Formation of very hard electron and gamma-ray spectra of flat spectrum radio quasar in fast-cooling regime

In external Compton scenario, we investigate the formation of the very hard electron spectrum in the fast-cooling regime, using a time-dependent emission model. It is shown that a very hard electron distribution $N'_{\rm e}(\gp)\propto\gp^{-p}$ with the spectral index $p\sim1.3$ is formed below the minimum energy of injection electron when inverse Compton scattering takes place in the Klein-Nishina regime, i.e., inverse Compton scattering of relativistic electrons on broad-line region radiation in flat spectrum radio quasars. This produces a very hard gamma-ray spectrum, and can reasonably explain the very hard \emph{Fermi}-LAT spectrum of the flat spectrum radio quasar 3C 279 during the extreme gamma-ray flare in 2013 December.