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Very high energy gamma-ray production inside the massive binary system Cyg X-1/HDE 226868

We apply the IC e$^\pm$ pair cascade model in order to investigate the possible $γ$-ray emission features from another compact massive binary of the microquasar type, Cyg X-1. We conclude that the observational constraints at lower energies (from MeV to GeV) suggest that the spectrum of electrons injected in the jet is likely steeper than in other TeV $γ$-ray microquasars. The cascade $γ$-ray spectrum produced by electrons with such spectrum in Cyg X-1 should be below the sensitivities of the MAGIC and VERITAS class Cherenkov telescopes. However, if the electron spectrum is flatter, then the highest TeV $γ$-ray fluxes are predicted at $\sim$7 hrs before and after the phase when the compact object is in front of the massive star. We suggest that Cherenkov telescopes should concentrate on these range of phases since the TeV flux can vary by a factor of $\sim$20 with the period of the Cyg X-1 binary system. Moreover, the model predicts clear anticorrelation of the GeV and TeV $γ$-ray emission. This feature can be tested by the future multiwavelength observations with the AGILE and GLAST telescopes in the GeV energy range and the MAGIC and VERITAS telescopes in the TeV energy range.