  Gravitational waves (GWs) 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. 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.
