Supermassive Black Hole Growth and Dark Matter Distribution in Massive Elliptical Galaxies: A Theoretical Framework with Applications to M87 and NGC 4889

We present a comprehensive theoretical framework for understanding the co-evolution of supermassive black holes (SMBHs) and dark matter (DM) distributions in massive elliptical galaxies. Building upon recent Event Horizon Telescope observations of M87 and high-precision kinematic measurements, we develop a self-consistent model that accounts for DM accretion onto SMBHs and its effect on the central density profiles. Our mathematical formalism, based on the Vlasov-Poisson equations with relativistic corrections, predicts a characteristic depletion of dark matter within the SMBH sphere of influence, quantified by Δ𝑀 DM ∼ 2 − 3 × 10 9 𝑀 ⊙ for M87-class galaxies. We derive analytical expressions for the modified Burkert profile in the presence of a growing SMBH and establish scaling relations between core radius 𝑟 0 , central density 𝜌 0 , and SMBH mass 𝑀 BH. The model predictions are consistent with observations of 47 massive elliptical galaxies, suggesting that DM accretion may contribute ∼ 15 − 30% of the final SMBH mass in the most massive systems. Our results provide testable predictions for future observations with the next-generation Event Horizon Telescope and the James Webb Space Telescope.