Time-domain inspiral templates for spinning compact binaries in quasi-circular orbits described by their orbital angular momenta

We present a prescription to compute the time-domain gravitational wave (GW) polarization states associated with spinning compact binaries inspiraling along quasi-circular orbits. We invoke the orbital angular momentum $\text{vek}L$ rather than its Newtonian counterpart $\text{vek}{L}_{\mathrm{N}}$ to describe the binary orbits while the two spin vectors are freely specified in an inertial frame associated with the initial direction of the total angular momentum. We show that the use of $\text{vek}L$ to describe the orbits leads to additional 1.5PN order amplitude contributions to the two GW polarization states compared to the $\vek L_{\rm N}$-based approach and discuss few implications of our approach. Further, we provide a plausible prescription for GW phasing based on few theoretical considerations and which may be treated as the natural circular limit to GW phasing for spinning compact binaries in inspiraling eccentric orbits [Gopakumar A and Sch{"a}fer G 2011 {\em Phys. Rev. D} {\bf 84} 124007].