A Parametrized Test of General Relativity for LISA Massive Black Hole Binary Inspirals

TL;DR

This work extends the Flexible Theory-Independent framework to LISA MBHB signals, introducing parametrized deviations in the inspiral PN phasing and ensuring a smooth transition back to GR before merger. Using Fisher and Bayesian analyses with two state-of-the-art waveform families and LISA’s A,E,T channels, the authors show that LISA can tighten constraints on agnostic beyond-GR deviations by at least two orders of magnitude relative to LVK results, particularly enhancing sensitivity to -2 PN (dipolar) terms. A careful reparameterization of the 0PN deviation removes strong degeneracies with the chirp mass, improving the accuracy of Fisher forecasts. The results also reveal that merger–ringdown information becomes increasingly valuable for high-mass MBHBs, underlining the importance of multimodal data in GR tests, while cautioning about the modeling assumptions in the inspiral-only deviation framework. Overall, the paper provides a robust, scalable pathway for GR tests with LISA and outlines future directions for waveform systematics and population-level analyses.

Abstract

Laser Interferometer Space Antenna (LISA) observations of massive black hole binaries (MBHBs) will provide long duration inspiral signals with high signal-to-noise ratio (SNR) data, ideal for testing general relativity (GR) in the strong-field and relativistic regime regime. We present an extension of the Flexible Theory-Independent (FTI) framework, adapted to gravitational waves (GWs) from MBHBs observed with LISA, to perform parametrized inspiral tests of GR. This approach introduces generic deviations to the post-Newtonian (PN) coefficients of the frequency-domain GW phase while accounting for the time- and frequency-dependent instrument response, thus effectively identifying potential deviations from GR by constraining modifications to the PN phasing formula. Complementary analyses using Fisher matrix and full Bayesian approaches confirm that LISA observations could improve constraints on agnostic, scale-independent deviations from GR by at least two orders of magnitude compared to the most recent LIGO-Virgo-KAGRA measurements. Since LISA's sensitivity to different GW phases -- inspiral, merger, and ringdown -- varies across the MBHB parameter space with masses between $10^4$ and $10^7M_{\odot}$, the optimal regime for testing agnostic deviations is not known a priori. Our results illustrate how the strength of these constraints depends significantly on both the total mass and the SNR, reflecting the trade-off between inspiral and merger-ringdown contributions to the observed signal. We also investigate the interplay between inspiral-only versus inspiral-merger-ringdown analyses in constraining these inspiral deviation parameters. This work contributes to the development of robust tests of GR with LISA, enhancing our ability to probe the nature of gravity and BHs with GW observations.

Figures (10)

• Figure 1: An example of the plus polarization, $h_+$, of the dominant $(2,2)$-mode GR waveform is shown in black, together with its non-GR counterparts in color for several values of $\delta\hat{\varphi}_2 \in [-1,1]$. The vertical red dashed line marks the time corresponding to the tapering frequency, $f^{\text{tape}}_{22} = 0.35 f^{\text{peak}}_{22}$, around which the non-GR deviations are gradually switched off. The shaded grey region indicates the window over which this smooth tapering is applied. By construction, the GR and non-GR waveforms coincide in the post-tapering region.
• Figure 2: The top panel shows the two-dimensional posterior distribution (blue) for the chirp mass $\mathcal{M}_c$ and the non-GR parameter $\delta\hat{\varphi}_0$ for an example system. Owing to the strong degeneracy between these parameters, as described by Eq. \ref{['eq:Mc_chi0_deg']}, the Fisher-predicted contours (black) are not visible—reflecting a significant underestimation of the uncertainties and the failure of the Fisher approximation to capture the parameter correlations. In contrast, the bottom panel shows the posterior distribution for $\mathcal{M}_c$ and the reparametrized deviation parameter $\delta\hat{\xi_0}$, where the Fisher contours are in good agreement with the full Bayesian posterior, demonstrating the improved performance of the Fisher approach under this reparametrization.
• Figure 3: Top panel: Distribution of $90\%$ upper bounds for non-GR deviation parameters $|\delta\hat{\varphi}_i|$ across different total mass systems ($M = 10^4, 10^5, 10^6, 10^7\,M_\odot$) at redshift $z=1$. Bottom panel: Distribution of $90\%$ upper bounds for non-GR deviation parameters $|\delta\hat{\varphi}_i|$ parameters across mass systems ($M = 10^5, 10^6, 10^7\,M_\odot$) with redshift adjusted to maintain fixed SNR $= 500$. Both analyses use 500 Fisher matrix computations with randomly sampled spins and mass ratio. Left , undashed (right, dashed) side of each distribution uses SEOBNRv5HM_ROM (IMRPhenomXHM) waveform model for the analysis.
• Figure 4: Characteric strain, $h_c$, of GW signals from MBHBs as a function of frequency. The systems analyzed all have $q=3$, $z=1$, and $\chi_{1,2} = 0.5, 0.2$, and are shown with their total detector-frame masses, $M$, indicated on top of each colored trace. The trace colors correspond to the accumulated SNR at each frequency. The dot-dashed black line represents LISA’s instrumental characteristic noise, $h_n$, the dashed blue vertical line denotes the nominal maximum frequency for LISA, $f = 0.5$ Hz, and the yellow triangle (salmon diamond, violet circle) represent the frequencies at (2,2)-peak (ISCO, (2,2)-tape) per each system.
• Figure 5: Distribution of $90\%$ upper bounds for each non-GR deviation parameter $|\delta\hat{\varphi}_i|$ for different total mass systems ($M = 10^4, 10^5, 10^6, 10^7 M_\odot$) using different frequencies cutoffs. The analysis includes four different frequency cutoffs: a fixed value of 0.5 Hz (turquoise), the peak frequency of the (2,2)-mode $f_{22}^{\text{peak}}$ (yellow), the innermost stable circular orbit frequency $f_{\text{ISCO}}$ (salmon), and the tapering-frequency of the (2,2)-mode $f_{22}^{\text{tape}} = 0.35f_{22}^{\text{peak}}$ (violet).