Analytical solutions of bound timelike geodesic orbits in effective-one-body frame

We derive the approximate analytical solutions of the bound timelike geodesic orbits in the effective-one-body (EOB) frame with extreme-mass ratio limit. The analytical solutions are expressed in terms of the elliptic integrals using Mino time $\lambda$ as the independent variable. Since Mino time decouples the $r$ and $\theta$-motion, we also give explicit expressions for three orbital frequencies ${\mathrm{\Omega }}_r,{\mathrm{\Omega }}_{\theta },{\mathrm{\Omega }}_{\phi }$ using the Fourier series expansion. With these analytical expressions at hand, we can perform Fourier expansions in Mino time $\lambda$ for any function expressed in terms of the coordinates $(r,\theta ,\phi )$. In particular, the observer’s time t is decomposed into Mino time $\lambda$, and the frequency-domain description is constructed from the $\lambda$-Fourier expansion and the expansion of t. These analytical expressions are quite simple to implement, and can be applicable for calculating gravitational waves (GWs) from extreme mass-ratio inspirals (EMRIs) with the frequency-domain Teukolsky equation.