A Density Functional Equation of State for Use in Astrophysical Phenomena

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

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 astrophysical 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 original 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 temperatures 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 astrophysical 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 astrophysical 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 positive. 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. Time is measured post-bounce. . . . . 93 5.4 Mass layer velocities in a simulation of core-collapse supernova from a 20 M progenitor model using the the NDL equation of state with the MSL0 Skyrme parameter set. Time is measured post-bounce, and velocity outwards as positive. 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...... 94

v TABLES

3.1 ADOPTED CONSTRAINTS ...... 62 3.2 SKYRME PARAMETER SETS ...... 64 3.3 SKYRME PARAMETER SETS ...... 65 3.4 SKYRME PARAMETER SETS ...... 66

vi CHAPTER 1

INTRODUCTION

This thesis work focuses on the development of an Equation of State (EoS) to describe the interactions of matter across a broad range of densities and temperatures. It allows for the possibility of a phase of matter consisting of (asymptotically-) freely interacting quarks and gluons, and incorporates several other sources suggested to be important in modeling successful supernova explosions. Chapter 2 will provide relevant background and describe current research. It begins with a review of supernovae and the diﬃculty of realistically producing an explosion in numerical simulations. This section will brieﬂy discuss neutron stars, as it is possible to use the supernova remnants to infer properties of the explosion mechanism. It also describes black hole formation, a natural endpoint of very massive (& 20 M ) progenitor stars. These dying giants have such a deep gravitational potential that the outer layers cannot escape and inevitably fall upon the proto- neutron star leading to the formation of a black hole. Subsequently, this chapter introduces the quark-gluon plasma, and summarizes current experiments and simulations that put constraints on any model describing it. It concludes by reviewing and comparing the current equations of state used in supernova simulations (Lattimer & Swesty and Shen et. al. EoSs), the equation of state previously used in Notre Dame’s supernova simulation (Bowers and Wilson EoS), and the new NDL equation of state developed as part of this dissertation. Chapter 3 presents the description of matter necessary to building the equation of state, incorporating the relevant physics at that energy regime. The chapter concludes

1 by listing the various constraints from nuclear and astrophysical sources that an equation of state must satisfy. Chapter 4 shows predictions of the NDL EoS, including the eﬀects pions have on the distribution of charge and pressure. A phase diagram of matter, separating out a ﬁrst-order phase transition to quark-gluon plasma is discussed. Finally, predictions for the maximum mass of a neutron star are provided for the NDL parameter set and the Skyrme models that meet all the nuclear experiment constraints listed in section 3.2.1. Chapter 5 shows the results of Notre Dame’s supernova simulation using the NDL equation of state, or one of several related Skyrme models. We use the original Bowers and Wilson EoS as a baseline for comparison. Finally, chapter 6 summarizes the conclusions of this thesis work.

2 CHAPTER 2

BACKGROUND

2.1 Supernovae

Fifty light-years away from the Solar System there was once a binary star system. One star was in its normal yellow-white phase, but the other had bloated up until it turned into a red giant, swallowing the planets around it. The nuclear fuel for the red giant ran out.... With its fusion-bomb center turned off, the energy the star needed to hold itself up against its self-gravitation was no longer available, and the star collapsed. At the center, the in-falling matter became denser under the terrific gravitational pressure until it turned almost completely into neutrons. The neutrons pressed closer and closer until they were packed radius to radius. Under these cramped conditions, the strong nuclear repulsion forces were finally able to resist the gravitational pressure. The inward rush of matter was quickly reversed, and the outward motion turned into an incandescent shock wave that traveled upward through the outer shell of the red giant. At the surface, the shock wave blew off the outer layers of the star in a supernova explosion that released more energy in one hour than the star had released in the previous million years. — Robert L. Forward, Dragon’s Egg

In 1930, Subrahmanyan Chandrasekhar calculated the maximum mass of a white dwarf, kept from gravitational collapse by the pressure of degenerate electrons [22]. Four years later (just a year after the discovery of the neutron [19]), Baade and Zwicky suggested that stars heavier than this Chandrasekhar limit (∼1.4 solar masses) overcome the electron-degeneracy pressure, and collapse to form a neutron star [4]. Baade and Zwicky coined the term supernovae for the observed special class of “much more luminous novae” [88] possibly associated with the formation of a neutron star.

3 1 In the current understanding , massive stars (∼8 M . M . 100 M ) burn their nuclear fuel quickly compared to cosmic timescales: ∼100 million − 1 million years2, respectively. Once nuclear burning stops in the core, the star will have evolved to a red or blue supergiant, with a radius of ∼1012 cm. The core is comprised of iron- group nuclei (or O/Ne nuclei at the low end of massive stars). It is approximately (3 − 10) × 103 km in radius, and supported against gravity’s pull primarily by the degeneracy pressure of relativistic electrons. Surrounding it are onion-like layers of material that are usually of lower mean atomic weight. Nuclear burning in the outer shells adds mass to the core, until it eventually exceeds the Chandrasekhar mass and collapses. This gravitational collapse is accelerated by the loss of pressure due to the capture of electrons on nuclei and free protons, as well as the photodissociation of heavy nuclei into alpha particles and nucleons. In just a few hundred milliseconds, the inner core will have obtained densities from 109 − 1010 g cm−3 (compared to 7.9 g cm−3, the density of iron on earth) to nuclear density (∼2.6 × 1014 g cm−3). At this point, the nuclear equation of state stiﬀens, due to the short-range repulsive nature of the strong nuclear force. This stabilizes the inner core. The inner core, however, overshoots its new equilibrium, then rebounds into the still collapsing outer core. This core bounce results in the formation of a hydrodynamic shock at the interface of the inner and outer core. The shock propagates outward in radius and mass through the in-falling outer core material, leaving behind the unshocked inner core (of ∼0.5 − 0.7 M ) which forms the interior core of the hot proto-neutron star (PNS).

1 There are two basic scenarios of stellar death: thermonuclear runaway at degenerate conditions, and the implosion of stellar cores. The former is associated with the destruction of white dwarf stars in type Ia supernovae. The latter is associated with core-collapse supernovae of types II and Ib/c and hypernovae. My dissertation will focus on this latter type of core-collapse supernovae.

2 9 Stellar evolution time scales approximately as tevol ≈ 7.3×10 years (M?/M )/(L?/L ), where 3.5 M? is the stellar mass and the stellar luminosity is L? ≈ (M?/M ) L [47]. The solar mass is 33 33 −1 M = 1.989 × 10 g and solar luminosity is L = 3.85 × 10 erg s . −2.5 This yields a stellar lifetime τ? ≈ (M/M ) τ .

4 The proto-neutron star rapidly accretes mass at a rate of about a solar mass per second and radiates away its gravitational energy (∼1051 erg) in neutrinos as it evolves to a neutron star on a timescale of tens of seconds. If the outer stellar envelope is not blown away, as in a failed shock scenario or in very massive progenitor stars, accretion continues and will push the proto-neutron star over its maximum mass limit. This results in another gravitational collapse, leading to the formation of a black hole. Finding observational evidence for the supernova mechanism is diﬃcult because the pre-explosion dynamics occur deep inside the stellar core, and are veiled from view. Telescopes can only detect secondary observables, such as progenitor type and mass, explosion morphology and energy, ejecta composition, neutron star properties and birth kick velocity. The primary indicators of ‘live’ information from deep inside the supernova engine are from neutrinos and gravitational waves, which interact very little with matter along the way from their emission site to terrestrial observatories. In early supernova models, the stellar mantle and envelope were believed to be ejected via the prompt initial core-bounce shock [46]. However, as the theory and simulations advanced to include more complete and accurate treatments of the Equa- tion of State (EoS), along with neutrino physics and transport, it was found that the prompt shock fails [47]. Essentially, the shock is drained of energy from the photodissociation of in-falling heavy nuclei into nucleons (at a cost of ∼8.8 MeV per baryon, for iron-group nuclei) and from neutrinos streaming away from the optically thin post-shock region. The shock slows down, stalls and turns into an accretion shock at a radius of 100 − 200 km, about 10 − 20 ms after bounce [90]. For a successful core-collapse supernova explosion to occur, there must be some method to revitalize the shock. Discerning the mechanism of shock revival has been the crux of core-collapse supernova theory for the past thirty-some years.

5 2.1.1 Types of Supernova Explosions

The classification of types of supernova explosions is mostly historical in nature, and is based upon observed optical spectra. This data, paired together with other observations and current theory, can reveal information concerning the explosion, such as the ejecta’s chemical composition, the distance to the supernova, and the explosion mechanism [37]. While there are several types of spectroscopically-distinct varieties of supernovae, these are split into two main classes: Type I and Type II supernovae (SNe). The differentiator is the presence of hydrogen in the spectra: Type I have no hydrogen lines3, whereas Type II SN do have hydrogen lines, although there is much variation in the Hα emission line. Further delineation of classes of supernovae is done according to the spectra. Type Ia have a strong Si II line at 625 nm, Type Ib have a strong helium line, and Type Ic lack strong helium lines. These groupings relate to the various physical processes of determining which physical mechanism is responsible. Types Ib and Ic are only seen in spiral galaxies near recent star formation, whereas Type Ia are found in all galaxies [18]. Luminosity is another marker of the various explosion mechanisms. All the Type I supernovae have similar rates of decline in brightness after the maximum brightness. Type II supernova typically rise rapidly in brightness reaching a maximum typically 1.5 mag dimmer than Type Ia’s. Type II supernovae are classified further as Type II-P or Type II-L, referring to whether the light curve contains a plateau or is linear, respectively. The Type II-L occur only about a tenth as often as the Type II-P [18]. There are exceptions to these classifications that give hints of a connection be-

3except for possible contamination from superimposed HII regions.

6 tween groupings, but the Type 1a are still fundamentally diﬀerent from the others. In this work, we will limit ourselves to the other three types which involve the collapse of a massive, evolved star; collectively known as core-collapse supernovae.

2.1.2 Explosion Mechanisms

After decades of research a number of mechanisms have been identiﬁed that contribute to the explosion of a massive star [47]:

1. thermonuclear mechanism (or carbon detonation),

2. electron capture on O-Ne-Mg supernovae,

3. prompt bounce-shock mechanism,

4. reheating via neutrinos,

5. magnetorotational instability,

6. SASI acoustic mechanism, and

7. new phase of matter.

We discuss these mechanisms individually, but it is most certainly the case that a combination of two of more of these are needed to cause an explosion. The ﬁrst method, the thermonuclear mechanism, is most often found in the lightest of massive, supernova-progenitor stars, which are not heavy enough to create iron cores. The general idea is that the energy released from carbon thermonuclear burning provides the energy to gravitationally unbind the outer shells and envelop from the star. This is also called a carbon detonation supernova. Simply igniting the free-falling shells from compression heating, however, is insuﬃcient to blow the matter outward. It was proposed that neutrinos from the collapsing stellar core radi- ate to heat the degenerate C and O shell to ignite the thermonuclear burning front,

7 but simulations could not verify this suggestion; either the shells fall inward before being exposed to enough neutrinos, or the shells are at a large radius and the ﬂux of neutrinos is insuﬃcient to trigger the reaction [47]. The second method, electron capture on O-Ne-Mg supernovae, is also for low- mass progenitors (M = 8 − 10M ). They undergo carbon burning and develop oxygen-neon-magnesium (O-Ne-Mg) cores. However, before gravitational forces are enough to ignite the neon, the star is already supported by electron degeneracy pressure. The Fermi-energy for electrons increases, and enables electron captures, triggering gravitational collapse and resulting in an electron-capture SN. This collapse mechanism ejects little carbon and oxygen and very little nickel, and results in a relatively faint supernova. These types of collapse could contribute to as much as 20% - 30% of all supernovae [47]. The third method, the bounce-shock mechanism, begins as a shockwave is launched at the moment of core bounce with a prompt ejection of the stellar mantle and envelope. This is maybe a viable explosion mechanism for very low mass progenitors [109] however it tends to be a weak explosion. Much supernova research has been devoted to the fourth mechanism, neutrino heating. A neutrino-heated layer naturally develops as the proto-neutron star con- tracts (radiating energy in the form of neutrinos) and its surrounding accretion layer compacts (increasing the density) right after bounce. The temperature rises in the contracting proto-neutron star, increasing the mean energies of the escaping neutrinos. Unfortunately, realistic neutrino heating alone has created successful supernova explosions only in low-mass progenitor simulations. In more massive progenitors, hydrodynamic instabilities have been key in determining the success of the explosion [47]. 2D and 3D simulations produce successful supernova explosions but are computationally intensive and are of low energy.

8 Another interesting approach to increasing the heating from neutrinos is through neutrino oscillations. Oscillations from lower energy νe andν ¯e to hotter muon and tauon flavors could enhance the neutrino heating behind the shock [92, 33]. However, this has been shown to not have an impact during the post bounce accretion phase [29, 98]4. If, instead, electron-neutrinos could oscillate to a sterile neutrino flavor, this could revitalize the shock [113]. Non-radial, turbulent flows during supernovae can also aid neutrino reheating through a process called neutrino heated convection. They can increase the amount of time matter spends in the gain layer of the star, where neutrinos deposit more energy than they dissipate [47, 51, 48]. A standing accretion shock instability (SASI) can similarly enhance the explosion energy. It was noted in [10] that low-mode (l = 1, 2) perturbations of a shock surface grow exponentially in a spherically accreting flow. If this instability forms in this standing accretion shock, it can increase the energy deposited by neutrinos with non-radial mass flows and shock expansion, but also leads to secondary shocks, strengthening the convection and giving an additional push to the shock [47]. Decreasing symmetry constraints on the degrees of freedom (i.e. going to higher- dimensional simulations) is also promising. The critical neutrino luminosity for an explosion has been found in some studies [26, 65] to be lower in the multi-dimensional case, some times by several tens of percent. However, other studies [110] have not reproduced this trend and see the opposite effect. Nevertheless, there remains hope that this could make it easier for the mechanisms listed above to increase the neutrino flux sufficiently to trigger a successful explosion. Ultimately, however, many processes affect the neutrino production, diffusion and absorption. A sophisticated neutrino transport method is required to obtain reliable answers [47].

4See [33] for a review of neutrino ﬂavor transformation in core-collapse supernovae.

9 The fifth mechanism, magnetorotational mechanism, uses a highly magnetized proto-neutron star to extract the energy needed to violently expel the outer stellar layers. The proto-neutron star and supernova plasma have low resistivity, hence, the magnetic fields lines are frozen in the flow. During collapse these compress, increasing the magnetic energy density, which in turn can be transferred to expelling the matter via magnetic pressure and magnetoturbulence, magnetic bounce, or gas heating due to the dissipation of rotational energy through turbulent magnetic viscosity. Magnetic fields are likely to play some role in all supernova cores, but it seems currently that they are crucial only in very rapidly spinning stars with a high progenitor magnetic field [47]. The sixth method, the acoustic mechanism, uses non-linear oscillations from the forming proto-neutron star (driven by turbulence and accretion from downstream), to emit sound waves into the region behind the shock, injecting heat and reviving the stalled shock. However, the acoustic mechanism leads to explosions that are perhaps too late, and too weak; bleeding the oscillation power from the proto-neutron star, which may diminish its relevance [115]. Also, it is not as important in 3D as it is in 2D simulations. Finally, the seventh mechanism, involves a phase-transition to a new quark-gluon phase of matter. This is the general category of shock-revival that this thesis explores. By allowing the collapsing core to transition to quark matter as it accretes matter and heats from contraction, a second shock can be triggered. This second bounce shock catches up and merges with the stalled primary shock, thereby gaining enough energy to throw off the outer stellar layers [97, 39]. Moreover, the burst of neutrinos, emitted once the second shock breaks out of the neutrinospheres, may be ultimately detectable by future well time-resolved neutrino observatories [28]. Problems that arise with this method are that the equation of state of quark-gluon plasma tends to be very soft (the matter is easily compressible) and the associated

10 maximum neutron star mass may not be large enough to explain the observations of neutron stars as heavy as 2 M [31, 2]. Varying the theory’s parameters to stiﬀen the equation of state can prevent a transition to quark-matter during the collapse, prevent a second shock from forming, or prevent an observational signal. Finding a way of combining a quark-gluon plasma phase with hadronic matter in such a way as to cause an observable secondary shock to revive the ﬁrst stalled shock, while also allowing massive neutron stars after the explosion, is one goal of this thesis research.

2.1.3 Black Hole Formation

Up to this point, we have been referring to a neutron star (NS) as the remnant of a core-collapse supernova (CCSN). However, a black hole5 (BH) can be formed if (1) back-falling accretion from a successful explosion causes the nascent NS to exceed its mass limit, (2) the cooling or phase transitions of the proto-neutron star reduces the pressure support, or (3) the explosion mechanism fails to throw oﬀ enough of the stellar envelope so that continued accretion pushes the proto-neutron star over the mass limit [86]. The last of these options could be due to a failure to revive the stalled shock, or (especially for more massive progenitor stars) insuﬃcient energy to overcome the pull of gravity. This failed supernova is observationally distinguished from the other channels to black hole formation by the faint electromagnetic signal – for the most part the original star simply vanishes. In 2008, Kockanek et. al. [57] proposed a survey to determine the fraction of stars that ended with this fate. Given the unique challenges of building massive star catalogs large enough for a non-trivial event rate, and of recognizing the nature of the death of a star from observational signatures, this survey has been impractical until quite recently.

5Typically, these black holes are on the order of several stellar masses – not to be confused with 5 10 supermassive black holes like those at the center of galaxies which can range from 10 − 10 M .

11 2.2 Quark-Gluon Plasma

In 1984, Witten suggested that a stable state of matter, composed of freely interacting quarks and gluons like in a plasma, could exist. Pockets of such matter could possibly be left over after the big bang, fragments of disrupted neutron stars, or in other dense or hot regions of stars [120]. This is quite a radical proposal considering quarks are notoriously short-lived (∼ 10−24 s in the laboratory), whereas stellar lifetimes are exceedingly long (∼ 106 − 1011 years). Within Witten’s proposal was the concept of “strange stars” - exotic stars with interiors comprised of up, down and strange quarks - which has raised much interest for astrophysicists and particle physicists as well.

2.2.1 Phase Diagram of Dense, Hot Matter

One goal of this thesis research has been to map out a new phase diagram for hot, dense matter matter as is formed in core-collapse supernovae. At suﬃciently high temperatures, one would expect hadrons to “melt”, thereby deconﬁning their constituent quarks and gluons. Alternatively, once can imagine compressing nuclear matter until the hadronic constituents overlap and can interact with others in adja- cent nuclei. Both describe a state in which the degrees of freedom are those of quarks and gluons instead of nucleons. Insights from Lattice QCD, and experiments such as AGS, SPS, RHIC and LHC have pushed forward our understanding of matter under these extreme conditions. We will discuss these advancements further in subsections 2.2.4 and 2.2.5, respectively.

2.2.2 First-Order Phase Transition to a Quark-Gluon Plasma

A plasma of freely interacting quarks and gluons is theorized to exist at high densities and temperatures. A phase transition from hadronic to quarkonic matter, was envisioned as early as the 1970s [40]. However, the nature of the transition from

12 hadronic matter to a Quark-Gluon Plasma (QGP) is largely unknown in the high density and temperature regime. It is unlikely [60] that the transition is a simple cross-over (a zeroth order phase transition) from hadronic to quark matter, although this is believed to be the case at high temperature and low density. Here, the order of phase transition refers an order parameter for which the derivative is discontinuous. A first-order phase transition seems likely at high densities, and suggested by lattice QCD [30, 60, 7] and supported by results from the Relativistic Heavy Ion Collider (RHIC) [111, 123, 105]. A transition to a QGP phase during the collapse can have a significant impact on the dynamics and evolution of the nascent proto-neutron star. In [44] it was first shown that a first order phase transition to a deconfined QGP phase resulted in the formation of two distinct but quickly coalescing shock waves. More recently, it has been shown [38] that if the transition is first order, but global conservation laws are invoked, then the two shock waves can be time separated by as much as ∼ 150 ms. Neutrino light curves showing such temporally separated spikes might even be resolvable in modern terrestrial neutrino detectors [38].

The observation of a 2.01 ± 0.04 M neutron star, however, constrains the possibility of a ﬁrst order phase transition to a quark gluon plasma taking place inside the interiors of stable cold neutron stars [2, 31]. Nevertheless, for initial stellar masses beyond & 20 M every phase of matter must be traversed during the formation of stellar mass black holes. Hence at the very least, this transition to QGP may have an impact [84] on the neutrino signals during black hole formation as well as its possible impact on core-collapse supernovae.

2.2.3 The MIT Bag Model

Quantum Chromodynamics is notoriously hard to calculate, due to the UV- asymptotic nature of the strong force. On small length scales (equivalently high

13 energies, temperatures or densities), quarks and gluons are essentially free to move and interact. This approximates asymptotic freedom. As two quarks moves further away, however, the field lines collapse so that the magnitude of the strong force increases until enough energy is stored to create new particles from the vacuum so that free quarks cannot exist. In 1974, a group from MIT [23] proposed a simple, yet surprisingly robust model for these hadron interactions, wherein the strongly interacting quarks and gluons are confined to a finite region of space, termed a “bag”. This is brought about in a Lorentz-invariant way by assuming that the bag possesses a constant, positive vacuum energy per unit volume, B, which is added to the stress-energy tensor:

µν µν µν T = Tﬁelds − g B. (2.1)

Inside the bag, quarks are free to move around, and outside the bag T µν vanishes. By imposing linear boundary conditions and using the equations of motion for free fermions and gluons, one can solve for the quark energies and make predictions for quantities such as the gyromagnetic ratio and the mean square charge radius of the proton. Other methods have been proposed, such as an inﬁnite, spherical “square well” potential of ﬁxed radius [11], or an EoS based on an interquark potential [32], but the MIT Bag Model is far more common, perhaps because of its simplicity.

2.2.4 Lattice QCD

Quantum Chromodynamics (QCD) gives us a description of strongly interacting matter, with which one can formulate a phase diagram. At a simple level – assuming massless quarks, non-interacting massless deconﬁned quarks and gluons – the pressure can be written as a function of temperature, and it depends only on the number of

14 degrees of freedom [105]:

P π2 7 = 2(N 2 − 1) + N N , (2.2) T 4 90 C 2 C f

where NC is the number of colors and Nf the number of quark flavors. Here and throughout we use natural units, ~ = c = 1. The first term is the contribution from gluons, and the second from quarks. To go beyond this simple description – e.g. incorporating color interactions, non- zero quark masses and chemical potentials, and predicting the location and nature of the transition to quark matter – calculations can be done on a space-time lattice (LQCD). Computing power has always been a limitation in calculating physically meaning- ful quantities from Lattice QCD. However, several predictions emerge consistent as inputs can be varied. There is indeed a phase transition, between hadronic matter to quark-gluon plasma, that happens around TC ∼ 170 MeV in the limit that the chemical potential goes to zero (low density). In this limit, the most realistic calculations have a zero-th order (crossover) phase transition. This is highly dependent on the mass of the quarks and the number of flavors (two light (u, d) and one heavier (s) quark flavor). There is a suggestion [30, 105] that at higher chemical potential the transition becomes first-order, and that there is a critical point (µB ≈ 350−700 MeV) where the transition itself changes from a crossover to the first order phase transition. It is also interesting to note that in most calculations, the deconfinement transition is also accompanied by a chiral symmetry restoration transition [105].

2.2.5 Collider Experiments

There are a few experiments probing dense nuclear matter, and a quark gluon plasma. The 1980s saw the ﬁrst heavy ion beams at relativistic energies at the

15 Alternating Gradient Synchrotron (AGS) at Brookhaven National Lab in the US, and at the Super Proton Synchrotron (SPS) at CERN in Europe. The center of mass energies were 5 and 18 GeV per nucleon pair, respectively. These experiments paved the way for the first heavy ion collider, RHIC, which was also built at Brookhaven. In 2000, the Relativistic Heavy Ion Collider (RHIC) saw the first Au-Au collisions at a center-of-mass energy of 130 GeV for the nucleon pair, and a year later reached its nominal energy of 200 GeV. Most recently in 2010, the Large Hadron Collider (LHC) had its first heavy ion collisions of Pb beam at 2760 GeV, and there are plans to double that energy [70]. The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory started as four experiments: BRAHMS, PHENIX, PHOBOS and STAR. Currently only STAR and PHENIX are still operating, and a new experiment, pp2pp, to ex- plore proton-proton elastic scattering. The various experiments at RHIC roughly correspond to the different detectors, which are stationed about the beam. The BRAHMS (Broad RAnge Hadron Magnetic Spectrometers) Experiment studied reaction mechanisms of the relativistic heavy ion collisions over a wide range of rapidity and transverse momentum. The PHENIX (Pioneering High Energy Nuclear Interac- tion eXperiment) measures particles such as electrons, muons, and photons, as direct probes of the collisions. The primary goal of PHENIX is to discover and study the quark gluon plasma. It is also the largest of the four experiments that have taken data at the Relativistic Heavy Ion Collider. The PHOBOS detector is designed to examine and analyze a very large number of gold-gold collisions, looking for rare and interesting collisions with readily identifiable new physics. Like PHENIX, the STAR experiment seeks to study the formation and characteristics of the quark gluon plasma. The former focuses on a small solid angle, making high precision measurements of the particles that come out in a small range. STAR, on the other hand, tries to measure everything that comes out, with slightly less precision.

16 The Large Hadron Collider (LHC) at CERN near Geneva, Switzerland is now gathering data on heavy-ion collisions. Three of the four LHC experiments participate in the heavy ion program: ALICE, ATLAS and CMS. ALICE is the only LHC experiment devoted to the study of QGP. All three programs, ALICE, ATLAS and CMS have conﬁrmed the phenomenon of jet quenching [1] related to the formation of QGP.

2.3 Equation of State

To incorporate relevant physics into a supernova explosion, one needs an equation of state (EoS). The EoS plays an important role in many high-density phenomena, such as relativistic heavy-ion collisions, neutron stars, supernova explosions, and black-hole formation. Currently there are two equations of state (EoSs) most commonly used in supernova simulations: that due to Lattimer and Swesty [63], using the liquid drop model; and that due to Shen et. al. [100, 101], using a relativistic mean field theory approach. Many other equations of state exist [62], but these are the two primarily adapted to supernova simulations because they include the relevant temperature dependence, along with a broad range for density and electron fraction. The equation of state currently employed by the supernova simulation [118] used at Notre Dame was originally based on the model of Bowers and Wilson [13]. It is a more phenomenological approach to the liquid drop model. This EoS was sufficient at first, but as my thesis work progressed we found that we needed to include 3-body forces, and part of my thesis was to develop a new equation of state based upon a Skyrme model to describe hadronic matter at high densities, including effects of three-body interactions. I will discuss these models in turn, emphasizing the pros and cons and listing the state of current research. The equations of state agree on treating the leptons

17 and photons as ideal Fermi & Bose gases (respectively), plus a transport scheme for neutrinos. Hence, here I focus the discussion on the treatment of baryonic matter. First, however, we give a brief review of current available supernova equations of state.

2.3.1 The Lattimer & Swesty Liquid-Drop Model

The main motivator for this model is to improve on the bulk fluid approach by including the effects of finite nuclear size via a compressible, liquid-drop [64]. The free energy density then contains terms corresponding to a volume (or bulk matter), surface, coulomb, and translational effects [63]:

FN = Fbulk,i + FS + FC + FH . (2.3)

The free energy per baryon for bulk matter is written as an expansion in density, n:

1 F (n, x, T ) ' −B − ∆x + K (1 − n/n )2 + S (1 − 2x)2 − a T 2 + ... (2.4) bulk 18 s s v v where x is the charge fraction (x = Z/A), T is the temperature, B is binding energy of

1 1 saturated symmetric nuclear matter (B = −fbulk(ns, 2 , 0) − 2 ∆ ' 16 MeV), ∆ is the neutron-proton mass diﬀerence, Ks is the incompressibility of bulk nuclear matter, ns is the saturation density of symmetric nuclear matter, Sv is the symmetry energy parameter of bulk nuclear matter, and av is the bulk level density parameter given in terms of the nucleon eﬀective mass for saturated symmetric matter, m∗:

2m∗ π 2/3 av = 2 . (2.5) ~ 12ns

The surface contributions represent the surface tension between two inﬁnite slabs of

18 nuclear matter, either in a dense phase for nuclei or light phase representing nucleon vapor. Several approximations were implemented to describe the free energy per baryon. First, the surface term, fs, is written as:

3σ(xi,T ) Fs = (2.6) rN ni

4 3 where rN is the nuclear radius deﬁned by VN ≡ 3 πrN , and the temperature and compositional dependence of the surface tension is approximated by

T 16 + q σ(x ,T ) = σ(1/2, 0)h MeV fm−2 (2.7) i −3 −3 TC (xi) xi + q + (1 − xi) where the temperature dependence is diminished at high temperature according to

  2 2 T [1 − (T/TC (xi)) ] T ≤ TC (xi); h = (2.8) TC (xi)  0 T > TC (xi) and q is related to the surface symmetry coeﬃcient by [63]

2 2/3 q = 384πrN σ(1/2, 0)/(SsA ) − 16 . (2.9)

The Coulomb contribution to the free energy is derived from a Wigner-Seitz cell approximation [63] and is expressed in terms of the electron charge e and an eﬀective

3 1/3 1 nuclear radius correction, D(u) ≡ 1 − 2 u + 2 u:

3 Z2e2 FC = D(u) (2.10) 5A rN

In the transition between free nuclei and nuclear matter, bubbles are parameter- ized in terms of a volume ﬁlling factor u = Vi/V , where Vi is the volume of the average bubble of nuclear matter and V is the volume of the Wigner Seitz cell. This

19 ﬁlling parameter simulates the gradual transition from nuclei to bubbles as in the Ravenhall et. al. scheme [94]. Finally, the total translational free energy density is approximated [63]:

" ! # u(1 − u)n u(1 − u)n F = i h T ln i − T (2.11) H A 5/2 0 nQA0

2 3/2 with nQ = (mT/2π~ ) . Together these deﬁnitions can be used with the other free energies, thermodynamic relations and the constraints of baryon number and charge conservation to construct a three-dimensional table of the relevant thermodynamic quantities as a function of the input density, temperature and charge fraction. This table has been widely in use since its creation. It has not been updated to include the eﬀects of hyperons, pions or kaon condensation or a quark-hadron phase transition, all of which seemed extraneous physics at the time of this EoSs creation but have recently come to be of more interest.

2.3.2 Relativistic Mean Field Theory

The starting point for the Shen et. al. equation of state [100] is a Lagrangian containing nucleonic and mesonic degrees of freedom:

1 − τ L =ψ¯ iγ ∂µ − M − g σ − g γ ωµ − g γ τ ρaµ − eγ 3 Aµ ψ µ σ ω µ ρ µ a µ 2

1 µ 1 2 2 1 3 1 4 + ∂µσ∂ σ − mσσ − g2σ − g3σ 2 2 3 4 (2.12) 1 1 1 − W W µν + m2 ω ωµ + c (ω ωµ)2 4 µν 2 ω µ 4 3 µ 1 1 1 − Ra Raµν + m2ρa ρaµ − F F µν 4 µν 2 ρ µ 4 µν

20 where

W µν = ∂µων − ∂νωµ, (2.13)

aµν µ aν ν aµ bµ cν R = ∂ ρ − ∂ ρ + gρabcρ ρ , (2.14)

F µν = ∂µAν − ∂νAµ (2.15) where ω, ρ, σ are meson fields and Aµ is the electromagnetic vector potential. This Lagrangian includes the non-linear σ and ω self-coupling terms which are needed, among other reasons, to get a reasonable value for the incompressibility [100]. Then, a relativistic mean field approximation is made whereby the coupling con- stants and unknown meson masses are effective parameters that are adjusted to fit nuclear data. The Shen EoS uses the TM1 parameter set from [108] which is the best at reproducing the properties of finite nuclear over the the wide mass range of the periodic table as well as neutron-rich nuclei.

2.3.3 Bowers and Wilson Model

The Bowers and Wilson equation of state [13] was introduced in 1982 and is a more phenomenological approach to describing the behavior of the matter in a stellar collapse and in dense neutron star. While not as universal as the Lattimer & Swesty or Shen et. al. EoSs, it was the EoS used in the first successful modeling of a supernova collapse and explosion [117]. It also serves as a starting point for the low density description of the NDL EoS. The definition of the Bowers and Wilson equation of state begins by describing the free energy of the constituent parts: photons, electrons, positrons, pions, free neutrons, free protons, and atomic nuclei. Everything is calculated in terms of three input variables which uniquely define the state: temperature, matter rest mass density and the charge per baryon [116].

21 During the collapse, matter is optically thick to photons, so the description for photons is grouped together with electrons, positrons and pions and remains the same throughout the wide range of densities and temperatures. Baryonic matter, however, is sectioned up into several regimes: baryons below nuclear saturation density which are not in nuclear statistical equilibrium (NSE), baryons below nuclear saturation density which are in nuclear statistical equilibrium, a transitional phase between subnuclear to supranuclear densities, and baryons above nuclear saturation density. Each regime is further broken into contributions: below nuclear saturation density and not in NSE, the free energy includes an ideal gas component and a Coulomb term; once NSE is reached, the free energy is counted from free neutrons, free protons, alpha particles and a representative heavy nucleus. The transition to supranuclear densities is smooth and described by a weighting

2 factor W = (1 − ρ/ρN ) where ρ is the baryonic matter density and

14 −3 5/2 ρN = 2.66 × 10 g cm 1 − (1 − 2Ye) (2.16) is the nuclear saturation density where nuclear matter becomes a uniform sea of

− nucleons, which depends on Ye, the excess of electrons and π . Above this saturation density, the free energy per baryon contains contributions, respectively, from symmetric nuclear matter, an asymmetry energy and a temperature contribution:

Fsupranuclear = F1(ρ) + F2(ρ, Yp) + F3(ρ, T ) . (2.17)

The phenomenological nature of the Bowers and Wilson EoS can be better seen in the ﬁrst two terms, which are formatted in terms of the measured binding energy per nucleon at nuclear saturation density (E0), compressibility parameter also at

22 saturation (K0) and adiabatic index (Γ):

1 ηΓ − 1 − Γ(η − 1) F (η = ρ/ρ ) = 8.79 MeV + E + K , (2.18) 1 N 0 9 0 ηΓ(Γ − 1) 72 MeV F (ρ, Y ) = η 16 MeV + (1 − 2Y )2 . (2.19) 2 p 1 + 4η p

The thermal contribution is simply the diﬀerence in energy between a ﬁnite temperature gas of neutrons, protons and delta particles and that of the zero temperature energy, for a given density:

F3(ρ, T ) = Θ(ρ, T ) − Θ(ρ, 0) , (2.20) where

Z 2 X 4πgidpipi µi Θ(ρ, T ) = 3 − kT ln(Di) , (2.21) Di i=N,∆ ~

−(i−µi)/kT Di = e + 1 , q 2 ∗ 2 ∗ mi i = pi + (mi ) , mi = , 1 + 0.27ρ/ρN

gi is the spin/isospin degeneracy factor, and mi is the average rest mass of the nucleon or delta particle. Once the Helmholtz free energies are stated, the remaining state variables can be calculated using thermodynamic relations:

∂F ∂F µn = , µp = , (2.22) ∂Yn ∂Yp ρ2 ∂F P = , (2.23) m ∂ρ B (T,Xi,Ye) T 2 ∂F = − . (2.24) m ∂T B (ρ,Xi,Ye)

23 for the neutron and proton chemical potentials, the pressure and energy density, respectively. A full description of this equation of state can be found in [116, 13].

2.3.4 The Skyrme Model

A Skyrme model is not an equation of state, rather a model of eﬀective nucleon- nucleon interactions. A description is included here, because it serves as the basis for the work of this thesis, the Notre Dame-Livermore Equation of State. The Skyrme model is a density functional theory involving two- and three-body interactions of nucleons. This is reﬂected in the potential:

X (2) X (3) V = vij + vijk (2.25) i

E = hφ, (T + V )φi (2.26) 2 X p 1 X 1 X = i i + hij|vij|iji + hijk|vijk|ijki . (2.27) 2m 2 6 i ij ijk

The initial parameterization of Vautherin and Brink [112] from the 1970s obtained this expression for the energy per nucleon:

E 3 3 1 3 = T + t ρ + t ρ2 + (3t + 5t )ρk2 (2.28) A 5 F 8 0 16 3 80 1 2 F

2 2 where TF = ~ kF /2m is the kinetic energy of a particle at the Fermi surface, kF is the Fermi momentum and {t0, t1, t2, t3} are Skyrme parameters that can be varied to create other models that better ﬁt experimental data. For this work, however, we will use the more modern version of Dutra et. al. [34]

24 that generalizes the formulation so that it can include subsequent work:

E 3 t = T H + 0 n[2(x + 2) − (2x + 1)H ] A 5 F 5/3 8 0 0 2 3 1 X + t nσi+1[2(x + 2) − (2x + 1)H ] 48 3i 3i 3i 2 (2.29) i=1 3 3π2 2/3 + n5/3 aH + bH 40 2 5/3 8/3 where

a = t1(x1 + 2) + t2(x2 + 2), (2.30) 1 b = [t (2x + 1) − t (2x + 1)] , (2.31) 2 2 2 1 1 m−1 m m Hm(Yp) = 2 [Yp + (1 − YP ) ] . (2.32)

This same paper lists modern constraints that rule out most of the 240 original parameterizations, leaving 16 models which are consistent with all eleven constraints. We use these 16 models, as well as our own parameterizations, for the Skyrme description of baryonic matter.

25 CHAPTER 3

THE NOTRE DAME-LIVERMORE EQUATION OF STATE

A wide variety of numerical simulations; like those that model stars, neutron stars, high energy collider experiments, etc.; need an eﬃcient way to model the eﬀects of forces on the constituent parts. In general, due to the large numbers involved, ab initio calculations are not well suited to this environment. A common solution is to use an equation of state (EoS) that relates a few input characteristic state variables to the other state variables needed for the calculation. Because of the general statements used in the creation of the EoS, it can be used in a wide variety of simulations, so long as it describes the needed input variable regimes. Because of the general nature of the EoS, once made, the EoS can be used in that wide variety of simulations. A simple example of an equation of state would be to assume an ideal gas and use the EoS “P = nRT/V ” to describe the pressure response to changes in density [18]. More realistic models incorporate a variety of particle interactions, relating many state variables (e.g. pressure, internal energy, entropy, etc.) for a set of input state variables. This is typically done by creating a table with values of the state variables as a function of various physical variables, and interpolating the values when needed in the numerical simulation. I discuss here the new Notre Dame-Livermore Equation of State (NDL EoS), and highlight the additions to it that were a part of this thesis work. At low densities, this EoS builds upon with the original framework of Bowers and Wilson [116, 13]. At high densities, however, the equation of state implements a density functional Skyrme

26 description of baryonic matter at intermediate densities and temperatures, and allows for a quark-gluon plasma phase of matter at high temperatures and densities. The independent state variables used as inputs to the table are (1) temperature, T , in units of MeV; (2) the matter number density, n, in units of fm−3; and (3) the net

1 charge per baryon , Ye = ne/nB. From these three independent state variables, the EoS specifies the dependent state variables: pressure [MeV fm−3], internal energy density [MeV fm−3], entropy per baryon (in units of Boltzmann’s constant), various chemical potentials [MeV], etc. To adequately describe the system, several variable definitions are common to the field. Because of the nature of the simulations, dimensionless quantities are used when possible. This facilitates comparisons between different layers of the star where conditions can vary greatly. The mass and charge fraction for a given particle type (indexed by i) are defined, respectively, as

mi mass fraction: Xi = (3.1) mB ni charge fraction: Yi = (3.2) nB

where nB is the total sum over all the species that carry baryon number; e.g. in a purely hadronic phase nB = nn + np + nα + nhAi, whereas in purely quark matter nB = (1/3)(nu + nd + ns). Notice that baryon conservation requires that

X Xi = 1 . (3.3) i

Densities are also often described in terms of the unitless parameter: η ≡ n/n0 =

ρ/ρ0, where n0 is the saturation density of bulk symmetric nuclear matter, typi-

1 Often referred to as the “electron fraction” or “excess charge fraction”, Ye includes not just the negative charge from electrons, but also from muons and taus as well. At high densities and/or temperatures, when the EoS includes the creation of pions and other higher mass particle resonances, their charge is also included in Ye.

27 −3 cally n0 = 0.16 fm [66].

3.1 Constructing the EoS

The various types of matter that contribute to the matter equation of state include photons, electrons, positrons, pions and other mesons, free neutrons, protons and other baryons, and atomic nuclei. In the QCD phase we include all relevant hadronic states. Neutrinos are also included in the simulation, but due to their long mean free path their contribution to the matter pressure and energy density must be modeled separately from the EoS. We can, however, include photons in the matter EoS because the matter is optically thick to photons. Baryonic matter is handled diﬀerently in each of ﬁve regimes (roughly corresponding to increasing density):

1. baryons below saturation density, not in nuclear statistical equilibrium (NSE);

2. baryons below saturation density, in NSE;

3. baryons at and above saturation density, consisting of purely hadronic matter;

4. baryons above saturation density, consisting of mixed hadronic and quarkonic matter; and

5. baryons above saturation density, consisting of purely quarkonic matter.

We will describe the treatment of each of these matter components in the following sections.

28 3.1.1 Photons, Electrons and Positrons

The photons are treated like black-body radiation. Their contributions to the energy density and matter pressure are given by the usual Stephan-Boltzmann law,

4 Eγ = 3Pγ = aT , (3.4) where a = 1.37 × 1026 erg cm−3 MeV−4. Both the electrons and positrons are approximated as a uniform background non-interacting ideal Fermi-Dirac gas. The general form of the species number density is

Z ∞ 4πp2dp n = g (3.5) i i E−µi 0 1 ± e kB T where gi is the degeneracy, E is the energy, µi is the chemical potential, and the sign in the denominator is positive for fermions and negative for bosons. Their contributions are obtained from tables generated via numerical integrations over Fermi-Dirac distributions as a function of temperature and chemical potential.

3.1.2 Baryons Below Saturation Density

A useful approximation in modeling large quantities of matter undergoing nuclear fusion is nuclear statistical equilibrium (NSE). This provides a statistical prediction for the concentration of individual elements at a given energy. In NSE, the nuclear abundances, X, are given by the nuclear Saha equation [116]:

3 A−1 2 2 (A−1) g(Z,A) mµρ 2π B(Z,A) 5/2 ~ A−Z Z kT X(Z,A) = A A−X Z Aµ Xn Xp exp , (3.6) 2 mn mp mµkBT

29 where g(Z,A) is the nuclear partition function,

X Ei g(Z,A) = E (2J + 1) exp , (3.7) i i k T i B mµ is the atomic mass unit, while mp and mn are the proton and neutron rest masses, respectively. As usual, A and Z refer to the atomic number and charge. The quantity B(Z,A) is the nuclear binding energy:

B(Z,A) = (A − Z)mn + Zmp − Aµ(Z,A)mµ , (3.8)

A where Aµ(Z,A) = M(Z,A)/mµ is the atomic mass of a nucleus Z. Note that baryon number and charge conservation, respectively, require that

X Xi = 1, (3.9) i X Zi X = Y . (3.10) i A e i i

Matter is assumed to be in nuclear statistical equilibrium at densities below nuclear saturation density (n0), but above temperatures of T ≈ 0.5 MeV. This is because, at this temperature, equilibrium between photodisintegration and capture reactions is rapidly achieved. Below this temperature, the isotopic abundances must be evolved dynamically, whereas at higher temperatures one can assume abundances from the Nuclear Saha Equation [Eqn. (3.6)]. Below saturation density, regardless of whether the matter is in nuclear statistical equilibrium, the equation of state is constructed from the Helmholtz free energy per baryon [95]

F = −kBT ln Z, (3.11) where Z is the relevant partition function. The next two sections describe the form

30 of the Helmholtz free energy density, which diﬀers if the NSE approximation is valid. Given the deﬁnition of the Helmholtz free energy density, the baryonic pressure and energy per unit mass – which are needed to build the EoS – can be calculated from the usual thermodynamic relations [95]:

ρ2 ∂f P = (3.12) m m ∂ρ B (T,Xi,Ye) T 2 ∂f = − . (3.13) m m ∂T B (ρ,Xi,Ye)

3.1.2.1 Baryons Below n0, Not in NSE

In the regime below the saturation density n0, and at low temperature, the rates of photodissociation of nuclei into α-particles and the fusion of light elements are unequal. To track the populations of every isotope is computationally intensive. Fortunately, only a few reactions participate in most of the energy generation and loss. Therefore, we approximate the nuclear constituents by a nine element nuclear burn network consisting of n, p, 4He, 12C, 16O, 20Ne, 24Mg, 28Si, and 56Ni [13]. These elements, then, are the ones summed over in determining the state variables. The free energy per baryon can be described as a sum of an ideal gas contribution

Fg (Eqn 3.14), and a coulomb correction Fc (Eqn 3.15) [116].

! X T XinA Fg = ln (3.14) A 3/2 5/2 i i T Ai 1 F = − n1/3e2hAi2/3Y 2. (3.15) c 3 e

The relevant variables here are the nuclear mass fraction Xi, the local baryon number density n, T is the temperature, e is the electron charge, Ai is the atomic mass number, hAi is the representative heavy nucleus, the index i runs over the entire

31 nuclear reaction network, and A is the thermal wavelength per baryon given by

8π3 A = 3/2 . (3.16) e (2πmB)

(Note, that natural geometric units (~ = c = kB = G = 1) have been adopted here and throughout the rest of this thesis.) From these relations the baryonic pressure and energy per unit mass can be calculated, using the thermodynamics relations (Eqn 3.12 - 3.13), to be:

X Xi 1 2/3 P = nT − n4/3e2hAi Y 2 , (3.17) m A 3m e i i B

3 T X Xi 1 2/3 = − n1/3e2hAi Y 2 . (3.18) m 2 m A 3m e B i i B

3.1.2.2 Baryons Below n0, in NSE

The beauty of nuclear statistical equilibrium is that the composition, entropy, pressure, and so forth, can be solved for in terms of three variables: the density n, temperature T and electron fraction Ye [99]. Instead of using the nine-element burn chain of the previous section, baryonic matter is approximated as consisting of neutrons, protons, alpha particles and a representative heavy nucleus. Energy balance is guaranteed from the relevant nuclear binding energies. Using this nine-element chain is fairly accurate and allows for swift, analytic calculations of the matter state variables. The disadvantages of this method however, are an uncertainty due to

ﬂuctuations in Ye and an inability to properly describe thermonuclear burning rates in this density and temperature regime [13]. Detailed stellar simulations on high- resolution supercomputers [69] seem to require more elements in the nuclear burning chain, or using full NSE. Nevertheless, for speed and simplicity the nine-element burn chain is adopted for the spherical symmetric (1D) evolution described here. We choose the zero of binding energy to be the normal 56Fe ground state energy.

32 This is unlike most other equations of state for which their zero point is chosen relative to dispersed free nucleons. The reason for this choice is that it avoids the numerical complication of negative internal energies in the hydrodynamic state variables at low temperature and density due to the binding energy of nuclei. The energy per nucleon

56 required to dissociate Fe into free nucleons is p0 = 8.37 MeV for protons, while for neutrons it is n0 = 9.15 MeV. In nuclear statistical equilibrium, the thermodynamic quantities are again found from the Helmholtz free energy per baryon, but we consider the contributions from each of the types of matter:

These contributions can be written analytically as [116]:

( " !#) 1 + p1 + ζ2 3 p 2 n Fn = XBYn n0W + N (1 − W ) + T 1 + ζn − ln , 2 βζn

(3.20) ( " !#) q 1 + p1 + ζ2 3 2 p Fp = XBYp p0W + N (1 − W ) + T 1 + ζp − ln , 2 βζp (3.21) T X nm α F = X W + (1 − W ) + ln α B , (3.22) α α α0 N 4 T 3/245/2

FhAi = Fbulk + FSurface + FCoulomb + Fthermal 1/3 1 ρ 2 2/3 2 2 = XhAi N (1 − W ) − e hAi YA + SE(YF e − YA) (3.23) 3 mB 3 4/3 2 T XAnmBα + ρ YAb(Ye) + ln 3/2 5/2 , 4 A gAT A

Following the notation of Eqn. (3.1), the various mass fractions are as follows: XB

33 4 refers to free baryons; Xα to He nuclei; and XhAi for the average heavy nucleus.

Similarly, following Eqn. (3.2), Yp and Yn are the relative number fractions of free baryons in protons or neutrons, respectively. Thus, Yp + Yn = 1. The quantity YhAi is the average Z/A for heavy nuclei. The representative heavy nucleus mass is a function of density, and is obtained from a ﬁt of a liquid drop representation of the average most stable nucleus. The mass of the representative heavy nucleus, hAi, is expanded in powers of the density parameter X:

2 2 3 hAi = 194.0(1 − Ye) (1 + X + 2X + 3X ) , where (3.24) ρ 1/3 X ≡ . (3.25) 7.6 × 1013 g cm−3

This expression allows for the attainment of extended heavy nuclei as one approaches nuclear matter density [116]. The quantity W in Eqns. (3.21) - (3.23) is a weighting factor that interpolates between the low and high density regimes. This transition, from subnuclear to supranuclear density, is expected to be smooth and continuous. The reason is that, as the density increases, progressively larger nuclei are former, more or less continuously. When a relativistic Thomas-Fermi representation of the electrons is evaluated at subnuclear density, the electron energy is lowered by more than ∼ 1 MeV [116]. The electrostatic nuclear energy also increases in magnitude. Similarly the transformation of nuclei from spheres to other, more exotic shapes (e.g. pasta nuclei, etc.) [61, 94] also lowers the energy of the medium by about 1 MeV. The net result is that the pressure and energy are very smooth functions of density near the nuclear saturation density. Hence, the weighting factor,

ρ 2 W ≡ 1 − , (3.26) ρN

34 is chosen to approximate this smooth transition.

The quantity ρN is the density at which nuclear matter becomes a uniform sea of nucleons. This was found by ﬁtting the saturation density of nuclear matter [i.e.

PM (ρ, T = 0,Ye) = 0] as a function of ρ and Ye. The zero temperature result was chosen to simplify the problem of making a smooth transition between the three equation of state regimes. The result is

14 −3 5/2 ρN = 2.66 × 10 g cm 1 − (1 − 2Ye) . (3.27)

The quantities ζn and ζp in Eqns. (3.21) and (3.20) are a measure of the degeneracy of the free baryons. They are deﬁned by

B(ρY X )2/3 B(ρY X )2/3 ζ = n B ; ζ = p B , (3.28) n kT p kT

2/3 where B(ρYiXB) is the energy per baryon of a zero temperature non-relativistic ideal fermion gas and the constant B is

3 3 2/3 h2 B = . (3.29) 10 8π 5/3 mB

The dimensionless constant β appearing in Eqns. (3.21) and (3.20) is determined such that the translational part of fp and fn reduces to the correct non-degenerate limit (T → ∞, ζi → 0). That is,

" !# 1 + p1 + ζ2 3 p 2 n XBρYiA kT 1 + ζn − ln −→ kT ln 3/2 . (3.30) 2 βζn T

This requirement implies

A2/3 3 β = = 0.781 , (3.31) 2 eB

35 where A is the thermal wavelength per baryon given in Eqn. (3.16).

The function for b(Ye) in Eqn. (3.23) is determined by the condition that the

Coulomb contribution to the pressure at ρ = ρN is canceled by the term proportional to b(Ye). This requires,

1 ! 3 " # e2 hAi2 1 ∂ ln hAi b(Y ) = + 2 . (3.32) e 18 m ρ ∂ρ B N ρN

The expression for the statistical weight of the heavy nucleus gA appearing in Eqn. (3.23) is taken to be

 s  " s # 1 3  T 2 T T T  ln gA = 1 − 1 +  + ln + 1 + , (3.33) A 2  TS TS TS TS  where

ρ TS = 8 MeV 1 + 2 . (3.34) ρN

In Eqn. (3.23) the constant SE = 120 MeV is derived for a symmetry energy of

56 30.4 MeV per nucleon. The constant YF e = 0.464 is the fraction of protons in Fe. This describes all the variables used in the deﬁnition of the free energy, Eqns. (3.19) - (3.23).

3.1.2.3 Treatment of Nuclear Pasta Phases

During core-collapse supernova simulations, the transition from subnuclear density (∼ 0.1 n0) to nuclear matter density (n0) occurs quite rapidly (∼ 1 ms) in the

2 the core. Hence, the simple quadratic extrapolation in W ≡ (1 − n/n0) of the formulation of [13] is adequate for the collapse. Nevertheless, there is a great deal of interesting nuclear physics in this regime at low temperatures in which the interplay

36 between the Coulomb and surface energies lead to various forms of “pasta” nuclei, with growing mass number and geometries varying from spherical to sheet-like to cylinder-like geometries [94]. Moreover, although this regime is not important during the collapse itself it does matter for the nascent proto-neutron star. This is because convection near the surface and in this density regime of the star can have a significant impact on the early (∼ 0.1−0.5 sec) transport of neutrino flux and its associated heating of material behind the shock. In this regard the EoS of [13] was useful as it leads to the development of a dendritic “neutron fingers” instability near the surface that enhances the explosion. This is gone further into detail in section (5.2.6). For that reason, therefore, we keep this as one option in the current EoS. Nevertheless, in the interest of providing a deeper physical underpinning of the current EoS we also add an option to include the possibility to treat the transition among the pasta phases. A great deal of effort [61, 94] has gone into describing this interesting regime, however, in the spirit of the current phenomenological Skyrme-force approach of the current work, we can follow the Wigner-Seitz cell derivation of [63, 94], updated to self-consistently transition the current Skyrme parameters of the EoS employed here. This approach was based upon an adoption of the Skyrme Interaction, but is applicable to a broad class of density functionals such as the ones of interest here, and hence is a natural means to extend the model developed here. Within the Wigner-Seitz cell one begins as above by dividing the nuclear free energy into contributions from the formation of very large bulk heavy nuclei that occupy a fraction of the volume in addition to an exterior fluid composed of neutrons protons and alpha particles. As in Eq. (12), heavy nuclei are then characterized by a bulk energy plus surface and coulomb energies. Hence the free energy in Eq. 3.23, except now we replace W in Eqns. (3.20) - (3.23) with u = VN /Vc where VN is the volume of heavy nuclei and Vc is the cell volume. The nuclear volume is expressed

37 3 VN = (4/3)πrN with rN the eﬀective nuclear radius corrected for various shapes as described below.

In this case, the free energy of the heavy nucleus FhAi is given by the bulk energy of the parametrization of nuclear matter as described above plus modiﬁed surface and coulomb terms due to the exotic shapes. That is, we write as before:

FhAi = Fbulk + FS + FC + Fthermal . (3.35)

However, for the formation of nuclear pasta phases in bulk nuclear matter in the

Wigner-Seitz cell approximation one can express [63] the sum of FS + FC during the passage through this transition as a simple analytic function of the volume parameter u, charge to mass ratio xi = (Z/A)i for the average nucleus i, and the temperature T as:

2 1/3 FS + FC = β[c(u)s(u) ] /n = βD(u)/n , (3.36) where D(u) is given in [63], using a ﬁt to the Thomas-Fermi Skyrme-force calculations of [94]: (1 − u)D(u)1/3 + u (D(1 − u))1/3 D(u) = u(1 − u) , (3.37) u2 + (1 − u)2 + αu2(1 − u)2 where D(u) ≡ 1 − (3/2)u1/3 + (1/2)u is a Coulomb correction for spherical bubbles in the Wigner-Seitz approximation, and α = 0.6 is a parameter adjusted to optimize the ﬁt to the Thomas-Fermi calculations of [94]. The normalization factor β then contains the dependence upon the charge-to-mass ratio YA and temperature T. This can also be written analytically

πσ(Y ,T )2e2Y 2n2 1/3 β = 9 A A i . (3.38) 15

Here, e is the electronic charge, ni is the nuclear number density, while σ(YA,T ) is the temperature dependent surface free energy per unit area. For a broad range of

38 density functionals can be written [63]:

16 + q σ(Y ,T ) = σ(0.5, 0)h(T ) , (3.39) A −3 −3 YA + q + (1 − YA) where σ(0.5, 0) = 1.15 MeV fm−2 is the surface tension of cold symmetric nuclear matter deduced [63] from ﬁts to individual nuclei. The temperature dependence of the surface tension is taken to diminish up to a critical temperature according to:

 22  T  1 − T ≤ Tc(YA)  Tc(Y ) h(T ) = A (3.40)  0 T > Tc(YA), where the critical temperature above which nuclear pasta phases do not exist is related the frequency of the giant monopole resonance [63]. Here, we express this in terms of the nuclear compressibility parameter K and density n as discussed below,

1/2 −1/3 Tc(YA) = 2.4344K n YA(1 − YA) MeV. (3.41)

The quantity q in Eq. (3.39) relates to surface symmetry energy S0 for symmetric nuclear matter discussed in the next section,

2 q = 384πr0σ(0.5, 0)/S0 − 16 , (3.42)

1/3 with r0 = (3/4πn0) is the nuclear radius parameter here written in terms of the nuclear saturation density n0.

This speciﬁes the transition to pasta nuclei in terms of u, YA, and T . What remains is to specify the dependent variables u and YA in terms of the EoS variables n and Ye. This is obtained from the conditions of mass and charge balance. In terms

39 of our notation we have

n − n u = 1 + i , ni − nn − np + nα(Vα(nn + np) − 4)

nYe (nA − (nn0 + np0)(1 − nανα) − rnα) − (np0(1 − nανα) + 2nα)(nA − n)) XA = nA (n − (nn0 + np0)(1 − nανα) − 4nα)

where n is the baryon number density and XB is the free baryon mass fraction, while

−3 vα = 24 fm is the volume of an alpha particle. The quantities np = Ypn, nn = Ynn, and nα = Yαn are the fractions of unbound protons, neutrons, and α particles, respectively. These quantities are determined from the minimization of the free energy as described below. This then provides a treatment of pasta phases consistent with the Skyrme parametrization above saturation density which we described in the previous section 3.1.2.2.

3.1.2.4 Chemical Potentials

The chemical potentials are found from the free energy via [95]:

∂F Yp ∂F µn = − , (3.43) ∂XB XB ∂Yp ∂F Yn ∂F µp = + , (3.44) ∂XB XB ∂Yp ∂F µα = 4 , (3.45) ∂Xα ∂F YA ∂F µnA = − , (3.46) ∂XA XA ∂YA ∂F (1 − YA) ∂F µpA = + , (3.47) ∂XA XA ∂YA

where µp, µn and µα are the chemical potentials of free protons, neutrons, and alpha particles. The quantities µnA and µnA are the chemical potentials of neutrons and protons within heavy nuclei. These quantities can be related by conditions of charge

40 and baryon conservation in equilibrium:

2µn + 2µp = µα (3.48)

2µnA + 2µpA = µα (3.49)

µnA − µpA = µn − µp ≡ µ.ˆ (3.50)

In the original Livermore formulation [13, 116], an analytical approximation was used to determine the average heavy nucleus mass fraction, XhAi. In the current implementation, the three chemical potential constraints combined with charge and baryon number conservation are solved self consistently to determine the matter composition. This leads to a 20% increase in the mass fraction of heavy nuclei when compared to the original approximation scheme [13, 116].

3.1.3 Baryons Above Saturation Density, Purely Hadronic Matter

As part of this thesis work a number of new features have been developed to describe nuclear matter above saturation density. We retain the deﬁnitions of electrons, photons and low density baryonic matter, as described in the previous sections, but the high density baryonic matter energy and thermal contributions described in this section are new as part of this thesis work.

3.1.3.1 Skyrme Density Functional Theory

Above nuclear matter density, but below the onset of a mixed phase, the baryons are treated as a continuous ﬂuid. In this regime, the free energy per nucleon is given in the form

f = f1 (n, Yp) + f2 (n, T ) + 8.79 MeV. (3.51)

The free energy is broken into three parts: a zero-temperature contribution, a thermal contribution, and an additional 8.79 MeV to set the zero of the free energy to be

41 the ground state of iron nuclei2. A further divination can be made to the zero- temperature free energy, by considering the energy from isospin symmetric matter

(where equal numbers of neutrons and protons yield a proton fraction of Yp = 1/2) from the energy due to a departure from that symmetry. The latter is called the symmetry energy, denoted S(n, Yp).

E f (n, Y ) = (n, Y = 1/2) + S (n, Y ) . (3.52) 1 p A p p

(2) (3) Above saturation density, f1(n, Yp) includes both 2-body (vij ) and 3-body (vijk) interactions in the many-nucleon system. The Hamiltonian of this system is thus given by ˆ X ˆ X (2) X (3) H = ti + vij + vijk, (3.53) i i

1 0 v(2) = t 1 + x Pˆ δ (r − r ) + t δ (r − r ) kˆ2 + kˆ 2δ (r − r ) (3.54) 12 0 0 s 1 2 2 1 1 2 1 2 ˆ2 ˆ ˆ0 ˆ + t2k · δ (r1 − r2) k + iW0 (σˆ1 + σˆ2) · k × δ (r1 − r2) k, (3.55)

2We choose the zero of energy to correspond to the most tightly bound nuclei (56Fe), with a binding energy of 8.79 MeV. This prevents negative energies, as compared to setting the zero to free baryons.

42 ˆ where Ps is the spin exchange operator, r1 and r2 are the position vectors in the two-

ˆ ˆ0 body potential, x0 is the coefficient for the isospin exchange operator, k and k are the momentum and conjugate momentum operators, and W0 is the coefficient of the two- body spin orbit interaction. We will discuss the Skyrme coefficients t0, t1, t2, t3, and σ in the following sections. For this Skyrme potential the high density behavior can be dominated by a 3- body repulsive interaction. This term is taken to be a zero range interaction of the form v123 = t3δ (r1 − r2) δ (r2 − r3). If the assumption is made that the medium is spin-saturated, which is valid for neutron star matter and nuclei [96], the three-body term is equivalent to a density dependent two-body interaction given by [112]

1 r + r v(3) = t 1 + Pˆ δ (r − r ) ρ 1 2 . (3.56) 12 6 3 s 1 2 2

In the present formulation we generalize this potential to a modiﬁed Skyrme interaction that replaces the linear dependence on the density by a power-law index σ. This modiﬁed Skyrme potential can then be written as [68]

0 1 r + r v(3) = t 1 + Pˆ δ (r − r ) ρσ 1 2 . (3.57) 12 6 3 s 1 2 2

This modification has been introduced [14] to increase the compressibility of nuclear matter at high densities. A value of σ = 1/3 is a common choice [58, 59]. However, in the present approach we choose to treat σ as a free parameter to be determined by constraining the third derivative of the energy per particle (e.g. skewness coefficient) from observed neutron-star properties [2, 31]. The main advantage of the Skyrme density functional is that the variables that characterize nuclear matter can be expressed as analytic functions. Also, its connection to the nuclear many-body Hamiltonian is an improvement over the liquid drop model [63] or the relativistic mean field approach [100, 101].

43 The isospin symmetric contribution is described by a Skyrme density functional with a modiﬁed three-body interaction term σ. We use TF to denote the kinetic energy of a particle at the Fermi surface

2 2 2/3 ~ 3π 2/3 TF = n (3.58) 2mB 2 where again, ~ is Planck’s constant, mB is the mass of a baryon, and n is the number density. In a similar fashion, we deﬁne TF 0 as TF when n = n0. Calculating the expectation value of the Hamiltonian [Eqn. (3.53)] in a Slater determinant and setting N=Z, the energy per nucleon for nuclear matter is [112] shown in Eqn. (3.59). All quantities and coeﬃcients for nuclear matter are obtained from the energy per particle [34]:

E 3 t = T H + 0 n[2(x + 2) − (2x + 1)H ] A 5 F 5/3 8 0 0 2 3 1 X + t nσi+1[2(x + 2) − (2x + 1)H ] 48 3i 3i 3i 2 i=1 3 3π2 2/3 + n5/3(aH + bH ) (3.59) 40 2 5/3 8/3 3 3π2 2/3 1 + n5/3+δ t (x + 2)H − t (x + )H 40 2 4 4 5/3 4 4 2 8/3 3 3π2 2/3 1 + n5/3+γ t (x + 2)H + t (x + )H 40 2 5 5 5/3 5 5 2 8/3 where

a = t1(x1 + 2) + t2(x2 + 2), (3.60) 1 b = [t (2x + 1) − t (2x + 1)], (3.61) 2 2 2 1 1 m−1 m m Hm(Yp) = 2 [Yp + (1 − Yp) ]. (3.62)

44 The variables t1, t2, t3i, t4, t5, x1, x2, x3i and σi are Skyrme parameters, varied to match experimental data. Typically standard parameterizations do not use t4, x4, t5, x5 which are used by [20], nor the summation over i in the term with t3. We also ignore these terms, and choose one of the standard models. From this expression for the energy per particle, we can calculate the properties describing increasingly higher derivatives of the energy: the pressure, volume compressibility of symmetric nuclear matter, and the skewness coeﬃcient. The relevant equations [95] are:

∂ E Pressure: P = n2 (3.63) ∂n A ∂2 E ∂P Compressibility: K = 9n2 = 9 (3.64) ∂n2 A ∂n ∂3 E Skewness Coeﬃcient: Q = 27n3 (3.65) ∂n3 A

These then obtain the following deﬁnitions, which with Eqn (3.59) completely describe the properties of nuclear matter:

2 3π2 2/3 t P = ~ n5/3H + 0 n2[2(x + 2) − (2x + 1)H ] 5m 2 5/3 8 0 0 2 3 1 X + t (σ + 1)nσi+2[2(x + 2) − (2x + 1)H ] 48 3i 1 3i 3i 2 i=1 1 3π2 2/3 + n8/3(aH + bH ) (3.66) 8 2 5/3 8/3 1 3π2 2/3 1 + (5 + 3δ)n8/3+δ t (x + 2)H − t (x + )H 40 2 4 4 5/3 4 4 2 8/3 1 3π2 2/3 1 + (5 + 3γ)n8/3+γ t (x + 2)H + t (x + )H 40 2 5 5 5/3 5 5 2 8/3

45 3 2 3π2 2/3 9t K = ~ n2/3H + 0 n[2(x + 2) − (2x + 1)H ] m 2 5/3 4 0 0 2 3 3 X + t (σ + 1)(σ + 2)nσi+1 [2(x + 2) − (2x + 1)H ] 16 3i 1 1 3i 3i 2 i=1 3π2 2/3 + 3 n5/3(aH + bH ) (3.67) 2 5/3 8/3 3 3π2 2/3 1 + (5 + 3δ)(8 + 3δ)n5/3+δ t (x + 2)H − t (x + )H 40 2 4 4 5/3 4 4 2 8/3 3 3π2 2/3 1 + (5 + 3γ)(8 + 3γ)n5/3+γ t (x + 2)H + t (x + )H 40 2 5 5 5/3 5 5 2 8/3 12 2 3π2 2/3 Q = ~ n2/3H 5m 2 5/3 3 9 X + t σ (σ + 1)(σ − 1)nσi+1 [2(x + 2) − (2x + 1)H ] 16 3i i i i 3i 3i 2 i=1 3 3π2 2/3 − n5/3(aH + bH ) (3.68) 4 2 5/3 8/3 3 3π2 2/3 1 + (2 + 3δ)(5 + 3δ)(3δ − 1)n5/3+δ t (x + 2)H − t (x + )H 40 2 4 4 5/3 4 4 2 8/3 3 3π2 2/3 1 + (2 + 3γ)(5 + 3γ)(3γ − 1)n5/3+γ t (x + 2)H + t (x + )H 40 2 5 5 5/3 5 5 2 8/3

There are two special cases of note: pure nuclear matter (Yp = 0) and symmetric matter (Yp = 1/2). Notice that for symmetric matter Hm = 1 for all values of m. The symmetry energy, S(n), is particularly important in modeling nuclear matter and ﬁnite nuclei because it probes the isospin part of the Skyrme interaction.

1 ∂2(E/A) S(n) = (3.69) 2 8 ∂YP n,Yp=1/2 1 t S(n) = T − 0 (2x + 1)n 3 F 8 0 3 2/3 1 X 1 3π2 − t nσi+1(2x + 1)nσi+1 + (a + 4b)n5/3 48 3i 3i 24 2 (3.70) i=1 1 3π2 2/3 1 3π2 2/3 − t x n5/3+δ + t (5x + 4)n5/3+γ 8 2 4 4 24 2 5 5

46 It is common to deﬁne four quantities, J = S(n0), L, Ksym, and Qsym, related to the symmetry energy and its

∂S L = 3n0 (3.71) ∂n n=n0 ∂2S 2 Ksym = 9n0 2 (3.72) ∂n n=n0 ∂3S 3 Qsym = 27n0 3 (3.73) ∂n n=n0

The density dependence of the symmetry energy (when Yp = 1/2) can be expanded as a function of x ≡ (n − n0)/3n0:

2 3 4 S = J + Lx + 12Ksymx + 16Qsymx + O(x ) . (3.74)

Several models exist with various Skyrme coeﬃcients achieved from ﬁts to nuclear data. I will employ a few representative models, as well as create a new parameter set using constraints from astrophysics, e.g. the observational constraint requiring the maximum mass of a neutron star to exceed 2.01 ±0.04 M [2, 31].

3.1.3.2 Thermal Contribution to Hadronic NDL EoS

To this point, we have not included any eﬀects from temperature in the equation of state. However, to use this EoS in hot environments like during supernovae, black hole formation, and neutron star mergers, we must specify a thermal contribution. The approach we take follows that of [116, 76]. We assume a degenerate gas of quasi-nucleons that includes the nucleon ground state and all 215 baryonic and mesonic excited states listed in the particle database [8] at the time of this paper, as a function of temperature T and baryon density n. Since the zero temperature contribution to the free energy is already properly taken into account by the Skyrme energy contributions, only the thermal portion needs to be added. We also assume

47 that the nucleons and the other particle resonances are in chemical equilibrium. To simplify the notation, we deﬁne h(p, α) as the usual Fermi distribution function:

1 h(pi, αi) = (3.75) exp[(i − αi)/T ] + 1 where

q m = p2 + m2 ; m = i , (3.76) i i e i e i 1 + 0.27η using me i for an effective particle mass deduced from fits to results from relativistic Bruckner Hartree-Fock theory [81]. We derive the expression for the thermal contribution first from the expression of the grand potential density [95]:

Z ∞ 4 Ω X gi p ω ≡ = − (h(p, α) + h(p, −α)) dp (3.77) V 6π2 i 0 i

Then expressing the Helmholtz free energy as F = αN + Ω, and subtracting oﬀ the zero temperature limit (denoted by 0 subscripts) which has already been accounted for in the deﬁnition of f1, we have the expression for the thermal contribution to the free energy per baryon:

1 f (n, T ) = α − α + (ω − ω ) . (3.78) 2 0 n 0

The zero-temperature limit of the grand potential density in Eqn. (3.77) is

" p 2 2 !# X gi q 5 3 α0 + α + m ω = − α α2 − m2 α2 − m2 + m4 ln 0 e i . (3.79) 0 2π2 0 0 e i 0 2 e i 2 e i m i e i

48 The α and α0 in Eqn. (3.78) are essentially chemical potentials, dependent on density and temperature, and only used for the thermal contribution. The chemical potentials are calculated from baryon number conservation:

Z ∞ X gi n = h(p , α ) − h(p , −α ) p2 dp (3.80) 2π2 i i i i i i i 0

X gi 2 23/2 −−−→ α0 − mi (3.81) T →0 2π2 e i

It is important to distinguish α from the true chemical potentials:

∂f ∂f µ = f + n − Y (3.82) n ∂n p ∂Y T,Yp p T,n ∂f ∂f µ = f + n + Y (3.83) p ∂n n ∂Y T,Yp p T,n obtained via Eqns. (3.43) - (3.44), using derivatives of the Helmholtz free energy given in Eqn. (3.51). Here again the charge fractions of neutrons and protons add to one: Yn + Yp = 1.

3.1.3.3 Pion Condensate

The impact on the hadronic EoS from the lightest mesons (i.e. the pions) has been constrained [77] from a comparison between relativistic heavy-ion collisions and one-ﬂuid nuclear collisions. The formation and evolution of the pions was computed in the context of Landau-Migdal theory [82] to determine the pion eﬀective energy and momentum. In this approach the pion energy is given by a dispersion relation [82]

2 2 2 π = pπ + me π , (3.84)

49 where me π is the pion “eﬀective mass” deﬁned to be

p me π = mπ 1 + Π (π, pπ, n) . (3.85)

Following [76] and [41] the polarization parameter Π can be written,

2 2 pπΛ (pπ) χ (π, pπ, n) Π(π, pπ, n) = 2 0 2 2 . (3.86) mπ − g mπΛ (pπ) χ (π, pπ, n) where the denominator is the Ericson-Ericson-Lorentz-Lorenz correction [36]. The

2 2 quantity Λ ≡ exp(−pπ/b ) with b = 7mπ, is a cutoﬀ that ensures that the dispersion relation [Eq. (3.84)] asymptotically approaches the high momentum limit,

q 2 2 ∞ ≡ (pπ → ∞, n) = m∆ + pπ − mN . (3.87)

Following [36] we take the polarizability to be

4a∞n χ (π, pπ, n) = − 2 2 , (3.88) ∞ − π

2 where a = 1.13/mπ. This form for the polarizability ensures that the eﬀective pion mass is always less than or equal to the vacuum rest mass mπ. A key quantity in the above expressions is the Landau parameter g0. This is an eﬀective nucleon-nucleon coupling strength. To ensure consistency with observed Gamow-Teller transition energies a constant value of g0 = 0.6 was used in [41]. How- ever, in [77], Monte-Carlo techniques were used to statistically average a momentum dependent g0 with particle distribution functions. It was found that g0 varies linearly with density and is approximately given by

0 g = g1 + g2η . (3.89)

50 A value of g1 = 0.5 was chosen to be consistent with known Gamow-Teller transitions.

A value for g2 was then obtained [77] by optimizing ﬁts to a range of pion multiplicity measurements obtained at the Bevlac [45]. These data were best ﬁt for a value of g2 = 0.06. The pions are assumed to be in chemical equilibrium with the surrounding nuclear matter. We consider the pion-nucleon reactions:

p ↔ n + π+ , n ↔ p + π− . (3.90)

This leads to the following relations among the chemical potentials for neutrons, protons, and pions

µp = µn + µπ+ , µn = µp + µπ− . (3.91)

These equilibrium conditions let us express the pion chemical potentials in terms of the neutron and proton chemical potentials:µ ˆ ≡ µn − µp = µπ− = −µπ+ . Using the deﬁnitions of µn and µp from Eqs. (3.82) - (3.83), the expressions for the pion chemical potentials are found to be

∂f µπ− = −µπ+ = − (3.92) ∂Yp T,n

For a given temperature (T ) and number density (n) the pion number densities are given by the standard Bose-Einstein integrals

Z ∞ p2 dp ni = , (3.93) 2 (π−µi)/T 0 2π e − 1

+ − 0 0 where i sums over {π , π , π }, and π is given by Eq. (3.84). Note that the π chemical potential is taken to be zero, since these particles can be created or destroyed without charge constraint. Following our standard notation, the charge fraction per baryon for the charged

51 pions is Yπ− = nπ− /n. From Eq. (3.92) we can calculate the pion number densities from the pion chemical potentials. Then, electric charge conservation gives,

Ye = Yp − Yπ− + Yπ+ . (3.94)

Thus, we can solve Eq. (3.94) for the unknown quantity Yp.

Once Yp is determined, the pionic energy densities and partial pressures can be calculated from Z ∞ 2 p dp π Ei = 2 , (3.95) 0 2π exp [(π − µi) /T ] − 1 and Z ∞ 2 p dp (1/3)p(∂π/∂p) Pi = 2 . (3.96) 0 2π exp [(π − µi) /T ] − 1

3.1.4 Baryons Above Saturation Density, Hadronic and Quark Matter

With the possibility of a phase transition to a Quark Gluon Plasma (QGP), we allow for both hadronic and quarkonic matter to exist simultaneously. A transition to a QGP phase during the collapse can have a significant impact on the dynamics and evolution of the nascent proto-neutron star. In [44] it was first shown that a first order phase transition to a deconfined QGP phase resulted in the formation of two distinct but quickly coalescing shock waves. More recently, it has been shown [38] that if the transition is first order, but global conservation laws are invoked, then the two shock waves can be time separated by as much as ∼ 150 ms. Neutrino light curves showing such temporally separated spikes might even be resolvable in modern terrestrial neutrino detectors [38].

52 stellar mass black holes. Hence at the very least, this transition to QGP may have an impact [84] on the neutrino signals during black hole formation as well as its possible impact on core-collapse supernovae. We ﬁrst specify the formulation of a pure QGP phase in the next section, 3.1.4.1, before addressing their coexistence phase in section 3.1.4.2.

3.1.4.1 Pure Quark Matter

For the description of quark matter we use the MIT bag model (see subsection 2.2.3 for a summary), and include eﬀects from quarks and gluons up to 2-loop. Whereas for hadronic matter, we used the Helmholtz free energy, F (T,V,N), as the starting point for calculating the needed state variables; for the quark-gluon plasma, it is more convenient to use the grand potential, Ω(T, V, µ). Both descriptions are P equivalent, and are related by a Legendre transform: Ω = F − i µiNi. The grand potential for the quark-gluon plasma takes the form:

X i i Ω = (Ωq0 + Ωq2) + Ωg0 + Ωg2 + BV. (3.97) i

th Here q0 and g0 denote the 0 -order bag model thermodynamic potentials for quarks and gluons, respectively, while q2 and g2 denote the 2-loop corrections. In most calculations suﬃcient accuracy is obtained by using ﬁxed current algebra masses

(e.g. mu ∼ md ∼ 0 GeV, ms ∼ 0.1 − 0.3 GeV). For this work we chose the strange quark mass to be ms = 150 MeV. We also adopt a value of αs = 0.33 for the strong coupling constant, as this is a representative value for the energy regime under consideration [8]. The quark contribution to the thermodynamic potential is given [78] in terms of a sum of the ideal gas contribution plus a two loop correction from phase-space

53 integrals over Feynman amplitudes [53]:

Z ∞ d3p i −β(Ei−µi) −β(Ei+µi) Ωq0 = − 2NcT 2 ln 1 + e + ln 1 + e (3.98) 0 (2π ) " Z ∞ 3 Z ∞ 3 3 0 i 2 1 d p Ni(p) d p d p 1 0 Ωq2 =αsπ Nc − 1 2 + 2 2 0 [Ni(p)Ni(p ) + 2] 3 0 (2π ) Ei(p) 0 (2π ) (2π ) Ei(p)Ei(p ) # N +(p)N +(p0) + N −(p)N −(p0) N +(p)N +(p0) + N −(p)N −(p0) × i i i i + i i i i , 0 2 0 2 0 2 0 2 (Ei(p) − Ei(p )) − (p − p ) (Ei(p) − Ei(p )) − (p − p ) (3.99)

± where the Ni denote Fermi-Dirac distributions:

± 1 Ni (p) = . (3.100) eβ(Ei(p)∓µi) + 1

In a similar fashion to that of the quarks, the one- and two-loop gluon and ghost contributions to the thermodynamic potentials can be evaluated:

Z ∞ 3 2 d p −β|p| Ωg0 =2(Nc − 1)T 2 ln 1 − e 0 (2π ) π = − N 2 − 1 T 4 (3.101) 45 c π Ω = α N N 2 − 1 T 4 (3.102) g2 36 s c c

For the massless quarks, Eqns. (3.98) & (3.99) are easily evaluated to give

N 7π2 µ4 Ωi = − c T 4 + µ2T 2 + i (3.103) q0 6 30 i 2π2 (N 2 − 1)α 5π2 µ4 Ωi = c s T 4 + µ2T 2 + i . (3.104) q2 8π 18 i 2π2

For the massive strange quark Eqn. (3.98) can be easily integrated. Eqn. (3.99), however, cannot be integrated numerically, due to the divergences inherent in it. We therefore, approximate [78] the two loop strange quark contribution with the zero

54 mass limit. This may over estimate the contribution due to a finite strong coupling constant, but given that the quark mass is relatively small compared to its chemical potential, this is a reasonable approximation. To determine a range of values for the MIT bag constant B, we turn to results from Lattice QCD and the early universe. At some point in the evolution of the early Universe there must have occurred a transition from quark-gluon plasma to confined hadronic matter. In the early universe when temperatures are above ∼ 1 GeV, quarks and gluons are unconfined. But, the temperature lowers as the universe expands, until a coexistence temperature TC where the quark-gluon plasma could exist in pressure and chemical equilibrium with a dense and hot gas of hadronic states (pions, neutrons, protons, deltas, etc.) where color charges are confined. Once the first generation of nucleated bubbles of hadronic phase appears, the release of latent heat from the QCD vacuum energy reheats the universe to TC , and further nucleation of hadronic phase is inhibited. The quark-gluon plasma phase and the confined, hadronic phase now coexist in pressure equilibrium. As the universe expands, the temperature is held at

TC , by the liberation of latent heat as the confined phase grows at the expense of the unconfined phase. This nearly-isothermal evolution may continue until all of the Universe has been converted to the confined phase [52]. One can find an approximate critical temperature for this early universe transition by equating the pressures in the hadronic phase (as a pion-nucleon gas) and the QGP phase (as a non-interacting relativistic gas of quarks and gluons) [43]:

901/4 T ≈ (g − g )−1/4 B1/4 , (3.105) C q h π2

where the statistical weight gq for a low-density high-temperature QGP gas with three relativistic quarks is gq ≈ 51.25, while gh ≈ 17.25 was found for the hadronic phase by summing over all known meson data.

55 Adopting the lattice gauge theory results (see subsection 2.2.4, as well as [60]) that 145 . Tc . 170 MeV then Eqn. (3.105) implies that a reasonable range for the QCD vacuum energy is

1/4 165 MeV . B . 240 MeV. (3.106)

This provides an initial range for the QCD vacuum energy. We will further constrain this parameter by requiring that the maximum mass of a neutron star exceed

2.01 ± 0.04 M [2, 31].

3.1.4.2 Mixed Hadronic and Quarkonic Matter

It is generally expected [78] that for sufficiently high densities and/or temperature, a transition from hadronic matter to quark-gluon plasma (QGP) can occur. Recent progress [60] in lattice gauge theory has shed new light on the transition to a QGP in the low baryo-chemical potential, high-temperature limit. It is now believed that at high temperature and low density a deconfinement and chiral symmetry restoration occur simultaneously at the crossover boundary. In particular, at low density and high temperature, it has been found [60] that the order parameters for deconfinement and chiral symmetry restoration changes abruptly for temperatures of T = 145 − 170 MeV [12, 7]. However, neither order parameter exhibits the characteristic change expected from a first order phase transition. An analysis of many [3, 6] thermodynamic observables confirms that the transition from a hadron phase to a high temperature QGP is a smooth crossover. At lower temperatures, like those in supernovae or neutron stars, the transition is first order, involving a coexistence phase, which we discuss here. We use a Gibbs construction [17] for the description of a first order phase transition. In this case, the two phases are in equilibrium when the chemical potentials,

56 temperatures and the pressures are equal. Throughout the mixed phase regime, all thermodynamic quantities vary in proportion to the volume fraction of the two phases (superscripts Q and H will refer to the quark and hadronic phases, respectively): V Q χ ≡ . (3.107) V Q + V H

For the description of the phase transition from hadrons to quarks this construction can be written

µp = 2µu + µd (3.108)

µn = 2µd + µu (3.109)

µd = µs (3.110)

TH = TQ (3.111)

H H Q Q P T,Ye, {µi } = P T,Ye, {µi } , (3.112)

H Q where {µi } ≡ {µn, µp, µe, µν} and {µi } ≡ {µu, µd, µs, µe, µν}. The Gibbs construction ensures that a uniform background of photons and leptons exists within the diﬀering phases. Therefore, the contribution from the photon, neutrino, electron, and other lepton pressures cancel out in phase equilibrium. Also from this, we ﬁnd that the two conserved quantities vary linearly in proportion to the degree of completion of the phase transition, i.e.

H H Q Q nBYe = (1 − χ) nB Yc + χnBYc (3.113)

H Q nB = (1 − χ) nB + χnB, (3.114)

H Q Q where we have deﬁned Yc = Yp + Yπ+ − Yπ− and nBYc = 1/3 (2nu − nd − ns). The internal energy and entropy densities likewise vary in proportion to the degree of

57 phase transition completion

= (1 − χ) H + χQ (3.115)

s = (1 − χ) sH + χsQ. (3.116)

3.2 Constraints on the EoS

Experimental and observational evidence can be used to constrain the parameter space of the defined Equation of State. At low density, nuclear data places constraints on the Skyrme parameters of the zero-temperature baryonic contribution to the free energy. But since this EoS is to be used in astrophysical situations, it also must meet the rather rigid constraint that it produce neutron stars as heavy as two solar masses [2, 31]. To this end, we will find a set of Skyrme parameters (NDL set 1) which meet both criteria, as well as use the sixteen models that satisfy the first constraints.

3.2.1 Nuclear Experimental Constraints

We use the notation of [34] to label the four constraints from symmetric nuclear matter;

SM1: K0 = 230 ± 30 MeV, (3.117)

SM2: K = 700 ± 500MeV, (3.118) 1 SM3: P n, Y = matches particle ﬂow data [27], (3.119) p 2 1 SM4: P n, Yp = matches kaon [67], 2 (3.120) and giant monopole resonance data [42]

58 two constraints of pure (Yp = 0) nuclear matter;

PNM1: E n, Yp = 0 is within the bands from [35], (3.121) PNM2: P n, Yp = 0 matches particle ﬂow data in [27], (3.122)

1 and ﬁve more constraints from both symmetric (Yp = 2 ) and pure (Yp = 0) nuclear matter. But ﬁrst we introduce some new variables relating to the symmetry energy (S(n)) and its derivatives evaluated at saturation density (n0):

J ≡ S(n0) (3.123) ∂S L ≡ 3n0 (3.124) ∂n n=n0 2 2 ∂ S Ksym ≡ 9n (3.125) 0 ∂n2 n=n0 3 3 ∂ S Qsym ≡ 27n (3.126) 0 ∂n3 n=n0

With these deﬁnitions, the symmetry energy can be expanded as a function of x =

(n − n0)/3n: 1 1 S = J + Lx + K x2 + Q x3 + .... (3.127) 2 sym 6 sym

The energy for symmetric nuclear matter (Yp = 1/2) is often times also expanded in x: 1 1 E(x, Y = 1/2) = E + K x2 + Q x3 + ... (3.128) p 0 2 0 6 0

59 where

1 E ≡ E(n , Y p + ) (3.129) 0 0 2 2 1 2 ∂ E(n, Yp = 2 ) K0 ≡ 9n (3.130) 0 ∂n2 n=n0 3 1 3 ∂ E(n, Yp = 2 ) Q0 ≡ 27n . (3.131) 0 ∂n3 n=n0

Finally, we deﬁne the volume term (Kτ,ν) of the isospin incompressibility of asymmetric nuclear matter, which plays an important role in analysis of data from giant monopole resonances and can further constrain our Skyrme parameter space. We now present the remaining constraints on the Skyrme parameters:

MIX1: 30 < J < 35 MeV (3.132)

MIX2: L = 58 ± 18 MeV (3.133)

MIX3: − 760 ≤ Kτ,ν ≤ −372 MeV (3.134) S(n /2) MIX4: 0.57 < 0 < 0.83 (3.135) J 3P (n ,Y = 0) MIX5: 0 p = 1 ± 0.1 . (3.136) Ln0

When these constraints are applied to 240 diﬀerent Skyrme interaction parameter sets, only 16 satisfy all these constraints. These are listed in Tables (3.2) - (3.3).

3.2.2 Observation of 2 M Neutron Stars

In 2010, Demorest et. al. [31] measured the mass of a neutron star and its binary companion to high precision. The pulsar of J1614–2230’s implied mass of

1.97 ± 0.04 M is signiﬁcantly higher than the typical value of 1.4 M for neutron stars, and provides a very stringent restraint on allowed EoSs for neutron stars.

In 2013, Antoniadis et. al. [2] measured an even heavier pulsar of 2.01 ± 0.04 M .

60 These measurements of the highest neutron star mass measured with such certainty, eﬀectively rules out the presence of hyperons, bosons, or free quarks at densities comparable to the nuclear saturation density [91]. It provides a strict bound for predictions of an equation of state, especially those containing quark matter.

3.2.3 Successful Parameter Sets

Dutra et. al. in [34] reviewed and analyzed 240 Skyrme interaction parameter sets, and find only 16 sets which satisfy the constraints listed in section 3.2.1. The various criteria are derived from macroscopic properties of nuclear matter in the vicinity of nuclear saturation density at zero temperature as well as from experiments with giant resonances and heavy-ion collisions. These parameter sets, reproduced here in Tables (3.2) - (3.4), can be used in the Skyrme energy definition of Eqn. (3.59) to extend the fits of terrestrial experiments to predictions of astrophysical phenomena. Conversely, we can develop our own Skyrme parameter set, utilizing the unique vantage of having astrophysical observables to bear on the predictions of the parameter set. That is to say, the usual approach is to choose a set of data, e.g. resonances, nuclear masses, charge radii, etc., to find a best set of parameters for a Skyrme

model. From these Skyrme coeﬃcients then, the quantities n0, TF0 , E0, K0, and Q0 in Eqs. (3.137) - (3.140) are deduced. For our purposes, however, we choose a more empirical approach: we adopt inferred values of n0, E0, K0, Q0 from the literature (see Table 3.1) and use these to determine the Skyrme model parameters. We also demand that these parameters allow neutron star masses greater than 2.01 ± 0.04 M [2, 31]. In this way our EoS relates directly to inferred properties of nuclear matter in the literature and is easily adaptable to improved experimental and theoretical determinations.

To make such a parameter set, which I will refer to as the NDL ﬁducial set, we start

61 TABLE 3.1

ADOPTED CONSTRAINTS

Parameter value ref.

−3 n0 0.16 ± 0.01 fm [66]

E0 -16 ± 1 MeV [66]

K0 240 ± 10 MeV [24]

Q0 -390 ± 90 MeV [79, 72, 73, 87]

Adopted constraints on properties of nuclear matter.

with symmetric matter (remember Hm(Yp = 1/2) = 1) at saturation density (n = n0):

3 3 3 3 E = T + n t + nσ+1t + cn5/3(3t + 5t ) (3.137) 0 5 F 0 8 0 0 48 0 3 80 0 1 2 2 P0 ~ 3 3 σ+1 1 5/3 0 = = c + n0t0 + (σ + 1)n0 t3 + cn0 (3t1 + 5t2) (3.138) n0 5m 8 48 16 3 2 27 9 3 K = ~ cn2/3 + n t + (σ + 1)(σ + 2)nσ+1t + cn5/3(3t + 5t ) (3.139) 0 m 0 4 0 0 16 0 3 2 0 1 2 12 2 36 3 Q = ~ cn2/3 + σ(σ + 1)(σ − 1)nσ+1t − cn5/3(3t + 5t ), (3.140) 0 5m 0 16 0 3 8 0 1 2 where I’ve deﬁned c ≡ (3π2/2)2/3 to simplify notation. In an eﬀort to generate a simple model, so we have only assumed one set of t3 and one σ (i.e. t32 = t33 = σ2 =

σ3 = 0) and not used the additional t4, x4, t5, x5 parameters. For x0, x1, x2, and x3 we chose the average of the 16 parameter sets that satisfy all the nuclear constraints.

This leaves a system of four equations in terms of the quantities t0, (3t1 + 5t2), t3, and σ. Solving Eqs. (3.137) - (3.140) for σ then yields

9 5 TF0 − 2K0 + Q0 − 45E0 σ = 27 . (3.141) 3K0 + 45E0 − 5 TF0

The expressions for the remaining variables are more complicated, so here we just

62 present their numerical values, using the central values from Table 3.1:

3 t0 = −1653.28 MeV fm (3.142)

5 (3t1 + 5t2) = −947.775 MeV fm (3.143)

3σ+3 t3 = 15448.1 MeV fm (3.144)

σ = 0.422 . (3.145)

The full set of Skyrme parameters for the NDL ﬁducial model is found in Ta- ble (3.4).

The next section evaluates the eﬀectiveness of describing heavy neutron stars of these models. It also presents other predictions of the EoS, such as the eﬀects of pions and a quark-gluon matter phase.

63 TABLE 3.2

SKYRME PARAMETER SETS

Skyrme parameter GSkI GSkII KDE0v1 LNS MSL0 NRAPR

t0 -1855.5 -1856.0 -2553.1 -2485.0 -2118.1 -2719.7

t1 397.2 393.1 411.7 266.7 395.2 417.6

t2 264.6 266.1 -419.9 -337.1 -64.0 -66.7

t31 13858.0 13842.9 14603.6 14588.2 12875.7 15042.0

t32 -2694.1 -2689.7 – – – –

t33 -319.9 – – – – –

x0 0.12 0.09 0.65 0.06 -0.07 0.16

x1 -1.76 -0.72 -0.35 0.66 -0.33 -0.05

x2 -1.81 -1.84 -0.93 -0.95 1.36 0.03

x31 0.13 -0.10 0.95 -0.03 -0.23 0.14

x32 -1.19 -0.35 – – – –

x33 -0.46 – – – – –

σ1 0.33 0.33 0.17 0.17 0.24 0.14

σ2 0.67 0.67 – – – –

σ3 1.00 – – – – –

A list of Skyrme parameters that satisfy the constraints listed in the [34] review 3 paper. All parameterizations have t4 = x4 = 0 and t5 = x5 = 0. t0 is in [MeV fm ], 5 3+3σi t1 and t2 are in [MeV fm ], and t3i is in [MeV fm ]. The x0, x1, x2, x3i, and σi are all dimensionless. Section 4.4.2 speciﬁes which of the Skyrme models predict a neutron star ≥ 2M , satisfying the [2, 31] observation.

64 TABLE 3.3

SKYRME PARAMETER SETS

Skyrme Parameter Ska25s20 Ska35s20 SKRA SkT1 SkT2 SkT3

t0 -2180.5 -1768.8 -2895.4 -1794.0 -1791.6 -1791.8

t1 281.5 263.9 405.5 298.0 300.0 298.5

t2 -160.4 -158.3 -89.1 -298.0 -300.0 -99.5

t31 14577.8 12904.8 16660.0 12812.0 12792.0 12794.0

t32 ––––––

t33 ––––––

x0 0.14 0.13 0.08 0.15 0.15 0.14

x1 -0.80 -0.80 0.00 -0.50 -0.50 -1.00

x2 0.00 0.00 0.20 -0.50 -0.50 1.00

x31 0.06 0.01 0.00 0.09 0.09 0.08

x32 ––––––

x33 ––––––

σ1 0.25 0.35 0.14 0.33 0.33 0.33

σ2 ––––––

σ3 ––––––

65 TABLE 3.4

SKYRME PARAMETER SETS

Skyrme Parameter Skxs20 SQMC650 SQMC700 SV-sym32 NDL ﬁducial

t0 -2885.2 -2462.7 -2429.1 -1883.3 -1653.28

t1 302.7 436.1 371.0 319.2 350

t2 -323.4 -151.9 -96.7 197.3 -399.555

t31 18237.5 14154.5 13773.6 12559.5 15448.1

t32 –––––

t33 –––––

x0 0.14 0.13 0.10 0.01 0.136

x1 -0.26 0.00 0.00 -0.59 -0.44

x2 -0.61 0.00 0.00 -2.17 -0.42

x31 0.05 0.00 0.00 -0.31 0.06

x32 –––––

x33 –––––

σ1 0.17 0.17 0.16 0.30 0.422

σ2 –––––

σ3 –––––

66 CHAPTER 4

PREDICTIONS OF THE NDL EQUATION OF STATE

In this section, I highlight features of the Notre Dame-Livermore Equation of State (NDL EoS), such as the eﬀects of including pionic matter and quarkonic matter with a mixed phase.

4.1 Comparison to Existing Equations of State

We begin by comparing the properties of the new NDL EoS with the two most commonly employed equations of state used in astrophysical collapse simulations – those of Lattimer & Swesty [63] – as well as the original EoS of Bowers & Wilson [13]. Fig. 4.1 shows the total internal energy as a function of baryon density. We compare the Lattimer & Swesty EoS [63] at two diﬀering compressibilities (K0 = 180 MeV and

K0 = 220 MeV) with the ﬁducial NDL EoS at a ﬁxed electron fraction of Ye = 0.3 and temperature of T = 10 MeV. A steep rise in the energy per baryon at high densities occurs for larger values of the compressibility as expected. Similarly, Fig. 4.2 depicts the pressure vs. density for the Shen EoS [102], the Lattimer & Swesty EoS [63] and the NDL EoS. The Shen EoS consistently leads to higher pressure. This results from the use of the TM1 parameter set that contains relatively high values for both the symmetry energy at saturation and the nuclear compressibility typical of RMF approaches.

67 1500 NDL EoS LS180 LS220 1000

500 Energy per Particle [MeV]

0 0 0.5 1 1.5 2 -3 Baryon Number Density [fm ]

Figure 4.1. The energy per particle as a function of density comparing the Lattimer & 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.

4.2 Pion Eﬀects on the NDL EoS

The solution to the pion dispersion relation, Eq. (3.84), does not produce conditions within the supernova core to generate a pion condensate. The pions con- sidered here are thermal pionic excitations calculated from the pion propagator in Eq. (3.86). In very hot and dense nuclear matter the number density of pionic excitations is greatly enhanced by the πN∆ coupling [41]. Hence, it becomes energetically favorable to form pions in the nuclear ﬂuid when the chemical balance shifts from electrons to negative pions. This allows the charge states to equilibrate with these newly formed bosons. Since the pions are assumed to be in chemical equilibrium

68 500

L&S K0 = 180 MeV L&S K0 = 220 MeV

] 400 Shen et al. -3 NDL EoS 300

200

Pressure [MeV fm 100

0 0 0.2 0.4 0.6 0.8 -3 Baryon Number Density [fm ]

Figure 4.2. The pressure as a function of density comparing two EoSs from Lattimer & 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. with the surrounding nuclear fluid, this has a profound effect on the proton fraction within the medium, particularly for low electron fractions, (see Fig. 4.3). The pion charge fraction as a function of baryon number density is shown in

Fig. 4.4. For a low ﬁxed Ye the charge fraction of negative pions can actually become greater than the electron fraction, and the negative pions essentially replace the electrons in equilibrating the charge. From the solution to the pion chemical potentials [Eq. (3.92)], one ﬁnds that as the density increases, negative pions are created due to the chemical potential

69 0.4

0.3 p Y

Ye = 0.1

0.2 Ye = 0.2

Ye = 0.3

Ye = 0.4

0.1 0.2 0.4 0.6 0.8 -3 Baryon Number Density [fm ]

Figure 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. constraints and the dispersion relation [Eq. (3.84)]. At the same time the number density of positively charged pions remains negligible due to the fact that it has a negative chemical potential. Due to its dependence on both the symmetry energy and the isospin asymmetry parameter I = (1 − 2Yp), the pion chemical potential [Eq. (3.92)] increases linearly with respect to density but decreases linearly with respect to Yp. Therefore, for high electron fractions the pion chemical potential will remain small and charge equilibrium can be maintained solely among the electrons and protons.

70 0.3

Yp 0.2 Ye

Charge Fractions 0.1 − Yπ

0 0.2 0.4 0.6 0.8 -3 Baryon Number Density [fm ]

Figure 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.

It should be noted, however, that as treated here, pions would not exist in the ground state conﬁguration of a cold neutron star. As the temperature approaches zero the pionic eﬀects diminish, until only the nucleon EoS contributes to the neutron star structure. In the hot dense medium of supernovae, however, these pions tend to soften the hadronic EoS since they relieve some of the degeneracy pressure due to the electrons. We have found that the reduction in pressure is relatively insensitive to the temperature of the medium and is lowered by approximately 10% for all representative temperatures found in the supernova environment, as shown in Fig. 4.5.

71 NDL EoS w/o Pions 300 NDL EoS w Pions ] -3

200

100 Pressure [MeV fm

0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -3 Baryon Number Density [fm ]

Figure 4.5. Pressure versus baryon number density showing a 10% reduction in the pressure at high densities due to the presence of pions.

This will aﬀect SN core collapse models since it allows collapse to higher densities and temperatures without violating the requirement that the maximum neutron star mass exceed 2.01 ± 0.04 M for cold neutron stars.

4.3 Mixed Phase of Hadrons and Quark-Gluon Plasma

The constraint of global charge neutrality exploits the isospin restoring force ex- perienced by the conﬁned hadronic matter phase. This portion of the mixed phase becomes more isospin symmetric than the pure phase because charge is transferred from the quark phase in equilibrium with it.

72 Fig. 4.6 shows the charge fractions of the mixed phase and hadronic phases. From this we see that the internal mixed phase region of a hot, proto-neutron star contains a positively charged region of nuclear matter and negatively charged regions of quark

−3 matter until a density of n0 ∼ 1.0 fm . The presence of the isospin restoring force causes the thermal pionic contribution to the state variables to be negligible. This is due to the dependence of the pion chemical potential on the isospin asymmetry parameter (1 − 2Yp). As the hadronic phase becomes more isospin symmetric, the pion chemical potential remains small compared to its eﬀective mass. Since stars contain two conserved quantities, electric charge and baryon number, the coexistence region cannot be treated as a single substance, but must be evolved as a complex multi-component ﬂuid. It is common in nature to have global conservation laws and not necessarily locally conserved quantities. Hence, within the Gibbs construction the pressure is a monotonically increasing function of density.

Fig. 4.7 shows pressure versus density for various values of Ye, through the mixed phase region into a phase of pure QGP. One of the features shown is that as the density increases through the mixed region the slope of the pressure decreases slightly. This becomes more evident when the adiabatic index, Γ, is analyzed as a function of density as shown in Fig. 4.8. Here, we find that the EoS softens abruptly upon entering the mixed phase due to the fact that increasing density leads to more QGP rather than an increase in pressure. If this occurs while forming a proto-neutron star, its evolution will be affected as Γ falls below the stability point of Γ < 4/3 for n ∼ 1.1 fm−3. The collapse simulations of [38, 44] show that as Γ falls below 4/3, a secondary core collapse ensues. The matter sharply stiffens upon entering the pure quark phase at n ∼ 1.4 fm−3, and a secondary shock wave is generated. As this shock catches up to the initially stalled accretion shock, a more robust explosion ensues.

Another feature seen in Fig. 4.7 is the Ye dependence of the onset density of the

73 1/4 αs = 0; B = 165 MeV; T = 20 MeV; Ye = 0.3 0.6

0.4

χ = 0.9 0.2 χ = 0.7

χ = 0.5 0 χ = 0.3 Charge Fractions Yp -0.2 χ= 0.1 Yq

-0.4 0.2 0.4 0.6 0.8 1 -3 Baryon Number Density [fm ]

Figure 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. mixed phase. Fig. 4.9 shows a phase diagram indicating the mixed phase transition temperature as a function of density for two values of Ye. For higher temperatures the onset happens at lower densities as would be expected. However, for high electron fractions (Ye ∼ 0.3) such as those that can be found deep inside the cores of a proto-

−3 neutron star, the transition density remains quite high nc ∼ 0.6 fm . It is also of note that the coexistence region slightly decreases as the electron fraction is increased.

74 300

Ye = 0.1 250 Ye = 0.2 ]

-3 Ye = 0.3 200

150

100 Pressure [MeV fm 50

0 0 0.2 0.4 0.6 0.8 1 -3 Baryon Number Density [fm ]

Figure 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.

4.4 Neutron Stars

As stated previously, the observation of a 2.01 ± 0.04 M neutron star [2] has ruled out many neutron star models [62]. Equations of state which are “soft” (e.g., the (in)compressibility, K, from Eqn. (3.64) is low) are favored for supernova collapse because the shockwave after bounce has more energy. However, to predict a neutron star as heavy as two solar masses, “harder” (high K) equations of state are better to support more mass. It becomes increasingly diﬃcult to ﬁnd models that produce both a successful supernova explosion and a heavy neutron star.

75 3 T = 10 MeV T = 15 MeV T = 20 MeV

Γ 2.5

Adiabatic Index 1.5

1 0.2 0.4 0.6 0.8 1 -3 Baryon Number Density [fm ]

Figure 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.

We can infer some properties; such as mass, spin rate, magnetic ﬁeld, age; from observations of pulsars. It is diﬃcult to measure both mass and radius simultaneously, although some progress is being made [9, 106]. The precision of the 2 M measurements of [2, 31] is what makes the maximum mass constraint so restrictive. The masses and radii of neutron stars are determined from the equation of state of cold dense matter using the Tolman-Oppenheimer-Volkov (TOV) relativistic stellar structure equations described in the next section.

76 80

70 Ye = 0.1 Y = 0.3 60 e

50 Pure Quark 40 Mixed Phase 30

Temperature [MeV] 20 Pure Hadronic 10

0 0 0.5 1 1.5 2 -3 Baryon Number Density [fm ]

Figure 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 temperatures it is seen that the onset density is highly Ye dependent.

4.4.1 Tolman-Oppenheimer-Volkoﬀ Equation

To predict the global structure of neutron stars the only relevant part of the equation of state is the relationship between pressure and density. This is because, just a few tens of seconds after birth, neutrons stars are cold (the temperature is much less than the Fermi energy) and deleptonized (meaning it is in beta equilibrium with no trapped neutrinos). As a result, for a given EoS, the mass and radius of the star depend only on the central density [106].

77 2.5

Antoniadis (2013) 2

1/4 B = 165 MeV 1/4 B = 180 MeV ☉1.5 1/4 B = 200 MeV 1/4 B = 220 MeV M/M 1

0.5

0 10 15 20 Radius [km]

Figure 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.

To extract the pressure vs density relationship from the EoS, we presume beta equilibrium. That is, the β-decay process

n → p + e +ν ¯ (4.1) does not occur, so that

µˆ = µn − µp = µe . (4.2)

The value of the proton fraction Yp that satisﬁes this constraint is consequently

78 also the minimum of the energy per baryon, with respect to Yp, referring back to

Eqns. (3.43) and (3.44). Notice that because of charge conservation, changing Yp alters the electron chemical potential µe(n, Ye). For purely hadronic matter with no pions, Ye = Yp; the addition of pions modiﬁes the charge conservation equation to

Ye = Yp + Yπ+ − Yπ− . In a similar manner, charge conservation in the mixed quark phase alters the electron chemical potential. The Tolman-Oppenheimer-Volkoﬀ (TOV) equation itself is a general relativistic equation describing hydrostatic equilibrium of an isotropic material, given as

dp(r) G ρ(r)c2 + p(r)M(r)c2 + 4πr3p(r) = − (4.3) dr c4r 2GM(r) 1 − c2r where p is the pressure, r is the radius, G is the gravitational constant, c is the speed of light, ρ is the matter density, and M is the enclosed gravitational mass.

1 The predicted neutron star masses are found by varying the central density ρc and integrating the TOV equation (4.3), with the mass conservation equation

dM(r) = 4πρ(r)r2 , (4.4) dr

from the core (r = 0, M = 0, ρ = ρc and p = pc) to the surface (r = Rtot, M = Mtot, ρ = 0 and p = 0). The maximum mass must reach two solar masses for the equation of state to meet the observation of [2, 31].

4.4.2 Maximum Mass of Neutron Star

Using the TOV equation to ﬁnd predicted masses, we see in Figure (4.11) that the Bowers & Wilson and LS180 equations of state are ruled out. The higher com-

1Equivalently, one could instead vary the central pressure, integrating to the surface where the pressure drops to zero.

79 pressibility Lattimer & Swesty LS220 model does, however, clear the 2.01 ± 0.04 M astrophysical constraint, as do the Shen and the NDL EoS.

Figure (4.12) shows the same neutron star mass predictions for the sixteen Skyrme parameter sets that are not ruled out by the nuclear constraints listed in Section 3.2.1.

Seven of the original sixteen Skyrme models are ruled out by the 2.01 ± 0.04 M restriction: GSkII, LNS, SKRA, Skxs20, SQMC650, SQMC700, and SV-sym32. Thus we can further restrict our models using astrophysical constraints.

80 3

2.5

Antoniadis (2013) 2 NDL EoS ☉ Shen EoS 1/4 NDL EoS B = 220 MeV 1.5 LS180 EoS LS220 EoS M/M Simple QGP Crossover 1 Bowers & Wilson

0.5

0 10 15 20 25 30 Radius [km]

Figure 4.11. Mass-radius relation for the Shen (longer dashed line), Lattimer & Swesty 180 & 220 (dash-dotted line and shorter dashed line), Bowers & Wilson 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 astrophysical constraint, with the exception of the Bowers & Wilson and LS180 equations of state.

81 2.5

1: GSkI Antoniadis (2013) 2: GSkII 2 3: KDE0v1 4: LNS 5: MSL0 6: NRAPR 7: Ska25s20 1.5 8: Ska35s20 ☉ 9: SKRA 10: SkT1 11: SkT2

M/M 12: SkT3 1 13: Skxs20 14: SQMC650 15: SQMC700 16: SV-sym32 0.5

0 8 10 12 14 16 Radius [km]

Figure 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 astrophysical constraint [2] is drawn in for reference.

82 CHAPTER 5

SUPERNOVA SIMULATION

5.1 Summary of the Core Collapse Process

During the lifetime of a star, gravitational collapse is prevented using the energy from nuclear fusion. Massive stars between ∼ 10 M to 30 M evolve until the formation of an iron core. At this point, no more energy can be gained from fusion, and support comes from electron degeneracy pressure until the core exceeds the

Chandrasekhar mass limit of ∼ 1.3 − 1.4 M [22, 21]. Neutrino emission, electron capture and photo-disintegration cool the inner ∼ 1 M , and remove pressure support from the core, causing the central density to rise.

9 −3 When the central density ρc approaches ρc & 10 g cm , neutrino emission is so large that collapse can exceed the speed of sound. As the core collapses, the inner

∼ 0.7 M collapses homologously [5, 25]. Once the core density exceeds nuclear

14 −3 density, ρc > 2.5×10 g cm , pressure rises rapidly and collapse is halted. However, matter continues to fall inward so an outward moving shock wave is formed, called the core bounce [116]. Photodissociation removes most of the energy from this shock, though; tearing apart nuclei in the remainder of the iron core ﬁrst, and then elements with lower atomic number above the core. Behind the shock, the material is composed of nucleons and alpha particles with an entropy per baryon of s ≈ 10 kB. This disintegration of heavy nuclei removes kinetic energy from the shock. If the shock breaks through with enough remaining kinetic energy to unbind the outer parts of the star, the explosion would occur by the “prompt” supernova mechanism [5, 25]. In most calculations,

83 however, the shock slows before it reaches the outer boundary of the iron core and becomes an accretion shock (moves outward in mass but not in radius). After just ≈ 0.1 s there is a proto-neutron star of ∼ 500 km in radius surrounded by hot gas, extending out to the shockwave at ∼ 500 km. Beyond that matter is falling inward at roughly sonic speed. The inner core is cooled by radiating energetic neutrinos, which release gravitational binding energy from the accreting material. The hot gas region is both being heated by the neutrinos from the hot proto-neutron star and is cooling by neutrino emission via

− p + e → n + νe (5.1)

+ n + e → p +ν ¯e , (5.2) and a small contribution from

− + e + e → νe +ν ¯e . (5.3)

The rate of change of the internal energy, E, then is the diﬀerence between heating from neutrino absorption and cooling from neutrino emission, and is approximately

T 6 R E˙ = acσ ν ν − T 6 , (5.4) 0 4 r

where a is, c is the sound speed, σ0 is the average zero-temp absorption cross-section for the processes in Eqns. (5.1) - (5.3), Rν is the radius of the neutrinosphere, and T is the temperature. As energy is deposited, the temperature rises and the entropy increases, leading to the production of numerous electron-positron pairs which can themselves scatter neutrinos, resulting in yet more energy deposition. Material behind the shock accretes back onto the newly formed proto-neutron star creating an evacuated region. The entropy per baryon rises as the bubble is further evac-

84 uated [117, 75, 76]. Although this region behind the shock is at low density, the entropy is so high that tremendous pressure builds up and revives the slowed shock’s outward motion. If the revived shock manages to break through the remainder of the initial core, a “delayed” supernova explosion occurs. The region behind the shock has high entropy and provides a tantalizing site [80, 121] for the production of heavy nuclei via the r-process (rapid neutron capture) [71], so long as the entropy is high. In the model of Mayle & Wilson [117, 75, 76] the entropy per baryon reaches s ≈ 400 kB, while some approximate “wind” models

(e.g. [93, 89]) lead to much lower entropy (s ∼ 100 kB). The supernova mechanism is very ineﬃcient; the binding energy of the neutron star is ∼ 3×1053 erg while the kinetic energy of the resulting explosion is only ∼ 1051 erg. It is important to test the simulation with a variety of Equations of State and progenitor models [85], to understand the extent to which uncertainties in the EoS can propagate to quantities like neutrino luminosity and explosion energy.

5.2 Numerical Routines for Simulation of Core-Collapse Supernovae

To simulate the core-collapse supernova we use the model developed by Mayle and Wilson [117, 75, 76], and the computer program of the Livermore group [118]. I refer the reader to Wilson and Mathews’ textbook [116] for an introduction to the subject and more in-depth description of modeling the collapse mechanisms. I summarize here highlights of the numerical routines used to model supernovae.

5.2.1 The Metric

A fully general relativistic description does not distinguish between space and time dimension. To be able to describe the star’s evolution in time, the model uses the ADM or (3+1) formalism. Time evolution of the metric is expressed as ﬁrst-order time derivatives, while the Einstein tensor Gµν contains second-order time derivatives.

85 The hydrodynamics are Lagrangian in nature, but the metric is general relativistic (and spherically symmetric) following May and White [74]:

ds2 = −a2dt2 + b2dm2 + R2(dθ2 + sin2 θdφ2) . (5.5)

Here m is a Lagrangian rest mass, and the metric coeﬃcients a, b, and R are functions of the rest mass m and time t. Angular coordinates are standard spherical coordinates on the unit sphere. The state variables are radial position of a mass zone (R), the radial component of the ﬂuid 4-velocity (U), the matter internal energy (M ), the net charge per baryon (Ye), and six energy distribution functions for

the neutrinos, (Fνe ,Fν¯e ,Fνµ ,Fν¯µ ,Fντ ,Fν¯τ ). The metric coeﬃcient a is related to the gravitational redshift and is given by a ≡ 1/U t where U t is the zero component of the four-velocity. The coeﬃcient b is inversely proportional to the square of the mass radius and the matter density: 1 b = . (5.6) 4πR2ρ

During each cycle, the state variables are remapped onto an updated grid to maintain good zoning [116].

5.2.2 Energy Momentum Tensor

Neutrinos occupy a unique role, compared to the rest of matter, so their contribution to the internal energy and pressure, respectively, is tracked separately:

E = ρ = ρM + Eν (5.7)

P = PM + Pν . (5.8)

The neutrinos are not necessarily in thermodynamic equilibrium near the neutrinosphere, so one must explicitly evolve distribution functions of neutrino energy and

86 3 angle Fi(E, θ) = fi(E, θ)E . The average energy Eν, pressure Pν and neutrino ﬂux Φν are then deﬁned by energy and angular integrations over this distribution function:

6 X Z Eν = Fi dEdΩν (5.9) i=1 6 X Z Φν = Ficos(θ) dEdΩν (5.10) i=1 6 Z X 2 Pν = Ficos (θ) dEdΩν , (5.11) i=1 where Ων = 2π sin θdθ is the neutrino solid angle. With these deﬁnitions, the energy-momentum tensor becomes

 2  ρ(1+) Φν 4πR ρ 2 0 0  a a   Φ 4πR2ρ   ν P (4πR2ρ)2 0 0  µν  a  T =   (5.12)  0 0 P +Wν 0   R2    P +Wν 0 0 0 R2 sin2 theta where we’ve introduced an auxiliary pressure correction variable:

E − 3P W = ν ν . (5.13) ν 2

5.2.3 Matter Evolution

The evolution of matter is given by:

Γ ∂R −1 ρ = (5.14) 4πR2 ∂m 1 ∂ρ 1 ∂ 1 = −ρ (R2U) + ρRΦ (5.15) a ∂t R2 ∂R 2Γ ν where we’ve deﬁned 2M 1/2 Γ ≡ 1 = U 2 − . (5.16) R

87 The matter internal energy evolves as:

6 Z 1 ∂M 1 ∂ 1 1 X = −P − Λ dEdΩ , (5.17) a ∂t M a ∂t ρ ρ i ν i=1 where PM is the matter pressure and the Λi are neutrino source terms and sink terms. The hydrodynamic equations are solved using a Lagrangian formulation for

i i the case of spherical symmetry and Vg = V . The speed of sound is very high in the developing proto-neutron star. This greatly restricts the time step, which must be chosen such that the partial derivatives with respect to time can be approximated as ﬁnite diﬀerences. This restriction is referred to as Courant-Friedrichs-Lewy (CFL) condition. For a given run of the supernova code, the central Courant time step is typically several microseconds. Thus, to run a collapse calculation for long timescales like ∼ 15 sec, this then can be prohibitive. Several considerations are made to computationally speed up the calculation by varying the time scale [116]. Following each step forward in time, after redistributing the matter according to the evolution equations, the zones of the star are remapped to a new spatial grid. This best represents the physical structure of the star with a limited number of zones (typically 300). All of the physical matter and neutrino variables are then remapped onto the new grid.

5.2.4 Neutrinos

A very critical component of the supernova model are the determination of the neutrino distribution functions and neutrino-matter interactions. While it is very important to the simulation code, a detailed description is not relevant for this thesis, so we refer the reader to [116] for the neutrino description.

88 5.2.5 Equation of State

In numerical simulations, the equation of state (EoS) is used to update temperature and pressure as the matter rest-mass density ρ, matter internal energy M , and electron fraction Ye are evolved. To achieve this, a Newton-Raphson iteration technique is utilized to ﬁnd the temperature consistent with a given ρ, M , and Ye. The temperature is updated in all zones at least once per iteration cycle. An estimate of the temperature and pressure is obtained between calls to the EoS by means of a speciﬁc heat (CV ) and an ideal index Γideal such that M = CV T and

PM = (Γideal − 1)ρM . During subroutine calls where M changes in value a new estimate of the temperature is thus easily obtained by

T ≈ M . (5.18) CV

Similarly, if the density changes from ρ1 to ρ2, then the new pressure can be approximated by

Γideal P2 ≈ P1(ρ2/ρ1) . (5.19)

The equation of state itself is calculated separately as a table, indexed by the density n, temperature T and electron fraction Ye. We have already discussed the deﬁnitions of the old Bowers and Wilson EoS, as well as the new NDL EoS that is the focus of this thesis, in Sections 2.3.3 and 3, respectively.

5.2.6 Convection

Convection can be crucial to the success of the neutrino heating mechanism in a supernova explosion. The supernova code, being a one-dimensional model, approximates convection in two separate regions; above and below the neutrinosphere radius Rν, which become convectively unstable at diﬀerent times. Above Rν quasi-

Ledoux convection occurs. Below Rν, convection can be induced by a doubly diﬀusive

89 phenomenon. The convective flows in this instability have been dubbed neutron fingers [104], as it is similar to a salt finger instability [107]. The role of neutron fingers is a controversial topic. A detailed investigation of neutrino transport properties at the conditions inside nascent neutron stars [15] revealed no evidence for neutron finger instability to trigger neutrino-driven explosions by convectively enhanced neutrino fluxes. This is problematic, as the Livermore supernova code fails to explode if the neutron finger instability is switched off [119]. However, there is evidence that another doubly diffusive instability may play a larger role. This arose while looking at full-scale hydrodynamic models of nascent neutron stars in two dimensions with a simple description of grey, radial neutrino diffusion and its effects on lepton number and energy transport. It showed the rapid development of convection inside the proto-neutron star [16] and its persistence in a growing volume until at least one second in non-rotating and rotating cases [56, 54, 55, 50, 49]. Further investigation [54] revealed that the exchange of energy – and in particular of lepton number between moving fluid elements and their surroundings – was important, so that the convective mode could not be described by ideal Ledoux convection. To properly investigate these interesting mysteries in these one- and two-dimensional simulations, one should perform a full relativistic Boltzmann neutrino transport in 3-dimensions.

5.3 Simulation with Former Bowers and Wilson EoS

Using a 20 M progenitor model from [114, 122], we simulate a core-collapse supernova using the Bowers and Wilson [13] equation of state that was original to this code. Figure 5.1 shows trajectories of several mass layers during the simulation. The inner mass layers collapse to form the proto-neutron star, while the outer layers start expanding outwards.

90 1e+10

1e+09

1e+08

Radius [cm] 1e+07

1e+06

0 0.1 0.2 0.3 0.4 0.5 Timepb [s]

Figure 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.

The formation of the outward expanding shock can be seen looking at the velocity graph in ﬁgure 5.2. 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 from neutrino reheating to propel layers outward in an explosion.

5.4 Simulation with Hadronic-only NDL EoS

Using the same progenitor model as before, we simulate a core-collapse supernova using the new NDL EoS with purely hadronic matter. The simulation shown in

91 2e+09

1e+09

-1e+09

-2e+09

-3e+09 Velocity [cm/s] -4e+09

-5e+09

-6e+09 0 0.1 0.2 0.3 0.4 0.5 Timepb [s]

Figure 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 positive. 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.

Figs. 5.3 - 5.4 use the Skyrme parameter set MSL0, one of the models that passes all the constraints listed in section 3.2.1 and is not ruled out by the 2.01 ± 0.04 M maximum neutron star mass constraint [2, 31]. We have graphed again the mass layer trajectories and velocities, as a function of time. Both collapse simulations display a successful supernova explosion. The collapse is followed by a delayed shockwave, due to neutrino heating. The time between the collapse and the outward moving shock diﬀer between the two equation of state runs,

92 1e+10

1e+09

1e+08

Radius [cm] 1e+07

1e+06

-0.1 0 0.1 0.2 0.3 Time [s]

Figure 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. Time is measured post-bounce. as well as the magnitude of the velocity of the shock. Further comparisons with, for example, the other Skyrme and the NDL ﬁducial parameter sets, can solidify the extent to which uncertainties in the equation of state lead to uncertainties in the neutrino luminosities, the explosion energy and ultimately the ﬁnal outcome of the death throes of these massive stars.

93 3e+09

2e+09

1e+09

-1e+09

-2e+09 Velocity [cm/s]

-3e+09

-4e+09

0 0.1 0.2 0.3 Timepb [s]

Figure 5.4. Mass layer velocities in a simulation of core-collapse supernova from a 20 M progenitor model using the the NDL equation of state with the MSL0 Skyrme parameter set. Time is measured post-bounce, and velocity outwards as positive. 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.

94 CHAPTER 6

CONCLUSION

The creation of a new equation of state to describe nuclear matter in astrophysical situations allows for comparison between equations of state in simulation; judging if predictions from the simulation are due to the EoS or to other physics of the simulation [85]. The Lattimer and Swesty [63] and the Shen et. al. [100, 101, 102] equations of state provide means for comparison, but the Notre Dame-Livermore Equation of State adds a Skyrme description of matter, heretofore never implemented in an astrophysical setting. The advantages of this allow for using nuclear experimental data to constrict low energy parameterization, and reciprocally, use astrophysical constraints to improve nuclear Skyrme models. This thesis work includes the formation of the Notre Dame-Livermore Equation of State. The NDL EoS features a density functional theory approach to describing hadronic matter at high densities. It allows for a transition to a quark-gluon plasma phase of matter at high density, and an improved Bowers and Wilson [13] formulation at low densities which includes the formation of pasta nuclei. Additionally, it allows for the creation of a pion condensate at high density and for pair production of the 215 currently 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). The formulation of the equation of state allows using any set of Skyrme parameters to make similar models with diﬀerent predictions for, among other quantities, the energy and pressure. I highlighted the small variations between the sixteen Skyrme

95 models that had passed all the nuclear constraints from Section 3.2.1. Furthermore, an additional six of those models are ruled out because they fail to predict the heaviest of observed neutron stars.

Finally, I showed simulations of a core-collapse supernova from a 20 M progenitor model with the routines of Mayle and Wilson [118]. I used both the NDL EoS and the Bowers and Wilson EoS for comparison, both of which produced a successful supernova explosion. We plan to make several variants of the NDL EoS table publicly available, including various Skyrme parameter sets. Instead of just the liquid drop model of Lattimer and Swesty [63], and the mean ﬁeld theory of Shen et. al. [100, 101, 102], now there exists a density functional approach to describing matter in the equations of state.

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