Efficient trajectory calculations for extreme mass-ratio inspirals using near-identity (averaging) transformations

Future space based gravitational wave detectors, such as the Laser Interferometer Space Antenna (LISA) will allow for the detection of previously undetectable gravitational wave sources. These include extreme mass ratio inspirals (EMRIs) which consist of a stellar mass compact object spiralling into a massive black hole (MBH) due to gravitational radiation reaction. These sources are of particular interest for their ability to accurately map the spacetime of the MBH, allowing for unprecedentedly accurate measurements of the MBH's mass and spin, and tests of general relativity in the strong field regime. In order to reach the science goals of the LISA mission, one requires waveform models that are (i) accurate to within a fraction of a radian, (ii) extensive in the source's parameter space and (iii) fast to compute, ideally in less than a second. This thesis focuses on the latter criteria by utilising techniques that will speed up inspiral trajectory calculations as well as extending prior models to include the MBH's spin. To this end, we develop the first EMRI models that incorporate the spin of the MBH along with all effects of the gravitational self-force (GSF) to first order in the mass ratio. Our models are based on an action angle formulation of the method of osculating geodesics (OG) for generic inspirals in Kerr spacetime. For eccentric equatorial inspirals and spherical inspirals, the forcing terms are provided by an efficient pseudo-spectral interpolation of the first order GSF in the outgoing radiation gauge. For generic inspirals where sufficient GSF data is not available, we construct a toy model from the previous two models. However, the OG method is slow to evaluate due to the dependence of the equations of motion (EOM) on the orbital phases. Therefore, we apply a near-identity (averaging) transformation (NIT) to eliminate all dependence of EOM on the orbital phases while maintaining all secular effects to post-adiabatic order. This inspiral model can be evaluated in less than a second for any mass-ratio as we no longer have to resolve all $\sim 10^5$ orbit cycles of a typical EMRI. This work marks the first time this technique has been applied in Kerr spacetime for eccentric, spherical, and generic inspirals. In the case of a non-rotating MBH, we compare eccentric inspirals evolved using GSF data computed in the Lorenz and radiation gauges. We find that the two gauges produce differing inspirals with a deviation of comparable magnitude to the conservative GSF correction. This emphasizes the need to include the (currently unknown) second order GSF for gauge independent, post-adiabatic waveforms. For spherical orbits, we perform a second averaging transformation to parametrise the averaged EOM in terms of Boyer-Lindquist time instead of Mino time, which is much more convenient for LISA data analysis. We also implement a two-timescale expansion of the EOM and find that both approaches yield self-forced inspirals can be evolved to sub radian accuracy in less than a second. We further improve our spherical inspiral model by incorporating high precision gravitational wave flux calculations and find that without making this modification, the final waveform would be out of phase by as much as $10 - 10^4$ radians for typical LISA band EMRIs. For generic inspirals, one can encounter transient orbital resonances where the standard NIT procedure breaks down. We use the standard NIT when far from these resonances and then we average all phases apart from the resonant phase when in their vicinity. This results in the fastest model to date which includes includes all resonant effects. Our preliminary results demonstrate that accurately modelling only the two lowest order resonances costs 10s of seconds for a typical EMRI, but the resulting waveforms are sufficiently accurate for LISA data science.