Testing gravitational wave polarizations with LISA

Abstract: In this paper we quantify the ability of the Laser Interferometer Space Antenna (LISA) to test the presence of non-tensorial polarizations as well as modifications to the tensor ones in gravitational waves emitted from massive black hole binaries. We employ the Parametrized Post-Einsteinian (PPE) formalism to model deviations from General Relativity (GR) for tensor, vector, and scalar polarizations. Our PPE parametrization is inspired by post-Newtonian waveforms from four modified gravity theories: Horndeski, Einstein-aether, Rosen's bimetric, and Lightman-Lee. We consistently implement these modifications across the inspiral, merger, and ringdown phases, ensuring proper waveform alignment and tapering. Subsequently, we perform Fisher forecasts to derive expected constraints on deviations from General Relativity and map these constraints to the parameter spaces of the four gravity theories. For tensor polarizations, LISA achieves constraints on amplitude modifications ranging between $\sim 10<sup>{-4}-10<sup>{-2}$ precision level, depending on the frequency evolution of the modifications, for systems with $10^5-10⁷ {\, \rm M}<em>\odot$ at $z = 1$. We find that LISA can distinguish breathing and longitudinal scalar polarizations only for relatively light binaries with $M \lesssim 10⁴ {\, \rm M}</em>\odot$, beyond which these modes become degenerate in the detector response. Importantly, constraints on vector polarizations are approximately 2-3 times more precise than for scalar polarizations. For both vector and scalar modes, amplitude measurements reach precisions ranging between $\sim 10<sup>{-8}-10<sup>{-2}$, depending on the frequency evolution of the modifications, for systems with $10^5-10⁷ {\, \rm M}_\odot$ at $z = 1$. These results demonstrate LISA's potential to probe gravity in the strong-field regime via gravitational wave polarizations.

• The paper provides the first quantitative assessment of LISA's capabilities to detect and constrain six gravitational wave (GW) polarizations using massive black hole binary (MBHB) mergers, demonstrating superior constraints on vector and tensor modes compared to scalar modes. This includes distinguishing between breathing and longitudinal modes at higher frequencies, a feature critical for heavy MBHBs.
• LISA's unique response to different GW polarizations, derived through a one-way Doppler shift calculation, enables the distinction between modes that are degenerate in ground-based detectors. The analysis shows significant improvements in constraints on non-tensorial polarizations and exceptional precision in tensor phase measurements, better than current LVC bounds and comparable to or exceeding those of next-generation ground-based detectors.
• Frequency scaling of modifications and mass dependence strongly influence the ability to distinguish scalar modes. With certain conditions met, vector polarizations can be measured more precisely than scalar polarizations.

Motivation and scope

The paper quantifies the ability of the Laser Interferometer Space Antenna (LISA) to test gravitational wave (GW) polarizations using massive black hole binary (MBHB) mergers. While General Relativity (GR) predicts only two transverse-traceless tensor polarizations, metric theories of gravity can in principle support up to six: two tensor ($+$, $\times$), two vector ($v_1$, $v_2$), and two scalar modes (breathing $b$ and longitudinal $l$). Ground-based detectors have so far provided only weak bounds on non-tensorial polarizations because of low signal-to-noise ratios (SNRs) and the need for detector networks. The authors argue that LISA is uniquely positioned for this test: MBHB signals last weeks to months and reach high SNRs, and LISA's orbital motion modulates its response to different polarizations over the observation. The paper fills a gap in the literature by providing the first comprehensive forecast for detecting and constraining extra polarizations from MBHBs with LISA, including a physically motivated frequency evolution of the modifications across inspiral, merger, and ringdown.

Polarization content of metric theories

The paper carefully distinguishes between dynamical degrees of freedom (DoFs) of a modified gravity theory and GW polarizations as defined through the detector response. Working in synchronous gauge in the far-away wave zone, the six independent components of the spatial metric perturbation directly define the six possible polarizations via the geodesic deviation equation. Using a scalar-vector-tensor decomposition, the authors show that these six modes can be written entirely in terms of gauge-invariant quantities ($E^{\rm TT}_{ij}$, $\Phi$, $\Theta$, $\Xi_i$), establishing them as physical observables. They assign spin weights 2, 1, 0 to tensor, vector, and scalar sectors respectively, enabling spin-weighted spherical harmonic decompositions for non-precessing binaries. An important conceptual point, illustrated explicitly for Horndeski gravity in an appendix, is that additional DoFs do not automatically imply additional polarizations: e.g., scalar Gauss-Bonnet gravity propagates a scalar DoF but excites no scalar polarization, while Brans-Dicke excites only the breathing mode because its scalar is massless.

LISA's response to different polarizations

A key technical result is the derivation of the one-way Doppler shift for all six polarizations, valid beyond the geodesic-deviation approximation that breaks down when the GW wavelength becomes comparable to LISA's arm length. The authors then compute sky-averaged response functions for the TDI $\times$0, $\times$1, and $\times$2 channels and provide, for the first time, analytic low-frequency expansions. In the $\times$3 channels, responses are degenerate at leading order between tensor-vector pairs and between breathing-longitudinal pairs, with differences appearing only at order $\times$4; the $\times$5 channel responds at order $\times$6. Because LISA operates at frequencies up to $\times$7 Hz where wavelengths become comparable to its million-kilometer arms, it can in principle distinguish breathing from longitudinal modes — something ground-based detectors cannot do. This capability, however, turns out to be mass-dependent: only binaries with total mass $\times$8 merge at frequencies high enough to break the degeneracy.

Parametrized post-Einsteinian framework and theory mapping

The analysis employs the Parametrized Post-Einsteinian (PPE) formalism, extended to all polarizations and both the $\times$9 and $v_1$0 angular harmonics. A central structural finding is that phase modifications of all polarizations are tied together: in the four theories considered, $v_1$1 and the frequency exponent $v_1$2 is shared across sectors, reducing the parameter space to eight free parameters. The PPE parametrization is mapped onto four theories with known PN waveforms for extra polarizations:

• Horndeski gravity: scalar-tensor; no vector modes; two branches depending on whether dipole or quadrupole corrections dominate ($v_1$3 or $v_1$4).
• Einstein-Æther: Lorentz-violating vector-tensor theory; all six polarizations present, with longitudinal proportional to breathing.
• Rosen's bimetric and Lightman-Lee: conservative bimetric theories; both lack smooth GR limits (highlighted in red boxes), meaning they predict definite non-vanishing deviations, and require tailored non-GR injections rather than GR injections.

The waveform implementation uses IMRPhenomXHM within lisabeta, with non-GR modifications tapered off near merger via a Planck window applied to the second phase derivative, following the Flexible-Theory-Independent approach. The choice of tapering frequency affects heavy systems most; inspiral-only injections are also presented as a more conservative alternative, yielding comparable constraints for $v_1$5 but degraded ones for heavier systems.

Forecasted constraints

Fisher forecasts over a grid of 645 points in $v_1$6, with 100 randomized realizations per point, yield the following headline results for sources at $v_1$7:

Quantity	Frequency scaling	Precision
Tensor amplitude $v_1$8	$v_1$9	$v_2$0
Tensor amplitude $v_2$1	$v_2$2	$v_2$3
Tensor phase $v_2$4	$v_2$5	$v_2$6 rad
Scalar amplitudes	$v_2$7	$v_2$8
Vector amplitudes	$v_2$9	$b$0

Several implications follow directly. First, the frequency scaling of the modification matters enormously: constraints for $b$1 are one to two orders of magnitude tighter than for $b$2, since low-frequency deviations accumulate over long inspirals. Second, vector polarizations are constrained 2–3 times more precisely than scalar polarizations at fixed frequency scaling, owing to LISA's stronger antenna response to vector modes. Third, the exception is the longitudinal mode with $b$3, whose steep frequency dependence yields constraints as tight as $b$4, outperforming vector bounds despite the weaker response. Fourth, judging detectability by SNR alone would be pessimistic: LISA can constrain amplitude parameters below the values needed for SNR = 10 detection of the extra mode alone. Fifth, the projected tensor-phase constraints are 2–3 orders of magnitude better than current LVK bounds and comparable to or better than forecasts for third-generation ground detectors.

On distinguishing the two scalar modes, joint injections of $b$5 and $b$6 show correlation factors growing strongly with mass; for $b$7 the median correlation is $b$8, dropping below 0.5 in favorable cases, whereas for $b$9 the modes become effectively degenerate. Independent measurement of both scalar modes would constrain distinct theory parameters — e.g., the scalar mass in Horndeski and the combination $l$0 in Einstein-Æther — but whether LISA will detect sufficiently light MBHBs depends on the unknown seed-formation channel.

Two further analyses strengthen the framework. A joint amplitude-plus-phase injection shows $l$1 and $l$2 are essentially uncorrelated (relative error change $l$3), which enables independent constraints on combinations such as $l$4 versus $l$5 in Einstein-Æther. Allowing complex $l$6, the phase $l$7 is measurable only when the real part is large: precisions of $l$8–$l$9 for $E^{\rm TT}_{ij}$0 at $E^{\rm TT}_{ij}$1, degrading by more than an order of magnitude for weaker modifications.

Mapping back to specific theories, representative median $E^{\rm TT}_{ij}$2 uncertainties at $E^{\rm TT}_{ij}$3 include $E^{\rm TT}_{ij}$4 in Horndeski to $E^{\rm TT}_{ij}$5–$E^{\rm TT}_{ij}$6, $E^{\rm TT}_{ij}$7 in Einstein-Æther to $E^{\rm TT}_{ij}$8–$E^{\rm TT}_{ij}$9, and $\Phi$0 in Lightman-Lee to $\Phi$1. For Rosen and Lightman-Lee, the quadrupole-driven phase branch does not admit a GR limit, so those bounds come from dedicated non-GR injections. An appendix on Einstein-dilaton Gauss-Bonnet finds that MBHBs constrain $\Phi$2 only weakly (bounds of $\Phi$3–$\Phi$4 km), since the relevant combinations scale as $\Phi$5; LVK, EMRI, and 3G forecasts remain competitive there.

Limitations and open questions

The paper is explicit about several caveats. The Fisher-matrix approach covers only a restricted mass–redshift window ($\Phi$6, $\Phi$7); depending on the population model, only $\Phi$8–$\Phi$9 of expected events fall within it, and a hierarchical Bayesian analysis over the full population remains to be done. Only one PPE deviation is varied at a time (with one proof-of-concept two-parameter case), so correlations among multiple modified-gravity parameters are unexplored. All polarizations are assumed to propagate at the same speed, excluding massive or dispersion-modified propagation effects that could themselves carry information. The merger-ringdown treatment assumes GR-like dynamics with tapered modifications, which is internally inconsistent for Rosen's theory (which admits no black hole solutions) and unverified for the others; inspiral-only bounds are therefore the most robust for that theory. Screening mechanisms that could suppress deviations near compact objects are deliberately ignored, and higher-order PN corrections — potentially biasing recovery at LISA's SNRs — await computation. Finally, the breathing-versus-longitudinal discrimination claim rests on Fisher estimates; a Bayesian model-comparison analysis is stated to be ongoing.

Conclusion

This work provides the first systematic forecast of LISA's capacity to test GW polarization content using MBHBs within a theory-motivated PPE framework spanning all six metric-theory polarizations. Its principal quantitative findings — amplitude constraints of $\Theta$0–$\Theta$1 on extra-polarization parameters, 2–3 times stronger sensitivity to vector than scalar modes, tensor-phase precision of $\Theta$2–$\Theta$3 rad, and mass-dependent distinguishability of the two scalar modes below $\Theta$4 — establish LISA as a strong-field probe complementary to ground-based networks. The open questions left by the paper concern population-level hierarchical inference, global-fit treatment of overlapping signals, propagation-speed effects, and self-consistent merger-ringdown modeling in modified gravity.