The Black Hole-Neutron Star Binary Merger in Full General Relativity: Dependence on Neutron Star Equations of State

185 Pages·2013·4.417 MB·English

by  Koutarou Kyutoku

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Koutarou Kyutoku
The Black Hole–Neutron
Star Binary Merger in Full
General Relativity
Dependence on Neutron Star
Equations of State
Doctoral Thesis accepted by
Kyoto University, Japan
123
Author (Current address) Supervisor
Dr. KoutarouKyutoku Prof.Masaru Shibata
Cosmophysics Group KyotoUniversity
KEK, IPNS,Theory Center Kyoto
Ibaraki Japan
Japan
ISSN 2190-5053 ISSN 2190-5061 (electronic)
ISBN 978-4-431-54200-1 ISBN 978-4-431-54201-8 (eBook)
DOI 10.1007/978-4-431-54201-8
SpringerTokyoHeidelbergNewYorkDordrechtLondon
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Parts of this thesis have been published in the journal articles:
1. K. Kyutoku, M. Shibata, K. Taniguchi, Phys. Rev. D 82, 044049 (2010).
2. K. Kyutoku, H. Okawa, M. Shibata, K. Taniguchi, Phys. Rev. D 84, 064018
(2011),andarereprintedwithpermission.Copyright2010-2011bytheAmerican
PhysicalSociety.
Supervisor’s Foreword
One of the most exciting events in general relativity and astrophysics in this
decade will be the first direct detection of gravitational waves. This will be
achieved by 2020. After their first detection, gravitational-wave astronomy will
begin andwillbeanew toolforobserving general relativisticobjectswhich have
been poorly explored. The black hole–neutron star binaries, which have not been
observed yet, are among the most promising sources of gravitational waves. The
firstdetectionofthemwillbeachievedbygravitational-waveobservation,andwe
expectthatitwillproviderichinformationforblackhole–neutronstarbinariesand
properties of the neutron stars.
The merger remnant of black hole–neutron star binaries is also the leading
candidateforthecentralenginesofshort-durationgamma-raybursts,forwhichthe
progenitor has not been determined yet. A coincident observation of gravitational
waves and gamma-ray bursts could provide the definite answer for this unsolved
issue. In addition, a class of black hole–neutron star binaries is likely to eject
neutron-rich material through the tidal disruption event of the neutron star. The
ejected material is a promising transient source of UV, optical, and radio signals,
which also have not yet been detected. Observation of these electromagnetic
signals is one of the exciting unsolved issues in astronomy, and will be an
important method for exploring the black hole–neutron star binaries.
Innear-futureobservationsbygravitational-wavedetectorsandelectromagnetic
telescopes,blackhole–neutronstarbinarieswillbeexploredindetail.However,to
extractphysicalandastrophysicalinformationfromtheobservationaldata,atheory
for their merger process is necessary. Numerical relativity is probably a unique
approach for determining the nature of the merger of black hole–neutron star
binaries.
Numerical relativity is the field in which Einstein’s equation and matter
equationsarenumericallyandaccuratelysolvedincomputers.Becausethemerger
processes are highly dynamical and general relativistic, numerical relativity is
required.Numericalrelativityhasbeensignificantlydevelopedinparticularinthe
past decade. Now, it is feasible to perform numerical-relativity simulations for a
varietyofproblemssuchasmergersofbinaryneutronstarsandbinaryblackholes.
vii
viii Supervisor’sForeword
Amongmanyothers,thesimulationforblackhole–neutronstarbinarieswasanew
topic in this field. The first simulation had been performed in 2006. Since then,
significant progress has been achieved in this community and Dr. Koutarou
Kyutoku, the author of this volume, has been a central person in this progress.
Dr. Kyutoku is one of the first persons who performed numerical-relativity
simulationsforthemergerofblackhole–neutronstarbinariessystematically.Inthis
problem,thereareseveralfreeparameters:massesoftheblackholeandneutronstar
andblack-holespin.Inaddition,theequationofstateofneutronstarsisstillpoorly
known.Thus,thesimulationhastobeperformedforavarietyofpossibleequations
of state. Dr. Kyutoku performed a large number of simulations for a variety of
parametersetsandequationsofstate,andclarifiedthenatureofthemergerprocess,
mergerremnants,andemittedgravitationalwavesinacomprehensivemanner.This
volume presents the results of such a systematic investigation. In particular, he
pointedoutthepossibilitythattheequationofstateforneutronstars,whichispoorly
known, could be strongly constrained by the observation of high-frequency
gravitational waves for the first time. He also clarified that for a class of black
hole–neutronstarbinaries,theremnantiscomposedofarapidlyspinningblackhole
surroundedbyamassiveanddensetorus.Sucharemnantisapromisingcandidate
for the central engine of short-duration gamma-ray bursts. These findings give
readersnewinsightsingeneralrelativityandhigh-energyastrophysics.
Kyoto, September 2012 Masaru Shibata
Acknowledgments
IamdeeplygratefultoProf.MasaruShibata,mysupervisor,forhiscontinuoushelp
andsupportinalltheaspectsofmyresearch.Iamalsogratefultohimforproviding
meplentifulcomputationalresources.
IthankProf. KeisukeTaniguchiforhelpingmewithcomputingquasiequilibrium
statesofbinarieswithLORENE.IalsothankDr. TetsuroYamamotofordeveloping
the code SACRA, and Dr. Hirotada Okawa for the update and parallelization of
SACRAaswellasfororganizingourcomputationalenvironments.
IwouldliketoexpressmygratitudetoProf. TakashiNakamura,Prof. Takahiro
Tanaka,Prof. MasakiAndo,andProf. NaokiSetofordiscussionsofgravitational-
wave astronomy on a weekly basis. I am especially indebted to Prof. Naoki Seto
for careful reading of the manuscript.
I am grateful to Dr. Yuichiro Sekiguchi, Dr. Kenta Kiuchi, Dr. Yudai Suwa,
and Mr. Kenta Hotokezaka for in-depth discussions of numerical relativity, high-
energyastrophysics,andphysicsofneutronstars.IamalsogratefultoDr. Atsushi
Naruko and Dr. Kentaro Tanabe for daily discussions of a wide range of physics.
It is a pleasure for me to acknowledge Prof. John L. Friedman, Prof. Eric
Gourgoulhon, Dr. Nicolas Vasset, and Dr. Benjamin D. Lackey for their helpful
discussion. I had an extraordinary time when I visited their institutes.
Kyoto, January 2012 Koutarou Kyutoku
ix
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Gravitational Waves from Compact Binaries. . . . . . . . . . . . . . . 1
1.1.1 The Quadrupole Formula and the Inspiral Phase. . . . . . . 3
1.1.2 The Estimated Detection Rate . . . . . . . . . . . . . . . . . . . 5
1.2 Electromagnetic Emission from the Compact Binary Merger . . . 7
1.2.1 Short-Hard Gamma-Ray Bursts and the Merger
Scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 The r-Process and the Kilonova . . . . . . . . . . . . . . . . . . 10
1.3 The Mass-Shedding Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Black Hole–Neutron Star Binaries in Numerical Relativity. . . . . 14
1.5 The Purpose of This Thesis and Convention. . . . . . . . . . . . . . . 17
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Equations of State of Neutron Star Matter. . . . . . . . . . . . . . . . . . 25
2.1 Formation and Cooling of (Proto-)Neutron Stars. . . . . . . . . . . . 25
2.2 Neutron Stars in Spherical Equilibria. . . . . . . . . . . . . . . . . . . . 28
2.2.1 The Tolman-Oppenheimer-Volkoff Equation . . . . . . . . . 29
2.2.2 The Tidal Love Number and Deformability. . . . . . . . . . 30
2.3 Current Constraints on the Equation of State . . . . . . . . . . . . . . 33
2.3.1 The Maximum Mass. . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.2 The Minimum Rotational Period. . . . . . . . . . . . . . . . . . 36
2.3.3 The Radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Piecewise Polytropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 The Cold-Part Equation of State:
Piecewise Polytropes. . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.2 Thermal Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 44
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
xi
xii Contents
3 Computing Initial Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1 Assumption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 The Initial Value Problem of General Relativity. . . . . . . . . . . . 50
3.2.1 The 3?1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.2 Solving the Initial Value Problem. . . . . . . . . . . . . . . . . 52
3.2.3 The Puncture Framework. . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Hydrostatic Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.1 The Hydrostatics in a Spacetime. . . . . . . . . . . . . . . . . . 56
3.3.2 The Hydrostatics in the Initial Value Problem . . . . . . . . 57
3.4 Free Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.1 Parameters Associated with the BH. . . . . . . . . . . . . . . . 60
3.4.2 Parameters Associated with the NS. . . . . . . . . . . . . . . . 62
3.4.3 Parameters of the Binary . . . . . . . . . . . . . . . . . . . . . . . 63
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Methods of Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1 The BSSN-Puncture Formalism. . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 BSSN Variables and Evolution Equations . . . . . . . . . . . 68
4.1.2 BSSN Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.3 The Moving Puncture Gauge Condition. . . . . . . . . . . . . 71
4.2 Hydrodynamic Evolution Equations. . . . . . . . . . . . . . . . . . . . . 71
4.2.1 Evolution Equations in a Conservative Form . . . . . . . . . 72
4.2.2 Recovery of Primitive Variables. . . . . . . . . . . . . . . . . . 76
4.2.3 An Artificial Atmosphere. . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Adaptive Mesh Refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.1 The Grid Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.2 Boundary Conditions and Data in the Buffer Zone . . . . . 79
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5 Diagnostics for Numerical Simulations . . . . . . . . . . . . . . . . . . . . . 83
5.1 Gravitational Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1.1 Extracting Gravitational Waves . . . . . . . . . . . . . . . . . . 84
5.1.2 The Fixed-Frequency Integration Method. . . . . . . . . . . . 86
5.1.3 The Taylor-T4 Formula. . . . . . . . . . . . . . . . . . . . . . . . 88
5.2 Quantities of Merger Remnants. . . . . . . . . . . . . . . . . . . . . . . . 90
5.2.1 Quantities of the Remnant Disk . . . . . . . . . . . . . . . . . . 90
5.2.2 Quantities of the Remnant Black Hole. . . . . . . . . . . . . . 91
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6 The Merger of Nonspinning Black Hole–Neutron Star Binaries. . . 93
6.1 Models and Setup of AMR Grids . . . . . . . . . . . . . . . . . . . . . . 93
6.2 Orbital Evolution and General Merger Process . . . . . . . . . . . . . 95
6.3 Gravitational Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.4 Gravitational-Wave Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 101