1. GZK cutoff and constraints on the Lorentz invariance violation Bi Xiao-Jun (IHEP) 2011/5/9

2. GZK cutoff We know that the UHECRs (E > 10 18 eV) are originated from the extragalactic sources. When propagate in the intergalactic space they interact with CMB photons, which results in energy and flux depletion. In particular, the photomeson process (pγ->pπ) will induce a suppression in the spectrum above (3–6) ×10 19 eV and lead to the well-known Greisen-Zatsepin-Kuzmin (GZK) cutoff. The spectrum of UHECRs can be calculated by assuming the source distribution and injection energy spectrum at the sources.

3. Introduction of LIV To account for the AGASA data beyond the GZK cutoff, LIV has been introduced by Coleman and Glashow, Even a LIV parameter as small as ~3 × 10 -23 may lead to the removal of the GZK cutoff

4. Cutoff confirmed by HiRes and Auger The HiRes Collaboration confirms the GZK cutoff with a 5σ standard deviation The Pierre Auger Collaboration gives results consistent with HiRes and rejects a single power-law spectrum above 10 19 eV at the 6 σ confidence level The two sets of data can set constraints on the LIV parameters introduced by Coleman and Glashow

5. Propagation of UHECRs We assume the composition of UHECRs is pure proton In the propagation we consider the adiabatic energy loss by the universe expansion and photopion production and e + e - pair production when interacting with CMB At z=0, σ is the interaction cross section, K is the average fraction of energy loss We have the threshold

6. At z, We employ two assumptions: (1) proton sources are distributed homogeneously in the Universe without the evolution effect; (2) the source spectrum is a power law with index γ g

7. The observed spectrum Solving the above equation with the initial condition E(z=0)=E 0, we get the observed spectrum is [Berezinsky 06] J(E 0 )ΔE 0 = F(E g) Δ E g, this means (Eg, Eg + dEg) at redshift z contribute to the detected energy interval (E 0, E 0 + d E 0 ); the sum of all redshifts gives the total flux L0 is the total luminosity of UHE protons and is determined by matching the calculated spectrum to the observational data.

8. Predicted spectrum and fit of γ g Angle is explained by e+e- pair GZK is due to photopion production

9. LIV takes effects The inelasticity K=E π /E p, is given [Alfaro 2003] The equation is solved numerically and averaged over the scattering angle.

11. Inelasticity with(out) the LIV ξ = +-1 ×10 -23 Generally the LIV surpress the inelasticity at high energies.

12. The modified spectra of UHE protons for several values of LIV parameters are shown in Fig. 3, together with the unmodified spectrum from the standard model.We see that the GZK suppression effect becomes less significant for LIV cases. For very high energies or for large magnitudes of LIV parameters, the source spectra tend not to be distorted; i.e., the photopion production process pgamma -> Npi does not play an important role anymore.

13. After considering the LIV For very high energies or for large magnitudes of LIV parameters, the source spectra tend not to be distorted

14. Fit power index and LIV parameter For the HiRes monocular spectra and the Auger combined spectrum, respectively We see

16. For iron component Auger shows that the UHECR are iron-like particles, we can also set constraints on the LIV To have the cutoff the reaction 56 Fe+CMB-> 55 Mn+p is only possible for ξ > -2 ×10 -25. On the other hand, spontaneous fragmentation 56 Fe- > 55 Mn+p should be forbidden for iron energy less than 3×10 20 eV, this gives ξ < 1.2 × 10 23 Finally we get -2 ×10 -25 < ξ < 1.2 × 10 23

17. Summary HiRes and Auger observed the GZK cutoff. This observation give constraints on the LIV effect. Calculate the UHECR propagation in the intergalactic space and the interaction with CMB, we give a global fit to the UHECR power index and the LIV parameter ξ