Radiation reaction in the 2.5PN waveform from inspiralling binaries in circular orbits
aa r X i v : . [ g r- q c ] J un COMMENT
Radiation reaction in the 2.5PN waveform frominspiralling binaries in circular orbits
Lawrence E. Kidder , Luc Blanchet , Bala R. Iyer Center for Radiophysics and Space Research,Cornell University, Ithaca, New York, 14853 U.S.A. G R ε C O , Institut d’Astrophysique de Paris —C.N.R.S.,98 bis boulevard Arago, 75014 Paris, France Raman Research Institute, Bangalore 560 080, IndiaE-mail: [email protected] , [email protected] , [email protected] Abstract.
In this Comment we compute the contributions of the radiation reactionforce in the 2.5 post-Newtonian (PN) gravitational wave polarizations for compactbinaries in circular orbits. (i) We point out and correct an inconsistency in thederivation of [1]. (ii) We prove that all contributions from radiation reaction in the2.5PN waveform are actually negligible since they can be absorbed into a modificationof the orbital phase at the 5PN order.PACS numbers: 04.25.Nx, 0.4.30.-w, 97.60.Jd, 97.60.Lf
Submitted to:
Class. Quantum Grav.
1. Introduction
The second and a half post-Newtonian (2.5PN ∼ c − ) waveform for inspiralling compactbinaries moving in quasi-circular orbits was computed by Arun et al [1]. Starting fromthe expressions of the radiative multipole moments given by Eqs. (3.4)–(3.6) of [1],and of the source moments given by (3.16)–(3.18) of [1], the wave form is made of theinstantaneous terms (Eqs. (5.1)–(5.4) of [1]), the hereditary memory-type contributions(Eqs. (4.24) of [1]), and the tail contributions (Eqs. (4.38) corrected by the publishedErratum [1]). In this Comment we are concerned with the piece of the instantaneouswaveform that is the contribution from radiation reaction (RR) at 2.5PN order in thedynamics of the inspiralling compact binary. We point out and correct an inconsistencyin the derivation of RR terms in [1], but argue that in fact the RR terms are negligiblein the 2.5PN waveform, since they can be absorbed into a 5PN order contribution tothe orbital phase evolution. adiation reaction in the 2.5PN waveform
2. Quasi-circular inspiral at 2.5PN order
Adopting the conventions of [1], r = | x | is the binary separation with x = y − y the vectorial separation between the particles, v = v − v is the relative velocity, m = m + m is the total mass of the binary system (distinct from the mass monopolemoment M given by the ADM mass of the system), and ν = m m /m is the reducedmass divided by the total mass.In modeling the orbital motion of the binary at 2.5PN order, Ref. [1] stressedthe importance of including the radiation-reaction force in the 2.5PN expression of thebinary acceleration. Consider an arbitrary orbit confined to the x-y plane. The relativeposition, velocity, and acceleration are given by x = r n , (1) v = ˙ r n + rω λ , (2) a = (¨ r − rω ) n + ( r ˙ ω + 2 ˙ rω ) λ . (3)Here λ = ˆz × n where ˆz is the unit vector along the z-direction. The orbital frequency ω is related as usual to the orbital phase φ via ω = ˙ φ .Through 2PN order, it is possible to model the motion of the binary as a circularorbit with the solution ¨ r = ˙ r = ˙ ω = 0 and rω = − n · a . Detailed calculations of the2PN equations of motion for a circular orbit (in harmonic coordinates) yield ω = Gmr ( − ν ) Gmrc + (cid:18) ν + ν (cid:19) (cid:18) Gmrc (cid:19) ) . (4)At 2.5PN order, however, the effect of the inspiral motion must be taken intoaccount. The leading order contribution to the inspiral can be obtained by examiningthe Newtonian orbital energy, E = − Gm νr , (5)and the leading order gravitational-wave luminosity, − dEdt = 325 G m ν r c . (6)Here we assume that the energy radiated by the gravitational waves is balanced by thechange in the orbital energy. This yields˙ r = dE/dtdE/dr = − G m νr c . (7)Similarly the orbital frequency changes by (using Gm = r ω at leading order)˙ ω = dE/dtdE/dω = 965 Gmνr (cid:18) Gmrc (cid:19) / , (8)while the orbital phase φ = R ωdt , found by integrating (8), is φ = − ν (cid:18) Gmrc (cid:19) − / . (9) adiation reaction in the 2.5PN waveform r ∼ O ( c − ) is of the order ofthe square of RR effects, the expressions for the 2.5PN inspiral relative velocity andacceleration in harmonic coordinates are obtained, v = rω λ − G m νr c n , (10) a = − ω x − G m νr c v , (11)where ω is given by (4). As we shall detail the 2.5PN RR terms in both the inspiralvelocity (10) and acceleration (11) should be substituted into the gravitational waveformat 2.5PN order.
3. Computation of polarization waveforms
In the leading quadrupole approximation the gravitational waveform is given by h TT ij = 2 Gc R P ijkl ( N ) h ¨ I kl + O ( c − ) i , (12)where R is the distance to the observer and P ijkl is the transverse-traceless (TT)projection operator P ijkl = P ik P jl − P ij P kl , with P ij = δ ij − N i N j and N = ( N i ) theradial direction from the source to the observer. Here ¨ I kl is the second time-derivativeof some appropriate source quadrupole moment defined from a general post-Newtonianmultipole moment formalism. The remainder O ( c − ) indicates higher PN correctionscoming notably from all the higher multipolar orders.Defining an orthonormal triad ( N , P , Q ) where P and Q are unit polarizationvectors transverse to the direction of propagation, the polarization waveforms h + and h × are computed from the waveform (12) by h + = ( P i P j − Q i Q j ) h TT ij , (13) h × = ( P i Q j + P j Q i ) h TT ij . (14)If the orbital plane is chosen to be the x-y plane with the orbital phase φ measuring thedirection of the unit vector n = x /r along the relative separation vector, then n = cos φ ˆx + sin φ ˆy . (15)Ref. [1] has chosen the polarization vector P to lie along the x-axis and the observer tobe in the y-z plane with N = sin i ˆy + cos i ˆz , (16)where i is the orbit’s inclination angle (0 ≤ i ≤ π ). With these definitions P lies alongthe intersection of the orbital plane with the plane of the sky in the direction of theascending node, and the orbital phase φ is the angle between the ascending node andthe direction of body one (say). The rotating orthonormal triad ( n , λ , ˆz ) describing themotion of the binary is then related to the polarization triad ( N , P , Q ) by n = cos φ P + sin φ ( c i Q + s i N ) , (17) λ = − sin φ P + cos φ ( c i Q + s i N ) , (18) ˆz = − s i Q + c i N , (19) adiation reaction in the 2.5PN waveform c i = cos i and s i = sin i .
4. Radiation reaction contributions to the waveform
All contributions arising from the gravitational RR are contained in the leading orderquadrupolar waveform given by (12). They have three different origins. The first typeof RR term is issued directly from the expression of the (post-Newtonian) quadrupolemoment I ij at order 2.5PN, and reads (see Eq. (3.16a) of [1]) I ij = νmx + · · · + 487 G m ν rc x . (20)The first term is the usual expression of the Newtonian quadrupole moment (the bracketssurrounding indices refer to the symmetric-trace-free projection). The dots indicate the1PN and 2PN conservative terms, and terms strictly higher than 2.5PN. Taking twotime-derivatives of (20) we get¨ I ij = 2 νm ( v + x ) + · · · − G m ν r c x . (21)The second type of RR term comes from inserting the expression of the acceleration a given by (11), which produces another RR term which modifies the term in (21) as¨ I ij = 2 νm (cid:16) v − Gmr x (cid:17) + · · · − G m ν r c x . (22)The sum of these RR contributions is exactly what has been computed in [1], where itwas denoted by ρ (5 / ij and given by Eq. (5.2). We now replace (22) into the waveform (12)and compute the two GW polarizations according to (13)–(14), hence h + , × = 2 Gc R P i P j − Q i Q j P i Q j + P j Q i ! h νm (cid:16) v i v j − Gmr x i x j (cid:17) + · · · − G m ν r c x i v j i . (23)For ease of notation, in Eq. (23) above and similar equations later the first row (line)corresponds to + and the second row (line) to × . The RR term in (22) has alreadybeen included in the final result of [1]. Its contribution to the polarization waveforms isgiven by δ h + , × = 2 Gνmxc R c i ) νx / sin 2 φ, − c i ) νx / cos 2 φ, (24)where we pose x = ( Gmω/c ) / . With the notation δ we remind that this term wasmade of two distincts pieces.However, let us show there is also another contribution to RR that has beenoverlooked in [1]. Indeed the instantaneous terms in the waveform h TT ij (Eqs. (5.1)–(5.4) of [1]) are given in terms of the relative position n and velocity v of the binary,as well as the PN parameter γ = Gm/ ( rc ). In computing the polarization waveforms, i.e. projecting out h TT ij to get h + , × , Ref. [1] has substituted, at the last stage of the adiation reaction in the 2.5PN waveform v = rω λ in the waveform to obtain Eqs. (5.9)–(5.10) of [1]. This is correctfor all the terms except the two leading order “Newtonian” terms in (23) for which onemust use the true expression of the velocity (10) including RR inspiral. We find thatsubstituting the inspiral velocity (10) into the leading terms of the waveform (23) yieldsthe following additional contribution to the polarization waveforms, δ h + , × = 2 Gνmxc R
645 (1 + c i ) νx / sin 2 φ, −
645 (2 c i ) νx / cos 2 φ. (25)Notice that both results (24) and (25) have the same structure. Since thecontribution (25) has not been taken into account in [1] we find that the followingterms in the 2.5PN polarization waveforms of [1] (Eqs. (5)–(6) of the Erratum) are tobe changed from H (2 . , × | old = · · · + sin 2 ψ h −
95 + 145 c i + 75 c i + ν (cid:16) − c i − c i (cid:17)i + · · · , · · · + c i cos 2 ψ h − c i + ν (cid:16) − c i (cid:17)i + · · · , (26)to H (2 . , × = · · · + sin 2 ψ h −
95 + 145 c i + 75 c i + ν (cid:16)
32 + 565 c i − c i (cid:17)i + · · · , · · · + c i cos 2 ψ h − c i + ν (cid:16) − c i (cid:17)i + · · · . (27)We recall that the phase variable ψ differs from φ and is given by Eq. (5.6) of [1]. Thedifference between ψ and φ is at order 4PN at least (see [1] for discussion). Note that thenew correction (25) is in terms that vanish in the limit ν →
0, and so is still consistentwith the results of black hole perturbation theory [2].
5. Phase modulation due to radiation reaction
We are now going to show that all the RR contributions are in fact negligible in the2.5PN waveform. From the sum of the previous results (24) and (25) we end up withthe following total contribution due to RR in the GW polarizations: δ RR h + , × = 2 Gνmxc R c i ) νx / sin 2 φ, − c i ) νx / cos 2 φ. (28)Comparing this result with the GW polarizations at Newtonian order, h (N)+ , × = 2 Gνmxc R − (1 + c i ) cos 2 φ, − (2 c i ) sin 2 φ, (29) adiation reaction in the 2.5PN waveform φ + 807 νx / . (30)This means that the 2.5PN waveform will take exactly the same expression as if wehad neglected the RR contributions (28), but with phase variable Φ in place of φ . Nowthe point is that the added term in (30) represents an extremely small modification ofthe phasing. Indeed, we take into account the expression for the phase as a function offrequency at the leading order ( i.e. due to leading order RR effects), which has beencomputed in (9) and whose order ∼ c is the inverse of the order ∼ c − of RR effects.Thus we find that the redefined phase is equivalent toΦ = − x − / ν h · · · − ν x i . (31)This means that the RR terms seen as modulations of the phase evolution, contribute tothe phase much beyond the 2.5PN order, namely at order 5PN ∼ x beyond the leadingphase evolution. Indeed there is in principle no point in including these terms becausethey are comparable to 5PN terms in the phase that are unknown — only the phaseevolution up to 3.5PN order is known.We conclude therefore that the RR terms can in fact be neglected in the 2.5PNwaveform for circular orbits. This is similar to what happens in the energy fluxfor circular orbits where we know that the 2.5PN radiation reaction gives finally acontribution only at 5PN order [3] (the 2.5PN terms in the energy flux are only due totails). Hence we can by the redefinition of the phase variable (30) completely removethe RR contribution given by (28). However we can also decide to include such termsin the 2.5PN waveform (as usual there are many different ways of presenting PN resultsat a given order of approximation). Here we simply propose to include the RR termsas they are in the templates of binary inspiral, i.e. by correcting the inconsistencyin [1] (which has by the above argument no physical effect at 2.5PN) and keeping thecorrected waveform in the form given by Eqs. (27). For completeness, we note that ifwe choose to include all the RR terms into the phase redefinition Φ, Eq. (27) will thenbe modified into H ′ (2 . , × = · · · + sin 2Ψ h −
95 + 145 c i + 75 c i + ν (cid:16) − c i − c i (cid:17)i + · · · , · · · + c i cos 2Ψ h − c i + ν (cid:16) − c i (cid:17)i + · · · , (32)where Ψ is equal to ψ plus the added RR contribution given by the second term in (30),and where ψ itself differs from φ by Eq. (5.6) in [1]. (All the other terms in the waveformare then to be expressed with the same phase variable Ψ.) Acknowledgments
This work was supported in part by grants from the Sherman Fairchild Foundation,NSF grants PHY-0354631, DMS-0553677 and NASA grant NNG05GG51G at Cornell.LB and BRI thank the Indo-French Collaboration (IFCPAR) for its support. adiation reaction in the 2.5PN waveform References [1] Arun K G, Blanchet L, Iyer B R, and Qusailah M S S 2004
Class. Quantum Grav. Prog. Theor. Phys. Phys. Rev. D71