The origin of ultrahigh-energy cosmic rays (UHECRs) is a half-century-old enigma1. The mystery has been deepened by an intriguing coincidence: over ten orders of magnitude in energy, the energy generation rates of UHECRs, PeV neutrinos and isotropic sub-TeV γ-rays are comparable, which hints at a grand unified picture2. Here we report that powerful black hole jets in aggregates of galaxies can supply the common origin for all of these phenomena. Once accelerated by a jet, low-energy cosmic rays confined in the radio lobe are adiabatically cooled; higher-energy cosmic rays leaving the source interact with the magnetized cluster environment and produce neutrinos and γ-rays; the highest-energy particles escape from the host cluster and contribute to the observed cosmic rays above 100 PeV. The model is consistent with the spectrum, composition and isotropy of the observed UHECRs, and also explains the IceCube neutrinos and the non-blazar component of the Fermi γ-ray background, assuming a reasonable energy output from black hole jets in clusters.
The origin of UHECRs is still unknown3. Measurements by the Pierre Auger Observatory (Auger)4 and the Telescope Array (TA)5 find a power-law spectrum, Φ ∝ E−2.6 − E−2.7 (where E is particle energy and Φ is the diffuse intensity in units of particles per energy, area, time and solid angle), with a decline above 6 × 1019 eV, probably due to the interaction of UHECRs with cosmic radiation backgrounds such as the cosmic microwave background (CMB) or an upper limit of the particle energy reachable by the accelerator. Small-scale anisotropy in their arrival directions has not been established4,5.
The IceCube Observatory recently discovered high-energy cosmic neutrinos6,7, which have been anticipated to provide crucial clues to this age-old mystery. An astrophysical flux in the 0.1–1 PeV range is found at the level of 10−8 GeV cm−2 s−1 sr−1 flavour6,7,8,9, which is consistent with expectations of cosmic-ray ‘reservoir’ models10,11,12,13. The arrival directions of the observed events present no significant clustering, and indicate that the sources are extragalactic6,8,9.
A γ-ray counterpart is expected from the hadronic processes that are responsible for neutrino production. If the source environment is transparent to γ-rays, these side products should show up at 1–100 GeV energies after cascading in the extragalactic background light (EBL). A significant fraction of the extragalactic γ-ray background (EGB)14,15 measured by the Fermi Gamma-Ray Space Telescope may be explained by neutrino sources13. Despite the unknown origins of these multi-messenger emissions and fine structures in their data (such as a possible excess in the 10–100 TeV neutrino spectrum6), it is remarkable that over ten orders of magnitude in energy, the energy generation rates of UHECRs, IceCube neutrinos and Fermi EGB are all comparable2,13.
Recent UHECR observations have revealed additional characteristic features of extragalactic cosmic rays. First, a hardening in the spectrum of light particles is seen around 100 PeV16,17, right in the energy range where a steepening in the spectrum of heavy primary particles is observed (which is often called the ‘second knee’). Second, a transition from light elements to medium-to-heavy elements around 1019 eV is suggested by the Auger data4, and a heavy UHECR composition is also supported by the non-detection of cosmogenic neutrinos18. Although the interpretation of the UHECR composition is still debated, direct measurements of indicators of the particle mass seem to be consistent between different experiments. These features were not considered in the simplest convergence theory2; here we provide a concrete astrophysical model in which black hole jets embedded in large-scale structures reconcile these observations.
Relativistic jets from accretion onto supermassive black holes provide promising sites for UHECR acceleration. The Hillas condition suggests that an active galactic nucleus (AGN) can accelerate a particle with charge Z to a maximum energy, Emax ≈ Z 1019 eV in jets or at external shocks that are known to be sites of leptonic emissions3,19. The energy spectrum of particles accelerated by the Fermi mechanism can be described by a power law, , with an index sacc ≈ 2–2.5. Radio observations often find extended lobes (sometimes referred as bubbles or cocoons, which are plasma cavities inflated by the jet), with 10–100 kiloparsec scales20 and 0.1–10 microgauss-level magnetic fields21. Particles with energies below Elobe,c = Z e Blobellobe,c ≈ 1.4 × 1018Z(Blobe/5 μG)(llobe,c/0.3 kpc) eV have a Larmor radius rL = E/(Z e B) that is much smaller than the coherence length of their magnetic structure, where Blobe and llobe,c ≈ (0.01 − 0.1)rlobe are the magnetic field strength and the coherence length, rlobe is the lobe size and c is the velocity of light. Unlike relativistic electrons that cool inside jets or lobes, high-energy ions diffuse for ≈ 6.1(rlobe/10 kpc)2 (E/Z 1 PeV)−1/3 (llobe,c/0.3 kpc)−2/3 (Blobe/5 μG)1/3 Myr and can enter the intracluster medium (ICM). Meanwhile, these high-energy ions suffer from adiabatic losses due to the expansion of the cocoon. The characteristic cooling time is tad ≈ 4.9 (rlobe/10 kpc) (vlobe/2000 km s−1)−1 Myr, where vlobe is the typical expansion velocity at source ages of 0.1–10 Myr22. Particles with energies above Elobe,c are less impacted, escaping semi-diffusively with . Considering the competition between diffusion and cooling, we approximate the spectrum of cosmic rays leaking into the cluster to be .
Radio-loud AGN activity that leads to cosmic-ray injections would preferentially reside in the centres of rich clusters23. A cluster with a halo mass of M = 1014M14M⊙, where M⊙ is the mass of the Sun, has a virial radius . The distribution of thermal gas is often described using the β model as nICM(r) ∝ [1 + (r/r c )2]−3β/2, where β ≈ 0.8 and r c ≈ 0.1 rvir is the core radius24. Turbulent magnetic fields in the ICM, which are probably induced by accretion shocks and other cluster dynamics, typically have a strength of a few microgauss in the cluster centre24. Assuming flux conservation and that the field traces the baryon distribution, we adopt a magnetic field profile B(r) = B0[1 + (r/r c )2]−β with B0 ≈ 5 μG.
Cosmic rays leaving the acceleration site and lobe enter the ICM of the host cluster (which functions as a cosmic-ray reservoir10,11). The highest-energy ions travel in a straight line through the ICM. Particles reaching an energy E c ≈ 2 × 1019 Z B−6(l c /20 kpc) eV have a gyro-radius comparable to the typical scales of magnetic field fluctuations in massive clusters, with l c about 1–10% of the virial radius24. Ions with energies well below E c propagate diffusively in the turbulent magnetic field of the cluster. The confinement, which could last for roughly 1–10 Gyr depending on the particle energy, leads to efficient interactions of cosmic-ray nuclei with baryons and infrared background photons in the cluster, producing pions that decay into neutrinos and γ-rays via and π0 → 2γ, respectively. Finally, particles that leave the cluster propagate to the Earth through the intergalactic medium and extragalactic magnetic fields. UHECRs from sources beyond the energy-loss horizon are depleted via photodisintegration, photomeson production and Bethe–Heitler pair production processes with the CMB and the EBL, producing cosmogenic neutrinos that peak around EeV energies and γ-rays that cascade down to GeV–TeV energies.
We numerically simulate the propagation of cosmic rays in the magnetized ICM and from the source to the observer. We assume that a jetted source as a cosmic-ray accelerator can be anywhere in the core of a cluster with equal probability. We inject five representative groups of elements: hydrogen (1H), helium (4He), nitrogen (14N), silicon (28Si) and iron (56Fe) according to the abundances of elements in Galactic cosmic rays (see Supplementary Information for details), and let each group follow the same power-law spectrum with a cutoff above the maximum rigidity, , where R = E/Ze is the rigidity, sacc = 2.3 and Rmax = 2 × 1021/26 V. We assume that ions are confined up to tinj = 2 Gyr, given that the peak period of AGN activity effectively lasts for around 2–3 Gyr (see Supplementary Information for discussions on model uncertainties and details). The redshift evolution of the source density is taken to be F(z) = (1 + z)3 up to z c = 1.5, but its moderate variations barely impact our results. The cumulative flux10 is obtained by:
where n is the halo number density, dn/dM is the halo mass function, H(z) is the Hubble parameter at redshift z, is the production rate of neutrinos (or propagated cosmic rays) from a given cluster with a redshifted energy E′ = (1 + z) E. We consider clusters with a halo mass above Mmin = 5×1013 M⊙ (corresponding to ~1011 M⊙), which present higher radio-loud AGN fractions23. For the intergalactic propagation, we assume that cosmic rays from a galaxy cluster have 50% chance of encountering magnetic structures with an average strength of 2 nG and a coherence length of 1 Mpc.
Figure 1 shows the integrated spectra of UHECRs and neutrinos from overdense regions with black hole jets. The normalization of the spectra is determined by a combined fit to the Auger spectral and data above 1018.45 eV, and the IceCube data above 2 × 1014 eV. The goodness fit results in χ2 = 44.5 for 30 degrees of freedom, corresponding to a P value of 0.043 for this fiducial case. The cosmic-ray confinement in the lobe and the host cluster makes the injection spectrum harder below the second knee10,13. The spectral shape is in agreement with measurements by both Auger and TA above 1018 eV. Primary and secondary cosmic-ray particles received by the observer are divided into two composition groups: light (including H and He) and intermediate/heavy (including CNO, Si, Mg, Fe), with the two crossing around 1019.5 eV. The mean of the maximum depth of an air shower, , which depends on the mass of the UHE nucleon or nucleus, is shown in Fig. 2. The trend follows the data measured by Auger. Below 1018 eV, accounting for a Galactic contribution with Φ ∝ E−3.4, the predicted cosmic-ray spectrum matches the light component of the KASCADE-Grande data16.
The neutrino spectrum is composed of two parts. Between 1014 eV and 1017 eV, it is mostly contributed by particle interactions in the ICM. It agrees with the IceCube measurements above 1014 eV. The low-energy neutrino spectrum is harder than that of accelerated cosmic rays, and the spectral steepening above 1015 eV results from the faster escape of higher-energy cosmic rays. Above 1018 eV, the neutrino flux is dominated by the cosmogenic neutrinos produced when UHECRs interact with the CMB and the EBL, and is consistent with the IceCube constraints at extremely high energies18. Likewise, the observed sub-TeV γ-rays are produced both in the ICM and during intergalactic propagation2. Thanks to the hard injection spectrum, the total γ-ray flux largely originates from electromagnetic cascades, and is consistent with the non-blazar component of the EGB15. In addition to the hard γ-ray spectrum, our model also predicts a dominance of low-mass clusters, and the γ-ray and radio limits from individual clusters25 can be satisfied.
The chance of previously or currently having active jets in a cluster, fjet, and the average cosmic-ray luminosity of contained active galaxies per cluster, LCR, are left as free parameters. Assuming LCR ~ 1044–1045 erg s−1, we obtain fjet ~ 10–100%. This is consistent with duty cycles of the accretion-driven evolution of black holes26. The number density of clusters and groups with a mass above 5 × 1013 M⊙ is a few 10−5 Mpc−3. This satisfies the muon neutrino limits on the neutrino source density derived from the absence of multiplets2, as well as the lower bounds on the UHECR source density derived from the lack of strong anisotropy in the UHECR data27. Our model predicts an association between the directions of neutrino events and the low-mass clusters of supermassive black holes that have past and ongoing jet activities. Alongside multiplet signals from nearby candidate sources, this correlation can be tested2 with future neutrino observations by experiments such as IceCube-Gen2.
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We thank R. Alves Batista, M. Bustamante, M. Coleman Miller, C. Reynolds and M. Unger for helpful comments. This work made use of supercomputing resources at the University of Maryland. We gratefully acknowledge support from the Eberly College of Science of Penn State University and the Institute for Gravitation and the Cosmos. The work of K.M. is supported by Alfred P. Sloan Foundation and NSF grant No. PHY-1620777.