We study the effect of torques on circular inspirals of intermediate-mass
    black hole binaries (IMBHBs) embedded in gas discs, wherein both BH masses are
    in the range $10^2$-$10^5~\rm{M}_\odot$, up to redshift $z = 10$. We focus on
    how torques impact the detected gravitational wave (GW) waveform in the
    frequency band of the Laser Interferometer Space Antenna (LISA) when the binary
    separation is within a few hundred Schwarzschild radii. For a sub-Eddington
    accretion disc with a viscosity coefficient $\alpha=0.01$, surface density
    $\Sigma\approx10^5$ g cm$^{-2}$, and Mach number $\mathcal{M}_{\rm
    a}\approx80$, a gap, or a cavity, opens when the binary is in the LISA band.
    Depending on the torque’s strength, LISA will observe dephasing in the IMBHB’s
    GW signal up to either $z\sim5$ for high mass ratios ($q\approx0.1$) or to
    $z\sim7$ for $q\approx10^{-3}$. We study the dependence of the measurable
    dephasing on variations of BH masses, redshift, and accretion rates. Our
    results suggest that phase shift is detectable even in high-redshift ($z = 10$)
    binaries, provided that they experience super-Eddington accretion episodes. We
    investigate if the disc-driven torques can result in an observable
    `time-dependent’ chirp mass with a simplified Fisher formalism, finding that,
    at the expected signal-to-noise ratio, the gas-induced variation of the chirp
    mass is too small to be detected. This work shows how perturbations of vacuum
    waveforms induced by gas should be strong enough to be detected by LISA for the
    IMBHB in the early inspiral phase. These perturbations encode precious
    information on the astrophysics of accretion discs and galactic nuclei.
    High-accuracy waveform models which incorporate these effects will be needed to
    extract such information.

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