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The very high energy (VHE) $\gamma$-ray spectra of some TeV Blazars, afterbeing corrected for absorption in the extragalactic background light (EBL),appear unusually hard, which poses challenges to conventional acceleration andemission models. We investigate the parameter space that allows the productionof such hard TeV spectra within time-dependent leptonic models, both forsynchrotron self-Compton (SSC) and external Compton (EC) scenarios. In thecontext of interpretation of very hard $\gamma$-ray spectra, time-dependentconsiderations become crucial because even extremely hard, initial electrondistributions can be significantly deformed due to radiative energy losses. Weshow that very steep VHE spectra can be avoided if adiabatic losses are takeninto account. Another way to keep extremely hard electron distributions in thepresence of radiative losses, is to assume stochastic acceleration models thatnaturally lead to steady-state relativistic, Maxwellian-type particledistributions. We demonstrate that in either case leptonic models can reproduceTeV spectra as hard as $E_{\gamma} dN/dE_{\gamma} \propto E_{\gamma}$.Unfortunately this limits, to a large extend, the potential of extracting EBLfrom $\gamma$-ray observations of blazars.

Formation of hard very-high energy spectra of blazars in leptonic models