Astrophysical origins of TeV features in the cosmic-ray lepton spectrum

The paper revisits the astrophysical origins of TeV-scale features in the cosmic-ray lepton spectrum, motivated by precise measurements from PAMELA, AMS-02, ATIC, and DAMPE that reveal spectral structures beyond a single smooth power-law background. The authors construct a smooth diffuse electron background using GALPROP models calibrated by cosmic-ray nuclei and diffuse gamma-ray observations, retaining 64 models after discarding those with halo heights z = 8 kpc and z = 10 kpc that fail to reproduce 9Be/10Be data. The models are normalized to AMS-02 all-electron flux at 100 GeV, and the median spectrum is parameterized with a smoothly broken power law: Φbg(E) = Φ0 (E/E0)(−γ1) (1 + (E/Eb)(1/s)) / (1 + (E0/Eb)(1/s)), with γ1 = 3.12, γ2 = 4.52, Eb = 1.37 TeV, and s = 0.206. The background softens smoothly across the TeV range, approaching an effective index of approximately 4.4–4.5 at higher energies, without generating narrow bumps or abrupt cutoffs.

For nearby pulsar contributions, the authors use a Green-function solution for discrete sources with radiative losses, considering both burst-like and continuous injection histories. The injected spectrum is Q(Es) = Q0 Es(−γ) exp[−(Es/Ecut) β], with diffusion coefficient D(E) = D0 (E/E0) δ (D0 = 4.3 × 10 28 cm2 s−1, E0 = 4 GeV, δ = 0.415) and cooling rate b(E) = b0 E2 (b0 = 10−16 GeV−1 s−1). For burst-like injection, the propagated spectrum develops a peak followed by a decline near the cooling boundary, with the cooling time t loss(E) = 1/(b0 E) ≃ 3.2 × 10 5 yr (1 TeV/E). The authors implement a stochastic inverse-Compton Monte Carlo using the Popescu et al. ISRF model along the Geminga line of sight, with path-averaged energy densities ρ opt = 0.435 eV cm−3, ρ IR = 0.504 eV cm−3, and ρ CMB = 0.260 eV cm−3, and a uniform magnetic field B = 3 µG. They find that stochastic inverse-Compton cooling broadens the sharp edge obtained with deterministic continuous losses, producing smoother post-peak declines and more extended high-energy tails. For the burst-like case with d = 500 pc and ages from 30 kyr to 1000 kyr, the effective post-peak sharpness parameter βeff (from fits of AE(−p) exp[−(E/Eeff) βeff] with p = 1.5) ranges from 1.29 ± 0.31 at 30 kyr to 3.61 ± 0.14 at 1000 kyr, showing that the sharpness varies with source age and is not a stable edge.

For continuous injection from a young pulsar with a hard, sharply cut off spectrum Q(Es) ∝ Es(−1) exp[−(Es/500 GeV)2] at d = 100 pc, the falling feature is mainly inherited from the intrinsic source cutoff, with βeff ≃ 2 for ages from 100 yr to 10 kyr. The energetics are examined: for burst-like mature pulsars at d = 500 pc normalized to 10% of the diffuse background at 1 TeV, the required total injected pair energy is We ≃ 5 × 10 47 erg for T = 100 kyr and We ≃ 5 × 10 48 erg for T = 300 kyr, with the 100 kyr case being energetically reasonable and the 300 kyr case demanding. For young continuous injection at d = 100 pc normalized at 500 GeV, T = 10 kyr requires Ẇe ≃ 1.1 × 10 34 erg s−1 (We ≃ 3.4 × 10 45 erg), while T = 300 yr requires Ẇe ≃ 7.3 × 10 37 erg s−1 (We ≃ 6.9 × 10 47 erg), which is difficult to realize given typical pulsar spin-down powers of 10 35–10 37 erg s−1.

The authors then use the two highest-energy DAMPE data points as an illustrative case to compare possible local-source contributions. They normalize three templates to the positive residuals relative to the median GALPROP background: a burst-like pulsar (d = 500 pc, T ≃ 20 kyr, We ≃ 8 × 10 46 erg), a continuous-injection pulsar (d = 100 pc, T = 3 kyr, Ecut = 5 TeV, Ẇe ≃ 6 × 10 33 erg s−1, We ≃ 5 × 10 44 erg), and an SNR-like burst source (γ = 2.4, Ecut = 20 TeV, d = 300 pc, T ≃ 10 kyr, We ≃ 4 × 10 47 erg, with βeff ≃ 2.7). The SNR-like template is broader and less endpoint-like due to its softer injection spectrum and simple exponential cutoff.

The paper concludes that nearby pulsars can produce structured high-energy electron spectra under favorable conditions, but a rapid endpoint-like falloff requires additional assumptions: in the burst-like case, stochastic inverse-Compton cooling softens the apparent cooling edge, while in the continuous-injection case, the rapid decline is inherited from the intrinsic cutoff. The spectra evolve continuously with source age rather than producing a stable, source-age-independent sharp edge. A very sharp and stable edge-like feature would be more difficult to accommodate with ordinary pulsar propagation and would strengthen the case for alternative origins such as dark matter. Future high-energy-resolution observations, including increased DAMPE exposure and HERD, will be needed to determine whether a putative TeV feature is a smooth astrophysical cutoff, a cooling-broadened local-source contribution, or a sharper edge-like structure more naturally associated with particle physics.

Improvements to AI Systems Based on This Paper:

1. Physics-Informed Spectral Fitting with Uncertainty Propagation

• Improvement: Train a generative model to produce cosmic-ray lepton spectra using the full 64-model GALPROP ensemble (with halo-height and 9Be/10Be constraints) rather than a single median. Use Bayesian neural networks or normalizing flows to output a distribution of background parameters (γ1, γ2, Eb, s) with correlated uncertainties.

• What the improved AI can do: Automatically fit any new cosmic-ray electron/positron dataset (e.g., from DAMPE, HERD) while marginalizing over astrophysical background uncertainties, yielding robust significance estimates for spectral features without manual model selection.

2. Stochastic Cooling Emulator for Pulsar Propagation

• Improvement: Replace the expensive stochastic inverse-Compton Monte Carlo with a neural operator or a diffusion-model surrogate that learns the mapping from (source age, distance, injection spectrum, ISRF energy densities, magnetic field) to the propagated electron spectrum, including the stochastic broadening of the cooling edge. Train on the paper’s Monte Carlo outputs.

• What the improved AI can do: Rapidly generate thousands of pulsar contribution templates in seconds, enabling real-time scans of parameter space (age, distance, cutoff energy) to match observed residuals—without rerunning full simulations.

3. Automated Source Classification and Anomaly Detection

• Improvement: Build a classifier (e.g., a transformer or CNN) that takes a measured TeV lepton spectrum and predicts the most likely origin: smooth diffuse background, burst-like pulsar, continuous-injection pulsar, SNR-like source, or dark-matter-like sharp edge. Use the paper’s templates (with βeff, energetics, and spectral shapes) as labeled training data.

• What the improved AI can do: Given a new spectrum from DAMPE or HERD, instantly output a posterior probability over source classes, flagging whether the feature is “cooling-broadened” (astrophysical) or “edge-like” (exotic), and estimate required source energetics (We, Ẇe) to assess physical plausibility.

4. Energy-Budget Constraint Enforcement

• Improvement: Integrate a physics-constrained loss function into any generative or regression model that predicts pulsar contributions. The loss penalizes predictions requiring total injected energy > 10 48 erg or spin-down power > 10 37 erg/s, using the paper’s energetics limits.

• What the improved AI can do: Automatically reject or down-weight physically implausible source parameters during fitting, ensuring that any claimed spectral feature is consistent with known pulsar power budgets—reducing false positives from overfitting.

5. Time-Evolution Forecasting of Spectral Sharpness

• Improvement: Train a recurrent or neural-ODE model on the paper’s age-dependent βeff values (from 30 kyr to 1000 kyr) to predict how the post-peak sharpness evolves for arbitrary source ages and distances.

• What the improved AI can do: Predict whether a newly discovered TeV feature will sharpen or broaden over time (or with better statistics), allowing observers to design follow-up observations to distinguish between a young continuous-injection source (βeff 2) and an old burst-like source (βeff 3.6).

6. Multi-Messenger Joint Inference

• Improvement: Develop a hierarchical Bayesian model that jointly fits cosmic-ray lepton data (this paper’s spectra) with gamma-ray observations (from the same pulsars or SNRs) and cosmic-ray nuclei data (to constrain diffusion parameters). Use the paper’s GALPROP calibration and ISRF model as priors.

• What the improved AI can do: Provide a unified, self-consistent picture of local astrophysical sources—simultaneously explaining TeV electrons, gamma-ray halos (e.g., Geminga), and boron-to-carbon ratios—reducing degeneracies in source parameters.

7. Automated Template Library Generation

• Improvement: Use a variational autoencoder to compress the paper’s family of propagated spectra (burst-like, continuous, SNR-like) into a low-dimensional latent space, then decode any point in that space to a full spectrum.

• What the improved AI can do: Instantly generate a continuous manifold of physically valid spectral templates for any combination of source age, distance, injection index, and cutoff—enabling fast, differentiable fitting to residuals without precomputing a discrete grid.

8. Sharpness-Metric Regression for Dark Matter Discrimination

• Improvement: Train a regression model to predict the βeff parameter (post-peak sharpness) from raw spectral data, using the paper’s fits as ground truth. Then compare predicted βeff to the theoretical value for dark-matter annihilation (which produces a very sharp, stable edge).

• What the improved AI can do: Quantitatively assess whether a measured TeV feature is consistent with a cooling-broadened pulsar (βeff varying with age) or a dark-matter-like edge (βeff large and stable), providing a direct statistical test for new physics.

Precise measurements of high-energy cosmic-ray electrons and positrons have revealed spectral structures that are difficult to capture with a single smooth power-law background. The rising positron fraction measured by PAMELA and AMS-02, together with the all-electron excess reported by ATIC and the high-precision all-electron spectrum measured by DAMPE, has motivated interpretations ranging from nearby astrophysical accelerators to dark-matter annihilation or decay. In this work we revisit the conventional diffuse electron background and the possible contribution from nearby pulsars in a common propagation framework. The diffuse component is modeled with GALPROP configurations calibrated by cosmic-ray nuclei and diffuse gamma-ray observations. We then use the Green-function solution for nearby discrete sources with radiative losses to study pulsar contributions with both burst-like and continuous injection histories, including the effect of stochastic inverse-Compton cooling on the propagated spectra. We also use the highest-energy DAMPE data points as an illustrative case to compare possible local-source contributions from pulsars and supernova-remnant-like burst sources. The spectral shape of such features provides a useful diagnostic for distinguishing physically plausible nearby-source features from more exotic interpretations.