tokens: be directly detected. In this case, T may be adopted as a small value . Our observations require that the maximum γ -ray luminosity at the relevant age T is smaller than the upper limits, i.e.,where , and is the eight-year limit presented in the last row in Table 1 . Given a set of magnetar parameters P 0 and B p , the spin-down luminosity depends on the age of the magnetar, T . Since the earliest burst, FRB 121102, was detected in late 2012, the magnetar needs to be at least four years old until now. We consider two possibilities. First, if the pulsar was born between 2009 January 1 and 2012 November 2, one should use the data during that period of time to constrain magnetar parameters assuming different transparent times by setting in Equation ( 1 ). The transparent time depends on the amount of ejecta and opacity, which may range from minutes (for NS–NS mergers) to ∼ years (for supernovae). In our calculation, we use 10 2 s, 10 4 s, 0.1 years, 0.5 years, and 1 year, respectively. Since the upper limits are slightly different in different time bins, we use the average upper limits to derive the constraints as presented in Figure 2 . Figure 2. Constraints of the pulsar parameter by the LAT upper limit. Solid lines: cases of dotted lines: cases of dashed lines: cases of . The horizontal line denotes the break-up limit P 0 = 0.6 ms. The parameters are constrained to the regions that are above the curves/lines. The second possibility is that the magnetar was born before 2009 January 1. For these cases, the time T should be adopted as the age of the magnetar when the first observation started (in 2009 January 1). When the birth time is much earlier than 2009 January 1, one can use the tighter upper limit over a much longer time (e.g., four-year or eight-year) limit to perform the constraint. For this case, we consider T = 10 years as an example. The results for , 10 4 s, 0.1 years, 0.5 years, 1.0 year, and 10 years are plotted in the plane ( and assumed) as shown in Figure 2 . For each T , we consider , , and . The region above each curve is the parameter space allowed for the given T and η value. Since the minimum spin period of a millisecond pulsar would be the break-up limit ms (e.g., Vink Kuiper 2006 ), our results suggest that there is essentially no constraint from the upper limits if the pulsar age in the case of . This means that the current limit cannot constrain the magnetar parameter if it was born before Fermi observations started. On the other hand, for and if the magnetar age is 4–8 years and the transparent time shorter than 0.1 years, the upper limits in the first four years could place some constraints on B p of the pulsar. In particular, if the transparent time is short enough, the non-detections exclude high magnetic field strengths. As a result, if FRB 121102 progenitor was indeed born after 2009 January 1 and has a magnetar origin, it must have been born in a dirty environment with a delayed transparent time (say years) in
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