Dark matter halos modeled by polytropic spheres influenced by the relict cosmological constant and trapping polytropes forming supermassive black holes

Research Centre for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava, Bezručovo nám. 13, 746 01 Opava, Czech Republic

^⋆ Corresponding author: jan.novotny@physics.slu.cz

Received: 13 March 2025
Accepted: 11 July 2025

Abstract

Aims. We study dark matter halos modeled by general relativistic polytropic spheres in spacetimes with the repulsive cosmological constant representing vacuum energy density, governed by a polytropic index, n, and a relativistic (cosmological) parameter, σ (λ), determining the ratio of central pressure (vacuum energy density) and central energy density of the fluid.

Methods. To give mapping of the polytrope parameters for matching the extension and mass of large dark matter halos, we study the properties of the polytropic spheres and introduce an effective potential of the geodesic motion in their internal spacetime. Circular geodesics enable us to find the limits of the trapping polytropes with central regions containing trapped null geodesics; supermassive black holes can be formed due to the instability of the central region against gravitational perturbations. The stability of the polytropic spheres relative to radial perturbations is determined. We match the extension and mass of the polytropes to the ones of dark matter halos related to large galaxies or galaxy clusters, with an extension of 100 < ℓ/kpc < 5000 and gravitational mass of 10¹² < M/M_⊙ < 5 × 10¹⁵. The velocity radial profiles of circular geodesics in the polytrope spacetimes are numerically compared to the observed velocity profiles.

Results. The observed velocity profiles simulated by the phenomenological dark matter halo density profiles can also be well matched by the velocity profiles of the exact polytrope spacetimes. The matching is made possible by the nonrelativistic polytropes for each value of n, with a relativistic parameter of σ ≤ 10⁻⁴ and a very low central energy density. Surprisingly, the matching works for “spread” relativistic polytropes with n > 3.3 and σ ≥ 0.1 when the central density can be much larger. The trapping polytropes forming supermassive black holes must have n > 3.8 and σ > 0.667. We thus explain the mass and structure of large galaxies and galaxy clusters, their extension limited by the cosmic repulsion, and the existence of black holes with mass M > 10¹⁰ M_⊙ in very large galaxies; we suggest black holes with M ∼ 10¹² M_⊙ in large galaxy clusters.