Systematic biases in parameter estimation on LISA binaries. II. The effect of excluding higher harmonics for spin-aligned, high-mass binaries

The Laser Interferometer Space Antenna (LISA) will observe massive black hole binaries (MBHBs) with astoundingly high signal-to-noise ratio, leaving parameter estimation with these signals susceptible to seemingly small waveform errors. Of particular concern for MBHBs are errors due to neglected higher-order modes. We extend Yi et al. [arXiv:2502.12237] to examine errors due to neglected higher-order modes for MBHBs with nonzero (aligned) progenitor spins and total mass up to $10^8,M_\odot$. For these very massive systems, there can be regions of parameter space in which the $(\ell, |m|)=(2,,2)$ modes are no longer dominant with respect to higher-order ones. We find that the extent of systematic bias can change significantly when varying the progenitor spins of the binary. We also find that for the heaviest, and therefore shortest, MBHB signals, slight systematic errors can cause severe mis-inference of the sky localization parameters. We propose an improved likelihood optimization scheme with respect to previous work as a way to predict these effects in a computationally efficient manner.

💡 Research Summary

This paper investigates systematic parameter‑estimation biases that arise when higher‑order gravitational‑wave modes (HOMs) are omitted from waveform models used to analyze massive black‑hole binary (MBHB) signals expected in the Laser Interferometer Space Antenna (LISA) data. Building on the authors’ earlier work (referred to as “Paper 1”), which considered non‑spinning binaries up to a detector‑frame total mass of 10⁶ M⊙, the present study extends the analysis to much higher masses (10⁵–10⁸ M⊙) and includes aligned spins on the progenitor black holes. The authors use the IMR‑PhenomXHM model, which incorporates the dominant (ℓ,m) = (2,2) mode together with several sub‑dominant harmonics ((3,3), (4,4), (2,1), (3,2)), and evaluate signal‑to‑noise ratios (SNRs) across a grid of mass ratios (q = 1–10), inclination angles (ι = π/12, π/3, 5π/12), and spin magnitudes (χ₁, χ₂ = +0.8, 0, −0.8).

Key findings:

1. Dominance of higher modes at high mass – For total masses above ≈10⁶ M⊙ the (2,2) mode no longer dominates the total SNR; in many regions of the parameter space it contributes less than 60 % of the total power. The (3,3) and (4,4) modes can even exceed the (2,2) contribution, especially for large mass ratios, moderate inclinations, and positive aligned spins.

2. Spin dependence – Positive aligned spins boost the ℓ = m modes, while anti‑aligned configurations redistribute power more evenly among ℓ = m and ℓ ≠ m harmonics. The secondary spin has a diminishing effect as q grows, but the primary spin remains a primary driver of mode hierarchy.

3. Magnitude of systematic bias – Using a “direct likelihood optimization” approach, the authors compare parameter estimates obtained with a (2,2)‑only template against the true parameters of a full‑HOM signal. For high‑SNR (ρ ≈ 10³–10⁴) MBHBs, the bias can be several times larger than the statistical uncertainty, reaching factors of 5–20 in mass and spin, and up to 180° in sky‑location angles. The bias is most severe for the heaviest, shortest‑duration signals, where the merger occurs near the low‑frequency edge of LISA’s sensitivity and the (2,2) mode alone cannot capture the full phase evolution.

4. Improved likelihood optimization scheme – The authors enhance the previous method by employing multi‑start optimization, a hybrid BFGS/Nelder‑Mead algorithm, and a non‑linear residual correction for the missing higher modes. This approach automatically detects multimodal likelihood surfaces, re‑initializes the search when necessary, and reproduces the full‑HOM bias with far fewer waveform evaluations than a brute‑force Markov‑Chain Monte‑Carlo.

5. Practical implications – The study demonstrates that LISA data‑analysis pipelines must incorporate higher‑order modes for any MBHB with total mass ≳10⁶ M⊙, regardless of spin alignment. Neglecting them leads to substantial systematic errors, particularly in sky localization, which could jeopardize electromagnetic follow‑up strategies. The new optimization framework offers a computationally cheap way to forecast these biases and to guide the construction of template banks that include the most relevant harmonics for a given region of parameter space.

In conclusion, the paper provides a thorough quantification of how omitted higher‑order modes bias LISA MBHB parameter estimation, shows that spin and mass dramatically affect the relative importance of those modes, and presents an efficient likelihood‑optimization technique that can predict and mitigate these biases. These results are essential for achieving LISA’s scientific goals of precision tests of general relativity, cosmology, and multimessenger astrophysics.