Research Project: 

Dark matter (DM) is one of the mysteries of the standard model of  cosmology, i.e. Lambda-CDM model, which assumes the existence of the cold dark matter (CDM). Despite many theoretical efforts to understand the nature of DM, its existence has not yet been  confirmed. On the other hand, there are only few works that address this problem mathematically rigorously.

The Lambda-CDM model  faces a number of challenges. This project has been motivated by one of these challenges, i.e., the so-called core-cusp problem. Early  N-body cosmological simulations showed that there exists a cusp in the centers of CDM halos, whereas observations do not support this phenomenon. To explain this discrepancy many other hypothetical forms of DM has been proposed. The so-called fuzzy dark matter (FDM) model is one of them.  The FDM is a special case of a more general DM model, i.e., the scalar field dark matter model which is described by the Einstein--Klein--Gordon (EKG) system. 
The dynamics of the FDM is governed by the Schr\"odinger--Poisson (SP) system that is the non-relativistic limit of the EKG system.

It is well known that the CDM can be described by the collisionless Boltzmann equation, i.e., the Vlasov equation because it is assumed that the CDM particles interact only via gravity. Therefore, it seems natural to couple the Vlasov matter to the EKG system resulting in the EVKG system. However, there is no mathematical result on the EVKG system in the static spherically symmetric case.

The goal of this project is to show the existence of a solution to the EVKG system in a static spherically symmetric spacetime. Then we will show some properties of these solutions numerically. This project would provide the first mathematical result for the EVKG system in a static spherically symmetric spacetime. 
 

Stays: October 1 - November 30, 2021 and January 1 - March 1, 2022