Gravitational waves (GWs) directly measure the luminosity distance to the merger, which, when combined with an independent measurement of the sources redshift, provides a novel probe of cosmology. The proposed next generation of ground-based GW detec
tors, Einstein Telescope and Cosmic Explorer, will detect tens of thousands of binary neutron stars (BNSs) out to cosmological distances ($z>2$), beyond the peak of the star formation rate (SFR), or cosmic noon. At these distances, it will be challenging to measure the sources redshifts by observing electromagnetic (EM) counterparts or statistically marginalizing over a galaxy catalog. In the absence of an EM counterpart or galaxy catalog, Ding et al. showed that theoretical priors on the merger redshift distribution can be used to infer parameters in a $w$CDM cosmology. We argue that in the BNS case, the redshift distribution will be measured by independent observations of short gamma ray bursts (GRBs), kilonovae, and known BNS host galaxies. We show that, in addition to measuring the background cosmology, this method can constrain the effects of dark energy on modified GW propagation. We consider the simple case in which the BNS rate is textit{a priori} known to follow the SFR. If the SFR is perfectly known, $mathcal{O}(10,000)$ events (to be expected within a year of observation with Cosmic Explorer) would yield a sub-tenth percent measurement of the combination $H_0^{2.8}Omega_M$. Fixing $H_0$ and $Omega_M$, this method may enable a 5% measurement of the dark energy equation of state parameter. Fixing the background cosmology and probing modified GW propagation, the running of the Planck mass parameter $c_M$ may be measured to $pm0.02$. Although realistically, the redshift evolution of the merger rate will be uncertain, prior knowledge of the peak redshift will provide valuable information for standard siren analyses.
