Gravitational Waves from Perturbed Kerr Black Holes: Two Investigations

This thesis includes two main projects. The first project is a study of the effect of black hole spin on the accuracy of the post-Newtonian approximation. We focus on the gravitational energy produced by the quasicircular, equatorial, extreme mass-ratio inspiral of a compact object into a Kerr black hole of mass $M$ and spin $J$. For a given dimensionless spin $a \equiv$ J/M$^2$ (in geometrical units G = c = 1), the energy flux depends only on the orbital velocity $v$ or (equivalently) on the Boyer-Lindquist orbital radius $r$. We investigate the formal region of validity of the Taylor post-Newtonian expansion of the energy flux (which is known up to order $v^8$ beyond the quadrupole formula) by comparing the expansion to numerical calculations of the flux. We find that, at any fixed post-Newtonian order, the edge of the region of validity (as measured by $v/v_{\rm{ISCO}}$, where $v_{\rm{ISCO}}$ is the orbital velocity at the innermost stable circular orbit) is only weakly dependent on $a$. Independently of $a$, the inclusion of angular multipoles up to and including $\ell$ = 5 in the numerical flux is necessary to achieve the level of accuracy of the best-known ($v^8$) expansion of the energy flux. In the second project we study the excitation of Kerr black holes produced by infalling particles. Such a study requires an accurate knowledge of the Green’s function describing the response of the black hole to external perturbations. Relying on the formalism developed by Mano, Suzuki and Takasugi, we improve and extend previous calculations of the contribution to the Green’s function coming from quasinormal mode residues in the complex frequency plane (“excitation factors $B_q$”). Using these results we compute the “excitation coefficients” $C_q$ (the mode amplitudes) when the source of the perturbations is a particle falling into the black hole along the symmetry axis. We compare this calculation with numerical integra- tions of the perturbation equations, and we show quantitatively how the addition of higher overtones improves the agreement with the numerical waveforms.