Persistent radio sources associated with fast radio bursts: Implications from magnetar progenitors

The rare association of three persistent radio sources (confirmed PRS1 and PRS2, candidate PRS3) with repeating fast radio bursts (FRB 20121102A, 20190520B, 20201124A) offers a unique probe into their magneto-ionic environments. PRSs are attributed to synchrotron emission from relativistic charged particles of magnetar wind nebula (MWN) powered by spin-down magnetohydrodynamic wind or internal magnetic field decay. Using a multizone hydrodynamic model, we track MWN evolution to constrain magnetar progenitor properties. For PRS1 and PRS2, we find an equipartition radius Rₑq ~ 0.1 pc that is consistent with the radio scintillation estimates ($> 0.03$ pc) and radio imaging limits ($<0.7$ pc). This compact size favors low expansion speeds and large initial spin periods, Pᵢ 10 ms, ruling out millisecond magnetar progenitors. Given Pᵢ 10 ms, a current size of ~ 0.1 pc, a supernova kinetic energy E_ SN ~ 10⁵⁰-10⁵¹ erg and an ejecta mass M ~ 3-10 \; M_, the PRS age is t ~ 10-10² yr. PRSs with $t>20$ years require an internal field (Bᵢnt ~ 10¹⁶-1016.5\;G) with a decay timescale td ~ 10-102.5 yr. The slowest field decay (td,max ~ 500 yr) favors sub-energetic supernovae (ESN ~ 10⁵⁰ erg) with massive ejecta (M 10\; M_) and low ionization fraction (~ 3%). For the sub-energetic scenario for the confirmed PRSs, we predict a cooling break at $100-150$ gigahertz at 20-40 \; μJy and self-absorption near 200 megahertz at 180\; μJy. For PRS3, a rotation-powered MWN is viable only if t ~ 10 yr; an inverted spectrum beyond 150 gigahertz would rule out this scenario.