Fast radio burst - persistent radio source systems III. The relation between PRS luminosity and FRB rotation measure

Abstract

Fast radio bursts (FRBs) are millisecond-duration radio transients of extragalactic origin whose physical origin remains uncertain. A small fraction of FRBs are known to repeat, and some of them are associated with persistent radio sources (PRSs), interpreted as synchrotron-emitting nebulae surrounding the FRB source. In the context of magnetar-based models, the rotation measure (RM) of an FRB is expected to correlate with the spectral luminosity of its associated PRS, providing a probe of the physical properties and evolution of the nebula. We investigate the relation between FRB RM and PRS spectral luminosity, constrain the characteristic size of the PRS nebulae, and use the intrinsic scatter of the relation to investigate evolutionary scenarios for FRB--PRS systems. We analyse a sample of 50 FRB sources with known RM, including both PRS detections and spectral luminosity upper limits, using a Bayesian MCMC framework that jointly accounts for detections and non-detections. We model the luminosity--RM relation as $L_ν\propto ζ_e γ_c^2 R^2 {\rm RM}^β$, constraining the characteristic nebular size $R$ and the RM scaling index $β$. For the full sample, we obtain $R = 0.016^{+0.115}_{-0.014}$ pc at $1σ$ confidence for a free $β$, while fixing the RM dependence to the canonical linear scaling, $β= 1$, gives $R = 0.008^{+0.058}_{-0.007}$ pc. The data mildly favour super linear RM scalings, although the inferred values of $β$ remain consistent with $β= 1$ within $2σ$. We find a substantial intrinsic scatter in the luminosity--RM relation, significantly larger than previous estimates based on confirmed PRSs alone. Interpreting this scatter within evolutionary models, our results favour scenarios involving efficient particle acceleration and/or rapid nebular expansion.