Critical Phenomena Associated with Boson Stars

Here we present a synopsis of related work [1, 2] describing a study of black hole threshold phenomena for a self-gravitating massive complex scalar field in spherical symmetry. Studies of models of gravitational collapse have revealed structure which can arise near the threshold of black hole formation. The solutions in this regime are known as " critical solutions," and their properties as " critical phenomena ". These solutions can arise generically, even in simple models such a massless scalar field in spherical symmetry [3]. Critical solutions can be constructed dynamically via numerical simulations, in which one considers continuous one-parameter families of initial data with the following " interpolating " property: for sufficiently large values of the family parameter, p, the evolved data describes a spacetime containing a black hole, whereas for sufficiently small values of p, the matter-energy in the spacetime disperses to large radii at late times, and no black hole forms. Within this range of parameters, there will exist a critical parameter value, p = p ⋆ , which demarks the onset, or threshold, of black hole formation. Over the past decade, numerical and closed-form studies of collapse in various matter models have enlarged the picture of critical phenomena [4, 5, 6, 7], so that we now have a more complete understanding of the relevant dynamics. (Interested readers should consult the reviews by Gundlach [8, 9] for a more comprehensive discussion of critical phenomena .) Black-hole-threshold solutions are attractors in the sense that they are almost completely independent of the specifics of the particular family used as a generator. Up to the current time, the only initial data dependence which has been observed in critical collapse occurs in models for which there is more than one distinct black-hole-threshold solution. Critical solutions are by construction unstable, having precisely one unstable mode [10, 11]. Thus letting p → p ⋆ amounts to minimizing or " tuning away " the initial amplitude of the unstable mode present in the system. These solutions also possess additional symmetry which, to date, has either been a time-translation symmetry, in which the critical solution is static or periodic, or a scale-translation symmetry (homethetic-ity), in which the critical solution is either continuously or discretely self-similar (CSS or DSS). These symmetries are indicative of the two principal types of critical behavior that have been seen in black hole threshold studies (with some models exhibiting both types of …