  We explore inferences on ultrahigh energy cosmic ray (UHECR) source
environments -- constrained by the spectrum and composition of UHECRs and
non-observation of extremely high energy neutrinos -- and their implications
for the observed high energy astrophysical neutrino spectrum. We find
acceleration mechanisms producing power-law CR spectra~$\propto E^{-2}$ are
compatible with UHECR data, if CRs at high rigidities are in the
quasi-ballistic diffusion regime as they escape their source environment. Both
gas-dominated and photon-dominated source environments are able to account for
UHECR observations, however photon-dominated sources give a better fit.
Additionally, gas-dominated sources are in tension with current neutrino
constraints. Accurate measurement of the neutrino flux at $\sim 10$ PeV will
provide crucial information on the viability of gas-dominated sources, as well
as whether diffusive shock acceleration is consistent with UHECR observations.
We also show that UHECR sources are able to give a good fit to the high energy
portion of the astrophysical neutrino spectrum, above $\sim$ PeV. This common
origin of UHECRs and high energy astrophysical neutrinos is natural if air
shower data is interpreted with the Sibyll2.3c hadronic interaction model,
which gives the best-fit to UHECRs and astrophysical neutrinos in the same part
of parameter space, but not for EPOS-LHC.
