Recoil velocity at second post-Newtonian order for spinning black hole binaries

Abstract

We compute the flux of linear momentum carried by gravitational waves emitted from spinning binary black holes at 2PN order for generic orbits. In particular we provide explicit expressions of three new types of terms, namely next-to-leading order spin-orbit terms at 1.5 PN order, spin-orbit tail terms at 2PN order, and spin-spin terms at 2PN order. Restricting ourselves to quasi-circular orbits, we integrate the linear-momentum flux over time to obtain the recoil velocity as function of orbital frequency. We find that in the so-called superkick configuration the higher-order spin corrections can increase the recoil velocity up to a factor ∼ 3 with respect to the leading-order PN prediction. Whereas the recoil velocity computed in PN theory within the adiabatic approximation can accurately describe the early inspiral phase, we find that its fast increase during the late inspiral and plunge, and the arbitrariness in determining until when it should be trusted, makes the PN predictions for the total recoil not very accurate and robust. Nevertheless, the linear-momentum flux at higher PN orders can be employed to build more reliable resummed expressions aimed at capturing the non-perturbative effects until merger. Furthermore, we provide expressions valid for generic orbits, and accurate at 2PN order, for the energy and angular momentum carried by gravitational waves emitted from spinning binary black holes. Specializing to quasi-circular orbits we compute the spin-spin terms at 2PN order in the expression for the evolution of the orbital frequency and found agreement with Mikóczi, Vasúth and Gergely. We also verified that in the limit of extreme mass ratio our expressions for the energy and angular momentum fluxes match the ones of Tagoshi, Shibata, Tanaka and Sasaki obtained in the context of black hole perturbation theory.

Spin effects on gravitational waves from inspiraling compact binaries at second post-Newtonian order

“…The instantaneous accelerations (3.1) and (3.5) are projected onto a triad consisting of the following unit vectors: n = x/r, the vector ℓ = L N /|L N | orthogonal to the instantaneous orbital plane, where L N = mν x × v denotes the Newtonian orbital angular momentum, and λ = ℓ × n. The orbital separation r and angular frequency ω are decomposed into their orbit averaged piece, indicated by an overbar, and remaining fluctuating pieces, r =r + δr and ω =ω + δω. Projecting the equations of motion along λ yields the equality 2ωṙ +ω r or, equivalently [93] …”

“…The details of the derivation of the modified Kepler law relating the orbit-averaged orbital angular frequency ω and the orbitaveraged orbital separation are discussed in Ref. [93]. The instantaneous accelerations (3.1) and (3.5) are projected onto a triad consisting of the following unit vectors: n = x/r, the vector ℓ = L N /|L N | orthogonal to the instantaneous orbital plane, where L N = mν x × v denotes the Newtonian orbital angular momentum, and λ = ℓ × n. The orbital separation r and angular frequency ω are decomposed into their orbit averaged piece, indicated by an overbar, and remaining fluctuating pieces, r =r + δr and ω =ω + δω.…”

Radial infall of two compact objects: 2.5-post-Newtonian linear momentum flux and associated recoil

“…The importance of this phenomenon has been realized widely in astrophysics community and there have been numerous analytical or semi-analytical [4][5][6][7][8][9][10][11][12][13][14][15][16] and numerical studies [17][18][19][20][21][22][23][24] to compute this effect. All these studies compute the recoil effects due to the loss of linear momentum from compact binary systems (which either have massasymmetry and/or have non zero spin) moving in quasi-Keplerian or in quasi-circular orbits.…”

“…The most recent related PN work [8] gives 2PN accurate expressions for the instantaneous part of the linear momentum and hence one can use the above transformations to write the 2PN expression for the instantaneous part of the linear momentum flux in terms of z andż. In the present work, we not only calculate the instantaneous part of the flux explicitly for the head on case to a higher order (2.5PN as compared to previous 2PN calculations) but also compute additional terms contributing at the 1.5PN order and 2.5PN order (tail contribution) whose nature has been discussed in more detail in the next section.…”

“…Some other (relatively recent) analytical/numerical works [25,29,30] compute the recoil effects taking in to account the asymmetry in mass and/or in the spin. As far as PN calculations are concerned, although, the recoil effects in a head-on collision case have not been investigated explicitly, one can use expressions for the linear momentum flux from nonspinning inspiralling compact binary systems moving in general orbits [4,5,8] to write equivalent expressions for the head-on case by using the following transformations 1 (as suggested in [31,32]):…”