Characterization of the gravitational wave emission of three black holes

Abstract

We study the gravitational wave emission of three compact objects using post-Newtonian (PN) equations of motion derived from the Arnowitt-Deser-Misner Hamiltonian formulation, where we include (for the first time in this context) terms up to 2.5 PN order. We perform numerical simulations of a hierarchical configuration of three compact bodies in which a binary system is perturbed by a third, lighter body initially far away from the binary. The relative importance of the different PN orders is examined. We compute the waveform in the linear regime considering mass quadrupole, current quadrupole and mass octupole contributions. Performing a spherical harmonic decomposition of the waveforms we find that from the l = 3 modes it is possible to extract information about the third body, in particular, the period, eccentricity of its orbit, and the inclination angle between the inner and outer binary orbits.Comment: 17 pages, 18 figures and 5 tables. Published versio

About gravitational-wave generation by a three-body system

“…However, by doing so, one obtains unphysical results such as those reported in fig. 18 of [19], as we will now explain. Indeed, we have reproduced the same behavior by applying the quadrupole and "quadrupole-octupole" formulae in the CoM reference frame of a series of triple systems with mass ratios m 2 /m 1 = 0.5 and m 3 /(m 1 + m 2 ) = 0.05, whose trajectories are computed with the code of [52] (which includes the 1PN and 2PN conservative triple dynamics, and the leading order dissipative dynamics).…”

“…Indeed, the coherent piling up of the perturbation induced by the third body may be disrupted due to relativistic precession effects appearing at 1PN order and beyond [22,27,43]. Finally, the GW emission from systems of three BHs has been studied in detail by means of numerical techniques [19,31,[44][45][46] in the event that they form and radiate in sufficiently large number to provide a sizable population for GW detectors. For particular configurations, they have also been investigated analytically [47,48] so as to gain some insight on their dynamics.…”

“…The observer is located in the xz plane of a fixed spatial frame (x, y, z), with spherical coordinates θ = π/4, φ = 0. To be compared with [19], fig. 18.…”

“…However, its general-relativistic dynamics can be obtained within the PN approximation scheme, just like in the twobody case. Indeed, one can write a PN-expanded, time-dependent Hamiltonian [16][17][18][19] that describes the conservative dynamics of a system of three non-spinning bodies up to the 2PN order (i.e., through order (v/c) 4 beyond the leading order Newtonian dynamics), as well as its dissipative dynamics (i.e., the back-reaction due to GW emission) to leading order in v/c, which corresponds to a contribution of 2.5PN order, or O(v/c) 5 , in the equations of motion. The PN dynamics of three body systems is not just an academic curiosity.…”

Post-Newtonian effects in N -body dynamics: conserved quantities in hierarchical triple systems

“…Galactic star clusters with massive central black holes [1,2], triple systems with a relativistic compact inner binary [3,4,5,6,7,8], binary black hole coalescence in the presence of a third body [9,10,11], and even the stability of the solar system [12] have been studied using combinations of N-body techniques and relativistic dynamics.…”