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6.7 Synchrotron radiation

Jacobson et al. [153Jump To The Next Citation Point] noticed that the synchrotron emission from the Crab nebula (and other astrophysical objects) is very sensitive to n = 3 modified dispersion relations. The logic behind the constraint in [153Jump To The Next Citation Point] is as follows. The Crab nebula emits electromagnetic radiation from radio to multi-TeV frequencies. As noted in Section 6.5.5, the spectrum is well fit over almost the entire frequency range by the synchrotron self-Compton model. If this model is correct, as seems highly probable, then the observed radiation from the Crab at 100 MeV is due to synchrotron emission from high energy electrons and/or positrons. The authors of [153Jump To The Next Citation Point], in the context of effective field theory, argued that the maximum frequency wc of synchrotron radiation in the presence of Lorentz violation is given by
3eBg3 wc = -------, (103) 2 E
where e is the charge, B is the magnetic field, and g and E are the gamma factor and energy of the source particle, respectively. The derivation of Equation (103View Equation) was challenged in [78]. More detailed calculations [222111221] show that in the case of the Crab nebula Equation (103View Equation) is correct, although the argument of [153Jump To The Next Citation Point] does not necessarily hold in general.28 Assuming the source particles are electrons, if f(e3) < 0 then there is a maximum electron velocity for any energy particle and hence a maximum possible value for wc. The maximum frequency must be above 100 MeV in the Crab, which leads to a constraint of either f (3) e,R or f(3) > - 7 × 10-8 e,L, i.e. at least one of the electron parameters must be above this value. The analysis of [153] does not take into account the possibility that the high energy synchrotron emission could be due to positrons, which may also be generated near the pulsar. This is an important possibility, since in the EFT that gives rise to f (3) terms for electrons, Equation (40View Equation), the positron has an opposite dispersion modification. Hence there is always some charged particle in the Crab with a dispersion modification that evades the synchrotron constraint. The possibility that there are two different populations, one electrons and one positrons, that contribute to the overall spectrum would be a departure from the synchrotron-self Compton model, which presupposes only one population of particles injected into the nebula. However, such a possibility cannot be ruled out without more detailed modelling of the Crab nebula and a better understanding of how the initial injection spectrum of particles from the pulsar is produced.29 The possible importance of positrons in the Crab implies that the synchrotron constraint is always satisfied when considered by itself. However, the synchrotron constraint can be combined with Čerenkov constraints to create a two sided bound [155Jump To The Next Citation Point]. Essentially, the SSC model is such that whatever species of particle is producing the synchrotron spectrum must also be responsible for the inverse Compton spectrum. Hence at least one helicity of electron or positron must satisfy both the vacuum Čerenkov and synchrotron constraints. This is not automatically satisfied in EFT and constitutes a true constraint. For more discussion see [155Jump To The Next Citation Point]. The combined synchrotron, threshold, birefringence, and time of flight constraints are displayed in Figure 4View Image (taken from [155]). In this plot j± is equal to fe(3()R,L) (not to be confused with the jR,L in Equation (40View Equation) which differs by a factor of two).
View Image

Figure 4: Constraints on LV in QED at n = 3 on a log-log plot. For negative parameters minus the logarithm of the absolute value is plotted, and region of width -10 10 is excised around each axis. The constraints in solid lines apply to q and both j±, and are symmetric about both the q and the j axis. At least one of the two pairs (j± ,q) must lie within the union of the dashed bell-shaped region and its reflection about the q axis. Intersecting lines are truncated where they cross.
Finally, note that if the effective field theory is CPT conserving then positrons/electrons have the same dispersion relation. So, for n = 2 dispersion the 100 MeV synchrotron radiation from the Crab yields a parallel constraint of roughly (2) f e,e+ > - 10-20 for at least one helicity of electron/positron.
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