Propagation of very-high-energy $\gamma$-rays from distant blazars

We re-derive the possible dependence of the redshift with very high energy (VHE) $\gamma$-ray photon index. The results suggest that the universe to VHE $\gamma$-rays is becoming more transparent than usually expected. We introduce the extragalactic background light (EBL) plus the photon to axion-like particle (ALP) oscillations to explain this phenomenon. We concentrate our analysis on 70 blazars up to redshift $z \simeq 1$. Assuming this correlation is solely the result of photon-photon absorption of VHE photons with the EBL, which finds the deviations between the predictions and observations, especially at redshifts $0.2 < z < 1$. We then discuss the implications of photon-ALP oscillations for the VHE $\gamma$-ray spectra of blazars. A strong evidence shows that: 1) the EBL attenuation results that the VHE $\gamma$-ray photon index increases non-linearly at the ranges of redshift, $0.03 < z < 0.2$; 2) the photon-ALP oscillation results in a attractive characteristic in the VHE $\gamma$-ray photon index at the ranges of redshift, $0.2 < z < 1$. We suggest that both the EBL absorption and photon-ALP oscillation can influence on the propagation of VHE $\gamma$-rays from distant blazars.