Joint cosmological inference of standard sirens and gravitational wave
  weak lensing
  We present the first joint inference of standard sirens and gravitational
wave weak lensing by filtering of the same dataset. We imagine a post-LISA
scenario emerging around the late 2030s when LISA will have accumulated a
number of detections at high redshift; LIGO-VIRGO will have finished observing
at low redshift, and Einstein Telescope will have started making new
observations out to redshifts possibly overlapping with LISA. Euclid and other
cosmological probes will have provided constraints at percent level by then,
but mostly exhausted their ability to improve any further. We derive forecasts
assuming $\sim1\,\text{deg}^{-2}$ detected sources, in conjunction with a
spectroscopic follow-up (e.g. by Euclid, DESI, or ATHENA). Thanks to the
statistical power of standard sirens as a geometry probe -- lifting key
degeneracies in the gravitational wave weak lensing -- and no external priors
assumed, the constraints on dark matter and its clustering, namely $\Omega_m$
and $\sigma_8$, could be achieved to $2\%$ and $3\%$. The Hubble constant could
be constrained to better than $1\%$ in all cases; the dark energy density,
$\Omega_\Lambda$, to $2\%$, and curvature, $\Omega_K$, to $0.02$; the amplitude
and spectral tilt of the scalar fluctuations, $\ln(10^{10}A_s)$ and $n_s$, to
$2\%$ and $7\%$. As a completely independent cosmological probe, with less
calibration requirements, the joint inference of standard sirens and
gravitational wave weak lensing might help solve the tensions currently
observed between other cosmological probes, such as CMB, galaxy lensing, and
Type Ia SNs, and distinguish between residual systematics and new physics.
