Quark Matter in the Core of Neutron Stars

UNIVERSIDADE FEDERAL DO RIO DE JANEIRO INSTITUTO DE F´ISICA

Quark matter in the core of neutron stars

Jos´eCarlos Jim´enezApaza

Disserta¸cãode Mestrado apresentada ao Programa de Pós-Gradua¸cãoem F´ısicado Instituto de F´ısicada Uni- versidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necessáriosàobten¸cãodo t´ıtulode Mestre em Ciências(F´ısica).

Advisor: Eduardo Souza Fraga

Rio de Janeiro Mar¸code 2016

iii

T266p JiménezApaza, JoséCarlos Quark matter in the core of neutron stars (Materia de quarks no núcleode estrelas de nêutrons)/JoséCarlos JiménezApaza - Rio de Janeiro: UFRJ/IF, 2016. 93 f. : Il. ; 30 cm. Orientador: Dr. Eduardo Souza Fraga Disserta¸cão(Mestrado em F´ısica) - Programa de Pós- gradua¸cãoem F´ısica,Instituto de F´ısica,Universidade Federal do Rio de Janeiro. ReferênciasBibliográficas:f. 67-74. 1.Diagrama de fases quiral. 2. Ponto cr´ıtico. 3. Colisões de ´ıonspesados. 4. F´ısica-Teses. iv

Resumo

Materia de quarks no n´ucleode estrelas de nˆeutrons

Jos´eCarlos Jim´enezApaza

Orientador: Eduardo Souza Fraga

Resumo da Disserta¸cão de Mestrado apresentada ao Programa de Pós- Gradua¸cãoem F´ısicado Instituto de F´ısicada Universidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necessários àobten¸cãodo t´ıtulode Mestre em Ciências(F´ısica).

O estudo das intera¸cõesfortes em regimes de temperatura e potencial qu´ımicoar- bitrárioséainda um problema aberto. Em particular, o regime de altos potenciais qu´ımicose temperaturas muito baixas do diagrama de fase da QCD éde grande relevância, uma vez que pode provavelmente ser encontrado no núcleodas estrelas de nêutrons.

Lá,os métodos perturbativos da QCD (pQCD) fornecem resultados com razoável precisão,e resultados mais recentes da pQCD in-medium produziram uma forma mais confiável para a equa¸cãode estado (EoS) para matériade quarks em altas densidades, incluindo uma estimativa do erro. Usamos estes resultados para calcular a rela¸cãomassa-raio e comparar com as recentes observa¸cõesastronômicasde massas de pulsars. Consideramos estrelas h´ıbridas, ou seja, supor que o núcleoda estrela de nêutronséfeita de quarks em equil´ıbriobeta e que este núcleoérodeado por um envoltóriode matériahadrônica.Diferentes EoSs para cada fase (quark e hadrônica)podem ser combinadas e comparadas com os dados astrônomicos.

Palavras-chave: Estrelas de nˆeutrons,diagrama de fases da QCD, transi¸c˜oesde fase. v

Abstract

Quark matter in the core of neutron stars

Jos´eCarlos Jim´enezApaza

Advisor: Eduardo Souza Fraga

Abstract da Disserta¸cão de Mestrado apresentada ao Programa de Pós- Gradua¸cãoem F´ısicado Instituto de F´ısicada Universidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necessários àobten¸cãodo t´ıtulode Mestre em Ciências(F´ısica).

The study of strong interactions in arbitrary regimes of temperature and chemical potential is still an open problem. In particular, the regime of high chemical potentials and very low temperatures of the QCD phase diagram is of great relevance, since it can probably be found in the core of neutron stars. There, perturbative methods of QCD (pQCD) provide results with reasonable accuracy, and state-of-the-art in-medium pQCD have recently produced a more reliable form for the equation of state (EoS) for quark matter at high densities, including an estimate of the error. We use these results to compute the mass-radius relation and compare to the recent astronomical observations of pulsar masses. We consider hybrid stars, i.e. assume that the core of the neutron star is made of quarks in beta equilibrium and that this core is surrounded by a mantle of hadronic matter. Diﬀerent EoSs for each phase (quark and hadronic) can be matched and compared to astronomical data.

Keywords: Neutron stars, QCD phase diagram, phase transitions. vi

Acknowledgements

First of all, I want to thank to all the members of my family: my parents Roman and Angelina, my systers Claudia and Alejandra and my nephews Kiara y Felipe . To my parents Roman and Angelina because they always taught me how to be as a person, respecting always people around me and, in the end of the day, to myself too. Their example of life continues being important to me. The way I see the world and life is based on their example: keeping always optimistic eyes when things get complicated. Thank you to my sisters Claudia and Alejandra. Their support and encouragement at every stage of my life in fundamental since I met them when I was a child. All the things that I have learned from both of them are always in my mind. Thanks also to my nephews which are my inspiration to continue working. This thesis is dedicated to you.

I would like to give immense thanks to my supervisor Eduardo. It was a pleasure to work with you. Thank you for your patience, dedication and time you had given to me in the last 2 years. I have learned many things about physics and the path to be a good physicist. I am truly grateful for his attention and advices when I got in troubles. There are a lot of good physicists around the world but few good persons which at the same time are good physicists. You are one of them.

I can not forget of my colleagues and friends Daniel Kroﬀ, Elvis and Mauricio from who I learned a lot of Physics, Portuguese and Life in the usual conversations of every day. Thank you for the creation of a very good enviroment adequeate to study and do vii research. Your presence in the creation of this thesis was very important. Thank you very much my friends.

I could not forget you Patricia. Thank you to be present in all the development of this thesis. From you I have learned about life, friendship, feelings and the most deep misteries that my personality have with people, starting at you. My work, day after day, was not too diﬃcult with your company. You had given to me one of the most important gifts I received in all my life up to now: You. This thesis is also dedicated to you.

Thanks to all the ICE group, for the atmosphere and all activities.

To the staﬀ of the secretary of Pos-Graduation, my gratitute for all the help and patience.

Finally I would like to thank CAPES for ﬁnancial support. viii

Contents

Table of Contents vi

List of Figures ix

1 Introduction1

2 Relativistic stellar structure4

2.1 White dwarfs...... 4

2.1.1 Chandrasekhar limit...... 6

2.2 Cold catalyzed matter...... 7

2.3 Matter in curved space-time...... 8

2.3.1 Remarks...... 10

2.3.2 Perfect ﬂuid energy-momentum tensor...... 14

2.4 Tolman-Oppenheimer-Volkoﬀ equations...... 15

2.4.1 A simple example: constant density case...... 16

2.4.2 Classiﬁcation of EoSs...... 19

2.5 Stability conditions and Mass-Radius relation...... 20

2.5.1 Necessary stability condition...... 21

2.5.2 Suﬃcient stability condition...... 22

2.5.3 Mass-Radius relation...... 23

3 Nuclear and quark matter at high densities 25 ix

3.1 Introduction...... 25

3.2 Thermodynamics for Neutron stars...... 25

3.3 General constrains in the NS-EoS...... 30

3.3.1 Superluminal and ultrabaric EoSs...... 30

3.3.2 Mass bounds...... 30

3.3.3 Constraints in the nuclear matter equation of state...... 32

3.3.4 Constraints on the quark matter EoS...... 35

3.4 Phase transition...... 36

4 Matching Equations of State 40

4.1 Nuclear matter EoSs...... 41

4.1.1 General problem...... 41

4.1.2 “Minimal” model...... 43

4.1.3 Solving the many-body problem...... 43

4.1.4 NM-EoS1: Akmal-Pandharipande-Ravenhall (APR)...... 45

4.1.5 NM-EoS2: TM1 parametrization (TM1)...... 49

4.1.6 Diﬀerences in techniques...... 57

4.2 Unpaired Quark matter...... 59

4.2.1 The bag model...... 60

4.2.2 Fraga-Kurkela-Vuorinen ﬁtting for cold pQCD...... 63

4.3 Scaling the TOV equations...... 65

4.4 Solving the TOV* equations...... 67

4.4.1 TOV* for Nuclear matter-EoSs...... 68

4.4.2 TOV* for Quark matter EoSs...... 70

4.5 Matching the EoSs: Hybrid star...... 72

4.6 Crossover: hybrid APR-FKV...... 73

4.7 First-order transition: hybrid TM1-FKV (X=1.52)...... 77 x

5 Final Remarks and Perspectives 83

Bibliographic References 84 xi

List of Figures

1.1 Process of stellar evolution giving rise, in the end, to compact objects like neutron stars. Extracted from Ref. [1]...... 2

1.2 Possible internal structure of neutrons stars. Extracted from Ref. [2]....3

2.1 Typical mass-radius relation of white dwarfs according to Chandrasekhar’s

theory. The arrows indicate the direction into which a non-equilibrium conﬁguration is pushed if the gravitational force (Gr.) is larger or smaller than the pressure gradient (Pr.). Extracted from Ref. [3]...... 7

2.2 Stable (continuous curves) and unstable (dashed curves) regions for stellar matter in the mass-radius diagram. Extracted from [3]...... 20

2.3 Stable conﬁgurations for white dwarfs and neutron stars varying with the allowed central densities for the EoS of dense matter. The continuous black

lines mean regions of stability and dotted-lines mean unstable conﬁgura- tions. Extracted from [4]...... 22

2.4 Generic illustrations of mass-radius relations for diﬀerent kinds of neutron stars. Extracted from [3]...... 24 xii

3.1 This plot shows nonrotating masses versus physical radii for several typical

EoS [5]: blue, nucleons; pink, nucleons plus exotic matter; green, strange quark matter. The horizontal bands show the observational constraint from the pulsar mass measurement J1614−2230 presented in [6] of (1.97±

0.04)M , similar measurements for two other millisecond pulsars ([7,8]) and the range of observed masses for double neutron star binaries ([9]). The grey regions show parameter space that is ruled out by other theoretical or

observational constraints, like general relativity (GR) and spin period (P). Extracted from [6]...... 31

3.2 Energy per nucleon vs baryon number density for diﬀerent values of δ.

Extracted from [10]...... 33

3.3 Feynman diagrams describing the most important meson exchange processes which contribute to the NN interaction. Time goes upwards. Thin vertical lines: nucleons. Thick vertical segments: ∆(= Nπ) resonance in

an intermediate state. Extracted from [10]...... 34

4.1 Feynman diagrams for three possible mesonic interactions. Extracted from [10]...... 42

4.2 Proton fraction xp as a function of the baryonic density ρ (this notation only for the original article of [11]). Extracted from [11]...... 46

4.3 APR-EoS for pure hadronic matter. See [11]...... 47

4.4 Normalized APR-EoS for pure hadronic matter...... 48

4.5 Proton fraction for the TM1 parametrization in the mean-ﬁeld theory for diﬀerent temperatures. In this thesis we only need the case of T = 0.

Extracted from [12]...... 55

4.6 Nuclear matter (NM) equations of state that will be used in this thesis. There are only tabulated tables for both EoSs...... 56 xiii

4.7 Normalized Nuclear-EoSs...... 57

4.8 Gravitational mass vs central energy density for pure hadronic matter in the APR description [11] and the TM1 parameterization of the mean-ﬁeld

theory. The APR-EoS reproduces bigger maximum masses than the TM1- EoS...... 69

4.9 Total gravitational mass versus radii relations for nuclear matter equations of state in the context of the APR and TM1 descriptions...... 70

4.10 Gravitational mass vs central energy density for pure quark matter in the context of the models used in this thesis: MIT bag model and FKV-EoS.. 71

4.11 Total gravitational mass-radius relation for pure quark matter using the MIT bag model for some value of the bag constant and the FKV-EoS

for some particular value of the renormalization scale. Both models can surpass easily the two solar mass limit...... 71

4.12 The matching of perturbative results for the quark matter EoS to hadronic EoSs at high temperature and density. The pressure at µ = 0, obtained

3/2 from resummed O(αs ) pQCD [13] and compared with the result of summing up the eﬀects of all hadron resonances with masses smaller than 2GeV [14, 15]...... 72

4.13 Normalized EoS for the APR-EoS (red curve), FKV for X= 3.205 (blue curve) and the matching of both EoS (black curve). The matching is carried

out smoothly...... 74

4.14 Equation of state for the matching of the APR and FKV with X= 3.205.

The matching was made continuosly...... 74

4.15 Total gravitational mass versus central energy density for the matching of

the APR and FKV for X= 3.205...... 75

4.16 Total gravitational mass versus radius for the matching of APR and FKV for X= 3.205...... 76 xiv

4.17 Normalized TM1 EoS and FKV for X= 3.205. They do not intersect at

any baryonic chemical potential, then the matching can not be carried out. Extracted from[16]...... 77 4.18 Critical chemical potentials for the matching of the TM1 and FKV EoSs

for given X values. Extracted from [16]...... 78 4.19 Normalized EoSs for the matching of the TM1 parameterization and FKV for X = 2.5. Extracted from [16]...... 79

4.20 Total gravitational mass-radius relation for the TM1+FKV with X = 2.5. This matching is ruled out because does not satisfy the recent experimental observations [6, 17]. Extracted from [16]...... 79

4.21 Normalized EoSs for the matching of the TM1-EoS and FKV for X = 1.52. Extracted from [16]...... 80 4.22 Matched EoS of the TM1-EoS and FKV for X = 1.52. Extracted from [16]. 81

4.23 Total mass vs the central pressure for the TM1-EoS and FKV with X = 1.52. Extracted from [16]...... 81 4.24 Mass-Radius relation for the matching of the nucleonic TM1-EoS and with

the bulk quark matter FKV EoS with X = 1.52. For comparison it also shown the matching of the APR+FKV EoS with X = 3.205. Extracted from [16]...... 82 1

Chapter 1

Introduction

In this thesis we discuss the properties of matter under extreme conditions of baryonic density and at negligible temperature. This scenario is expected to be found in neutron stars: very compact astrophysical objects [4,2, 18, 19].This thesis is completely written in this scenario of extreme densities.

From the process of stellar evolution shown in Fig. 1.1, one knows that the end of that evolution could be a small and dense object. Landau in 1932, shortly after the discovery of the neutron, postulated the existence of astrophysical objects called neutron stars as hypothetical objects that are not forbidden to exist by the laws of stellar evolution known at the time [20]. Then, in 1934, based on observations, Baade and Zwicky wrote [21]: “With all reserve we advance the view that supernovae represent the transition from ordinary stars to neutron stars, which in their ﬁnal stages consist of closely packed neutrons”.

After that, in 1939, Oppenheimer and Volkoﬀ [22] studied the problem quantitatively, and solved the corresponding general relativictic equations of stellar equilibrium. 2

Figure 1.1: Process of stellar evolution giving rise, in the end, to compact objects like neutron stars. Extracted from Ref. [1].

Today one knows that neutron stars are the remnant of a dying star after a supernova explosion. There have been many observations and theories to describe its equilibrium properties and the microscopic dynamics which gives rise to its global properties, like its mass and radius which are astronomical observables.

In Fig. 1.2 one can see a cross section of the possible composition of a neutron star (NS). All the possible phases of matter within the star are presented and each one depends on how deep one is inside its bulk. In this ﬁgure we can also see explicitly the presence of some regions which suﬀer phase transitions. Since, by construction, we assume (and need) global charge neutrality and β equilibrium in the NS, we are left with a competition involving the gravitational and strong interactions. By modelling the microscopic interactions, we obtain a Equation of State (EoS) for a given NS. To describe the gravitational interaction we must use the Eintein’s theory of General Relativity (GR), since the densities involved are so high. The balance between these two fundamental interactions gives rise to these astrophysical objects [19]. 3

Figure 1.2: Possible internal structure of neutrons stars. Extracted from Ref. [2].

This thesis is organized as follows. We present in Chapter 2 a review of the fundamental ideas from General Relativity needed to describe the hydrostatic equilibrium between matter and gravity. In Chapter 3 we present the fundamental ideas needed to find an equation of state which could describe microscopically matter at such high energy densities, as well as some general constraints. Chapter 4 contains the application of all the ideas presented in the preceeding chapters 2 and 3. There we also present our final results for the observables of neutron stars. Finally, we conclude by presenting some perspectives and final remarks in Chapter 5.

Throughout this thesis we use natural units [4],

~ = c = kB = 1. (1.1) and the Minkowski metric has the form:

ηµν = diag[+1, −1, −1, −1]. (1.2) 4

Chapter 2

Relativistic stellar structure

The eﬀects of gravity for compact stars are important. These compact objects could be black holes, neutron stars or white dwarfs [18]. In this thesis we will be involved only with neutron stars (NS). To treat them one has to take into account the curvature of space-time, essentially because the energy density is very high compared to the small region of space it occupies. Along this chapter we will elucidate the main ingredients of General Relativity (GR) that will be used to understand correctly the observables related to NS which can be compared with current observations.

2.1 White dwarfs

We review ﬁrst some general concepts of compact stars using a simple case: White dwarfs. They are usually modeled as a gas of free electrons in a medium made of nucleons. Free degenerate electrons generate the pressure which will counterbalance gravity. The total energy density is given by [18]:

A = n m + (ke ), (2.1) e N Z elec F where ne is the electron number density, mN the nucleon mass, A the baryon number, Z the proton number and elec, the electronic energy density, is given by

(ke ) = 0 (2x3 + x)(1 + x2)1/2 − sinh−1(x) , (2.2) elec F 8 5

e where kF is the electron Fermi momentum, the mass of the electron, me, and

m4 = e , x = ke /m . (2.3) 0 π2 F e

The degenerate pressure for electrons is given by

P (ke ) = 0 (2x3 − 3x)(1 + x2)1/2 + 3sinh−1(x) . (2.4) F 24

Now, since the rest masses of nucleons are bigger of electrons, we realize that the energy density of the nucleons will dominate pressure. This gives us the following hierarchy to describe the global structure of white dwarfs [4]:

2M(r) << 1, (2.5) r 4πr3P (r) << M(r), (2.6)

P (r) << ε(r), (2.7) where r is a radial coordinate for the white dwarf from its center and M is the mass enclosed up to the radius r. Up to this point, we see the expected Newtonian behaviour in the solutions for the equations of hydrostatic equilibrium (see Eq. (2.5)). The meaning of the above inequalities is as follows: The inequality in Eq. (2.5) comes from the fact that no region of a star can lie within its Schwarzschild radius [23], inequality in Eq. (2.7) comes from the EoS for that system and ﬁnally, the inequality in Eq. (2.6) follows from

Eq. (2.7) since: 4π M(r) ∼ r3,¯ (2.8) 3 so that P (r) 4πr3P (r) ∼ 3 M(r) << M(r), (2.9) ¯ where ([4]) g MeV ¯ ∼ 1015 ∼ 564 , (2.10) cm3 fm3 6 is the ﬁducial energy density in compact stars adapted to give stellar masses and radii of the order given by astronomical observations (see [6]):

M ≈ 2M , (2.11)

R ≈ 10km. (2.12)

Therefore, even for nonrelativistic compact objects (in this case white dwarfs), there exists a maximum limit inherent to general relativity on the masses and radii.

2.1.1 Chandrasekhar limit

S. Chandrasekhar was the ﬁrst to study quantitatively the above limit in white dwarf masses using Fermi statistics to ﬁnd that EoS and in the Newtonian approximation adequate for white dwarfs [24, 25]. He did that using some relativistic polytropic approxima-

e tions depending on the dimensionless momentum x = kF /me with increasing density and polytropic index Γ = 4/3, where Γ is given by Γ = d log(P )/d log(). In the nonrelativistic case, the degenerate electron gas is modeled as a polytropic EoS with Γ = 5/3 [18]. Although we know that the transition from nonrelativistic to relativistic EoSs does not occur abruptly but smoothly, we can imagine that an idealized stellar model consisting of degenerate matter can be constructed by ﬁtting these two regions smoothly: a relativistic polytrope core surrounded by an nonrelativistic mantle [4]. Chandrasekhar did that with the assumption that the mantle is small near the surface and the unique relevant part is the relativistic polytrope where the only possible maximum mass is [3]

MCh ≈ 1.46M , (2.13) called the Chandrasekhar limiting mass. We can see this limit in figure 2.1 which shows the mass-radius relation for a typical family of white dwarfs. If we continue increasing the central density we get a finite mass and the radius approaches zero. All astronomical measurements of white dwarf masses respect this limit up to today. In general, we are 7 tempted to say that this result is unrealistic since the EoS is modified by different effects at very high densities near the core. However, complete numerical integrations in the EoSs for the hydrostatic equilibrium give the same results [26].

Figure 2.1: Typical mass-radius relation of white dwarfs according to Chandrasekhar’s theory. The arrows indicate the direction into which a non-equilibrium conﬁguration is pushed if the gravitational force (Gr.) is larger or smaller than the pressure gradient (Pr.). Extracted from Ref. [3].

2.2 Cold catalyzed matter

The interior of NSs is not made of ordinary matter because one can encounter there high densities. Rather, matter in NSs must be very compressed allowing the existence of exotic phases of nuclear matter [2]. This matter has been fully processed by nuclear combustion, so that all available energy has been extracted at each density, the hadronic matter is in its ground state, and is called cold catalyzed matter [4]. The surface of a star (any star) is characterized by having P = 0. Such layer is made of 8 atoms of Fe-56, arranged in a Coulomb lattice. At increasing density (getting into the star), the atoms become progressively more ionized, and the electrons ﬁll the interstices [4]. A Coulomb lattice arrangement of nuclei in the electron gas minimizes the energy. At

p 2 2 higher density (such that the electron chemical potential µe≡ me + ke is large enough to induce inverse beta decay), a lower-energy state is achieved through the capture of energetic electrons by nuclei. Neutrinos and photons produced diﬀuse out of the star, thus lowering this energy state. The nuclei become increasingly neutron-rich by this neutronization process [19].

Neutronization sets at a density for which the electron Fermi energy µe equals the threshold for the reaction. The threshold density for neutronization is ∼ 9 × 1011g/cm3 [19]. At such high densities, the most weakly bound neutrons drip out of the nuclei, and a gas of neutrons and electrons occupies the interstices [18]. The drip density is high compared to the highest density in stable white dwarfs, but occurs in the crust of NSs. At still higher

14 3 densities, around the saturation density of nuclear matter, 0 = 2.51 × 10 g/cm , nuclei disassemble into a uniform charge-neutral mixture of baryons and leptons [18]. This is what one refers to as neutron star matter, i.e. nuclear matter that is charge-neutral and in its lowest-energy state compatible with that constraint. It contains as many baryon species at each density as is required for equilibrium, so that no particle transformations by beta decay or otherwise will take place [11].

2.3 Matter in curved space-time

In neutron stars, gravity compresses matter (made up of particles) and the geometry of space-time is changed considerably from flat space-time. So, the compression is very high. Therefore models for such stars must be constructed in the framework of Einstein’s general theory of relativity (GR) combined with theories of superdense matter that will take into account the microscopic interactions between them. The effects of matter on space-time and (the inverse) are included by introducing the 9 energy-momentum tensor for matter fields into Einstein’s field equations. The generally covariant Lagrangian density is [23]:

L = LM + LG, (2.14)

where the dynamics of particles composing the star is encoded in LM and the gravitational

Lagrangian density LG is given by

√ √ µν LG = −gR = −gg Rµν, (2.15)

where g is the determinant of the metric tensor gµν, R is the Ricci scalar and Rµν is the Ricci tensor given by

σ σ σ κ σ κ Rµν = ∂νΓµσ − ∂σΓµν + ΓκνΓµσ − ΓκσΓµν. (2.16)

The connection deﬁned as

1 Γσ = gσλ(∂ g + ∂ g − ∂ g ). (2.17) µν 2 ν µλ µ νλ λ µν

Microscopic interactions will be manifested through the energy-momentum tensor. The fundamental interactions that appear in the interior of neutron stars are the gravitational and the strong nuclear forces. We would have to use GR and Quantum Chromodynamics

µν µν (QCD). Therefore TNS ≡ TQCD. So, in principle, this balance among forces must be written in the Einstein’s equations:

1 G ≡ R − g R = 8πT QCD(, P ()), (2.18) µν µν 2 µν µν QCD which couples the Einstein curvature tensor, Gµν, to the QCD energy-momentum tensor,

µν TQCD, of stellar matter. This last one contains the ﬁrst principles QCD EoS, PQCD(), of the stellar matter, discussed later. 10

2.3.1 Remarks

Normally one defines the thermodynamics of the system under study in the frame of reference of the thermal bath from (and in) which we can define the statistical ensembles. This frame of reference could be an inertial coordinate system where we can define the laws of Special Relativity (SR). It can be done if we assume our position to be some useful place within the system, like the center-of-mass of the body or any other appropiate place. Technically, we have formed a Lorentz frame at that small region of spacetime where the metric is given by the Minkowski metric ηµν. Usually one does particle and nuclear physics using this spacetime metric because the curvature of spacetime is not important and we ignore it.

In this (global) Lorentz frame of special relativity we have the evolution in spacetime of the particles composing the gas (which models our neutron star). They will follow paths within the light-cone,we can deﬁne our ensembles and therefore our EoS. Then, a natural question arises: what would happen if this group of particles (the gas) were so dense to create a global curvature of spacetime? [4]. The particles would follow the geodesics in that curved spacetime. Each one of them would couple to spacetime through the Einstein’s equation complicating the equations because, in general, Einstein’s equations are nonlinear coupled second-order diﬀerential equations. We would have to replace

β the Minkowski metric by the general gµν(x ) in all the terms of the theory dependent on the spacetime point xβ. However, we can still deﬁne Local Lorentz frames in this global curved spacetime. In order to simplify this complication for our special case of NS one uses ideas from nuclear and particle physics in Minkowski spacetime as the important

ﬁrst step towards the solution of the general problem.

Interestingly, curved spacetime will inﬂuence very little the EoS for a neutron star 11

(NS). One can see that statement as follows: taking the extreme case of a NS as massive as a black hole (BH), with mass in the order of M ∼ 3M and RSchw ∼ R which means that this star is in the limit of gravitational collapse where we can take a safe value of compactness (which will be deﬁned later), say, C = 0.3, which does not violate Buchdahl’s limit (to be discussed later).

Let us introduce this values in the metric tensor for the space outside the star [23], 2GM −1 gout(r > R) = − 1 − , (2.19) 11 r and for the interior of the star where we have the metric tensor given by 2GM(r)−1 gint(r≤R) = − 1 − , (2.20) 11 r where M(r) is obtained from [23]: dM(r) = 4πr2(r). (2.21) dr Using this result one ﬁnds at the origin, r = 0:

int g11 (r = 0) = −1. (2.22)

Then, calculating their ratio

out −1 g11 (R) −1 6 int = (1 − 2C) = 1 − = 2.5, (2.23) g11 (0) 10 which tells us that the metric is changing by a factor 2.5 from the origin to some quasi- critical radius. Therefore, the metric from the center of the NS up to its surface has only changed more or less three times i.e. macroscopically this change is not really big enough to expect a microscopic chang in the EoS [4].

A natural question at this point would be: how this macroscopic change in the spacetime metric affects the EoS, i.e. the pressure, which fundamentally depends on the distances among particles (hadrons and/or quarks)? The relative distance D of nucleons in 12 the star is 2r d D ∼ 0 = = 2A−1/3 ≈ 10−19. (2.24) R R where r0 is the fiducial nucleon radius, R is the nuclear radius and A the baryonic number in NS which are of the order of 2.6 × 1057 [4]. The metric would change by a mere 1 part in 109 across a shell in the radial direction with thickness of, say, 1010 nucleon spacings. So, even with the high densities found in NSs we have a negligible change over a distance spanning many times the nucleon distance d. Then, our volume under study does not change significantly for every region of the star with this global curvature. However, the pressure differs in radial locations by an appreciable amount. Therefore we can define a global volume where the curvature of spacetime is not important. GR is not acting in this volumes locally. However, such local volumes are not inertial regions.

Interestingly, the curvature of spacetime is finite though uniform in each such volume and our EoS will depend on this volume. The notion of an EoS in the interior of the star is possible because one can define a locally inertial frame where all volumes are equivalent having the same number of baryons and energy density [27]. The energy density in all small regions of the star is therefore uniquely specified by the baryon density:

= (nB), (2.25)

where denotes the energy density and nB the baryon density.

Enlarging the volume of study which falls freely (in the context of the Equivalence

Principle [19]) and occupies an inertial frame allow us to have uniqueness in the deﬁnition an EoS in a global Lorentz frame: = (nB). Thus, each small volume in the star, which, however, is large enough to be uniform in density, can be described by the laws of physics of Special Relativity to high accuracy. We make a negligible error for neutron stars by solving the Einstein’s equations for matter 13 in the absence of gravity in the creation of the energy-momentum tensor for matter. For example, in Minkowski spacetime one can construct the energy-momentum tensor that is diagonal in a comoving Lorentz frame, like for the perfect ﬂuid case

T µν = diag(, P, P, P ). (2.26)

In summary, Einstein’s equations and the many-body equations (for each particle composing the star) were to be solved simulataneously since the baryons and quarks move in a curved ST whose geometry, determined by Einstein’s equations, is coupled to the total mass-energy density of matter. In the case of neutron stars, as for all astrophysical situations for which the long-range gravitational forces can be cleanly separated from the short-range forces, the deviation from ﬂat ST over the length scale of the strong interaction, ∼ 1fm, is however practically zero up to the highest densities reached in the cores of such stars (some 1015g/cm3). This is not to be confused with the global length scale of a neutron star, ∼ 10km, for which M/R ∼ 0.3, depending on the star’s mass. So, gravity curves spacetime only on a macroscopic scale but leaves it ﬂat to a very good approximation on a microscopic scale.

To achieve an appreciable curvature on a microscopic scale set by the strong interaction, mass densities greater than ∼ 1040g/cm3 would be necessary [28]. This circumstance divides the construction of models of compact stars into two distinct problems. First, the effects of the short-range nuclear forces on the properties of matter are described in a comoving proper reference frame(local inertial frame), where spacetime metric is flat, by the parameters and laws of (special relativistic) many-body physics. Second, the coupling between the long-range gravitational field and matter is taken into account by solving Ein- stein’s equations for the gravitational field described by the general relativistic curvature of spacetime, leading to the global structure of the stellar configuration. 14

2.3.2 Perfect ﬂuid energy-momentum tensor

For the reasons given above we can model neutron star matter as a perfect ﬂuid (PF), which is a good approximation for this kind of systems. The energy-momentum tensor of such a ﬂuid is given by [19]:

µν µ ν µν TPF = u u ( + P ) − g P, (2.27) where uµ is the four-velocity deﬁned as

duµ uµ ≡ . (2.28) dτ

The perfect ﬂuid in the frame of reference at rest, in the center of the body, takes the simpler form

µν TPF = diag(, P, P, P ). (2.29)

The components of uµ form the macroscopic velocity of any small portion of the stellar matter with respect to the inertial coordinate system that is being used to derive the stellar equilibrium equations. The production of curvature by mass of the star is speciﬁed by Einstein’s equations. Finally, we need to specify the metric of a nonrotating body in General Relativity (GR). From GR we know that for a coordinate system having spherical symmetry, the metric has the following general form, independent of being within or out of the body of the star [19]

ds2 = e2ν(r)dt2 − e2λ(r)dr2 − r2dθ2 − r2sin2θ(r)dφ2, (2.30) where ν(r) and λ(r) are radially varying metric functions. It has the nonvanishing connection components [19]:

r ν(r)−λ(r) 0 t 0 r 0 θ −1 φ −1 Γtt = e ν (r), Γtr = ν (r), Γrr = λ (r), Γrθ = r , Γrφ = r ,

r −2λ(r) φ r 2 −2λ(r) θ Γθθ = −re , Γθφ = cotθ, Γφφ = −rsin θe , Γφφ = −sinθcosθ, (2.31) where primes denote diﬀerentiation with respect to the radial coordinate. 15

2.4 Tolman-Oppenheimer-Volkoﬀ equations

Neutron stars (NS) are fully relativistic objects. Therefore, we need to use General

Relativity (GR) in for its correct description. As a ﬁrst, and important, approximation to model a NS we will assume that it has spherical symmetry and is nonrotating. This allows us to use the Schwarzschild-like invariant interval for the interior of the star [4]:

ds2 = e2ν(r)dt2 − e2λ(r)dr2 − r2dθ2 − r2sin2θdφ2, (2.32)

2 α µ ν ds = gµν(x )dx dx , (2.33) where the components of the metric tensor for the interior are given by

2ν(r) 2λ(r) 2 2 2 g00 = e , g11 = −e , g22 = −r , g33 = −r sin θ. (2.34)

After introducing this metric components and the perfect fluid energy-momentum tensor into the Einstein’s equations, we find a pair of equations obtained by Oppenheimer, Volkoff and Tolman in 1939 ([29], [22]):

dP (r) GM(r)(r) P (r) 4πr3P (r) 2GM(r)−1 = − 1 + 1 + 1 − , (2.35) dr r2 (r) M(r) r together with dM(r) = 4πr2(r), (2.36) dr

−2 where G = Mp is Newton’s constant in natural units in terms of the Planck mass Mp

19 given by Mp = 1.22×10 GeV. These are called the Tolman-Oppenheimer-Volkoff (TOV) equations. When the NS problem is solved for the microscopic interactions, we obtain the equation of state (EoS), P = P (), which serves as to solve closely the TOV equations above. In that way, we have a one-parameter family of stellar models corresponding to a specific EoS. There exists several EoS corresponding to different degrees of completeness in the understanding of the interior of a NS. The TOV equations are solved with an initial 16

condition, say, the central density (r = 0) = c (or central pressure P (r = 0) = Pc) and M(r = 0) = 0. The equations have to be integrated until the pressure becomes zero, which means that we have reached the surface of the star. Then the radius R and the mass M(R) = M(R) for the chosen c are found. Choosing a succession of increasing values for c corresponds to developing a sequence of stars of increasing mass, until the mass limit for a given EoS is reached [4].

2.4.1 A simple example: constant density case

The TOV equations and an appropiate (realistic) EoS are usually solved numerically since, in general, they are not analitically tractable. There are very few cases where one can solve the TOV equations analitically and which are, at the same time, important. This is the case of a star with [19]

(r) ≡ 0, (2.37) i.e. constant and uniform energy density. Obviously, this assumption is an idealization which does not occur in Nature because this matter would have to be incompressible to maintain a constant energy density. However, this case is interesting because from it we can classify our physical EoSs since the energy distribution they give within a NS near the maximum mass limit is rather uniform, though certainly not constant [19].

Remarkably, this idealization also illustrates that, even in the hypothetical (extreme) case of incompressible matter, there is a limit on the maximum possible mass of a stable, compact, relativistic star. For simplicity we will assume that our star is ﬁnite and has spherical symmetry. From the second TOV equation for M(r) we ﬁnd that the mass is given by:

4 3 1. M(r) = 3 π0r for r < R and

4 3 2. M(r = R) ≡ M = 3 π0R for r ≥ R. 17

With the aid of the ﬁrst TOV equation for P (r) we have 4 − 16 Gπ r 0 dP = 3 0 dr. (2.38) 8 2 (0 + P )(0 + 3P ) 1 − 3 Gπ0r We can restate the left hand side of our last equation as 4 3 1 0 dP = 2 − dP. (2.39) (0 + P )(0 + 3P ) 0 + 3P 0 + P

If we combine our two last equations and integrate the result from P (r = 0) ≡ Pc to P (r = r0) = 0 and r = 0 to r = r0, respectively, we obtain 0 + 3P 0 + P 8 2 2 log + log = log 1 − Gπ0R , (2.40) 0 + P 0 + 3P 3 or, equivalently, r 0 + 3P 0 + 3Pc 1 2 = 1 − Gπ0R . (2.41) 0 + P 0 + Pc 8 With Eq. 2.40, Eq. 2.41 results in

q 02 2GMr 0+Pc 1 − 3 − R 0+3Pc P = 0 . (2.42) q 02 0+Pc 2GMr 3 − 1 − 3 0+3Pc R Finally, we apply the condition of vanishing pressure at r0 = R, yielding r + P 2GM 0 c = 1 − , (2.43) 0 + 3Pc R or, equivalently, the mass and radius of this hypothetical star are linearly dependent 2GM + P 2 = 1 − 0 c . (2.44) R 0 + 3Pc In other words, the ratio M/R is upper bounded. Finally, the solution (renaming r0 → r) is q 2GMr2 q 2GM 1 − R3 − 1 − R P (r) = 0 , (2.45) q 2GM q 2GMr2 3 1 − R − 1 − R3 which means that although the energy density is uniform within the star, the pressure is not. To study the behaviour of this solution we consider the central pressure

q 2GM 1 − 1 − R Pc = 0 , (2.46) q 2GM 3 1 − R − 1 18 while the stellar radius is simply the position at which P (r = R) = 0, i.e. s 2 3 (0 + Pc) R = 1 − 2 . (2.47) 8π0 (0 + 3Pc) Since the energy density ¯ = 3M/4πR3 is a constant, the larger the radius of a uniform density star, the more massive it is, and the higher Pc must be to support this mass.

Thus, since at r = 0 (and in all the star interior) we have Pc < 0, if one makes Pc very large (but not inﬁnity and yet less than 0) and omiting the energy density terms, since they would only appear in the equality, we can create an upper-bound independent of 0 which would be the absolute limit, given by [19]:

2GM 12 8 < 1 − = (2.48) R 3 9

Therefore, our equation for Pc tells us that it only depends on the stellar mass and radius, but also it diverges for GM 4 −→ . (2.49) R 9 In other words, if we deﬁne the stellar compactness as the dimensionless ratio

GM C ≡ , (2.50) R then an inﬁnite pressure is necessary to support a star with a compactness larger than the

4 critical one, Ccrit ≡ 9 ' 0.444. If the stellar matter is compressed to reach a compactness

Ccrit or larger, it can only collapse to produce a black hole [18]. We can easily compute the corresponding radius and mass for this limit recalling that

4π M 1 = , (2.51) 3 0 R R2 and using the absolute critical value GMcrit/Rcrit = 4/9 to ﬁnd 1 Rcrit = , (2.52) 3πG0 and 4 GM = R , (2.53) crit 9 crit 19 which are the critical values for stars made of incompressible matter. This limit is intrinsic to GR. The importance of this result, known as the Buchdahl’s limit [19], is that, although we have derived it for a constant energy-density star, holds for any EoS, so that the radius of any relativistic star must be larger than the critical one

9 9 R > R ≡ R = GM, (2.54) crit 8 S 4 where RS ≡ 2GM is the Schwarzschild radius. This last condition may appear as a very restrictive one, but this is really the case only for compact stars. To ﬁx ideas, the critical radius for the sun approximately 3km, while it is as small as approximately 1cm for the Earth. In practice, most realistic EoSs lead to NS models with compactness

0.1 ≤ C ≤ 0.2 . (2.55)

17 3 When we consider 0 = 10 kg/m , of the order of the saturation nuclear density n0, we

31 get Rlim = 21km and Mlim = 4 × 10 kg= 20M . These values match the typical values for a compact star [19].

It is important to keep in mind that if these, and any other compact star, would be perturbed, they could suﬀer dynamical instabilities, leading to gravitational collapse [18].

2.4.2 Classiﬁcation of EoSs

From the example above, taken as an extreme case of compressed matter, we can classify

EoSs by how far from being incompressible they are. Every NS has a maximum critical mass Mmax. Even in the idealization of incompressible matter, the structure of the TOV equations is such as to impose a limit to its mass (see section above). Therefore, we realize that the maximum critical mass of a model for a sequence of stars depends on the compressibility of matter, determined in the EoS. So,

1.a soft EoS is relatively more easily to be compressed than 20

2.a stiﬀ one, which tends asymptotically to be incompressible.

One EoS is said to be stiffer than another if its pressure at every energy density is greater. The stiffer the EoS, the larger the mass that can be sustained against gravitational collapse. The cases that will be discussed in this thesis involve the quark matter and hadronic mater whose EoSs compete for different densities.

2.5 Stability conditions and Mass-Radius relation

Solutions of the TOV equations correspond to stellar conﬁgurations that are in hydrostatic equilibrium, by construction. We know that equilibrium not necessarily implies stability.

Equilibrium conﬁgurations may correspond either to a maximum or to a minimum in the energy (density) with respect to radial compression or dilation. So, we need a general, and practical, criterion to select solutions (stellar conﬁgurations) that are in stable equilibrium.

Figure 2.2: Stable (continuous curves) and unstable (dashed curves) regions for stellar matter in the mass-radius diagram. Extracted from [3]. 21

In ﬁgure 2.2 we see a continuum of conﬁgurations in equilibrium which correspond to an

EoS that is smooth and satisﬁes the condition of microscopic stability of matter

dP > 0, (2.56) d

the so-called Le Chatelier’s principle [4]. White dwarfs and neutron stars correspond to two regions of stable equilibrium in this figure. At the same time, there is a vast region in this figure where there are unstable configurations. Stars in the unstable region between white dwarfs and neutron stars are subject to vibrational modes that will either cause their disassembling or else their collapse to black holes. Above the stable NS sequence, there are no additional regions of stability if the EoS is of polytropic form, or more generally if it is sufficiently smooth [19].

2.5.1 Necessary stability condition

The necessary (but not suﬃcient) condition for stability of stars along the sequence of equilibrium conﬁgurations is based on the following inequality [4]:

∂M( ) c > 0 . (2.57) ∂c

From the last inequality we can conclude that an extremal point in a plot for gravitational mass versus central energy density means a potential point for the inset instabilities in the stellar configuration. In figure 2.3 one can see different behaviours for different values of density. 22

Figure 2.3: Stable conﬁgurations for white dwarfs and neutron stars varying with the allowed central densities for the EoS of dense matter. The continuous black lines mean regions of stability and dotted-lines mean unstable conﬁgurations. Extracted from [4].

The passage from stability to instability along a sequence of equilibrium conﬁgurations occurs only at the critical points of M(c). The maximum in the mass is known as the mass limit, or limiting mass. White dwarfs and neutron star sequences both have a limiting mass but, naturally, for diﬀerent reasons.

2.5.2 Suﬃcient stability condition

Astrophysical objects are not only in hydrostatic equilibrium but also stable against radial oscillations. If these objects are perturbed, one expects vibrations. To analyze these vibrations we would have to Fourier decompose them in normal radial modes. After some

2 calculations we would ﬁnd the star’s oscillatory eigenfrequency spectrum, ωn, from an

2 Sturm-Liouville problem for the radial perturbation [4]. It is known that ωn form an inﬁnite discrete sequence

2 2 2 ω0 < ω1 < ω2 < .... (2.58) 23

If any of these is negative for a particular star, the frequency ωn is purely imaginary and, therefore, the radial perturbation of the star, r(t), will grow exponentially in amplitude of oscillation as e+|ω|t. Such stars are said to be unstable. As a consequence of the ordering of the eigenfrequencies and the fact that one mode of oscillation changes in stability at every stationary point, we may infer the following: if the fundamental mode n = 0 becomes unstable at the maximum NS mass, and it does, then at the next minimum in the sequence either stability is restored to the fundamental mode or the next (n = 1) mode becomes unstable, and so on [19].

2.5.3 Mass-Radius relation

The most important representation of a NS sequence is its mass-radius relation. It is uniquely related to the underlying EoS. While both mass and radius are not known si- multaneouly so far for any pulsar, it is, in principle, possible to determine them in some cases. Masses can often be determined for a pulsar that is in a binary orbit with a com- panion [6]. In principle, radius determinations could be made through the measurement of the Doppler shift of known spectral transitions giving M/R [4]. The gemetrical form of this mass-radius relation is generic to compact stars that are bound by the gravitional interaction as illustrated in ﬁgure 2.4. 24

Figure 2.4: Generic illustrations of mass-radius relations for diﬀerent kinds of neutron stars. Extracted from [3]. 25

Chapter 3

Nuclear and quark matter at high densities

3.1 Introduction

In general, we should find in the interior of neutron stars (NS) cold matter under extreme conditions: intense gravitational fields, of the order of ∼ 1010g (where g is Earth’s surface gravity), high densities, above and of the order of nuclear saturation density n0, and intense magnetic fields of the order of 1015G[4]. For superdense matter, governed by the strong nuclear interaction, the fundamental degrees of freedom must be quarks and gluons. As one is getting far from the center, the baryon densities decrease and there must exist a point where the hadronic degrees of freedom become important [30]. Quarks must undergo color confinement at this transition. This is a difficult and open problem.

We can relax this diﬃculty using diﬀerent EoSs for each phase of matter: one for quark matter and other for hadronic matter and then match both at some critical energy density using some thermodynamic construction, like that given by Maxwell [31].

3.2 Thermodynamics for Neutron stars

In this thesis we only study neutron stars (NS) which, in principle, must be governed microscopically by the nuclear strong interaction described by QCD at high µB with van- 26 ishing temperature (T → 0). The QCD’s ﬁrst principles are well known, in the sense that one has a well-deﬁned prescription for calculating any quantum-mechanical amplitude [32]. However, phenomena which result from the application of these principles are complicated.

Asymptotic freedom (very short distances) indicates that perturbative QCD (pQCD) is a natural language for weakly interacting quarks and gluons [33]. On the other hand (for large distances), this description becomes inadequate in several ways. First, the running coupling grows large and perturbative estimates are no longer reliable. Second, the assumption that physical quantities should be analytic functions of the coupling becomes untenable [34]. Quarks and gluons are not weakly fluctuating degrees of freedom at these length scales. Other dynamical variables, perhaps instantons, color-bearing monopoles, and flux tubes, provide a more relevant description of the physics in this energy sector. These facts indicate that reliable calculations in non-perturbative QCD are needed but they are beyond our existing mathematical tools [35]. In this thesis, we consider extreme enviroments and assume there is thermodynamic equilibrium for finite densities of matter or energy. The primary mathematical object which encompasses all the thermodynamic information in the theory is the thermodynamic partition function Z. This function can be written as a functional integral in terms of the classical QCD action. The calculation of this integral from first principles is a very difficult mathematical task [33]. However, there are limiting cases like, for example, QCD in the limit of very large density, for which this calculation can be done in an expansion of a controllably small parameter: the effective strong coupling at the relevant energy scale [34, 36]. Another example is the limit of low density and small quark masses(the chiral limit), for which a controllable expansion in the coupling and quark masses can be done in a framework called chiral perturbation theory [37, 38]. Lattice simulations allow us, in principle, to calculate the partition function (or its derivatives) for any values of parameters [39]. The QCD partition function is an infinite-fold integral that can 27 be rendered finite by limiting the number of quantum-field degrees of freedom through an ultraviolet (UV) cutoff (the lattice spacing) and by an infrared(IR) cutoff(the lattice volume) [40]. Although the results obtained in this way are numerical, in many cases they serve as a basis for our intuition on the behaviour of QCD in parameter regimes that are inaccessible analitically [41]. The most basic phenomena, confinement and chiral symmetry breaking, are prime examples [33]. Lattice QCD simulations often play a role similar to an experiment. Analytic calculations based on controllable expansions as well as lattice calculations based on discretization of the partition function are often referred to as first-principles calculations [13]. The system of study is characterized by global, bulk quantities, such as the total volume, total energy, and total value of a conserved charge (electric, baryon number, and so on).

The grand canonical partition function Z(T, µ) serves as a generating function for all thermodynamic quantities. The logarithmic derivatives of it with respect to T and µ give the average energy and charge:

∂logZ hE − µNi = −T 2 , (3.1) ∂T ∂logZ hNi = T . (3.2) ∂µ It is convenient to introduce the following thermodynamic potential:

Ω(T, µ) = −T logZ. (3.3)

For a macroscopic system we can write Ω in the form

Ω = +E − TS − µN. (3.4)

All these quantities have in common that they are macroscopic. To involve the macroscopic constituition of matter we ﬁrst introduce the statistical deﬁnition of entropy

S = logZ, (3.5) 28 where Z is the partition function, which is a measure of the number of microstates corresponding to the observed thermodynamic macrostate. For example, the entropy can be expressed in terms of Ω (and, therefore, in terms of the generating functional Z),

∂Ω S = − . (3.6) ∂T

For neutron stars one works at T = 0 and then Ω = E − µN is the total energy, corrected by the chemical potential, of the ground state at a given µ. The ground state is the state with the minimal value of E −µN. We can think of the chemical potential as a parameter that controls the total charge (like baryon number) in the system. The grand canonical ensemble corresponds to a situation in which our system can exchange charge, and energy, with the reservoir, which is appropiate for relativistic systems. The chemical potential and the temperature of our system are equal to those in the reservoir.

The derivative of −Ω with respect to the volume produces the pressure [34]

∂Ω P = − . (3.7) ∂V

For a macroscopic translation-invariant system, Ω is proportional to the volume V . There- fore, ∂Ω/∂V = Ω/V , which implies that [34]

Ω = −PV. (3.8)

Finally, the energy of the system is

E = TS + µN − PV, (3.9) which shows that it can be increased by supplying heat(with N and V fixed), by adding particles(with S and V fixed), or by squeezing it(with S and N fixed). From the relations above one finds the following useful thermodynamic relations. The internal energy density for a fixed volume of a grand canonical ensemble is given by the fundamental Euler 29 integral [42] X = −P + T s + µini, (3.10) i where s is the entropy density, µi is the chemical potential and ni is the number density of particle i and denotes the possibility of several kinds of contributing particles. According to the Gibbs-Duhem equation X nidµi + sdT − dP = 0, (3.11) i where d denotes the differential of the corresponding quantity. Also, the first law of thermodynamics for a given volume reads X d = T ds − dP + µidni. (3.12) i Temperature, pressure, chemical potential and densities are intensive properties of matter, whereas the internal energy U, the entropy S, the volume and the particle number are extensive properties. Also, according to the Gibbs-Duhem equation, these extensive properties are not independent. With these quantities we can define the enthalpy H, the

Helmholtz free energy F , the Gibbs free energy G and the grand potential Ω as

H = U + PV,F = U − TS, (3.13)

G = U + PV − TS, Ω = U − TS − µN = −PV. (3.14)

Their diﬀerentials may be derived via Legendre transformations [34]. A physical quantity of interest can be deduced from the diﬀerentiation of the corresponding relation with respect to an appropiate variable.

From the deﬁnition of the grand potential Ω, one obtains diﬀerent equilibrium thermodynamical quantities, such as ∂Ω niV = − (3.15) ∂µi and ∂Ω sV = . (3.16) ∂T 30

3.3 General constrains in the NS-EoS

There are two important constraints that any realistic EoS must obey. So, the EoS for neutron star matter that we will use later must satisfy these conditions. Moreover, there are constraints that come from observations [17].

3.3.1 Superluminal and ultrabaric EoSs

Since the EoS for baryon number densities very high compared to the nuclear saturation density, nB n0, is very uncertain, it is important to impose model-independent bounds. The basic requirement is that any EoS should respect Lorentz invariance and causality. So, any physically reasonable EoS can be neither superluminal nor ultrabaric, where [4]

1. Ultrabaric EoS: P > ;

2. Superluminal EoS: dP/d > 1.

It is important to note that an excited medium can be superluminal without breaking

Lorentz invariance.

3.3.2 Mass bounds

Neglecting nucleon-nucleon interactions, Oppenheimer and Volkoﬀ [22] have found that the maximum allowable mass for a NS is given by Mmax = 0.71M , which is (approximately) half of the Chandrasekhar mass limit for white dwarfs. Today it is known that the mass of the pulsar in the Hulse-Taylor binary is, to high pre- cision, 1.44M [43]. So, nuclear strong interactions are very important. Moreover, the maximum masses for an EoS produced by some model must satisfy the following relation:

max Mmax(EoS) ≥ Mobs . (3.17)

For example, in ﬁgure 3.1 it is shown mass-radius relations for diﬀerent EoSs from [6]. Any EoS curve that does not intersect the J1614−2230 band is ruled out by this measurement. 31

In particular, most EoS curves involving exotic matter, such as kaon condensates [44] or hyperons [45], tend to predict maximum masses well below 2.0M and are therefore presently ruled out. If one includes the eﬀect of rotation, then it is known that this increases the maximum possible mass for each EoS [4]. More recent measuments by

Antoniadis et. al [17] of the PSR J0348 + 0432 give a NS mass of M = (2.01 ± 0.04)M . Notice that the realistic minimal model for neutron stars made of neutrons, protons, electrons and muons also survives this test.

Figure 3.1: This plot shows nonrotating masses versus physical radii for several typical EoS [5]: blue, nucleons; pink, nucleons plus exotic matter; green, strange quark matter. The horizontal bands show the observational constraint from the pulsar mass measurement J1614 − 2230 presented in [6] of (1.97 ± 0.04)M , similar measurements for two other millisecond pulsars ([7,8]) and the range of observed masses for double neutron star binaries ([9]). The grey regions show parameter space that is ruled out by other theoretical or observational constraints, like general relativity (GR) and spin period (P). Extracted from [6]. 32

3.3.3 Constraints in the nuclear matter equation of state

The models for the EoS that will be used to describe nuclear matter must respect some very important properties up to the nuclear saturation density n0. Nuclei are droplets of self-bound nuclear matter (NM). NM is an ideal inﬁnite system with baryon number A =

N +Z −→ ∞, being N and Z the number of neutrons and proton, respectively. Coulomb forces have been switched off. It is also a uniform medium, with nucleon baryon density nB = nn + np and asymmetry parameter δ = (nn − np)/nB, so that nn = (1 + δ)nB/2, np = (1 − δ)nB/2. Charge symmetry of nuclear forces implies that E(nB, δ) = E(nB, −δ) [10]. In figure 3.2 we can see the energy per nucleon versus baryon number density for symmetric nuclear matter (δ = 0), asymmetric nuclear matter with δ = 0.4 (such an asymmetry corresponds to the neutron-drip point in a neutron star crust and to a central core of a newly born protoneutron star), and pure neutron star matter (δ = 1). Minima of the E(nB) curves are indicated by filled dots [10]. Dotted segments correspond to negative pressure. Calculations are performed for the SLy4 model of effective nuclear Hamiltonian,

−3 which was used to calculate the SLy EoS. It yields n0 = 0.16fm and E0 = −16.0MeV [10].

Nuclear matter properties

Since the binding energy B0 = −E0 = maximum binding energy per nucleon in nuclear matter (where at saturation point means nB = n0, δ = 0), then, at the vicinity of the saturation point, one has

2 2 K0 nB − n0 E(nB, δ) ' E0 + S0δ + , (3.18) 9 n0 where S0 and K0 are the nuclear symmetry energy and incompressiblity at the saturation point, respectively, and are given by:

1 ∂2E ∂2E S = ,K = 9 n2 . (3.19) 0 2 ∂δ2 0 B ∂δ2 nB =n0, δ=0 nB =n0, δ=0 33

For these properties experiments give us [10]:

−3 n0 = (0.16 ± 0.01)fm ,B0 = (16.0 ± 1.0)MeV, (3.20)

S0 = (32 ± 6)MeV,K0 ≈ 230MeV. (3.21)

Figure 3.2: Energy per nucleon vs baryon number density for diﬀerent values of δ. Ex- tracted from [10].

Nuclear experiments

In the next chapter we will use some speciﬁc hadrons to model the EoS for nuclear matter and, therefore, it is useful to review some of their fundamental properties. The experimentally important meson measured masses are [10]:

mπ = 138MeV, mη = 548MeV, mρ = 769MeV, mω = 783MeV. (3.22)

The scalar σ meson plays an important role in the relativistic mean field theory that we will see later. It represents a scalar state of an exchanged pion pair (ππ), and its mass is 34 found from fitting the meson exchange model to nucleon - nucleon (NN) scattering data which gives mσ = 550MeV [10]. One can see these meson exchanges in figure 3.3.

Figure 3.3: Feynman diagrams describing the most important meson exchange processes which contribute to the NN interaction. Time goes upwards. Thin vertical lines: nucleons. Thick vertical segments: ∆(= Nπ) resonance in an intermediate state. Extracted from [10].

Apart from the experimental meson masses, the meson exchange model contains coupling constants determined by ﬁtting experimental data [10].

Finally, in order to account for ﬁnite sizes of interacting hadrons, one has to introduce form factors at every meson-nucleon vertex. The form factors describe the eﬀect of shortest- range strong interactions, which depend on the quark structure of baryons, and are not calculable within the meson exchange model [10].

Nuclear beta and electric charge equilibrium

The hadronic EoS is constrained to satisfy the following weak nuclear processes:

n −→ p + e +ν ¯e, p + e −→ p + µ +ν ¯µ, (3.23) 35

n −→ p + µ +ν ¯µ, p + µ −→ n + νµ. (3.24) which will induce β-stability. Each one of these processes has an associated baryon number density. This is arbitrary. However, as we are dealing with neutron stars then they must be electrically charge neutral. This is done in the standard way (see [10]). We consider that the core of the neutron stars is transparent to neutrinos as soon as [4]

T ≤ 109 ∼ 1010K, (3.25) so they will not aﬀect the thermodynamics.

3.3.4 Constraints on the quark matter EoS

Perturbative QCD gives us the thermodynamic potentials Ω. Apart from thermodynamical consistency checks, we need some additional constraints to apply them to the physics of neutron stars.

Weak interaction equilibrium

Electromagnetic interactions are negligible for the EoS, and weak interactions enter the problem only indirectly by opening some channels for reaching the ground state of matter. Chemical equilibrium is reached via the following weak processes where (for reasons of the energy scale for quark matter in neutron stars) we only need to keep the quarks up (u), down (d) and strange (s) [46]:

d −→ u + e +ν ¯e, u + e −→ d + νe (3.26)

s −→ u + e +ν ¯e, u + e −→ s + νe, (3.27)

s + u d + u, (3.28) which imply the following conditions:

µs = µd ≡ µ, µu = µ − µe, (3.29) 36

with µe being the electron chemical potential. Neutrinos escape quickly as their mean free path is considerably larger than the physical size of the system. So, we put µν = 0.

Notice that we can restric ourselves to two the quark chemical potentials: µ, and µe [46].

Charge neutrality

Local charge neutrality relies on the presence of electrons and leads to the relation

2 1 1 n − n − n − n = 0, (3.30) 3 u 3 d 3 s e

3 2 where ne = µe/(3π ) is the electron number density. Finally, solving the two last equations

ﬁxes the electron chemical potential µe as a function of µ and then one can express all the quark chemical potentials only in terms of µ.

3.4 Phase transition

The picture presented by lattice QCD for T ≥ 0, µB = 0 cannot be easily extended to the case µ 6= 0, the reason being that standard Monte Carlo simulations can only be applied to the case where either µ = 0 or is purely imaginary. Simulations with µ 6= 0 are hindered by the sign-problem [41], though some mathematical extensions of lattice techniques [47] can probe this region. On the other hand, a number of different-model approaches indicate that the transition along the µ axis, at T = 0, is strongly first order [48, 49, 50, 51, 52, 53, 54, 55]. Since the first order line originating at T = 0 cannot end at the µB = 0 axis which corresponds to the starting point of the cross-over line, it must terminate somewhere in the middle of the phase diagram. This point is generally referred to as the Critical End Point (CEP). The location and observation of the CEP continue to be at the center of efforts to understand the properties of strongly interacting matter under extreme conditions. Lattice results predicts only a possible CEP and a possible first order line [56]. 37

So, if someone would explore in some way the interior of a NS from the surface this person would discover ﬁrst a purely hadronic phase. Going deeper into the star means increasing the chemical potential and, at some radius, there would be a critical chemical potential (therefore critical pressure and critical radius) where a new phase of mater begins, the phase of deconﬁned quark matter.

The EoS for cold quark matter ([36, 57]) was calculated for the bulk, i.e. the case when one considers the volume of the system (neutron stars) very small compared to the volume taken at the thermodynamic limit. Then if Rsystem ∼ RNS is the radius of the system we have 4π R3 V , (3.31) 3 NS TL where RCS is the typical radius of a neutron star and VTL is the volume of the thermodynamic limit. The reason of doing this is to avoid surface effects that would appear in systems of finite size. This way of modeling the problem serves as a starting point where additional effects would be considered later. The degrees of freedom of the deconfined phase are not localized into individual hadrons, instead they have wave functions that range over the entire size of the system.

In this work we investigate if there are energy density windows that allow for a smooth or discontinuous match of quark matter and hadronic matter EoSs. In general, the matching of two EoS at T 6= 0 and µ 6= 0 requires some general conditions. First, let us introduce the symbol Θ to stand for either ”T “ or ”µ“ so that we may discuss the matching in full generality. To perform the matching, we impose that at the phase transition point the pressures of the two phases are equal. After that we use the thermodynamic constraint that the pressure of a system must increase with Θ:

P (Θ + ∆Θ) ≥ P (Θ), (3.32) 38 and that above a phase transition point, the physical phase is the one with higher pressure.

The pressure can also depend on other variables, X, for one of the two phases. In the plots of Chapter 4, for the matchings we solve the following set of two equations with two unknowns Θ0 and some physical scale which now we only call X for generality:

Pphase1(Θ0) = Pphase2(Θ0,X0), (3.33)

dP (Θ ) ∂P (Θ ,X ) phase1 0 = phase2 0 0 , (3.34) dΘ ∂Θ for some critical values Θ0 and X0. The second of these equations amounts to assuming that the phase transition could be of second-order. Less restrictive schemes can be im- plemented by allowing for a ﬁrst order phase transition where the only equation that is valid the ﬁrst one.

In this work, the matching of a hadronic (phase 1) and quark matter (phase 2) EoS relies on taking Θ ≡ µB, where µB is the baryonic chemical potential, and T = 0, which is useful to discuss neutron stars. To be clear we impose the following conditions [36]:

1. At the matching point µc, Phadron(µc) = Pquark(µc).

2. Both the hadronic and quark matter phases are locally charge neutral.

p 3. The speed of sound cs ≡ dP/d has to be less than the speed of light in both phases.

4. The energy density, , has to increase monotonically with µB.

For a more general treatment:

1. Criterion (2) can be easily relaxed by considering a two-component system that is only globally charge neutral (see [58]). This is, however, only a minor eﬀect

in comparison with the other uncertainties in the calculation and an unnecessary accuracy to perform a detailed analysis of the two-component phase transition [59]. 39

2. The last two criteria are, on the other hand, meant to impose naturaless on the

resulting EoS: cs < 1 is required to maintain causality [19], and it would be quite

bizarre if for any given µB matter composed of nuclei could have an energy density higher than that of quark matter (except if nuclei did not correspond to the true

ground state of hadronic matter).

3. Together with criterion (1), the monoticity of the energy density also implies that,

above µc, the physical phase is always the one with larger pressure. This is an important consistency veriﬁcation of our matching treatment [4]. 40

Chapter 4

Matching Equations of State

From the beginning of this thesis we have emphasized that neutron star is not an accurate name for this kind of astrophysical objects because they are not composed only of neutrons but also have other particles to be charge neutral and in β-equilibrium. This is well known since the work of Oppenheimer and Volkoff [22]. With the appearence of QCD we have learnt that at high densities quarks and gluons could become the adequate degrees of freedom. So, it is natural to investigate how these effects could be included in the EoS for neutron-star matter. It was first postulated that the cores could have high enough densities to possess a quark-gluon plasma and, as one moves out from the center, the density decreases up to a point where a phase transition could occur from quark to hadronic matter. From this radius up to the surface we would have to use our normal EoS for nucleonic matter. Therefore, we can model accurately a “neutron star” an hybrid star, i.e. a star whose core has quarks and gluons and an exterior mantle of nucleonic matter.

We can not deduce this from ﬁrst principles since one does not know how to treat the transition exactly. The next possible approach is to match the EoSs for each phase of matter. This will aﬀect the mass-radius relation. First, we will review some of the important ingredients to construct the EoS for each phase of matter, then the hybrid case. 41

4.1 Nuclear matter EoSs

Usually the nuclear matter phase EoS (NM-EoS) is calculated by doing microscopic many- body calculations using phenomenological mean-ﬁeld theories (MFT) or by means of realistic nucleon-nucleon interaction potentials. The potentials have to provide an accurate description of the measured nucleon-nucleon (NN) scattering data from low-energy nuclear physics. The technical tools to calculate the NM-EoS are the variational [11] or quantum Monte-Carlo techniques [60]. Variational techniques rely on the parametrization of the nuclear wavefunctions combined with correlated interactions between pairs of nucleons, arising because of Fermi statistics. In contrast, quantum Monte Carlo techniques are not limited by the form of the variational wavefunctions and can potentially include all possible correlations in the many-body system i.e. it is more complicated computationally because the number of required operations grows roughly as 2AA!/(N!Z!), where A, N and Z are the baryon, neutron and proton numbers, respectively. This method suﬀers, in principle, from the sign problem [61].

4.1.1 General problem

There are many terms which we have to include into the nucler matter Hamiltonian to make our model more realistic. These terms are the following:

Three-body interactions

Normally, as a ﬁrst step, two-body hadronic interactions are introduced. However, they give only a part of the hadronic Hamiltonian of dense matter. For baryonic densities near the saturation density, nB ∼ n0, interactions of three or more hadrons might be important. The three-nucleon (NNN) potentials are necessary to reproduce properties of H-3 and He-4 and to obtain correct parameters of symmetric nuclear matter at saturation density. Our experimental knowledge of three-body interaction is restricted to nucleons [10]. 42

Meson-exchange of NN interaction

Also important are the nucleon-nucleon (NN) interactions. They result from the exchange of virtual mesons, with range of 1/mπ ' 1.4 fm. As it is shown in figure 4.1, at lowest order we have one pion exchange and, therefore, a one pion exchange potential. Current models treat nucleon fields coupled to meson fields in meson exchange models (MEM).

The mesons involved are:

1. Pseudoscalar mesons π, η (J P = 0−),

2. Scalar mesons σ, δ (J P = 0+),

3. vector mesons ρ, ω (J P = 1+), where J P denotes the meson spin J and parity P [10].

Figure 4.1: Feynman diagrams for three possible mesonic interactions. Extracted from [10]. 43

4.1.2 “Minimal” model

In view of such a high degree of ignorance at densities of the order of saturation density, it seems reasonable to start with a model which is the simplest, and not obviously wrong.

Such a “minimalistic” approach consists in extending the neutron, proton, electron and muon model to nB ≥ 2n0. The calculated EoS has to be confronted with observations to see whether it is suﬃcient to explain observational data. After fulﬁlling this minimal procedure, we can try richer models, including hyperons and exotic phases of hadronic matter. Whatever model of dense matter one assumes, one should calculate its ground state as a function of energy density [10].

4.1.3 Solving the many-body problem

The general formula for the ground-state energy per baryon of a system of A baryons is

D ˆ E Ψ0|HB|Ψ0 EB = , (4.1) A hΨ0|Ψ0i

ˆ where HB is a baryon Hamiltonian operator and Ψ0 is a ground-state wavefunction of the system. In our case EB should be calculated in the thermodynamic limit, i.e. A −→ ∞ and V −→ ∞.

In the simplest (“Minimal” approach) case of nucleon matter, the calculation will give

EB as a function of the neutron and proton number densities, nn and np, respectively.

The knowledge of EB(nn, np) is suﬃcient for calculating the EoS of matter consisting of nucleons and leptons. Leptons contribute in the form of on EoS for Fermi free gases. The technical methods to solve these are the following: 44

Variational solution

Based on the variational principle of Quantum Mechanics [62] we can get the following solution for the ground state of the above problem:

D ˆ E Ψvar|HB|Ψvar (var) (exact) EB = ≥ EB , (4.2) A hΨvar|Ψvari where Ψvar is a trial wave function. This method consists in minimizing the energy

(var) functional EB within a set of trial wave functions, which should be suﬃciently rich ˆ in their structure, reﬂecting the structure of HB. There has been many monumental computational achievements (like that of [11]).

Eﬀective baryon Hamiltonians or Lagrangians

We can do simple calculations provided that we have at hand eﬀective Hamiltonians or Lagrangians (see, for example, [12]).

1. Nonrelativistic case: The eﬀective Hamiltonian is based partly on experimental

data and partly on selected many body results for pure neutron matter:

D ˆ eff E ΨHF |H |ΨHF (HF ) B (exact) EB = ' EB . (4.3) A hΨHF |ΨHF i

where ΨHF means Hartree-Fock trial wave functions [10].

2. Relativistic mean-field approximation: In this model the correlations are ne- glected. Usually it is only used the Hartree approximation. One uses the relativistic mean field theory. The parameters of the model are fixed by comparison with nu-

clear saturation properties provided by experimental data [10].

EoS for neutron-star matter

We have shown that we need to know the energy density for pure neutron-star matter and symmetric nuclear matter to calculate the composition for each one of them at a given 45 baryon number density. The pressure P is then given by the law of thermodynamics [4]

2 ∂ (nB, xp) P = nB , (4.4) ∂nB nB

where xp is the proton fraction, i.e. the ratio of the proton number density and the baryonic number density. In our models, the EoS comes as a table relating energy density

, pressure P and baryon number density nB. In the following two subsections we will present two EoSs that will be use along the rest of this chapter.

4.1.4 NM-EoS1: Akmal-Pandharipande-Ravenhall (APR)

∗ This EoS was originally (and also technically) called A18 + δv + UIX , and was introduced by Akmal, Pandharipande and Ravenhall [11]. This is the typical representant of variational theory which includes asymmetric nuclear matter properties. It reproduces the properties of cold catalyzed matter for nB > (1/2)n0. For this thesis we will use a tabulated APR-EoS that takes into account, for lower densities the usual Baym et. al. EoS for nuclear matter [63]. This EoS describes β-stable and charge-neutral nuclear matter. It includes correlations among nucleons and solve them using the variational chain summation. Akmal et. al. [11] employ a realistic nonrelativistic Hamiltonian with the

Argonne v18 two-body potential, the Urbana IX three-nucleon interaction and a relativistic boost correction δv. This EoS is written as a function of baryon density nB and proton fraction xp. It is obtained by interpolating between the extreme results of pure neutron matter(xp = 0) and symmetric nuclear matter(xp = 0.5) using a generalized Skyrme interaction containing momentum- and density-dependence via delta function interactions

(see ﬁgure 4.2). The energy density is evaluated for a variational wave function that takes into account many-body correlation eﬀects. 46

Figure 4.2: Proton fraction xp as a function of the baryonic density ρ (this notation only for the original article of [11]). Extracted from [11].

The APR-EoS exhibits a transition from a low-density phase to a high-density phase having spin-isospin order, possibly due to neutral pion condensation in pure neutron matter at a density of ∼ 0.2 fm−3 and in symmetric nuclear matter at ∼ 0.32 fm−3 [11]. The variational chain summation calculations of the energy of pure neutron matter and symmetric nuclear matter are extrapolated to general values of xp using a function of the form [31]:

1 1 N (nB, xp) = + f(nB, xp) τp + + f(nB, 1 − xp) τn+ 2mN 2mN 2 2 g(nB, xp = 0.5)[1 − (1 − 2xp) ] + g(nB, xp = 0)(1 − 2xp) , (4.5)

motivated by a generalized Skyrme interaction. Here N is the total nuclear energy density,

τn,p are the neutron and proton Fermi gas kinetic densities, f(nB, xp) and f(nB, 1 − xp) are functions that parameterize the eﬀective mass of nucleons, and g(nB, xp = 0) and g(nB, xp = 0.5) are potential energy terms. In ﬁgure 4.3 we present the APR-EoS in a form useful to solve the TOV equations. 47

5 APR ]

3 4

2 P [ GeV / fm

0 0 1 2 3 4 ϵ[GeV/fm 3] Figure 4.3: APR-EoS for pure hadronic matter. See [11].

In general, the energy density of nuclear matter is used to determine the allowed EoS of hybrid stars for the case of a possible sharp or smooth transition to quark matter and to determine the allowed phases for a mixed transition to quark matter. Figure 4.4 represents the same APR-EoS but normalized with respect to the Stefan-Boltzmann gas

PSB defined as [34]: µ4 P (µ) = N N , (4.6) SB c f 12π2 where Nf is the number of massless quark flavours, Nc is the number of color degrees of freedom and µ is the quark chemical potential where µ per quark is the same as the Fermi momentum. Since the APR - EoS, PAP R = PAP R(µB), is written in terms of the baryon chemical potential, then we define it in terms of the quark chemical potentials

X µB = µNf . (4.7)

So, for this thesis Nc = Nf = 3, corresponding to three color degrees of freedom and three quark ﬂavours: up (u), down (d) and strange (s). Then

µB = µu + µd + µs, (4.8) 48 where since we are treating the massless case we have [46]:

µu = µd = µs ≡ µ, (4.9)

then µ = µB/3. Thus, our SB-EoS is

3 µ 3 P (µ ) = B (4.10) SB B 4π2 4 which is written in terms of baryonic chemical potential µB to make the normalization of the APR-EoS easily [57]. We will use this same SB-EoS to normalize another EoSs in the following sections.

1.0

0.8 APR

0.6

P / Psb 0.4

0.2

0.0 0 1 2 3 4 5 μB[GeV] Figure 4.4: Normalized APR-EoS for pure hadronic matter. 49

4.1.5 NM-EoS2: TM1 parametrization (TM1)

This EoS is based on calculations in the context of the relativistic mean field theory which is an effective field theory of QCD for many-body nuclear systems and whose goal is to describe nuclear matter near and above the nuclear saturation density [4]. From the beginning, this model has many free parameters and, depending on their values one has slightly different models for the problem at hand. In this thesis we will use the so-called TM1 parameterization used by Shen et. al [12] and references therein. The original (renormalizable) Walecka model [64] contains nucleons interacting through the exchange of the Lorentz isoscalar σ and vector ω mesons. However, this model predicts a very large (unrealistic) value for the compression modulus K0 = 545 MeV. In order to improve the incompressibility and finite nuclei results of the Walecka model, one has to supplement it with additional cubic and quartic scalar self-interaction terms, the so-called non-linear σ-ω model. As a result a reduced value of the compression modulus, consistent with the isoscalar giant monopole resonance in Pb-208, has been obtained [65].

The ω meson quartic self-interaction term has been considered by Bodmer [66] to obtain the positive value of the quartic scalar self-interaction coeﬃcient as its negative-value may mean that the energy spectrum has no lower bound. The inclusion of this vector meson quartic self-interaction term leads to a softening of the EoS at high density [10].

The Lagrangian of nuclear matter, constructed on the basis of the original Walecka model and extended by the inclusion of nonlinear scalar and vector self-interacting terms, is given by (in the remainder of this section we follow [4, 64]):

1 1 L = ψ¯ [iγµD − (m − g σ)] ψ + ∂ σ∂µσ − Ω Ωµν− N µ N σN N 2 µ 4 µν 1 1 1 1 Ra Raµν + m2 ω ωµ + m2ρa ρaµ − U(σ) + c (ω ωµ)2, (4.11) 4 µν 2 ω µ 2 ρ µ 4 3 µ 50 where p ψ = (4.12) N n is the bispinor of the nucleon ﬁeld, Dµ denotes the covariant derivative deﬁned as

a a Dµ = ∂µ + igωN ωµ + igρN I3N τ ρµ, (4.13) and the scalar potential function U(σ) has a well-known polynomial form

1 1 1 U(σ) = m2 σ2 + g σ3 + g σ4. (4.14) 2 σ 3 3 4 4

The nucleon mass is denoted by mN (N=n,p) whereas mi (i= σ, ω, ρ) are the masses

a assigned to the meson ﬁelds. The ﬁeld tensors Rµν and Ωµν have the following forms

a a a b c Rµν = ∂µρν − ∂νρµ + gρN εabcρµρν, (4.15)

Ωµν = ∂µων − ∂νωµ. (4.16)

The parameters that enter into the Lagrangian, are determined by ﬁtting to the bulk properties of nuclear matter (see [4]). The Euler-Lagrange equations in the presence of the interactions can be written as

µ 2 2 3 ¯ (∂ ∂µ + mσ)σ(x) + g3σ (x) + g4σ (x) = gσN ψN (x)ψN (x), (4.17)

1 ∂ Ω (x) + m2 ω (x) + c (ω (x)ω (x))ων(x) = g ψ¯ (x)γµψ (x), (4.18) ν µν ω µ 6 3 µ µ ωN N N 1 ∂ Ra (x) + m2ρa (x) = g ψ¯ (x)γµτ aψ (x), (4.19) ν µν ρ µ 2 ρN N N 1 γµ i∂ − g ω (x) − g γµτ aρa (x) − (m − g σ(x)) ψ (x) = 0. (4.20) µ ωN µ 2 ρN µ N σN N The Eq. (4.17) is the Klein-Gordon equation with a scalar source term and nonlinear self- interactions. Equation (4.18) is the equation for a massive vector ﬁeld with the conserved baryon current

µ ¯ µ a JB = ψN γ ψN ≡ (nB,JB), (4.21) 51 satisfying

µ ∂µJb = 0. (4.22)

0 ¯ 0 Its time component JB≡nB = ψN γ ψN is the baryon density. The equation of motion of the ρ field (4.20) is similar to Eq. (4.17). The difference is connected with the isovector character of the ρ meson field. The conserved isospin current for nucleons has the form 1 J µa = ψ¯ γµτ aψ . (4.23) 2 N N µ3 ¯ µ 3 Its 3-component can be written as J = (1/2)ψN γ τ ψN and, for the τ3 representation

τ3 = diag(+1, −1), (4.24) the isospin density has the form

1 J 0 = (ψ†ψ − ψ† ψ ). (4.25) 3 2 p p n n

Solutions of the equations of motion can be found within the relativistic mean field approximation for which meson fields are separated into classical mean field values and quantum fluctuations which are not included in the ground state

σ(x) =σ ¯(x) + s0, (4.26)

ωµ(x) =ω ¯µ(x) + hwµi , (4.27) and

a a 3a ρµ(x) =ρ ¯µ(x) + hrµi δ . (4.28)

In the relativistic mean-ﬁeld approach the equations of motion turn out to be

2 2 3 ¯ mσs0 + g3s0 + g4s0 = gσN ψN ψN , (4.29)

2 3 ¯ 0 mω hw0i + c3 hw0i = gωN ψN γ ψN , (4.30) 52

2 3 ¯ k mω hwki + c3 hwki = gωN ψN γ ψN , (4.31) 1 m2 hr i = g ψ¯ γ0τ 3ψ , (4.32) ρ 0 2 ρN N N 1 m2 hr i = g ψ¯ γkτ 3ψ . (4.33) ρ k 2 ρN N N In the momentum space the Dirac equation has the following form:

1 γµ(k − g hw i − g τ 3 hr i) − (m − g s ) ψ (k) = 0. (4.34) µ ωN µ 2 ρN µ N σN 0 N

The system which is considered is assumed to be the isotropic, i.e. inﬁnite matter in its ground state. For inﬁnite isotropic matter the space components of hωµi and hρµi vanish

(hωµi = hρµi = 0 for µ 6= 0). The classical mean ﬁeld values can be written as

hωµi = w0δµ0 (4.35) and

hρµi = r0δµ0. (4.36)

The resulting ﬁeld equations for this approximation have a reduced, simpler form

2 2 3 ¯ mσs0 + g3s0 + g4s0 = gσN ψN ψN , (4.37)

2 3 ¯ 0 mωw0 + c3w0 = gωN ψN γ ψN , (4.38) and 1 m2r = g ψ¯ γ0τ 3ψ . (4.39) ρ 0 2 ρN N N For inﬁnite, isotropic matter the energy eigenvalues are given by the following relation

1 E (k) = g + g r ± k2 + (m − g s )21/2 , (4.40) ± ωN 2 ρN 0 N σN 0 where the eﬀective nucleon mass meff,N is deﬁned as

meff,N = mN − gσN s0. (4.41) 53

¯ 0 The ground state expectation values of the baryon density ψN γ ψN , scalar density ¯ ¯ 0 3 ψN ψN and isospin density ψN γ τ ψN are given by the following relations

N Z kF ¯ X 1 mN − gσN s0 2 ρs≡ ψN ψN = k dk, (4.42) π2 p 2 2 N=n,p 0 k + (mN − gσN s0)

kN X 1 Z F X k3 n ≡ ψ¯ γ0ψ = k2dk = N , (4.43) B N N π2 3π2 N=p,n 0 N=p,n k3 k3 ρ ≡ ψ¯ γ0τ 3ψ = p − n , (4.44) 3 N N 3π2 3π2 where kN is the Fermi momenta for nucleons. The nucleon Fermi energy µN = E(k) and thus the eﬀective nucleon chemical potential can be deﬁned as

νN = µN − gωN w0 + I3N gρN r0. (4.45)

In order to calculate the energy density and pressure of nuclear matter the energy- momentum tensor Tµν, which is given by the relation

∂L ν Tµν≡ ∂ φi − ηµνL, (4.46) ∂(∂µφi) has to be used. In this last equation φi denotes boson and fermion ﬁelds.

The energy density equals < T00 > whereas the pressure P is related to the statistical average of the trace of the spatial component Tij of the energy-momentum tensor. Calcu- lations done for the considered model lead to the following explicit formula for the energy density [10]: 1 3 1 = m2 w2 + c w4 + m2r2 + U(s ) + , (4.47) 2 ω 0 4 3 0 2 ρ 0 0 N where kN X 2 Z F = k2dkpk2 + (m − g s )2, (4.48) N π2 N σN 0 N=p,n 0 and the pressure 1 1 1 P = m r2 + m w2 + c w4 − U(s ) + P , (4.49) 2 ρ 0 2 ω 0 4 3 0 0 N 54 where

kN X 1 Z F k4 PN = . (4.50) 3π2 p 2 2 N=n,p 0 k + (mN − gσN s0)

The form of the EoS (see [67], [68]) determines the physical state and composition of matter at high densities. In order to construct the NS model through the entire density span, it is necessary to add the EoS characteristic for the inner and outer core, relevant to lower densities. Thus, a more complete and more realistic description of a neutron star requires taking into consideration not only the interior region of a neutron star, but also its remaining layers. In these calculations the composite EoS for the entire neutron star density span was constructed by joining together the EoS of the neutron rich matter region, the Negele-Vautherin EoS [69] and Bonn [70] for the relevant density range between 1014 and 5 × 1010 g/cm3 and the Haensel-Pichon EoS [71] for the density region 9.6 × 1010 g/cm3 to 3.3 × 107 g/cm3. Since the density drops steeply near the surface of a NS, these layers do not contribute significantly to the total mass. The inner neutron-rich region up to the density ρ ∼ 1013 g/cm3 influences decisively the neutron star structure and evolution. In figure 4.5 we also present the proton fraction xp behaviour for different baryonic densities for this parameterization. 55

Figure 4.5: Proton fraction for the TM1 parametrization in the mean-ﬁeld theory for diﬀerent temperatures. In this thesis we only need the case of T = 0. Extracted from [12].

In this thesis we use this EoS in a tabulated form. The parametrization also includes the low density sector used to describe the crust of the neutron star. This EoS is presented in ﬁgure 4.6. 56

APR 5

] TM1 3 4

2 P [ GeV / fm

0 0 1 2 3 4 ϵ[GeV/fm 3] Figure 4.6: Nuclear matter (NM) equations of state that will be used in this thesis. There are only tabulated tables for both EoSs.

Finally, for purposes of future comparison we present in ﬁgure 4.7 both normalized

EoSs for hadronic matter and see its behaviour with increasing baryonic chemical potential. 57

1.0

0.8 TM1 APR

0.6 SB

P / 0.4

0.2

0.0 0 1 2 3 4 5

μB[GeV] Figure 4.7: Normalized Nuclear-EoSs

4.1.6 Diﬀerences in techniques

1. The mean-ﬁeld theory (in the TM1 parameterization) EoS is stiﬀer than the vari-

ational one (see e.g. [11]) at low densities and softer at higher densities (see ﬁgure 4.6), leading in the next sections to diﬀerent predictions of maximum masses of NSs.

2. The typical proton fraction, xp, in the mean-ﬁeld theory (in the TM1 parameterization) EoS is higher than in the variational at high densities (see ﬁgures 4.2 and

4.5). Since the diﬀerences can in part be attributed to diﬀerences in the density dependence of the nuclear symmetry energy.

As one increases the energy density and approaches the deconfinement transition region, the quantitative accuracy of these models begins to suffer problems. The primary uncertainties in the NM-EoS are coming from the unknown composition of matter, that is, the conjectured presence of hyperons or kaon condensation, for example. Secondary 58 uncertainties are details of the calculations such as the exact form of the respective variational ansatz or the effect of neglecting the simulatenous interactions of more than two nuclei (see [11, 45, 44]). 59

4.2 Unpaired Quark matter

To describe bulk quark matter (QM) at T = 0 1 (which could be found at the core of neutron stars) one has to use EoSs for cold superdense matter, i.e. relatively high baryonic chemical potentials µB ∼ 1 GeV. However, the phase diagram of QCD for this order of

5 µB is poorly known [32]. Only at asymptotically large µB, of the order of 10 GeV, the physical phase is known, the so-called color-ﬂavor-locked (CFL) superconducting phase (CSC), as the ground state of three-ﬂavour matter (very similar to the BCS theory of usual superconductivity) [76]. Although it may persist to much lower chemical potentials, at these high chemical potentials one can consider the quarks up (u), down (d) and strange

(s) eﬀectively as massless. At lower µB the strange quark mass ms becomes nonnegligible

CFL relative to the chemical potential. At some value of chemical potential µB there must exist a transition to unpaired (normal) three-ﬂavour quark matter [77]. If one includes nuclear matter, then there is a three-way competition between nuclear matter, unpaired quark matter, and CFL quark matter for relatively high µB. The modeling would consist in adding to the total pressure a term accounting for the condensation energy of Cooper pairs in the CFL phase [76, 78, 77]:

∆2µ2 P ≡ B (4.51) CSC 3π2 where the baryon chemical potential is µB ≡ µu + µd + µs, and the gap parameter ∆ approaches at asymptotically high densities the form [79]:

2 √ bµ −3π / 8παs ∆ = 5/2 e (4.52) (4παs) with b a constant and αs is the strong coupling constant of QCD which will run with the energy scale of the problem at hand. Along this thesis we will not take into account color superconducting eﬀects, since the EoS is not very sensitive to this eﬀect [36].

1Strictly speaking, the temperature in the interior of neutron stars is not zero obviously, however as the typical baryonic chemical potentials are much bigger than the temperature scales one can consider as a good approximation the case of vanishing temperature. 60

4.2.1 The bag model

Color conﬁnement in hadrons has stimulated a minimalistic approach known as the bag model. The simplest and useful descriptions of the bag model are presented in the following lines.

Warmup: Bogoliubov bag model

The idea of this eﬀective nonperturbative QCD model is to give quarks an enormous mass. This would conﬁne the quarks by making them unable to move [80, 81]. However, this would seem to contradict the asymptotic freedom observed at a very short distances.

Bogoliubov solved this by confining the quarks within a spherical cavity of radius R in which they feel an attractive field of strength m. The quark masses were also set to m, and then one lets m −→ ∞. This resulted in Bogoliubov’s bag-model, where the quarks could move freely inside the bag but were completely confined within it. Although Bogoliubov’s model was a very simple one, it still gave some accurate predictions [82]. The MIT bag-model is a continuation of this simple model.

MIT bag model

This is a very elegant model, although naive but which serves us as a toy model [81], treats all hadrons like small drops of another perturbative phase of QCD. All nonperturbative physics is included in one universal quantity, the famous bag constant, B, which describes the difference in energy density between the perturbative (interior of the bag) and nonperturbative (physical) vacua. This effective model tries to describe confinement in a very crude way. The Hamiltonian of the model is a sum of three terms

H = Hkinetic + Hspin−spin + BV, (4.53) where the first term describes the kinetic energy of the quarks confined in the bag, the second describes the spin-spin interaction, and the last “bag” term contains the bag constant B times the volume of the bag interior. If the quarks are (nearly) massless, it 61 is clear, for dimensional reasons, that the only answer for the first term can be E∼1/R where R is the bag radius; it tries to expand the bag. Solving the Dirac equation in a cavity with boundary conditions one gets Equark = 2.04/R [81]. More elaborate versions of the bag model included the kinetic energy not only of the confined valence quarks, but also of the virtual quark and gluon colored fields confined in a bag, the so-called Casimir energy; this too must be ECasimir ∼ 1/R for the same dimensional reasons [82]. The bag term, B(4π/3)R3, tries to contract the bag; as a result the equilibrium is reached. This bag constant (the only dimensional quantity of the model) determines the masses and sizes of all hadrons. The fit to masses of the “usual” hadrons (the “unusual” ones like pions, η0 or scalars are not reproduced) made in the original MIT bag model was rather successful, producing the following value of the bag constant [83]:

3 BMIT = 56MeV/fm . (4.54)

Theoretically speaking, the MIT bag model has a number of problems which subsequent developments have tried to cure. One serious flaw is that chiral symmetry is badly broken, which is seen from the fact that the axial current is not conserved at the bag boundary [81]. Indeed, when a quark is reflected from the bag wall, its momentum is flipped but the spin is not; thus a left-handed quark can become right-handed [33]. The other problem is that the universality of B, however beautiful it may be as a concept, is not supported by observations. In particular, the MIT bag model predicts the same scale of mass for glueballs as for ordinary mesons. Solutions of the Yang-Mills equations in a cavity bring in about the same kinetic energies as the Dirac equation, modulo numerical coefficients of the order (and really far from) unity [33]. QCD phase transitions at high T further constrain the real bag constant, defined as the difference between the vacuum and the quark-gluon plasma (QGP) ground energy, and have been shown to be indeed around a much larger value [33],

3 B = 500 − 1000 MeV/fm BMIT . (4.55) 62

Furthermore, the smallness of the MIT bag constant thus reﬂects an important generic point: all known hadrons are not at all the small drops of the new phase, but rather relatively small perturbations of the QCD vacuum. Finally, one important problem of this model is that if hadrons are small bubbles of a new phase of matter, one may expect them to behave as all other bubbles do and coalesce if they meet each other. If the nonzero surface tension is introduced it becomes more obvious, but it is true even with only the volume bag term. The nucleons in the nuclei do not coalesce like that. If one puts a pair of nucleons on top of each other a rather strong repulsive core is observed, as directly seen by the scattering phases of NN scattering [33]. Spin forces in the MIT bag model have ﬁxed this problem for the NN case, but not in general.

Therefore, bulk quark matter (in this view) is that of an ideal relativistic Fermi gas complemented by the bag constant. For Nf massless quark ﬂavors its pressure is [33]:

µ4 P (µ) = N − B, (4.56) MIT f 4π2 where µ is the quark chemical potential. It is more conveniently written in the following form [84]: 1 P () = ( − ) (4.57) MIT 3 vac where vac = 4B is the oﬀset due to the vacuum energy density. Otherwise this relativistic EoS does not lead to stable conﬁgurations [19]. 63

4.2.2 Fraga-Kurkela-Vuorinen ﬁtting for cold pQCD

The calculation of a perturbative EoS for the bulk cold quark matter in the framework of QCD (pQCD) throught the calculation of a thermodynamical potential of a plasma of massless quarks (up, down and strange) and gluons dates back to the works of Freedman

2 2 and McLerran [85, 85] and Baluni [86] to O(αs) where the strong coupling αs = g /4π is given in terms of the QCD coupling g (see also [87, 88, 89]). Addionally, they have also included eﬀects of the strange quark mass ms up to order O(αs), dropping the mass entirely

2 at O(αs). This calculation was made in the context of the so-called momentum-space substraction (MOM) scheme. However, the inclusion of the modern renormalization group

(RG) running effects for the quark mass and strong coupling is complicated and not man- ifestly gauge invariant in this scheme [90]. Fraga, Pisarki and Schaffner-Bielich [91] have found the same results but in the context of the Modified Minimal-Substraction scheme (MS). In this scheme one can easily include the effects mentioned above. They considered the massless case for the three flavors of quarks. Some years later after that work, Fraga and Romatschke [92] considered the case of a massive quark flavor up to O(αs) including RG effects on the mass and αs up two loops. The state-of-the-art perturbative QCD EoS result which include the RG effects up

2 to O(αs) in the quark mass (strange) and the strong coupling up to three loops for unpaired quark matter was calculated by Kurkela et al. [36, 59]. This calculation is relevant for neutron star physics because most importantly it takes into account the nonzero value of the strange quark mass. Morever, like all perturbative results evaluated to a finite order in the coupling αs, also the quark matter EoS is a function of an unphysical parameter: the scale of the chosen renormalization scheme (here MS scheme), the renormalization scale Λ.¯ This dependence, which diminishes order by order in perturbation theory, offers a convenient way to estimate the contribution of the remaining, undetermined orders, and thus serves as quantitative measure of the inherent uncertainty in the result. This error is 64 commonly estimated by choosing a reasonable fiducial scale and varying Λ¯ around it by a factor of two. In this work we follow this procedure, choosing as the fiducial (central) ¯ scale the commonly used value Λ = (2/3)µB [91, 92]. Kurkela et al. [36] fixed the strong coupling constant and the strange quark mass at arbitrary reference scales (using ex- ¯ ¯ perimental and lattice data), αs(Λ = 1.5GeV) = 0.326 and ms(Λ = 2GeV) = 0.938 GeV [93, 39], and then let them evolve as functions of the MS scale Λ[¯ 59]. Although this EoS is important and relevant for our studies, this result was offered in the form of a complicated numerical EoS. However, Fraga et al. [57] demostrated that this state-of-the-art result can be cast in the form of a simple, easy-to-use fitting function for the pressure in terms of the baryon chemical potential µB. All relevant observables depending on the pressure and its first and second derivatives (such as energy density as a function of pressure) are faithfully described by the fit. This fitting function, which we call FKV-EoS, obeys local charge neutrality and β-equilibrium. The FKV-EoS have the following compact form [57]:

a(X) PQCD(µB,X) = PSB(µB) c1 − , (4.58) (µB/GeV ) − b(X) where

−ν1 −ν2 a(X) = d1X , b(X) = d2X , (4.59) where we have denoted the pressure of Nf = 3 noninteracting massless quark ﬂavors and same number of colors Nc = 3 (up (u), down (d) and strange (s)) i.e. the Stefan- Boltzmann (SB) gas as: 3 µ 4 P (µ ) = B . (4.60) SB B 4π2 3 ¯ ¯ The functions a(X) and b(X) will inherit the dependence on Λ through X ≡ 3Λ/µB, which will be modulated from 1 to 4 depending on what we want to extract from the

ﬁt. The values of the constants (c1, d1, d2, ν1, ν2) are ﬁxed by minimizing the value of the following merit function [57]:

2 2 2 2 2 χ = [∆P (µB,X)] + [∆N(µB,X)] + [∆cs(µB,X)] , (4.61) 65

2 where ∆P , ∆N, and ∆cs are the diﬀerences between the values of the pressure, quark number density and speed of sound squared obtained from the ﬁt and from the corresponding full perturbative expressions of [36], normalized to the corresponding SB values.

The best ﬁt (per cent accuracy for µB < 6 GeV) occurs when [57, 94]

c1 = 0.9008, d1 = 0.5034, d2 = 1.452, ν1 = 0.3553, ν2 = 0.9101. (4.62) p We have a very good ﬁt ( χ2 . 0.01) in the region deﬁned by the conditions [94]:

µB < 2GeV,P (µB) > 0, and X ∈ [1, 4]. (4.63)

The upper limit in µB is taken to use this result in compact star physics.

4.3 Scaling the TOV equations

In this section we will review the main ingredients to solve the Tolman-Oppenheimer-

Volkoﬀ equations (TOVs) numerically.

In the TOV equations (see Sec. 2.3)

dM(r) = 4πr2(r), (4.64) dr

dP (r) GM(r)(r) P (r) 4πr3P (r) 2GM(r)−1 = − 1 + 1 + 1 − , (4.65) dr r2 (r) M(r) r it is easy to see that they contain dimensionful quantities. For purposes of calculation, it is better to transform these equations into a scale invariant form. There are two reasons for such change: one reason is that the computational treatment of differential equations benefits from a dimensionless format; the other reason is that a scaled equation needs to be solved only once. As soon as the general solution is found one can just rescale it by appropiate (dimensionful) factors to get the result for specific (astro)physical cases. The TOV equations can be scaled in the following ways: 66

1. One observes that the three correction factors in the second TOV equation (from

general relativity) are already in a dimensionless form.

2. From the ﬁrst factor, (1 + P/), one can scale pressure and energy density by a

constant common factor ”0“ as

0 0 P (r) ≡ 0P (r), (r) ≡ 0 (r), (4.66)

where P 0 and 0 are the dimensionless pressure and energy density, respectively.

3. Similarly, one deﬁnes a dimensionless mass and radial coordinate via two dimen-

sionful parameters ”a“ and ”b“ in the following form:

M(r) ≡ bM0(r), r ≡ ar0, (4.67)

and therefore M0, P 0 and 0 will depend on the dimensionless radius r0. From now on, all primed quantities mean they are dimensionless.

4. Plugging these deﬁnitions into the second TOV gives us

dP 0 bM0(r) 0(r) P 0(r) 4πa3r03 P 0(r) 2GbM0(r)−1 0 = −G 0 1 + 0 1 + 0 1 − , 0 2 02 0 0 0 adr a r 0 (r) bM (r) ar (4.68)

where r = r(r0).

5. Using the scale invariance conditions for the third and second correction terms, respectively, we obtain:

Gb a3 = 1 and 0 = 1, (4.69) a b

giving for a and b the following expressions

1 1 a = √ and b = √ . (4.70) 3 G0 G 0 67

6. So, the second TOV takes the scale invariant form

dP 0(r0) M0(r0) P 0(r0) 4πr03P 0(r0) 2M0(r0)−1 = − 1 + 1 + 1 − . (4.71) dr0 r02 0(r0) M0(r0) r0

7. Doing the same for the ﬁrst TOV one gets

bdM0(r) = 4πa2r02 0(r0) (4.72) adr0 0

or (using the above deﬁnitions for a and b)

dM0(r0) = 4πr020(r0) (4.73) dr0

We will call these new equations (4.73) and (4.75) TOV* equations.

We also have to note that in doing this scaling our original dimensionful EoS turns out to be dimensionless: P 0 = P 0(0). We will denote this new EoS by EoS*. That means that quantities scale with the factor 0 to some power. In particular, Mmax and Rmin scale as √ 0. In practice: Solve the TOV* equations for a given type of EoS* and then rescale the √ results with 0 for physical values.

4.4 Solving the TOV* equations

In practice, the TOV* equations are solved by choosing a value of central pressure P 0(r0 =

0 0) = Pc at the center of the star (r = 0) and then integrating outwards up to the surface, where P 0(r0 = R0) = 0, being R0 the radius of the star. The resulting mass and radius can

0 be calculated for any Pc 6= 0 allowed by the EoS* under consideration which will differ markedly between different classes of EoSs*, i.e. bulk quark matter EoSs and hadronic EoSs. Numerically, we have to be aware of the initial condition for the first TOV* equation for the mass contained up to a certain radius. If one would define the mass to be zero at r0 = 0, then this would cause a singularity in the TOV (TOV*) equations. Hence, for 68 numerical stability, we define the mass to be zero at some small radius very close to zero.

This should not aﬀect the result because it will be small compared to the radius of the star. Now we solve the TOV* equations numerically for various central pressures for bulk quark (QM) and nuclear matter (NM) and calculate the radius and the mass thereby. In general, the result implies masses which are related to two diﬀerent radii. That is the reason why one call mass-radius relation, i.e. because it is not a function M = M(R). The relation between the mass and radius of a star is highly sensitive to the details of the underlying EoS of high density nuclear and/or quark matter. We perform all the calculations using Mathematica [95].

4.4.1 TOV* for Nuclear matter-EoSs

After solving the TOV* equations and reinserted the dimensionful parameters in the solutions we need to study the behaviour of the solution for the allowed values for the energy density given by the EoS. To do this we make a plot of the total gravitational mass of the star versus the central energy densities. We have discussed this test in Chapter 2. For the nuclear matter EoSs we present them in ﬁgure 4.8, where one can see that the APR-EoS is more sensitive to gravitational collapse for lower values of central energy density than in the case of the relativistic mean ﬁeld theory with TM1 parametrization. 69

2.0

APR TM1

] 1.5 ⊙

1.0 Mass [ M 0.5

0.0 0 1 2 3 4 5 ϵc[GeV/fm3] Figure 4.8: Gravitational mass vs central energy density for pure hadronic matter in the APR description [11] and the TM1 parameterization of the mean-ﬁeld theory. The APR-EoS reproduces bigger maximum masses than the TM1-EoS.

The important observables of our theory of dense matter are the mass-radius relations. We can see that in this model one can easily pass the value of two solar masses which is the maximum value which have at hand observationally.

In figure 4.9 we present the APR mass-radius relation in comparison with the other model that we will use in the next sections, the relativistic mean-field theory in the TM1 parameterization. Although they reproduce approximately the same maximum masses we see from this figure that the radius for the TM1 model is larger than for the APR. We can also see that in the APR description one can easily surpass the value of two solar masses which is the maximum value obtained observationally [17]. So, the APR-EoS predicts neutron star masses in accordance with observations but which are more compact than the TM1 family of stars. 70

2.0

APR TM1 ] 1.5 ⊙

1.0 Mass [ M 0.5

0.0 0 2 4 6 8 10 12 14 Radius[Km] Figure 4.9: Total gravitational mass versus radii relations for nuclear matter equations of state in the context of the APR and TM1 descriptions.

However, observational evidence for the radius measurements are diﬃcult to obtain. It is expected that in the following years the band of errors for those radii will be reduced and a better comparison with theory could be made.

4.4.2 TOV* for Quark matter EoSs

Since the EoSs for quark matter that we use in this work have an analytic representation, it is not necessary to make plots of them again. After solving the TOVs numerically, we ﬁnd the mass versus central energy density presented in ﬁgure 4.10. For lower central energy densities the FKV-EoS produces stars with much higher masses than the MIT bag model with the standard value of the bag constant B1/4 = 145 MeV. 71

2.0 MIT bag model(B1/4 =145MeV) ] 1.5 ⊙ FKV(X=3.205) 1.0 Mass [ M 0.5

0.0 0 2 4 6 8 10 3 ϵc[GeV/fm ] Figure 4.10: Gravitational mass vs central energy density for pure quark matter in the context of the models used in this thesis: MIT bag model and FKV-EoS.

Finally, the expected mass-radius relation is shown in ﬁgure 4.11 for pure quark matter (QM) EoSs. We have used the value of X = 3.205 for the next sections in the FKV-EoS as will be discussed later. We can see that both QM-EoSs easily surpass the limit of two-solar masses.

2.0 MIT bag model(B 1/4 =145MeV) ]

⊙ 1.5 FKV(X=3.205)

1.0 Mass [ M

0.5

0.0 0 2 4 6 8 10 12 14 Radius[Km] Figure 4.11: Total gravitational mass-radius relation for pure quark matter using the MIT bag model for some value of the bag constant and the FKV-EoS for some particular value of the renormalization scale. Both models can surpass easily the two solar mass limit. 72

4.5 Matching the EoSs: Hybrid star

We will now consider a more realistic model to describe a neutron star: the hybrid star. This star is composed, from the core up to some radius, by bulk quark matter (QM) and then a mantle of hadronic matter covering this QM up to the surface. The general idea to match EoSs for different degrees of freedom corresponding to different densities of matter: hadronic, quark and gluon, lead us to estimate, implicitly but not completely, the location of the confinement / deconfinement transition of matter at some critical chemical potential

µc (or range of chemical potentials) for which the matching can be carried out. While it is clear that this approach can never capture the details of the confinement process itself, our results could be used to get a reasonable estimate for µc. Something similar happens for the case of high temperatures and small µB, where the matching between the hadronic and quark-gluon plasma EoSs suggests a range where one would find the location for the deconfinement transition, which agrees reasonably well with lattice data. This is shown in figure 4.12.

Figure 4.12: The matching of perturbative results for the quark matter EoS to hadronic EoSs at high temperature and density. The pressure at µ = 0, obtained from resummed 3/2 O(αs ) pQCD [13] and compared with the result of summing up the eﬀects of all hadron resonances with masses smaller than 2GeV [14, 15]. 73

4.6 Crossover: hybrid APR-FKV

Let us start with the matching of the APR-EoS for nucleon matter onto quark matter described by the FKV EoS for some renormalization scale X. To do that we use the normalized EoSs as shown in figure 4.13. In this figure we see three functions: the red line which represents the nuclear matter EoS [11], the blue one which represents the normalized quark matter EoS for X= 3.205 and the black one which represents the matched EoS. Fraga et al. [57] have chosen that value of renormalization scale because it is the unique which coincides smoothly at some critical baryonic chemical potential with the APR EoS. As usual, we can use both EoSs to generate one which includes values of pressure corresponding to nuclear matter before a certain critical baryonic chemical potential µc, and a pressure corresponding to quark matter for µB > µc. To do the matching we use the conditions of Sec. 3.4. The critical baryon chemical potential takes the value of µc = 1.27 GeV. The matching shows continuity in the pressure and also in the baryon number density (first derivative of the pressure) and sound velocity ( second-order derivative of the pressure). The procedure is as follows [57]:

1. We vary the renormalization scale X between 1 and 4 to ﬁnd equal pressures at some critical chemical potential. Interestengly, for the matching value of X = 3.205

there is also continuity in the derivatives with respect to µB.

2. The complete APR-EoS works only up to values of µB that respect the superluminal

limit, i.e. µc = 1.27 GeV, which is a value near the matching.

3. The matching process carries controlled quantitative uncertainties due to the perturbative nature of the EoS. 74

0.8 FKV(X=3.205)

0.6 APR SB 0.4 P /

0.2 FKV(X=3.205)+APR

0.0 0 1 2 3 4

μΒ[GeV] Figure 4.13: Normalized EoS for the APR-EoS (red curve), FKV for X= 3.205 (blue curve) and the matching of both EoS (black curve). The matching is carried out smoothly.

To solve the TOV equations we need P = P (). This is shown in ﬁgure 4.14, where one can see that for low energy densities it increases approximately quadratically and from some critical value of it increases approximately linearly.

0.8

APR-FKV(X=3.205)

] 0.6 3

0.4

P [ GeV / fm 0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ϵ[GeV/fm 3] Figure 4.14: Equation of state for the matching of the APR and FKV with X= 3.205. The matching was made continuosly. 75

After solving the TOV equations, it is important to study the stability of the solutions. This is made by making a plot of the total mass versus the central energy density in the range allowed by the matched EoS. In ﬁgure 4.15 it is shown clearly up to what central energy density our solutions will be stable for given masses in our next plot for mass versus radius.

1.5 APR-FKV(X=3.205) ] ⊙

1.0

Mass [ M 0.5

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3 ϵc[GeV/fm ] Figure 4.15: Total gravitational mass versus central energy density for the matching of the APR and FKV for X= 3.205.

Finally, the main observable related to this hybrid star is the total gravitational mass of the sequence of stars versus their radii. In ﬁgure 4.16 we see a typical behaviour of a nucleonic star but in this case, it contains quark matter. This indicates that some neutron stars can contain quark matter in their cores, but the measurement of mass and radius can be similar to that of a regular nucleonic star. 76

2.0

1.5 APR-FKV(X=3.205) ] ⊙

1.0 Mass [ M 0.5

0.0 0 2 4 6 8 10 12 14 Radius[km] Figure 4.16: Total gravitational mass versus radius for the matching of APR and FKV for X= 3.205. 77

4.7 First-order transition: hybrid TM1-FKV (X=1.52)

Now we consider the case of a first-order phase transition. We use a value of X = 1.52 for the FKV-EoS which represents a strong first-order phase transition from nucleonic matter described in terms of the TM1 parameterization in the relativistic mean field theory. To obtain the matched EoS, we follow the steps below:

1. For comparison, we present in ﬁgure 4.17 the normalized FKV-EoS for the X =

3.205, which was used in the previous section, and the normalized TM1-EoS. The matching only works if both EoSs (normalized) meet at some critical chemical potential. However, from this ﬁgure we can see that they do not intersect at any chemical

potential and therefore we can conclude that X = 3.205 is not an appropriate value for the matching. X must be lower than this value.

1.0

FKV(X=3.205) 0.8 TM1

0.6 SB

P / 0.4

0.2

0.0 0 1 2 3 4 5

μB[GeV] Figure 4.17: Normalized TM1 EoS and FKV for X= 3.205. They do not intersect at any baryonic chemical potential, then the matching can not be carried out. Extracted from[16]. 78

2. Now we can run the value of X searching for intersections and ﬁnding the corre-

sponding values of critical chemical potential µc. This is presented in ﬁgure 4.18.

3. We will consider extreme cases: X = 1.52 and X = 2.5, yielding high and low critical chemical potentials, respectively.

2.0 [ GeV ]

c 1.8

1.6

1.4

1.2

1.0 Critical chem. pot. μ 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 Renorm. Scale X Figure 4.18: Critical chemical potentials for the matching of the TM1 and FKV EoSs for given X values. Extracted from [16].

In ﬁgure 4.19 the normalized EoSs intersect at µB ≈ 1.1 GeV for X= 2.5. Now we proceed systematically in the process of getting the matched EoS. 79

1.0

FKV(X=2.5) 0.8 TM1

0.6 SB

P / 0.4

0.2

0.0 0 1 2 3 4 5

μB[GeV] Figure 4.19: Normalized EoSs for the matching of the TM1 parameterization and FKV for X = 2.5. Extracted from [16].

We present in ﬁgure 4.20 the mass-radius relation for this matched EoS. The bad news about this plot is that it does not reproduce the maximum masses of two-solar masses found in observations recently [6, 17]. So, a soft ﬁrst-order phase transition built from the TM1 with the FKV is ruled out.

2.0

1.5 TM1-FKV(X=2.5) ] ⊙

1.0 Mass [ M 0.5

0.0 0 2 4 6 8 10 Radius[km] Figure 4.20: Total gravitational mass-radius relation for the TM1+FKV with X = 2.5. This matching is ruled out because does not satisfy the recent experimental observations [6, 17]. Extracted from [16]. 80

Now let us consider the other extreme case: X = 1.52. This EoS is shown in ﬁgure 4.21.

There we can see that the latent heat is much bigger than in the previous case. Then, this represents a strong ﬁrst-order phase transition at µc ≈ 1.8 GeV.

1.0

FKV(X=1.52) 0.8 TM1

0.6 SB

P / 0.4

0.2

0.0 0 1 2 3 4 5

μB[GeV] Figure 4.21: Normalized EoSs for the matching of the TM1-EoS and FKV for X = 1.52. Extracted from [16].

To ﬁnd the EoS, it is more convenient to compute:

= (P ) (4.74) and then solve the TOV equations. This form of the EoS is shown in ﬁgure 4.22.

In this matched EoS we use the Maxwell construction, typical of ﬁrst-order phase transitions, for the jump in the energy density. Numerically, the implementation of this construction is straighforward when one expresses the EoS in this form. 81

TM1 FKV X 1.52 3 - ( = ) ] 3

2 ϵ [ GeV / fm 1

0 0.0 0.2 0.4 0.6 0.8 P[GeV/fm3] Figure 4.22: Matched EoS of the TM1-EoS and FKV for X = 1.52. Extracted from [16].

Introducing this matched EoS into the TOVs will tell us about the stability region of the central pressures (for this hybrid case). This is show in ﬁgure 4.23.

2.0 TM1-FKV(X=1.52) ]

⊙ 1.5

1.0 Mass [ M 0.5

0.0 0.0 0.2 0.4 0.6 0.8 3 Pc[GeV/fm ] Figure 4.23: Total mass vs the central pressure for the TM1-EoS and FKV with X = 1.52. Extracted from [16]. 82

Finally we obtain our mass-radius plot (ﬁgure 4.24) and compare these results with the mass-radius relation for the matched APR-FKV (X = 3.205). In ﬁgure 4.24 we also see explicitly that the behaviour of our hybrid star mimics that of a nucleonic star. For high enough densities it has a linear behaviour typical of superdense stars [16].

2.0 APR-FKV(X=3.205) ]

⊙ 1.5

TM1 FKV X 1.52 1.0 - ( = ) Mass [ M

0.5

0.0 6 8 10 12 14 Radius[km] Figure 4.24: Mass-Radius relation for the matching of the nucleonic TM1-EoS and with the bulk quark matter FKV EoS with X = 1.52. For comparison it also shown the matching of the APR+FKV EoS with X = 3.205. Extracted from [16]. 83

Chapter 5

Final Remarks and Perspectives

In this thesis we have studied the possibility of having quark matter in the interior of neutron stars using eﬀective hadronic models and state-of-the-art results from perturbative QCD depending how deep we are in the neutron star.

We have started with a review of the necessary ingredients of General Relativity which are needed to describe correctly the hydrostatic equilibrium for this kind of compact stars. In doing that we have presented the Tolman-Oppenheimer-Volkoﬀ equations, their interpre- tation, constraints and solutions. Then we continued to presenting some tools required to choose correctly the observables that can be compared with astronomical observations, for example, the total gravitational mass and the equatorial radius. We have discussed the standard EoSs used in the literature to model hadronic and quark matter phases in the interior of hybrid stars. Finally, we have analyzed the particular case of the matching of the TM1 hadronic EoS with the FKV EoS which models quark matter in the core and near the core of the neutron star. Their matching produced masses which surpass the maximum masses found by astronomical observations up to now and can represent a new family of neutron stars. There is a lot of indications that the majority of neutron stars are hybrid stars with varying composition depending on the scale of density [31]. 84

The work presented here can be improved and extended in a number of ways. The simplest of them would be to analyze the case of a mixed phase, i.e. a two-component system of quark and hadronic matter in hybrid stars using the Glendenning construction

[58,4], instead of the Maxwell construction.

Another work would be related to the study of the eﬀects of the state-of-the-art results for the perturbative EoS in the process of nucleation of quark matter and stable strange quark matter after the bounce of a supernova explosion. This problem has been analyzed some years before by Mintz et al. [96] but the new maximum mass measurements in binary pulsar and results from pQCD could change the rate of creation for that kind of matter.

Likewise, it would be straightforward to include the effects of a color superconducting phase in the EoS using effective models based on a mechanism similar to the BCS theory of superconductivity [97, 36, 76, 78], even though these effects are expected to be minor.

At the same time, astrophysical results for the measuraments of the radius of neutron stars will be more precise in the next years and should put stringent constraints in the EoS [98]. So, we believe that the perturbative EoS [36, 57] should be improved to four

3 loops. Although the full αs seems quite demanding, a meaningful and considerably more straightforward challenge would be to determine an expansion of the EoS up to the order

3 O(αs log(αs)) at T = 0 [36]. 85

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