  The upcoming Laser Interferometer Space Antenna (LISA), set for launch in the
mid-2030s, will enhance our capability to probe the universe through
gravitational waves (GWs) emitted from binary black holes (BBHs) across a broad
range of cosmological distances. LISA is projected to observe three classes of
BBHs: massive BBHs (MBBHs), extreme mass-ratio inspirals (EMRIs), and stellar
mass BBHs. This study focuses on MBBHs, which are anticipated to occur in
gas-rich environments conducive to producing powerful electromagnetic (EM)
counterparts. This capability to precisely localize MBH events in the sky
positions them as excellent candidates for bright sirens. By combining GW
luminosity distance measurements from these bright sirens with Baryon Acoustic
Oscillation (BAO) measurements derived from galaxy clustering and sound horizon
measurements from the Cosmic Microwave Background (CMB), we propose a
data-driven model-independent method to reconstruct deviations in the variation
of the effective Planck mass (in conjunction with the Hubble constant) as a
function of cosmic redshift. Using this multi-messenger technique, we achieve
precise measurements of deviations in the effective Planck mass variation with
redshift (z), with a precision ranging from approximately $2.4\%$ to $7.2\%$
from redshift $z=1$ to $z=6$ with a single event. Additionally, we achieved a
measurement of the Hubble constant with a precision of about $1.3\%$,
accounting for variations in the effective Planck mass over 4 years of
observation time ($T_{\mathrm{obs}}$). This assumes that EM counterparts are
detected for $75\%$ of the events. This precision improves with observation
time as $T_{\mathrm{obs}}^{-1/2}$. This approach not only has the potential to
reveal deviations from General Relativity but also to significantly expand our
understanding of the universe's fundamental physical properties.
