Associating Fast Radio Bursts with Extragalactic Radio Sources: General Methodology and a Search for a Counterpart to FRB 170107

The discovery of a repeating fast radio burst has led to the first precise localization, an association with a dwarf galaxy, and the identification of a coincident persistent radio source. However, further localizations are required to determine the nature of FRBs, the sources powering them, and the possibility of multiple populations. Here we investigate the use of associated persistent radio sources to establish FRB counterparts, taking into account the localization area and the persistent source flux density. Due to the lower areal number density of radio sources compared to faint optical sources, robust associations can be achieved for less precise localizations as compared to direct optical host galaxy associations. For generally larger localizations which preclude robust associations, the number of candidate hosts can be reduced based on the ratio of radio-to-optical brightness. We find that confident associations with $\sim 0.01-$1 mJy sources, comparable to the luminosity of the persistent source associated with FRB 121102 over the redshift range $z\approx 0.1-1$, require FRB localizations of $\lesssim 20’'$. In the absence of a robust association, constraints can be placed on the luminosity of an associated radio source as a function of localization and DM. For DM $\approx 1000\mathrm{p}\mathrm{c}\mathrm{c}{\mathrm{m}}^{-3}$, an upper limit comparable to the luminosity of the FRB 121102 persistent source can be placed if the localization is $\lesssim {10}^{\mathrm{\prime }\mathrm{\prime }}$. We apply our analysis to the case of the ASKAP FRB 170107, using optical and radio observations of the localization region. We identify two candidate hosts based on a ratio of radio-to-optical brightness of $\gtrsim 100$. We find that if one of these is associated with FRB 170107, the resulting radio luminosity ($1 \times 10^{29} - 4 \times 10^{30} \ \rm erg
s^{-1} \ Hz^{-1}$) is comparable to the luminosity of the FRB 121102 persistent source.