Recent studies of fast radio bursts (FRBs) have led to many theories associating them with young neutron stars. If this is the case, then the presence of supernova ejecta and stellar winds provide a changing dispersion measure (DM) and rotation measure (RM) that can potentially be probes of the environments of FRB progenitors. Here we summarize the scalings for the DM and RM in the cases of a constant density ambient medium and of a progenitor stellar wind. Since the amount of ionized material is controlled by the dynamics of the reverse shock, we find the DM changes more slowly than in previous simpler work, which simply assumed a constant ionization fraction. Furthermore, the DM can be constant or even increasing as the supernova remnant sweeps up material, arguing that a young neutron star hypothesis for FRBs is not ruled out if the DM is not decreasing over repeated bursts. The combined DM and RM measurements for the repeating FRB 121102 are consistent with supernova ejecta with an age of $sim10^2-10^3,{rm yrs}$ expanding into a high density ($sim100,{rm cm^{-3}}$) interstellar medium. This naturally explains its relatively constant DM over many years as well. Other FRBs with much lower RMs may indicate that they are especially young supernovae in wind environments or that their DMs are largely from the intergalactic medium. We therefore caution about inferring magnetic fields from simply by dividing an RM by DM, because these quantities could originate from distinct regions along the path an FRB propagates.
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