Gravitational-wave Phase Shifts in Eccentric Black Hole Mergers as a Probe of Dynamical Formation Environments

Abstract We here quantify the gravitational-wave (GW) phase shift appearing in the waveform of eccentric binary black hole (BBH) mergers formed dynamically in 3-body systems. For this, we have developed a novel numerical method where we construct a reference binary, by evolving the post-Newtonian ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">P</mml:mi> <mml:mi>N</mml:mi> </mml:math> ) evolution equations backwards from a point near merger without the inclusion of the third object, that can be compared to the perturbed binary that evolves under the influence from the third BH. From this, we quantify how the interplay between dynamical tides, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">P</mml:mi> <mml:mi>N</mml:mi> </mml:math> -effects, and the time-dependent Doppler shift of the eccentric GW source results in unique observable GW phase shifts that can be used to probe the dynamical assembly mechanism of individual GW sources. We further find an analytical expression for the GW phase shift, which has a universal functional form that only depends on the time-evolving BBH eccentricity. The normalization scales with the BH masses and initial separation, which can be linked to the underlying astrophysical environment. GW phase shifts from a chaotic 3-body BH scattering taking place in a cluster, and from a BBH inspiraling in a disk migration trap near a supermassive BH, are also shown for illustration.