Coherence length of cosmic background radiation enlarges the attenuation length of the ultra-high energy proton

It is pointed out that an agreement of the one particle energy spectrum of the cosmic background radiation (CMBR) with Plank distribution of 2.725 [K] does not give a strong constraint on the coherence length of CMBR if the mean free path of CMBR is very long. The coherence length in this situation is estimated as a few times of $k_BT$. Due to this finite coherence length, the attenuation length of ultra-high energy cosmic rays (UHECR) is reduced in the $\Delta $ resonance region,i.e., around $10^{20}$ [eV]. The small attenuation length makes the suppression of the flux of cosmic rays in this energy region less prominent than the naive estimation