  Standard sirens are the gravitational wave (GW) analog of the astronomical
standard candles, and can provide powerful information about the dynamics of
the Universe. In this work, we simulate a catalog with 1000 standard siren
events from binary neutron star mergers, within the sensitivity predicted for
the third generation of the ground GW detector called Einstein telescope. After
correctly modifying the propagation of GWs as input to generate the catalog, we
apply our mock data set on scalar-tensor theories where the speed of GW
propagation is equal to the speed of light. As a first application, we find new
observational bounds on the running of the Planck mass, when considering
appropriate values within the stability condition of the theory, and we discuss
some consequences on the amplitude of the running of the Planck mass. In the
second part, we combine our simulated standard sirens catalog with other
geometric cosmological tests (Supernovae Ia and cosmic chronometers
measurements) to constrain the Hu-Sawicki $f(R)$ gravity model. We thus find
new and non-null deviations from the standard $\Lambda$CDM model, showing that
in the future the $f(R)$ gravity can be tested up to 95\% confidence level. The
results obtained here show that the statistical accuracy achievable by future
ground based GW observations, mainly with the ET detector (and planed detectors
with a similar sensitivity), can provide strong observational bounds on
modified gravity theories.
