Constraining anisotropic diffusion between Geminga and Earth with the cosmic-ray electron and positron spectrum

The gamma-ray halo surrounding Geminga suggests a notable reduction in cosmic-ray diffusion. One potential explanation for this phenomenon is the projection effect of slow diffusion perpendicular to the average magnetic field (represented by the diffusion coefficient $D_\perp$) within an anisotropic diffusion framework. In this context, the diffusion coefficient parallel to the mean field ($D_\parallel$) may remain substantial, allowing electrons and positrons ($e^\pm$) generated by Geminga to effectively propagate towards Earth along magnetic field lines, potentially leading to an observable $e^\pm$ flux. This study initially establishes the fundamental parameters of the anisotropic model based on the morphology and spectral observations of the Geminga halo, and subsequently forecasts the $e^\pm$ flux generated by Geminga at Earth's location. Our findings indicate that the $e^-+e^+$ spectrum obtained by DAMPE can provide critical constraints on the anisotropic diffusion model: to ensure that the projected spectrum does not surpass the observational data, the Alfv\'en Mach number of the turbulent magnetic field ($M_A$) should not fall below 0.75, corresponding to $D_\parallel/D_\perp\lesssim3$ given $D_\perp=D_\parallel M_A^4$. This suggests that a substantial reduction in $D_\parallel$ relative to the Galactic average may still be necessary. Additionally, our analysis reveals that within the anisotropic diffusion framework, Geminga could generate a distinct peak around 1 TeV in the $e^-+e^+$ spectrum, potentially accounting for the anomalous 1.4 TeV excess tentatively detected by DAMPE.