Copyright by Athena Ranice Stacy 2011 - The University of Texas at ...

corresponding threshold, which is ɛGZK = 5 × 10 19 eV in today’s Universe.
This cutoff has recently been observed by the HiRes experiment (Abbasi et al.
2008).
In the high-redshift Universe, however, the GZK cutoff will be some-
what lower, as can be seen as follows: While in today’s Universe the average
energy of a CMB photon is ɛCMB = 2.7kBTCMB = 6 × 10 −4 eV, at higher red-
shifts this energy will be larger by a factor of (1 + z). In the CR rest frame,
the CMB photon energy is
ɛ ′ CMB ≈ γ 6 × 10 −4 eV (1 + z) , (6.10)
where γ is the Lorentz factor of a CR proton. Equating ɛ ′ CMB with ɛt, we find
γ ≈
2 × 1011
(1 + z)
. (6.11)
We can now calculate the CR energy for which the threshold for photo-pion
production is reached:
ɛGZK(z) = γmHc 2 ≈ 3 × 1020 eV
, (6.12)
(1 + z)
where mH is the mass of a proton. A more precise calculation, carrying out
an integration over the entire Planck spectrum and over all angles, yields
ɛGZK(z) = 5 × 1019 eV
1 + z
≈ 2 × 10 18 −1 1 + z
eV
. (6.13)
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Thus, at redshifts of 10 or 20, the GZK cutoff is around an order of magnitude
smaller than in today’s Universe, giving a robust upper limit to the CR energy.
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