Fast radio burst counterparts and their implications for the central engine

While the radiation mechanism of fast radio bursts (FRBs) is unknown, coherent curvature radiation and synchrotron maser are promising candidates. We find that both radiation mechanisms work for a neutron star (NS) central engine with $B\gtrsim 10^{12}$ G, while for the synchrotron maser, the central engine can also be an accreting black hole (BH) with $B\gtrsim 10^{12}$ G and a white dwarf (WD) with $B\sim 10^8-10^9$ G. We study the electromagnetic counterparts associated with such central engines, i.e., nebulae for repeating FRBs and afterglows for non-repeating FRBs. In general, the energy spectrum and flux density of the counterpart depend strongly on its size and total injected energy. We apply the calculation to the nebula of FRB 121102 and find that the persistent radio counterpart requires the average energy injection rate into the nebula to be between $2.7\times10^{39}~{\rm erg/s}$ and $1.5\times10^{44}~{\rm erg/s}$, and the minimum injected energy be $6.0\times10^{47}~{\rm erg}$ in around $7$ yr. Consequently, we find that for FRB 121102 and its nebula: (1) WD and accretion BH central engines are disfavored; (2) a rotation-powered NS central engine works when $1.2\times10^{12}~{\rm G}\lesssim B\lesssim 7.8\times10^{14}~{\rm G}$ with initial period $P<180$ ms, but the radio emission must be more efficient than that in typical giant pulses of radio pulsars; and (3) a magnetic-powered NS central engine works when its internal magnetic field $B\gtrsim 10^{16}$ G. We also find that the radio-emitting electrons in the nebula could produce a significant rotation measure (RM), but cannot account for the entire observed RM of FRB 121102.