Including higher harmonics in gravitational-wave parameter estimation and cosmological implications for LISA

Massive black holes (MBHs) are crucial in shaping their host galaxies. How the MBH co-evolves with its host galaxy is a pressing problem in astrophysics and cosmology. The valuable information carried by the binary MBH is encoded in the gravitational waves (GWs), which will be detectable by the space-borne GW detector LISA. In the GW data analysis, usually only the dominant (2,2) mode of the GW signal is considered in the parameter estimation for LISA. However, including the higher harmonics in parameter estimation can break the degeneracy between the parameters, especially for the inclination angle and luminosity distance. This may enable the identification of GW signals without electromagnetic counterparts, known as "dark sirens." Thus, incorporating higher harmonics will be beneficial to resolve the Hubble tension and constrain the cosmological model. In this paper, we investigate the role of higher harmonics in the parameter estimation for GWs emitted by binary MBHs. We demonstrate that including the (3,3) mode can lead to a 10³-times improvement in angular resolution and a 10⁴-times improvement in luminosity distance. Meanwhile, our results indicate that considering higher harmonics increases the probability of identifying over 70% host galaxies from 10^-2 Gpc³ cosmological volume threshold (corresponding 10⁵ host galaxies), while the probability is less than 8% for only the (2,2) mode. Thus, our results underscore the importance of including higher modes in the GW signal from binary MBHs, for LISA at least (3,3) mode.