Scatter in the Relation between Persistent Radio Source Luminosity and Fast Radio Burst Rotation Measure: A Window into Circum-burst Environments

The association of persistent radio sources (PRSs) with repeating fast radio bursts (FRBs) offers unique insights into their circum-burst environments. Building upon the physical link between PRS luminosity ($L_{\nu }$) and FRB rotation measure (RM), we introduce a novel diagnostic framework utilizing the intrinsic scatter of the $L_{\nu }-\mathrm{\mid }\mathrm{R}\mathrm{M}\mathrm{\mid }$ relation as a physical probe of nebula dynamics. We demonstrate that this scatter encodes critical information regarding the temporal evolution of the nebula radius ($R\propto t^{\alpha }$). By deriving a generic scaling $L_{\nu }\propto R^{\epsilon }\mathrm{\mid }\mathrm{R}\mathrm{M}\mathrm{\mid }$ and analyzing the residuals of the five confirmed FRB-PRS systems, we constrain the nebula’s evolutionary index to $\alpha \mathrm{\mid }\epsilon \mathrm{\mid }=1.5\pm 0.7$ (1$\sigma$ uncertainty). This measurement provides a powerful diagnostic tool for distinguishing among different astrophysical scenarios. Its value deviates from the expectations for supernova remnants (SNRs) in the Sedov-Taylor phase ($\alpha \mathrm{\mid }\epsilon \mathrm{\mid }\sim 0.2-0.4$), reverse shocks during the free-expansion phase of SNR/interstellar medium (ISM) interactions ($\alpha \mathrm{\mid }\epsilon \mathrm{\mid }\gtrsim 3.5$), and pulsar wind nebulae (PWNe) powered by a decaying wind ($\alpha \mathrm{\mid }\epsilon \mathrm{\mid }\sim 0-0.15$). Instead, it is more consistent with forward shocks in the free-expansion phase of both SNR/ISM and PWN/SNR systems ($\alpha \mathrm{\mid }\epsilon \mathrm{\mid }\sim 2.0-2.8$), and young PWNe driven by a nearly constant wind ($\alpha \mathrm{\mid }\epsilon \mathrm{\mid }\sim 1$). These findings suggest that active repeaters are powered by dynamically young, rapidly expanding nebulae. While currently limited by the small sample size, this framework establishes a robust methodology for future population studies to constrain the physical origin of PRSs.