External magnetic field influence on massive binary black hole inspiral gravitational waves and its similarity with environmental effects

Abstract: Magnetic fields represent a critical component of astrophysical research, laying the foundation for interpreting high-energy astrophysical activity across galactic scales. In this work, we investigate the parametrized post-Einsteinian (ppE) waveform imprints induced by the external magnetic fields of Bertotti-Robinson and Bonnor-Melvin black holes, with the aim of distinguishing such magnetic effects from environmental influences--particularly for massive black holes posited to reside at galactic centers. We first compute the ppE frequency-domain waveform for a small black hole inspiraling into a massive Kerr-Bertotti-Robinson (KBR) black hole, which corresponds to a Kerr black hole embedded in an external magnetic field. We find that the leading-order correction arising from the magnetic field is at the $-2$ post-Newtonian (PN) order relative to the quadrupole term, while the next-leading-order correction is at $-1.5$ PN, originating from the spin of the black hole. We further examine the effects of a spinning Kerr-Bonnor-Melvin (KBM) black hole, whose leading-order magnetic correction is at $-3$ PN (consistent with the preceding result), whereas its spin-induced correction is also at $-1.5$ PN. The leading-order ppE corrections for both KBR and KBM black holes do not appear degenerate with any modified theory of gravity effects; nonetheless, we demonstrate that they resemble the gravitational pull contributions from additional matter with power-law distributions of index $γ=1$ and $γ=0$, respectively. As a result, future gravitational wave (GW) observations detecting $-3$ or $-2$ PN order corrections will infer their origin as either magnetic field effects or matter environmental influences.