Critical Collapse of a Massive Scalar Field in Newtonian Gravity - Numerical Investigations of the Schodinger-Poisson System Inwood, Andrew
Critical collapse in General Relativity (GR) has been studied extensively, both numerically and analytically, for a wide variety of matter models including massive and massless scalar fields coupled to gravity, as well as a variety of gas models. Comparatively less work has been done towards the investigation of critical collapse in Newtonian gravity, although some progress has been made using an isothermal gas model. In this paper we numerically investigate critical collapse of a massive scalar field in spherical symmetry, coupled to gravity in the Newtonian limit. The evolution of such a field is described by the Schrodinger-Poisson (SP) system. A Crank-Nicolson finite-differencing approximation is coded using the Rapid Numerical Prototyping Language (RNPL), and the updates are coded in FORTRAN. The SP system is investigated using Gaussian initial data, centred at the origin. By varying the amplitude of the Gaussian data, no evidence for critical behaviour is found, and so the SP system is modified through the gravitational coupling in an attempt to induce critical behaviour. This modification is realized by making the exponent in the Poisson equation variable. Type I critical behaviour is found using the Gaussian initial data centred at the origin for an exponent epsilon = 3. By performing a bisection search the critical value of the amplitude, A, is found to 15 digits of precision. The critical solution is found to be periodic in time, characteristic of Type I phenomena, with a period of T = 0.02. By surveying the 1-dimensional parameter space of A, the Lyapunov exponent associated with the growing mode in Type I phenomena is found to 3 significant digits to be lambda = 163.
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