Astrophysics of Bertrand space-times

Abstract

We construct a model for galactic dark matter that arises as a solution of Einstein gravity, and is a Bertrand space-time matched with an external Schwarzschild metric. This model can explain galactic rotation curves. Further, we study gravitational lensing in these space-times, and in particular we consider Einstein rings, using the strong lensing formalism of Virbhadra and Ellis. Our results are in good agreement with observational data, and indicate that under certain conditions, gravitational lensing effects from galactic dark matter may be similar to that from Schwarzschild

Galactic space-times in modified theories of gravity

“…Here, we have critically examined the delicate nature of the definition of circular velocity in a GR context, and showed that conventional definitions that are often used in the literature may not be very useful from a practical point of view. We argued that a phenomenological definition given in [8], [9] might be more effective in comparing with experimental data, in the context of Bertrand space-times, which we propose as a viable space-time metric seeded by galactic dark matter. We also commented on the spectroscopic interpretation of the formula for the rotation curve for BSTs.…”

“…However it should be clear from the preceding discussion that on doing this, fitting with observational data for galactic rotation curves becomes difficult (for example, the maximum value of the circular velocity in this case becomes c/(2 √ 2), an unrealistically large number). The resolution of this problem is to match an internal BST with an external Schwarzschild solution, as discussed in section 2 of [8]. However, this often results in the presence of a thin shell of matter near the matching radius, and in such situations, analytical handle on the problem might pose problems.…”

“…One such situation was envisaged in [8], [9] where galactic space-times were modeled by a Bertrand space-time (BST) metric of the form…”

“…If BSTs are thought of as viable spacetime models of dark matter, then we naturally move out of the paradigm of weak gravity. Although this new approach has not been very popular in existing literature, such a scenario cannot be ruled out, as experimental data can be well fitted within this framework [8], [9]. It is therefore important and instructive to extend the analysis on such theories, in particular in the framework of modified theories of gravity, and this is one of the tasks that we undertake in this work.…”

“…In particular, we focus on a class of space-time models proposed as viable models of galactic dark matter in [8], [9], which were originally discovered by Perlick [10] and called Bertrand space-times (BSTs). In these works, it was shown that in the framework of GR, BSTs can provide excellent fits to experimental data on galactic rotation curves.…”