Spin effects in Spherical Harmonic Modes of Gravitational Waves from Eccentric Compact Binary Inspirals

We compute the leading and sub-leading spin effects through the second post-Newtonian order (2PN) in spherical harmonic modes of gravitational waveforms from inspiralling compact binaries in non-circular orbits with non-precessing components. The two spin couplings, linear-in-spin (spin-orbit; SO) and quadratic-in-spin (spin-spin; SS), that appear in 2PN waveforms are computed with desired accuracy and explicit expressions for relevant modes are derived. The modes that have spin corrections through 2PN include $(\ell, |m|)$=$((2,2),\,(2,1),\,(3,3),\,(3,2),\,(3,1),\,(4,3),\,(4,1))$ modes. Closed form expressions for these modes for compact binaries in general orbits as well as in elliptical orbits are being provided. While the general orbit results can be used to study signals from binaries in orbits of arbitrary shape and nature, elliptical orbit results are applicable to systems with arbitrary eccentricities. We also express the elliptical orbit results as leading eccentric corrections to the circular results. Our prescription represents, the first, fully analytical treatment that combines spins, eccentricity and higher modes together and completes computation of spin effects through 2PN order. These should find immediate applications in inspiral-merger-ringdown modelling for eccentric mergers including the effect of non-precessing spins and higher modes as well as in parameter estimation analyses employing inspiral waveform