Effects of the Cosmological Constant as the Origin of the Cosmic-Ray Paradox

We show that a tiny but non-zero positive cosmological constant, which has been strongly suggested by the recent astronomical observations on supernovae and CMBR, may change notably the behaviors of the ultrahigh energy cosmic ray interacting with soft photons. The threshold anomaly of the ultrahigh energy cosmic ray disappears naturally after the effects of the cosmological constant are taken into account.