Studying Compact Star Equation of States with General Relativistic Initial Data Approach (Springer Theses) 9811541507, 9789811541506

Citation preview

Springer Theses Recognizing Outstanding Ph.D. Research

Enping Zhou

Studying Compact Star Equation of States with General Relativistic Initial Data Approach

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses are accepted into the series by invited nomination only and must fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder. • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to scientists not expert in that particular field.

More information about this series at http://www.springer.com/series/8790

Enping Zhou

Studying Compact Star Equation of States with General Relativistic Initial Data Approach Doctoral Thesis accepted by Peking University, Beijing, China

123

Author Dr. Enping Zhou Max-Planck Institute for Gravitational Physics (Albert-Einstein Institute) Potsdam, Germany

Supervisors Prof. Renxin Xu Kavli Institute for Astronomy and Astrophysics, Department of Astronomy State Key Laboratory of Nuclear Science and Technology and School of Physics Peking University Beijing, China Prof. Luciano Rezzolla Institute for Theoretical Physics Frankfurt Institute of Advanced Studies Frankfurt, Germany

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-15-4150-6 ISBN 978-981-15-4151-3 (eBook) https://doi.org/10.1007/978-981-15-4151-3 © Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Supervisor’s Foreword

The ultimate target of physics study is to understand the fundamental units of matter and the interaction between them. A standard model of particle physics has been established, thanks to the great efforts of many physicists in the twentieth century. In the standard model, basic Fermions make up the matter, whereas gauge Bosons mediate the fundamental interactions between the Fermions. There are four interactions currently known, including strong, weak, electromagnetic, and gravitational ones. Among them, gravity and strong force are more subtle to deal with although both of them are described in delicate framework theoretical, i.e., general relativity (GR) for gravity and quantum chromodynamics (QCD) for strong one. The difficulty in dealing with GR arises from the non-linearity of the field equation, but for QCD the non-perturbative property at low energy level. In this sense, compact star (or pulsar in observations) is a perfect laboratory for us to study both the strong and gravitational interactions. On one hand, as the densest object in the Universe, the property and internal structure of compact stars is tightly related to low temperature/high chemical potential QCD. Certainly, figuring out the equation of state of such dense object will greatly enrich our knowledge about strong interaction. On the other hand, being such a massive and compact object, they provide a way for us to test GR in strong field regime and they themselves could be gravitational wave sources as well. The accurate periodic signal from pulsars could also help us detect low-frequency gravitational waves (nHz). Additionally, compact star is the key for one to understand different manifestations of high energy phenomena in astrophysics. This is the reason why any science related to pulsar is fascinating for my research group at Peking University. In particular, me, my students, and colleagues are interested in constraining the equation of state of pulsars with all possible observational phenomenon. The study of compact degenerate Fermion stars dates back to early twentieth century. In 1926, Fowler has pointed out degenerate pressure of electrons might be the support against gravity for white dwarfs. He also realized that stars with density as high as [1014 gcm 3 could exist. Later Chandrasekhar realized that when considering relativistic energy–momentum relation, there would be a mass limit

v

vi

Supervisor’s Foreword

beyond which degenerate pressure of electrons can no longer support the star, which is now so-called Chandrasekhar limit of white dwarfs. With such backgrounds, Lev Landau began to consider what happens to stars beyond this limit, inevitably shrinking to a “gigantic nuclei”. He believes that in the core of such stars, there will be pathological region where electron and proton become particle-like with charge neutrality. With the discovery of neutron by Chadwick later on, Landau’s speculation of “gigantic nuclei” then evolved into the very original idea of “neutron star”, which then became a hot topic in astrophysics and nuclear physics. The original idea of neutron star is created when the strong interaction is poorly understood, let alone the quark model. As our knowledge about strong interaction gets richer, a conjecture that three flavor quark matter is the absolute stable state of dense matter has thus been suggested. Based on the conjecture, strange star (which is composed of deconfined or confined u; d; s quarks) model has been suggested. Most of the colleagues in my group are interested in testing strange star model or understanding puzzling observations in such a model, but Enping showed great interest in general relativity when he just started his Ph.D. study. Therefore, he chose to focus his Ph.D. study on the topic of distinguishing between strange and neutron stars with numerical relativity calculations and to seek for possible observational tests in the gravitational wave era. A joint training program is approved for Enping to study this topic, starting from 2014 in the group of Prof. Luciano Rezzolla. The long-term goal of his research is to obtain accurate gravitational waveform of strange stars, either in binary merger or from fast rotating single source, and to determine the ejecta properties of binary strange star merger so as to infer the electromagnetic counterpart behavior. Nevertheless, to perform numerical relativity evolution, he had to first obtain initial data of strange stars. Enping spent most of the time in Frankfurt working on constructing initial data of strange stars. By the time of October 2017, when the first multimessenger observation of a binary neutron star merger (GW170817, GRB170817A, and AT2017gfo) is announced, he has obtained a few results by initial data study. For instance, tidal deformability of strange star models has been explored and compared with the constraint from GW170817. Configurations of uniformly or differentially rotating strange stars are also studied and compared with the case of conventional neutron stars to indicate possible differences in observation. Such studies are the main content of his Ph.D. thesis. Since the detection of GW170817, more binary merger events involving neutron stars have already been detected as the sensitivity is increased in the new operational runs of LIGO and VIRGO. In the future, more constraints on the equation of state of compact star will be obtained as more facilities are going to join the multimessenger astronomical observation. At the same time, Enping has also begun to tackle the problem of evolving strange stars in numerical relativity. Hopefully, in

Supervisor’s Foreword

vii

the near future, we could justify the possibility of strange stars, as well as constraining the equation of states in the strange star scenario. It will be good news if we could finally know the inner structure of neutron star in this multimessenger era, with the knowledge improvement of both gravitational and strong forces. Beijing, China January 2020

Prof. Renxin Xu

Preface

On September 14, 2015, after the efforts of several decades, a direct detection of gravitational wave signal from a binary black hole system is finally achieved [3]. With this detection, another prediction of Einstein’s general relativity has proved true and the theory itself has been tested in strong field regime. This detection has opened a brand new window for human to observe the universe and announced the start of gravitational wave astronomy era. Throughout the history of man kinds, the level of civilization is always closely related to the depth of the understanding about the universe. Motion of celestial objects was recorded on oracle bones thousands of years ago and the annual plan of agriculture has been fixed according to it by our ancestors. The position of moon and stars allows captains to figure out the location of their ship and make intercontinental sailing possible hundreds of years ago. Four hundred years ago, the craters on the moon, the phase change of Venus as well as the motion of Jupiter moons have been discovered by Galileo’s telescope, which eventually frees the study of natural science from religions and pushed human beings into industrial age. Now, the gravitational wave window has come to our aid to observe the universe. This will undoubtedly impact the modern astronomy and physics. Indeed, 2 years later, on August 17, 2017, advLIGO and VIRGO managed to detect a gravitational wave signal from an inspiraling binary neutron star system, GW170817 [4]. Unlike the binary black hole systems which are almost purely manifestation of gravity, binary neutron star merger will result in much richer observations due to the presence of matter. With the efforts of more than 70 observatories, the electromagnetic counterparts of this merger event have also been detected [5], including a relatively dim short gamma ray burst and a kilonova. This has announced the birth of multimessenger astronomy.

ix

x

Preface

The detection of the binary neutron star merger event has significantly enriched our knowledge about the central engine of short gamma ray bursts as well as the origin of heavy elements in the Universe [1, 2]. To make the detection possible as well as to interpret the observations, theoretical models for binary neutron star merger events are essential. However, even the motion of two point masses (i.e., binary black hole systems) could not be solved analytically, needless to say for binary neutron star system which is far more complicated. Especially when the two neutron stars are close to each other or post to merger, the star might be significantly deformed, neutrino emissions might be important due to the increase of the temperature, and even strong interaction phase transition could take place. Numerical relativity, which is to solve general relativistic hydrodynamics with numerical methods, is thus essential in modeling binary mergers. Due to the complicity in mathematics, numerics as well as in physics, the numerical relativity method is still quite limited in many aspects. For instance, the description of viscosity and heat conductivity is not properly implemented, which could play a very important role during the merger. It’s also computationally extremely expensive if we want to perform a long-term simulation with high resolution. As a consequence, the understanding of many post-merger behaviors, such as the jet formation, is quite indirect. And due to the non-perturbative property of strong interaction at low energy scale, the equation of state we should apply for the fluid in the simulation is also with high uncertainty. Nevertheless, every challenge is also an opportunity. Indeed, constraining the equation of state of dense matter, with the aid of numerical relativity method, is also a hot topic in the multimessenger astronomy era. It’s worth noting that, apart from the conventional neutron star model, it has been shown that other theoretical models such as strange quark star model could also explain the observation of GW170817 as well as its electromagnetic counterparts. However, quark stars possess a very large surface density (i.e., as large as several times of nuclear saturation density), due to the self-bound nature by strong interaction. This discontinuity on the surface makes it quite difficult to directly perform numerical simulations with such type of equation of state. In the past, binary quark star mergers have been studied merely via smooth particle hydrodynamics which could handle this sharp discontinuity. Hence, it’s not easy to fully explore the possibility of quark star models as well as interpret the observations in the scheme of quark stars. Considering this issue for quark star models, as well as the time consumption of numerical relativity simulations, constructing quasi-equilibrium solutions (or in another word, calculating initial data) still plays an important role in understanding the merger observations, particularly the post-merger stage. In particular, I have constructed quasi-equilibrium models of uniformly rotating, differentially rotating as well as triaxial rotating quark stars and compared the results with the solutions of neutron stars to seek for possible way to distinguish between them. This thesis will be organized as follows: in the first chapter, I will make a brief introduction to numerical relativity methods, initial data calculation as well as equation of state models; in the second chapter, I will introduce the calculation of tidal deformability for compact stars and demonstrate how the observation could be

Preface

xi

interpreted differently for neutron star and quark stars; in the third chapter, I will introduce the calculation of critical mass for uniformly and differentially rotating compact stars; in the fourth chapter, results of triaxially rotating compact stars will be introduced; a conclusion will be given in the last chapter. Potsdam, Germany June 2019

Enping Zhou

References 1. Eichler D, Livio M, Piran T, Schramm DN (1989) Nucleosynthesis, neutrinobursts and gamma-rays from coalescing neutron stars. Nature 340:126–128 2. Narayan R, Paczynski B, Piran T (1992) Gamma-ray bursts as the death throesof massive binary stars. Astrophys J Lett 395:L83–L86 3. The LIGO Scientific Collaboration and the Virgo Collaboration (2016) Observation of gravitational waves from a binary black hole merger. Phys Rev Lett 116(6):061102 4. The LIGO Scientific Collaboration and The Virgo Collaboration (2017) GW170817: Observation of gravitational waves from a binary neutron star inspiral. Phys Rev Lett 119:161101 5. The LIGO Scientific Collaboration, the Virgo Collaboration, Abbott BP, Abbott R, Abbott TD, Acernese F, Ackley K, Adams C, Adams T, Addesso P et al (2017) Multi-messenger observations of a binary neutron star merger. Astrophys J Lett 848(2):L12. http://stacks.iop.org/ 2041-8205/848/i=2/a=L12

Parts of this thesis have been published in the following journal articles: [1] Two types of glitches in a solid quark star model, E. P. Zhou, J. G. Lu, H. Tong, R. X. Xu, MNRAS 443 2705 (2014) [2] Uniformly rotating, axisymmetric and triaxial quark stars in general relativity, E. P. Zhou, A. Tsokaros, L. Rezzolla, R. X. Xu, K. Uryu, PRD 97 023013 (2018) [3] Constraints on interquark interaction parameters with GW170817 in a binary strange star scenario, E. P. Zhou, X. Zhou, A. Li, PRD 97 083015 (2018) [4] Merging Strangeon Stars, X. Y. Lai, Y. W. Yu, E. P. Zhou, Y. Y. Li, R. X. Xu, RAA 18 024 (2018) [5] Causal propagation of signal in strangeon matter, J. G. Lu, E. P. Zhou, X. Y. Lai, R. X. Xu, SCPMA 61 89511 (2018) [6] Neutron Star Equation of State from the Quark Level in Light of GW170817, Z. Y. Zhu, E. P. Zhou, A. Li, ApJ 862, 98 (2018) [7] Strangeons constitute bulk strong matter: Test using GW 170817, X. Y. Lai, E. P. Zhou, R. X. Xu, EPJA, 55, 60 (2019) [8] To Constrain Neutron Star’s equation of state by GRB X-ray Plateau, S. Du, E. P. Zhou, R. X. Xu, ApJ 886, 2 (2019) [9] Constraint on the maximum mass of neutron stars using GW170817 event, M. Shibata, E. P. Zhou, K. Kiuchi, S. Fujibayashi, PRD 100 023015 (2019) [10] Differentially rotating strange star in general relativity, E. P. Zhou, A. Tsokaros, K. Uryu, R. X. Xu, M. Shibata, PRD 100 043015 (2019)

xiii

Acknowledgements

I would like to sincerely thank the support of all the teachers, friends, and my family during my study and writing this thesis. Among them, Prof. Renxin Xu, who is my Ph.D. supervisor as well as my tutor during my undergraduate years in Peking University, is the most important guide for me, not only for training my academic abilities but also for impacting me with his own behavior on how to become a fair and nice person. He encouraged me to follow the topic which I’m more interested in and recommended me to study abroad with scholarship during my Ph.D. Without him, I wouldn’t be able to get into the field of numerical relativity. I’m also extremely grateful for Prof. Dr. Luciano Rezzolla in University Frankfurt, who always spares time with me to instruct me although he has quite a large group and quite a lot of administrative stuffs to deal with as the head of the institute. He also generously provided me the same study environment and international travel opportunities as the other students, although I was only a visiting student. Even after I finished the visit and went back to Peking University, he still supports me to have several short-term visits to Frankfurt as well as allowing me to still use the supercomputer in the group. Without his help, I would not be able to get familiar with this field so fast, nor would I be able to finish any projects mentioned in this thesis. In addition to them, Dr. Antonios Tsokaros who is one of the creators of COCAL is essentially the closest teacher for me in Frankfurt. I was not even familiar with Linux command lines when I first arrived in Frankfurt, and it’s him that instructed me for any type of technical questions and eventually led me by hand to the field of initial data calculation. We have kept in contact regularly even until today and he is always willing to help me not only scientifically but also for daily lives and career plans. Professor Zhoujian Cao and Prof. Runqiu Liu from Chinese Academy of Science have also given me significant instructions in the field of numerical relativity. Additionally, Prof. Masaru Shibata has kindly allowed me to take some time for translating this thesis into English after I joined his division as a Postdoc. I’m also grateful to him for making this book possible as well as for his instructions which greatly improves my academic abilities in this field. xv

xvi

Acknowledgements

Apart from all the kind teachers, I should also thank my friends. Dr. Jiguang Lu, with his great talent in physics and mathematics, always gave me inspirations in my own research and reminded me to be humble. Dr. Lijing Shao, my academic elder brother, is always ready to discuss questions with me and try to find solutions to the problems which I encountered in daily life. All the Chinese and foreign friends that I met during my visit to Frankfurt have also made both my researches and life more joyful in Germany. Mentioning the visit to Germany, I surely should thank the support of China Scholarship Council, who made my life there financially possible. I should also mention that it’s my parents who have been offering the most important support to me during my studies. Although they are not very well educated themselves due to the restrictions of their generation, they are quite open-minded and supported me to pursue my own dream in astronomy since my childhood. They always share the joy of success with me and encourage me when there are difficulties ahead. In Chinese saying, people always use the analogy of “frozen window” to describe the difficulties of studies. Nevertheless, with the support of my parents, my “window” is much warmer and I’m finally able to reward them with this thesis. Last but not least, I have to thank Xiaojie Wu, my wife, who has patiently waited for me for many years when we have to separate because of my study and who have always tolerated me for the lack of romance. Her love, across all the distances between Germany and China, through all the time since 2014, is always the most important support and motivation for me to carry on.

Contents

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

1 1 4 6 8 9 11 13 14

2 Tidal Deformability of Compact Stars . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Calculation of Tidal Deformability . . . . . . . . . 2.2.2 K Constraint of GW170817 . . . . . . . . . . . . . . 2.2.3 Constraint on NS Models . . . . . . . . . . . . . . . . 2.3 Constraint on Strangeon Star Model with GW170817 . 2.3.1 Tidal Deformability of Strangeon Star . . . . . . . 2.3.2 Test of Solid Strangeon Star Model . . . . . . . . 2.4 Constraint on MIT Bag Model with GW170817 . . . . . 2.4.1 The EoS Model . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Constraint on Strange Quark Mass . . . . . . . . . 2.4.3 Constraint on Beff and a4 . . . . . . . . . . . . . . . . 2.4.4 Constraint on D Parameter . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

17 17 18 18 21 22 23 23 24 27 27 29 31 33 35

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 History of Numerical Relativity . . . . . . . . . . . . . . . 1.2 3+1 Decompositon of the Spacetime . . . . . . . . . . . 1.3 3+1 Decomposition of the Einstein Field Equations 1.4 Initial Data Problem . . . . . . . . . . . . . . . . . . . . . . . 1.5 Hydrostatic Equilibrium for Initial Data Problem . . 1.6 Equation of State of Compact Stars . . . . . . . . . . . . 1.7 Numerical Code: COCAL . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

xvii

xviii

Contents

... ...

37 37

...

37

. . . . . .

. . . . . .

39 42 44 49 49 50

...

54

... ...

56 60

. . . . .

. . . . .

63 64 65 68 71

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 74

Appendix: Accuracy and Convergence Tests . . . . . . . . . . . . . . . . . . . . . .

75

3 Axisymmetric Rotating Compact Stars . . . . . . . . . . . . . . . . . . . . 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Fate of the Merger Remnant and Maximum Mass of Compact Stars . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Constraints on NS EoS According to EM Counterparts of GW170817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 A More Consistent Constraint on NS MTOV . . . . . . . . . . . . . . 3.3 Uniformly Rotating Strange Stars . . . . . . . . . . . . . . . . . . . . . 3.4 Maximum Mass of Rotating Strange Stars . . . . . . . . . . . . . . . 3.4.1 Rotation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Critical Mass of Differentially Rotating Strange Stars . 3.4.3 Critical Mass of Constant Angular Momentum Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Configuration Types of Differentially Rotating Strange Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Triaxially Rotating Strange Stars . . . . . . . . . . . . . . . . . . . . . 4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Constructing Triaxially Rotating Strange Stars with COCAL 4.3 Comparison with Triaxially Rotating NS . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . . .

. . . . .

Chapter 1

Introduction

In this chapter, a brief history of the development of numerical relativity method, the 3+1 decompostion, initial data problem as well as equations of state of compact stars will be introduced. For more detailed introductions, one can find, for example, in [14, 26].

1.1 History of Numerical Relativity Among four fundamental interactions, gravitational interaction is the one first studied in details by human. Four centuries ago, Newton came up with his theory of gravity, that matter attracts each other with a force proportional to their masses and inverse separation squared. In 1915, Einstein submitted his paper on the theory of general relativity, describing gravity as the curvature effect of the spacetime manifold with the presence of energy and momentum being the source of this curvature. Newton’s law of gravity then was included in general relativity as the approximation for weak field limit, which is no longer valid for systems with strong gravity, such as binary black hole or binary neutron star mergers. Applying general relativity, is essential in understanding binary compact star mergers. Although the most basic equation for general relativity, i.e., the Einstein equation has a very simple tensorial form, G μν = 8πTμν ,

(1.1)

to solve it is by no means easy. The first exact solution came up one year later in 1916 by Schwarzchild, in the most symmetric case (a static spherically symmetric solution in vacuum). And with just a little bit less symmetry, i.e. going from spherical symmetry to axial symmetry, the solution of Einstein’s field becomes way more

© Springer Nature Singapore Pte Ltd. 2020 E. Zhou, Studying Compact Star Equation of States with General Relativistic Initial Data Approach, Springer Theses, https://doi.org/10.1007/978-981-15-4151-3_1

1

2

1 Introduction

difficult. Almost half a century later, the exact solution for a static axisymmetric vacuum spacetime was finally given by Kerr in 1963. And people couldn’t derive this solution until another ten years later. The non-vacuum case yields even less solutions. In 1939, Tolman, Oppenheimer and Volkoff derived the equation (hence called TOV equation) for solving the structure of a static spherically symmetric star. And for the general cases, this equation has no analytic solution and has to be solved numerically. In addition, most of those solutions obtained are static or stationary solutions, which are not evolving with time. When it comes to binary compact star systems which are highly dynamical, the solutions and the methods to obtain those solutions can provide very little help. Since the mid-20th century, with the improvement of the calculation speed of computers, to solve Einstein equation with computers has become a possibility and research field for studying the dynamics of compact stars. This is the topic of numerical relativity, to solve Einstein equation with computers. General relativity is a great leap of our understanding of spacetime. The idea of Galileo and Newton’s absolute time and space are replaced by a brand new model based on a four dimensional space-time manifold. Nevertheless, this leap brings inconvenience for numerical calculation. When we perform simulation to study the evolution of a certain physical problem on computers, a common strategy is to reformulate it into a typical Cauchy problem. For the case of Einstein’s field equation, which in essence is a partial differential equation system, we need to split those equations that contains second order derivative with respect to time (which are called evolution equations) and those not (which are called constraint equations). After that, we can solve for initial data on a spacelike hypersurface which satisfies the constraint equations, and then we can evolve the data to future time with the evolution equations. This procedure is called 3+1 decomposition. In 1952, Yvonne Choquet-Bruhat proved the local existence and uniqueness of the solutions of a 3+1 decomposed Einstein’s filed equation [10]. In 1962, Arnowitt, Deser and Misner came up with way to decompose Einstein’s filed equation, which was later improved by York and widely known as the ADM formalism [4, 40]. York has also developed a way for calculating initial data [39]. Many attempts have been done to perform numerical simulations with the ADM formalism, which turns out to be successful only in very limited cases. The reason is that ADM formalism is not very suitable for numerical simulations as this formalism is weakly hyperbolic and therefore the numerical solutions are not guaranteed to be stable. With this concern, Nakamura tried to fix this undesirable weak hyperbolicity starting from the standard ADM formalism [23]. Later on, Shibata and Nakamura improved the formalism based on a conformal decomposition formulation [28]. Subsequently, Baumgarte and Shapiro proposed a new variable to have a slightly simpler form compared with that derived by Shibata and Nakamura [6]. This is called the BSSN formalism. Although BSSN formalism improved the stability of numerical calculations, it still doesn’t automatically result in a successful simulation by just simply applying the formalism. Another crucial ingredient for those numerical relativity methods based on spacetime foliation is the gauge choice, or in another word, how do you choose the

1.1 History of Numerical Relativity

3

spacetime coordinate structures during the evolution. It turns out many physically intuitive choice of coordinates will not be so useful in numerical simulations and may even cause the code to crash at some moment. A bad choice, for example, might not be able to avoid the physical singularity and causes ‘NAN’ (not a number) in numerical simulations or cause the actual resolution to decrease in the physical relevant regions. Seeking for a good gauge condition has become a concern in the first several years since BSSN formalism was proposed. In 1999, Shibata and Uryu has made the first successful binary neutron star merger simulation in 3D [27] with the BSSN formalism. However, in order to acquire a good coordinate choice during the simulation, they chose to solve elliptic equations to impose certain properties of the slicing which are known to be helpful for numerical simulations. Solving elliptic equations are computationally more expensive than other evolution equations which could be solved by simple time integration methods. To solve elliptic equations, one usually have to iterate for one timestep for the result to converge. Additionally, binary neutron star systems do not contain singularity during most time of the simulations. Thus this gauge choice cannot be simply used by, for example, binary black hole simulations. In 2003, Alcubierre et al. has proposed a gauge choice which is called moving puncture method which could avoid singularity during the simulation [1]. The gauge choice is described by time evolution of some variables thus very easy to solve. They have made the first successful numerical relativity simulation for single black hole with BSSN formalism and their gauge choices. Remarkable progresses have been made in the year of 2005. Pretorius et al. have applied the so called generalized harmonic formalism and successfully performed the numerical evolution of a binary black hole merger [25]. Later on, Baker et al. and Campanelli et al. have also independently obtained successful simulation of binary black hole merger with BSSN formalism [5, 11]. Compared with Alcubierre et al. original attempt in 2003, the most important improvement is that they have applied fourth order finite differencing scheme instead of second order, which improves the accuracy. Moreover, they modified the original BSSN formalism with some variable replacement such that it works better with the moving puncture method. Those progresses has announced the success of numerical relativity with the spacetime foliation. Once a stable numerical relativity evolution is achieved, the accuracy of it has become the new concern as an accurate gravitational wave template is what everyone expects numerical relativity to give us, which is also quite essential for gravitational wave detection as well as interpreting the observation. The accuracy of numerical relativity could be quantified be the so-called constraint violation. As mentioned before, part of Einstein equations (4 out of 10) doesn’t contain second order time derivatives. They are called constraint equations and should be satisfied during the simulation. In principle, they should automatically be satisfied provided the initial data is satisfying these equations. Nevertheless, due to the finite resolution and accuracy of numerical methods, these equations will not be strictly satisfied thus constraint violation is

4

1 Introduction

produced. In this sense BSSN formalism is not a very ideal method. For instance, the characteristic propagation speed of Hamiltonian constraint violation is 0, which means whatever violation created during the evolution will stay in the computational domain and accumulate. The so-called Z4 formalism is proposed to fix this problem, together with its conformal versions such as CCZ4 and Z4c [2, 8, 9, 15, 16]. Such formalisms have been applied by various groups for simulating binary compact star mergers and turns out to be very successful and more accurate. Compared with binary black holes, which might be numerically more difficult to simulate due to the presence of singularities, binary neutron star systems contain more complicated physical processes due to the presence of matter. Viscosity of the fluid and effect of finite temperature, which do not exist for binary black hole systems, will play a very important role in binary neutron star mergers instead, in affecting the amount and the velocity of the ejecta. The merger remnant is also quite hot and a sufficient neutrino emitter which will affect the electron fraction of the ejecta. All those properties, the amount, velocity and electron fraction of the ejecta are closely related the electromagnetic counterparts of the merger event and thus very important if we can figure it out with numerical relativity simulations. At the same time, neutron stars usually possess a strong magnetic field which would be further enhanced by the differentially rotating post-merger remnant. This strong magnetic is a crucial element in understanding the central engine of short gamma ray bursts. Consequently, adding more ingredients, such as magnetohydrodynamics, neutrinos, viscosity as well as finite temperature equation of state, into the numerical relativity recipe is very important and has become the goal of numerical relativity research in the recent years.

1.2 3+1 Decompositon of the Spacetime A globally hyperbolic spacetime can be foliated into a set of spacelike hypersurfaces (Cauchy surfaces) t parameterized by global function t. Let n μ be the future directing normal vector, hence n μ n μ = −1. Then, with the spacetime metric gμν , a projection tensor can be induced γμν = gμν + n μ n ν γ μν = g μν + n μ n ν .

(1.2)

This tensor works as a projection tensor for mapping a spacetime tensor to a purely spatial tensor on the spacelike hypersurface. The three dimensional spatial part of this projection tensor γ i j , works as the metric tensor on the hypersurface and could be used to lower and raise indices for tensors on the hypersurface.

1.2 3+1 Decompositon of the Spacetime

5

It’s worth noting that, the direction of the time axis ((∂/∂t)μ ) is not necessarily parallel to the norm vector of the hypersurface. Let t μ be a timelike vector field in the spacetime which is tangent to the time axis, and t μ ∇μ t = 1. Then we can decompose it into two components as (see the illustration in Fig. 1.1) α := −t μ n μ , β μ := t ν γνμ ,

(1.3)

in which we have the so called lapse function α that describes the proper distance between two neighboring hypersurfaces a timelike normal vector, i.e., the speed that time elapses; and the purely spatial shift vector β μ = (0, β i )1 which represents the difference between the time axis and the normal vector of the hypersurface. With this setup, the line element can always be written in the following form: ds 2 = −(α2 − β i βi )dt 2 + 2βi d x i dt + γi j d x i d x j .

(1.4)

Using these elements, the spacetime metric components can be written as gμν = and g

μν

=

  2 −α + β k βk β j , βi γi j  − α12 βi α2

βj α2

γi j −

(1.5)

 βi β j α2

.

(1.6)

From the above expression it could also be found that the determinant of the spacetime metric g := |gμν | is equal to −α2 γ where γ is the determinant of the spatial metric γi j . With this foliation, it’s also possible to define the three dimensional covariant derivative operator as (1.7) Dν := γνμ ∇μ = (δνμ + n μ n ν )∇μ . Just as ∇μ is compatible with the four dimensional spacetime metric gμν , it can be proven that Dν is also compatible with the three dimensional spatial metric γμν as Dσ γαβ = γσρ γαμ γβν ∇ρ (gμν + n μ n ν ) = γσρ γαμ γβν (n μ ∇ρ n ν + n ν ∇ρ n μ ) = 0,

(1.8)

where we have applied the metric compatibility condition ∇ρ gμν = 0 and the fact that n μ is the normal vector such that n μ γαμ = 0. This covariant derivative operator is very important in splitting the Einstein field equations as can be found the in the section below.

1 For the indices of a tensor, the notation in this thesis is that Greek letters are for the four dimensional

time and space indices (for example, μ vary from 0 to 3) and the Latin letters are for three dimensional spatial indices (for example i vary from 1 to 3).

6

1 Introduction

Fig. 1.1 An illustration of the 3+1 foliation of the spacetime manifold. t and t+dt are two neighboring spacelike hypersurfaces specified by constant time coordinate t and t + dt, respectively. The 4-vector t represents the direction of time, which could be decomposed into two parts, the fully spatial part β and the timelike part αn. The scalar function α describes the rate at which time elapses, and is called lapse function. The vector field β describes the relative shift of the spatial coordinate between two neighboring hypersurfaces, thus called shift vector

1.3 3+1 Decomposition of the Einstein Field Equations The nature of Einstein field equations is a tensorial equation, with the preparation in the previous section, we can also try to decompose Einstein equations with the projection tensors given. Here in this introduction I will only briefly go through the most basic way of the decomposition, i.e., the ADM formalism. It’s worth noting that the tensors involved in Einstein equations, namely the Riemann tensor, metric tensor and energy momentum tensor, are rank 2 tensors. As a consequence, the projection tensors have to be applied twice for decomposing the equations. Considering this, there are 3 ways in total to do the projections: purely spatial projection onto the spacelike hypersurface, purely timelike projection to the direction perpendicular to the hypersurface, and a mixed projection of the two mentioned above. By projecting Einstein equation to the spacetimelike hypersurface we can obtain: ∂t K i j = −Di D j α + β k ∂k K i j + K ik ∂ j β k + K k j ∂i β k (3)

+ α(

Ri j + K K i j −

2K ik K kj )

+ 4πα[γi j (S − E) − 2Si j ],

(1.9)

1.3 3+1 Decomposition of the Einstein Field Equations

7

in which D is the 3 dimensional covariant derivative operator compatible with the metric on the hypersurface, (3) Ri j is the Ricci tensor on the hypersurface and K i j is the extrinsic curvature of the hypersurface defined as Ki j = −

1 (∂t − Lβ )γi j . 2α

(1.10)

It’s worth noting that extrinsic curvature contains first order time derivative of the metric tensor and Eq. (1.9) contains the first order time derivative of extrinsic curvature. According to the symmetry of K i j , Eqs. (1.9) and (1.10) each has 6 components. Therefore, Eqs. (1.9) and (1.10) together represent the 6 components in Einstein equation which contain second order time derivatives. By introducing K i j as a variable for the evolution system, six differential equations with second order time derivative is now re-written as twelve differential equations with first order time derivative, hence much more straight forward for time evolution. Those twelve equations (Eqs. 1.9 and 1.10) are called evolution equations. Like in classic mechanics, given initial position and velocity, the motion of a particle could then be predicted. In numerical relativity, given the initial metric and its first order time derivative (i.e., the metric tensor and extrinsic curvature of the hypersurface), Eqs. (1.9) and (1.10) could then be used to evolve them. Nevertheless, initial metric and curvature used for the evolution shouldn’t be given arbitrarily. Some constraints, which are encoded in the other 4 components of Einstein equation without second order time derivatives, must be satisfied. Those are called constraint equations which could be obtained by the other projections of Einstein equation. By a mixed projection, we can obtain D j (K i j − γ i j K ) = 8πS i .

(1.11)

This equation has 3 components and is called the momentum constraint. The projection purely to the direction perpendicular to hypersurface is (3)

R + K 2 − K i j K i j = 16π E.

(1.12)

This is called the Hamiltonian constraint. Equations (1.11) and (1.12) are the constraints that the metric and extrinsic curvature have to satisfy. Together with the evolution equations (Eqs. 1.9 and 1.10), they represent all the components in Einstein equation. There is one thing worth mentioning here: the space-like hypersurface could be totally described by 12 independent variables (6 for metric and 6 for extrinsic curvature) which needs to satisfy 4 constraint equations. Hence, there are 12 − 4 = 8 degrees of freedom, 4 out of which come from the choice of coordinates t, x, y, z. The remaining 4 degrees of freedom accounts for the 2 polarization modes of gravitational wave. In addition, there are hydrodynamic quantities on the right hand side of those equations, which are also obtained from projection of energy momentum tensor Tαβ :

8

1 Introduction

Sαβ = γ μα γ νβ Tμν Sα = −γ μα n ν Tμν S = S μμ

(1.13)

E = n μ n ν Tμν , in which Sα and E correponds to the momentum and energy density of the fluid as observed by an observer with n as the four velocity. This is also the reason why Eqs. (1.11) and (1.12) are called momentum constraint and Hamiltonian constraint, respectively.

1.4 Initial Data Problem The initial data problem of numerical relativity is to choose the coordinates for a initial spacelike hypersurface and determine the constraint-satisfying metric and extrinsic curvature of it according to the astrophysical problem that one is interested in (for example, rotating compact star or binary compact star systems). By the discussion in the previous section, it could be seen that it’s not enough to obtain all the 12 variables needed by only solving 4 constraint equations. With more detailed analysis it could be seen, Hamiltonian constraint is one constraint on γi j whereas momentum constraint are 3 constraints for K i j . Two polarization modes of gravitational wave are 2 constraints for both γi j and K i j . As a consequence, the gauge condition (i.e. the choice of the coordinates or in another word, the way to foliate the initial hypersurface) will make 3 constraints for γi j and one for K i j . In practice, we usually have to apply some particular assumptions when foliating the spacetime to obtain the initial hypersurface by considering the symmetry of the physical process that we are interested in to reduce the degrees of freedom in γi j and K i j , hence having enough constraint equations to solve the initial data problem. One widely used assumption is the so-called conformally flat approximation, which treats the metric on the initial hypersurface as a conformal transformation of a flat 3-metric: (1.14) γi j = ψ 4 δi j . With this assumption, the line element in the spacetime becomes ds 2 = −α2 dt 2 + ψ 4 δi j (d x i + β i dt)(d x j + β j dt),

(1.15)

which is then fully determined by 5 variables: α,β i and ψ the conform factor. The conformally flat approximation is applied in many formalisms to calculate initial data, including one called IWM formulation (Isenberg-Wilson-Mathews (IWM)

1.4 Initial Data Problem

9

formulation) [18, 19, 36] which we will further introduce below.2 In such a form, one evolution equation has to be used together with 4 constraint equations to determine the 5 variables (α,β i and ψ), which is 1 (G μν − 8πTμν )(γ αβ + n α n β ) = 0. 2

(1.16)

Nevertheless, as this additional equation is a component of evolution equations, it contains second order time derivative (i.e. time derivative of extrinsic curvature). As a result, a slicing condition K = 0 = ∂t K has to be imposed to drop the terms with time derivative. By doing this, this equation together with 4 constraint equations could be written as ∇2ψ = −

ψ5 Ai j Ai j − 2πψ 5 E, 8

(1.17)

1 ψ6 + 16παS i , ∇ 2 β i + ∂ a ∂b β b = −2α Ai j ∂b ln 3 α 7 ∇ 2 (αψ) = αψ 5 Ai j Ai j + 2παψ 5 (E + 2S), 8

(1.18) (1.19)

in which Ai j = K i j = ψ −4 (∂ i β j + ∂ j β i − 23 δ i j ∂k β k )/2α. The initial data problem of numerical relativity in IWM formulation, is then to use those 5 Poisson equations to solve 5 variables of α, β i , ψ with proper boundary condition. Considering that the spacetime is asymptotically flat in spatial infinity, the condition below could be used: lim ψ = 1 ,

r →∞

lim α = 1 ,

r →∞

lim β i = 0 .

r →∞

(1.20)

1.5 Hydrostatic Equilibrium for Initial Data Problem In this thesis, as the main focus is to study the equation of state of compact stars by constructing quasi-equilibrium models, I will not go into details about the hydrodynamic formalisms used for numerical relativity evolutions. Instead, I will give a brief introduction about the formulation of hydrostatic equilibrium calculation which is used for the purpose of constructing initial data. There are terms related to hydrodynamic quantities on the right hand side of Einstein equations, hence in the projected components as well. In order to construct 2 It’s

worth noting that it’s not always valid to assume that the 3-metric on the hypersurface is conformally flat. In particular, large error may be induced by this approximation in the case when the compactness of the star (M/R) is large or when the star is rotating very fast. In such cases, one normally has to apply more components of Einstein equation (thus less assumptions) for more accurately determining the initial data. In a method which is called waveless formulation, all the components of Einstein equation are used for calculating initial data [33].

10

1 Introduction

an initial data, we need to also make sure those quantities satisfy the hydrostatic equilibrium. This could be obtained from the conservation laws for hydro quantities, namely, the conservation of rest mass, momentum and energy. Such conservation laws could be written in the covariant form as: ∇μ (ρu μ ) = 0 ∇μ T μν = 0

(1.21)

The energy momentum tensor Tμν , under a perfect fluid approximation, is T μν = (e + p)u μ u ν + pg μν

(1.22)

in which e and p is the energy density and pressure of the fluid, respectively. We could introduce specific enthalpy as h = e+ρ p , in which ρ is rest mass density. Thus the left hand side of energy momentum tensor conservation could be written as ∇β Tαβ = ρ[u β ∇β (hu α ) + ∇α h] + hu α ∇β (ρu β ) − ρT ∇α s,

(1.23)

in which s is the specific entropy of the fluid. The existence of s is by applying the first law of thermal dynamics, i.e., dh = T ds + dp/ρ By substituting the conservation of rest mass as well as the adiabatic condition u α ∇α s = 0, we can obtain u β ∇β (hu α ) + ∇α h = Lu (hu α ) + ∇α h = 0.

(1.24)

When considering a particular physical process, we could further simplify this equation with the symmetry of the physical problem. For instance, when considering a rotating compact star, the motion of the fluid follows a helical symmetry in four dimensional spacetime, i.e., u α = u t k α = u t (t α + φα ).

(1.25)

With such a condition, there exists a constant integral for the motion of the fluid which can be obtained from Eq. (1.24): h exp [ ut

 j ()d] = const

(1.26)

in which j () = u t u φ is called the differential rotation law which we will further discuss and introduce in later chapters. An equilibrium configuration of a rotating compact stars could be obtained by solving the equation above.3 By far, we have already obtained the equations for solving the initial data of spacetime metric (Eqs. 1.17, 1.18 and 1.19) as well as the equation for hydrostatic 3 For

more detailed discussions c.f. [33].

1.5 Hydrostatic Equilibrium for Initial Data Problem

11

equilibrium (Eq. 1.24). One more piece is needed to connect all the thermal dynamic variables of the fluid before we can solve those equations, this final piece is given by the equation of state (EoS).

1.6 Equation of State of Compact Stars Due to the non-perturbative property of strong interaction at the energy scale of compact stars, it is not possible to obtain the EoS of compact stars, nor can we answer the question that whether it should be a neutron star or strange quark star. In spite of such difficulty, many attempts have been carried out to model the EoS of dense matter with either phenomenological models or effective approaches. Once such model is given, one can then close the system of general relativistic hydrodynamics and obtain solutions of it. By comparing such solutions with observations, constraint on the EoS models could be made. This has become a common way of studying the EoS of compact stars. The simplest one, for instance, is to calculate the hydrostatic equilibrium configuration of a non-rotating compact star (by solving TOV equation) and obtain the maximum mass of such a non-rotating configuration (MTOV ), and then compare with the current measurement on the mass of massive pulsars [3, 13]. One popular way to model neutron star equation of state is the so-called piecewise polytrobe parameterization: pi = κi ρi = κi ρ1+1/ni ,

i = 1, 2, . . . , N .

(1.27)

With this relation, following first law of thermal dynamics we can obtain de =

e+ p dρ, or d ρ

  e p = 2 dρ . ρ ρ

(1.28)

We can then derive the dependence of energy density and specific enthalpy on rest mass density, κi ρi + ρi ei = i − 1 (1.29) ei + pi i κi i −1 ρ hi = = +1 ρi i − 1 which allows us to compute all the quantities needed for solving the general relativistic hydro equation system.4 However, under the conjecture that 3-flavor quark matter (with u, d and s quark) might be the absolute stable state of dense matter [7, 37], there could exist another the term ρ in the expression of energy density. From the perspective of physics, this could be understood as the contribution of rest mass energy. From mathematical point of view, this is actually the integration constant when we calculate e from de/dρ. We will turn back to this point when we talk about strange quark stars, which has a different integration constant. 4 Note

12

1 Introduction

type of compact stars, namely strange quark star, which is self-bound by strong interaction and hence possesses a finite surface density. To describe the EoS of such kind of matter, a so-called MIT bag model is widely used [12]: p = σ(e − es ) ,

(1.30)

in which σ and es are two constants. When the mass of strange quark can be neglected, σ is equal to 1/3. The non-zero constant es describes the finite surface energy density of strange quark stars and is related to the so-called bag constant as B = es /4. Another model that we will concentrate on in this thesis, is the strangeon star model [22] which is also composed of u, d, s quarks but quarks are still confined in so-called strangeon. One way to obtain the EoS model of strangeon star is to compare the interactions between clusters of quarks with the interaction of molecules and model this interaction with Lennard-Jones potential [21]. Considering the fact that the residual strong interaction between strangeons could still be large enough at such energy scale, it’s suggested that strangeons in strangeon stars could be crystallized. Within such a model, the EoS is given as 1 1 4 p = 4U0 (12.4r012 n 5 − 8.4r06 n 3 ) + (6π 2 ) 3 cn 3 , 8

(1.31)

in which U0 and r0 are two model parameters which characterize the depth of the interaction potential and the length scale of the interaction, respectively. Besides, the number of quarks in each strangeon is needed for converting the number density n to rest mass density ρ. In this thesis, we mainly focus on the model with U0 = 50 MeV and Nq = 18. It is worth noting that, strictly speaking, it’s not appropriate to calculate strangeon star model with perfect fluid model, as strangeon star is suggested to be in solid state [38]. Nevertheless, according to the calculation in [42], the critical strain of solid strangeon star is really tiny. Whenever the relative difference between the actual eccentricity of the star and that of the equilibrium configuration is larger than 10−6 , starquake will be induced to bring the star back to the equilibrium configuration. As a consequence, we believe that configuration calculated with perfect fluid approximation should be almost the same as the case when elastic structure is considered. Similar to strange quark stars, strangeon star also possesses a non-zero surface density. Apart from this, the integration constant when calculating energy density by integrating the first law of thermal dynamics, is no longer equal to ρ. To account for such differences with neutron stars, we have found that the following polynomial form of EoS could be a good description of strange stars5 :

5 In this thesis, the concept of ‘compact star’ includes neutron star, strange quark star and strangeon

star models. And due to the similarity of strange quark star and strangeon star model, they will all be called ‘strange star’ when it’s not necessary to refer to a particular model of them.

1.6 Equation of State of Compact Stars

13

p=

N 

κi ρi .

(1.32)

i=1

The energy density and specific enthalpy are then given as e=

N  i=1

κi ρi + ρ(1 + C) , i − 1

 + p  i κi i −1 +1+C. = ρ ρ i − 1 i=1

(1.33)

N

h=

(1.34)

Note that ρ(1 + C), the integration constant, for neutron star is ρ, namely C = 0. Whereas for MIT bag model, when mass of strange quark could be neglected, C = −1.

1.7 Numerical Code: COCAL To obtain initial data, we have to solve the above equations (i.e., equations for determining gravitational field Eqs. (1.17)–(1.19), equations for hydrostatic equilibrium Eq. (1.24) as well as EoS for the fluid) numerically. To do this, the so-called Compact Object CALculator (cocal) is employed for most of the numerical calculations done in this thesis. cocal code is capable for calculating configurations of binary compact star systems such as binary black hole (BBH), binary neutron star (BNS) and black hole-neutron star binaries (BH-NS) [29, 31, 32, 34], as well as 3 dimensional configurations of rotating compact stars [17, 33, 41, 43]. Calculations of self-gravitating disc system around a black hole as well as magnetized rotating neutron stars have been recently implemented into the code as well [30, 35]. The basic idea of solving for initial data with cocal originates from the method to calculate structure of stars in Newtonian scheme in [24], which was then adopted to general relativistic cases by Komatsu, Eriguchi and Hachisu [20]. In order to include the calculation of strange stars, the EoS part of the code is modified as well as the treatment for the surface of the star. The surface-fit coordinate employed in the code allows us to deal with the sharp surface discontinuity of strange stars. It’s worth noting that, the polynomial form to deal with strange star EoS is actually more general than that of the neutron star. In another word, we could in principle apply the polynomial form to describe any NS EoS as well once the surface density and the integration constant of first law of thermal dynamics is carefully treated. This allows us to test the accuracy and convergence behavior of our code, for which more details could be found in [41].

14

1 Introduction

References 1. Alcubierre M, Brügmann B, Diener P, Koppitz M, Pollney D, Seidel E, Takahashi R (2003) Gauge conditions for long-term numerical black hole evolutions without excision. Phys Rev D 67(8):084023 2. Alic D, Bona-Casas C, Bona C, Rezzolla L, Palenzuela C (2012) Conformal and covariant formulation of the Z4 system with constraint-violation damping. Phys Rev D 85(6):064040 3. Antoniadis J, Freire PCC, Wex N, Tauris TM, Lynch RS et al (2013) A massive pulsar in a compact relativistic binary. Science 340:448 4. Arnowitt R, Deser S, Misner CW (2008) Republication of: the dynamics of general relativity. Gen Relat Gravit 40:1997–2027 5. Baker JG, Joan C, Dae-Il C, Michael K, van James M (2006) Gravitational wave extraction from an in spiraling configuration of merging black holes. Phys Rev Lett 96:111102 6. Baumgarte TW, Shapiro SL (1999) Numerical integration of Einstein’s field equations. Phys Rev D 59(2):024007 7. Bodmer AR (1971) Collapsed Nuclei. Phys Rev D 4:1601–1606 8. Bona C, Ledvinka T, Palenzuela C, Zácek M (2003) General-covariant evolution formalism for numerical relativity. Phys Rev D 67(10):104005 9. Bona C, Ledvinka T, Palenzuela C, Zácek M (2004) Symmetry-breaking mechanism for the Z4 general-covariant evolution system. Phys Rev D 69(6):064036 10. Bruhat Y (1952) Theoreme d’existence pour certains systemes d’equations aux derivees partielles non lineaires. Acta Mathematica 88:141–225 11. Campanelli M, Lousto CO, Marronetti P, Zlochower Y (2006) Accurate evolutions of orbiting black-hole binaries without excision. Phys Rev Lett 96:111101 12. Chodos A, Jaffe RL, Johnson K, Thorn CB, Weisskopf VF (1974) New extended model of hadrons. Phys Rev D 9:3471–3495 13. Demorest PB, Pennucci T, Ransom SM, Roberts MSE, Hessels JWT (2010) A two-solar-mass neutron star measured using Shapiro delay. Nature 467:1081–1083 14. Gourgoulhon E (2012) 3+1 formalism in general relativity. Vol 846 of Lecture Notes in Physics, Berlin Springer 15. Gundlach Carsten, Martin-Garcia Jose M, Calabrese G, Hinder I (2005) Constraint damping in the Z4 formulation and harmonic gauge. Class Quantum Grav 22:3767–3774 16. Hilditch D, Bernuzzi S, Thierfelder M, Cao Z, Tichy W, Brügmann B (2013) Compact binary evolutions with the Z4c formulation. Phys Rev D 88(8):084057 17. Huang X, Markakis C, Sugiyama N, Ury¯u K (2008) Quasi-equilibrium models for triaxially deformed rotating compact stars. Phys Rev D D 78:124023 18. Isenberg JA (2008) Waveless approximation theories of gravity. Int J Modern Phys D 17:265– 273 19. Isenberg J, Nester J (1980) Canonical gravity. In: Held A (ed) General relativity and gravitation. Vol. 1. One hundred years after the birth of Albert Einstein. Edited by A. Held. New York, NY: Plenum Press, p. 23, 1980, Vol 1 p 23 20. Komatsu H, Eriguchi Y, Hachisu I (1989) Rapidly rotating general relativistic stars. I— Numerical method and its application to uniformly rotating polytropes. Mon Not R Astron Soc 237:355–379 21. Lai XY, Xu RX (2009) Lennard-Jones quark matter and massive quark stars. Mon Not R Astron Soc 398:L31–L35 22. Lai XY, Xu RX (2017) Strangeon and Strangeon star. J Phys Conf Ser 861:012027 23. Nakamura T, Oohara K, Kojima Y (1987) General relativistic collapse to black holes and gravitational waves from black holes. Progr Theor Phys Suppl 90:1–218 24. Ostriker JP, Mark JW-K (1968) Rapidly rotating stars. I. The self-consistent-field method. Astrophys J 151:1075–1088 25. Pretorius F (2005) Evolution of binary black hole spacetimes. Phys Rev Lett 95:121101 26. Rezzolla L, Takami K (2013) Black-hole production from ultrarelativistic collisions. Class Quantum Grav 30(1):012001

References

15

27. Shibata M, Ury¯u K (2000) Simulation of merging binary neutron stars in full general relativity: =2 case. Phys Rev D 61(6):064001 28. Shibata M, Nakamura T (1995) Evolution of three-dimensional gravitational waves: harmonic slicing case. Phys Rev D 52:5428–5444 29. Tsokaros A, Ury¯u K, Rezzolla L (2015) New code for quasiequilibrium initial data of binary neutron stars: corotating, irrotational, and slowly spinning systems. Phys Rev D 91(10):104030 30. Tsokaros A, Ury¯u K, Shapiro SL (2019) Complete initial value spacetimes containing black holes in general relativity: application to black hole-disk systems. Phys Rev D 99(4):041501 31. Tsokaros A, K¯oji U (2012) Binary black hole circular orbits computed with cocal. J Eng Math 82(1):133–141. https://doi.org/10.1007/s10665-012-9585-6 32. Ury¯u K, Tsokaros A (2012) New code for equilibriums and quasiequilibrium initial data of compact objects. Phys Rev D 85(6):064014 33. Ury¯u K, Tsokaros A, Galeazzi F, Hotta H, S Misa, Taniguchi K, Yoshida S (2016) New code for equilibriums and quasiequilibrium initial data of compact objects. III. Axisymmetric and triaxial rotating stars. Phys Rev D D93(4):044056 34. Ury¯u K, Antonios T, Philippe G (2012) New code for equilibriums and quasiequilibrium initial data of compact objects. II. Convergence tests and comparisons of binary black hole initial data. Phys Rev D D86:104001 35. Uryu K, Shijun Y, Eric G, Charalampos M, Kotaro F, Antonios T, Keisuke T, Yoshiharu E (2019) New code for equilibriums and quasiequilibrium initial data of compact objects. IV. Rotating relativistic stars with mixed poloidal and toroidal magnetic fields. arXiv e-prints arXiv:1906.10393 36. Wilson JR, Mathews GJ (1989) Relativistic hydrodynamics, pp 306–314 37. Witten E (1984) Cosmic separation of phases. Phys Rev D 30:272–285 2 38. Xu RX (2003) Solid quark stars? Astrophys J Lett 596:L59–L62 39. York JW (1973) Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial value problem of general relativity. J Math Phys 14:456 40. York JW (1979) Kinematics and dynamics of general relativity. In: Smarr LL (ed) Sources of gravitational radiation. Cambridge, UK: Cambridge University Press pp 83–126 41. Zhou E, Tsokaros A, Rezzolla L, Xu R, Ury¯u K (2018) Uniformly rotating, axisymmetric, and triaxial quark stars in general relativity. Phys Rev D 97(2):023013 42. Zhou EP, Lu JG, Tong H, Xu RX (2014) Two types of glitches in a solid quark star model. Mon Not R Astron Soc 443:2705–2710 43. Zhou E, Tsokaros A, Ury¯u K, Renxin X, Shibata M (2019) Differentially rotating strange star in general relativity. Phys Rev D 100(4):043015

Chapter 2

Tidal Deformability of Compact Stars

2.1 Introduction During the inspiral stage of a binary merger event, due to the energy dissipation of the gravitational wave (GW) emission, the orbital separation shrinks and orbital frequency increases with time. Consequently, the frequency and amplitude of the GW increases as time. According to the frequency and frequency derivative of the GW signal, the so-called chirp mass of the system, which is a combination of each component masses M = (m 1 m 2 )3/5 (m 1 + m 2 )−1/5 , could be determined very precisely. In the early inspiral stage, the binary separation is large and the motion is relatively slow, the finite size and detailed structure of the component compact stars could be neglected and treated as point masses. Whereas in the late inspiral, as the relativistic effect becomes important, more information could be extracted from GW signals in high frequencies, e.g. the finite size effect of the compact stars. Studying the mass-radius relation, or in another word the size of compact stars, has been widely recognized as a crucial and convincing way to distinguish between different EoS models of compact stars, and is a very interesting topic for ground based GW observatories as well. In the recent decades, works has been done in determining the influence of the radius of NSs on the GW template by constructing binary NS (BNS) initial data [16], as well as analyzing the spectrum properties of GW emission from dynamical evolution of different BNS configurations and with various EoS [33]. To constrain NS radius from BH-NS merger events has also been studied [37]. This dream of using GW observation to constrain the size of NSs has finally come true in August 17th 2017, with the detection of the inspiral GW signal from a BNS system GW170817 [35]. The so-called tidal deformability parameter has been measured and constrained from the GW signal [17, 18]. Due to the presence of matter, NS in a binary system will be deformed by the tidal field of its companion. In the late inspiral phase, this tidal deformation is phase-coherent with the orbital motion, hence contributes to additional energy dissipation through GW emission and accelerate the coalescence. As a result, this tidal deformation leaves a detectable © Springer Nature Singapore Pte Ltd. 2020 E. Zhou, Studying Compact Star Equation of States with General Relativistic Initial Data Approach, Springer Theses, https://doi.org/10.1007/978-981-15-4151-3_2

17

18

2 Tidal Deformability of Compact Stars

signature in the high frequency GW signal. In general, a NS with a larger size will be more likely to be deformed by its companion, as the gravitational bound is relatively weaker compared to a smaller NS with the same mass. Hence, the tidal deformability constraint obtained from the observation of GW170817 (initially as (1.4) < 800 +390 [2, 35]). for low spin priors and later improved to (1.4) = 190−120 In this chapter, I will talk about 3 models we have considered to interpret the tidal deformability constraint: strange quark star model, strangeon star model as well as a NS model based on quark mean field models. Due to the finite surface density of strange stars, the calculation of tidal deformability requires a surface correction for strange stars, which results in significant difference when one wants to translate the tidal deformability constraint into constraint on radius or maximum mass of compact stars.

2.2 Background 2.2.1 Calculation of Tidal Deformability In order to test EoS models with the tidal deformability constraint from GW observations, we have to first obtain the tidal deformability for any given EoS with theoretical calculations. A quadrupole moment Q i j will be induced if the star is affected by an external field Ei j , the ratio of which characterizes the tidal deformability λ = −Q i j /Ei j . This ratio is related to the so-called tidal love number as k2 = 23 λR −5 . In practice, a dimensionless tidal deformability is widely used  = λ/M 5 = 23 k2 (R/M)5 . In order to calculate k2 theoretically, we usually add l = 2 perturbations on to the spacetime of a static spherically symmetric star.1 In this section, we will introduce the way to calculate k2 and hence  based on [18]. The structure of a static spherical star as well as its spacetime could be obtained by solving the Tolman-Oppenheimer-Volkhoff Equation (TOV equation). After introducing the l = 2 mode perturbation to the spacetime metric, both the Einstein tensor on the left hand side of Einstein equations and the energy momentum tensor will be altered by the perturbation: G μν = G (0) μν + δG μν (2.1) (0) Tμν = Tμν + δTμν , in which, G (0) and T (0) already solve Einstein equation as they are solutions of the TOV equation. As a result, the perturbed part needs to satisfy: δG μν = 8π δTμν , 1 This is also the origin of the lower index ‘2’ in k

(2.2)

2 . In principle, one can also try to calculate higher mode tidal deformability, c.f. [12] for a more comprehensive discussion.

2.2 Background

19

which could be considered as the perturbed Einstein equations. After simplifying the equation with symmetry conditions, the non-zero part of the equation yields:   2 λ 2m(r ) +e + 4πr ( p − e) H +H r r2     e+ p 6eλ λ 2 − ν = 0, + H − 2 + 4π e 5e + 9 p + r dp/de 





(2.3)

in which H characterizes the perturbation in the metric and  denotes the derivative with respect to the radial direction d/dr . λ and ν in this equation are related to the spacetime metric components obtained from a TOV solution.2 Variables related to quantities of the star (m(r ), p and e) should also be obtained by solving the TOV equation. Equation (2.3) could be solved in the interior and exterior of the star. In the interior, we need to pay special attention to the boundary condition for solving this differential equation. We usually start the numerical integration at r = 0 for solving TOV equation. However, it’s not possible for Eq. (2.3) as r appears in the denominator multiple times so we have to start from a finite r . In addition, H is arbitrarily introduced to describe the perturbation of the tidal field so the absolute amplitude doesn’t really matter in this calculation. Nevertheless, Eq. (2.3) is a second order differential equation. Therefore we have to provide H and H  consistently at the boundary to obtain the correct result. Noticing that when r approaches 0, there is an approximate solution to Eq. (2.3):   2π e(0) + p(0) 2 (5e(0) + 9 p(0) + )r + O(r 3 ) H (r ) = a0 r 2 1 − 7 (dp/de)(0) hence

H  ∼ 2a0 r ∼ 2H/r.

(2.4)

(2.5)

In the case of finite and small r , any combination of H and H  that satisfies Eq. (2.5) will become an appropriate boundary condition for numerically solve Eq. (2.3). Given the boundary condition near the center of the star, we can then numerical solve Eq. (2.3) to the surface of the star. In the exterior of the star, due to the absence of terms related to the fluid, Eq. (2.3) reduces to a much simpler form as H  +

2 When



   λ 2 6e 2 H = 0, − λ H  − + λ r r2

(2.6)

solving for the static spherical spacetime, the line element could be generally written as ds 2 = −eν(r ) dt 2 + eλ(r ) dr 2 + r 2 (dθ 2 + sin2 θdφ 2 ).

20

2 Tidal Deformability of Compact Stars

which has general solution in the following form   ⎡ ⎤ 2 2     r 2  2M ⎣ M (M − r ) 2M + 6Mr − 3r r 3 ⎦+ 1− H = c1 + log − M r 2 r − 2M r 2 (2M − r )2   r 2  2M . 1− 3c2 M r

(2.7) For sufficiently large r , this general solution could be approximated as 8 H= 5



M r



3 c1 + O

M r

4 +3

r   r 2 . c2 + O M M

(2.8)

In addition, we also know that the tt component of the spacetime metric approaches the following for sufficiently large r : 3Q i j M 1 − gtt =− − 2 r 2r 3



1 n n − δi j 3 i

j



1 + Ei j x i x j . 2

(2.9)

Now that H is the perturbation on top of TOV solutions, we could compare the above two approximations and find the correspondence between the them. From the correspondence we could deduce 15 1 E λE 8 M3 1 c2 = M 2 E. 3 c1 =

(2.10)

Therefore, the information about λ the tidal deformability is encoded in the ratio of c1 and c2 . Now let’s consider again the solution we obtained in the interior of the star by numerically solving Eq. (2.3). This interior solution should match the solution of the exterior on the surface of the star, hence helping us determine the ratio between c1 and c2 according to the quantities on the surface. Following this logic, with a little bit more efforts figuring out the algebra, we obtain k2 =

8C 5 (1 − 2C)2 [2 + 2C(y − 1) − y] × 5   {2C(6 − 3y + 3C(5y − 8)) + 4C 3 13 − 11y + C(3y − 2) + 2C 2 (1 + y) + 3(1 − 2C)2 [2 − y + 2C(y − 1)] log(1 − 2C)}−1 ,

in which C = M/R is the compactness of the star and y = R H  (R)/H (R).

(2.11)

2.2 Background

21

Now it becomes clear why the calculation of tidal deformability for strange stars is different. Due to the non-zero surface density, a correction has to be made to match the interior and exterior solutions on the surface [12, 30].3 After estimating H  and H on the surface of the star by solving Eq. (2.3), the correct y is given as y = R H  (R)/H (R) −

es , M/4π R 3

(2.12)

in which es is the energy density on the surface and M, R are the mass and radius of the star, respectively.

2.2.2  Constraint of GW170817 As mentioned before, after knowing the way to theoretically calculate tidal deformablity, we also need to compare the results with the observation. In Aug 17th 2017, LIGO and VIRGO detected a signal from a BNS inspiral stage. The source is localized in an area of around 28 square degrees and the distance is determined to be roughly 40 Mpc. A short gamma ray burst (sGRB) was detected 1.7 s after the merger inferred from the GW signal. Hours later, the kilonova emission [24] which is powered by the radioactivities of the heavy elements synthesized in the neutron rich ejecta 4 has been detected as well in optical/UV/IR bands. The host galaxy of this BNS merger event has also been confirmed accordingly: NGC 4993. Roughly ten days later, the counterparts in X-ray band and radio band have been observed as well [36]. According to the GW emission in the inspiral stage, the chirp mass of the binary system is determined as 1.188 solar mass, with mass ratio between 0.7 and 1.0. According to such information, the total mass of the binary is found to be between 2.73–3.29 solar masses, with the mass of the components being 0.86–2.26 solar masses. This is consistent with the double NS systems observed in the Galaxy. The detection of the electromagnetic (EM) counterparts also supports this event is indeed caused by a BNS merger. The late inspiral GW signal has put constraints on the tidal deformability of the merger binary. The parameter which is directly constrained by the observation is the mass-weighted binary tidal deformability: ˜ = 

3 Note

16 (m 1 + 12 m 2 ) m 41 1 + (m 2 + 12 m 1 )m 42 2 . 13 (m 1 + m 2 )5

(2.13)

that in [30] there is a sign mismatch for this correction.

4 Note that other models could also explain the kilonova observation, for example, by the spin down

power injection of the central remnant [21, 25].

22

2 Tidal Deformability of Compact Stars

˜ is an upper limit of 800 in the low spin case (|χ | < 0.05) and The constraint on  700 in the high spin case (|χ | < 0.89). According to the study in [9], it’s not possible to tidally synchronize the spin of the component NSs with realistic viscosity during the late inspiral stage. Therefore, the low spin scenario is more likely to be valid and we will only consider the constraint in the low spin prior in this thesis. By expanding ˜ around a given NS mass, say (1.4), this constraint could be further translated as  (1.4) < 800 in the low spin prior. In addition, according to the EM counterparts in the optical/UV/IR bands, the mass of the ejecta could be inferred [1], which has been found to be positively correlated to the tidal deformability of the merging binary. According to the this relation, a lower bound for the tidal deformability is derived (1.4) > 400, according to the minimum amount of ejecta to explain the observation of EM counterparts [31]. Nevertheless, in a more comprehensive investigation on the EoS parameterization as well as mass ratio of the binary, such a lower limit is found to be not very reliable [19].

2.2.3 Constraint on NS Models The observations mentioned in the previous section favors more compact EoSs by only looking at the tidal deformability measurement, such as APR4 or SLy [3, 15]. Nevertheless, such EoS models are all constructed within certain nuclear physics models and might not be the real EoS that nature uses to build NSs. A more comprehensive study could be done by treating (1.4) as a functional of the EoS (which is a function of p = p(ρ) in the simplest case). Such interpretation are done in e.g. [2, 6, 27]. For instance, in [2], the logarithm of the adiabatic indexes are treated as a polynomial of the pressure, namely  = ( p; γi ) in which γi = (γ0 , γ1 , γ2 , γ3 ) are freely chosen as parameters. For low density (say below half the nuclear saturation density), the EoS is connected to the SLy EoS [15]. Sampling of the EoS models is then to uniformly sample γi in certain intervals. For each of the EoS samples in this systematic study, the mass radius relation and tidal deformability could be theoretically calculated and then constrained by the observation of both the tidal deformability and mass measurement of massive pulsars [7, 14]. According to such a systematic study, the radius of the merging NSs of GW170817 is found to be R = 11.9+1.4 −1.4 km. Similar analysis is done in e.g. [6], which constrains the radius of a 1.4 solar mass NS to be in the range of [9.9, 13.8] km. However, it’s worth noting that such analysis might be affected by choices of EoS priors. In [27], it has been pointed out that when the prior for possible twin star (for which there is hadronquark phase transition inside the star) branch EoSs is considered, the radius get less constrained, i.e., R1.4 ∈ [8.53, 13.74] km. Following such systematic investigations, some suggestions have been made to explore the correlation between tidal deformability and radius of NSs (such as in [41]). Intuitively, for NSs with a certain mass, it is true that the smaller the radius, the stronger the bound by self-gravity and thus the less likely to be deformed by external tidal fields. In the most extreme limit, the compact star with smallest radius, i.e. a BH,

2.2 Background

23

Table 2.1 Properties of NS models investigated in [43] with the symmetry energy slope L being the EoS parameter. Properties of 1.4 solar mass stars as well as TOV maximum mass with four different choices of L are listed, including radius, compactness, k2 and tidal deformability  L [MeV] R [km] M/R k2  MTOV [M ] 20 40 60 80

11.725 11.829 12.011 12.512

0.17499 0.17481 0.17216 0.16526

0.12064 0.08816 0.07404 0.06889

471 331 326 373

2.088 2.078 2.068 2.070

has tidal deformability of zero. Nevertheless, from the introduction in the previous section, it could be seen that the dependence of (1.4) and R is not necessarily monotonic. We have explored this correlation based on quark mean field (QMF) models (for more details see [43]) and demonstrated that even for a certain nuclear physical model, there is no monotonic relation between (1.4) and R. Therefore, one has to be careful in investigating or employing such correlations. Generally speaking, for symmetric nuclear matter (namely, matter with β = n n −n p = 0), we have quite good understanding both from theoretical studies and n n +n p from experiments [13]. While for NSs which is composed of highly asymmetric nuclear matter, our knowledge is relatively less, due to the uncertainty of, for example, the slope of the symmetry energy (L(n) = d E sym /dn). In this work, we try to model NS EoS in the QMF scenario in which L is an important parameter which helps us better understand the dependence of tidal deformability on radius and L. We have studied four cases with L = 20, 40, 60 and 80 MeV, respectively. According to Table 2.1, the slope of symmetry energy L has very significant influence on the radius of the NSs but not on the TOV maximum mass. Specifically, the larger L is, the larger the size of the NS is. This is known as the famous L-versus-R relation [22]. However, the tidal deformability of a NS with 1.4 solar mass is found to be not monotonic with the choice of L, hence not monotonic with radius of the NS ss well. As a consequence, it’s not possible to directly translate the tidal deformability constraint into the constraint on radius of the NS, at least within this EoS model.

2.3 Constraint on Strangeon Star Model with GW170817 2.3.1 Tidal Deformability of Strangeon Star In the case of NS models, an EoS model is usually referred as a ‘soft’ EoS if the tidal deformability is small.5 Intuitively, for NSs with a certain mass, if the EoS is softer, the self-gravity tends to make the star more compact thus more difficult to be 5 As

a consequence, many literatures claim that the observation of GW170817 favors ‘soft’ EoS models (c.f. [6]).

24

2 Tidal Deformability of Compact Stars

deformed by external fields. In the limit of extremely soft EoS, the NS will not be able to fight self-gravity and collapse to BH, which has a tidal deformability of zero. At the same time, the stiffness of the EoS is also related to the TOV maximum mass of NSs (MTOV ). It has been shown that there is some correlation between (1.4) and MTOV . According to the studies in [6], for example, the upper limit of 800 for (1.4) will exclude NS EoS models with MTOV larger than roughly 2.8 solar masses. Provided the fact that strangeon star model predicts very large MTOV (larger than 3 solar masses), one might simply conclude that strangeon star model could not pass the test of tidal deformability constraint with GW170817. However, as mentioned before, the most significant difference between strange stars and NSs is that NSs are bound by gravity whereas strange stars are self-bound by strong interaction. For strange stars, the relation between the stiffness of the EoS models and the compactness of a given mass is not the same as NSs. As mentioned in previous section, the self-bound nature of strange stars result in a finite surface density which requires special correction when calculating the tidal deformability. We have done the analysis for strangeon star with model parameter U0 = 50 MeV and Nq = 18. Although the TOV maximum mass is as high as 3 solar masses, the dimensionless tidal deformability for a 1.4 solar mass strangeon star is found to be only 381.9, which is similar to ‘soft’ NS EoSs such as APR4 and SLy. More details about this comparison is shown in Fig. 2.1. Although both strangeon star model and conventional NS models (such as APR4 and SLy) satisfy the tidal deformability constraint of GW170817, the TOV maximum mass of SLy is only slightly larger than 2 M (2.05 M ). While APR4 EoS model only includes {npeμ} particles and was initially proposed for exploring whether the central density of NSs is large enough for excitation of hyperons. As a consequence, the hyperon puzzle is unavoidable for such EoS models [11]. In order to understand how we can constrain the strangeon star model with future observations, we have also made an investigation on the parameters (i.e., U0 and ρs ) of the strangeon star model. This result is shown in Fig. 2.2. According to the results shown in the figure, it could be seen that the correlation between (1.4) and MTOV still holds for strangeon star qualitatively, as the case for NSs. Nevertheless, the quantitative result changed a lot. The largest possible MTOV is 4 M while the tidal deformability constraint is still satisfied. Future mass measurement of very massive pulsars (i.e., pulsars with mass ∼3 M ) or EM counterparts of mass gap merger events (component mass ∈ [3, 5] M ) might be a smoking gun for the existence of strangeon star.

2.3.2 Test of Solid Strangeon Star Model Another significant difference between strangeon star model and NS model is that the latter is suggested to be entirely solid [40] whereas NSs only possess a solid crust. It’s not very accurate to calculate the tidal deformability with the energy momentum tensor of perfect fluid for a solid strangeon star. Indeed, due to the existence of shear

2.3 Constraint on Strangeon Star Model with GW170817

25

3000 2500

Λ2

2000 1500

50

500

%

0

strangeon star APR4 SLy MPA1

% 90

1000

0

500

1000

1500

Λ1

2000

2500

3000

3000 2500

Λ2

2000 1500

50%

500 0

strangeon star APR4 SLy MPA1

90%

1000

0

500

1000

1500

Λ1

2000

2500

3000

Fig. 2.1 Comparison of the tidal deformability of strangeon star model with the observation of GW170817. The upper panel shows the results for low spin prior (|χ| < 0.05) and the lower panel for high spin prior (|χ| < 0.89). The dashed lines are the 50% (the one closer to the bottom left corner) and 90% posterior contour [35]. Results of strangeon star model (in grey areas) as well as 3 other NS models are shown. Strangeon star model is still consistent with the observation. This figure is adopted from [21]

26

2 Tidal Deformability of Compact Stars

=8

00

100

Λ(1 .4)

90 80

M

V TO

=

60

0

4M

1. 4)

60

Λ(

U0 [MeV]

70

=

50

M 3.5 = V

40

M TO

00

=4

Λ

30 20

4) (1.

M TOV 1.5

1.6

1.7

ρs [ρnuc]

1.8

1.9

= 3M 2.0

Fig. 2.2 Parameter space exploration for the strangeon star model. By fixing values for U0 (y-axis) and ρs (x-axis), we could obtain an EoS model and solve the TOV maximum mass and (1.4) for each EoS model. Contours of (1.4) (blue dashed lines) and MTOV (black solid lines) are shown to indicate possible parameter space. It could be seen that the TOV maximum mass of strangeon star model could be as high as 4 M without violating the tidal deformability constraint. This figure is adopted from [20]

module as well as the accumulation of elastic energy, it will be more difficult to deform a solid compact star compared with the case of a star composed of perfect fluid. Consequently, the real tidal deformability of strangeon star should be even smaller than the results calculated in the previous sections. According to the discussion in [30], if the induced mass quadrupole moment of a solid star by a tidal field exceeds the critical value that the elastic structure of the star could stand, such a solid star will crack. In another word, the solid star will be tidally melt. For NSs, the solid crust only contributes very tiny amount of matter to the entire star. The critical mass quadrupole of the solid crust is thus very small. Therefore, BNS could be entirely treated as perfect fluid in the sensitive frequency band of LIGO and VIRGO, as the tidal field will be sufficient to destroy the solid crust of NSs during the merger at large orbital separations hence at low GW frequency (i.e., when GW frequency is smaller than 10 Hz). However, for strangeon star which is entirely solid, details need to be considered. As discussed in [30], in a binary merger event, when the tidal field is too strong for the solid structure of the star to stand, the GW frequency radiated by the binary is

2.3 Constraint on Strangeon Star Model with GW170817

27

  1/4  1 Q 22max 2 f br = 3 π λ 1/2  −1/2  λ Q 22max = 20 × Hz, 1040 g cm2 2 × 1036 g cm2 s2

(2.14)

in which Q 22max is the critical mass quadrupole moment that the solid star could stand. For a strangeon star with 1.4 solar mass, the typical value of  is 2 × 1036 g cm2 s2 . The critical mass quadrupole moment of a solid compact star is estimated as [29]: μ 4 × 1032 erg cm−3



6 

−1

σmax g cm2 , 0.01 (2.15) in which μ is the shear module with a typical value of 4 × 1032 erg cm−3 as suggested by [40] and σmax is the breaking strain for the solid structure. Combining the above two equations, it could be derived that the tidal melting for a binary strangeon star merger occurs when the frequency of the GW emitted is roughly 100 Hz. This is lower than the frequency when information on tidal deformability could be effectively extracted. As a result, the perfect fluid assumption is in fact valid for calculating the tidal deformability of strangeon stars to compare with the observations. View it differently, due to the lack of knowledge on the EoS of dense matter, there is huge uncertainty in the shear module and breaking strain for solid strangeon stars. If the real value of μ could be one or two magnitudes larger, the tidal melting will happen at f GW ∼ 500 Hz for binary strangeon star mergers. This actually provides an opportunity for us to study the solid structure of strangeon star with future observations once the sensitivity of LIGO and VIRGO is further improved. For instance, if there is the hint that tidal effect is negligible when GW frequency is lower than a certain value and tidal effect becomes finite after reaching this GW frequency, it could be inferred as evidence of solid structure of the merging binary. And according to this critical GW frequency, parameters such as shear module and breaking strain could be constrained. Q 22max = 2.8 × 1041

R 10 km

M 1.4 M

2.4 Constraint on MIT Bag Model with GW170817 2.4.1 The EoS Model Apart from strangeon star, we have also considered a conventional strange star model: MIT bag model [4]. MIT bag model was originally proposed to understand hadrons phenomenologically. If constraints on the parameters of MIT bag model could be made by the observation of GW170817, it would greatly enrich our knowledge

28

2 Tidal Deformability of Compact Stars

about strong interaction. Moreover, such constraint is totally independent of nuclear physical theories as well as nuclear experiments. This is the motivation of such a parameter study. In the description of MIT bag model, strange star is composed of {u, d, s, e} particles. The following weak interactions is important for understanding the EoS of strange quark star: d → u + e + ν˜ e , u + e → d + νe , s → u + e + ν˜ e , u + e → s + νe , s + u → d + u.

(2.16) Charge neutrality and β equilibrium should be preserved in the star: μs = μd = μu + μe 2 1 1 n u − n d − n s − n e = 0. 3 3 3

(2.17)

The grand canonical potential of such a thermaldynamic system is free =



i0 +

i

 μ 4 3 b (1 − a ) + Beff , 4 2 4π 3

(2.18)

in which i0 is the thermaldynamic potential of each species of the particles (the lower index i corresponds to 4 species of u, d, s, e particles) assuming that they are ideal relativistic Fermi gases. The second term on the right hand side represents the perturbative QCD corrections due to gluon mediated interaction between quarks up to O(αs2 ). The parameter a4 stands for the degree of such a correct, with a4 = 1 for no correction. The term μb = μu + μd + μs is the baryon chemical potential.6 The third term Beff is the effective bag constant, which is a phenomenological parameter characterizing the non-perturbative QCD effects. EoS of such a strange star could be obtained once the grand canonical potential is given. The energy density could be derived as e=



μi n i + .

(2.19)

i

Pressure and number density could then be obtained from first law of thermal dynamics:   e ∂ (2.20) P = n 2b ∂n b n b

6 Note

that in MIT bag model, the strange star is composed of deconfined quarks. Therefore, there is no baryons inside the star. The concept of “baryon chemical potential” is just for the convenience of calculation.

2.4 Constraint on MIT Bag Model with GW170817

 μ 3 μq3   1 b n q = − ∂/∂μq V = 2 − 2 (1 − a4 ) π π 3 nb =

1 a4  μb 3 nq ∼ 2 . 3 q π 3

29

(2.21) (2.22)

In practice, the mass of u, d quarks is much smaller than that of strange quark and we can neglect the rest mass of u, d quarks. If one also neglects the mass of s quark, the EoS of strange star in MIT bag model will be simplified as the form of Eq. (1.30). With such a form, different choices of a4 will not affect the relation between pressure and energy density. As a consequence, the mass radius relation will be totally independent on the value of a4 [8, 23]. In the case of finite s quark mass, a4 will actually affect the EoS. It indicates that strange quark mass is also a parameter of the EoS, although not explicitly included in the above equations. To conclude, for any given {m s , a4 , Beff }, we could construct an EoS model for strange quark star. Then we can solve TOV equation and the perturbed TOV equation to solve for the TOV maximum and tidal deformability of strange quark stars, respectively. By comparing the results with the observation of massive pulsars [7, 14] and GW170817 [35], constraint could be made on {m s , a4 , Beff }.

2.4.2 Constraint on Strange Quark Mass Among those 3 parameters, m s is the one with the best constraint from nuclear physics studies (m s = 95 ± 5 MeV [28]). Therefore, we decide to first fix the choice of Beff and a4 and study the influence of m s on the EoS models. If the impact of m s is not large within its possible range, we can then confidently fix the value of m s in the following studies and focus only on Beff and a4 , making this problem simpler. In practice, we have chosen widely used value for Beff and a4 ((138 MeV)4 and 0.61) and studied three EoS models with different choice of m s (0, 90 and 100 MeV). The details could be seen from Fig. 2.3 and Table 2.2. In particular, finite strange quark mass will soften the EoS model, reducing the TOV maximum mass and tidal deformbaility at the same time. This gives a hint on a similar correlation between MTOV and  as in the case of NSs and strangeon stars, which we will introduce in more details in the next subsection. Moreover, the model with zero strange quark mass predicts a tidal deformability of 791.4 for a 1.4 solar mass star, which marginally satisfies the initial analysis [35] but will be excluded by the later results [2]. This provides information on the lower limit of strange quark mass independently from nuclear physics studies, assuming that GW170817 originates from a binary strange star merger.

30

2 Tidal Deformability of Compact Stars 3000 2500

Λ2

2000 1500

% 90

1000

0

APR4 SLy MPA1

% 50

500

0

500

1000

1500

2000

2500

3000

Λ1 Fig. 2.3 Investigating the influence of finite strange quark mass on the tidal deformability of strange quark star models. Comparison is made with the observation of GW170817 (with the two dashed lines denoted by 50 and 90% representing the 50 and 90% contours of the posterior probability for the derived tidal deformability of component stars for GW170817 based on a low spin prior (|χ| < 0.05)) [35] as well as 3 NS models (APR4, SLy and MPA1). Grey, pink and green areas correspond to the result of MIT bag model with stange quark mass of 0, 90 and 100 MeV, respectively. Beff is chosen as (138 MeV)4 and a4 is equal to 0.61 for all 3 models. In order to plot the areas, we have employed the derived component masses (M1 , M2 ) within 90% posterior probability. It could be seen that in the previous constrained range of strange quark mass (i.e., from 90 to 100 MeV) or even neglecting strange quark mass, the results all pass the test of GW170817 with this certain choice of {a4 , Beff } [42] Table 2.2 Properties of strange quark stars with 1.4 solar masses in MIT bag model with three different choices of strange quark mass, including central number density (n c ), radius (R), compactness (M/R), tidal love number k2 , dimensionless tidal deformability  as well as TOV maximum mass MTOV . More details could be found in [42] m s [MeV] n c [fm−3 ] R [km] M/R k2  MTOV [M ] 0 90 100

0.327 0.355 0.361

11.814 11.478 11.415

0.17499 0.18016 0.18115

0.19510 0.18357 0.18133

792.8 644.9 619.7

2.217 2.101 2.079

Although increasing strange quark mass will result in strange stars with smaller radius hence smaller tidal deformability, the difference in both tidal deformability and TOV maximum mass is negligible for possible range of m s and they all pass the test of observation of massive pulsars and GW170817. Considering this, we will fix m s to 100 MeV and study the effects of other parameters in the following analysis.

2.4 Constraint on MIT Bag Model with GW170817

31

2.4.3 Constraint on Beff and a4 Once m s is fixed as 100 MeV, the EoS model will be uniquely determined for a given pair of Beff and a4 . In order to do this parameter study, we have varied a4 uniformly 1/4 between the range of (0.5, 1.0) and Beff in (134, 142) MeV. For each of those EoS models, we could determine their TOV maximum mass and tidal deformability. By interpolating the results onto the entire parameter space we could find out the possible parameter space for Beff and a4 according to nuclear stability, observation of massive pulsars and GW170817. Such a parameter space exploration is shown in Fig. 2.4 with 5 selected models shown in Table 2.3. Apart from constraint on MTOV and , we have also considered the stability condition of nuclear physics (green curves in the Figure) [38]. Two flavor line is derived according to the fact that two flavor quark matter (matter composed of deconfined {u, d} quarks) should have higher energy compared with nuclei, otherwise nucleus will spontaneously decay into two flavor quark matter. On the other hand, by requiring 3 flavor quark matter is more stable than nucleus (thus strange star exists), we could obtain the constraint given by the ‘3 flavor line’ (this is in essence the Bodmer-Witten conjecture [10, 39]). The first property we can find in the results is that (1.4) increases as MTOV , which is consistent with the case of NSs (c.f. [6]), strangeon stars [20] as well as when

144 142

2.01 M

1/4

Beff [MeV]

140

Λ(1.4) = 600

138 136

Λ(1.4) = 800

134

2.2 M

132 130 0.5

0.6

0.7

a4

0.8

0.9

1.0

Fig. 2.4 Constraint on the parameter space of Beff and a4 parameter in MIT bag model. The mass of strange quark is chosen to be 100 MeV for all the EoS models. Grey area is all the permitted parameter space considering the constraint of GW170817 (red curves), mass measurement of pulsars (black curves) and nuclear stability conditions (green curves). (1.4) = 600 and MTOV = 2.2 M curves are also included to indicate possible constraints from future observations. This figure is made according to the results in [42]

32

2 Tidal Deformability of Compact Stars

Table 2.3 Properties of 1.4 solar mass strange quark star in MIT bag model. The first two columns show the model parameters with the other columns showing the properties of the star including radius (R), compactness (M/R), tidal love number (k2 ) as well as dimensionless tidal deformability (). Mass of strange quark is fixed to be 100 MeV. This table is produced according to the results in [42] 1/4 a4 Beff [MeV] R [km] M/R k2  0.61 0.61 0.61 0.72 0.83

133 136 138 138 138

12.046 11.662 11.415 11.453 11.482

0.17166 0.17731 0.18115 0.18055 0.18008

0.19973 0.18865 0.18133 0.18262 0.18367

893.4 717.7 619.7 634.5 646.6

we vary m s in MIT bag model. Similar to the case of strangeon star, the quantitative dependence of MTOV on (1.4) is different from that of NSs. Additionally, MTOV and (1.4) both increases monotonically as a4 increases. In another word, perturbative QCD corrections also tend to soften the EoS models. Nevertheless, quantitatively, this dependence is very weak hence contour lines of MTOV and (1.4) are almost horizontal. Opposite to the weak influence of a4 , the effect bag constant significantly alters the stiffness of the EoS models hence is best constrained. Combining all the observations, we have figured out the possible range for model 1/4 parameters of the MIT bag model: Beff ∈ (134.1, 141.4) MeV and a4 ∈ (0.56, 0.91). Those parameters will be better constrained with future observations. In addition, noticing that the contour curves for TOV maximum mass and those for tidal deformability are almost parallel to each other (indicating that their dependence on a4 is highly similar), it could be inferred that there might be a correlation between MTOV and (1.4). We have fitted the results with different model parameters and found a power law relation between MTOV and (1.4). Even the models with different choice of m s follow this relation (see the results in Fig. 2.5):  (1.4) = 510.058 ×

MTOV 2.01 M

5.457 .

(2.23)

Such a relation means that, within MIT bag model, we could in principle translate the tidal deformability constraint by GW170817 directly to the constraint on TOV maximum mass. To be specific, (1.4) not larger than 800 sets an upper limit of 2.18 solar masses for MTOV . On the other hand, the observation of 2 M pulsars indicates that (1.4) should not be smaller 510.1 in MIT bag model.7 This means we could totally rule out strange quark models if pulsars with larger mass is observed or a more stringent tidal deformability upper limit is found in future observations.

7 As

a reference, (1.4) for APR4 EoS is 255.8.

2.4 Constraint on MIT Bag Model with GW170817 1200

a4 = 0.57 a4 = 0.585 a4 = 0.61 a4 = 0.72 a4 = 0.83 ms = 90

1000

800

Λ(1.4)

33

600

400

1.9

2.0

2.1

2.2

2.3

MTOV [M ] Fig. 2.5 The correlation between TOV maximum mass of tidal deformability in MIT bag model. Details about the model parameters are denoted in the top left. The variance of strange quark mass is also included in this figure, with the black dots for the case of m s = 90 MeV and the other dots for m s = 100 MeV. The black line is the fitted relation according to the results. Note that all the dots with different colors and symbols all locate closely to the fitted curve, indicating that this relation is insensitive to the choice of a4 and m s . This figure is adopted from [42]

2.4.4 Constraint on  Parameter In the previous section we considered only the simplest MIT bag model. In more detailed researches [5, 26, 32, 34], it has been pointed out that quarks in bulk quark matter could form Copper pairs and be in color-superconducting state. In such a situation, one additional term needs to be considered in the grand canonical potential of quark matter: 3 (2.24) CFL = free − 2 2 μ2b , π in which  represents the energy gap on the Fermi surface due to the formation of quark pairs. Because of the lack of theoretical studies and experimental constraints, the energy gap parameter has very large uncertainties and is believed to be in the range of (0, 100) MeV. Similar to the methods introduced in the previous section, we have fixed m s as 100 MeV and a4 as 1 or 0.61 and varied Beff and  uniformly in certain range to obtain different EoS models and then compare with the observations and nuclear stability conditions to make constraints on the energy gap parameter. The results are shown in Fig. 2.6. A lower limit of 40 MeV is found for the cases with a4 = 1 where as almost no constraint is found for  for the more realistic case of a4 = 0.61. Additionally, the correlation between MTOV and (1.4) becomes less

34

2 Tidal Deformability of Compact Stars 154 152

1/4

Beff [MeV]

150

= .4) Λ(1

1M

148

2.0

146

= .4) Λ(1

144

800

M 2.2

142 140 10

600

20

30

40

50

60

70

80

90

100

Δ [MeV] 150 148

1 2.0

144

2.2

M 4) (1.

142

=

M

0

60

Λ

1/4

Beff [MeV]

146

= 4) (1.

0

80

Λ

140 138 136 134 10

20

30

40

50

60

70

80

90

100

Δ [MeV] Fig. 2.6 Similar to Fig. 2.4 but with EoS models of color-superconducting quark matter. Observations of GW170817 and massive pulsars as well as nuclear stability conditions are considered for constraining the effective bag constant and energy gap parameter. Upper panel shows the case for a4 = 1 while lower panel for a4 = 0.61. This figure is adopted from [42]

2.4 Constraint on MIT Bag Model with GW170817

35

strict than the case of normal quark matter without pair formation. As a consequence, the largest possible MTOV in the permitted parameter space is increased to roughly 2.32 M (for both a4 = 1 and a4 = 0.61). On the other hand, the least possible (1.4) now becomes less than 400 and still possible to reach MTOV ∼ 2 M . Namely, considering color-superconducting quark matter makes the available parameter space larger. There is also possibilities to better constrain all the parameters with future observations or totally exclude strange star models.

References 1. Abbott BP, Abbott R, Abbott TD, Acernese F, Ackley K, Adams C, Adams T, Addesso P, Adhikari RX, Adya VB et al (2017) Estimating the contribution of dynamical ejecta in the Kilonova associated with GW170817. Astrophys J Lett 850:L39 2. Abbott BP, Abbott R, Abbott TD, Acernese F, Ackley K, Adams C, Adams T, Addesso P, Adhikari RX, Adya VB et al (2018) GW170817: measurements of neutron star radii and equation of state. Phys Rev Lett 121(16):161101 3. Akmal A, Pandharipande VR, Ravenhall DG (1998) Equation of state of nucleon matter and neutron star structure. Phys Rev C 58:1804–1828 4. Alcock C, Farhi E, Olinto A (1986) Strange stars. Astrophys J 310:261–272 5. Alford M, Rajagopal K, Wilczek F (1999) Color-flavor locking and chiral symmetry breaking in high density QCD. Nuclear Physics B 537:443–458 6. Annala E, Gorda T, Kurkela A, Vuorinen A (2018) Gravitational-wave constraints on the neutron-star-matter equation of state. Phys Rev Lett 120(17):172703 7. Antoniadis J, Freire PCC, Wex N, Tauris TM, Lynch RS et al (2013) A massive pulsar in a compact relativistic binary. Science 340:448 8. Bhattacharyya S, Bombaci I, Logoteta D, Thampan AV (2016) Fast spinning strange stars: possible ways to constrain interacting quark matter parameters. Mon Not R Astron Soc 457:3101– 3114 9. Bildsten L, Cutler C (1992) Tidal interactions of inspiraling compact binaries. Astrophys J 400:175–180 10. Bodmer AR (1971) Collapsed Nuclei. Phys Rev D 4:1601–1606 11. Bombaci I (2017) The hyperon puzzle in neutron stars. In: Proceedings of the 12th international conference on Hypernuclear and strange particle physics (HYP2015), id.101002, 8 pp. p 101002 12. Damour T, Nagar A (2009) Relativistic tidal properties of neutron stars. Phys Rev D 80(8):084035 13. Danielewicz P, Lacey R, Lynch WG (2002) Determination of the equation of state of dense matter. Science 298:1592–1596 14. Demorest PB, Pennucci T, Ransom SM, Roberts MSE, Hessels JWT (2010) A two-solar-mass neutron star measured using Shapiro delay. Nature 467:1081–1083 15. Douchin F, Haensel P (2001) A unified equation of state of dense matter and neutron star structure. Astron Astrophys 380:151–167 16. Faber JA, Grandclément P, Rasio FA, Taniguchi K (2002) Measuring neutron-star radii with gravitational-wave detectors. Phys Rev Lett 89(23):231102 17. Flanagan É, Hinderer T (2008) Constraining neutron-star tidal Love numbers with gravitationalwave detectors. Phys Rev D 77:021502. https://doi.org/10.1103/PhysRevD.77.021502 18. Hinderer T (2008) Tidal love numbers of neutron stars. Astrophys J 677:1216–1220 19. Kiuchi K, Kyutoku K, Shibata M, Taniguchi K (2019) Revisiting the lower bound on tidal deformability derived by AT 2017gfo. Astrophys J Lett 876(2):L31 20. Lai X, Zhou E, Xu R (2018) Strangeons constitute strong matter in bulk: to test using GW170817. arXiv e-prints

36

2 Tidal Deformability of Compact Stars

21. Lai X-Y, Yu Y-W, Zhou E-P, Li Y-Y, Xu R-X (2018) Merging strangeon stars. Res Astron Astrophys 18:024 22. Lattimer JM, Prakash M (2004) The physics of neutron stars. Science 304:536–542 23. Li A, Zhu Z-Y, Zhou X (2017) New equations of state for postmerger supramassive quark stars. Astrophys J 844:41 24. Li L-X, Paczynski B (1998) Transient events from neutron star mergers. Astrophys J 507:L59 25. Li S-Z, Liu L-D, Yu Y-W, Zhang B (2018) What powered the optical transient AT2017gfo associated with GW170817? Astrophys J Lett 861(2):L12 26. Lugones G, Horvath JE (2002) Color-flavor locked strange matter. Phys Rev D 66(7):074017 27. Most ER, Weih LR, Rezzolla L, Schaffner-Bielich J (2018) New constraints on Radii and Tidal Deformabilities of neutron stars from GW170817. Phys Rev Lett 120(26):261103 28. Olive KA, Particle Data Group (2014) Review of particle physics. Chin Phys C 38(9):090001 29. Owen BJ (2005) Maximum elastic deformations of compact stars with exotic equations of state. Phys Rev Lett 95(21):211101 30. Postnikov S, Prakash M, Lattimer M (2010) Tidal Love numbers of neutron and self-bound quark stars. Phys Rev D 82(2):024016 31. Radice D, Perego A, Zappa F, Bernuzzi S (2018) GW170817: joint constraint on the neutron star equation of state from Multimessenger observations. Astrophys J Lett 852:L29 32. Rajagopal K, Wilczek F (2001) Enforced electrical neutrality of the color-flavor locked phase. Phys Rev Lett 86:3492–3495 33. Rezzolla L, Takami K (2016) Gravitational-wave signal from binary neutron stars: a systematic analysis of the spectral properties. Phys Rev D 93(12):124051 34. Rischke DH (2004) The quark-gluon plasma in equilibrium. Prog Part Nucl Phys 52:197–296 35. The LIGO Scientific Collaboration and The Virgo Collaboration (2017) GW170817: observation of gravitational waves from a binary neutron star Inspiral. Phys Rev Lett 119:161101. https://doi.org/10.1103/PhysRevLett.119.161101 36. The LIGO Scientific Collaboration, the Virgo Collaboration, Abbott BP, Abbott R, Abbott TD, Acernese F, Ackley K, Adams C, Adams T, Addesso P et al (2017) Multi-messenger observations of a binary neutron star merger. Astrophys J Lett 848(2):L12. http://stacks.iop. org/2041-8205/848/i=2/a=L12 37. Vallisneri M (2000) Prospects for gravitational-wave observations of neutron-star tidal disruption in neutron-star-black-hole binaries. Phys Rev Lett 84:3519–3522 38. Weissenborn S, Sagert I, Pagliara G, Hempel M, Schaffner-Bielich J (2011) Quark matter in massive neutron stars. Astrophys J 740:L14 39. Witten E (1984) Cosmic separation of phases. Phys Rev D 30:272–285 40. Xu RX (2003) Solid quark stars? Astrophys J Lett 596:L59–L62 41. Zhang N-B, Qi B, Wang S-Y (2019) The key factor to determine the relation between radius and tidal deformability of neutron stars: slope of symmetry energy. arXiv e-prints arXiv:1909.02274 42. Zhou E-P, Zhou X, Li A (2018) Constraints on interquark interaction parameters with GW170817 in a binary strange star scenario. Phys Rev D 97(8):083015 43. Zhu Z-Y, Zhou E-P, Li A (2018) Neutron star equation of state from the quark level in light of GW170817. Astrophys J 862(2):98

Chapter 3

Axisymmetric Rotating Compact Stars

3.1 Background In the previous chapter, we have discussed about the tidal deformability measurement of compact stars with GW170817 and how to apply this measurement to constrain compact star EoSs and hence on models of strong interaction. Such a constraint is made mainly via the inspiral GW signal. In fact, important clues could also be found in the post-merger phase by understanding the evolution of the merger remnant and its relation with the EM counterparts. On one hand, the post-merger GW spectrum is closely related to the instabilities of the remnant star and hence to the EoS model [6, 42]. In addition, many oscillations might be triggered due to the fact that the remnant is extremely rapidly rotating and become important GW sources [1, 14]. On the other hand, the properties of the EM counterparts is determined by the type of the merger remnant and the lifetime of it before collapsing to a BH, which is related to the rotational properties (i.e., uniform rotation or differential rotation) of the remnant and MTOV . Considering the above two points, it’s thus very important to study the equilibrium solutions of rotating compact stars which we will focus in this chapter.

3.1.1 Fate of the Merger Remnant and Maximum Mass of Compact Stars Depending on the TOV maximum mass and the total mass of the merging binary, there might be 4 different outcomes after the merger: • If the total mass is much larger than the TOV maximum mass, the merger remnant will promptly collapse to a BH. This critical mass for prompt collapse to happen is also called the threshold mass (Mthres ). When prompt collapse happens, the ejecta is mainly caused by the tidal torque in the inspiral stage and will be relatively less compared with other cases. The amount of ejecta is determined mainly by the © Springer Nature Singapore Pte Ltd. 2020 E. Zhou, Studying Compact Star Equation of States with General Relativistic Initial Data Approach, Springer Theses, https://doi.org/10.1007/978-981-15-4151-3_3

37

38

3 Axisymmetric Rotating Compact Stars

mass ratio of the binary. The larger difference in the component masses, the more matter will be ejected for both BNSs and binary strange stars [7, 34]. It is worth noting that whether or not prompt collapse will happen, depends not only on the TOV maximum mass but also compactness of the star with a certain mass. BHs are objects with the largest compactness (M/R = 0.5). The larger the compactness of the merging star, the more likely the remnant will promptly collapse to a BH. If we could determine whether or not a prompt collapse happens for a certain binary merger observation, constraints can be made on the EoS models [9]. • If total mass of the binary is smaller than the threshold mass but larger than the maximum mass of rigidly rotating stars, the merger remnant then has to be supported by differential rotation to avoid collapse to a BH (such stars are called hypermassive). This differential rotation is normally believed to be dissipated in a time scale of 10–100 ms, through mechanisms such as magnetorotational instability or viscosity. The magnetic field is believed to be enhanced by the differential rotation as well. When differential rotation is dissipated, the remnant will collapse to a BH. In this case, due to the merger shock, the viscosity in the disc as well as the neutrino irradiation of the central remnant, more matter will be ejected compared with the prompt collapse case (c.f. a recent review in [41]). Such a delayed BH formation with enhanced magnetic field is believed to be the key for explaining sGRB [35]. • If the total mass of the binary is even smaller, to be specific, smaller than the maximum mass of the uniform rotation case. The remnant could still stably exist as a rigidly rotating compact star after dissipating the differential rotation (such stars are called supramassive). Angular momentum dissipation for such a remnant will happen in a much longer time scale (i.e., from seconds to years) mainly through magnetic dipole radiation, similar to the case of radio pulsars. Such a remnant will still collapse to a BH eventually (although lifetime is much longer compared with the previous cases) after losing enough angular momentum and reach the marginally stable solutions of uniformly rotating compact stars (which is also called the turning points). In the magnetar model for sGRB, the spin down power of such a massive remnant with strong magnetic field is needed for explaining many observations associated with sGRBs [30, 47]. In the case of a hypermassive remnant and supramassive remnant, due to the high temperature of the merger remnant, neutrino irradiation will be very important in determining the weak interaction equilibrium of the ejecta and changing the electron fraction, which ultimately change the observation of the EM counterparts. Consequently, the fate of the merger remnant could be inferred from the observations and then be used to constraint the maximum mass of compact stars. • In the case that the total mass of the binary is even smaller than MTOV , the remnant will then become a stable star. Such scenario is more likely to happen for EoS models that predicts large MTOV , such as the strangeon star model. Pulsars with uncommonly large mass, i.e., M > 2 M might be such a remnant resulted from a merger event in the past. Searching for such massive pulsars will be quite a strong evidence favoring such stiff EoS models.

3.1 Background

39

In principle, according to the GW radiation of the massive remnant, the fate of the merger remnant and the time of collapsing to a BH could be determined precisely. Nevertheless, the post-merger GW radiation frequency is too large for the current ground based GW observatories to detect [43]. This denies the possibility of robustly constrain the TOV maximum mass of the EoS models. However, according to what we have mentioned above, clues about the merger remnant, hence about the EoS, could still be found by the EM counterparts.

3.1.2 Constraints on NS EoS According to EM Counterparts of GW170817 Although there is no direct detection of the GW signal from the merger remnant and ringdown of the BH for the case of GW170817, it is still widely accepted that the merger remnant of GW170817 is a hypermassive NS or short-lived supramassive NS which collapses to a BH after sufficient amount of angular momentum is dissipated [33, 35, 39, 43]. There are mainly 2 reasons for drawing such a conclusion. Firstly, both the BH formation as well as the enhanced magnetic fields are required to explain the launching of the relativistic and collimated jet, which is essential for the observation of sGRB (GRB170817A). In the case of a prompt collapse scenario, the magnetic field is not strong enough for the launching of the jet, whereas if MTOV is too large and the lifetime of the remnant is too long, it’s hard to explain the 1.7 s delay between the merger and the sGRB observation [28, 35]. Secondly, the neutron rich matter ejected during the merger event is an important site for r-process nucleosynthesis. The radioactive decay of many isotopes synthesized in the ejecta will then inject energy into the ejecta, which powers not only the adiabatic expansion of the ejecta but also the radiation in optics/UV/IR band, namely, the kilonova [27, 31]. According to the difference in the electron fraction of the ejected matter, the abundance of synthesized elements will be quite different. If the remnant promptly collapse to BH, the matter ejection will be dominated by the tidal driven ejecta in the late inspiral, which consists of very neutron rich matter (i.e., electron fraction Ye is low) and leads to the formation of lanthanides. Whereas in the case of a short-lived massive remnant NSs, additional matter will be ejected in the polar direction due to the shock as well as in the disc due to the viscous transport process and (neutrino) wind of the remnant. This part of the ejecta will be less neutron rich (or higher Ye ) due to high temperature and neutrino irradiations and the abundance of lanthanides will be much less than in the neutron rich ejecta. As a result, the opacity of the tidal driven ejecta will be much higher and thus kilonova looks redder and the shock-driven ejecta and wind-driven ejecta will be relatively bluer. The observation of the transient AT2017gfo indicates the presence of both components [13] hence disfavors the prompt collapse scenario. On the other hand, if the remnant lives for too long time, the energy injection from the spinning down

40

3 Axisymmetric Rotating Compact Stars

of the central remnant will significantly accelerate the expansion of the ejecta and change the luminosity, which is also inconsistent with the observations [29]. According to the two points above, we could conclude that the remnant of GW170817 neither promptly collapses to a BH, nor lives too long before collapse. Each of these 2 conclusions provides important insight in the EoS models of NSs: (1) No prompt collapse By performing smooth particle hydrodynamics (SPH) simulations of binary mergers with various EoS models and various binary parameters, the threshold mass is found to be approximately determined by the following relation [9]: G Mmax + 2.38)Mmax , (3.1) Mthres = (−3.606 2 c R1.6 in which Mmax is the TOV maximum mass of the given EoS and R1.6 being the radius of the NS with 1.6 solar masses. Given any value for R1.6 , threshold mass then becomes a quadratic function of MTOV , which has a maximum value. If this maximum value is smaller than the total mass of the binary for GW170817, which is 2.74 M , then a prompt collapse is inevitable. On the other hand, knowing prompt collapse didn’t happen for GW170817, we could seek out all the possible range for R1.6 , which is R1.6 > 10.68 km. In principle, the fitting formula is not unique and could be expressed with the radius of any fixed mass. Another way of deriving threshold mass is suggested by [9] with Rmax which is radius of the NS at TOV maximum mass: Mthres = (−3.38

G Mmax + 2.43)Mmax . c2 Rmax

(3.2)

Following a similar logic, Rmax is constrained to be larger than 9.6 km. Future observations with different total masses will give us further information on NS radius by applying this analysis. (2) short-lived remnant massive NS The maximum mass of a rotating NS will be increased compared, due to the centrifugal force fighting against gravity. The maximum mass that a rotating star could reach with certain amount of angular momentum, is called the critical mass for this angular momentum (Mcrit , such a solution is also called the turning point defined as ∂ M/∂ρc | J = 0). Critical mass increases as angular momentum of the rotating star increases. Nevertheless, the angular momentum in the star could not be increased infinitely and the largest possible mass that a rotating star could reach is the mass limit for rigid rotating star (denoted as MKep as this is the corresponding Keplerian mass limit in the Newtonian case). The corresponding angular momentum is then denoted as JKep . Early studies of rotating NSs in general relativity [12] reveals that the maximum mass of a rotating NS is roughly 20% larger than MTOV . A more detailed study [11] has considered 28 EoS models for NSs and figured out that with proper rescaling, the relation between critical mass and the corresponding angular momentum depends only very weakly on EoSs:

3.1 Background

41

2.4 2.2

M [M ]

2.0

APR SFH0 DD2 ALF2

1.8 1.6 1.4 1.2 1.0

9

10

11

12

13

14

15

16

R [km] Fig. 3.1 A summary of all the constraints one can make on NS mass-radius relation according to GW170817 and its EM counterparts with the results of several selected NS EoS models. The grey areas are excluded by the tidal deformability measurement and the red areas by the fact that this merger event didn’t result in prompt collapse. Solid horizontal line is set by the observation of massive pulsars [3] and dashed horizontal lines are given according to the fate of the merger remnant for GW170817

Mcrit J 2 J 4 = 1 + a2 ( ) + a4 ( ) , MTOV Jkep Jkep

(3.3)

in which a2 = 0.1316 and a4 = 0.07111 by fitting all the results. A direct conclusion from such a relation is that at Keplerian angular momentum, the mass of the NS is Mmax = Mcrit (J = Jkep ) = (1 + a2 + a4 )MTOV ∼ (1.203 ± 0.022)MTOV , (3.4) which is consistent with the results in [12]. Knowing this 20% factor and the assumption that the remnant of GW170817 collapses to BH after differential rotation is dissipated, one can then constrain the TOV maximum mass of NSs [29, 33, 35] to be smaller than roughly 2.16–2.17 solar masses. A summary of all the constraint that one can make on NS EoS models according to GW170817 and its EM counterparts is shown in Fig. 3.1. It should be kept in mind that the above constraint is valid only for NSs but not for strange stars. Due to the self-bound nature of strange quark stars, the structure of rotating strange stars is quite different from the case of NSs. We will demonstrate those differences in this chapter by solving the structure of rotating strange stars and also show how to interpret the observations differently for constraining EoS of strange stars. Additionally, it is still with doubt whether or not the 2.16 solar mass

42

3 Axisymmetric Rotating Compact Stars

upper limit of MTOV is reliable even for NSs, due to the uncertainties in the angular momentum of the rotating NS at the time of collapsing to BH. We will also introduce a more consistent analysis in this chapter to improve the constraint on NS MTOV .

3.2 A More Consistent Constraint on NS MTOV The MTOV 1.0, f uni

(3.13)

for strange stars, in which f uni denotes Mcrit for a certain J for uniform rotation case and f Aˆ for j-constant rotation law case. More interestingly, as can be seen from both 5 Although

in [10] it has been shown that this universal relation cannot be extended even for the ˆ case of uniformly rotating SSs, it’s still quite useful to verify whether the A-insensitive relationship still holds for strange stars.

56

3 Axisymmetric Rotating Compact Stars 2.0

M [M ]

1.8

1.6

1.4

1.2

1.0 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040

ρc

Fig. 3.10 Mass versus central density diagram for  = 2, K = 100 NS EoS. The green curve and red curve are for TOV sequence and the mass shedding limit of rigid rotation case. Black curve shows the relation for uniformly rotating case with a specific angular momentum (J = 1.17) and blue curve is the relation for a monotonically increasing differential rotation profile case with the same angular momentum. As can be seen from the figure, the critical mass for this type of differential rotation profile is almost identical with the rigid rotation case

panels of Fig. 3.11, the solutions with the new rotation law (denoted by filled square and star symbols) are also found to follow this relation. Additionally, we have also applied a monotonically increasing differential rotation law for NS models and the result is positive for NSs as well (c.f. Fig. 3.10). This result indicates that the critical mass dependents mainly on the angular momentum contained in the rotating star, but not on how these angular momentum are distributed. Hence, the outcome of a binary merger event can be inferred without having to know the details of the rotational profile in the merger remnant.

3.4.4 Configuration Types of Differentially Rotating Strange Star For rigidly rotating relativistic stars or differentially rotating stars with weak differential rotation, the solution sequences terminate at the so-called mass-shedding limit with a finite axis ratio Rz /Rx when the star is still ellipsoidal. With relatively strong differential rotation degrees and small axis ratio, the star no longer looks ellipsoidal in the x–z plane and the maximum density migrates from the center to a ring with finite radius in the star. Solutions of this type (classified as type C solution) could be

3.4 Maximum Mass of Rotating Strange Stars

57

1.30 1.25

Mcrit/MTOV

1.20 1.15 1.10 1.05 1.00 0.0

0.2

0.4

0.6

0.8

1.0

1.2

J/JKep

1.8

Mcrit/MrmT OV

1.6

1.4

1.2

1.0 0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

J/JKep Fig. 3.11 The relationship between critical mass Mcrit normalized by MTOV and angular momentum J normalized by JKep for strangeon stars (upper panel) and MIT bag model stars (lower panel). Both rigid rotating case (solid blue line) and differentially rotating case (green dots for Aˆ = 3.0 and red dots for Aˆ = 1.0) are shown. The 1% error range for the relationship of the rigid rotating case is shown in dashed blue lines for comparison purpose. We have also labeled the results from the new differential rotation law with black markers

58

3 Axisymmetric Rotating Compact Stars

1.0

z/Rx

0.5

0.0

−0.5

−1.0 −1.0

−0.5

0.0 x/Rx

0.5

1.0

ρ/ρc Ω/Ωc

Q/Qc

1.0

0.5 −1.0

−0.5

0.0 x/R

0.5

1.0

Fig. 3.12 Stellar surface and rest mass density contour of a differentially rotating strangeon star (upper panel) and its density and angular velocity profile (lower panel). Details about the solution shown in this figure can be found in the ‘DR-LX-4’ model in Table 3.2. This figure is adopted from [48]

3.4 Maximum Mass of Rotating Strange Stars

59

found even for vanishing z axis. Identifying such solutions for HMSSs will help us better understand the influence of a certain differential rotation rate. For NSs models, this transit comes to exist at Aˆ  1.0 [19] with more precise value dependent on the central density of the solution. As a comparison, properties of selected type C solutions for differentially rotating SSs are listed in Table 3.2. It turns out that type C emerges at much larger Aˆ for strange stars. For both MIT bag model and strangeon star model, toroidal solutions with Rz /Rx = 0 is already found for the entire central density range for Aˆ = 3.0. This is another evidence that differential rotation has more impacts on structure of strange stars. Apart from the toroidal limit, we have also tried to understand the influence of differential rotation by looking at the onset of the transit to toroidal class. This could be done by searching for the first solution in a sequence whose maximum density is larger than central density, or identically the first solution the surface of which is no longer elliptical in x–z plane. For solutions with Aˆ = 1.0, such transition occurs at axis ratio very close to 1.0 (in another word, with very little the angular momentum). Such a solution is listed as ‘DR-LX-3’ in Table 3.2.

Table 3.2 Quantities of selected solutions for rotating strange stars. In the above, R x is the coordinate (proper) equatorial radius and Rz /Rx is the ratio of coordinate (proper) polar to the equatorial radius. ρc is the central rest-mass density and ρmax the maximum rest-mass density in the star. c , MADM , J , and T /|W | are the central angular velocity, Arnowit-Deset-Misner mass, angular momentum and ratio between kinetic energy and gravitational potential. Definitions can be found in the Appendix of [45]. In this table, ‘UR-SS’ and ‘UR-MIT’ labels the maximum mass solution of uniformly rotating strangeon star and MIT bag model star, respectively. ‘DR-SS-1’ and ‘DR-MIT-1’ are the maximum mass solutions for differentially rotating strangeon star and MIT bag model star with Aˆ = 1 j-const law. ‘DR-SS-2’ is the maximum mass solution for the new differential rotation law with Rz /Rx = 0.25 for the strangeon star model. ‘DR-SS-3’ and ‘DR-SS-4’ are two selected type-C solutions with j-const law and the new differential rotation law, respectively. This table is adopted from [48] Model

Rx

Rz /R x

ρc

ρmax

c

UR-SS

4.82 (15.1)

0.53125 (0.584)

1.56 × 10−3

1.56 × 10−3

0.0603 4.39

16.4

0.222

DR-SS-1

4.36 (12.4)

0.015625 (0.0190)

8.68 × 10−4

1.51 × 10−3

0.382

3.78

10.3

0.183

DR-SS-2

4.07 (14.4)

0.25 (0.295)

1.20 × 10−3

1.40 × 10−3

0.110

4.49

17.6

0.290

DR-SS-3

4.83 (10.9)

0.9375 (0.947)

1.51 × 10−3

1.51 × 10−3

0.0638 3.25

2.28

0.0135

DR-SS-4

4.26 (12.8)

0.50 (0.553)

1.46 × 10−3

1.51 × 10−3

0.0945 3.92

11.9

0.203

UR-MIT

8.23 (15.1)

0.484375 (0.523)

1.76 × 10−3

1.76 × 10−3

0.0433 3.17

8.56

0.198

DR-MIT-1 6.79 (13.9)

0.015625 (0.0172)

6.07 × 10−3

1.34 × 10−3

0.163

10.8

0.236

MADM J

3.60

T /|W |

60

3 Axisymmetric Rotating Compact Stars

We have done similar analysis for the solutions with the new differential rotation law to verify the influence of rotation laws on configuration of the solutions. We manage to find toroidal solutions with vanishing z-axis for the low central density sequence. For relatively large central density sequence, the root-finding for adopting A and B parameter in the rotation law becomes numerically more difficult as axis ratio decreases. Nevertheless, we can still try to identify the transition to toroidal class by looking at the stellar surface and density profile of the star. The result shows that for axis ratio Rz /Rx = 0.5, the onset of the transition already occurs for all the central density range (an example can be found in Fig. 3.12). Hence, we conclude that type C toroidal class should be a common configuration differentially rotating relativistic stars with strong differential rotation degree.

References 1. Andersson N (1998) A new class of unstable modes of rotating relativistic stars. Astrophys J 502:708 2. Ansorg M, Gondek-Rosi´nska D, Villain L (2009) On the solution space of differentially rotating neutron stars in general relativity. Mon Not R Astron Soc 396:2359–2366 3. Antoniadis J, Freire PCC, Wex N, Tauris TM, Lynch RS et al (2013) A massive pulsar in a compact relativistic binary. Science 340:448 4. Baiotti L, Rezzolla L (2017) Binary neutron star mergers: a review of Einstein’s richest laboratory. Reports Prog Phys 80(9):096901 5. Baumgarte TW, Shapiro SL, Shibata M (2000) Astrophys J 528:L29 6. Bauswein A, Janka H-T, Hebeler K, Schwenk A (2012) Equation-of-state dependence of the gravitational-wave signal from the ring-down phase of neutron-star mergers. Phys Rev D 86(6):063001 7. Bauswein A, Janka H-T, Oechslin R, Pagliara G, Sagert I, Schaffner-Bielich J, Hohle MM, Neuhäuser R (2009) Mass ejection by strange star mergers and observational implications. Phys Rev Lett 103(1):011101 8. Bauswein A, Stergioulas N (2015) Unified picture of the post-merger dynamics and gravitational wave emission in neutron star mergers. Phys Rev D 91(12):124056 9. Bauswein A, Just O, Janka H-T, Stergioulas N (2017) Neutron-star radius constraints from GW170817 and future detections. Astrophys J Lett 850:L34 10. Bozzola G, Stergioulas N, Bauswein A (2018) Universal relations for differentially rotating relativistic stars at the threshold to collapse. Mon Not R Astron Soc 474:3557–3564 11. Breu C, Rezzolla L (2016) Maximum mass, moment of inertia and compactness of relativistic stars. Mon Not R Astron Soc 459:646–656 12. Cook GB, Shapiro SL, Teukolsky SA (1994) Rapidly rotating neutron stars in general relativity: realistic equations of state. Astrophys J 424:823 13. Cowperthwaite PS, Berger E, Villar VA, Metzger BD, Nicholl M, Chornock R, Blanchard PK, Fong W, Margutti R, Soares-Santos M et al (2017) The electromagnetic counterpart of the binary neutron star merger LIGO/Virgo GW170817. II. UV, Optical, and near-infrared light curves and comparison to Kilonova models. Astrophys J Lett 848(2):L17 14. Cutler C (2002) Gravitational waves from neutron stars with large toroidal B fields. Phys Rev D 66(8):084025 15. Demorest PB, Pennucci T, Ransom SM, Roberts MSE, Hessels JWT (2010) A two-solar-mass neutron star measured using Shapiro delay. Nature 467:1081–1083 16. Dietrich T, Bernuzzi S, Ujevic M, Brügmann B (2015) Numerical relativity simulations of neutron star merger remnants using conservative mesh refinement. Phys Rev D 91(12):124041

References

61

17. Espino P, Paschalidis V (2019) Revisiting the maximum mass of differentially rotating neutron stars in general relativity: Übermassive stars with realistic equations of state 18. Friedman JL, Ipser JR, Sorkin RD (1988) Turning-point method for axisymmetric stability of rotating relativistic stars. Astrophys J 325:722–724 19. Gondek-Rosi´nska D, Kowalska I, Villain L, Ansorg M, Kucaba M (2017) A new view on the maximum mass of differentially rotating neutron stars. Astrophys J 837:58 20. Hanauske M, Takami K, Bovard L, Rezzolla L, Font JA, Galeazzi F, Stöcker H (2017) Rotational properties of hypermassive neutron stars from binary mergers. Phys Rev D 96(4):043004 21. Hotokezaka K, Kiuchi K, Kyutoku K, Muranushi T, Sekiguchi Y-I, Shibata M, Taniguchi K (2013) Remnant massive neutron stars of binary neutron star mergers: evolution process and gravitational waveform. Phys Rev D 88(4):044026 22. Kastaun W, Galeazzi F (2015) Properties of hypermassive neutron stars formed in mergers of spinning binaries. Phys Rev D 91(6):064027 23. Kiziltan B, Kottas A, De Yoreo M, Thorsett SE (2013) The neutron star mass distribution. Astrophys J 778:66 24. Komatsu H, Eriguchi Y, Hachisu I (1989) Rapidly rotating general relativistic stars. I— Numerical method and its application to uniformly rotating polytropes. Mon Not R Astron Soc 237:355–379 25. Komatsu H, Eriguchi Y, Hachisu I (1989) Rapidly rotating general relativistic stars. II— Differentially rotating polytropes. Mon Not R Astron Soc 239:153–171 26. Li A, Zhang B, Zhang N-B, Gao H, Qi B, Liu T (2016) Internal X-ray plateau in short GRBs: Signature of supramassive fast-rotating quark stars? Phys Rev D 94(8):083010 27. Li L-X, Paczy´nski B (1998) Transient events from neutron star mergers. Astrophys J 507:L59– L62 28. LIGO Scientific Collaboration, Virgo Collaboration, F. Gamma-Ray Burst Monitor and INTEGRAL (2017) Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A. Astrophys J Lett 848(2):L13. http://stacks.iop.org/2041-8205/ 848/i=2/a=L13 29. Margalit B, Metzger BD (2017) Constraining the maximum mass of neutron stars from multimessenger observations of GW170817. Astrophys J Lett 850:L19 30. Metzger BD, Quataert E, Thompson TA (2008) Short-duration gamma-ray bursts with extended emission from protomagnetar spin-down. Mon Not R Astron Soc 385:1455–1460 31. Metzger BD, Martínez-Pinedo G, Darbha S, Quataert E, Arcones A, Kasen D, Thomas R, Nugent P, Panov IV, Zinner NT (2010) Electromagnetic counterparts of compact object mergers powered by the radioactive decay of r-process nuclei. Mon Not R Astron Soc 406:2650–2662 32. Radice D, Perego A, Bernuzzi S, Zhang B (2018) Long-lived remnants from binary neutron star mergers. Mon Not R Astron Soc 481(3):3670–3682 33. Rezzolla L, Most ER, Weih LR (2018) Using gravitational-wave observations and quasiuniversal relations to constrain the maximum mass of neutron stars. Astrophys J Lett 852:L25 34. Rezzolla L, Takami K (2013) Black-hole production from ultrarelativistic collisions. Class Quantum Grav 30(1):012001 35. Ruiz M, Shapiro SL, Tsokaros A (2018) GW170817, general relativistic magnetohydrodynamic simulations, and the neutron star maximum mass. Phys Rev D 97(2):021501 36. Shibata M, Taniguchi K, Ury¯u K (2005) Merger of binary neutron stars with realistic equations of state in full general relativity. Phys Rev D 71(8):084021 37. Shibata M, Taniguchi K, Ury¯u K (2005) Merger of binary neutron stars with realistic equations of state in full general relativity. Phys Rev D 71(8):084021 38. Shibata M, Ury¯u K (2000) Simulation of merging binary neutron stars in full general relativity:  = 2 case. Phys Rev D 61(6):064001 39. Shibata M, Fujibayashi S, Hotokezaka K, Kiuchi K, Kyutoku K, Sekiguchi Y, Tanaka M (2017) Modeling GW170817 based on numerical relativity and its implications. Phys Rev D 96(12):123012 40. Shibata M, Zhou E, Kiuchi K, Fujibayashi S (2019) Constraint on the maximum mass of neutron stars using GW170817 event. Phys Rev D 100(2):023015

62

3 Axisymmetric Rotating Compact Stars

41. Shibata M, Hotokezaka K (2019) Merger and mass ejection of neutron star binaries. Ann Rev Nucl Part Sci 69(1):annurev 42. Takami K, Rezzolla L, Baiotti L (2014) Constraining the equation of state of neutron stars from binary mergers. Phys Rev Lett 113(9):091104 43. The LIGO Scientific Collaboration, the Virgo Collaboration, Abbott BP, Abbott R, Abbott TD, Acernese F, Ackley K, Adams C, Adams T, Addesso P, et al (2017) Search for post-merger gravitational waves from the remnant of the binary neutron star merger GW170817. ArXiv e-prints 44. Ury¯u K, Tsokaros A, Baiotti L, Galeazzi F, Taniguchi K, Yoshida S (2017) Modeling differential rotations of compact stars in equilibriums. Phys Rev D 96(10):103011 45. Ury¯u K, Tsokaros A, Galeazzi F, Hotta H, Sugimura M, Taniguchi K, Yoshida S (2016) New code for equilibriums and quasiequilibrium initial data of compact objects. III. Axisymmetric and triaxial rotating stars. Phys Rev D D93(4):044056 46. Weih LR, Most ER, Rezzolla L (2018) On the stability and maximum mass of differentially rotating relativistic stars. Mon Not R Astron Soc 473:L126–L130 47. Zhang B (2013) Early X-ray and optical afterglow of gravitational wave bursts from mergers of binary neutron stars. Astrophys J 763:L22 48. Zhou E, Tsokaros A, Ury¯u K, Renxin X, Shibata M (2019) Differentially rotating strange star in general relativity. Phys Rev D 100(4):043015

Chapter 4

Triaxially Rotating Strange Stars

In the previous Chapter, we have discussed about the similarities and differences between uniformly rotating and differentially rotating neutron stars and strange stars, with axisymmetry. Nevertheless, in reality, due to the high mass, kinetic energy as well as strong magnetic fields of the merger remnant, the axisymmetry might be easily broken. Under such conditions, the remnant might be significant gravitational wave sources, which could provide us valuable insights about the internal structure of the massive remnant as well as many microscopic properties of matter at such density. In general, the following mechanisms could lead to a non-axisymmetirc merger remnant: • The magnetic field of the remnant will be significantly enhanced by differential rotation and toroidal components could develop. If the energy density of the enhanced magnetic field becomes comparable to that of the compact star matter (which normally requires the magnetic field strength to be at least as large as 1016 G), the massive remnant will be deformed by the magnetic field. If the spin axis and magnetic axis are not aligned, the remnant will no longer be axisymmetric and become a gravitational wave radiator [8]. • For rapidly rotating compact stars, oscillation modes with Coriolis force as restoring force (r-mode) might be excited [1, 10]. In addition, for differentially rotating relativistic star with large differential rotation rates, dynamical instabilities which will lead to non-axisymmetric deformations of the star may easily be triggered [5, 25–27]. • For rotating stars with larger kinetic energy to potential energy ratio (T /|W | > 0.1375 in Newtonian case), the star will spontaneously break its axisymmetry, evolve into triaxial configuration which has lower energy than axisymmetric case by viscous and GW dissipation [6, 24]. If reasonable angular momentum is stored in the vortexes in the star, then such configurations are called Dedekind ellipsoid, otherwise it is called Jacobian ellipsoid. The evolution and GW emission properties of those two types of configurations are quite different from each other [18].

© Springer Nature Singapore Pte Ltd. 2020 E. Zhou, Studying Compact Star Equation of States with General Relativistic Initial Data Approach, Springer Theses, https://doi.org/10.1007/978-981-15-4151-3_4

63

64

4 Triaxially Rotating Strange Stars

In this chapter, we will focus on the Jacobian ellipsoid configuration of strange stars. R-mode instability could be suppressed significantly by viscosity and might not be important for strangeon star which is suggested to be in solid state [32]. For calculation of Dedekind ellipsoid, there is no helical symmetry for the fluid and we can no more use Eq. (1.26) as hydrostatic equilibrium equation. We will leave the calculation of Dedekind ellipsoid for a future study.

4.1 Background Equilibrium configurations of self-gravitating, uniformly rotating incompressible stars have been studied systematically in Newtonian scheme already by Chandrasekhar in 1969 [7]. As mentioned above, depending on the rotational kinetic energy, the most stable configuration could be either axisymmetric, i.e. Maclaurin ellipsoids, or non-axisymmetic, such as Jacobian or Dedekind ellipsoids which are triaxial. For compact stars which we are interested in, studies in full general relativistic scheme is needed. Relativistic rotating compact stars have also been studied for long [9, 21]. It has been found that rotating NSs will spontaneously break its axisymmetry if T /|W | ratio exceeds a critical value. Such high value of T /|W | is achievable by a newly born compact star during a core collapse supernova, a NS spun up by accretion or the remnant of a BNS merger event [2, 19, 22, 23, 30, 31]. Rotating relativistic stars with such a configuration might be very important GW sources and worth more detailed studies. Quasi-equilibrium configurations of triaxially rotating NSs have been studied in full GR by constructing initial data [15, 29]. The dynamical stability properties have also been investigated in [28] by evolving such initial data. Nevertheless, the bifurcation from axisymmetric to triaxial configurations happens very close to mass shedding limit for uniformly rotating NSs. Particularly, the larger the compactness of the NS, the larger T /|W | value is required for the bifurcation. As a consequence, for soft EoS models and with large compactness, the triaxial sequence could totally vanish as the required T /|W | is too large to be reached even at mass shedding limit [3, 4, 14, 16, 17]. Moreover, the existence of supramassive triaxial NSs (namely, triaxially rotating NSs with mass larger than MTOV ) has also been discussed in [29]. The result confirms the existence of such solutions, but only for extremely stiff EoS models. Considering that the total mass of all the currently known BNS systems are all larger than TTOV , it’s then also with doubt whether such configurations could really play an important role in real astrophysical scenarios with realistic EoS models. Differently from NSs, rotating strange star has larger moment of inertia hence larger kinetic energy, due to the finite surface density [12, 13]. Therefore, triaxial bar mode instability could play a more important role for rotating strange stars. This is verified by previous studies for MIT bag model in full relativistic schemes [11]. In this investigation, perturbations on lapse function are imposed on axisymmetric solutions

4.1 Background

65

of rotating strange stars and series of triaxial solutions are constructed accordingly to see whether the perturbation is damped or grows. By such treatment, the onset points of triaxial instability is identified. In this chapter, we will introduce our research on triaxially rotating strange stars with both MIT bag model and strangeon star model in a different approach with cocal code.

4.2 Constructing Triaxially Rotating Strange Stars with COCAL Differently from the methods in [11], we do not need to input perturbations on top of axisymmetric solutions to induce triaxial deformation. Triaxial solutions will emerge spontaneously at large enough T /|W |, although not exactly at the bifurcation point. To find the entire triaxial sequence and to identify the quantities of the rotating strange star at the bifurcation point, we have to following the procedure below: (1) similar to the case mentioned in the previous chapter about constructing axisymmetric uniformly rotating stars, we have to start from a TOV solution and reduce the axis ratio Rz /Rx gradually. What is different from the previous case is that, we no longer impose axisymmetry condition for metric and fluid variables after solving the field equations and hydro equation in every iteration. (2) when Rz /Rx is large (i.e., rotation is slow), the solution is still axisymmetric although no symmetry condition is imposed in the calculation. Nevertheless, when Rz /Rx is small enough, triaxial deformation is spontaneously triggered, namely, R y becomes smaller than Rx .1 Such a configuration is energetically favored than the axisymmetric solution. (3) keep decreasing Rz /Rx ratio, R y /Rx ratio will also become smaller, which means triaxial deformation becomes larger for faster rotation. We could reach the mass shedding limit of triaxial sequence by decreasing axis ratio gradually. (4) when we calculate axisymmetric solutions, the 3 points we choose to determine , R0 and the constant in Eq. (3.8) are the center of the star, the surface of the star on positive z-axis and x-axis. Once the mass shedding limit of triaxial solution is achieved, we can choose the surface on positive y-axis instead of z-axis. By doing so, we will be able to create solutions with desired R y /Rx ratio. (5) take the mass shedding limit solution and increase R y /Rx towards 1.0 (which is the bifurcation point) gradually to obtain the entire triaxial sequence and quantities on the bifurcation point. (6) take TOV solutions with another compactness and repeat the above steps to investigate the influence of different compactness on triaxial solutions.

1 In principle, during such a spontaneous symmetry breaking, there should be no preferred direction

in the equatorial plane. However, we always define the longer axis to be x-axis for convenience of the calculation.

66

4 Triaxially Rotating Strange Stars

Table 4.1 Quantities at the point of bifurcation of triaxial sequences from axisymmetric ones for the two EoS considered. The compactness of the spherical star with the same rest mass C are the model parameters. In the above, R x is the coordinate (proper) equatorial radius, and Rz /Rx is the ratio of coordinate (proper) polar to the equatorial radius. c is the energy density at the center of the compact star,  is the angular velocity. In the last three lines we report the bifurcation point of simple polytropes with polytropic index n as computed in [15] for comparison. Note that we have chosen appropriate values for κ such that the TOV maximum mass for those polytropic EoS reach 2.5 M . To convert to geometric G = c = 1 or cgs units, use the fact that 1 = 1.477 km = 4.927 μs = 1.989 × 1033 g EOS M/R Rx Rz /Rx c  MADM J T /|W | MIT

0.1

MIT

0.15

MIT

0.2

Strangeon 0.1 Strangeon 0.15 Strangeon 0.2 n = 0.3

0.1

n = 0.3

0.2

n = 0.5

0.1

7.021 (8.077) 7.962 (9.922) 8.415 (11.43) 5.698 (6.557) 6.515 (8.130) 6.972 (9.528) 6.624 (7.634) 7.312 (9.979) 10.30 (11.83)

0.5647 (0.5693) 0.5565 (0.5640) 0.5478 (0.5590) 0.5644 (0.5689) 0.5566 (0.5639) 0.5469 (0.5574) 0.5634 (0.5693) 0.5394 (0.5535) 0.5461 (0.5536)

6.200 × 10−4

0.02808 0.6515

0.4580

0.1520

6.811 × 10−4

0.02985 1.169

1.293

0.1609

7.696 × 10−4

0.03199 1.731

2.649

0.1706

9.144 × 10−4

0.03451 0.5312

0.3039

0.1518

9.542 × 10−4

0.03630 0.9686

0.8850

0.1607

9.977 × 10−4

0.03838 1.481

1.932

0.1715

9.221 × 10−4

0.03180 0.5841

0.3708

0.1507

1.243 × 10−3

0.03926 1.435

1.835

0.1688

5.153 × 10−4

0.02197 0.8416

0.7644

0.1493

We have calculated triaxial sequence for both MIT bag model and strangeon star model with 3 different spherical compactness 0.10, 0.15 and 0.20. The quantities on the bifurcation point of those cases are listed in Table 4.1 together with the results for NSs in [15] as comparison. In Newtonian gravity, the bifurcation to triaxial sequence happens at T /|W | = 0.1375. In GR cases, there will be correction related to the compactness of the star, [11]: 

 in which

T |W |



 =

crit

T |W |

 + 0.126 C (1 + C) ,

(4.1)

crit,Newt



T |W | crit,Newt

is the Newtonian value of 0.1375. According to the results

listed in the table, this relation is preserved with relative error smaller than 3%. In Fig. 4.1, the angular velocity-eccentricity diagram is shown for the entire axisymmetric sequence as well as the triaxial sequence for all the models we have considered. One interesting feature is that once triaxial deformation is induced, the

4.2 Constructing Triaxially Rotating Strange Stars with cocal

67

0.07 0.06

ΩMADM

0.05 0.04 0.03 0.02 0.01 0.00

0.5

0.6

0.7

e

0.8

ML C = 0.1 ML C = 0.15 ML C = 0.2

0.9

1.0

JB C = 0.1 JB C = 0.15 JB C = 0.2

0.07 0.06

ΩMADM

0.05 0.04 0.03 0.02 0.01 0.00 0.5

0.6

0.7

ML C = 0.1 ML C = 0.15 ML C = 0.2

e

0.8

0.9

1.0

JB C = 0.1 JB C = 0.15 JB C = 0.2

Fig. 4.1 Plots of MADM versus eccentricity for strangeon star model (upper panel) and MIT bag model (lower panel) sequences. Solid curves are axisymmetric solution sequences, and dashed curves are triaxial solution sequences, that correspond to C = M/R = 0.2 (the top green curve), 0.15 (the middle red curve) and 0.1 (the bottom blue curve) respectively. Note that M is the spherical ADM mass

68

4 Triaxially Rotating Strange Stars

angular velocity of the rotating star actually decreases as the star gains more angular momentum. This because the change in moment of inertia exceeds the change in angular momentum in the triaxial sequence. This is also the reason why triaxial solution is energetically favored. For fixed angular momentum, the kinetic energy is given as E k = J 2 /2I , which decreases as moment of inertia I increases. For triaxial configurations, the rate of decrease in kinetic energy exceeds the increase in potential energy when I is increased by triaxial deformation, hence the total energy is smaller than axisymmetric case.

4.3 Comparison with Triaxially Rotating NS It has been mentioned that triaxial configurations might be more important for rotating strange stars compared with NSs, due to the finite surface density of strange stars. According to our results, there are indeed mainly three differences found: (1) Strange stars have longer triaxial sequences than NSs. In another word, the eccentricity is larger on the equatorial plane at the mass shedding limit of triaxial sequence for strange stars (hence R y /Rx is smaller as well). For instance, according to [15], for NSs with spherical compactness 0.20, the triaxial deformation is induced at e = 0.835 and the triaxial sequence terminates at e = 0.88. Whereas for strange stars with same compactness, the eccentricity at bifurcation point is e = 0.83 and e = 0.89 at the mass shedding limit, for both MIT bag model and strangeon star model. This could be seen more clearly from Fig. 4.2. Rotating strange stars could reach much higher T /|W | than NSs, due to the finite surface density. In the case of C = 0.20, the T /|W | ratio at the mass shedding limit could be as high as 0.27 for rotating strange stars, while only 0.20 for NSs. (2) There exists supramassive triaxial solutions for strange star model. In Table 4.2, we have listed the most massive traixial solutions we have obtained for MIT bag model and strangeon star model, respectively. For both EoS models, the mass exceeds their TOV maximum mass. This, in essence, is again due to the larger T /|W | ratio strange star could reach. According to Eq. (4.1), we know that the T /|W | ratio needed for inducing triaxial deformation increased as compactness increases. At the same time, compactness of the star increases with mass. As a consequence, for supramassive rotating NSs, the T /|W | ratio required for bifurcating into triaxial branch is too large to be reached even at the mass shedding limit for realistic EoSs. Nevertheless, such high T /|W | ratio is not a problem for strange stars due to the self-bound nature. (3) Traxially rotating strange stars are more efficient GW sources. To obtain precise results of GW emission properties of triaxially rotating compact stars, one has to evolve the initial data and extract the GW signal in the simulations. Nevertheless, according to [28], the result obtained with quadrupole formula is consistent with

4.3 Comparison with Triaxially Rotating NS

69

0.30

T /|W |

0.25 0.20 0.15 0.10 0.05

0.5

0.6

0.7

e

0.8

0.9

ML C = 0.1

JB C = 0.1

ML C = 0.15

JB C = 0.15

ML C = 0.2

JB C = 0.2

1.0

0.30

T /|W |

0.25 0.20 0.15 0.10 0.05

0.5

0.6

0.7

e

0.8

0.9

ML C = 0.1

JB C = 0.1

ML C = 0.15

JB C = 0.15

ML C = 0.2

JB C = 0.2

1.0

 Fig. 4.2 Plots for T /|W | versus eccentricity e := 1 − (Rz /Rx )2 (in proper length) for strangeon star model (upper panel) and MIT bag model (lower panel) sequences. Solid curves are axisymmetric solution sequences, and dashed curves are triaxial solution sequences, that correspond, to C = M/R = 0.2 (the top green curve), 0.15 (the middle red curve) and 0.1 (the bottom blue curve) respectively. Note that M is the spherical ADM mass. Bottom panel zooms into the region of the triaxial solutions (marked points). Curves in grey color is the comparison case for NSs [15]

70

4 Triaxially Rotating Strange Stars

Table 4.2 Quantities of triaxial supramassive QS solutions with the largest triaxial deformation (smallest R y /Rx ratio) in our calculations. The above quantities are defined in the same way as in Table 4.1. The TOV maximum mass of each EoS is also shown as a comparison. Due to the limitation of IWM formulation, there might be 5%–10% errors on the quantities listed above. To convert to geometric G = c = 1 or cgs units, use the fact that 1 = 1.477 km = 4.927 μs = 1.989 × 1033 g EOS Rz /Rx R y /Rx Rx c  MADM J T /|W | MIT

0.4375 (0.4713) Strangeon 0.4375 (0.4912)

0.040

0.7657 (0.7938) 0.7586 (0.8104)

9.978 (16.32) 7.660 (16.49)

1.259 × 10−3 1.348 × 10−3

0.03870

2.862

6.847

0.1839

0.05001

3.727

11.30

0.1948

0.87

0.88

0.89

MIT StrangenonStar

Dh/MADM

0.035 0.030 0.025 0.020 0.015 0.010 0.85

0.86

e

Fig. 4.3 An estimation of the gravitational wave amplitude for the C = 0.2 triaxial sequence for both MIT bag model (blue solid curve) and strangeon star model (red dashed curve). The quantities are estimated according to the quadrupole formula. The top panel shows the GW strain for l = m = 2 mode normalized by the distance (D) and the ADM mass (MADM ) of the source. And the GW luminosity is shown in the bottom panel. Shown in black triangles are comparison models of NSs obtained in [28]

the results obtained with full numerical relativity simulations. Therefore, we have also estimated the the GW strain and power of triaxially rotating strange stars with quadrupole formula and compared with the results of NSs. As can be seen from Fig. 4.3, triaxially rotating strange stars are more effective GW emitters. This, once again, is due to the fact that strange star possesses a finite surface density due to the self-bound nature, hence larger quadrupole moment at the same axis ratio R y /Rx . Nevertheless, according to the constraint in [20], the GW strain from either triaxially rotating NSs or strange stars is below the upper limit of the post-merger GW signal for GW170817. Therefore, we could

4.3 Comparison with Triaxially Rotating NS

71

not distinguish whether the merger remnant of GW170817 is a NS or strange star yet according to the non-detection of the post-merger GW signal. In addition, it’s worth noting that the angular velocity of the rotating star at the bifurcation point is different for different EoS models. And according to Fig. 4.1, the angular velocity at the bifurcation point is actually the upper limit of the entire sequence. Considering that for solid strangeon star models, r-mode instability might be suppressed, a strangeon star might be able to reach such high angular velocity by accretion. Searching for fast rotating pulsars (such as sub-millisecond pulsars) in the future would also be helpful for constraining EoS models.

References 1. Andersson N (1998) A new class of unstable modes of rotating relativistic stars. Astrophys J 502:708 2. Bildsten L (1998) Gravitational radiation and rotation of accreting neutron stars. Astrophys J L 501:L89–L93 3. Bonazzola S, Frieben J, Gourgoulhon E (1996) Spontaneous symmetry breaking of rapidly rotating stars in general relativity. Astrophys J 460:379 4. Bonazzola S, Frieben J, Gourgoulhon E (1998) Spontaneous symmetry breaking of rapidly rotating stars in general relativity: influence of the 3D-shift vector. Astron Astrophys 331:280– 290 5. Centrella JM, New KCB, Lowe LL, David Brown J (2001) Dynamical rotational instability at low T /W . Astrophys J 550:L193–L196 6. Chandrasekhar S (1970) The effect of gravitational radiation on the secular stability of the maclaurin spheroid. Astrophys J 161:561 7. Chandrasekhar S (1969) Ellipsoidal figures of equilibrium. Silliman Foundati Lect New Haven, CT: Yale Univ. Press. https://cds.cern.ch/record/207046 8. Cutler C (2002) Gravitational waves from neutron stars with large toroidal B fields. Phys Rev D 66(8):084025 9. Friedman JL, Stergioulas N (2013) Rotating relativistic stars 10. Friedman JL, Morsink SM (1998) Axial instability of rotating relativistic stars. Astrophys J 502:714 11. Gondek-Rosi´nska D, Gourgoulhon E, Haensel P (2003) Are rotating strange quark stars good sources of gravitational waves? Astron Astrophys 412:777–790 12. Gondek-Rosinska D, Haensel P, Zdunik JL, Gourgoulhon JL (2000) Rapidly rotating strange stars. In: Kramer M, Wex N, Wielebinski R (eds) IAU Colloq. 177: Pulsar Astronomy—2000 and Beyond. Vol. 202 of Astronomical Society of the Pacific Conference Series p 661 13. Gondek-Rosi´nska D, Bulik T, Zdunik L, Gourgoulhon E, Ray S, Dey J, Dey M (2000) Rapidly rotating compact strange stars. Astron Astrophys 363:1005–1012 14. Hachisu I, Eriguchi Y (1982) Bifurcation and fission of three dimensional, rigidly rotating and self-gravitating polytropes. Progress Theoret Phys 68:206–221 15. Huang X, Markakis C, Sugiyama N, Ury¯u K (2008) Quasi-equilibrium models for triaxially deformed rotating compact stars. Phys Rev D D 78:124023 16. James RA (1964) The structure and stability of rotating gas masses. Astrophys J 140:552 17. Lai D, Rasio FA, Shapiro SL (1993) Ellipsoidal figures of equilibrium—compressible models. Astrophys J Suppl Ser 88:205–252 18. Lai D, Shapiro SL (1995) Gravitational radiation from rapidly rotating nascent neutron stars. Astrophys J 442:259–272

72

4 Triaxially Rotating Strange Stars

19. Lai D, Shapiro SL (1995) Gravitational radiation from rapidly rotating nascent neutron stars. Astrophys J 442:259–272 20. LIGO Scientific Collaboration, Virgo Collaboration, F. Gamma-Ray Burst Monitor and INTEGRAL (2017) Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A. Astrophys J Lett 848(2):L13. http://stacks.iop.org/2041-8205/ 848/i=2/a=L13 21. Meinel R, Ansorg M, Kleinwächter A, Neugebauer G, Petroff D (2008) Relativistic figures of equilibrium. Cambridge University Press 22. Piro AL, Ott CD (2011) Supernova fallback onto magnetars and propeller-powered supernovae. Astrophys J 736:108 23. Piro AL, Thrane E (2012) Gravitational waves from fallback accretion onto neutron stars. Astrophys J 761:63 24. Roberts PH, Stewartson K (1963) On the stability of a maclaurin spheroid of small viscosity. Astrophys J Lett 137:777 25. Saijo M, Baumgarte TW, Shapiro SL (2003) One-armed spiral instability in differentially rotating stars. Astrophys J 595:352–364 26. Shibata M, Karino S, Eriguchi Y (2002) Dynamical instability of differentially rotating stars. Mon Not R Astron Soc 334:L27 27. Shibata M, Karino S, Eriguchi Y (2003) Dynamical bar-mode instability of differentially rotating stars: effects of equations of state and velocity profiles. Mon Not R Astron Soc 343:619 28. Tsokaros A, Ruiz M, Paschalidis V, Shapiro SL, Baiotti L, Ury¯u K (2017) Gravitational wave content and stability of uniformly, rotating, triaxial neutron stars in general relativity. Phys Rev D 95(12):124057 29. Ury¯u K, Tsokaros A, Baiotti L, Galeazzi F, Sugiyama N, Taniguchi K, Yoshida S (2016) Do triaxial supramassive compact stars exist? Phys Rev D 94(10):101302 30. Watts AL, Krishnan B, Bildsten L, Schutz BF (2008) Detecting gravitational wave emission from the known accreting neutron stars. Mon Not R Astron Soc 389:839–868 31. Woosley S, Janka T (2005) The physics of core-collapse supernovae. Nature Phys 1:147–154 32. Xu RX (2003) Solid quark stars? Astrophys J Lett 596:L59–L62

Chapter 5

Conclusion

In this thesis, we have tried to study EoS of compact stars by calculating tidal deformability and constructing initial data of uniformly, differentially and triaxially rotating compact stars. In particular, we have focused on the difference between the results of NS models and two strange star models (MIT bag model and strangeon star model). After the observation of GW170817 and its EM counterparts, many researches have been done on how to interpret the observation as constraint on NSs: the radius constraint from tidal deformability measurement and the exclusion of prompt collapse scenario; the constraint on TOV maximum mass according to the fate of the merger remnant which can be inferred from EM counterparts. However, we have demonstrated that the constraint on compact star EoSs by the multi-messenger observation of GW170817/GRB170817A/AT2017gfo is quite different for NSs and strange stars, from both aspects. Firstly, due to the finite surface density of strange stars, the calculation of tidal deformability requires a correction on the surface. This leads to a different interpretation of the tidal deformability measurement. Particularly, strangeon star could reach MTOV as high as 4 solar mass without violating the tidal deformability constraint from GW170817. We have also applied the results of GW170817 to make constraints on interquark interaction parameters. Secondly, we have employed a self-consistence analysis based on conservation laws to infer the maximum mass of NSs according to the EM counterparts of GW170817. A more reliable range of 2.10−2.35 M for MTOV is derived for NS models. For strange star models, again due to the finite surface density, the enhancement in the maximum mass by uniform rotation is more than that of NS: 40% versus 20%. Therefore, the remnant will most likely be long lived if the remnant is a strange star. Thirdly, we have found that the relation between critical mass and angular momentum of differentially rotating compact stars doesn’t depend on the rotation law. Especially, this relation also holds for strange stars. We did find that strange stars are affected more by differential rotation. Particularly, the structure of strangeon star, which has a smaller ratio between central density and surface density, is affected by © Springer Nature Singapore Pte Ltd. 2020 E. Zhou, Studying Compact Star Equation of States with General Relativistic Initial Data Approach, Springer Theses, https://doi.org/10.1007/978-981-15-4151-3_5

73

74

5 Conclusion

differential rotation than MIT bag model. For instance, the drop in critical mass and transition into type C configurations happens at smaller differential rotation rate for strangeon stars. In the end, we have also studied the triaxially rotating sequence of strange stars. We have found that the critical T /|W | ratio for triggering triaxial deformation follows the same relation as in the case of NSs while we did also find 3 main differences: strange stars have longer triaxial sequences; supramassive triaxially rotating strange stars do exist and triaxially rotating strange stars are more efficient GW radiation sources. Those difference, again, is a result of the finite surface density of strange stars. Combining the above properties of rotating strange stars, we find that strange star works quite well with the magnetar scenario. On one hand, for the remnant mass of GW170817, the remnant will most likely be very long-lived for both MIT bag model and strangeon star model. Therefore, the BH central engine model for sGRB could not be applied for strange star model. At the same time, it’s quite uncertain about the electron fraction of the ejecta from a binary strange star merger and whether or not the r-process nucleosynthesis could be sufficient to produce significant amount of lanthanides. As a consequence, a remnant magnetar is required for the binary strange star merger scenario, both to explain the sGRB as well as the red components of the kilonova (details about how to explain the kilonova observation within binary strangeon star scenario could be found in [2]). On the other hand, the larger momentum of inertia of massive strange star might lead to more energy dissipation by GW in the early post-merger phase. In addition to this, the differential rotational configuration of strange stars might lead to low T /|W | instabilities [1, 3–5], which will dissipate the kinetic energy of the remnant as a competition of enhancing the dipole magnetic fields. GW dissipation in the early post-merger and a relatively low dipole magnetic field strength is also quite important for the magnetar scenario, to avoid too much energy injection into the ejecta which is not consistent with the EM counterpart observations. Last but not least, magnetar scenario will require a relatively large TOV maximum mass without violating the tidal deformability constraint and strange star could easily satisfy both conditions.

References 1. Centrella JM, New KCB, Lowe LL, David Brown J (2001) Dynamical rotational instability at low T /W . Astrophys J 550:L193–L196 2. Lai X-Y, Yu Y-W, Zhou E-P, Li Y-Y, Xu R-X (2018) Merging strangeon stars. Res Astron Astrophys 18:024 3. Saijo M, Baumgarte TW, Shapiro SL (2003) One-armed spiral instability in differentially rotating stars. Astrophys J 595:352–364 4. Shibata M, Karino S, Eriguchi Y (2002) Dynamical instability of differentially rotating stars. Mon Not R Astron Soc 334:L27 5. Shibata M, Karino S, Eriguchi Y (2003) Dynamical bar-mode instability of differentially rotating stars: effects of equations of state and velocity profiles. Mon Not R Astron Soc 343:619

Appendix

Accuracy and Convergence Tests

Most of the contents mentioned in this thesis is based on the results of the cocal code which is modified for the numerical calculation of strange stars. It’s essential to test the code before applying the result for any scientific implications. We will provide the information of our code tests in this section which contains mainly two aspects: the comparison with results of previous studies to show the accuracy and the convergence test. cocal is originally designed for calculation of initial data of NSs and BHs. During the development of the code, the accuracy and convergence behavior have already been studied several times (in, for example, [1, 2]). Hence, one first thing we should do is to compare the results of the modified code and the original code. As mentioned before, the modification of the code is mainly related to the EoS part. For calculation of strange stars, the polynomial form of EoS (Eqs. 1.32–1.34) is used to account for the finite surface density. However, if we specify the number of polynomial terms to be 1 and the integration constant C = 0, this polynomial type of EoS reduces to a polytropic EoS which could be used to describe NSs. We could calculate a configuration for such an EoS model with both the original code and modified version and compare the results. Generally speaking, the larger the polytropic index, the more difficult it is for the result to converge. Considering this, we have chosen a model with very large polytropic index  = 13/3 for this comparison. Two different compactness C = 0.1 and 0.2 are considered and we have constructed uniformly rotating solutions for this comparison. The result is shown in Table A.1 and as can be seen, the relative difference in all quantities of the solution is below 10−6 . In the iterations of the cocal code, the criterion of claiming converged result is that the relative difference between two adjacent iterations is below 10−6 . Hence, we could claim the modified version of the code is consistent with the original one. Besides the comparison between different versions of the code, we have also compared with results of previous studies to validate the accuracy of IWM formulation, in particular, in the case of high compactness and angular velocity. It’s worth © Springer Nature Singapore Pte Ltd. 2020 E. Zhou, Studying Compact Star Equation of States with General Relativistic Initial Data Approach, Springer Theses, https://doi.org/10.1007/978-981-15-4151-3

75

0.890625 0.890625 0.640625 0.640625

0.890625 0.890625 0.640625 0.640625

RQS RNS RQS RNS

RQS RNS RQS RNS

Rz /Rx

5.7530609 (8.0550374) 5.7530610 (8.0550374) 6.7217295 (9.2447957) 6.7217305 (9.2447923)

5.4358552 (6.3246085) 5.4358547 (6.3246049) 6.2854795 (7.2557171) 6.2854788 (7.2557128)

Rx n = 0.3 0.018070655 0.018070641 0.029921434 0.029921411 n = 0.3 0.022534551 0.022534504 0.036914579 0.036914571



1.4003781 1.4003720 1.4246605 1.4246549

0.57855257 0.57855224 0.58272647 0.58272620

MADM C = 0.1 0.032288017 0.032288112 0.11808879 0.11808922 C = 0.2 0.035430027 0.035430056 0.12800151 0.12800175

T /|W |

0.14805818 0.14805720 0.31583322 0.31583109 0.72801675 0.72801037 1.5166206 1.5166095

1.4458805×10−3 1.4458758×10−3 1.3119873×10−3 1.3119837×10−3

J

1.0291791×10−3 1.0291779×10−3 9.5401758×10−4 9.5401648×10−4

c

0.30254712 0.30254469 0.32247605 0.32247356

0.12180662 0.12180622 0.12912362 0.12912319

Zp

Table A.1 Rotating NS solutions obtained from the rotating QS solver are compared with those obtained from the original rotating NS solver in cocal. EoS model parameters of these solutions in the top panel are n = 0.3, C = 0.1, while C = 0.2 for the bottom panel. R x is the equatorial radius in coordinate length while the values in parenthesis are calculated in proper length.  is the angular velocity and c is the energy density in the center of the star. J is the angular momentum and Z p is the polar redshift

76 Appendix: Accuracy and Convergence Tests

Appendix: Accuracy and Convergence Tests

77

noting that the configurations of MIT bag model star applied in this work scales very simply as the bag constant. Therefore, we could easily compare our results with other calculations for MIT bag model, such as [3]. According to the comparison of differentially rotating MIT bag model stars in [4], the relative difference indeed increases as the configurations become more massive or the angular velocity gets larger. Nevertheless, even for the most massive case, the difference is 1% for mass, 2% for radius and 5% for central angular velocity, which is within our expectation for employing the conformal flatness approximation. Besides the quantitative tests for the accuracy of the modified cocal code. We have also investigated the convergence behavior of the code for the calculation of strange stars. Five different resolutions are applied for this convergence test, denoted as H2.0, H2.5, H3.0, H3.5 and H4.0. The resolution of the grids gradually increase. From H2.0 to H4.0, the grid intervals r , θ and φ is 2/3, 3/4, 2/3, 3/4 times of the previous grid. In principle, as the grid resolution increases, the numerical results should approach the exact solution. The larger the resolution, the smaller the grid intervals, the smaller error is between the numerical solution and the real solution. This is the convergence behavior of numerical calculations. For the same physical quantity, the results obtained in two different numerical resolutions should satisfy  fμ − fν ≈ A

μ ν

n

 − 1 nν ,

(A.1)

in which A is a constant, and μ is the grid interval in resolution H μ (in our case, H μ varies from H2.0 to H4.0). If we consider f H 3.0 − f H 2.0 , f H 3.5 − f H 2.5 and f H 4.0 − f H 3.0 , we always have μ /ν = 1/2. We could then study the convergence order n, namely, how the numerical error decreases as resolution increases. We have calculated the uniformly rotating solution with axis ratio Rz /Rx = 0.75 in all five resolutions for both MIT bag model and strangeon star model. Quantities such as angular velocity, ADM mass, angular momentum, T /|W | ratio and eccentricity of the star to investigate the convergence behavior. The test could be found in [5]. As can be seen from the fit, ADM mass, angular velocity and eccentricity converges at second order and angular momentum, T /|W | ratio and baryonic mass at first order.

References 1. Ury¯u K, Tsokaros A, Galeazzi F, Hotta H, Sugimura M, Taniguchi K, Yoshida S (2016) New code for equilibriums and quasiequilibrium initial data of compact objects. III. Axisymmetric and triaxial rotating stars. Phys Rev D D93(4):044056 2. Tsokaros A, Mundim BC, Galeazzi F, Rezzolla L, Ury¯u K (2016) Initial-data contribution to the error budget of gravitational waves from neutron-star binaries. arXiv:1605.07205 3. Szkudlarek M, Gondek-Rosi´nska D, Villain L, Ansorg M (2019) Maximum mass of differentially rotating strange quark stars. arXiv e-prints arXiv:1904.03759

78

Appendix: Accuracy and Convergence Tests

4. Zhou E, Tsokaros A, Ury¯u K, Renxin X, Shibata M (2019) Differentially rotating strange star in general relativity. Phys Rev D 100(4):043015 5. Zhou E, Tsokaros A, Rezzolla L, Xu R, Ury¯u K (2018) Uniformly rotating, axisymmetric, and triaxial quark stars in general relativity. Phys Rev D 97(2):023013