order. Here we sketch this derivation, which yields the phasing formula including SO tail effects through the same 3PN order. Those results can be employed to improve the accuracy of analytical templates aimed at describing the whole process of inspiral, merger, and ringdown

Blanchet, L.

Buonanno, A.

Faye, G.

MPG.PuRe

9th LISA Symposium, ParisASP Conference Series, Vol. 467G. Auger, P. Bine´truy and E. Plagnol, eds.c©2012 Astronomical Society of the PacificThird Post-Newtonian Spin-orbit Effect in the GravitationalRadiation Flux of Compact BinariesGuillaume Faye,1 Luc Blanchet,1 and Alessandra Buonanno21Institut d’Astrophysique de Paris, UMR 7095 CNRS,Universite´ Pierre &Marie Curie, 98bis boulevard Arago, 75014 Paris, France2Maryland Center for Fundamental Physics & Joint Space-Science Institute,Department of Physics, University of Maryland, College Park, MD 20742, USAAbstract. Gravitational waves contain tail effects that are due to the backscatteringof linear waves in the curved space-time geometry around the source. The knowledgeas well as the accuracy of the two-body inspiraling post-Newtonian (PN) dynamics andof the gravitational-wave signal has been recently improved, notably by computing thespin-orbit (SO) terms induced by tail effects in the gravitational-wave energy flux at the3PN order. Here we sketch this derivation, which yields the phasing formula includingSO tail effects through the same 3PN order. Those results can be employed to improvethe accuracy of analytical templates aimed at describing the whole process of inspiral,merger, and ringdown.1. IntroductionBy 2016 − 2018, the ground-based gravitational-wave detectors Virgo and LIGO will be up-graded to such a sensitivity that event rates for coalescing binary systems will increase by ap-proximately a factor one thousand, making likely the first detection of gravitational waves fromthose systems. The search for gravitational waves from coalescing binary systems and the ex-traction of source parameters require a rather accurate knowledge of the waveform of the incom-ing signal. The post-Newtonian (PN) expansion is the most powerful approximation scheme inanalytical relativity capable of describing the two-body dynamics and the gravitational-waveemission of inspiraling compact binary systems (Blanchet 2006).The presence of spin effects adds substantial complexity to the gravitational waveforms,making it indispensable to include them into search templates.We have recently improved (Blanchet et al. 2011) the knowledge and accuracy of thetwo-body inspiraling dynamics and gravitational-wave signal by computing the spin-orbit (SO)terms induced by tail contributions due to the back-scattering of linear waves in the curvedspace-time geometry around the source. Here we shall summarize this work, which is thecontinuation of Faye et al. (2006) and Blanchet et al. (2006) where we obtained the 2.5PN SOcontributions in the equations of motion and gravitational-wave energy flux.After briefly reviewing the post-Newtonian multipole moment formalism and discussingrelevant properties of tails, we describe how spin effects are included to our scheme and de-rive the binary’s evolution equations when black holes carry spins. Next, we investigate thetime evolution of the moving triad and solve the precessing dynamics at the relevant PN order.We then compute the tails, which depend on the recent past history of the source, restrictingourselves to quasi-circular adiabatic inspiral. At last, we derive the 3PN SO tail effects in theenergy flux and in the gravitational phasing.215216 Faye, Blanchet and Buonanno2. Wave generation formalismThe gravitational waveform hTTi j , generated by an isolated source described by a stress-energytensor T µν with compact support, and propagating in the asymptotic regions of the source, isthe transverse trace-free (TT) projection of the metric deviation at the leading-order 1/R in thedistance to the source. It is parametrized by symmetric trace-free (STF) mass-type momentsUL and current-type ones VL, referred to as radiative moments, which constitute observablequantities at infinity from the source. The general expression of the TT waveform, in a suitableradiative coordinate system Xµ = (c T,X), reads (Thorne 1980), when neglecting termsO(1/R2)with R = |X|hTTi j =4Gc2RPTTi jkl+∞∑ℓ=2NL−2cℓℓ![UklL−2 − 2ℓc(ℓ + 1) Nm εmn(k Vl)nL−2]. (1)The moments UL and VL are functions of the retarded time TR ≡ T − R/c. The integer ℓ refersto the multipolar order and N = X/R is the unit vector pointing from the source to the far awaydetector. The tensor PTTi jkl denotes the transverse-traceless (TT) projector Pi(kPl) j − Pi jPkl/2,where Pi j = δi j − NiN j is the projector orthogonal to N. The quantity εi jk is the Levi-Civitasymbol such that ε123 = 1. Round parentheses indicate symmetrization. After plugging Eq. (1)into the standard expression for the gravitational-wave energy flux, we getF =+∞∑ℓ=2Gc2ℓ+1(ℓ + 1)(ℓ + 2)(ℓ − 1)ℓ ℓ!(2ℓ + 1)!![U (1)L U(1)L +4ℓ2c2(ℓ + 1)2 V(1)L V(1)L], (2)where the superscript (1) stands for the first time derivative (Thorne 1980).In the multipolar-post-Minkowskian formalism, developed by Blanchet & Damour (1986,1988, 1992), the radiative moments are linked to six sets of multipole moments characterizingthe source, collectively called the “source” moments and denoted IL, JL, WL, XL, YL, ZL. Therelation between UL, VL and the former quantities encode all the non-linearities in the wavepropagation between the source and the detector. After being re-expanded in a PN way, they areseen to contain the contribution of the so-called gravitational-wave tails, due to backscatteringof linear waves onto the space-time curvature associated with the total mass of the source itself.The expression for UL(TR) at the 1.5PN order, where the tail effects first appear, isUL(TR) = I(ℓ)L (TR) +2GMc3∫ TR−∞dt I(ℓ+2)L (t) ln(TR − t2τ′0)+ O( 1c3)non−tail . (3)Here τ′0 is a freely specifiable time scale (Blanchet 1995). A similar equation links VL(TR) to JL.For the present application, WL, XL, YL and ZL, associated to a possible gauge transformationperformed at linear order, play no role. It will be sufficient to consider IL at 1.5PN and JL at0.5PN order. The 1.5PN mass moments are given by an integral extending over the mass densityσ ≡ (T 00+T ii)/c2 and the current density σi ≡ T 0i/c of the matter source (Blanchet et al. 2006):IL =∫d3x[xˆLσ +12c2(2ℓ + 3) xˆL|x|2 σ(2) − 4(2ℓ + 1)c2(ℓ + 1)(2ℓ + 3) xˆiL σ(1)i]+ O( 1c4). (4)The 0.5PN part of JL is obtained by substituting εab〈iℓ xL−1〉a σb to xˆL σ in the first term.To control the past behavior of the tail integral in Eq. (3), we assume that, at early times,the source was formed from a bunch of freely falling particles moving on some hyperbolic-like orbits, and forming at a later time a gravitationally bound system by emission of grav-itational radiation. This ensures that the integral UL(TR) ≡∫ TR−∞ dt I(ℓ+2)L (t) ln[(TR − t)/2τ′0]is convergent. To compute it, it is convenient to perform the Fourier decomposition IL(t) =∫+∞−∞ dΩ/(2π) I˜L(Ω) e−iΩt, and commute the tail integral with the Fourier one. We obtain aThird Post-Newtonian Spin-orbit Effect 217closed-form expression by resorting to standard mathematical formulae for the integrals ofe−iΩt ln t. The result reads (with s(Ω) ≡ Ω|Ω| and γE being the Euler constant)UL(TR) = i∫+∞−∞dΩ2π(−iΩ)ℓ+1 I˜L(Ω) e−iΩTR[π2s(Ω) + i(ln(2|Ω|τ′0) + γE)]. (5)3. Applications to spinning binaries3.1. Spin vectors for point-like objectsFollowing our previous works (Faye et al. 2006; Blanchet et al. 2006), we base our calculationson the model of point-particles with spins (Mathisson 1937; Papapetrou 1951; Tulczyjew 1959;Bailey & Israel 1980). The stress-energy tensor T µν of a system of spinning particles is the sumof a monopolar piece, made of Dirac delta-functions, plus the dipolar or spin piece, made ofgradients of delta-functions:T µν = c2∑A∫+∞−∞dτA{pµAuνAδ(4)(x − yA)√−gA− 1c∇ρ[S ρµA uνAδ(4)(x − yA)√−gA]}, (6)where δ(4) is the four-dimensional Dirac function, xµ is the field point, yµA is the world-line ofparticle A, uµA = dyµA/(cdτA) is the four-velocity such that gAµνuµAuνA = −1 (with gAµν ≡ gµν(yA)denoting the metric at the particle’s location), pµA is the linear momentum of the particle, andS µνA denotes its antisymmetric spin angular momentum. Our spin variables has been rescaled bya factor c (S µν = c S µνtrue) so as to have a non-zero Newtonian limit for fast rotating compact ob-jects. Imposing that the antisymmetric part of T µν should vanish yields the covariant equationsof evolution for the spin, while the covariant equations of motion follow from the conservationrelation ∇νT µν = 0 (Papapetrou 1951).In order to fix unphysical degrees of freedom associated with an arbitrariness in the defini-tion of S µν in the case of point particles (due to the freedom in the choice of the location of thecenter-of-mass worldline within the extended bodies), we adopt the covariant supplementaryspin condition S µνA pAν = 0 (Tulczyjew 1959). We restrict the computation to linear terms in thespins, neglecting O(S 2A). Then the mass mA ≡ (−pµA pAµ/c2)1/2 is conserved and S µνA is paralleltransported along the worldline of body A. As is standard, we introduce a spin vector variableS iA with constant magnitude (see e.g., Kidder 1995), for instance by projecting S µνA onto someorthonormal basis of the A-particle rest space and taking the Hodge dual with εi jk.The post-Newtonian expressions of the densities σ, σi entering the source moments (4),as well as the stress density σi j ≡ T i j, are obtained iteratively from the truncated componentsof the stress-energy tensor (6). They are used in turn to compute the metric at the next iterationstep of the perturbative scheme.Inserting the former matter densities into Eq. (4) (or its counterpart for VL), we see thatspins arise at 1.5PN order in IL and 0.5PN order in JL. By virtue of Eq. (2), leading SO termsin the gravitational-wave flux are thus seen to be of half-integer PN order. However, due tothe 1/c3 factor in front of the tail integrals, as shown in Eq. (3), terms of integer orders mayalso appear: (i) when the source moments are contained in the tail integrands, or (ii) when theyare multiplied by non-spin tail integrals. The leading 3PN terms coming from Ui j and Vi j areprecisely those we want to compute.3.2. 1.5PN dynamics with spin-orbit effectsTo reduce the accelerations generated by the time differentiations of Ii j and Ji j and, most im-portantly, to find the time dependence of I(4)i j and J(4)i j required for calculating integrals such asthe one in Eq (3), we shall rely on the 1.5PN dynamics. The corresponding PN equations ofevolution at this order are derived by inserting the 1.5PN metric into the covariant equations and218 Faye, Blanchet and BuonannodiscardingO(1/c4) remainders. Going to the center-of-mass frame, we denote by x = y1−y2 therelative position of the particles (and r = |x|, v = dx/dt). Following Kidder (1995) we introducean orthonormal moving triad {n, λ, ℓ} defined by n = x/r, ℓ = LN/|LN|, where LN ≡ mν x × vdenotes the Newtonian orbital angular momentum (with m = m1 + m2 and ν = m1m2/m2), andλ = ℓ × n. For general orbits, the time derivatives of those vectors may be written in the formdndt = ωλ,dλdt = −ωn − ωprec ℓ ,dℓdt = ωprec λ. (7)The angular frequency ω at which the separation n rotates in the instantaneous orbital plane isthe so-called orbital frequency, while the third equation (7) defines the precessional frequencyωprec, which gives thus the variation of ℓ in the direction of λ.Instead of writing the equations of motion in the form of a link between dv/dt and thekinematic variables, we give the expressions of ω, ωprec. Focusing from now on to the case ofquasi-circular motion, for which r is constant apart from the 2.5PN secular radiation damping,we have (see e.g., Faye et al. 2006)ω2 =G mr3{1 + γ (−3 + ν) + γ3/2 (−5sℓ − 3δσℓ)}+ O( 1c4), (8)ωprec = −ωγ3/2(7sn + 3δσn)+ O( 1c4), (9)where the dimensionless spin variables are defined by s = (S1 + S2)/(Gm2) and σ = (S2/m2 −S1/m1)/(Gm), whereas δ ≡ X1 − X2, sℓ ≡ s · ℓ and sn ≡ s · n; γ ≡ Gm/(rc2) is the PN parameter.The spin vectors SA satisfy the usual-looking precession equationsdSAdt = ΩA × SA , with Ω1 = ωγ[34+ν2− 34δ]ℓ+ O( 1c4), (10)showing that the spins, at the 1.5PN order, precess around the direction of ℓ, and at the quasi-constant rate ΩA = |ΩA|. To obtainΩ2 we simply have to change δ into −δ.4. Computation of the tailsThe only time dependence in the source moments composing the tail integrals occurs throughthe triad vectors and the spins at the 1.5PN order. We adopt the following strategy to integrateEqs. (7): (i) we express the three moving triad vectors in terms of the Euler angles that link thelatter triad to some orthonormal fixed triad whose third vectors points to the direction of thetotal angular momentum at some time t0; (ii) we express the appropriate brick combinations ofthose angles as a function of S An , S Aλ , S Aℓ and φ ≡∫dtω(t); (iii) we solve for the precessionequations (10) in the triad basis and insert the results into the formulae we got for n, λ, ℓ in theprevious step.Although the binary continuous spectrum of frequencies of IL(t) should contain all orbitalfrequencies at any epoch in the past, say ω(t) with t ≤ TR, differing from the current orbitalfrequency ω(TR) due to gravitational radiation damping, we can actually compute the tail inte-gral by considering only the current frequency ω(TR), modulo small error terms of negligibleorder O(ln c/c5). Similarly, we can regard the precession frequenciesΩA(t) of the two spins asconstant in the calculation. Thus, we may replace φ(t) by ω(t − t0) + φ(t0) and ΩA(t) by ΩA(TR)in the tail integrand. This takes the form of a sum of complex exponentials of the type eiωnpqt,with ωnpq = nω+ pΩ1 + qΩ2, times certain coefficients AnpqL . It is then straightforward to takethe Fourier transform of the source moments, e.g. IL(t), and apply formula (5), which leads toUL(TR) =∑n,p,qi AnpqL (−iωnpq)ℓ+1e−iωnpqTR[π2s(ωnpq) + i(ln(2|ωnpq|τ′0) + γE)]. (11)Third Post-Newtonian Spin-orbit Effect 2195. Energy flux and orbital phasingInserting the explicit expressions for Ui j and Vi j into the energy flux (2) yields the net SO tailcontribution at 3PN order, in terms of the gauge-invariant PN parameter x ≡ (Gmω/c3)2/3 andthe spins. The SO terms have to be included in either the instantaneous moments in front of thetail integrals, or in the tail integrals themselves. This gives several “direct” SO contributionscoming from tails at relative 1.5PN order (for the mass quadrupole tail) or 0.5PN order (forthe current quadrupole tail) which are then added together. In addition there is the crucialcontribution due to the reduction to circular orbits of the standard (non-spin) tail integral at1.5PN order, for which the relation between the orbital separation r and the orbital frequency ω[as given by the inverse of Eq. (8)] provides a supplementary SO term at relative 1.5PN order,which contributes in fine at the same 3PN level as the “direct” SO tail terms. Finally we obtainδF = 325c5Gx8 ν2[−16π sℓ − 31π6 δ σℓ], (12)Using an energy balance argument, we finally equate the averaged evolution of the bindingenergy of the binary reduced for quasi-circular orbits dE(x, sℓ, σℓ)/dt (see e.g., Blanchet et al.2006) to −F (x, sℓ, σℓ), where F is the sum of the 2PN SO flux obtained by Kidder (1995), the3PN non-spin contribution investigated by Blanchet et al. (2002, 2004) and the correction (12)we just calculated. The time derivative applied to E acts only to the frequency variable since, aswe have checked, sℓ and σℓ are secularly conserved (neglecting SS contributions). We find thefollowing secular variation of frequency, denoted ω˙ for simplicity, at the 3PN order includingSO effects:ω˙ω2=965 ν x5/2{1 + x(−743336 −114ν)+ x3/2(4π − 473 sℓ −254δσℓ)+x2(3410318144 +136612016 ν +5918ν2)+x5/2(−4159672 π −5861144sℓ − 80984 δσℓ + ν[−1898 π +100112sℓ +2818 δσℓ])+x3(16447322263139708800 +163 π2 − 1712105 γE −856105 ln(16 x) −188π3 sℓ −151π6 δσℓ+ν[−56198689217728 +45148 π2]+541896ν2 − 56052592ν3)}. (13)Integrating this, we get the “carrier” phase φ(t) =∫ω dt. The total phase Φ = φ − α + O(1/c4),where α is the precessional phase, can be computed numerically or manually.ReferencesBailey, I., & Israel, W. 1980, Ann. Phys., 130, 188Blanchet, L. 1995, Phys. Rev. D, 51, 2559— 2006, Living Rev. Rel., 9, 4. gr-qc/0202016Blanchet, L., Buonanno, A., & Faye, G. 2006, Phys. Rev. D, 74, 104034. Erratum Phys. Rev. D,75:049903(E), 2007— 2011, Phys. Rev. D, 84, 064041Blanchet, L., & Damour, T. 1986, Phil. Trans. Roy. Soc. Lond. A, 320, 379— 1988, Phys. Rev. D, 37, 1410— 1992, Phys. Rev. D, 46, 4304Blanchet, L., Damour, T., Esposito-Fare`se, G., & Iyer, B. R. 2004, Phys. Rev. Lett., 93, 091101Blanchet, L., Faye, G., Iyer, B. R., & Joguet, B. 2002, Phys. Rev. D, 65, 061501(R). ErratumPhys. Rev. D, 71:129902(E), 2005Faye, G., Blanchet, L., & Buonanno, A. 2006, Phys. Rev. D, 74, 104033220 Faye, Blanchet and BuonannoKidder, L. E. 1995, Phys. Rev. D, 52, 821Mathisson, M. 1937, Acta Phys. Polon., 6, 163Papapetrou, A. 1951, Proc. R. Soc. London A, 209, 248Thorne, K. 1980, Rev. Mod. Phys., 52, 299Tulczyjew, W. 1959, Acta Phys. Polon., 18, 393

Third post-Newtonian spin-orbit effect in the gravitational radiation flux of compact binaries

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