UNIVERSIDADE FEDERAL DO RIO DE JANEIRO INSTITUTO DE F´ISICA Quark matter in the core of neutron stars Jos´eCarlos Jim´enezApaza Disserta¸c~aode Mestrado apresentada ao Programa de P´os-Gradua¸c~aoem F´ısicado Instituto de F´ısicada Uni- versidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necess´arios`aobten¸c~aodo t´ıtulode Mestre em Ci^encias(F´ısica). Advisor: Eduardo Souza Fraga Rio de Janeiro Mar¸code 2016 iii T266p Jim´enezApaza, Jos´eCarlos Quark matter in the core of neutron stars (Materia de quarks no n´ucleode estrelas de n^eutrons)/Jos´eCarlos Jim´enezApaza - Rio de Janeiro: UFRJ/IF, 2016. 93 f. : Il. ; 30 cm. Orientador: Dr. Eduardo Souza Fraga Disserta¸c~ao(Mestrado em F´ısica) - Programa de P´os- gradua¸c~aoem F´ısica,Instituto de F´ısica,Universidade Federal do Rio de Janeiro. Refer^enciasBibliogr´aficas:f. 67-74. 1.Diagrama de fases quiral. 2. Ponto cr´ıtico. 3. Colis~oes de ´ıonspesados. 4. F´ısica-Teses. iv Resumo Materia de quarks no n´ucleode estrelas de n^eutrons Jos´eCarlos Jim´enezApaza Orientador: Eduardo Souza Fraga Resumo da Disserta¸c~ao de Mestrado apresentada ao Programa de P´os- Gradua¸c~aoem F´ısicado Instituto de F´ısicada Universidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necess´arios `aobten¸c~aodo t´ıtulode Mestre em Ci^encias(F´ısica). O estudo das intera¸c~oesfortes em regimes de temperatura e potencial qu´ımicoar- bitr´arios´eainda um problema aberto. Em particular, o regime de altos potenciais qu´ımicose temperaturas muito baixas do diagrama de fase da QCD ´ede grande relev^ancia, uma vez que pode provavelmente ser encontrado no n´ucleodas estrelas de n^eutrons. L´a,os m´etodos perturbativos da QCD (pQCD) fornecem resultados com razo´avel precis~ao,e resultados mais recentes da pQCD in-medium produziram uma forma mais confi´avel para a equa¸c~aode estado (EoS) para mat´eriade quarks em altas densidades, incluindo uma estimativa do erro. Usamos estes resultados para calcular a rela¸c~aomassa-raio e comparar com as recentes observa¸c~oesastron^omicasde massas de pulsars. Consideramos estrelas h´ıbridas, ou seja, supor que o n´ucleoda estrela de n^eutrons´efeita de quarks em equil´ıbriobeta e que este n´ucleo´erodeado por um envolt´oriode mat´eriahadr^onica.Diferentes EoSs para cada fase (quark e hadr^onica)podem ser combinadas e comparadas com os dados astr^onomicos. Palavras-chave: Estrelas de n^eutrons,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¸c~ao de Mestrado apresentada ao Programa de P´os- Gradua¸c~aoem F´ısicado Instituto de F´ısicada Universidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necess´arios `aobten¸c~aodo t´ıtulode Mestre em Ci^encias(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]............................