  Gravitational waves (GWs) from merging black holes and neutron stars directly
measure the luminosity distance to the merger, which, when combined with an
independent measurement of the source's redshift, provides a novel probe of
cosmology. The proposed next generation of ground-based GW detectors, 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. We argue
that even in the absence of an EM counterpart or galaxy catalog, the BNS
redshift distribution will be measured by independent observations of short
gamma ray bursts (GRBs), kilonovae, and known BNS host galaxies. As a simple
example, we consider the case in which the BNS rate is \textit{a priori} known
to follow the SFR and explore how combining this redshift distribution with
measurements of GW distances can constrain cosmology and modified gravity
theories. We find that $\mathcal{O}(10,000)$ events (to be expected within a
year of observation with Cosmic Explorer) would yield a sub-tenth percent
measurement of the combination $H_0^{2.8}\Omega_M$ in a flat $\Lambda$CDM
model. Beyond $\Lambda$CDM, this method would enable a 5\% measurement of the
dark energy equation of state parameter $w$ given a sub-percent prior
measurement on $H_0$ and $\Omega_M$. Alternatively, fixing the background
cosmology and instead probing modified GW propagation, we expect to constrain
the running of the Planck mass parameter $c_M$ to $\pm0.02$.
