Cosmology with the lights off: Standard sirens in the Einstein Telescope era

We explore the prospects for constraining cosmology using gravitational-wave (GW) observations of neutron-star binaries by the proposed Einstein Telescope (ET), exploiting the narrowness of the neutron-star mass function. This builds on our previous work in the context of advanced-era GW detectors. Double neutron-star (DNS) binaries are expected to be one of the first sources detected after ``first-light'' of Advanced LIGO. DNS systems are expected to be detected at a rate of a few tens per year in the advanced era, but the proposed ET could catalog tens, if not hundreds, of thousands per year. Combining the measured source redshift distributions with GW-network distance determinations will permit not only the precision measurement of background cosmological parameters, but will provide an insight into the astrophysical properties of these DNS systems. Of particular interest will be to probe the distribution of delay times between DNS-binary creation and subsequent merger, as well as the evolution of the star-formation rate density within ET's detection horizon. Keeping ${H}_{0}$, ${\ensuremath{\Omega}}_{m,0}$ and ${\ensuremath{\Omega}}_{\ensuremath{\Lambda},0}$ fixed and investigating the precision with which the dark-energy equation-of-state parameters could be recovered, we found that with ${10}^{5}$ detected DNS binaries, we could constrain these parameters to an accuracy similar to forecasted constraints from future $\mathrm{CMB}+\mathrm{BAO}+\mathrm{SNIa}$ measurements. Furthermore, modeling the merger delay-time distribution as a power-law ($\ensuremath{\propto}{t}^{\ensuremath{\alpha}}$) and the star-formation rate density as a parametrized version of the Porciani and Madau SF2 model, we find that the associated astrophysical parameters are constrained to within $\ensuremath{\sim}10%$. All parameter precisions scaled as $1/\sqrt{N}$, where $N$ is the number of cataloged detections. We also investigated how parameter precisions varied with the intrinsic underlying properties of the Universe and with the distance reach of the network (which is affected, for instance, by the low-frequency cutoff of the detector). We also consider various sources of distance-measurement errors in the third-generation era and how these can be folded into the analysis.