Exploring tests of the no-hair theorem with LISA

In this study, we explore the possibility of testing the no-hair theorem with gravitational waves from massive black hole binaries in the Laser Interferometer Space Antenna (LISA) frequency band. Based on its sensitivity, we consider LISA's ability to detect possible deviations from general relativity (GR) in the ringdown. Two approaches are considered: an agnostic quasinormal mode (qnm) analysis and a method explicitly targeting the deviations from GR for given qnms. Both approaches allow us to find fractional deviations from GR as estimated parameters or by comparing the mass and spin estimated from different qnms. However, the estimated deviations may vary depending on whether we rely on prior knowledge of the source parameters from a premerger or inspiral-merger-ringdown analysis. Under some assumptions, the second approach targeting fractional deviations from GR allows us to recover the injected values with high accuracy and precision. We obtain 5%, 10% uncertainty on $\ensuremath{\delta}\ensuremath{\omega},\ensuremath{\delta}\ensuremath{\tau}$ respectively, for the (3, 3, 0) mode, and 3%, 17% for the (4, 4, 0) mode. As each approach constrains different features, we conclude combining both methods would be necessary to perform a better test. In this analysis, we also forecast the precision of the estimated deviation parameters for sources throughout the mass and distance ranges observable by LISA.