Excitation of the non-resonant streaming instability around sources of Ultra-High Energy Cosmic Rays

1. Problem Statement

The observed spectrum and composition of Ultra-High Energy Cosmic Rays (UHECRs) by the Pierre Auger Observatory and Telescope Array present a significant puzzle:

• Spectral Hardening: The inferred source spectrum above the "ankle" ($\sim 10^{18.3}$ eV) is unusually hard ($\gamma \lesssim 1$, potentially inverted $\gamma < 0$), contradicting standard diffusive shock acceleration models which predict $\gamma \gtrsim 2$.
• Flux Suppression: There is a suppression of the UHECR flux below $\sim 1$ EeV.
• Existing Explanations: Previous models attribute this to:
  1. Confinement around sources: Heavy nuclei trapped near sources undergo photodisintegration, altering composition and spectrum.
  2. Magnetic Horizon Effect: Diffusion in strong extragalactic magnetic fields (IGMF) delays low-energy particles, creating a low-energy cutoff. However, this requires IGMF strengths near observational upper limits ($\gtrsim 10$ nG), which is difficult to reconcile with Faraday rotation and synchrotron constraints.

The authors propose a third mechanism: Self-induced confinement. They investigate whether the electric current carried by escaping UHECRs can excite plasma instabilities that generate the necessary magnetic turbulence to trap particles, thereby explaining the spectral features without requiring unrealistically strong pre-existing magnetic fields.

2. Methodology

The authors develop a semi-analytical model based on the non-resonant streaming instability (NRSI) (Bell 2004) and its non-linear saturation.

• Physical Setup:
  • A source (e.g., a galaxy or galaxy cluster) with radius $R$ and luminosity $L_p$ injects UHECRs into a medium with a pre-existing magnetic field $B_0$ and coherence length $\lambda_B$.
  • Escaping UHECRs create a positive electric current density $J_{CR}$.
• Instability Excitation:
  • The current drives the NRSI if the energy density of the current exceeds the magnetic energy density of the background field ($E_{J} \gtrsim B_0^2/4\pi$).
  • This leads to the growth of magnetic perturbations $\delta B$ at wavenumbers $k_{max} \gg 1/R_L$ (where $R_L$ is the Larmor radius).
• Saturation and Transport:
  • The instability saturates when $\delta B \sim B_{upper}$, creating a Bohm-like diffusion coefficient $D \propto E/\delta B$.
  • Timescales: The authors calculate the saturation time $\tau_{sat}$, diffusion time $\tau_{diff}$, and advection time $\tau_{adv}$.
  • Advection: They incorporate the effect of cosmic ray pressure gradients driving the background plasma (Alfvénic drift), which can eventually carry particles out of the confinement region if the source luminosity is too high.
• Population Modeling:
  • Instead of assuming identical sources, the authors model a population of sources with a luminosity function (LF) (power-law and broken power-law distributions typical of AGNs and starburst galaxies).
  • They calculate the emissivity of escaping protons by convolving the source LF with the energy-dependent escape probability (determined by $\tau_{esc} = \min(\tau_{diff}, \tau_{adv})$).
• Neutrino Calculation:
  • They calculate the cosmogenic neutrino flux (from escaping protons interacting with CMB/EBL) and the confined neutrino flux (from protons trapped near the source interacting with local radiation fields).

3. Key Contributions

1. Derivation of Confinement Conditions: The paper establishes the specific range of source luminosities ($L_p$) and pre-existing magnetic fields ($B_0$) required for self-confinement to occur at EeV energies.
   • Lower Limit on $B_0$: If $B_0$ is too weak, the instability grows on scales comparable to thermal ion Larmor radii, suppressing growth.
   • Upper Limit on $B_0$: If $B_0$ is too strong, the current cannot drive the instability.
   • Luminosity Bounds: There is a minimum luminosity ($L_{min}$) to excite the instability and a maximum ($L_{max}$) beyond which advection dominates and particles escape too quickly.
2. Spectral Hardening Mechanism: The authors demonstrate that self-confinement naturally produces a low-energy cutoff in the escaping spectrum. Because the most luminous sources confine particles up to higher energies, the convolution of the source luminosity function with the confinement effect results in an escaping spectrum that appears harder ($\gamma < 2$) than the injection spectrum, mimicking the Auger observations without requiring hard injection spectra.
3. Neutrino Constraints: The study quantifies the neutrino flux from confined protons. It shows that while confined protons produce a significant neutrino flux in the PeV range (exceeding the purely cosmogenic flux), the total flux remains consistent with current IceCube upper limits.

4. Key Results

• Confinement Energy Scale: For a source with $L_p \sim 10^{45}$ erg/s and $R \sim 1$ Mpc, particles with energies $E \lesssim 0.6$ EeV can be confined for times exceeding the age of the Universe ($\sim 10$ Gyr), provided $B_0 \sim 0.1 - 1$ nG and $\lambda_B \sim 10$ Mpc.
• Escape Spectrum: The escaping spectrum $Q_{src}(E)$ is truncated below an energy $E_D$ defined by:
  $$E_D \simeq 0.6 \text{ EeV } \left(\frac{L_p}{10^{45} \text{ erg/s}}\right)^{1/2} \left(\frac{R}{\text{Mpc}}\right)^{-1} \left(\frac{\lambda_B}{10 \text{ Mpc}}\right)^2$$
• Luminosity Function Impact:
  • For a power-law LF with slope $\beta < 2$, the escaping spectrum scales as $Q(E) \propto E^{-2 + 2(2-\beta)}$. This explains the observed hardening.
  • For broken power-law LFs (typical of AGNs), the hardening is driven by the high-luminosity end of the distribution.
• Neutrino Flux: The confined protons generate a neutrino flux that peaks in the 1–100 PeV range. While this component dominates over the cosmogenic flux in this energy band, the total predicted flux does not violate IceCube constraints, even under optimistic assumptions about the proton fraction.
• Magnetic Field Amplification: The model predicts that powerful UHECR sources are surrounded by "magnetic cocoons" extending tens of Mpc, where the magnetic field is amplified to $\sim$ nG levels. The estimated Faraday Rotation Measure (RM) from these cocoons is $\lesssim 10^{-9} \mu\text{G cm}^{-3}$, which is consistent with current observational limits.

5. Significance

This work provides a self-consistent physical framework to explain the UHECR spectral hardening and low-energy suppression observed by Auger without invoking:

1. Unusually hard injection spectra ($\gamma < 2$) which are difficult to justify with standard acceleration physics.
2. Strong, ubiquitous extragalactic magnetic fields that conflict with Faraday rotation data.

Instead, it attributes these features to self-generated turbulence driven by the UHECRs themselves. The mechanism is robust across a range of plausible source luminosity functions and environmental parameters. Furthermore, the model makes testable predictions regarding the diffuse neutrino flux (specifically an excess in the PeV range) and the magnetic structure of the intergalactic medium around UHECR sources, offering a promising avenue for multi-messenger astronomy validation.

The authors note that while the current analysis focuses on protons, the rigidity-dependent nature of the mechanism suggests it applies to mixed compositions, which will be explored in future work.