Time-dependent cosmic-ray escape from wind bubbles: hard spectra formation

Technical Summary: Time-dependent cosmic-ray escape from wind bubbles: hard spectra formation

Problem Statement
Wind-driven bubbles, generated by sources such as young massive stellar clusters (YMSCs), Wolf–Rayet stars, and super-Eddington accreting compact objects, are promising candidates for accelerating cosmic rays (CRs) to PeV energies (PeVatrons). While diffusive shock acceleration (DSA) at the inner wind termination shock is widely accepted as the primary acceleration mechanism, the time-dependent transport of these particles from the termination shock to the forward shock (the escape boundary into the interstellar medium) remains poorly explored. Previous studies often relied on stationary descriptions or estimated transport timescales rather than solving the full time-dependent transport equation. This work addresses the gap in understanding how the dynamic expansion of the bubble and the time-evolving distance between the termination and forward shocks impact the energy spectrum of escaping cosmic rays.

Methodology
The authors model the wind bubble as a spherically symmetric system where both the termination shock (TS) and the forward shock (FS) expand with time-dependent velocities ($R_{ts} \propto t^{2/5}$ and $R_{fs} \propto t^{3/5}$ during the deceleration phase). The transport of CRs is governed by a one-dimensional spherical time-dependent transport equation (Fokker-Planck equation), which includes advection by the shocked wind flow and diffusion.

To solve this equation, the authors transform it into a set of stochastic differential equations (SDEs) using Itô's lemma. These SDEs are integrated using a modified version of the open-source CRPropa3.2 framework. Key methodological features include:

• Injection: Particles are injected at the TS with a pre-accelerated power-law spectrum ($dN/dE \propto E^{-2}$) and a time-dependent high-energy cutoff determined by the Hillas limit and acceleration timescales.
• Advection: A radial velocity profile $u(r) \propto r^{-2}$ is assumed in the downstream region, normalized to the post-shock velocity.
• Diffusion: The study investigates three turbulence models: Bohm ($\delta=1$), Kraichnan ($\delta=3/2$), and Kolmogorov ($\delta=5/3$). The diffusion coefficient depends on the local magnetic field strength and the turbulence power spectrum.
• Magnetic Field: The magnetic field is estimated based on a fraction of the ram pressure. The study tests various radial dependencies of the downstream magnetic field ($B \propto r^{-\alpha}$ with $\alpha = 0, -1, -2$) and magnetization levels ($\epsilon_B$).
• Simulation Parameters: Simulations run up to 5 Myr, covering the wind-driven phase before supernovae dominate the energy budget.

Key Results
The time-dependent treatment reveals significant deviations from the standard expectation of escaping spectra:

1. Spectral Hardening: In specific physical configurations, the escaping CR spectra can be harder than the injected $E^{-2}$ spectrum. This hardening is most pronounced for turbulence models with stronger energy dependence (e.g., Bohm diffusion) and weaker magnetic field dependencies.
2. Low-Energy Suppression: A critical finding is the significant suppression of low-energy particles in the escaping flux. Because the forward shock (escape boundary) moves away from the source over time, low-energy particles with small diffusion coefficients cannot catch up to the escape boundary and remain trapped in the downstream region.
   • For Bohm diffusion, low-energy particles ($E < 1$ TeV) are almost completely depleted from the escaping flux.
   • For Kraichnan diffusion, a moderate hardening ($\Delta \gamma \approx 0.1-0.2$) and partial low-energy suppression are observed.
   • For Kolmogorov diffusion, the depletion is minimal, and the escaping spectrum closely resembles the injected spectrum, though the maximum energy remains lower ($\sim 20$ TeV).
3. Parameter Dependence: The degree of low-energy depletion and spectral hardening depends on the turbulence model, the radial dependence of the magnetic field (weaker dependence leads to harder spectra), and the magnetization level.
4. Observational Implications: When propagated to Earth (simulated for a source at 3 kpc), the low-energy suppression persists, though the hardening is partially smoothed out by Galactic diffusion over long timescales.

Significance and Claims
The paper claims that the time-dependent nature of the wind bubble expansion is a crucial factor in shaping the observed CR spectrum, independent of energy loss processes. The authors emphasize that:

• Hard Spectra Formation: Wind bubbles can naturally produce hard CR spectra (harder than $E^{-2}$) without requiring non-standard acceleration physics, provided the transport is time-dependent.
• Low-Energy Cutoff: The "trapping" of low-energy particles within the bubble is a robust prediction of this time-dependent model. This could lead to observable features in multi-messenger radiation (gamma rays and neutrinos) from within the bubble and affect the accumulated grammage of CRs.
• Distinction from SNRs: The resulting hard spectra and low-energy cutoffs differ from those typically expected from supernova remnants (SNRs), potentially helping to explain the CR spectrum around the "knee" or specific gamma-ray observations of stellar clusters (e.g., Westerlund 1).

The authors conclude that while their model focuses on the wind-driven phase, the trapped low-energy particles may later be re-accelerated by supernova shocks or escape when the forward shock ceases, potentially contributing to the low-energy CR population with a modified spectral shape. Future work will incorporate hadronic interactions to predict time-dependent gamma-ray and neutrino emissions.