Swift observations of the very intense flaring activity of Mrk 421
  during 2006: I. Phenomenological picture of electron acceleration and
  predictions for the MeV/GeV emission
  We present results from a deep spectral analysis of all the Swift
observations of Mrk 421 from April 2006 to July 2006, when it reached its
largest X-ray flux recorded until 2006. The peak flux was about 85 milli-Crab
in the 2.0-10.0 keV band, with the peak energy (Ep) of the spectral energy
distribution (SED) laying often at energies larger than 10 keV. We performed
spectral analysis of the Swift observations investigating the trends of the
spectral parameters in terms of acceleration and energetic features
phenomenologically linked to the SSC model parameters, predicting their effects
in the gamma-ray band, in particular the spectral shape expected in the Fermi
Gamma-ray Space Telescope-LAT band. We confirm that the X-ray spectrum is well
described by a log-parabolic distribution close to Ep, with the peak flux of
the SED (Sp) being correlated with Ep, and Ep anti-correlated with the
curvature parameter (b). During the most energetic flares the UV-to-soft-X-ray
spectral shape requires an electron distribution spectral index s about 2.3.
Present analysis shows that the UV-to-X-ray emission from Mrk 421 is likely to
be originated by a population of electrons that is actually curved, with a low
energy power-law tail. The observed spectral curvature is consistent both with
stochastic acceleration or energy dependent acceleration probability
mechanisms, whereas the power-law slope form XRT-UVOT data is very close to
that inferred from the GRBs X-ray afterglow and in agreement with the universal
first-order relativistic shock acceleration models. This scenario hints that
the magnetic turbulence may play a twofold role: spatial diffusion relevant to
the first order process and momentum diffusion relevant to the second order
process.
