Tesi di LAUREA SPECIALISTICA Titolo Numerical Simulations of Critical Phenomena in Neutron Star Collapse Data 2009-12-23 Autore/i Radice, David Relatore Magli, G. Relatore Rezzolla, L. Full text non disponibile Abstract In 1993, Choptuik [36] considered a one-parameter family of solutions, S[p], of the Einstein s equations for a massless scalar eld, such that for every p > p*, S[p] contains a black hole and for every p < p*, S[p] is a solution not containing singularities. Then he asked himself What happens when p approaches p* and, more importantly, What is the nature of the threshold solution, S[p*]? Choptuik studied this problem numerically and found that, as p approaches p*, the system undergoes a phase transition, where the solutions approach a universal solution, S[p*], not depending on the particular choice of the family of solutions. After the seminal work of Choptuik similar transitions were discovered for a wide range of systems, including massive scalar elds and ultra-relativistic uids [54]. All these phenomena have the common property that, as p approaches p*, S[p] approaches a universal solution S[p*] and that all the physical quantities of S[p] depend only on p - p*. These transitions were later classied as type I critical phenomena, with static or periodic critical solutions, or type II critical phenomena, with selfsimilar critical solutions, in analogy with critical phase transitions in statistical mechanics [55]. In this work we are concerned with the study of critical phenomena in the collapse of spheres of sti gas modeling neutron stars. In particular we will focus on type I phenomena as they have not been as well studied as type II, both from the analytical and the numerical point of view. From an analytical point of view we extended a work by Hara et al. [58] on type II critical phenomena to the type I case, in order to study its main properties. From a numerical point of view we developed a new 1D discontinuous Galerkin, spectral element method, code and we used it together with the Whisky2D code [74] to study the phase transition of a one-parameter family of solutions with relativistic compact star initial data. We found a new type I critical transition which was not observed before. In Chapter 1 we recall some basic thermodynamics, mainly in order to x the notation and then we treat the mathematical properties of spherically symmetric static congurations of gases and their linear oscillations. In Chapter 2 we cast the Einstein s equations as a dynamical system, making use of the so called 3+1 decomposition and the Hamiltonian formalism. This reformulation is the basic starting point for both our analytic studies and our numerical ones. In Chapter 3 we explain in detail the theory behind critical phenomena and review what is known about critical phenomena in neutron star collapse. Then we extend the theoretical work by Hara et al. [58] to study the fundamental properties of type I critical phenomena. In Chapter 4 we review some basic facts about discontinuous Galerkin methods and stabilization techniques for spectral methods in computational fluid dynamics. Then we explain in detail how we discretized the Einstein s equations in spherical symmetry with our code. In Chapter 5 we show some benchmark performed with our new code and we also discuss some of its weaknesses. In Chapter 6 we report in detail the results obtained in the numerical study of type I critical phenomena in neutron star collapse. Finally, in Chapter 7, we summarize our ndings and we discuss about possible future work directions. This work was carried out in collaboration with the Numerical Relativity Group of the Max Planck Institute for Gravitational Physics, Albert Einstein Institute, in Potsdam (Germany). All the computations were carried out on the Damiana supercomputer at the Albert Einstein Institute in Potsdam.