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Knee structure in high-energy inverse Compton scattering of CMB photons

We study the inverse Compton scattering of the CMB photons off nonthermal high-energy electrons. In the previous study, assuming the power-law distribution for electrons, we derived the analytic expression for the spectral intensity function $I(ω)$ in the Thomson approximation, which was applicable up to the photon energies of $ω<$ O(GeV). In the present paper, we extend the previous work to higher photon energies of $ω>$ O(GeV) by taking into account the terms dropped in the Thomson approximation, i.e., the Klein-Nishina formula. The analytic expression for $I(ω)$ is derived with the Klein-Nishina formula. It is shown that $I(ω)$ has a "knee" structure at $ω=$ O(PeV). The knee, if exists, should be accessible with gamma-ray observatories such as Fermi-LAT. We propose simple analytical formulae for $I(ω)$ which are applicable to wide photon energies from Thomson region to extreme Klein-Nishina region.