Steady-State or Not? The Evolution of Cosmic Ray Electron Spectra in Galaxies

TL;DR

This study tests the validity of steady-state assumptions for galactic CR electron spectra by comparing time-dependent Crest evolution on tracer particles in a Milky Way–mass AREPO disk to steady-state Crayon+ post-processing and simple analytic models. Crest solves the full momentum-space Fokker–Planck equation with adiabatic changes and radiative losses (synchrotron, IC, bremsstrahlung, Coulomb) on a live galaxy, while Crayon+ provides a per-cell steady-state benchmark, allowing direct assessment of where non-equilibrium effects matter. Globally, Crest reproduces the steady-state spectrum up to $p\sim10^6$ (roughly $E_{kin}\sim$500 GeV), with deviations tied to recently injected high-energy electrons and to outflows; regionally, spectra reveal non-equilibrium near star-forming sites and strongly cooled populations in winds, highlighting the importance of time-dependent modelling for high-energy and transport-dominated components. The findings support using steady-state models for bulk disk analyses but emphasize that time-dependent treatments are essential to capture high-energy tails, outflows, and local spectral variations that inform non-thermal emission predictions and interpretations of radio/gamma-ray data.

Abstract

Cosmic ray (CR) electrons are key tracers of non-thermal processes in galaxies, yet their spectra are often modelled under the untested assumption of steady state between injection and cooling. In this work, we present a time-dependent modelling of CR electron spectra in a galactic context using the CREST code, applied to magnetohydrodynamical simulations of an isolated Milky Way-mass galaxy performed with AREPO. CR electrons are injected at supernova sites and evolved with adiabatic changes and cooling processes on Lagrangian tracer particles, including losses from synchrotron, inverse Compton, bremsstrahlung, and Coulomb interactions. We compare these fully time-dependent spectra to local and global steady-state models computed with CRAYON+, as well as to one-zone analytic steady-state solutions. We find that the global CR electron spectrum in the simulated galactic disk closely resembles a steady-state solution up to energies of 500 GeV, with deviations only at higher energies where cooling times become shorter than injection timescales. High-energy electrons are dominated by recently injected populations that have not yet reached equilibrium, however, producing a steeper spectrum and lower normalisation than a steady-state model predicts. Spatially, the electrons modelled on-the-fly with CREST are more confined to the star-forming disk, in contrast to the more extended distributions from steady-state post-processing models. Our results demonstrate that while steady-state assumptions capture the bulk CR electron population in star-forming disks, a time-dependent treatment is essential to describe the high-energy and outflowing components.

Figures (16)

• Figure 1: Overview of the simulated galaxy at 0.5 Gyrs (first two rows) and 3.0 Gyrs (last two rows) of evolution, both face-on (first and third row) and edge-on (second and fourth row). From left to right, we show the gas surface density, gas density, CR proton energy density, magnetic field strength and ratio of the interstellar radiation field $\varepsilon_\star$ to the CMB energy densities. The gas surface density is projected along 44 kpc and the magnetic field strength is averaged over a thin slice of thickness 1 kpc, overlaid with a line integral convolution indicating its orientation 1993CabralLeedom. All other quantities are shown as thin slices through the centre.
• Figure 2: Distribution of CR electrons in the steady-state model with Crayon+ (upper four panels) and Crest (lower four panels), both face-on and edge-on at 1 and 100 GeV (left- and right-hand panels, respectively) at $t=3$ Gyr.
• Figure 3: Slices showing cooling timescales for CR electrons at 1 GeV (left) and 100 GeV (right), shown both face-on (upper panels) and edge-on (lower panels) at $t=3$ Gyr. The timescales are calculated from slices of the gas density, magnetic field strength and photon energy density.
• Figure 4: Left: CR electron spectra in the galactic disk ($R<18$ kpc, $|z|<1$ kpc) at $t=1$ Gyr. The Crest spectrum (black) closely follows the one-zone steady-state model (light gray) up to $p\approx 10^4$, then steepens slightly, and deviates noticeably at $p\gtrsim 10^6$. The cell-based steady-state spectrum without diffusion (dashed), obtained with Crayon+, matches the one-zone result, while diffusion (dotted) flattens the spectrum at low momenta ($p\lesssim 10^3$). Right: The Crest spectrum (black) decomposed by time since the last injection ($t_\mathrm{inj}$), as indicated by the colorbar. The individual components are volume averaged spectra, rescaled to the total volume. The total spectrum at $p=10^3$ is dominated by older electrons ($t_\mathrm{inj}\sim10^7-10^8$ yr), whereas the high-energy end is composed entirely of young electrons that have recently undergone energy injection ($t_\mathrm{inj}\lesssim10^5$ yr). Shaded regions show the 16th-84th percentile range of each electron population.
• Figure 5: Median ages of CR electrons contributing to the total spectrum shown in Fig. \ref{['fig:electron_spectra']} (solid lines) compared to their cooling timescales estimated from disk-averaged gas properties (dashed line). The shaded regions denote the 16th–84th percentile range and overlap closely with the cooling time up to $p\sim2\times10^6$. At higher momenta, the median age drops below the cooling time, indicating that these electrons were recently injected and have not yet reached a steady-state. The shaded horizontal bands correspond to the $t_\mathrm{inj}$ bins shown in Fig. \ref{['fig:electron_spectra']}.