The cosmological impact of future constraints on $H_0$ from
  gravitational-wave standard sirens
  Gravitational-wave standard sirens present a novel approach for the
determination of the Hubble constant. After the recent spectacular confirmation
of the method thanks to GW170817 and its optical counterpart, additional
standard siren measurements from future gravitational-wave sources are expected
to constrain the Hubble constant to high accuracy. At the same time, improved
constraints are expected from observations of cosmic microwave background (CMB)
polarization and from baryon acoustic oscillations (BAO) surveys. We explore
the role of future standard siren constraints on $H_0$ in light of expected
CMB+BAO data. Considering a $10$-parameters cosmological model, in which
curvature, the dark energy equation of state, and the Hubble constant are
unbounded by CMB observations, we find that a combination of future CMB+BAO
data will constrain the Hubble parameter to $\sim 1.5 \%$. Further extending
the parameter space to a time-varying dark energy equation of state, we find
that future CMB+BAO constraints on $H_0$ are relaxed to $\sim 3.0 \%$. These
accuracies are within reach of future standard siren measurements from the
Hanford-Livingston-Virgo and the Hanford-Livingston-Virgo-Japan-India networks
of interferometers, showing the cosmological relevance of these sources. If
future gravitational-wave standard siren measurements reach $1\%$ on $H_0$, as
expected, they would significantly improve future CMB+BAO constraints on
curvature and on the dark energy equation of state by up to a factor $\sim 3$.
We also show that the inclusion of $H_0$ constraints from gravitational-wave
standard sirens could result in a reduction of the dark energy figure-of-merit
(i.e., the cosmological parameter volume) by up to a factor of $\sim 400$.
