Systematic bias in LISA ringdown analysis due to waveform inaccuracy

Abstract

Inaccurate modeling of gravitational-wave signals can introduce systematic biases in the inferred source parameters. As detector sensitivities improve and signals become louder, mitigating such waveform-induced systematics becomes increasingly important. In this work, we assess the systematic biases introduced by an incomplete description of the ringdown signal from massive black hole binaries in the LISA band. Specifically, we investigate the impact of mode truncation in the ringdown template. Using a reference waveform composed of 13 modes, we establish a mode hierarchy and determine the minimum number of modes required to avoid parameter biases across a wide range of LISA sources. For typical systems with masses $\sim 10^6$--$10^7\,M_\odot$ at redshifts $z \sim 2$--$6$, we find that at least 3--6 modes are needed for accurate parameter estimation, while high-SNR events may need at least 10 modes. Our results are a window-insensitive lower bound on the minimum number of modes, as more modes may be needed depending on the choice of time-domain windowing of the post-merger signal.

Figures (11)

• Figure 1: Characteristic strain of three QNMs in the ringdown of a BBH with remnant mass $M = 1.44\times 10^6 M_\odot$ and dimensionless spin $a = 0.66$, observed at redshift $z = 0.1$. The modes $h_{220}$, $h_{330}$, and $h_{550}$ are shown in light blue, pink, and purple, respectively. Solid and dot-dashed lines indicate the mirroring technique ($h^{\rm mirr}_{\ell m n}$) and a Heaviside time window ($h^{\Theta}_{\ell m n}$), applied before taking the Fourier transform. The black dashed line represents the LISA strain sensitivity curve, while the thick grey vertical dashed line marks indicates the PhenomA $f_\text{cut}$. Observe that there is significant spectral leakage from the dominant $h_{220}$ mode into higher-frequency regions above $f_\text{cut}$.
• Figure 2: QNMs ordered by SNR from the loudest (up) to the quietest (down). We take 3 representative values of spin and mass ratio, fixing the primary mass to $10^6\,M_{\odot}$, the luminosity distance to $5 \text{Gpc}$, and the angles to $\theta = \psi=\iota=\pi/3$, $\phi=0$. The odd-$m$ modes are highlighted in blue. Note their absence in a mass-symmetric system, and their increasing relevance as we take a small $q$. Further observe that they tend to climb the ranking as we consider more spinning systems. Besides these two features, for highly spinning and mass-asymmetric systems, overtones tend to become more important. The most peculiar case comes with the $331$, which becomes louder than the fundamental $330$.
• Figure 3: Comparison of statistical and systematic error for mass and spin, as a function of the number of modes $N$ for an approximate template $h_{\text{AP}}^{(N)}$. The selected system has primary progenitor mass $10^6M_{\odot}$ and mass ratio $q = 0.5$ and it is located at a luminosity distance of $10 \,{\rm Gpc}$. The angles are fixed to the values $\theta = \psi = \iota = \pi/3$ and $\phi = 0$. The true value of the parameters is represented with the red dashed line, while the offset of the round points is given by the systematic error. Finally, the error bars represent the statistical error. The orange and blue arrows indicate respectively the points in which quadratic and linear retrograde modes are included. Observe that the inclusion of more modes tends to tame the systematic error.
• Figure 4: Minimum number of modes $N_{\text{min}}$, represented by the color code, as a function of primary mass and redshift. The mass ratio and the progenitor spins have been fixed to $q=0.5$ and $a_1=a_2=0$, while we have averaged over sky localization. Observe that at low redshift $z<1$ and for $M/M_{\odot}\sim \mathcal{O}\left(10^{6-7}\right)$, we need $N_{\min} \in [8,10]$. Further note that the contours track the behavior of the SNR, given the PSD of LISA.
• Figure 5: Relative waveform error (blue dots) and the maximum dephasing (orange dots) between $h_{\text{AP}}$ and the fiducial template, as a function of the number of modes. The system has primary mass $5\times 10^6\,M_{\odot}$ and is located at luminosity distance $5 \,{\rm Gpc}$, with fixed $\theta=\psi=\iota = \pi/3$ and $\phi = 0$. $N_{\text{min}}= 7$ is shown with the vertical dashed line. Observe that with increasing $N$, the dephasing and relative waveform errors decrease appreciably, allowing for the use of the linear signal approximation in estimating $N_{\min}$.