Can historic cosmic ray measurements explain spectral differences between Auger and Telescope Array?

The discrepancy the historic data are asked to arbitrate

Two modern observatories report very different fluxes at the highest energies. From an exposure of about 1.4×10^4 km^2 sr yr, largely northern-hemisphere, the Telescope Array Collaboration reported 18 events above 100 EeV, giving an integral flux of (1.3±0.3)×10^-3 km^-2 sr^-1 yr^-1. The Pierre Auger Observatory, from an exposure of 6×10^4 km^2 sr yr and largely southern-hemisphere observations, reported 24 events above 100 EeV and a flux of (2.5±0.7)×10^-4 km^-2 sr^-1 yr^-1 — roughly a factor of five lower.

The paper's question is whether older northern-hemisphere arrays side with one of these results. In the second half of the 20th century, Volcano Ranch, Haverah Park and AGASA reported 16 events above 100 EeV, an integral flux of (8±2)×10^-3 km^-2 sr^-1 yr^-1 from a combined exposure about seven times smaller than Telescope Array's and about 30 times smaller than Auger's. Taken at face value, that historic flux is even higher than Telescope Array's, which is why the authors do not simply compare raw numbers.

Why Auger is used as the reference spectrum

The comparison rests on an asymmetry between the two modern measurements. The Auger spectrum is data-driven: no assumption is made about primary mass or about features of hadronic interactions at the relevant energies. The northern-hemisphere ground arrays, by contrast, estimated energy using assumptions about shower development (Volcano Ranch) or mass-dependent models rooted in pre-LHC data (Haverah Park, AGASA). Telescope Array follows that same philosophy, but with models updated using post-LHC data.

The authors therefore take the Auger spectrum above 3 EeV as the best available reference, with a quoted systematic uncertainty of ±14%, and treat the historic measurements as the quantities to be re-evaluated. This choice is a stated assumption of the analysis, not a conclusion derived from it, and it is the assumption on which the rest of the argument depends.

Re-evaluating historic energies with EPOS-LHC

The first attempt re-simulated the historic arrays at energies of 10^18.5, 10^19.0, 10^19.5 and 10^20.0 eV and zenith angles of 0, 30 and 45 degrees using the EPOS-LHC model of hadronic interactions. The revised energy is written as E_mod = kE, where E is the original estimate. For proton and iron primaries respectively, the factors k are (0.72, 0.71) for Volcano Ranch, (0.73, 0.65) for Haverah Park and (0.66, 0.59) for AGASA, with uncertainties below 6% for proton and below 2% for iron.

Applied to the famous 100 EeV event reported in 1963 at Volcano Ranch, this would reduce its energy to about 70 EeV. But the correction does not reconcile the spectra: with an energy- and zenith-independent factor k, Auger and AGASA remain in poor agreement, as do the Volcano Ranch and Haverah Park samples. The corrections also assume proton primaries, while the authors note that arguments independent of hadronic-interaction assumptions appear to invalidate that assumption for northern-hemisphere data; adopting a heavier mean mass would enhance the disagreement. Even under the most favourable proton assumption, a significant disagreement with the Auger spectrum remains.

Residuals and the pre-ankle calibration

Because the model-based correction fails, the authors switch to a residual approach. They define Residual = (J/J_s − 1), where J is the measured flux in an energy interval and J_s is the flux of a reference spectrum, taken as an approximation to the Auger spectrum with four slopes and three breaks. For AGASA they also apply an approximate unfolding for energy uncertainty, using the ratio of observed to true event numbers; this typically reduces the event count by a factor between 0.88 and 0.94 depending on the energy interval.

The calibration exploits a physical expectation: below the ankle, around 5 EeV, the fluxes from the two hemispheres should be the same, supported by negligible anisotropy in arrival directions and by the independence of mass composition from declination. Fitting only data below the ankle with a chi-squared minimisation, the energy shifts for best alignment are (0.77±0.01) for AGASA and (0.94±0.01) for Telescope Array, with reduced chi-squared values of 0.35 and 1.7. The small Telescope Array shift agrees with an earlier report, which the authors treat as validation of the residual method. Notably, the AGASA shift is smaller than the pure-proton EPOS-LHC value of k = 0.66, and there is evidence that protons, helium and nitrogen dominate in this energy region.

Above 30 EeV the historic fluxes track Auger

Extending the residual comparison over the full energy range, the Telescope Array measurements begin to fall outside the Auger ±14% bounds above log E = 19.5, whereas the AGASA data lie within those bounds except for the last three points, corresponding to the 11 highest-energy events. The reduced chi-squared values over all data are 4.35 for AGASA and 7.01 for Telescope Array, but the authors emphasise that the fit in the highest-energy region, which they call the crucial one, is clearly better for AGASA.

The integral flux comparison makes the same point in numbers. Above 31.5 EeV, Auger measures 0.0210±0.0006 km^-2 sr^-1 yr^-1, Telescope Array 0.040±0.002, AGASA 0.022±0.004, and AGASA combined with Volcano Ranch and Haverah Park 0.023±0.003. Above 56 EeV, Auger measures 0.0045±0.0003, Telescope Array 0.014±0.01 and AGASA 0.007±0.002. The historic fluxes sit close to Auger and about a factor of two below Telescope Array. The authors state this conclusion is independent of assumptions about hadronic interaction models.

The 11 highest-energy AGASA events need a large shift

The exception to the agreement is the set of 11 highest-energy AGASA events. To match the Auger flux, their energies must be shifted by 0.58, confirming that a single energy-independent factor k cannot be applied and that, depending on energy range, the AGASA energy estimate should be reduced by roughly 0.8 to 0.6. The authors call this an extremely large adjustment, one that might imply trans-iron nuclei or exceptionally energetic events from sources unique to the northern hemisphere.

They then examine the four highest-energy AGASA events for which details are available, re-evaluating energies with EPOS-LHC and QGSJetII-04. Averaged over protons and iron, the reduction is about 73% for the two near-vertical events and about 47% for the events at 37 and 44.2 degrees, with the two models agreeing. The authors acknowledge that the hadronic model is known to be inadequate, but consider these the best evaluations currently possible. They also cite an unpublished presentation from around 2006 by Teshima, who stated that the 11 events should shift downward by about 15% and reduce to 5 to 6 events; applying that, only five events above 100 EeV remain. On that basis, the authors conclude no exotic explanation is necessary for the highest-energy events.

What the agreement does not resolve

The paper's conclusion is deliberately two-sided. Assuming the Auger fluxes are accurate and that fluxes below the ankle are the same in both hemispheres, the historic fluxes above 31 and 56 EeV agree better with Auger than with Telescope Array, and are a factor of two lower than the Telescope Array values. Because the historic arrays are all northern-hemisphere, this is a direct challenge to the Telescope Array result rather than a resolution of it.

The authors therefore suggest that the Telescope Array Collaboration might consider a different hadronic-interaction model and a heavier mass composition than the all-proton assumption currently chosen, and that more attention be given to matching model predictions with direct measurements of the energy spectrum where mass estimates are most reliable. One limitation is stated plainly: no attempt was made to account for the complex change in mass composition across the energy range, which would make the energy adjustment k vary with energy. The residual calibration itself also rests on the pre-ankle equal-flux assumption and on treating Auger as the reference.

Where the historic data carry weight

The practical reading of this paper is a split by energy range rather than a single verdict. Below the ankle, the historic data serve mainly as a calibration check: the small Telescope Array shift of 0.94±0.01 reproduces an earlier result, which is what validates the residual method. Above 30 EeV, the same data become evidence, and there the AGASA, Volcano Ranch and Haverah Park fluxes line up with Auger at both 31.5 and 56 EeV thresholds.

The decisive comparison is therefore not historic versus modern in general, but historic northern-hemisphere data versus the modern northern-hemisphere measurement. The historic fluxes are about a factor of two below Telescope Array's, and the authors argue this conclusion does not depend on hadronic-interaction assumptions. The remaining soft spot is the 11 highest-energy AGASA events, which require a shift of 0.58 and rest on energy assessments with considerable uncertainties for the four events with available detail. Readers weighing the two modern spectra should treat the sub-ankle calibration, the 31.5 and 56 EeV integral fluxes, and the 11-event tail as three separate pieces of evidence, because only the first two are in good agreement with Auger.

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