The Impact of the magnetic Horizon on the Interpretation of the Spectrum and Composition Data
One of the main questions that the Pierre Auger Observatory intends to answer is where and how ultra-high energy cosmic rays (UHECRs) are accelerated. The measurement of the spectrum and the distribution of the atmospheric depth of the maximum development of the air showers (Xmax) provide important clues to infer the mass composition and energy distribution of the particles accelerated at the sources. In their way from the sources to the Earth, UHECRs interact with the radiation backgrounds present in the intergalactic medium and are also deflected by the extragalactic magnetic fields (EGMF).
In a previous paper the spectrum and composition measurements above 0.6 EeV were fitted in a simple astrophysical model with two extragalactic populations of sources for the case that the EGMF effects are negligible. Each extragalactic population was assumed to have a mixed mass composition with a rigidity dependent power-law spectrum with a broken exponential cutoff. The data were well-described if the sources giving the main contribution at energies above the ankle emit a mix of intermediate and heavy mass nuclei with a very hard spectrum (∝ E2), far from the expectations from the diffusive shock acceleration mechanism, that is close to E−2.
In this paper we extend the analysis including the effect of a turbulent EGMF and exploring also sharper cutoffs for the injected spectrum, parametrized as sech((E/ZRcut)Δ). The presence of a non negligible EGMF between the observer and the closest sources can significantly modify the spectrum reaching the Earth, suppressing the flux of low-rigidity particles. This happens because the deflections of the particles’ trajectory results in longer travel times and, if these become larger than the age of the source, particles do not reach the Earth even from the closest sources. This is known as the magnetic horizon effect.
The magnetic horizon effect can be parametrized in terms of a suppression factor G(E) ≡ J(E)/J(E)ds→0, defined as the ratio between the observed flux from a discrete source distribution with inter-source distance ds = ns−3 (for a uniform density of sources ns) to the flux arriving from a continuous source distribution with the same emissivity per unit volume. Note that for a continuous source distribution, the magnetic fields have no suppression effect on the total flux reaching the Earth due to the so-called propagation theorem, which reflects the fact that the suppression of the faraway sources gets compensated by the diffusive enhancement of the nearby ones. The suppression factor depends on the EGMF root mean square amplitude Brms and the coherence length Lcoh through the critical energy Ecrit ≡ |e|ZBrmsLcoh as it is shown in Figure 1 for three values of the normalized intersource distance 
, with rH = c/H0 the Hubble radius.
We perform a combined fit to the spectrum and Xmax distributions taking as free parameters the elemental fractions, spectral index and rigidity cutoff for both source populations, as well as the critical energy and normalized inter-source distance characterizing the magnetic horizon suppression factor. We find that for a soft cutoff with Δ = 1 the best fit results for the case without magnetic horizon effect and a very hard spectrum of the high-energy population of sources. On the other hand, for sharper cutoffs, a solution with a significant magnetic horizon and a soft spectrum for the high-energy sources provides the best fit. As an example, Figure 1 (right panel) shows the injection spectra for both source populations for the case of a sharp cutoff with Δ = 3, assuming the EPOS-LHC hadronic model. The high-energy population, depicted in dotted lines, has a spectrum E−γ with γ = 1.43.
Figure 1: Left panel: Magnetic suppression factor as a function of energy over critical energy forthree values of the normalized intersource distance Xs. Right panel: Injection rate of particles at the source per logarithmic energy bin for H (red), He (gray), N (green), Si (cyan) and Fe (blue) assuming a EPOS-LHC model with magnetic horizon effect.
Figure 2: Flux at Earth (upper panel) and moments of the Xmax distribution (lower panel). Dotted lines in the spectrum plots represent the flux coming from the primary nuclei while solid lines correspond to the total flux (primary plus secondary) of each mass group.
Figure 2 shows the spectrum at Earth and the first two moments of the Xmax distribution for the same scenario as in the right-panel of Figure 1. The effect of the magnetic horizon is apparent, being the responsible of the hard observed spectrum of the different mass components of the high-energy population. The required mean inter-source distance is large, with Xs > 2 and the turbulent extragalactic magnetic field between the closest sources and the Earth, that is inside the Local supercluster, should be strong, Brms = O(10 to 200 nG). We see that the composition grows heavier for increasing energies, with each element peaking in a narrow energy range. Most protons and He nuclei are secondary particles from the disintegration of N nuclei. These He nuclei give rise to the instep feature of the spectrum at 15 EeV. Above the instep, the N contribution dominates
the flux until the high-energy suppression, above which we find mostly iron and silicon.
In conclusion, we have shown that if the source density is small and the extragalactic magnetic field is strong, the magnetic horizon effect can provide an alternative explanation of the very hard spectra of the individual mass components at the Earth which is inferred at the highest energies.
Related paper:
Impact of the Magnetic Horizon on the Interpretation of the Pierre Auger Observatory Spectrum and Composition Data
The Pierre Auger Collaboration, JCAP07(2024)094
[arxiv.org/abs/2404.03533] [doi: 10.1088/1475-7516/2024/07/094]
