Dark matter electron anisotropy: a universal upper limit

We study the dipole anisotropy in the arrival directions of high energy CR electrons and positrons (CRE) of Dark Matter (DM) origin. We show that this quantity is very weakly model dependent and offers a viable criterion to discriminate among CRE from DM or from local discrete sources, like e.g. pulsars. In particular, we find that the maximum anisotropy which DM can provide is to a very good approximation a universal quantity and, as a consequence, if a larger anisotropy is detected, this would constitute a strong evidence for the presence of astrophysical local discrete CRE sources, whose anisotropy, instead, can be naturally larger than the DM upper limit. We further find that the main source of anisotropy from DM is given by the fluctuation in the number density of DM sub-structures in the vicinity of the observer and we thus devote special attention to the study of the variance in the sub-structures realization implementing a dedicated Montecarlo simulation. Such scenarios will be probed in the next years by Fermi-LAT, providing new hints, or constraints, about the nature of DM.

💡 Research Summary

The paper investigates the dipole anisotropy (Δ) in the arrival directions of high‑energy cosmic‑ray electrons and positrons (CRE) as a diagnostic tool to distinguish a dark‑matter (DM) origin from local astrophysical sources such as pulsars. CRE propagation in the Galaxy is modeled with the standard diffusion‑loss equation, where the diffusion coefficient D(E)=D₀(E/E₀)^δ and the energy‑loss term b(E)≈b₀E² account for scattering on magnetic turbulence and synchrotron/Inverse‑Compton processes. The source term Q includes contributions from a smooth Galactic halo and from a population of DM sub‑haloes (sub‑structures). Sub‑haloes are generated according to a mass function dN/dM∝M^−α (α≈1.9) with a minimum mass around 10⁻⁶ M⊙, and each sub‑halo follows an NFW or Einasto density profile with a concentration‑mass relation taken from recent N‑body simulations.

The anisotropy is defined as Δ = (Φ_max − Φ_min)/(Φ_max + Φ_min), where Φ(θ, φ) is the CRE flux in a given direction. Two contributions are identified: (i) a global, very small anisotropy from the smooth halo (Δ≲10⁻⁴) and (ii) a potentially larger anisotropy caused by the stochastic spatial distribution of nearby sub‑haloes. The latter is termed “sample variance” and can boost Δ by an order of magnitude or more if a massive sub‑halo lies within a few hundred parsecs of the observer.

To quantify this effect, the authors perform extensive Monte‑Carlo simulations. In each realization they populate a spherical volume of radius 20 kpc with sub‑haloes drawn randomly from the prescribed mass, spatial, and concentration distributions, typically generating ≈10⁵ sub‑haloes per run. For each configuration they solve the diffusion‑loss equation using Green’s‑function techniques to obtain the directional CRE flux and compute Δ. By repeating the procedure thousands of times they derive the statistical distribution of Δ, extracting its mean ⟨Δ⟩ and standard deviation σ_Δ.

A striking result emerges: across a wide range of DM particle masses (100 GeV–10 TeV), annihilation or decay channels (e⁺e⁻, μ⁺μ⁻, etc.), and propagation parameters (different D₀ and δ values), the maximum anisotropy that DM can produce remains confined to a narrow band, roughly 10⁻³–10⁻². This “universal upper limit” is dictated primarily by the probability of having a massive sub‑halo close to the Solar System, a quantity set by the overall sub‑halo mass function and Galactic geometry rather than the microphysics of the DM model. Consequently, the anisotropy upper bound is essentially model‑independent.

For comparison, the paper models realistic nearby pulsars (e.g., Geminga, Monogem) as point‑like CRE injectors with power‑law spectra Q_psr(E)∝E^−γ exp(−E/E_cut). Using the same propagation framework, the authors compute the pulsar‑induced anisotropy Δ_psr. They find that a single nearby pulsar can easily generate Δ values of 10⁻²–10⁻¹, well above the DM universal bound. Thus, an observed anisotropy exceeding the DM limit would constitute strong evidence for a dominant astrophysical source.

The authors discuss observational prospects, noting that the Fermi‑LAT instrument already measures CRE anisotropies at the Δ≈10⁻² level and is expected to improve to Δ≈10⁻³ with several more years of data. If future measurements reveal Δ larger than the DM universal upper limit, the DM hypothesis would be strongly disfavored in favor of local discrete sources. Conversely, a non‑detection or a Δ below the bound would keep the DM scenario viable, though not uniquely identified.

In conclusion, the paper establishes that the dipole anisotropy of high‑energy CRE provides a robust, nearly model‑independent discriminator between dark‑matter and astrophysical origins. By explicitly accounting for the stochastic nature of sub‑halo realizations through Monte‑Carlo simulations, the authors derive a universal upper limit on DM‑induced anisotropy. This limit can be tested with current and upcoming γ‑ray observatories (Fermi‑LAT, CTA), offering a powerful indirect probe of the dark‑matter particle nature and its distribution in the Milky Way.