tokens: visible. 3. Physical and Observational Constraints of the Allowed Parameter Space Region that Enables the Production of FRBs The physical conditions in the region where the masing takes place can be constrained using the physics of the maser and constraints from observations, allowing us to place limits on the magnetic fields and number densities where the synchrotron maser can plausibly be the emission mechanism for FRBs. We consider a cataclysmic FRB progenitor. However, as the repeating burst FRB 121102 is the only one to have a known redshift, it is the only source that provides observational constraints for quantities such as the burst energy and the DM of the host galaxy. Therefore, we use the values it provides as representative limits for our calculations. The data enable us to obtain constraints linking the size and number density of the masing region to the magnetic field of the neutron star. These constraints are obtained from: (i) the energetics of the burst and size of the masing region, (ii) the efficiency of the maser mechanism, (iii) the dispersion measure of the burst, and (iv) the frequency of the signal. We consider that masing takes place in a spherical shell of thickness d , located a distance R from the central object. The maser will be activated by the formation of a population inversion in the shell. The magnetic fields and short timescale (≲ d/c Γ) required suggest that this object is a neutron star, though the timescale for maser emission is given by the duration of the maser itself (Ghisellini 2017 ). Assuming that the blast wave is relativistic, the shocked plasma has a width ∼ R /Γ (Blandford McKee 1976 ), where Γ is the Lorentz factor of the blast wave. This typical width provides the first constraint, on the thickness of the shell:The minimum thickness of the shell depends on the number of particles that contribute to the masing, , and their number density, n e . Here, E is the energy of the bursts (in the range 10 38 –10 40 erg for the repeater (Law et al. 2017 ; Tendulkar et al. 2017 )), η is the fraction of the electrons’ energy that contributes to the maser, and is the average energy of the masing electrons. These shocked electrons have a thermal energy of γ ≈ Γ. Constraint (ii) originates from the efficiency of the maser. For the maser to be a viable emission mechanism, the growth rate of the signal must be large enough to extract a fraction η of the particle energy. The maser will be quenched when the maser reaches saturation. The efficiency of the maser mechanism in simulations of relativistic shocks was shown to be η ≲ 10 −1 (e.g., Gallant et al. 1992 ; Sironi Spitkovsky 2009 ). In the case of AKR, the efficiency is in the range η ∼ 10 −1 –10 −3 (Wu 1985 ). The exact value, however, depends on the form of the particle distribution, which is uncertain. We therefore examine values of η in the range 10 −3 ≲ η ≲ 10 −1 in this Letter. Lu Kumar ( 2018 ) gave upper limits to the efficiency of η ≲ 10 −5 ,
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