Spin-induced Quadrupole Moment (SIQM) Test for Eccentric Compact Binaries

Spin-induced deformations of individual components of a binary can be quantified using the gravitational wave signal the binary emits. Such deformations are characterised by a parameter, $\kappa$, which takes a value of 1 for a black hole and thus its measurement can be used to test the no-hair conjecture. However, in practice, only a symmetric combination of this parameter for a binary ($\kappa_s$) can be measured, thus instead enabling a test for the no-hair conjecture in the context of a binary black hole system; see for instance, Krishnendu et al., Phys. Rev. Lett. 119, 091101 (2017), arXiv:1701.06318. While previous studies have focused on circular binaries, we extend this test to eccentric systems in a Fisher matrix based analysis. We find that the error in the measurement of the parameter $\kappa_s$ reduces from a value of about 18% (for the circular case) to close to 8% (4%) for a $10 M_{\odot}$ system with dimensionless component spins $>0.8$ and with a reference initial eccentricity ($e_0$) of 0.2 (0.4) evaluated at 5 Hz for a third generation detector, Cosmic Explorer (CE). Compared to the estimates obtained by using Advanced LIGO design sensitivity, eccentricity and the overall improved sensitivity of CE detectors together seem to improve these estimates almost by an order of magnitude.