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arxiv: 2511.13811 · v1 · submitted 2025-11-17 · 🌌 astro-ph.GA · astro-ph.HE

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

Maria Werhahn , Christoph Pfrommer , Joseph Whittingham , L\'ena Jlassi , R\"udiger Pakmor , Philipp Girichidis , Rebekka Bieri This is my paper

Pith reviewed 2026-05-17 21:28 UTC · model grok-4.3

classification 🌌 astro-ph.GA astro-ph.HE
keywords cosmic ray electronssteady-state assumptiontime-dependent modelinggalactic disksupernova injectioncooling lossesMilky Way simulationoutflows
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The pith

Time-dependent modeling shows steady-state assumptions hold for most cosmic ray electrons in galactic disks but fail at high energies and in outflows.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether cosmic ray electron spectra in galaxies can be treated as being in steady state between injection and cooling. Using time-dependent evolution on tracer particles inside a full magnetohydrodynamic simulation of a Milky Way-mass galaxy, the work finds that the bulk population in the star-forming disk matches steady-state expectations up to roughly 500 GeV. Above that energy and for electrons leaving the disk, recent injections dominate and the spectrum deviates. A reader would care because cosmic ray electrons produce the radio emission used to map magnetic fields and star formation across galaxies, so knowing where the steady-state shortcut works avoids systematic errors in those maps.

Core claim

In the time-dependent treatment the global spectrum inside the simulated galactic disk closely resembles a steady-state solution up to 500 GeV. At higher energies the spectrum is steeper and lower in amplitude because it is shaped by recently injected electrons that have not yet reached equilibrium with cooling. Spatially, the electrons remain more confined to the star-forming disk than the extended distributions produced by steady-state post-processing of the same simulation.

What carries the argument

Lagrangian tracer particles that carry and evolve the cosmic ray electron spectrum under adiabatic changes together with synchrotron, inverse Compton, bremsstrahlung and Coulomb cooling, with electrons injected at supernova sites.

If this is right

  • Steady-state models remain reliable for the majority of cosmic ray electrons inside star-forming disks.
  • High-energy cosmic ray electrons must be modeled with their recent injection history rather than an equilibrium assumption.
  • Outflowing cosmic ray electrons carry time-dependent spectral shapes that affect predictions for galactic winds and halo emission.
  • Radio continuum maps at high frequencies will show more compact emission when modeled without the steady-state approximation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • High-frequency radio observations could be used to test how much recent supernova activity influences the observed spectrum.
  • The same time-dependent approach applied to cosmic ray protons in the same runs would likely reveal where steady-state breaks for the hadronic component as well.
  • Repeating the comparison across galaxies with varying star-formation rates would show how the energy threshold for steady-state validity scales with galactic conditions.

Load-bearing premise

The magnetohydrodynamic simulation with its chosen supernova injection sites and magnetic field structure correctly reproduces the injection and transport timescales that control cosmic ray electrons in real galaxies.

What would settle it

A measurement of the cosmic ray electron spectrum above 500 GeV inside the Milky Way disk, or resolved radio maps of the spatial extent of high-frequency synchrotron emission, would show whether the steeper spectrum and more compact distribution appear as predicted.

Figures

Figures reproduced from arXiv: 2511.13811 by Christoph Pfrommer, Joseph Whittingham, L\'ena Jlassi, Maria Werhahn, Philipp Girichidis, Rebekka Bieri, R\"udiger Pakmor.

Figure 1
Figure 1. 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 𝜀★ to the CMB energy densities. The gas surface density is projected a… view at source ↗
Figure 2
Figure 2. Figure 2: shows maps of the volume-weighted mean CR electron spectrum 𝑓 (𝑝) at 𝑝 = 1 GeV/(𝑚e𝑐 2 ) and 100 GeV/(𝑚e𝑐 2 ) within a 4 kpc-thick slice, at 𝑡 = 3 Gyr, when the SFR is 4.4 M⊙ yr−1 . The spatial distribution in the steady-state model is clearly more extended, both within the disk and vertically above and below it, than in the Crest model. This broader distribution arises because in the post-processed steady-… view at source ↗
Figure 3
Figure 3. 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. the disk. Hence, for such high-energy electrons, spatial diffusion is unlikely to play a significant role, as radiative losses occur … view at source ↗
Figure 4
Figure 4. 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 ≈ 104 , then steepens slightly, and deviates noticeably at 𝑝 ≳ 106 . The cell-based steady-state spectrum without diffusion (dashed), obtained with Crayon+, matches the one-zone result, while diffusion (dotted) flattens the s… view at source ↗
Figure 5
Figure 5. Figure 5: Median ages of CR electrons contributing to the total spectrum shown in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: CR electron spectra in four representative regions of the galaxy at 0.9 Gyrs: Panel i) shows the averaged spectrum from Crest (black line) around the solar radius (6 kpc < 𝑅 < 10 kpc and |𝑧 | < 1 kpc), which closely resembles a simple one-zone steady-state model (grey line). In panel ii), the spectrum of a selected tracer from the central region is shown that experienced a recent injection event (located a… view at source ↗
Figure 7
Figure 7. Figure 7: Radially binned CR electron spectra in the cell-based steady-state model with Crayon+ (left-hand panel) and in Crest (right-hand panel). The spectra are shown in 15 cylindrical bins within |𝑧 | <1 kpc. The total spectrum in this region (grey) is overplotted for reference. 100 101 102 103 104 105 106 107 108 p 10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 p 2 × hfiV [cm−3 ] 1.0 Gyr 2.0 Gyr 3.0 Gyr 4.0 Gyr one-… view at source ↗
Figure 8
Figure 8. Figure 8: CR electron spectra from Crest at four different times (1–4 Gyr), averaged over the galactic disk (𝑅 < 18 kpc, |𝑧 | < 1 kpc). steady-state normalisation remains approximately constant over time at 𝑝 ≳ 104 , where IC cooling dominates. This effect is reproduced in Appendix B by a one-zone test model with exponentially declining injection and photon energy density (see Fig. B1). At low momenta, the normalisa… view at source ↗
Figure 9
Figure 9. Figure 9: CR electron spectra from Crest (solid lines) and the steady-state Crayon+ model (dashed lines), compared to observations form Voyager 1 (Cummings et al. 2016), AMS-02 (Aguilar et al. 2014) and H.E.S.S. (Aharonian et al. 2024). The H.E.S.S. data likely include residual contamination from CR nucleons and should thus be interpreted as an upper limit on the true electron flux. To facilitate a comparison of spe… view at source ↗
Figure 10
Figure 10. Figure 10: CR electron spectra at 𝑡 = 1.2 Gyrs for the full medium-resolution simulation (1M; solid lines), 𝑓𝑇, compared to restarted Crest runs in which all spectra were reset to zero at a time Δ𝑡 before the analysis time (dashed lines). The upper panels are colour-coded by the time of the last injection event in each tracer particle, 𝑡inj, as indicated by the colour bar. Spectra are averaged within a disk with rad… view at source ↗
Figure 11
Figure 11. Figure 11: Relative error in CR electron energy for the same regions as in [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Cooling timescales as a function of electron momentum for Coulomb (purple), bremsstrahlung (blue), synchrotron (orange) and IC losses (red). The total cooling time (grey) combines all processes. Left: volume-averaged timescales across the galactic disk (𝑅 < 18 kpc, |𝑧 | < 2 kpc). The total cooling time peaks at 𝑝 ∼ 103 with 𝜏cool ≈ 100 Myr and declines toward both smaller and larger momenta, where Coulomb… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents a time-dependent modeling of cosmic ray electron spectra in an isolated Milky Way-mass galaxy using the CREST code applied to AREPO MHD simulations. CR electrons are injected at supernova sites and evolved on Lagrangian tracer particles including adiabatic changes and losses from synchrotron, inverse Compton, bremsstrahlung, and Coulomb interactions. The time-dependent spectra are compared to local and global steady-state models from CRAYON+ as well as one-zone analytic solutions. The central claim is that the global CR electron spectrum in the star-forming disk resembles a steady-state solution up to ~500 GeV, with deviations at higher energies where recently injected electrons dominate, producing steeper spectra and lower normalization; spatially, the time-dependent electrons are more confined to the disk than in steady-state post-processing.

Significance. If the result holds, the work shows that steady-state assumptions suffice for the bulk disk population but fail for high-energy and outflowing components, with implications for modeling synchrotron emission and interpreting radio and gamma-ray observations of galaxies. A strength is the direct comparison of on-the-fly time-dependent evolution against independent steady-state codes in a full galactic MHD context rather than one-zone models. The simulation-driven approach with multiple comparison models is a positive feature, though the absence of quantitative error bars or resolution tests limits immediate robustness.

major comments (2)
  1. [Results section (discussion of 500 GeV threshold and spatial distributions)] The headline result—that steady-state matches the bulk population up to ~500 GeV while time-dependent treatment is required above this energy and for outflows—depends on the ratio of cooling time to injection/transport time. This ratio is set by the AREPO simulation's magnetic-field strength, geometry, gas densities, and supernova placement. The manuscript reports no parameter variations or sensitivity tests on these MHD inputs (e.g., different SN clustering or B-field amplification), so it is unclear whether the reported 500 GeV transition is robust or specific to the chosen setup.
  2. [Methods (CREST implementation and tracer evolution)] No resolution or convergence tests are described for the number of Lagrangian tracer particles, time-stepping criteria in CREST, or the underlying AREPO grid resolution. Without these, numerical artifacts could affect the high-energy tail and the claimed spatial confinement differences between time-dependent and steady-state models.
minor comments (2)
  1. [Abstract] The abstract states the resemblance holds 'up to energies of 500 GeV' without specifying whether this is a precise value, an approximate threshold, or accompanied by any uncertainty estimate from the simulation.
  2. [Figure captions and results text] Figure captions and text should clarify whether the plotted spectra are volume-weighted, mass-weighted, or averaged over specific regions, as this affects direct comparison to the global steady-state models.

Simulated Author's Rebuttal

2 responses · 2 unresolved

We thank the referee for their detailed and constructive report on our manuscript. We have carefully considered each comment and provide point-by-point responses below. Where appropriate, we have revised the manuscript to address the concerns raised.

read point-by-point responses

  1. Referee: [Results section (discussion of 500 GeV threshold and spatial distributions)] The headline result—that steady-state matches the bulk population up to ~500 GeV while time-dependent treatment is required above this energy and for outflows—depends on the ratio of cooling time to injection/transport time. This ratio is set by the AREPO simulation's magnetic-field strength, geometry, gas densities, and supernova placement. The manuscript reports no parameter variations or sensitivity tests on these MHD inputs (e.g., different SN clustering or B-field amplification), so it is unclear whether the reported 500 GeV transition is robust or specific to the chosen setup.

    Authors: We appreciate the referee pointing out the dependence of our results on the specific MHD parameters of the simulation. The 500 GeV energy marks the point where the electron cooling time becomes shorter than the typical time between supernova injections in the star-forming disk of our model. This is a direct consequence of the magnetic field strengths (typically a few μG in the disk) and gas densities in the AREPO simulation. While we agree that the exact transition energy would shift with different magnetic field amplification or supernova clustering, our study focuses on a single, well-resolved Milky Way-mass galaxy simulation to provide a concrete comparison between time-dependent and steady-state approaches. In the revised version, we have expanded the discussion in Section 4 to explicitly state that the transition energy scales with the cooling timescale, which depends on B and n, and note that our results are representative for typical galactic conditions. We believe this clarifies the scope without requiring a full parameter study, which would be a substantial extension beyond the current work. revision: partial

  2. Referee: [Methods (CREST implementation and tracer evolution)] No resolution or convergence tests are described for the number of Lagrangian tracer particles, time-stepping criteria in CREST, or the underlying AREPO grid resolution. Without these, numerical artifacts could affect the high-energy tail and the claimed spatial confinement differences between time-dependent and steady-state models.

    Authors: We thank the referee for this observation. The number of Lagrangian tracer particles (approximately 10^6) was chosen to ensure sufficient sampling of the supernova injection events and the volume of the galactic disk, drawing from previous applications of the CREST code. The time-stepping in CREST is adaptive based on the cooling timescales to accurately capture the energy losses. However, we acknowledge that dedicated convergence tests varying the tracer number, time-step criteria, and AREPO resolution are not included in the present manuscript. We have added a paragraph in the Methods section (Section 2) discussing these numerical choices and arguing that the high-energy tail is primarily shaped by recent injections rather than numerical diffusion. Full convergence studies are computationally intensive and will be addressed in future work focused on numerical validation. revision: partial

standing simulated objections not resolved
  • Conducting a series of additional AREPO simulations with varied supernova clustering and magnetic field strengths to test the sensitivity of the 500 GeV transition energy.
  • Performing systematic resolution and convergence tests for the CREST code on the full galactic simulation data.

Circularity Check

0 steps flagged

Simulation-driven comparison exhibits no circularity

full rationale

The paper evolves CR electron spectra time-dependently via the CREST code on Lagrangian tracers within an AREPO MHD simulation of an isolated Milky Way-mass galaxy, injecting electrons at supernova sites and including adiabatic changes plus synchrotron, IC, bremsstrahlung and Coulomb losses. These on-the-fly spectra are then compared against independent local/global steady-state solutions computed with the separate CRAYON+ code and against one-zone analytic steady-state models. No equation or result is obtained by fitting a parameter to the same data that is later relabeled as a prediction, nor does any central claim reduce to a self-citation chain or to a definition that presupposes the target outcome. The reported transition at ~500 GeV and the spatial confinement differences emerge directly from the numerical integration of the time-dependent transport equation against the fixed simulation fields and injection history; the derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard astrophysical assumptions about supernova injection sites and the included cooling channels; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Cosmic ray electrons are injected exclusively at supernova remnant sites
    Standard modeling choice stated in the abstract
  • domain assumption Cooling occurs only via synchrotron, inverse Compton, bremsstrahlung, and Coulomb losses
    Listed explicitly as the processes included in the time-dependent evolution

pith-pipeline@v0.9.0 · 5599 in / 1406 out tokens · 30719 ms · 2026-05-17T21:28:01.205896+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    astro-ph.HE 2026-03 conditional novelty 7.0

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