# GZK Horizons and the Anisotropy of Highest-energy Cosmic Ray Sources

GZK Horizons and the Anisotropy of Highest-energy Cosmic Ray Sources

Institute of Physics, National Chiao-Tung University, Hsinchu 300, Taiwan.

Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 106, Taiwan.

Motivated by recent Pierre Auger result on the correlation of the highest-energy cosmic rays with the nearby active galactic nuclei, we explore possible ultrahigh energy cosmic ray (UHECR) source distributions and their effects on GZK horizons. Effects on GZK horizons by local over-density of UHECR sources are examined carefully with constraints on the degree of local over-density inferred from the measured UHECR spectrum. We include the energy calibration effect on the Pierre Auger data in our studies. We propose possible local over-densities of UHECR sources which are testable in the future cosmic ray astronomy.

## 1 Introduction

Recently, Pierre Auger observatory published results on correlation of the highest-energy cosmic rays with the positions of nearby active galactic nuclei (AGN) . Such a correlation is confirmed by the data of Yakutsk while it is not found in the analysis by HiRes . In the Auger result, the correlation is maximal for the threshold energy of cosmic rays at eV, the maximal distance of AGN at Mpc and the maximal angular separation of cosmic ray events at . Due to increasing efforts on verifying the Auger result, it is worthwhile to examine the above correlation from a phenomenological point of view.

Since the angular scale of the observed correlation is a few degrees, one expects that these cosmic ray particles are predominantly light nuclei. The effect of GZK attenuations on these cosmic ray particles can be described by a distance scale referred to as “GZK horizon”. By definition, the GZK horizon associated with a threshold energy is the radius of a spherical region which is centered at the Earth and produce of UHECR events arriving on Earth with energies above .

Assuming a uniform distribution of UHECR sources with identical cosmic ray luminosity and spectral index , the GZK horizon for protons with EeV is about Mpc while the V-C catalog used by Pierre Auger for the correlation study is complete only up to Mpc. Such a deviation may arise from non-uniformities of spatial distribution, intrinsic luminosity and spectral index of local AGN as mentioned in . In addition, the energy calibration also plays a crucial role since the GZK horizon is highly sensitive to the threshold energy . Energy values corresponding to the dip and the GZK cutoff of UHECR spectrum were used to calibrate energy scales of different cosmic ray experiments . It has been shown that all measured UHECR energy spectra can be brought into good agreements by suitably adjusting the energy scale of each experiment . Furthermore, it has been shown that a different shower energy reconstruction method infers a higher UHECR energy than that determined by Auger’s fluorescence detector-based shower reconstruction .

In this presentation, we report our results on examining the consistency between Auger’s UHECR correlation study and its spectrum measurement. The impact by the local over-density of UHECR sources is studied. We also study the energy calibration effect on the estimation of GZK horizon and the spectrum of UHECR. Certainly a upward shift on UHECR energies reduces the departure of theoretically calculated GZK horizon to the maximum valid distance of V-C catalog . The further implications of this shift will be studied in fittings to the shifted Auger spectrum.

## 2 GZK horizons and the UHECR spectrum

GZK horizons corresponding to different local over-densities and are summarized in Table I. Within the same , local over-densities up to do not significantly alter GZK horizons. One could consider possibilities for higher local over-densities. However, there are no evidences for such over-densities either from astronomical observations or from fittings to the measured UHECR spectrum. We note that GZK horizons are rather sensitive to . Table I shows that GZK horizons are Mpc or less for EeV.

Fittings to the Auger spectrum have been performed in . In our work, we take into account the over-density of UHECR sources in the distance scale Mpc. The local over-density of UHECR sources affects the cosmic-ray spectrum at the highest energy, especially at energies higher than eV. Hence the degree of local over-density can be examined through fittings to the measured UHECR spectrum.

The left paenl in Fig. 1 shows our fittings to the Auger measured UHECR spectrum with and respectively. We take the red-shift dependence of the source density as with . We have fitted Auger data points beginning at the energy eV. We make a flux normalization at eV while varying the power index and the the degree of local over-density, . Part of values from our fittings are summarized in Table II. We found that gives the smallest value with . For the same , is ruled out at the significance level . For , is ruled out at the significance level . We note that, for both and , the GZK horizon with , EeV, and EeV is about Mpc. Since is clearly disfavored by the spectrum fitting, one expects a GZK horizon significantly larger than Mpc for EeV.

We next perform fittings to the shifted Auger spectrum. The results are shown in the right panel in Fig. 1 where the cosmic ray energy is shifted upward by . Part of values are summarized in Table III. The smallest value occurs approximately at , with . One can see that values from current fittings are considerably smaller than those from fittings to the unshifted spectrum. Given a significance level , it is seen that every local over-density listed in Table III except is consistent with the measured UHECR spectrum. It is intriguing to test such local over-densities as will be discussed in the next section. We note that, with a upward shift of energies, the cosmic ray events analyzed in Auger’s correlation study would have energies higher than EeV instead of EeV. The GZK horizon corresponding to EeV is Mpc for and Mpc for .

We have so far confined our discussions at . In the literature, has been taken as any number between and . It is demonstrated that the effect on UHECR spectrum caused by varying can be compensated by suitably adjusting the power index . Since GZK horizons are not sensitive to and , results from the above analysis also hold for other ’s.

## 3 Discussions and conclusions

We have discussed the effect of local over-density of UHECR sources on shortening the GZK horizon. The result is summarized in Table I. It is seen that such an effect is far from sufficient to shorten the GZK horizon at EeV to Mpc for a local over-density consistent with the measured UHECR spectrum. With a energy shift, each cosmic ray event in Auger’s correlation study would have an energy above EeV instead of EeV. GZK horizons corresponding to EeV then match well with the maximum valid distance of V-C catalog. Fittings to the shifted Auger spectrum indicate a possibility for the local over-density of UHECR sources.

We point out that the local over-density of UHECR sources is testable in the future cosmic ray astronomy where directions and distances of UHECR sources can be determined. Table IV shows percentages of cosmic ray events that come from sources within Mpc for different values of and . Although these percentages are calculated with , and EeV, they are however not sensitive to these parameters.

For EeV and , only of cosmic ray events come from sources within Mpc. For and the same , of cosmic ray events are originated from sources in the same region.

In conclusion, we have shown that the deviation of theoretically calculated GZK horizon to the maximum valid distance of V-C catalog can not be resolved by merely introducing the local over-density of UHECR sources. On the other hand, if Auger’s energy calibration indeed underestimates the UHECR energy, such a discrepancy can be reduced. More importantly, fittings to the shifted Auger spectrum indicate a possible local over-density of UHECR sources, which is testable in the future cosmic ray astronomy.

Acknowledgements

We thank A. Huang and K. Reil for helpful discussions. We also thank F.-Y. Chang, T.-C. Liu and Y.-S. Yeh for assistances in computing. This work is supported by National Science Council of Taiwan under the grant number 96-2112-M-009-023-MY3.

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