Cosmology with LIGO/Virgo dark sirens: Hubble parameter and modified
  gravitational wave propagation
  We present a detailed study of the methodology for correlating `dark sirens'
(compact binaries coalescences without electromagnetic counterpart) with galaxy
catalogs. We propose several improvements on the current state of the art, and
we apply them to the GWTC-2 catalog of LIGO/Virgo gravitational wave (GW)
detections, and the GLADE galaxy catalog, performing a detailed study of
several sources of systematic errors that, with the expected increase in
statistics, will eventually become the dominant limitation. We provide a
measurement of $H_0$ from dark sirens alone, finding as the best result
$H_0=67.3^{+27.6}_{-17.9}\,\,{\rm km}\, {\rm s}^{-1}\, {\rm Mpc}^{-1}$ ($68\%$
c.l.) which is, currently, the most stringent constraint obtained using only
dark sirens. Combining dark sirens with the counterpart for GW170817 we find
$H_0= 72.2^{+13.9}_{-7.5} \,{\rm km}\, {\rm s}^{-1}\, {\rm Mpc}^{-1}$.
  We also study modified GW propagation, which is a smoking gun of dark energy
and modifications of gravity at cosmological scales, and we show that current
observations of dark sirens already start to provide interesting limits. From
dark sirens alone, our best result for the parameter $\Xi_0$ that measures
deviations from GR (with $\Xi_0=1$ in GR) is $\Xi_0=2.1^{+3.2}_{-1.2}$. We
finally discuss limits on modified GW propagation under the tentative
identification of the flare ZTF19abanrhr as the electromagnetic counterpart of
the binary black hole coalescence GW190521, in which case our most stringent
result is $\Xi_0=1.8^{+0.9}_{-0.6}$.
  We release the publicly available code $\tt{DarkSirensStat}$, which is
available under open source license at
\url{https://github.com/CosmoStatGW/DarkSirensStat}.
