A DENSITY FUNCTIONAL EQUATION OF STATE FOR USE IN ASTROPHYSICAL PHENOMENA A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by J. Pocahontas Olson Grant Mathews, Director Graduate Program in Physics Notre Dame, Indiana December 2015 c Copyright by J. Pocahontas Olson 2015 All Rights Reserved A DENSITY FUNCTIONAL EQUATION OF STATE FOR USE IN ASTROPHYSICAL PHENOMENA Abstract by J. Pocahontas Olson In this thesis, I present a new equation of state for use in simulating supernovae, black holes and neutron star mergers. It is the first such equation of state for as- trophysical applications to use a density functional theory description for hadronic matter. The inclusion of thermal effects of matter enable nuclear Skyrme models, which have been highly tested and constrained at laboratory energy scales, to expand their domain to predictions of astronomical phenomena. Broadening the scope of these models can further confine parameter sets, using vastly different energy scales. The new equation of state, titled the Notre Dame-Livermore Equation of State (NDL EoS), allows for the creation of a pion condensate at high density and pair production of all known baryonic and mesonic states at high temperature. The description of matter also allows for the possibility of the formation of a net proton excess (Ye > 0:5). In addition to the density functional theory formulation for hadronic matter, the NDL EoS contains low and high density completions to better describe matter in these specific energy regimes. The low density description expands upon a Bowers and Wilson formulation, adding a transition through nuclear pasta phases, which are of particular importance in neutron star structure. These low density definitions have further been updated to include an improved treatment of the nuclear statistical J. Pocahontas Olson equilibrium and the transition to heavy nuclei as the density approaches nuclear matter density. At high densities, matter is allowed to transition to a quark-gluon plasma (QGP) either as a first-order Gibbs transition, or a smooth crossover. Notre Dame-Livermore I identify predictions of the NDL EoS, contrasting them to existing equations of state and various Skyrme models of the NDL EoS. The observation of a heavy (two solar masses) neutron star restricts many descriptions of matter, and rules out several Skyrme parameter sets that had heretofore been entirely within the bounds set by nuclear experiments. Finally, I present the results of core-collapse simulations, both with the origi- nal Bowers and Wilson equation of state and the NDL EoS, using the spherically symmetric Mayle and Wilson supernova code. CONTENTS FIGURES . iv TABLES . vi CHAPTER 1: INTRODUCTION . .1 CHAPTER 2: BACKGROUND . .3 2.1 Supernovae . .3 2.1.1 Types of Supernova Explosions . .6 2.1.2 Explosion Mechanisms . .7 2.1.3 Black Hole Formation . 11 2.2 Quark-Gluon Plasma . 12 2.2.1 Phase Diagram of Dense, Hot Matter . 12 2.2.2 First-Order Phase Transition to a Quark-Gluon Plasma . 12 2.2.3 The MIT Bag Model . 13 2.2.4 Lattice QCD . 14 2.2.5 Collider Experiments . 15 2.3 Equation of State . 17 2.3.1 The Lattimer & Swesty Liquid-Drop Model . 18 2.3.2 Relativistic Mean Field Theory . 20 2.3.3 Bowers and Wilson Model . 21 2.3.4 The Skyrme Model . 24 CHAPTER 3: THE NOTRE DAME-LIVERMORE EQUATION OF STATE 26 3.1 Constructing the EoS . 28 3.1.1 Photons, Electrons and Positrons . 29 3.1.2 Baryons Below Saturation Density . 29 3.1.2.1 Baryons Below n0, Not in NSE . 31 3.1.2.2 Baryons Below n0, in NSE . 32 3.1.2.3 Treatment of Nuclear Pasta Phases . 36 3.1.2.4 Chemical Potentials . 40 3.1.3 Baryons Above Saturation Density, Purely Hadronic Matter . 41 3.1.3.1 Skyrme Density Functional Theory . 41 3.1.3.2 Thermal Contribution to Hadronic NDL EoS . 47 3.1.3.3 Pion Condensate . 49 ii 3.1.4 Baryons Above Saturation Density, Hadronic and Quark Matter 52 3.1.4.1 Pure Quark Matter . 53 3.1.4.2 Mixed Hadronic and Quarkonic Matter . 56 3.2 Constraints on the EoS . 58 3.2.1 Nuclear Experimental Constraints . 58 3.2.2 Observation of 2 M Neutron Stars . 60 3.2.3 Successful Parameter Sets . 61 CHAPTER 4: PREDICTIONS OF THE NDL EQUATION OF STATE . 67 4.1 Comparison to Existing Equations of State . 67 4.2 Pion Effects on the NDL EoS . 68 4.3 Mixed Phase of Hadrons and Quark-Gluon Plasma . 72 4.4 Neutron Stars . 75 4.4.1 Tolman-Oppenheimer-Volkoff Equation . 77 4.4.2 Maximum Mass of Neutron Star . 79 CHAPTER 5: SUPERNOVA SIMULATION . 83 5.1 Summary of the Core Collapse Process . 83 5.2 Numerical Routines for Simulation of Core-Collapse Supernovae . 85 5.2.1 The Metric . 85 5.2.2 Energy Momentum Tensor . 86 5.2.3 Matter Evolution . 87 5.2.4 Neutrinos . 88 5.2.5 Equation of State . 89 5.2.6 Convection . 89 5.3 Simulation with Former Bowers and Wilson EoS . 90 5.4 Simulation with Hadronic-only NDL EoS . 91 CHAPTER 6: CONCLUSION . 95 BIBLIOGRAPHY . 97 iii FIGURES 4.1 The energy per particle as a function of density comparing the Lat- timer & Swesty EoS with compressibilities K0 = 180 MeV and K0 = 220 MeV with the fiducial NDL EoS, at a fixed electron fraction and temperature of Ye = 0:3 and T = 10 MeV. 68 4.2 The pressure as a function of density comparing two EoSs from Lat- timer & Swesty [63], the Shen EoS [102] and the fiducial NDL EoS of the present work, at a fixed electron fraction and temperature of Ye = 0:3 and T = 10 MeV. 69 4.3 The proton fraction above nuclear saturation showing the effects of pions in the hot dense supernova environment. For small electron fractions more pions are created due to the dependence of the chemical potential on the isospin asymmetry parameter I............. 70 4.4 Pion charge fraction versus density at a temperature T = 10 MeV. At a density of about n ≈ 0:65 fm−3, the pion charge fraction exceeds the electron fraction of the medium. 71 4.5 Pressure versus baryon number density showing a 10% reduction in the pressure at high densities due to the presence of pions. 72 4.6 Charge fractions of the mixed quark and hadronic phase (dashed and solid lines). The electron fraction is shown in a dotted line. Due to the redistribution of charge, the hadronic phase becomes isospin symmetric even exceeding Yp > 0.5. This has the effect of lowering the symmetry energy and thus reducing the pressure in the hadronic phase. 74 4.7 Pressure as a function of baryon number density through the mixed phase transition. The EoS softens significantly upon entering the mixed phase due to the larger number of available degrees of freedom. 75 4.8 The adiabatic index Γ as a function of baryon number density showing the softening of the EoS as it enters the mixed phase regime. The EoS promptly stiffens as it exits the mixed phase into the pure quark matter phase due to losing the extra degrees of freedom supplied by the nucleons. 76 iv 4.9 Density-temperature phase diagram showing the density range of the mixed phase coexistence region for two values of Ye (dashed line for Ye = 0:1 and dash-dotted line for Ye = 0:3). The onset of the mixed phase is indicated by the set of curves on the left, while the curves on the right show the completion of the mixed phase. For low tempera- tures it is seen that the onset density is highly Ye dependent. 77 4.10 Neutron star mass-radius relation for various values of the bag constant B1=4. We find that a first order phase transition is consistent with the maximum mass neutron star measurement for our adopted value of B1=4 = 220 MeV. 78 4.11 Mass-radius relation for the Shen (longer dashed line), Lattimer & Swesty 180 & 220 (dash-dotted line and shorter dashed line), Bowers & Wil- son EoS (dash-dash dotted line), the NDL EoS with (dash-dot-dotted line) and without (solid line) a mixed phase transition to quark gluon plasma as well as a simple crossover transition (dotted line) to a QGP. Note that all of these curves satisfy the 2:01 ± 0:04 M astrophysi- cal constraint, with the exception of the Bowers & Wilson and LS180 equations of state. 81 4.12 Mass-radius relation for the sixteen Skyrme parameter sets that are not already ruled out by nuclear constraints. The 2:01 ± 0:04 M as- trophysical constraint [2] is drawn in for reference. 82 5.1 Mass layer trajectories in a simulation of core-collapse supernova from a 20 M progenitor model using the Bowers and Wilson equation of state. Time is measured post-bounce. 91 5.2 Mass layer velocities in a simulation of core-collapse supernova from a 20 M progenitor model using the Bowers and Wilson equation of state. Time is measured post-bounce, and velocity outwards as pos- itive. The collapse is evident at tpb = 0 s, with mass layers rapidly falling towards the core. It is not until tpb ∼ 0:25 s that the shockwave gains enough energy to propel layers outward in an explosion. 92 5.3 Mass layer trajectories in a simulation of core-collapse supernova from a 20 M progenitor model using the NDL equation of state with the MSL0 Skyrme parameter set.