Assessing the detectability of the secondary spin in extreme mass-ratio inspirals with fully-relativistic numerical waveforms

Extreme mass-ratio inspirals~(EMRIs) detectable by the Laser InteferometricSpace Antenna~(LISA) are unique probes of astrophysics and fundamental physics.Parameter estimation for these sources is challenging, especially because thewaveforms are long, complicated, known only numerically, and slow to compute inthe most relevant regime, where the dynamics is relativistic. We perform atime-consuming Fisher-matrix error analysis of the EMRI parameters usingfully-relativistic numerical waveforms to leading order in an adiabaticexpansion on a Kerr background, taking into account the motion of the LISAconstellation, higher harmonics, and also including the leading correction fromthe spin of the secondary in the post-adiabatic approximation. We payparticular attention to the convergence of the numerical derivatives in theFisher matrix and to the numerical stability of the covariance matrix, whichfor some systems requires computing the numerical waveforms with approximately$90$-digit precision. Our analysis confirms previous results (obtained withapproximated but much more computationally efficient waveforms) for themeasurement errors on the binary's parameters. We also show that the inclusionof higher harmonics improves the errors on the luminosity distance and on theorbital angular momentum angles by one order and two orders of magnitude,respectively, which might be useful to identify the environments where EMRIslive. We particularly focus on the measurability of the spin of the secondary,confirming that it cannot be measured with sufficient accuracy. However, due tocorrelations, its inclusion in the waveform model can deteriorate the accuracyon the measurements of other parameters by orders of magnitude, unless aphysically-motivated prior on the secondary spin is imposed.