Measuring interstellar turbulence in fast radio burst host galaxies

Turbulence is a vital part of the interstellar medium (ISM) of galaxies, contributing significantly to galaxy energy budgets and acting as a regulator of star formation. Despite this, little is understood about ISM turbulence empirically. In the Milky Way, multiple tracers are used to reconstruct the density- and velocity-fluctuation power spectra over an enormous range of scales, but questions remain on the nature of these fluctuations at the smallest scales. Even less is known about the ISM of distant galaxies, where only a few tracers of turbulence, such as non-thermal broadening of optical recombination lines, are accessible. We explore the use of radio-wave scattering of fast radio bursts (FRBs) to add a second probe of turbulence in extragalactic galaxies on scales many orders of magnitude smaller than those probed by emission-line widths. We first develop the formalism to compare the scattering measures of FRBs to alternative probes of density and velocity fluctuations in the host-galaxy ISM. We then apply this formalism to three FRBs with detailed host-galaxy analyses in the literature, with the primary motivation of determining whether FRB scattering within the host galaxy probes the same turbulent cascade as the gas seen in H${\rm \alpha}$ emission. In all cases we consider, we find such an association plausible, although in one of these sources, FRB 20121102A, the radio-scattering limit on the turbulent energy is much less constraining than the H${\rm \alpha}$ line width. We anticipate that future FRB surveys, especially those at frequencies below 1 GHz, will find many FRBs that illuminate the small-scale properties of extragalactic ISM.