Mathematical reconstruction of dark matter density profiles in galaxies

Reconstructing the density profiles of dark matter in galaxies is essential for understanding the dynamics of galaxies, but observations have uncertainties and insufficient information on baryonic content.This study proposes a mathematical reconstruction framework to estimate the continuous dark matter density distributions that can be obtained from galaxy rotation curve and stellar kinematic observations by combining the Poisson equation, Jeans equation, Bayesian inverse modeling, and Tikhonov regularization.Nonlinear least-squares minimization, with adaptive regularization, is applied to solve the inverse optimization problem, and produce physically consistent, stable solutions.The performance is tested by density reconstruction error, the accuracy of the rotation curve, root mean square error (RMSE), mass conservation error and convergence stability.The proposed framework successfully provides a 96.8% accuracy of the rotation-curve fitting, decreases RMSE by 18.7%, decreases density reconstruction error by 21.4% and the mass conservation is maintained above 99.1%.The proposed inverse