*3.3. Small Values of e<sup>B</sup>*

For a long time, the typical value of *e<sup>B</sup>* has been considered to variate between 0.01 and 0.1, both on the basis of theoretical considerations on particle acceleration and findings by numerical simulations. Indeed, the present understanding of the micro-physics at weakly magnetized shocks invoke the existence of self-generated micro-turbulence both behind and in front of the shock, at a level corresponding to *e<sup>B</sup>* ∼ 0.01–0.1. This layer of intense micro-turbulence is expected on theoretical grounds and recently corroborated by numerical PIC simulations. Inferences of the value of *e<sup>B</sup>* from the early modeling of afterglow radiation were broadly consistent with these numbers, confirming the presence of large self-generated fields in ultra-relativistic weakly magnetized shocks. More recently, several independent methods have provided evidence for significantly lower values.

In particular, several studies on GRBs with GeV temporally extended emission detected by LAT arrive to the same conclusions that in order to explain GeV radiation as part of the synchrotron emission, multi-wavelength observations require *e<sup>B</sup>* = 10−6–10−<sup>3</sup> [24,109–112]. Similar values have been inferred from studies that are based on radio, optical and/or X-ray emission and do not make use of high energy emission, such as [67–69,113]. A smaller magnetic field in the region where most of the particle cooling occurs might increase the expected relevance of the SSC component, as supported by recent detections of TeV radiation by IACTs.

Such small values of *e<sup>B</sup>* may appear to be problematic [114], because strongly selfgenerated micro-turbulence must be present to ensure the scattering and acceleration of the particles, which otherwise would be simply advected away.

It was later pointed out that the inferred low values of magnetization might be indicative of a turbulence that is decaying on time-scales comparable with the electron cooling time [65,66]. From a theoretical perspective, indeed, the micro-turbulence is expected to decay beyond some hundreds of skin depths. This picture has been validated by PIC simulations, which however are still far from probing time-scales comparable with the dynamical time-scale of the system. Dedicated simulations the magnetic field does decay behind the shock, on a time-scale much longer than *cωpi*. Immediately behind the shock, the magnetic field carries a magnetization *<sup>e</sup><sup>B</sup>* <sup>∼</sup> 0.01, which decays in time after <sup>10</sup>2–10<sup>3</sup> plasma times. Eventually, the magnetic field will settle to the shock-compressed value 4Γ *Bu*, where *B<sup>u</sup>* is the magnetization of the upstream unperturbed medium. In this scenario, high-energy particles, which produce MeV-GeV photons, feel only the region close to the shock, where the magnetization is large, due to their short cooling time. Particles that cool on longer time-scales (and produce radio, optical and X-ray photons) cool on longer time-scales, and then in a region where the magnetic field has decayed.

The application of cooling in a decaying magnetic turbulence to four GRBs detected by LAT has proved to be very successful and even able to give indications on how fast the turbulence decays, being consistent with a power-law decay *e<sup>B</sup>* ∝ *t* <sup>−</sup>*α<sup>t</sup>* with *<sup>α</sup><sup>t</sup>* <sup>∼</sup> 0.5 [66].

To understand and constrain the value of the magnetic field relevant for the particle cooling is of great importance, since an incorrect assumption or prior affects the estimates of all the other afterglow parameters, and in particular the density of the external medium [67,106].

A low value of *e<sup>B</sup>* tends to increase the level of SSC luminosity for a given synchrotron luminosity. The recent detection of bright TeV emission from the afterglow of GRBs is an indication that this might indeed be the case. Existing and future TeV observations will shed a light on this issue, fostering a revision of our prejudice on the value of the magnetic field in the region where particles cool.
