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 $\sim 10^{-4}-10^{-2}$ precision level, depending on the frequency evolution of the modifications, for systems with $10^5-10^7 {\, \rm M}_\odot$ at $z = 1$. We find that LISA can distinguish breathing and longitudinal scalar polarizations only for relatively light binaries with $M \lesssim 10^4 {\, \rm M}_\odot$, 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 $\sim 10^{-8}-10^{-2}$, depending on the frequency evolution of the modifications, for systems with $10^5-10^7 {\, \rm M}_\odot$ at $z = 1$. These results demonstrate LISA's potential to probe gravity in the strong-field regime via gravitational wave polarizations.

Figures (25)

• Figure 1: Illustration of how the polarizations of a GW traveling towards $\hat{z}$ affect a ring of tests particles. The gray circle represents the unperturbed ring, while the blue and orange ellipses represent the perturbed ring at different moments in time, oscillating between orange and blue shapes. Figure adapted from the pioneering work Eardley:1973br (see also Will:2018bme).
• Figure 2: LISA consists of three spacecraft, labeled $A$, $B$, and $C$, arranged in an equilateral triangle with constant arm length $L$. Left: The solid line illustrates a single round-trip light path from spacecraft $A$ to $B$, as described in Section \ref{['subsec:LISA_Antenna_Response']}. Right: The solid and dashed lines represent the two round-trip light paths that form the Michelson interferometer signal, discussed in Section \ref{['subsec:aet_channels']}.
• Figure 3: Combination of sky-averaged response functions in the AE channels (left) and T channel (right) for different polarizations.
• Figure 4: Distribution of the absolute errors on $\alpha_T$ for different termination frequency (as reported in the legend) and different total mass $M$ and $z=1$. We assume a reference value of $\alpha_T=0$ and $a=-2$. Higher masses are more affected. For the rest of the paper, we adopt $\gamma=0.99$.
• Figure 5: Median SNR fractional variations due to $\alpha_T\not=0$, as a function of redshifted total mass $M_z$ and $\alpha_T$. Brighter colors indicate larger variations. In the left (right) panel we report the results for the case $a=-2$ ($a=0$). Note that the $y$-axis range and the color bars are different in the two panels. The values of $\alpha_T$ are chosen to cover the smallest and largest value of $\alpha_T$ LISA can constrain over all redshifted masses. Note the value of $\ell=2$ here is a reminder that we only modify the amplitude of the $(\ell = |m|=2)$ tensor angular harmonic.