In my first lecture, I will introduce the history of NSs discovery and theirgeneral structure going from the envelope through outer and inner crusts tothe core. I will also survey the latest astrophysical observational data on NSproperties.

The second lecture will be devoted to exploration of the NS Equation of State(EoS) which is still unknown and is a subject of an extensive research.Among the variety of theoretical and empirical models of the EoS currently inthe literature, I will describe in more detail the Quark-Meson-Coupling (QMC)Model, an effective relativistic mean-field model in which the forces betweenindividual baryons are self-consistently mediated by exchange of virtualmesons between the valence quarks in the baryons.

The neutron star merger (BNSM) and the related gravitational waves will bethe subject of the final, third lecture. This topic is currently most activelyexplored, using novel frameworks of multi-messenger techniques. Advantagesand disadvantages of this trend will be discussed.

LectureI

Macroscopyandobservation

History of NSs discovery, their general macroscopic features togetherwith the survey the latest astrophysical observation techniques.

Evolutionofthe(observablepart)oftheUniverse

fromtheBigBang(left)tothepresent(right).

CourtesyAnd

rewSteiner

Whitedwarf

Neutronstar

Blackhole

Supernovaa

PlanetaryNebula

Finalstageofstarswhicharenotverymassive(over97%ofstarsofourGalaxy):(main-sequencestaroflowormediummassbelow8solarmasses)https://astronomy.swin.edu.au/cosmos/h/hertzsprung-russell+diagram

I. Hydrogen-fusing– expandtoredgiant

II. Heliumfusingtocarbonandoxygen

inthecorebythetriplealphaprocess

III.IfTtoolowtofusecarbon,CandO

accumulateinthecore,outerenvelope

isshedsofftoformplanetarynebula

(theyreturnlightelementsbackto

interstellarmedium)andwhitedwarfisborn

IV: Whitedwarf:(CO),(ONeMg),(ONe),(He)

V: Whitedwarfissupportedfrom

gravitationalcollapse

byelectrondegeneracypressure

VI:Ifawhitedwarf(CO)massexceeds1.4Msol

itmayexpode astype1asupernova

(carbondetonation)

Absolutemagnitude isthemagnitudethestarwouldhaveifitwasplacedatadistanceof10parsecsfromEarth(VegainnorthernconstellationLyraistakenaszero)

Parse

c:Thedista

nceatw

hich

thera

diuso

fEarth

'sorbitsu

btendsa

nangleofo

nese

condofa

rc.

H- He

BeyondWhiteDwarfMasses(about1.4Msolar):

AfterChandrasekhar’smass(maximum)isexceeded(derivedfromelectrondegenerasytheory:

-Gravitationalcollapsecontinues-Nucleondensityisreached

-Ifmasslessthen2-3solarmasses:nucleondegeneracypressuremayhold

thecollapse

allprotonsareconvertedtoneutrons

viaelectroncapture(weakinteraction)

andaneutronstarisbornwitharadius

about500timessmallerthanthewhitedwarf

-Ifmassbiggerthan2-3solarmasses– thecollapsecontinuestoablackhole

Whitedwarfsandneutronsstarsaretheonlypossiblestableconfigurationsbetween

Matthias HempelRussbach, 14.3.2014

From progenitor stars via CCSNe to neutron stars

• what is the state of matter during all these stages?

progenitor star at onset of collapse

core-collapse supernova explosion

cold neutron star

15TakenfromHempeletal

Averagedensity1014-15g/cm3 (compare5.5g/cm3 oftheEarth)

1010 humansonEarth@50,000geach=5×1014gCompressthemallintoasugarcubeandwereachneutronstardensity!

Gravitationaccelerationg≈1012 ms-2 (compareabout10ms-2ontheEarth)

escapevelocityfromthesurfaceofnon-rotatingstarabout210,000km/s(compareabout11km/sfromtheEarth)

NASAX-raytelescopeontheSwiftSatellite– Soderberg andBerger(NatureMay22,2008)

ThefirsttimesupernovaburstwasdetectedInactioninNGC2770.

DetectionofabrightX-rayflash(left)

Followed- upopticalobservation(right)

2008

Jan7

2008

Jan8

SelectedtypeIIsupernovae

YearConstellationApp.Mag.Dist.(lY).HostPulsar

SN~11000agoVela+12.0815NGC2736Velanebula+B0833-45

SN386Sagittarius+1.514,700MilkyWay(G11,2-0.3+PSRJ1811-1926)

?

SN1054Taurus-66,500MilkyWay Crab Nebula+B0531+21

Cas ACassiopeia+511,000MilkyWayNotvisible– dust

(1680)thebrightestsourceinthesky

SN1885AAndromeda+5.852,600,000M31

SN1987ADorado+2.9160,000Gr.Mag.Cloud

Themostdistantsupernova everdetectedtookplace10.5billionyearsago,orthree-quarterstheageoftheUniverseitself(discoveredbyDarkEnergySurvey)

Smithetal,ApJ 854,37(2018)

StudyingtheUltravioletSpectrumoftheFirst

SpectroscopicallyConfirmedSupernovaatRedshiftTwo

DES16C2nm isclassifiedasabrightest,superluminous

supernova(about100times

brighterthancommonCCSN-II).

Physicaloriginnotyetclear– typeIa orIc (hydrogenfree)?

ONSUPER-NOVAEBYW.BAADEANDF.ZWICKY

Proc.Nat.Astr.Soc.

CommunicatedMarch19,1934

Aspecialclassofnovae(newstars)withextraordinarybrightnesswerecalled

Super-novaeforthefirsttime.

Analysisofdataon1885A–suggestionof

generationofacompactobject

Announced15-16December1933

publishedPhys.Rev.48,76(1934)

Thatobjectisaneutronstar

We are fully aware that our suggestion carries with it grave implications regarding the ordinary views about the constitution of stars and therefore will require further careful studies.

1893– 1960GermanGoettingen

1898-1974Swiss/CzechCaltech

AndromedaGalaxy

David(Deddy)Dayag2019, Wikimedia

Commons

,;(B+7(BQ7(,;(BB((

>"%0%O'*:0"%'(30-"3(

;*:0"%'(30-"3(};%'O"%0-N'/(

'*:0"%'(30-"3(A-$#%(P:&3-"37((AA8H,((

;*:0"%'(30-"3(#'((E%[OU-33(9O"-4(+#'-"#*3(<EC9\3@((

9.B;3( 89>3(,TA3( 55D3(

S%0(V($*'3*7(&#6*NU*(XYZ(3*Q%'$3(

Q%&$(V($*'3*7(NU*3Q-&*(+#&&#%'3(%6(4*-"3(

Magnetars

Classical neutron star composition in 1933 - neutrons only �

DanyPageUNAM

current

THEORY

Whatobservationaldatawehavetoidentifybasicpropertiesofneutronstars?

Mass,radius,temperature,dynamics……

Problem:- Noenergygenerationafterformation- Smallsurface>rapidinitialcoolandlowopticalluminosity

Hubbleseesaneutronstaraloneinspace.Veryhotandverysmall,lessthan28kmindiameter(Rosat X-rayandXUVobservatory1997)

Isolatedstars- atmospheres

Starsinbinaries- gravitationaleffects

Mergers– gravitationalwaves

Detection:electromagneticsignals

radio,X-rays,gamma-rays,UV,optical

Pulsedradioemissionfromobject

wayoutsidesolarsystem

- P=1.337s:

compactobject(WDorNS)

- Periodgraduallyincreasing:

rotationalperiodratherthan

oscillationperiod

CambridgepulsarCP1919(PSRB1919+21)

PSRJ1921+2153inVelpecula

1968:

CrabandVelapulsars – pulsarare

magnetised neutronstars

Thepulsarsinsidepowerthenebulae

Dame JocelynBellBurnell,

DBE,FRS,FRAS,1943-Cambridge,Bath,Oxford)

AntonyHewish,

1924– CambridgeNobelPrice1974

withMartinRyle

Vela– 12,000yago

Radio,optical,X- gamma-rayP=89.33ms

Crab– 1054

X-rayblue,optical,redP=33ms

Discoveryofpulsars1967

Assumedmechanismofpulsars(highlymagnetizedrotatingstars)

l Rotation induces strong E quadrupole field,

accelerates charged particles off surface into

magneto-sphere Fel/Fgr ~ 1012 – surface

particles are subject to elmg forces

l Conduction of material is strong along B lines,

low perpendicular to them: particles forced

into co-rotation-corotating plasma

l Light cylinder: v =RLω = c – particles forced

to cones above magnetic poles.

l Emission mechanism: probably curvature

radiation (radiation of a charged particle

accelerated along a curved field line)

l Coherent emission to explain high

brightness of small emitting region: bunches

of particles moving in the same direction and

radiating in phase

X-rays!−rays

Cordesetal.:https://doi.org/10.1016/j.newar.2004.09.040M.Kramer,arXiv:astro-ph/0405178,DOI:10.1007/b13178

Brakingindexn ofisolatedpulsars

Hamil,JRS:PRD91,063007(2015)TheLightHouseEffect

Puremagneticdipole:n=3

FixedmomentofinertiaI

VariableIRotationaxisRadiation

Artistsimpression

,;(B+7(BQ7(,;(BB((

>"%0%O'*:0"%'(30-"3(

;*:0"%'(30-"3(

};%'O"%0-N'/('*:0"%'(30-"3(

A-$#%(P:&3-"37((AA8H,((

;*:0"%'(30-"3(#'((E%[OU-33(9O"-4(+#'-"#*3(<EC9\3@((

9.B;3( 89>3(,TA3( 55D3(

S%0(V($*'3*7(&#6*NU*(XYZ(3*Q%'$3(

Q%&$(V($*'3*7(NU*3Q-&*(+#&&#%'3(%6(4*-"3(

Magnetars

XDINs:X-rayDimyoung

isolatedNS

SGRs:SoftGammarepeaters

AXPs:AnomalousX-raypulsars

CCOs:CentralCompactobjects

ThermalemittingNSPureblackbody

Spectrum- flux

" = $%&'(

)

Largevarietyofpulsarsobservedandclassified

Massandradiusofneutronstars

(usingpulsarsdataandGR)

NSmasscanbemeasureddirectly onlyinbinarysystems(BUTonly5-10%ofknownpulsarsareinbinaries)

NSgravitationalmassfromobservationseee.g.https://www3.mpifr-bonn.mpg.de/staff/pfreire/NS_masses.html :

PSRB1913+16pulsar+NSHulse-Taylor

ThefirstbinarysystemdiscoveredPeriodicchangesinradiopulsefrequencyledto

discoveryofanunknownNScompanion

PSRJ1614-2230NS+WD

(usede.g.Demorestetalhttps://doi.org/10.1038/nature09466)

Shapirodelay:GRincreaseinlighttraveltimethroughthecurvedspace-timenearamassive

bodydoi:10.1103/PhysRevLett.13.789

PSRJ0348+0432NS+WD(Antoniades 2013)DOI:10.1126/science.1233232

radio-timingobservationsofthepulsar

andphase-resolvedopticalspectroscopyofits

WDcompanion

PSRJ0740+6620.NS+WD

NatureAstronomy volume 4, pages72–76(2020)usingtheShapirodelay

PSRB1913+16NSbinary(Hulse-Taylor),P=59msMg =1.4414±0.0002M� (ApJ 195,1975)

PSRJ1903+0327NSonaneccentric orbit around MSstar;P=59ms

Mg=1.667±0.021M� (Freireetal,10.1111/j.1365-2966.2010.18109.x)

PSRJ0737-3039thefirstdoublepulsar(A,B);P=2.77s(B)Mg =1.249±0.001M� (10.1126/science.1094645)

PSRJ1614-2230NS+WDP=3.15ms

Mg =1.97±0.04M� (Demorestetal,Nature467,1081(2010)

later1.928 ± 0.017M� (Fonsecaetal2016),1.908 ± 0.016M� (Arzoumanian etal2018)

Selectedneutronstarmasses:(noradiiknown):

PSRJ0348+0432NS+WDP=39ms

Mg =2.03±0.03M� (Antoniades etal,10.1126/science.1233232

PSRJ0740+6620NS+WDP=2.89msMg =2.14±0.10M� Cromartieetal,https://doi.org/10.1038/s41550-019-0880-2Mg =2.14±0.07M� Fonsecaetal,E.Fonsecaetal 2021ApJL 915L12

Dependentondistance,atmosphere,diskarea,mediumcompositionbetween

thestarandtheobserverandandsomeestimateoftheNSmass(usuallytakenas1.4Msolar)

Ozel andFreire(Annu.Rev.Astron.Astrophys.2016,54,401)(X-ray) 10– 11.5kmSteineretal:Astrophys.J.2013,765,L5(X-ray) 104.- 12.9km

Annala etal:Phys.Rev.Lett.2018,120,172703(GW)13.6kmAbbottetal:Phys.Rev.Lett.2018,121,161101 (GW) 11.9+/- 1.4km

Burgio etal.,Nucl.Part.Phys.Proc. 2019,306–308,61 (GW) 11.8– 13.1.(1.5Msolar)

Capano etal.,Nat.Astron.2020,4,625 (GW) 11.0+0.9– 0.6kmAl-Mamun etal.,Phys.Rev.Lett.2021,126 (GW)9.8– 13.95km

ExtractionofNSradiifromobservations

TheX-raysareproducedbymatterfallingfrom

onecomponent,(normalstar),totheothercomponent,aneutronstar orblackhole.

Whenthein-fallingmaterialreachesthesurface

oftheNSandtheignitiontemperatureishigh

enough itcausesanX-rayburstrawdataforestimationoftheradiusoftheNS.

Low-Mass-X-ray-Binaries

XMMNewton.

Spaceobservatories

X-rayMulti-MirrorMission-Space

telescopelaunchedbyEuropeanSpaceAgency1999– 2022– inorbit.

Forthefirsttimetheinfluenceofthegravitationalfieldofaneutronstaronthelightitemits.

SensitivetoX-raysfromatmospheresMappedthegrowthof12,000

supermassiveblackholesinthecoresofgalaxiesandgalaxyclusters.

Nustar

NuSTAR (NuclearSpectroscopic

TelescopeArray)missionhasdeployedthefirstorbiting

telescopestofocuslightinthehighenergyX-ray(3- 79keV)LaunchedbyNASA2012–

NuSTAR consistsoftwoco-alignedgrazingincidence

telescopeswithspeciallycoatedopticsandnewlydeveloped

detectors.

Chandra

ChandraisanEarthsatelliteina64-

hourorbit,anditsmissionisongoingasof2021(launchedJuly23,1999).

ItisNASAflagshipmissionforX-rayastronomy.

DatafromNASA'sChandraX-rayObservatoryandpreviouslyunpublisheddatafromNASA'sNuclearSpectroscopicTelescopeArray(NuSTAR),incombinationwithdatafromthe

ground-basedAtacamaLargeMillimeterArray(ALMA)

IntriguingcollectionofevidenceforthepresenceoftheneutronstaratthecenterofSN1987A.

Thefirstsupernovavisiblewiththenakedeyeinabout400years,Supernova1987A

(orSN1987Aforshort)sparkedgreatexcitementamongscientistsandsoonbecameoneofthemoststudiedobjectsinthesky.ThesupernovaislocatedintheLargeMagellanic Cloud,

asmallcompaniongalaxytoourownMilkyWay,onlyabout170,000light-yearsfromEarth.

NICER– NeutronStarInteriorCompositionExplorer

NASA- launched2017placedontheInternationalSpaceStationSoftX-ray,similartoChandraandXMM– BUT

Timing-basedtechniquesforconstrainingMandRrelyonthepresenceofsurface

inhomogeneities,leadingtoemissionthatvariesperiodicallyasthestarrotates

(Watts,Rev.Mod.Phys.88,021001(2016)

NICERprovidessimultaneousfasttimingandspectroscopywithhighsignaltonoiseratiophotoncountingcapabilitywithin0.2-12keV X-rayband(has56X-rayconcentratoropticsandsilicon

driftdetectorpairs.Theprimarytarget:

PSRJ0437−4715with independentlyknow mass to5%should yield the radius

to2%

Other targets:

isolated radio millisecond pulsarsPSRJ0030+0451andJ2124−3358should

yield less precise results asthere is noprospect for measuring their masses.

Left:Astheneutronstarrotates,emissionfromasurfacehotspotgeneratesapulsation.Theobserverinclinationisi andhotspotinclinationisα.Right:Psaltis etal(2014)Thewaveformismodifiedfromapuresinusoid,andthe

temperature(indicatedbythecolor,whichistheratioofthenumberof

photonswithenergiesabovetothosebelowtheblackbodytemperature)

Wattsetal,Rev.Mod.Phys.88,021001

Hotspots:(Gourgouliatos):Anewbornneutronstardoesnotrotateuniformly-- Itwindsupandstretchesthemagneticfieldinsidethestarinawaythatresemblesatightballofyarn.Wefoundthatahighlywoundmagneticfieldisunstable.Itspontaneouslygeneratesknots,whichemergefromthesurfaceoftheneutronstarandformspotswherethemagneticfieldismuchstrongerthanthelarge-scalefield.Thesemagneticspots

producestrongelectriccurrents,whicheventuallyreleaseheat,inthesamewayheatisproducedwhenanelectriccurrentflowsinaresistor."Itispossibletogenerateamagneticspotwitharadiusofafewkilometresandamagneticfieldstrengthinexcessof10billionTesla.Thespotcanlastseveralmillionyears,evenifthetotalmagneticfieldoftheneutronstarhasdecayed.Evenneutronstarswithweakeroverallmagneticfieldscouldstillformvery

intensemagnetichotspots.Thiscouldexplainthestrangebehaviour ofsomemagnetars,forexampletheexoticSGR0418+5729,whichhasanunusuallylowspinrateandarelativelyweaklarge-scalemagneticfieldbuteruptssporadicallywithhigh-energyradiation.

Someresultssofar:

SummaryofLectureI

Weunderstandtheoriginandmechanicsofthebirthofneutronstars

Theincreasingsophisticationofobservationtechnologyanddataanalysis

yieldsevidenceofagreatvarietyofcompactobjectsandtheirmacroscopic

properties,masses,sizeanddynamics.

Toobtainthesedataweneedgeneralrelativity

Thesedataalonedonotdirectlygiveinformationabouttheinternal

structureofneutronstars

Thesensitivityoftheobservationaldatatothecompositionoftheinterior

remainsanopenquestionasdiscussedinLectureII.

LectureII

Microscopy

The second lecture will be devoted to exploration of the NSEquation of State (EoS) which is still unknown and Is asubject of an extensive research. Among the variety oftheoretical and empirical models of the EoS currently in theliterature, I will describe in more detail the Quark-Meson-Coupling (QMC) Model, an effective relativistic mean-fieldmodel in which the forces between individual baryons areself-consistently mediated by exchange of virtual mesonsbetween the valence quarks in the baryons.

F. Weber Prog.Part.Nucl.Phys. 54, 193 (2005)

Extremeconditionsinneutronstarsallowwidespeculationsabouttheirinternalstructure:WHICHAREREALLYTHERE?

TakenfromtheNuPECC LongRangePlan2017"PerspectivesforNuclearPhysics"

QCDdiagramofhighdensitymatter ChemicalpotentialsBaryon:

Isospin

μB andμS coupletobaryonnumber

andstrangeness,μI couplestothedifferenceofthenumberofprotonsandneutrons–

relevantparameterforsystemswithanasymmetrybetweenprotonsandneutrons.

Relevantforthecoreofneutronstarstheinitialstateofheavy-ioncollisions.

SinceneutronsdominateoverprotonsinμI <0.Atthesametime,thenonzerobaryonnumberimpliesμB >0.

Strange

BrandtandEndrodi arXiv:1611.06758

AnatomyofaNeutronStar

IronLatticesurroundedbydegenerateelectrons

>107 gcm-3: increasinglylargeneutronrich

nuclei,relativistically degenerateelectrons.

4x1011 gcm-3 outercrustends- neutrondrip

Innercrust:verylargeneutronrichnuclei,

neutrongasincreasinglydominatespressure

1014 gcm-3 innercrustboundary;

exoticnuclearshapes

Core:uniformnuclearmatter,90%n,10%p,e,μ

? Hyperons?Pion,kaoncondensates?

Quarkmatter,strangematter?

10km

0.6km

0.3km

Atmosphere

Afewcm

Chamel andHaensel,Physicsofneutronstarcrusts,Livingrelativity11,10(2008)

Ocean

(Very) schematic sequence of equilibrium phasesof nuclear matter as a function of density:

Nucleons+

heavy baryons+

leptons

Nuclei in Neutron

+Electron

gas

‘Pasta phase’

n,p,e,μ(β-equilibrium)

Quarks ???

Nuclei in electron

gas

~2x10-4 fm-3 ~0.06 fm-3<~2x10-4fm-3

~0.1 fm-3 0.3-0.5 fm-3 >0.5 fm-3

Innercrust– nuclearpasta

Deformationsandcracksinthecrustsofneutronstarshavebeenlinkedwithphenomena

suchasgravitationalwaves,burstsofgammarays,and“glitches”—eventswhereastar’sspinsuddenlyspeedsup.Theinnercrust’sstructureiskeytounderstandingtheseevents.

Intheinnercrust– thecompetitionbetweenCoulombrepulsionandsurfacetension

ofnucleons(nuclearattraction)resultsforthemattertoafrustratedphase-self-organized‘pasta’phase.3Dfullyself-consistentHartree-Fock:(Newton+,JRS arXiv:2104.11835(2021).

Thewholepastaregioncanoccupyupto70%ofthecrustbymassand40%bythickness,

andthelayerinwhichprotonsaredelocalizedcouldoccupy45(25)ofthecrustmass(thickness).

0.04-0.05fm-3

N~500-1000

0.07-0.09fm-3

N~500-1000

Soft solids: emulsions, foams, colloids, polymers, gels, liquid crystals, cytoplasmaFlexible internal structure, weak interactions, easily influenced by external conditionsFRUSTRATED MATTER

MaxPlankInstituteforDynamicsandSelf– Organisation

Liquid crystal

Granular matterunder stress

Geometry of fluid interfaces

HydrostaticequilibriumofasphericalobjectwithisotropicmassdistributioninGR:Tolman-Oppenheimer-Volkoff (TOV)equations:Tolman Phys.Rev.55,364(1939),OppenheimerandVolkoff,Phys.Rev.55,374(1939)

dPdr

= −GM (r)ε

r2(1+ P / εc2 )(1+ 4πr3P /M (r)c2 )

1− 2GM (r) / rc2

M (r) = 4πr ' 20

r

∫ ε(r ' )dr '

G- gravitationalconstant

M- gravitationalmass,P– presure

ε - totalenergydensity

Input:P(ε)Pressureasafunctionoftotalenergydensity- EQUATIONOFSTATE

Output:M(r)

Gravitationalmassasafunctionofthecorrespondingradius

LudwigBoltzmann

1844- 1906

TheEquationofState(EoS):Idealgas:

Nuclearmatter:

P = ε(ρ,T ) ε(ρ,T ) = EA

ρ,T( )ρ⎛⎝⎜

⎞⎠⎟f

∑f

µB = (P + ε ) / ρ

Twokeypoints:

I.TheEoS isdependentoncomposition(summingoverf)

CONSTITUENTS+INTERACTIONS

IIEnergyperparticleE/AandITSDENSITYandTEMPERATUREDEPENDENCE

mustbedeterminedbynuclearand/orparticlemodels.

IIIMacroscopicNSproperties(M,R)dependentonmicroscopicvariables

Aretheysensitiveenough?Howmanyconstraintsdoweneed?

SolutionoftheTOVEquation

Selectionofmass–radiuscurvesofcoldNSscontainingonlyn,p,eandμ inthecore.

TakenfromOertel etal2016

ThereisnogeneralconsensusoncompositionoftheNScore.

AtdensitiesatandbelowapproximatelythreetimesnuclearsaturationdensitythematterinacoldNSconsistsofnucleonsandleptonsinchemicalequilibrium.

Athigherdensities:Coldstars:hyperons appearnaturally(duetoPauliblocking),whentheirchemicalpotentialsexceedtheireffectivemassesandthestrangenessnon-conservingweakprocessesbecome possible.Hotstars:hyperons existatalldensities

Quarks?

Bosoncondensates

Mini-blackholes?

CommentsonEoS ofNScores(e.g.Oertel etal,Rev.Mod.Phys.89,015007(2017))

NuclearPhysicsinputtodensematterEoS:

fordetailseeJRS,Universe7,257(2021)andref.therein

Thefundamentalpropertiesoftheforces,actingbetweenbaryonsindensematterareunknown(nuclearmany-bodyproblem).Theeffectofmediumthe

interactionisnotunderstoodandmany-bodytechniqueshavelimitations.

I.Macroscopicmodels(SMF,liquid-drop)

vonWeizsacker,C.F.Zurtheoriederkernmassen.Z.FuerPhys.1935,96,431–458.

Bindingenergyperparticle(E/A)

fittedtoexperimentalnuclearmasses,(noexplicitknowledge

ofnuclearforcesneeded)-

avol volumetermasurf surfacetermac Coulombtermasym symmetryterm

ExtrapolationtoinfinitenumberofparticlesA– conceptofnuclearmatterand

itsdifferentstagesindependenceonN/Z(symmetric,asymmetric,pureneutron)BetheAnnu.Revs.Nucl.Sci.1971,21,93–244.

**MyersandSwiteckiAnn.Phys.1974,84,186.Molleretal.At.DataNucl.DataTables2016,109–110,1

IncombinationwithLiquid-dropmodels**

basicparametersofthesymmetricnuclearmatterformedasetofconstraints:

Atsaturationdensity*+ = +. -./012:Saturationenergy(E/A)=-16MeVSymmetryenergyJ

SlopeofJL

IncompressibilityK0 220– 315MeV(R=r0 A

1/3 r0 =1.2– 1.4fm fromexp)

Reed etal.,Phys.Rev.Lett. 126,172503,2021

Pbneutron skincorrelation with L

BlueSnneut.skinGreenHICRedGDRDarkIASWhitePb DP

Parityviolation

inelectronscattering

Saturationofnuclearforces:

TheAdependenceoftheindividualtermsisbasedontheassumptionthatthenucleusisaspherewithradiusR,containingcloselypackedsphericalnucleonswithradiir0

Experiment

InSNM(N=Z)B/Aisconstant=avolTheconstantdensityresultsfromabalancebetweenattractive andrepulsive components

ofthenuclearforcewhichareequilibratedatthatdensity– thesaturationdensity.

Atthisdensity,thenumberofsurrounding

nucleonsforeachnucleoninthematteris

thesame,regardlessofthepositioninspace.

Ifthenuclearforceisofashortrangecomparablewiththeinter-nucleondistance,eachnucleonwillinteractonlywithafewnucleonsinitsvicinity,resulting,onaverage,inthesamecontributiontothetotalbindingenergy

perparticle– thesaturationenergy.

Saturationpoint

II.Microscopicrealisticmodels:

MicroscopiccalculationoftheEoS ofhighdensitymatterrequirestheknowledgeofNNinteraction(potential)inmedium.

UseNNinteractioninfreespaceanduseamathematicaltechniquetoincludethemediumeffect

Potentials(10– 40parameters)perfectlyfitteddataonNN-scatteringphaseshifts

Soft-coreReid,Argonne+(non-relativistic),Nijmegen+,Bonn+(relativistic),………

Twoproblems:(i)ThebareNNinteractionswerenotadditiveinmany-bodysystems(as,forexample,theCoulombinteraction)

(ii)theywerephase-shiftequivalent– nouniqueidentificationpossible

Themediumeffects:Brueckner–Hartree–Fock (BHF),Dirac–Brueckner–Hartree–Fock (DBHF)

variational chainsummation,V_lowk,othermethodsincludinglimitationofhighmomentumcomponentsinthescatteringamplitudes

andrenormalizationschemes.

Results:Needforthree- (possibly)higherorderforcestoreproducecorrectly

thesaturationdensityandenergyofthesymmetricnuclearmatter.

BRIEF REPORTS PHYSICAL REVIEW C 74, 047304 (2006)

and single-particle energies in the Bethe-Goldstone equationhas been shown to introduce errors well below 1 MeV for thebinding energy at saturation [19].

Concerning the inclusion of three-body forces in the BHFapproach, we use the formalism developed in Refs. [5–7],namely a microscopic model based on meson exchange withintermediate excitation of nucleon resonances (Delta, Roper,and nucleon-antinucleon). The meson parameters in thismodel are constrained to be compatible with the two-nucleonpotential, where possible.

For the use in BHF calculations, this TBF is reduced toan effective, density-dependent, two-body force by averagingover the third nucleon in the medium, the average beingweighted by the BHF defect function g, which takes accountof the nucleon-nucleon in-medium correlations [6,8,20]:

Vij (r) = ρ

!d3rk

"

σk ,τk

[1 − g(rik)]2[1 − g(rjk)]2Vijk. (5)

The resulting effective two-nucleon potential has the operatorstructure

Vij (r) = (τ i ·τ j )(σ i ·σ j )V τσC (r) + (σ i ·σ j )V σ

C (r) + VC(r)

+ Sij (r)#(τ i ·τ j )V τ

T (r) + VT (r)$

(6)

and the components V τσC , V σ

C , VC, V τT , VT are density depen-

dent. They are added to the bare potential in the Bethe-Goldstone equation (1) and are recalculated together withthe defect function in every iteration step until convergenceis reached. This approach has so far been followed with theParis [6], the V14, and the V18 [7] potentials and the resultswill be shown in the following presentation of our results. Forcomplete details, the reader is refered to Refs. [5–7].

We begin in Fig. 1 with the saturation curves obtained withour set of NN potentials. On the standard BHF level (blackcurves) one obtains in general too strong binding, varyingbetween the results with the Paris, V18, and Bonn C potentials(less binding), and those with the Bonn A, N3LO, and IS(very strong binding). Including TBF (with the Paris, V14,and V18 potentials; red curves) adds considerable repulsionand yields results slightly less repulsive than the DBHF oneswith the Bonn potentials [16] (green curves). This is notsurprising, because it is well known that the major effect of theDBHF approach amounts to including the TBF correspondingto nucleon-antinucleon excitation by 2σ exchange within theBHF calculation [6,7]. This is illustrated for the case of the V18potential (open stars) by the dashed (red) curve in thefigure, which includes only the 2σ -exchange “Z-diagram”TBF contribution. The remaining TBF components are overallattractive and produce the final solid (red) curve in thefigure.

Figure 2 shows the saturation points of symmetric matterextracted from the previous results. Indeed there is a stronglinear correlation between saturation density and energy,confirming the concept of the Coester line. One can roughlyidentify three groups of results: The DBHF results with theBonn potentials as well as the BHF+TBF results with the Paris,V14, and V18 potentials lie in close vicinity of the empiricalvalue. The BHF results with Paris, V14, V18, and Bonn C forma group with about 1–2 MeV too-large binding and saturation

FIG. 1. (Color online) Energy per nucleon of symmetric nuclearmatter obtained with different potentials and theoretical approaches.For details see text.

at about 0.27 fm−3. The remaining potentials, in particular themost recent CD-Bonn, N3LO, and IS, yield strong overbindingat larger density, more than twice saturation density in thelatter cases. From a practical point of view, it would thereforeappear convenient to use the potentials of the former groupfor approximate many-body calculations, because the requiredcorrections are smaller, at least for Brueckner-type approaches.

Historically, there is the observation that the position ofa saturation point on the Coester line seems to be strongly

FIG. 2. (Color online) Saturation points obtained with differentpotentials and theoretical approaches. The (online blue) squareindicates the empirical region.

047304-2

BINDINGENERGYPER

PARTICLEINSYMMETRIC

NUCLEARMATTER

Lietal.,PRC74,047304(2006)

“Realistic”modelsobtained

fromfreenucleonscattering

andrenormalizedto

nuclearmedium.

NOTCALIBRATEDTO

SYMMETRICNUCLEAR

MATTERPARAMETERS.

E/AMeV

J.Carlsonetal.Rev.Mod.Phys.87,1065(2015)

Akmaleta

l.,PRC58,1084(1

998)

EoSofpureneutronmatterascalculatedindifferentmodels

Marisetal.,PRC87,054318(2013)

Gezerlis etal.,PRL111,032501(2013)

Effectof3BFQuMoCaw5678

50.0

A18

Ar18+9:

U14-DDI(FP)

A18+UIX

A18+9: + UIX

NUCLEARMATTERPROPERTIESFROMMEANFIELDMODELSWITHDENSITY

DEPENDENTEFFECTIVEINTERACTION:

Non-relativisticmodelsbasedontheSkyrme interaction- densitydependent

effectivenucleon-nucleonforcedependentonupto15adjustableparameters

nuclearmatter(noderivatives)finitenuclei

240setsoftheSkyrme parameterstested,5sets satisfiedalltheconstraints

Dutra,Lourenco,Martins,Delfino,JRS,Stevenson PRC85,035201(2012)

H = T + Ho + H3 + Heff

Ho =14 t0 (2+ x0 )ρ

2 − (2x0 +1)(ρ p2 + ρn

2 )⎡⎣ ⎤⎦H3 =

124 t3ρ

α (2+ x3)ρ2 − (2x3 +1)(ρ p

2 + ρn2 )⎡⎣ ⎤⎦

Heff =18 t1(2+ x1) + t2 (2+ x2 )⎡⎣ ⎤⎦τρ +

18 t2 (2x2 +1) − t1(2x1 +1)⎡⎣ ⎤⎦ τ pρ p +τ nρn( )

H fin =1

323t1 2 + x1( )− t2 2 + x2( )⎡⎣ ⎤⎦ ∇ρ( )2

− 132

3t1 2x 1+1( )− t2 2x2 +1( )⎡⎣ ⎤⎦ ∇ρp( )2+ ∇ρn( )2⎡

⎣⎢⎤⎦⎥

Hso =12W0 J.∇ρ + Jp.∇ρp + Jn.∇ρn⎡⎣ ⎤⎦

Hsg = − 116

t1x1 + t2x2( )J 2 + 116

t1 − t2( ) Jp2 + Jn2⎡⎣ ⎤⎦ + Hcoul + Hpair

Generalnon-linearfiniterangerelativisticmeanfieldmodelsbasedonmesonexchange(seerecentveryinformativeapplicationtoNS:Menezes,Universe7(8),267(2021))

Performanceof 263RMFmodelsweretestedagainst3slightlydifferentsetsofconstraintsofonpropertiesofnuclearmatterandthenumberofmodelsthatsatisfiedtheconstraints

SET1:2 modelsSET2a:4 models(30).SET2b:3 models(35)

Dutra,Lourenco,Avancini,Carlson,Delfino,Menezes,Providencia,Typel,JRS,PRC90,055203(2014)

Chiraleffectivefieldtheory

Weinberg,Phys.Lett.B1992,295,114–121.Phys.Lett.B1990,251,288–292

Hierarchy of chiral nuclear forces up to

N4LO.

The diagrams are organized according to

Weinberg power counting(1.10).

Dashed lines denote pions, solid lines

denote nucleons.

Solid dots, filled circles, filled rectangles,

filled diamonds, and open rectangles

depict vertices with ∆i=0,1,2,3,4,

respectively.

Blue-, orange-, and green-shaded

diagrams are available for calculations

Drichler,PhDThesis,2017

PhysicsofChiralEFTmodel:

• theχEFT usesnucleons,theirexcitations,andpions,insteadofquarksandgluons.

• systematiclow-momentumexpansionoflow- andmedium-rangeforcebetweennucleonsandpions,consistentwiththespontaneouslybrokenQCDchiralsymmetry.

• theshort-rangephysicsisparameterizedbycontacttermsconstrainedbyparity,time-reversalandtheusualconservationlaws,butNOT bychiralsymmetry.

• Thesetermshavetobefittedtodata.ThecontributionstotheχEFT expansionsareregulatedbyacutoffparameterthathastodeterminedfromacomparisonwithexperiment.

• modelshigh-densitymatteruptoapproximatelytwicenuclearsaturationdensity

Thepositiveaspectofthetheoryisthatitprovidesasystematichierarchyofnuclear

interactionsbyincludingthethree-bodyandhigher-orderforcesnaturallyonthesamefooting. Butitisagainaphenomenologicalmodel.

Bindingenergyperparticleforselectedlightnuclei B/A(E/A)forneutronandsymmetricNM

TakenfromJRSUniverse7,257(2021)

wheretheoriginalreferencescanbefound

FUNDAMENTALQUESTIONS:

1.Isthebaryonimmutable?

2.Whenimmersedtoanuclearmediumwithappliedscalarfieldwithstrengthoforderofhalfofitsmassisitreallyunchangeable?

3.Isthiseffectrelevanttonuclearstructure?

EnergyscalesinhadronandNuclei

Replaceinteractionbetweenbaryons

byinteractionbetweenvalencequarks

inindividual(non-overlapping)baryons

Lookforthemodificationofthequark

dynamicsinabaryonduetopresence

ofotherbaryonsto

ACCOUNTFORTHEMEDIUMEFFECT

History:Quark-MesonCouplingmodelOriginal:PierreGuichon(Saclay),PLB200,235(1988)SeveralvariantsdevelopedinJapan,Europe,Brazil,Korea,China

Latest:Guichon,JRSandThomas,Prog.Part.Nucl.Phys.100(2018)262–297

JRSetal.,Mon.Not.Royal.Astron.Soc.,(MNRAS)502,3476–3490(2021)

6

FIG. 10: Isosurface and surface plot of C(y) for a 10-sweepsmeared T-shape source with quark positions as in the seventhconfiguration of Table I. The maximum expulsion is 8.3% andthe isosurface is set to 4.4%. Further details are described inthe caption of Fig. 6.

FIG. 11: Isosurface and surface plot of C(y) for a 10-sweepsmeared Y-shape source with quark positions as in the seventhconfiguration of Table I. The maximum expulsion is 8.3% andthe isosurface is set to 4.4%. Further details are described inthe caption of Fig. 6.

FIG. 12: Isosurface and surface plot of C(y) for a 10-sweepsmeared L-shape source with quark separations of ℓ = 10.The maximum expulsion is 8.8% and the isosurface is set to4.4%. Further details are described in the caption of Fig. 6.

tive three-quark potential for the various quark positions,source shapes and Euclidean time evolutions. The vac-uum expectation value for W3Q is

⟨W3Q(τ)⟩ =∞!

n=0

Cn exp(−a Vn τ), (4)

where Vn is the potential energy of the n-th excited stateand Cn describes the overlap of the source with the n-th state. The effective potential is extracted from theWilson loop via the standard ratio

a V (r, τ) = ln

"

W3Q(r, τ)

W3Q(r, τ + 1)

#

. (5)

If the ground state is indeed dominant, plotting V as afunction of τ will show a plateau and any curvature canbe associated with excited state contributions. Statisticaluncertainties are estimated via the jackknife method [16].

Our results for the various quark positions and sourceshapes are shown in Fig. 16. All small shapes are stableagainst noise over a long period of time evolution andeven some of the largest shapes show some stability be-fore being lost into the noise.

Robust plateaus are revealed for the first four quarkpositions of Table I for the T and Y shape sources. Thissuggests the ground state has been isolated and indeedthe four lowest effective potentials of the T- and Y-shapesources agree. This result was foreseen in the qualita-tive analysis where Figs. 6 and 7 for the T- and Y-shapesources respectively displayed the same correlations be-tween the action density and the quark positions.

Conversely, the disagreement between Figs. 10 and 11indicates the ground state has not been isolated in oneor possibly both cases. Indeed the nontrivial slopes ofthe seventh effective potentials of Fig. 16 for the Y- andT-shape sources confirm this. On the other hand, thecurves are sufficiently flat to estimate an effective poten-tial at small values of τ , and given knowledge of the nodeposition from our qualitative analysis, one can make con-tact with models for the effective potential.

The expected r dependence of the baryonic potentialis [2, 4]

V3Q =3

2V0 −

1

2

!

j<k

g2CF

4πrjk+ σL , (6)

where CF = 4/3, σ is the string tension of the qq poten-tial and L is a length linking the quarks. There are twomodels which predominate the discussion of L; namelythe ∆ and Y ansatze.

In the ∆-ansatz, the potential is expressed by a sumof two body potentials [4]. In this case L = L∆/2 =3⟨dqq⟩/2 where L∆ is the sum of the inter-quark dis-tances. In the Y-ansatz [2, 6], L = LY = 3⟨rs⟩ is thesum of the distances of the quarks to the Fermat point.

LatticeQCDsimulationsofthestructureofanucleon

Schematicillustrationofamulti-baryonsystem:baryonscanappearcloseenoughto

toexchangemesonsbetweenquarks

throughdisturbanceintheQCDvacuum(Guichon)Bissey,PRD76,114512(2007)

Exchange

σ,ω,ρ

http://www.physics.adelaide.edu.au/theory/staff/leinweber/VisualQCD/Nobel/index.html

IllustrationofthemannerinwhichQCDvacuumfluctuationareexpelledfromthe

Interiorregionoftheproton.

Quarks– coloredspheres

Surfaceplot– reductionofthevacuum

actiondensityintheplane

throughthecentersofthe

quarks

Vectors– gradientofthereduction

Positionsinspacewherethefluctuationsthemaximallyexpelledfromtheproton–>fluxtubes

Startatabout0.5fm andremainapproximatelyconstanttomaximumseparationToexpelthefluctuationscostsenergy- >linearconfiningpotentialbetweenquarks

Guichon,JRS,ThomasProg.Part.Nucl.Phys.2018JRS, https://doi.org/10.3390/universe7080257

Baryon(MITbag)witharadiusRBinameandensitydependentscalarfieldApproximatethesolutionofthebagequationsbyadynamicalnucleoneffectivemass

<=($) = <= - ?=@AB$ +C)?=D @AB$ )

Thelasttermrepresentstheresponseofanucleontothescalarfieldwithbeingthescalarpolarizability–theoriginofMANY-BODYforcesinQMC(noadditionalparameter–discalculatedintermsofthebagradius)

OUTLINEOFTHEQMCMODEL

Guichonetal.,Nucl.Phys.A772,1,(2006),GuichonandThomasPRL93,132502(2004)

Guichon,Stone,Thomas:ProgressinParticleandNuclearPhysics100(2018)262–297

gσN = 3gσ

q drBag∫ qq(r ), gωN = 3gω

q , gρN = gρq

Thequark-mesoncouplingsarerelatedtothenucleoncouplingstos, w andr mesonsinfreespace

ThescalarcouplinggsN isdensitydependentanddecreaseswithincreasingdensitySolveself-consistentlyforthemesonfieldsusingthecondition

∂E / ∂σ = 0 ∂E / ∂ω = 0 ∂E / ∂ρ = 0

ConstructaquantizedHamiltonian/Lagrangian foragivensystem(non-rel)forfinitenucleiandrelativisticfornuclearmatter.ThesedependonlyonnucleondynamicsandaresolvedbystandardHartree-Fock methodstodetermineobservablesofinterest.

-densitydependenceoftheEDFismicroscopicallycalculated

- multi-bodyforcesareautomaticallyincluded

-exchangetermsarealwaysincluded

- heavybaryonscanbeincludedwithoutincreaseoftheNoofparameters

- spin-orbittermappearsnaturallyinbothNRandRmodels

- protonandneutrons.p.potentialsarecalculated– noneedforfitting.

Parameters(verylittlemaneuveringspace):

I.3nucleon-mesoncouplingconstantsinvacuumWedefine(forconvenience)

gσN ,gωN ,gρN

GσN = gσN2 /mσ

2 GωN = gωN2 /mω

2 GρN = gρN2 /mρ

2

II.Mesonmasses:ω,ρ,πkeeptheirphysicalvalues650MeV<Mσ <700MeV

III.Bagradius(freenucleonradius):1fm(limitedsensitivitywithinchange+/- 20%)

Allotherparameterseithercalculatedwithinthemodelorfixedbysymmetry

QMC-Ain2020:Coldandhothyperonic matterinQMC-AmodelMNRAS5023476(2021)

Chiralrelativisticmean-fieldmodel(CMF)– Dexheimer K=300MeVRMFmodelwithdensitydependentcouplings(DD2/DD2-T)– Typel K=243MeVQMC-A- JRS,Guichon,ThomasK=292MeV

2Msolar

1.4Msolar

Ovals:ConstraintsonradiifromNICERmissionRiley2019Miller2019

Populationinunitsofn0EoS

ColdNeutronStars

MvsR

FixedYL =0.4S/A=1 Equil – S/A=2

HOTNEUTRONSTARS

WarmNSdonothavewelldefinedsurface

ThereisNOdensitythresholdforappearanceofhyperons

Searchforsurface– solidlinesnpYdashedlinenp

QMC-A

CMF

DD2

11 11.5 12 12.5 13 13.5 14R [km]

0

0.5

1

1.5

2

2.5

3

M /M

sola

r

2.08+/-0.7 Msolar

1.44+/-015 Msolar

M.C.Miller et al, arXiv:2105.06997PSR J0740+6620 from NICER and XMM Newton Data

PSR J0039+0456

PSR J0740=6620

LatestcoldneutronstarQMCmodelJune2021npY (includingoverlap)

K=301MeV

L=543

nc =2.4n0

nc=5.1n0

Finitenuclei:

Non-relativistic extensionoftherelativisticQMCmodelFullyself-consistentHartree-Fock +BCSapproximationExtensivepredictionofgroundstatepropertiesofeven-evennucleifrom16Oto270Ds.

2016:PRL118,092501QMC

2019:PRC100,024333QMC-pII

2020:PRC102,034304QMC-pIII andpIII-T

Calculation:bindingenergies,shapes,chargeandmatterdistributions,neutronskin,spectraofsingle-particleenergies,two-particleseparationenergies,shellgaps,a-decayQ-values,GMRenergies,even-evensuperheavy nuclei

Comparedwith:availableexperimentaldata,mean-fieldSkyrme andmacro-micromodels

Conclusion: Excellentagreementwithdata,comparablewithpredictionsofothermodelsbutwithsignificantlysmallernumberofvariablewellconstrainedparametersandnolocaladjustment.

Hypernuclei inQMC

Guichon etal.Nucl.Phys.A81466(2008)KazuoTsushima

PredictedboundΞ-hypernuclei butunboundΣ-hypernuclei

TosummarizeLectureII,theexamplesshownhereofvariousattemptsusedtounderstandtheeffectofnuclearmediumonthebareNNinteractionhavenotreachedthe“theholygrail’asyettoasatisfactoryconclusion.

Themodelsdonothaveenoughsensitivitytointerprettheirdifferencesinrelationtothephysicstheyarebasedon.

IstheQMCshowingthewayforward?

SummaryofLectureII

LectureIII

The heavy ion collisions provide another data on densematter. Is it useful for neutron stars?

The neutron star merger (BNSM) and the relatedgravitational waves will be the subject of the final, thirdlecture. This topic is currently most actively explored, usingnovel frameworks of multi-messenger techniques.Advantages and disadvantages of this trend will bediscussed.

HeavyIonCollisions

Measurement: Beamenergy35AMeV– 5.5ATeVCollisions(Au,Au),(Sn,Sn),(Cu,Cu)butalso(p,p)foracomparisonTransverseandEllipticalparticleflow

Calculation:Transportmodels-- empiricalmeanfieldpotentialsFittodataà energydensityà P(ε)à theEoS(extrapolationtoequilibrium,zerotemperature,infinitematter)(e.g Danielewicz etal.,Science298,2002,Bao-AnLietal.,Phys.Rep.464,2008)

QuantumMolecularDynamics(e.g.Yingxun Zhang,Zhuxia Li,AkiraOno)

HeavyIoncollisions:

GSI,MSU,TexasA&M,RHIC,LHCexistingFAIR(GSI),NICA(Dubna,Russia)planned

er deflections.) The open and solid points inFig. 2 show measured values for the directedtransverse flow in collisions of 197Au projec-tile and target nuclei at incident kinetic ener-gies Ebeam/A, ranging from about 0.15 to 10GeV per nucleon (29.6 to 1970 GeV totalbeam kinetic energies) and at impact param-eters of b ! 5 to 7 fm (5 " 10#13 to 7 "10#13 cm) (13–16). The scale at the top ofthis figure provides theoretical estimates forthe maximum densities achieved at selectedincident energies. The maximum density in-creases with incident energy; the flow dataare most strongly influenced by pressurescorresponding to densities that are somewhatless than these maximum values.

The data in Fig. 2 display a broad maxi-mum centered at an incident energy of about2 GeV per nucleon. The short dashed curvelabeled “cascade” shows results for the trans-verse flow predicted by Eq. 1, in which themean field is neglected. The disagreement ofthis curve with the data shows that a repulsivemean field at high density is needed to repro-duce these experimental results. The othercurves correspond to predictions using Eq. 1and mean field potentials of the form

U ! $a% " b%&)/[1'(0.4%/%0)&–1] ' (Up

(5)

Here, the constants a, b, and & are chosen toreproduce the binding energy and the satura-tion density of normal nuclear matter whileproviding different dependencies on densityat much higher density values, and (Up de-scribes the momentum dependence of themean field potential (28, 33, 34) (see SOMtext). These curves are labeled by the curva-

ture K § 9 dp/d%)s/% of each EOS about thesaturation density %0. Calculations with largervalues of K, for the mean fields above, gen-erate larger transverse flows, because thosemean fields generate higher pressures at highdensity. The precise values for the pressure athigh density depend on the exact form chosenfor U. To illustrate the dependence of pres-sure on K for these EOSs, we show thepressure for zero temperature symmetricmatter predicted by the EOSs with K ! 210and 300 MeV in Fig. 3. The EOS with K !300 MeV generates about 60% more pres-sure than the one with K ! 210 MeV atdensities of 2 to 5 %0 (Fig. 3).

Complementary information can be ob-tained from the elliptic flow or azimuthalanisotropy (in-plane versus out-of-planeemission) for protons (24, 25, 36). This isquantified by measuring the average value*cos2+,, where + is the azimuthal angle ofthe proton momentum relative to the x axisdefined in Fig. 1. (Here, tan+ ! py/px , wherepx and py are the in-plane and out-of-planecomponents of the momentum perpendicularto the beam.) Experimental determinations of*cos2+, include particles that, in the cen-ter-of-mass frame, have small values for therapidity y and move mainly in directionsperpendicular to the beam axis. Negative val-ues for *cos2+, indicate that more protonsare emitted out of plane (+ - 90°or + -270°) than in plane (+ - 0°or + - 180°), andpositive values for *cos2+, indicate thereverse situation.

Experimental values for *cos2+, for in-cident kinetic energies Ebeam/A ranging from0.4 to 10 GeV per nucleon (78.8 to 1970 GeVtotal beam kinetic energies) and impact pa-rameters of b ! 5 to 7 fm (5 x 10#13 to 7 "10#13 cm) (17–19) are shown in Fig. 4. Neg-ative values for *cos2+,, reflecting a pref-erential out-of-plane emission, are observedat energies below 4 GeV/A, indicating thatthe compressed region expands while the

spectator matter is present and blocks thein-plane emission. Positive values for*cos2+,, reflecting a preferential in-planeemission, are observed at higher incident en-ergies, indicating that the expansion occursafter the spectator matter has passed the com-pressed zone. The curves in Fig. 4 indicatepredictions for several different EOSs. Cal-culations without a mean field, labeled “cas-cade,” provide the most positive values for*cos2+,. More repulsive, higher-pressureEOSs with larger values of K provide morenegative values for *cos2+, at incident en-ergies below 5 GeV per nucleon, reflecting afaster expansion and more blocking by thespectator matter while it is present.

Transverse and elliptic flows are also in-fluenced by the momentum dependencies(Up of the nuclear mean fields and the scat-tering by the residual interaction within thecollision term I indicated in Eq. 1. Experi-mental observables such as the values for*cos2+, measured for peripheral collisions,where matter is compressed only weakly andis far from equilibrated (28), now providesignificant constraints on the momentum de-pendence of the mean fields (21, 28). This isdiscussed further in the SOM (see SOM text).The available data (30) constrain the mean-field momentum dependence up to a densityof about 2 %0. For the calculated resultsshown in Figs. 2 to 4, we use the momentumdependence characterized by an effectivemass m* ! 0.7 mN, where mN is the freenucleon mass, and we extrapolate this depen-dence to still higher densities. We also makedensity-dependent in-medium modificationsto the free nucleon cross-sections followingDanielewicz (28, 32) and constrain these

Fig. 2. Transverse flow results. The solid andopen points show experimental values for thetransverse flow as a function of the incidentenergy per nucleon. The labels “Plastic Ball,”“EOS,” “E877,” and “E895” denote data takenfrom Gustafsson et al. (13), Partlan et al. (14),Barrette et al. (15), and Liu et al. (16), respec-tively. The various lines are the transport the-ory predictions for the transverse flow dis-cussed in the text. %max is the typical maximumdensity achieved in simulations at the respec-tive energy.

Fig. 3. Zero-temperature EOS for symmetricnuclear matter. The shaded region correspondsto the region of pressures consistent with theexperimental flow data. The various curves andlines show predictions for different symmetricmatter EOSs discussed in the text.

Fig. 4. Elliptical flow results. The solid and openpoints show experimental values for the ellip-tical flow as a function of the incident energyper nucleon. The labels “Plastic Ball,” “EOS,”“E895,” and “E877” denote the data of Gutbrodet al. (17), Pinkenburg et al. (18), Pinkenburg etal. (18), and Braun-Munzinger and Stachel (19),respectively. The various lines are the transporttheory predictions for the elliptical flow dis-cussed in the text.

R E S E A R C H A R T I C L E S

22 NOVEMBER 2002 VOL 298 SCIENCE www.sciencemag.org1594

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A.Fire,Y.Kohara,and

M.Driscollforreagents

andstrains;

J.Parrish

fordiscussion;

andY.C.Wu,B.

Boswell,andX.D.Liuforhelpful

comments

onthis

manuscript.This

workwassupported

inpartbySearle

ScholarAwards

(D.X.andY.S.),

aRita

AllenScholar

Award(Y.S.),

theBurroughs

Wellcom

eFund

CareerAward(D.X.),and

grantsfrom

theU.S.Departm

entof

Defense

(D.X.)andNIH(D.X.and

Y.S.).

SupportingOnline

Material

www.sciencem

ag.org/cgi/content/full/298/5598/1587/DC1

Materials

andMethods

ReferencesandNotes

Figs.S1andS2

16July2002;accepted

27Septem

ber2002

Determ

inationof

theEquation

ofState

ofDense

Matter

Paweł

Danielew

icz, 1,2Roy

Lacey, 3William

G.Lynch

1*

Nuclear

collisionscancompress

nuclearmatter

todensities

achievedwithin

neutronstars

andwithin

core-collapsesupernovae.

Thesedense

statesof

matter

existmomentarily

beforeexpanding.W

eanalyzed

theflowofmatter

toextract

pressuresinexcess

of1034pascals,

thehighest

recordedunder

laboratory-controlledconditions.

Using

theseanalyses,

weruleoutstrongly

repulsivenuclear

equationsofstate

fromrelativistic

meanfieldtheory

andweakly

repulsiveequations

ofstatewithphase

transitionsatdensities

lessthan

threetimesthat

ofstable

nuclei,butnotequations

ofstate

softenedathigher

densitiesbecause

ofatransform

ationtoquark

matter.

The

nucleon-nucleoninteraction

isgenerally

attractiveat

nucleon-nucleonseparations

of(r)

!1

to2

fm(1

"10

#13

cmto

2"

10#

13

cm)

butbecom

esrepulsive

atsm

allsepara-

tions($

0.5fm

),m

akingnuclear

matter

difficulttocom

press.As

aconsequence,m

oststable

nucleiare

atapproxim

atelythe

same

“saturation”density,%

0&

2.7"

1014

g/cm3,

intheir

interiors,andhigher

densitiesdo

notoccur

naturallyon

Earth.

Matter

atdensities

ofup

to%

!9

%0

may

bepresent

inthe

interiorsof

neutronstars

(1),and

matter

atdensities

upto

about%!

4%

0m

aybe

presentin

thecore

collapseof

typeII

supernovae(2).

The

relationshipbetw

eenpressure,

density,and

temperature

describedby

theequation

ofstate

(EO

S)of

densem

attergoverns

thecom

-pression

achievedin

supernovaeand

neutronstars,

asw

ellas

theirinternal

structureand

many

otherbasic

properties(1–5).

Models

thatextrapolate

theE

OS

fromthe

propertiesof

nucleinear

theirnorm

aldensity

andfrom

nucleon-nucleonscattering

arecom

monly

ex-ploited

tostudy

suchdense

systems

(1,3–9).C

onsequently,it

isim

portantto

testthese

extrapolationsw

ithlaboratory

measurem

entsof

high-densitym

atter.

Nuclear

collisionsprovide

theonly

means

tocom

pressnuclear

matter

tohigh

densityw

ithina

laboratoryenvironm

ent.T

hepres-

suresthat

resultfrom

thehigh

densitiesachieved

duringsuch

collisionsstrongly

in-fluence

them

otionof

ejectedm

atterand

pro-vide

thesensitivity

tothe

EO

Sthatis

neededfor

itsdeterm

ination(10–19).

Fullequilibri-

umis

oftennotachieved

innuclearcollisions.

Therefore,itis

necessaryto

studyexperim

en-tal

observablesthat

areassociated

with

them

otionsof

theejected

matter

andto

describethem

theoreticallyw

itha

dynamical

theory(20–27

).T

orelate

theexperim

entalobservables

tothe

EO

Sand

theotherm

icroscopicsources

ofpressure,w

eapply

am

odelformulated

within

relativisticL

andautheory,

which

includesboth

stableand

excited(delta,

N*)

nucleons(thatis,baryons)

asw

ellaspions

(20,28).Itdescribes

them

otionof

theseparticles

bypredicting

thetim

eevolution

ofthe

(Wigner)

one-bodyphase

spacedistribution

functionsf(r,p,t)

forthese

particles,using

aset

ofB

oltzmann

equationsof

theform

'f't!

()p !*

"()

r f*#

()r !*

"()

pf*$

I

(1)

Inthis

expression,f(r,p,t)

canbe

viewed

semi-classically

asthe

probabilityof

findinga

particle,at

time

t,w

ithm

omentum

pat

positionr.

The

single-particleenergies

!in

Eq.

1are

givenin

alocal

frame

by

!$

KE

+U

(2)

where

KE

isthe

kineticenergy

andU

isthe

average(m

eanfield)

potential,w

hichde-

pendson

theposition

andthe

mom

entumof

theparticle

andis

computed

self-consistentlyusing

thedistribution

functionsf(r,p,t)

thatsatisfy

Eq.1

(20,28).The

particledensity

is%(r,t)

!,

dp"

f(r,p,t);theenergy

densitye

canbe

similarly

computed

from!

andf(r,p,t)

bycarefully

avoidingan

overcountingof

po-tential

energycontributions.

The

collisionintegral

Ion

theright-hand

sideof

Eq.

1governs

them

odificationsof

f(r,p,t)by

elasticand

inelastictw

o-bodycol-

lisionscaused

byshort-range

residualinter-

actions(20,

28).T

hem

otionsof

particlesreflect

acom

plexinterplay

between

suchcollisions

andthe

densityand

mom

entumdependence

ofthe

mean

fields.Experim

entalm

easurements

(12–19,29–31),theoreticalin-novations,

anddetailed

analyses(10,

20–29,32–34

)have

allprovided

important

insightsinto

thesensitivity

ofvarious

observablesto

two-body

collisions(29,

32)and

thedensity

andm

omentum

dependence(28,

33,34

)of

them

eanfields.

The

presentw

orkbuilds

onthese

earlierpioneering

efforts.Compression

andexpansion

dynamics

inenergetic

nucleus-nucleuscollisions.

Collision

dynamics

playan

important

rolein

studiesof

theE

OS.

Severalaspects

ofthese

dynamics

areillustrated

inFig.1

fora

colli-sion

between

two

Au

nucleiat

anincident

kineticenergy

of2

GeV

pernucleon

(394G

eV).

The

observablessensitive

tothe

EO

Sare

chieflyrelated

tothe

flowof

particlesfrom

thehigh-density

regionin

directionsperpendicular

(transverse)to

thebeam

axis.T

hisflow

isinitially

zerobutgrow

sw

ithtim

eas

thedensity

grows

andpressure

gradientsdevelop

indirections

transverseto

thebeam

axis.T

hepressure

canbe

calculatedin

theequilibrium

limit

bytaking

thepartial

deriv-ative

ofthe

energydensity

ew

ithrespect

tothe

baryon(prim

arilynucleon)

density%

P$

%2

"! '(e/%*"'% # $

s/%(3)

atconstant

entropyper

nucleons/%

inthe

collidingsystem

.T

hepressure

developedin

thesim

ulatedcollisions

(Fig.1)is

computed

microscopically

fromthe

pressure-stressten-

sorT

ij,which

isthe

nonequilibriumanalog

ofthe

pressure[see

supportingonline

material

1NationalSuperconducting

Cyclotron

Laboratoryand

Departm

entofPhysics

andAstronom

y,Michigan

StateUniversity,East

Lansing,MI48824

–1321,USA.

2Gesellschaft

furSchw

erionenforschung,64291

Darmstadt,

Germany.

3Departm

entofChemistry,

StateUniversity

ofNewYork,

StonyBrook,

NY

11794–3400,U

SA.

*Towhom

correspondenceshould

beaddressed.

E-mail:lynch@

nscl.msu.edu

RESEA

RCH

ARTICLES

22NOVEMBER2002

VOL298

SCIENCEwww.sciencem

ag.org1592

on March 17, 2012www.sciencemag.orgDownloaded from

Transportmodelswithparametersfittedtodataonellipticalandtransverseflow

Science298,1592(2

002)

Wethereforeconcludethatdatafromheavy-ioncollisionsinthisregimecannotdirectlyinferconstraintsonpropertiesofcompactobjects.Althoughtechnicallypossible,suchprocesscannotbejustifiedatthefundamentallevel.

Gravitationalwaves(observation):

TherearethreeGWeventsinvolvingNSreportedtodate.GW170817chirpmass1.186(1)M�,massratioq�[1,1.34],reducedtidalparameter.Λ�300andsmallerthan�800.gammarayburstGRB170817,1.7safterthecoalescence,opticalsignalAT2017gfo(kilonova)0.47–18.5daysaftertheevent.

GW190814 coalesce of two objects, 23.2 and 2.59 Msolarchirp mass 6.09 Msolarmass ratio q = 0.112no elmg couterparts

GW19025 twoneutronstarswithmassesrangingfrom1.12to2.52M� (spindependence)chirpmassandthetotalmassofthissystemarelargerthananypreviouslyknownbinaryNSsystem.Thus,apossibilitythatoneorbothcomponentsarelightblackholescannotberuledoutfromGWobservation.Noelmg counterparts

Dataanalysis:

Theinterpretationofgravitational-waveeventsandtheirelmg counterpartscruciallyreliesongeneral-relativisticmodelsofthemergerremnants.Quantitativemodelscanbeobtainedonlybymeansofnumericalrelativitysimulationsin3+1dimensionsincludingdetailedinputphysicsforthenuclearmatter,electromagneticandweakinteractions.Bernuzzi GeneralRelativityandGravitation(2020)52:108,PRL115,091101(2015)

Statisticalanalysisinthemulti-messengerera

ThefrustrationwithmanymodelsappliedtofewpiecesofdataledthefieldtoturnawayfromindividualmodelstoastatisticalapproachbasedonBaysiean theoremandrelatedtechniques.

ThomasBayes,1763

Conditionalprobability

Complicatedanalysis,necessaryshortcuts:

RealisticEoS replacedbypolytrophic,spectralorparameterizedEoS

Universal(EoS insensitive)relationbetweenobservables

Bernuzzi 2020: PhenomenologicalEOS-insensitiverelationbetweentheGW’s mainpeakpostmerger frequencyandthe(modified)tidalparameterξ(κ2T,ν)

ConstraintsontheEoS:S=32± 2MeV;themassesofthethreemostmassiveneutronstars;thetidaldeformabilityofGW170817;a hypothetical(M,R)=(1.4Me,12km)to5%precision;ahypotheticalmomentofinertiaofa1.338Mestarto10%precision;Ahypotheticalknowledgeofthebaryonicrestmassofastarto0.005Msolar.Precision;

ExampleofdeterminationoftheEoS inthepracticalBayesianapproach

M.C.Miller et al:The Astrophysical Journal, 888:12,2020 January 1

Al-Mamun etal,2021

Prior:2EoS – 3polytropesor4linessegments

DatafromLMXBDatafromNICERDatafromLIGO

Polytrop:P=KE(GHI)/GKconstantn– polytropic index

Summary: Focusingonthenuclearforceactingindensenuclearmatteranditsequationofstate,wediscussedseveralsnapshotsofevolutionofthetheoryofnuclearforces.

Startingfromoriginalideasinthe1930swemovedtoitsoverwhelmingdiversitytoday,supportedbymodernobservationalandterrestrialdatainthemulti-messengerera,aswellasbynewmathematicaltechniquesandcomputerpower.

Despitetheadmirableeffortbothintheoryandmeasurement,multiplemodelsdependentonalargenumberofcorrelatedparametersexist,whichcannotbeconstrainedbydata,notyetaccuratenorsensitiveenoughtoidentifythetheoryclosesttoreality.

Theroleofmicrophysicsinthetheoriesisseverelylimitedorneglected,mostlydeemedtobetoodifficulttotackle.Novelapproaches,basedonmorefundamentalideasandlessparameters,shouldbedevelopedtomakeprogress.

Back-upslides

CourtesyJamesZanotti

TWOPROBLEMS:THEEXPONENTIALWEIGHTINGFACTOROSCILLATESATFINITECHEMICALPOTENTIALS

OVERLAPPROBLEM(PLEASEGOOGLEFORDETAILS)

neutron stars: the journey from birth to death. - in2p3

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