Reconstruction of the dark matter density profile from cosmic positron anomaly data

In this work we continue our investigations of the possibility of explanation of the positron anomaly (PA) in cosmic rays with the help of annihilating or decaying dark matter (DM) component by varying its space distribution. In the contrast of our previous studies, where we first assumed some specific spatial distribution of DM component and looked at how it agrees with data, here we solve, in some sense, the inverse problem: we search for distribution, in a mathematical way, which satisfies observational data. A unique algorithm has been implemented which, using linear algebra and adaptive grid methods, adjusts distribution to the data. It allows telling in principle whether or not is possible to solve PA problem by variation of spatial distribution of DM sources. A positive result has been formally obtained. A class of solutions can be identified. Though the distributions obtained at the chosen injection spectra may seem slightly realistic, nonetheless it demonstrates a quite powerful possibility in explaining PA that could be realized in more realistic models.

💡 Research Summary

The paper tackles the long‑standing positron anomaly (PA) observed in cosmic‑ray measurements by asking a reverse question: rather than assuming a particular spatial distribution of dark‑matter (DM) sources and testing its compatibility with data, the authors ask what DM density profile would be required to reproduce the observed positron and gamma‑ray spectra. They develop a novel algorithm that treats the problem as a linear algebraic reconstruction using non‑negative least squares (NNLS) combined with adaptive grid refinement.

The Galactic volume is partitioned into a set of regions (D_i). Each region is assigned a unit density (\rho_i = 1) (in arbitrary units) and the corresponding positron and gamma‑ray spectra are simulated. Positron propagation is performed with the GLAPROP code, while gamma‑ray fluxes are calculated analytically. The initial particle spectra for the DM annihilation (or decay) channels are generated with PYTHIA. For each region the simulated spectra are normalized by the experimental uncertainties, forming a matrix (A) whose columns are the normalized spectra (\phi_i/\sigma_i). The experimental data vector (\hat b) consists of the measured AMS‑02 positron fluxes and Fermi‑LAT isotropic gamma‑ray background (IGRB) points, also divided by their errors. The reconstruction problem is then to find a non‑negative coefficient vector (\hat k) that minimizes (|A\hat k - \hat b|_2). Because the annihilation rate scales as the square of the density, the physical DM density in each region is taken as (\rho = \sum_i \sqrt{k_i},\rho_i).

A simple NNLS solution yields a unique set of coefficients that reproduces the data with the smallest possible norm, but it suffers from two issues: (1) the solution can be highly asymmetric due to numerical noise, and (2) the number of free regions can exceed the number of data points, making the statistical interpretation ambiguous. To address these, the authors introduce two extensions. First, they explore the entire family of solutions whose chi‑square lies within a small tolerance (\Delta\chi^2) of the minimum (typically 2–5 % of (\chi^2_{\rm min})). This generates a set of admissible density profiles rather than a single one. Second, they impose an additional linear constraint that minimizes the maximum squared density, (t = \rho_{\max}^2), effectively suppressing spurious peaks and selecting a more physically plausible profile. The constrained optimization is performed with convex‑programming tools such as CVXPY.

Two concrete geometries are examined. In the first “flat” configuration, a thin slab of the Galactic plane (thickness 0.6 kpc, extending (\pm20) kpc in (x) and (y)) is divided into an (8\times8) grid (64 cells). The DM particle mass is set to 1 TeV and annihilation is considered into either (e^+e^-) or (\mu^+\mu^-) final states. In the second “centrally symmetric” configuration, a cylindrical volume around the Galactic centre (radius 0–18 kpc, height (\pm3) kpc) is partitioned into 16 radial and 8 vertical bins (128 cells), with an adaptive refinement scheme to keep computational cost manageable. For each geometry the authors compute the full set of positron and gamma‑ray spectra, determine the NNLS coefficients, and then derive both an upper bound on the source density (by taking the largest coefficient in each row of the coefficient matrix) and a best‑fit density profile under the maximum‑density‑minimization constraint.

The resulting profiles share a striking feature: the reconstructed DM density is concentrated toward the outer regions of the Galaxy, far from the Solar position. This peripheral concentration allows the model to reproduce the rising positron fraction measured by AMS‑02 while staying below the IGRB limits measured by Fermi‑LAT. The authors note that such a distribution differs from previously studied disk‑like, spiral‑arm, or baryon‑like configurations, which generally failed to satisfy both data sets simultaneously.

The paper also discusses methodological limitations. The linear decomposition can involve more basis functions than independent data points, leading to an under‑determined system. Numerical precision can introduce asymmetries, and the physical plausibility of the required “small DM substructures” remains to be tested against galactic dynamics and structure‑formation simulations. The authors suggest future work incorporating Bayesian model selection, more sophisticated adaptive meshing, and additional astrophysical constraints (e.g., rotation curves, stellar kinematics) to refine the reconstruction.

In summary, this work demonstrates that a DM‑based explanation of the cosmic‑ray positron anomaly can be achieved by treating the spatial distribution as an inverse problem. By employing linear‑algebraic reconstruction with non‑negative constraints and adaptive grids, the authors show that there exists a class of DM density profiles—particularly those with enhanced peripheral sources—that can simultaneously fit AMS‑02 positron data and respect Fermi‑LAT gamma‑ray background limits. The approach opens a new avenue for exploring dark‑matter phenomenology in cosmic‑ray physics, providing a flexible framework that can be extended with more realistic particle physics models and tighter astrophysical constraints.