  Gravitational waves observation with electromagnetic counterparts provides an
approach to measure the Hubble constant which is also known as the bright siren
method. Great hope has been put into this method to arbitrate the Hubble
tension. In this study, we apply the simulation tool \GWT\, and modeling of the
aLIGO-design background to simulate the bright siren catalogues of
sub-threshold double neutron star mergers with potential contamination from
noise and dis-pairing between gravitational waves and electromagnetic
counterparts. The Hubble constant and other cosmology parameters are thus
inferred from the simulated catalogues with a Bayesian method. From our
simulation study, we reach the following conclusions: 1) the measurement error
of the $H_0$ decreases with a lower signal-to-noise ratio threshold (or
equivalently the $P_{\rm astro}$) in the region where $P_{\rm astro} \gtrsim $
0.1, while the inferred most probable $H_0$ trends to bias towards larger
values; and 2) other higher order cosmological parameters such as $\Omega_{m}$
remain unconstrained even with the sub-threshold catalogues. We also discuss
adding the network of the gravitational wave detectors to the simulation tool
and the electromagnetic counterparts follow-up efficiency simulation, which
will improve our work in the future.
