The metric describing a two-body system in general relativity can be approximated (at lowest order) as a known background solution due to the more-massive body. A first-order metric perturbation (and the associated self-force), proportional to the mass-ratio of the two bodies, can be found by solving the linearized Einstein equations. This first-order solution serves as (part of) the source of the second-order problem. In recent decades, great strides have been made toward solving the first-order problem in a wide variety of parameter space. In this talk, I will review the state of first-order gravitational self-force analysis with particular attention paid to practical calculations.

I will present the first result for calculating the first order self-force on generic bound geodesics in Kerr spacetime.

Time-domain evolutions of Lorenz-gauge metric perturbations (on black hole backgrounds) suffer from an $\ell=1$ gauge instability. I will describe this instability and review various techniques for avoiding it.

We calculate the first-order scalar self-force acting on a scalar-charged particle on a generic, bound geodesic in a background Kerr spacetime. To perform these calculations, we wrote a frequency domain code in Mathematica. By using spectral integration and the MST function expansion formalism, our code makes calculations with arbitrary numerical precision. We present scalar self-force results for several inclined, eccentric orbits, with inclinations ranging from ι = 0 to ι = π/3 and eccentricities ranging from e = 0 to e = 0.8. In these results, we also observe previously reported quasinormal mode excitations in the self-force for highly-eccentric orbits around a highly-spinning black hole.

We present ongoing work on the computation of the local force due to back-reaction effects caused by the spin of the lesser component in an extreme mass-ratio binary

Last year the European Space Agency selected the LISA mission for the L3 launch slot in its future science program. LISA will open the low-frequency spectrum for gravitational wave astronomy, listening to sources such as supermassive black holes and extreme mass ratio inspirals. The industrial Phase-A study has just begun to prepare for a launch in the early 2030s.

In this talk we will review the expected event rate for Extreme Mass Ratio Inspiral (EMRIs) in the LISA band. We discuss the astrophysical uncertainties which influence the event rate and distribution of source parameters. Then we describe the simulation of the LISA data with gravitational wave (GW) signal from EMRI system in the coming LISA data challenge and future plans. Finally, we will overview the main methods for detecting EMRI signal.