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Measurement of the cosmic-ray energy spectrum above 2.5 EeV using 19 years of operation of the Pierre Auger Observatory

T0 review · 0 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Nineteen years of Pierre Auger data resolve the cosmic-ray 'instep' at ~13 EeV with 5.5σ significance, firmly establishing the spectral hardening alongside the ankle and suppression.

desk verdict Auger's 19-year spectrum confirms the instep at 5.5 sigma with a careful, statistically clean combination; minor calibration caveats only. read the letter →

arxiv 2507.08573 v1 pith:TBQO3LAJ submitted 2025-07-11 astro-ph.HE

Diego Ravignani (for the Pierre Auger Collaboration) This is my paper
classification astro-ph.HE
keywords ultra-high-energycosmicrayscosmic-rayenergyspectrumPierreAugerObservatoryspectralinstepankleGZKsuppressioninclinedairshowerscalibration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Using the full 19-year Phase I dataset of the Pierre Auger Observatory, this paper combines vertical (zenith angle $< 60^\circ$) and inclined ($60^\circ$–$80^\circ$) events into a single cosmic-ray energy spectrum covering 2.5 EeV to beyond 100 EeV with a total exposure of 104,900 km² sr yr. The paper's central claim is that this spectrum establishes the previously reported 'instep' feature — a spectral hardening around 13 EeV — at discovery-level significance of 5.5σ, alongside the well-known ankle and high-energy suppression. If the claim is correct, viable models of ultra-high-energy cosmic-ray sources and propagation must reproduce a spectral break at ~13 EeV, not just the ankle and the cutoff. The paper also reports that the spectrum is consistent across declination bands except for the known dipolar anisotropy, and that a preliminary two-year spectrum from the upgraded observatory matches the Phase I result.

What carries the argument

The analysis rests on two independent energy estimators tied to a common fluorescence-detector energy scale: vertical events use the shower size $S_{38}$ derived from a lateral distribution fit with Constant Intensity Cut attenuation, $E = A S_{38}^B$ with $A = (186 \pm 3)$ PeV and $B = 1.021 \pm 0.004$; inclined events use a scaled signal-pattern fit with equivalent size $N_{68}$, $E = A N_{68}^B$ with $A = (5.29 \pm 0.06)$ EeV and $B = 1.046 \pm 0.014$. The two spectra are merged with a Poisson-likelihood fit of a smooth four-power-law flux model that has transition energies at the ankle, instep, and suppression, while floating the inclined calibration parameters $\delta A$, $\delta B$, and $\delta C$ (with $\delta B$ and $\delta C$ applying below and above 10 EeV) under penalty terms. An exposure of 104,900 km² sr yr is computed from per-second detector-status monitoring, and the 2.5 EeV threshold is set where trigger efficiency exceeds 97%. This machinery is what allows the instep to be tested at 5.5σ rather than the 3.9σ of the 2020 spectrum.

What would settle it

Re-fit the combined spectrum replacing the two slope corrections ($\delta B$, $\delta C$) with a flexible, energy-dependent energy-scale function (for example a spline in $\log E$): if the likelihood-ratio significance of the instep drops below 5σ, the claimed discovery-level status would not survive a more general calibration model. A second independent check: examine whether a bump near 13 EeV appears in the vertical-only and inclined-only spectra before any relative recalibration is applied; if the feature emerges only after the recalibration offsets are floated, it could be an artifact of the combination itself.

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Extended reading notes

Core claim

The authors report the combined spectrum of cosmic rays above 2.5 EeV measured with the 1500 m surface detector array over 19 years (2004–2023, pre-upgrade), totalling an exposure of 104,900 km² sr yr. Combining the vertical and inclined event sets with a simultaneous likelihood fit that allows small energy-scale recalibrations of the inclined data, they measure the ankle at $(5.1 \pm 0.1 \pm 1.1)$ EeV, the instep at $(13 \pm 1 \pm 2)$ EeV, and the suppression at $(48 \pm 2 \pm 5)$ EeV, with spectral indices $2.51 \pm 0.03$ and $2.99 \pm 0.03$ before and after the instep. The instep is assessed against a reference model with a smooth slow suppression in place of the instep: in $10^{8}$ simulations, only two produced a likelihood-ratio test statistic larger than the observed value ($\sim 35$), corresponding to a significance of 5.5σ. Within statistical uncertainties, the spectrum is independent of declination from the south celestial pole to $+44.8^\circ$, aside from the modulation expected from the previously measured dipole. A preliminary spectrum from the first two years of the upgraded Phase II array is statistically consistent with the Phase I result.

Load-bearing premise

The combination of vertical and inclined spectra treats any energy-dependent bias in the inclined energy reconstruction as a power-law correction with just two slopes, one below and one above 10 EeV; if the true bias bends differently with energy or depends on primary composition in an unmodelled way, the shape of the combined spectrum — and with it the position and significance of the instep — could be distorted.

Editorial extensions

If this is right

  • Source models for ultra-high-energy cosmic rays must produce a spectral hardening at about 13 EeV, in addition to the ankle near 5 EeV and the suppression near 50 EeV.
  • The measured instep position and the steepening from spectral index 2.51 to 2.99 give fits of galactic-to-extragalactic transitions and of possible new source populations a new observational anchor.
  • The consistency of the spectrum across declinations from the south celestial pole to +44.8° strengthens the conclusion that the only large-scale anisotropy at these energies is the previously reported dipole.
  • The agreement between the Phase I spectrum and the preliminary Phase II spectrum supports a continuous energy-scale across the AugerPrime upgrade, so the 19-year exposure can be extended with future data.
  • At 5.5σ, the instep becomes a well-defined feature for independent experiments observing the same energy range to confirm or refute.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the instep signals the onset of a new source population, one might expect the average mass composition to harden or change across the 10–20 EeV range; the paper does not test this, but the upgraded observatory's composition measurements could look for that correlation.
  • The quoted 5.5σ is a single-experiment significance for one feature; a fair global assessment would need a trial factor for the instep's energy position and a comparable test on independent data, neither of which is part of this analysis.
  • The two-slope recalibration of the inclined spectrum is the main systematic caveat: if the vertical–inclined energy bias is not a pure two-slope power law, the instep position could shift with additional data, and a spline-based calibration on this same dataset would make that risk explicit.
  • The Phase II spectrum offers a ready-made validation: once it reaches a few tens of percent of the Phase I exposure, the instep should reappear at the same energy and significance when analyzed with the same method; otherwise the feature may be tied to the vertical–inclined combination.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper reports the cosmic-ray energy spectrum above 2.5 EeV measured with the Pierre Auger Observatory's 1500 m surface array during 19 years of Phase I operation (2004-2023). Vertical (zenith angle <60°) and inclined (60°-80°) event sets are combined, reaching a total exposure of 104,900 km² sr yr, with a recalibration of the inclined energy scale via three nuisance parameters (δA, δB, δC). The combined spectrum is fitted with a smooth power-law model with three transitions, yielding the ankle at 5.1±0.1±1.1 EeV, the instep at 13±1±2 EeV, and the suppression at 48±2±5 EeV. The instep is claimed at 5.5σ significance from a Monte Carlo likelihood-ratio test (t_obs≈35, 2 of 10^8 simulations), and a preliminary Phase II spectrum is shown to be consistent with Phase I.

Significance. If correct, this is an important result: it is the highest-exposure spectrum from a single observatory at these energies, statistically establishes the instep as a distinct spectral feature, extends the measurement to declinations up to +44.8° using inclined events, and provides an independent cross-check with the first Phase II data. The analysis builds on previously published, well-documented reconstruction and calibration procedures, and the systematic uncertainties are treated in a transparent way. I specifically checked the concern that the two-parameter inclined recalibration (Section 3) could distort the instep; the fitted shifts are small (δA=160±39 PeV, δB=0.003±0.016, δC=−0.02±0.02), the induced spectral-index change is about 0.02 compared to the observed 0.48 change across the instep, and the combination deviance D=40.5 with p≈0.12 provides no indication of a residual energy-dependent bias. The Phase II consistency test (p=0.30) further supports the stability of the result.

minor comments (6)
  1. [Section 4] The reference model used for the instep significance test is described only verbally ('a slow suppression instead of the instep'); please provide its functional form or a precise pointer to the null model in reference [3] so that the 5.5σ claim can be reproduced.
  2. [Table 1] The entry in the second row, first column reads '0 .4' in the manuscript; this is presumably a typo for '0.4' and should be corrected.
  3. [Figures 1–5] The axis labels contain the placeholder symbol '□' (e.g., 'km□2 sr□1 yr□1'); the final version should use proper superscripts and symbols.
  4. [Section 3] The deviance D=40.5 is quoted with p≈0.12 but the number of degrees of freedom is not stated; reporting the ndf would let the reader assess the goodness of fit directly.
  5. [Abstract and Conclusion] The abstract says 'more than 5σ confidence' while the conclusion states '5.5σ'; these should be harmonized.
  6. [Section 2] The threshold is stated as log10(E/eV)=18.4 for vertical and 18.6 for inclined events; the abstract's 'above 2.5 EeV' matches only the vertical threshold, so the discussion should clarify that the combined spectrum starts at 2.5 EeV even though the inclined sample begins at a higher energy.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectrum measurement and 5.5-sigma instep significance are self-contained, with only contextual self-citations.

full rationale

The paper derives the combined spectrum from detector-level exposures, trigger-efficiency thresholds, and fluorescence-calibrated energy estimators; the instep significance is obtained by a Monte Carlo likelihood-ratio test against a reference model without an instep, not by any parameter fitted to the instep. The only jointly fitted quantities beyond the eight spectral parameters are three calibration offsets (delta-A, delta-B, delta-C), which are explicitly penalized and small (delta-B = 0.003 +/- 0.016, delta-C = -0.02 +/- 0.02), producing at most a few-percent energy shift that cannot generate the observed 0.48 spectral-index jump. The paper itself flags that the declination-band agreement with the dipole 'is expected since the anisotropy and spectrum data sets overlap considerably,' so that consistency check is not presented as independent confirmation. Citations [3] and [5] are prior Auger spectrum papers and the data release; they are contextual and not load-bearing for the 5.5-sigma claim. No equation reduces to its own input, and no fitted parameter is renamed as a prediction. The independent Phase II spectrum also provides an external consistency check. Therefore no circular step is present.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the assumed flux parameterization, the trigger-efficiency plateau, the fluorescence energy scale, the Constant Intensity Cut attenuation corrections, and the dipole model for declination dependence. The fitted nuisance parameters are the inclined calibration shifts and the energy-calibration constants; no new physical entities are introduced.

free parameters (4)
  • Inclined energy calibration shifts delta-A, delta-B, delta-C = delta-A=(160±39) PeV, delta-B=0.003±0.016, delta-C=-0.02±0.02
    Fitted in Section 3 during the vertical/inclined combination, with penalties from the FD calibration priors. They adjust the inclined energy scale by about +3% at 4-10 EeV and about -1% at 100 EeV, directly influencing the shape of the combined spectrum and the instep parameters.
  • Inclined energy calibration A, B = A=(5.29±0.06) EeV, B=1.046±0.014
    Calibrated with FD coincidence events (Section 2); the power-law relation E = A N68^B sets the absolute energy of all inclined events.
  • Vertical energy calibration A, B = A=(186±3) PeV, B=1.021±0.004
    Calibrated with FD coincidence events; sets the absolute energy of vertical events, with negligible statistical uncertainty.
  • Attenuation function coefficients (vertical a_ij and inclined f_att) = Vertical a_ij in Table 1; inclined f_att = 1 + (0.292-0.468y)x + (-4.96+0.79y)x^2
    Empirically derived via the Constant Intensity Cut method; they convert measured S(1000) and N19 to zenith-independent shower sizes S38 and N68, and affect all reconstructed energies.
assumptions (5)
  • domain assumption The true cosmic-ray flux is described by the smoothly-broken power law of Eq. 2 with exactly three transitions (ankle, instep, suppression) and fixed transition widths w_i=0.05.
    Invoked in Section 2 and used for all fits; if the true spectrum has additional structures or the widths are wrong, the fitted parameters and instep significance could be biased.
  • domain assumption Trigger efficiency exceeds 97% above 2.5 EeV (vertical) and 4 EeV (inclined), making the exposure time-dependent only through detector-status monitoring.
    Section 2, thresholds chosen at log10(E/eV)=18.4 and 18.6; the exposure calculation relies on this efficiency plateau.
  • domain assumption The fluorescence detector provides a calorimetric energy scale that is independent of hadronic-interaction simulations, with a 14% systematic uncertainty.
    Section 2, citing reference [12]; the entire energy scale of both spectra inherits this assumption.
  • domain assumption The Constant Intensity Cut method gives an unbiased attenuation correction for both vertical and inclined showers.
    Section 2, applied to derive S38 and N68; this assumes the flux is isotropic in zenith angle at fixed energy, which is approximately true after accounting for the dipole.
  • domain assumption The expected declination dependence of the flux is fully described by the dipolar anisotropy measured in reference [2].
    Section 3, used to predict band-to-band flux ratios; the anisotropy and spectrum datasets overlap considerably, making this partly a self-consistency check.

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Cite this review

Pith. "Pith review of Measurement of the cosmic-ray energy spectrum above 2.5 EeV using 19 years of operation of the Pierre Auger Observatory." pith.science (2026). https://pith.science/paper/TBQO3LAJ

@misc{pith2026250708573,
  author       = {Pith},
  title        = {Pith review of: Measurement of the cosmic-ray energy spectrum above 2.5 EeV using 19 years of operation of the Pierre Auger Observatory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TBQO3LAJ}},
  note         = {Machine review of arXiv:2507.08573}
}
abstract

We present the spectrum of cosmic rays with energies above 2.5 EeV measured at the Pierre Auger Observatory after 19 years of operation, covering the period before the AugerPrime upgrade. Two independent event sets from the surface array of 1500 m-spaced detectors are combined, yielding a total exposure of approximately 100,000 km$^2$ sr yr. The first set includes events with zenith angles less than 60$^\circ$, while the second consists of events between 60$^\circ$ and 80$^\circ$, for which azimuthal asymmetries must be accounted for in the energy estimator. The threshold energy is chosen to ensure a trigger efficiency of the surface detector greater than 97%, thus minimizing composition biases. The energy scale is determined using high-quality fluorescence measurements, providing calorimetric estimates without reliance on simulations. A statistically successful combination is achieved within the uncorrelated systematic uncertainties of the individual spectra. All spectra are consistent when analyzing potential declination dependences, except for a mild modulation expected from the previously reported dipolar anisotropy. In particular, this statement applies to the northernmost declination band [+25$^\circ$,+45$^\circ$], where only contribute events with zenith angles between 60$^\circ$ and 80$^\circ$. Beyond the firmly established ankle and suppression spectral features, the combined spectrum across declinations $-90^\circ$ to +45$^\circ$ provides high-precision measurements of the instep feature with more than 5$\sigma$ confidence.

Figures

Figures reproduced from arXiv: 2507.08573 by the authors.

Figure 1
Figure 1. Vertical spectrum using events arriving with a zenith angle less than 60◦ , and inclined spec￾trum from events with a zenith angle between 60◦ and 80◦ . The spectrum data corrected for the detec￾tor response and the fitted flux are shown. No cut in the declination of the arrival direction has been applied. 18.5 19.0 19.5 20.0 log10(E/eV) 0 2 4 6 E3 J(E) (eV2 km−2 sr−1 yr−1 ) ×1037 Vertical Inclined Combined [PITH_F… view at source ↗
Figure 3
Figure 3. Left panel: Spectra in five declination bands combining vertical and inclined events. The flux fitted in the common declination band [−84.8 ◦ , +24.8 ◦ ] is shown in red, and the flux modulated by the dipole of the arrival direction distribution is shown in green. Right panel: Residuals of the combined spectra and the fitted flux in each declination band clipped to [−0.25, 0.25]. inclined spectra by combining them i… view at source ↗
Figure 4
Figure 4. Combined spectrum using events arriving with zenith angle up to 80◦ observed at declinations from the south celestial pole up to +44.8 ◦ . The systematic flux uncertainty is shown as a shaded band. 18.5 19.0 19.5 20.0 log10(E/eV) 0 2 4 6 E3 J(E) (eV2 km−2 sr−1 yr−1 ) ×1037 Phase I Phase II [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

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Cited by 1 Pith paper

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Reference graph

Works this paper leans on

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