A numerical study of boson stars

In this thesis we present a numerical study of general relativistic boson stars in both spherical symmetry and axisymmetry. We consider both time-independent problems, involving the solution of equilibrium equations for rotating boson stars, and time-dependent problems, focusing on black hole critical behaviour associated with boson stars.

Boson stars are localized solutions of the equations governing a massive complex scalar field coupled to the gravitational field. They can be simulated using more straightforward numerical techniques than are required for fluid stars. In particular, the evolution of smooth initial data for a scalar field tends to stay smooth, in sharp contrast to hydrodynamical evolution, which tends to develop discontinuities, even from smooth initial conditions. At the same time, relativistic boson stars share many of the same features with respect to the strong-field gravitational interaction as their fermionic counterparts. A detailed study of their dynamics can thus potentially lead to a better understanding of the dynamics of compact fermionic stars (such as neutron stars), while the relative ease with which they can be treated numerically makes them ideal for use in theoretical studies of strong gravity.

In this last vein, the study of the critical phenomena that arise at the threshold of black hole formation has been a subject of intense interest among relativists and applied mathematicians over the past decade. Type I critical phenomena, in which the black hole mass jumps discontinuously at threshold, were previously observed in the dynamics of spherically symmetric boson stars by Hawley and Choptuik [1, 2]. We extend this work and show that, contrary to previous claims, the subcritical end-state is well described by a stable boson star executing a large amplitude oscillation with a frequency in good agreement with that predicted for the fundamental normal mode of the end-state star from linear perturbation theory.

We then extend our studies of critical phenomena to the axisymmetric case, studying two distinct classes of parametrized families of initial data whose evolution generates families of spacetimes that “interpolate” between those than contain a black hole and those that do not. In both cases we find strong evidence for a Type I transition at threshold, and are able to demonstrate scaling of the lifetime for near-critical configurations of the type expected for such a transition. This is the first time that Type I critical solutions have been simulated in axisymmetry (all previous general relativistic calculations of this sort imposed spherical symmetry).

In addition, we develop an efficient algorithm for constructing equilibrium configurations of rotating boson stars, which are characterized by discrete values of an angular momentum parameter, k (an azimuthal quantum number). We construct families of solutions for k = 1 and k = 2, and demonstrate the existence of a maximum mass in each case.