On the detectability of higher harmonics with LISA

Supermassive black hole binaries (SMBHBs) are expected to be detected by the future space-based gravitational-wave detector LISA with a large signal-to-noise ratio (SNR). This prospect enhances the possibility of differentiating higher harmonics in the inspiral-merger-ringdown (IMR) waveform. In this study, we test the ability of LISA to identify the presence of different modes in the IMR waveform from a SMBHB. We analyze the contribution of each mode to the total SNR for different sources. Higher modes, in particular the mode $(3, 3)$ and $(4, 4)$, can dominate the signal observed through the LISA detector for SMBHB of the order of $10^8 M_\odot$. With Bayesian analysis, we can discriminate models with different harmonics. While spherical harmonics are often considered orthogonal, we observe it is not the case in the merger-ringdown phase observed by LISA. Omitting harmonics not only diminishes the SNR but can also lead to biased parameter estimates. We analyze the bias for each model in a source example and quantify the threshold SNR where we can expect the parameter bias to be comparable to the statistical error. By computing the waveform model error with the Fisher approximation and comparing it with the posterior distribution from our sampler results, we can evaluate the veracity of the analytical bias, which converges with the sampler results as more harmonics are introduced. To conclude, SMBHB events with SNR of a few hundred, as expected in LISA, are required to use templates with at least modes $(2, 2)$, $(2, 1)$, $(3, 3)$, $(3, 2)$, $(4, 4)$, $(4, 3)$ to estimate all intrinsic parameters correctly. Our work highlights the importance of higher modes to describe the gravitational waveform of events detected by LISA.