Constraints on the Cosmic-Ray Ionization Rate in the $z\sim2.3$ Lensed Galaxies SMM J2135$-$0102 and SDP 17b from Observations of OH$^+$ and H$_2$O$^+$.

Cosmic rays are predominantly accelerated in shocks associated with star formation such as supernova remnants and stellar wind bubbles, so the cosmic-ray flux and thus cosmic-ray ionization rate, $\zeta_{\rm H}$, should correlate with the star-formation rate in a galaxy. Submillimeter bright galaxies (SMGs) are some of the most prolific star forming galaxies in the Universe, and gravitationally lensed SMGs provide bright continuum sources suitable for absorption line studies. Abundances of OH$^+$ and H$_2$O$^+$ are useful for inferring $\zeta_{\rm H}$ when combined with chemical models, and have been used for this purpose within the Milky Way. At redshifts $z\gtrsim2$ transitions out of the ground rotational states of OH$^+$ and H$_2$O$^+$ are observable with ALMA, and we present observations of both molecules in absorption toward the lensed SMGs SMM J2135$-$0102 and SDP 17b. These detections enable an exploration of $\zeta_{\rm H}$ in galaxies with extreme star formation and high supernova rates, both of which should significantly enhance cosmic-ray production. The observed OH$^+$ and H$_2$O$^+$ absorption is thought to arise in massive, extended halos of cool, diffuse gas that surround these galaxies. Using a chemical model designed to focus on the reaction network important to both species, we infer cosmic-ray ionization rates of $\zeta_{\rm H}\sim10^{-16}$-$10^{-14}$ s$^{-1}$ in these extended gaseous halos. As our estimates come from gas that is far away from the sites of cosmic-ray acceleration, they imply that cosmic-ray ionization rates in the compact regions where star formation occurs in these galaxies are orders of magnitude higher.