Cosmological inference using only gravitational wave observations of binary neutron stars

[Abridged] This study presents the first Bayesian investigation of the accuracy with which the cosmological parameters can be measured using information coming \emph{only} from the gravitational wave observations of binary neutron star systems by Einstein Telescope. We find, by direct simulation of $10^3$ detections of binary neutron stars, that, within our simplifying assumptions, $H_0,\Omega_m,\Omega_\Lambda,w_0$ and $w_1$ can be measured at the $95\%$ level with an accuracy of $\sim 8\%,65\%,39\%,80\%$ and $90\%$, respectively. We also find, by extrapolation, that a measurement accuracy comparable with current measurements by Planck is possible if the number of gravitational wave events observed is $O(10^{6-7})$. We conclude that, while not competitive with electro-magnetic missions in terms of significant digits, gravitational wave alone are capable of providing a complementary determination of the dynamics of the Universe.