Bound orbit domains in the phase space of the Kerr geometry

We derive the conditions for a non-equatorial eccentric bound orbit to exist around a Kerr black hole in two-parameter spaces: the energy, angular momentum of the test particle, spin of the black hole, and Carter's constant space ($E$, $L$, $a$, $Q$), and eccentricity, inverse-latus rectum space ($e$, $\mu$, $a$, $Q$). These conditions distribute various kinds of bound orbits in different regions of the ($E$, $L$) and ($e$, $\mu$) planes, depending on which pair of roots of the effective potential forms a bound orbit. We provide a prescription to select these parameters for bound orbits, which are useful inputs to study bound trajectory evolution in various astrophysical applications like simulations of gravitational wave emission from extreme-mass ratio inspirals, relativistic precession around black holes, and the study of gyroscope precession as a test of general relativity.