Galactic Dark Matter and Bertrand Space-times

TL;DR

This paper explores how certain dark matter distributions in galaxies can give rise to Bertrand space-times, providing a new perspective on galaxy rotation curves and dark matter profiles within Einstein's framework.

Contribution

It introduces a novel connection between dark matter distributions in galaxies and Bertrand space-times, including analytic solutions and a new Einstein's equation solution with BST and Schwarzschild metrics.

Findings

BSTs can model stable, closed orbits in galaxies.

The model predicts NFW dark matter profile for flat rotation curves.

A new Einstein's equation solution with BST and Schwarzschild metrics is presented.

Abstract

Bertrand space-times (BSTs) are static, spherically symmetric solutions of Einstein's equations, that admit stable, closed orbits. Starting from the fact that to a good approximation, stars in the disc or halo regions of typical galaxies move in such orbits, we propose that, under certain physical assumptions, the dark matter distribution of some low surface brightness (LSB) galaxies can seed a particular class of BSTs. In the Newtonian limit, it is shown that for flat rotation curves, our proposal leads to an analytic prediction of the NFW dark matter profile. We further show that the dark matter distribution that seeds the BST, is described by a two-fluid anisotropic model, and present its analytic solution. A new solution of the Einstein's equations, with an internal BST and an external Schwarzschild metric, is also constructed.