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 $R_\mathrm{eq} \sim 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, $P_\mathrm{i} \gtrsim 10$ ms, ruling out millisecond magnetar progenitors. Given $P_\mathrm{i} \gtrsim 10$ ms, a current size of $\sim 0.1$ pc, a supernova kinetic energy $E_\mathrm{ SN} \sim 10^{50}-10^{51}$ erg and an ejecta mass $M \sim 3-10 \; M_{\odot}$, the PRS age is $t \sim 10-10^{2}$ yr. PRSs with $t>20$ years require an internal field ($B_\mathrm{int} \sim 10^{16}-10^{16.5}\;$G) with a decay timescale $t_\mathrm{d} \sim 10-10^{2.5}$ yr. The slowest field decay ($t_\mathrm{d,max} \sim 500$ yr) favors sub-energetic supernovae ($E_\mathrm{SN} \sim 10^{50}$ erg) with massive ejecta ($M \gtrsim 10\; M_{\odot}$) and low ionization fraction ($\sim 3\% $). For the sub-energetic scenario for the confirmed PRSs, we predict a cooling break at $100-150$ gigahertz at $20-40 \; \mu \text{Jy}$ and self-absorption near 200 megahertz at $180\; \mu \text{Jy}$. For PRS3, a rotation-powered MWN is viable only if $t \sim 10$ yr; an inverted spectrum beyond 150 gigahertz would rule out this scenario.