Crater II with tidally stretched wave dark matter

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Understanding the ‘feeble giant’ Crater II with tidally stretched wave dark matter

Alvaro Pozo,

¹^,²^‹

Tom Broadhurst,

¹^,²^,³^‹

Razieh Emami

⁴

and George Smoot

²^,⁵^,⁶^,⁷^,⁸^‹

1DepartmentofPhysics,UniversityoftheBasqueCountryUPV/EHU,E-48080Bilbao,Spain

2DIPC,BasqueCountryUPV/EHU,E-48080SanSebastian,Spain

3Ikerbasque,BasqueFoundationforScience,E-48011Bilbao,Spain

4CenterforAstrophysics|Harvard&Smithsonian,60GardenStreet,Cambridge,MA02138,USA

5InstituteforAdvancedStudyandDepartmentofPhysics,IASTT&WFChaoFoundationProfessor,HongKongUniversityofScienceandTechnology,Clear WaterBay,Kowloon,999077HongKong

6EnergeticCosmosLaboratory,NazarbayevUniversity,53,KabanbaybatyrAve.,Nur-Sultancity,010000,RepublicofKazakhstan

7PhysicsDepartment,UniversityofCaliforniaatBerkeleyCA94720,Emeritus,USA

8ParisCentreforCosmologicalPhysics,APC,AstroParticuleetCosmologie,UniversitéParisDiderot,CNRS/IN2P3,CEA/lrfu,\\UniversitéSorbonneParis Cité,10,rueAliceDomonetLeonieDuquet,75205ParisCEDEX13,France

Accepted2022June29.Received2022June29;inoriginalform2022January10

ABSTRACT

Theunusuallylarge‘dwarf’galaxyCraterII,withitssmallvelocitydispersion,3kms⁻¹,defiesexpectationsthatlow-mass galaxiesshouldbesmallanddense.WecombinethelateststellarandvelocitydispersionprofilesfindingCraterIIhasaprominent darkcoreofradius0.71⁺₋⁰₀^._.⁰⁹₀₈kpc,surroundedbyalowdensityhalo,withatransitionvisiblebetweenthecoreandthehalo.We showthatthisprofilematchesthedistinctivecore-haloprofilepredictedby‘WaveDarkMatter’asaBose-Einsteincondensate, ψDM,wherethegroundstatesolitoncoreissurroundedbyatenuoushaloofinterferingwaves,withamarkeddensitytransition predictedbetweenthecoreandhalo. Similarcore-halo structureisseeninmostdwarfspheroidalgalaxies(dSphs),butwith smallercores,0.25kpcandhighervelocitydispersions,9kms⁻¹,andweargueherethatCraterIImayhavebeenatypical dSphthathaslostmostofitshalomasstotidalstripping,soitsvelocitydispersionislowerbyafactorof3andthesolitonis widerbyafactorof3,followingtheinversescalingrequiredbytheUncertaintyPrinciple.ThistidalsolutionforCraterIIin thecontextofψDMissupportedbyitssmallpericenterof20kpcestablishedbyGaia,implyingsignificanttidalstrippingof CraterIIbytheMilkyWayisexpected.

Keywords: DarkMatter– Galaxy:kinematicsanddynamics.

1 I N T R O D U C T I O N

Asurprisingdiversityofdwarfgalaxieshasbeensteadilyuncovered over the pastdecade,with many‘ultra-faint dwarfs’(Read etal.

2006;Koposovetal.2015;Munozetal.2018;Mutlu-Pakdiletal.

2018;Moskowitz&Walker2020)thataremuchsmalleranddenser thanthewellstudiedclassofdwarfspheroidalgalaxies(dSph),and other‘ghostly’dwarfsofverylowsurfacebrightnessthatarelarge andlowerindensity.Theaptlynamed‘feeblegiant’,CraterII,has beenparticularly puzzling,withitsdwarf-likevelocitydispersion of only3kms⁻¹(Torrealbaetal. 2016; Caldwellet al. 2017) andlargesize,over2kpc,whichstrainsthedwarfdefinitionin termsofsize.Despitethelow-velocitydispersion,themass-to-light ratioofCraterIIislarge,M/L30(Caldwelletal.2017;Sanders, Evans&Dehnen2018;Jietal.2021)andappearstohaveashallow, coredprofile(Sandersetal.2018;Jietal.2021) and anold,but notancientstellarpopulation,datedto10Gyrs(Torrealbaetal.

2016;Caldwelletal.2017),resemblingintheserespectsthecommon classofdSphforwhichdarkmatter-dominatedcoresarecommonly

E-mail:[email protected](AP);

[email protected](TB);[email protected](GS)

claimed.However,boththestellarsurface-brightnessandthevelocity dispersionofCraterIIaremuchlowerthanthetypicaldSphdwarfs.

Recently, it has becomeclear that the darkcores of the dSph galaxies match well an essential prediction for dark matter as a Bose-Einsteincondensate,forwhichastandingwave,solitonforms a prominent core in every galaxy (Schive, Chiueh& Broadhurst 2014a). The first cosmological simulations in this context have revealedpervasive wavestructureon thedeBrogliescale,termed ψDM,(Schiveetal.2014a,b;Schwabe,Niemeyer&Engels2016; Huietal.2017;Moczetal.2017).Thisincludesastablesolitoncore formedatthe centreofeachgalaxy,correspondingto theground state(Schiveetal.2014a),surroundedbyahaloofexcitedstates.

Abosonmassof10⁻²²eVprovidescoresof0.5kpctypicalof thedwarfspheroidal(Schiveetal.2014b;deMartinoetal.2018; Pozoetal.2020),wherethewidthofthesolitoncoreissetsimply bythedeBrogliewavelengthandisseentovarybetweengalaxiesin thesimulations,beinglargerforlower-massgalaxies,reflectingtheir lowermomentum.Here,thelocaldeBrogliescaleisλ_dB=h/(m_ψσ), whereσisthelocalvelocitydispersionthatsetsthemomentumscale togetherwiththebosonmassmψ.

ThisisascalarfieldinterpretationofDMdescribedsimplybya coupledSchrodinger-Poissonequationforthemeanfieldbehaviour thatevolvesonlyunderself-gravity,forwhichthebosonmassisthe

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Crater II tidal waves 2625

onlyfreeparameter.Themostuniquepredictionisthataprominent standingwave comprisesthe ground stateat the centre of every galaxy,surroundedbyanextensivehaloofslowlyvaryingdensity wavesthatfullymodulatethehalodensityonthedeBrogliescaleby self-interference(Schiveetal.2014a,b;Moczetal.2017;Veltmaat, Schwabe & Niemeyer 2019; Hui et al. 2021). Condensates are inherentlynon-relativistic,andhenceψDMisviableasdarkmatter, behavingas ColdDarkMatter(CDM)onlargescales, exceeding the de Broglie wavelength, as demonstrated by the pioneering simulations of Schive et al. (2014a). This ψDM interpretation makesseveraluniquepredictions,includingacoreprofilewiththe uniquesolitonprofile(seeSection2)andamoresignificantsoliton radius,r_sol,inlowermassgalaxiesm_gal,oflowermomentum,asa consequencethatthedeBrogliewavelengthsetsthesolitonradius.

Followingthisscalingrelation:r_sol∝m⁻_gal¹^/³foundbySchiveetal.

(2014b) and verified in independent simulations (Schwabe et al.

2016;Niemeyer2020).

TheeffectoftidalstrippinginψDMhasbeenexaminedrecently bySchive,Chiueh&Broadhurst(2020),Duetal.(2018)demonstrat- ingtheresilienceofthesoliton,whichisself-reinforcingsothathalo isstrippedfirst,withanabruptdisruptionofthesolitonpredictedto followifsignificanttidalstrippingofthesolitonfollows(Duetal.

2018).Asteadylossofthehaloisfound intheψDMsimulation ofSchiveetal.(2020),fora typicaldwarf of10⁹M,following acircularorbitof200kpcinaMilkyWaysizedhalo,showing thatthehalodensitydropsrelativetothesolitonoverseveralGyrs withlittlechangeintheslopeofthedensityprofileofthehalo.This behaviourhasbeenrecentlyclaimedtobrackettherangeofprofiles foundfordSphs,whichareobservedtofollowthepredictedform withaclearcoreandhalostructureseeninallthestellarprofilesof theclassicaldSphs(Pozoetal.2020),andthisisalsosupportedby thegenerallylowerlevelofthevelocitydispersionseeninthehalo relativetothecoreaspredictedforψDM(Pozoetal.2020).This phenomenonisreinforcedbySchiveetal.(2020)ashealreadyfound thattidalstrippingincreasesthedensitycontrastbetweenthecore andthehalo,asthehaloisrelativelyeasilystrippedcomparedtothe core.Asmassoutsidethetidalradiusisstripped,thecoreshould relax,becominglessmassiveandhencemoreextended,obeyingthe uncertainty principle.With increasedstripping suchthatthe tidal radiusbecomessimilarin sizeto thecoreradius,thecorewillbe disrupted,perhapsrather abruptly,leaving justextended tails(Du etal.2018).ArecentdedicatedψDMsimulationthatexploresthe propertiesoftheEridanusIIdwarfshowsthatundersteadymodest stripping,thecoreremainsstableasthehaloisreducedindensity, leadingto anenhanced contrastover timebetween the relatively densecoreandthe tenuoussurroundinghalo(Schiveetal.2020), andtherangeofcore-to-halovariationpredictedbythissimulation hasbeenshowntomatchwellthefamilyofprofilesofdSphgalaxies, indicatingmostarestrippedatsomelevel,relativetothemoredistant

‘isolated’dSphs(Pozoetal.2020).

Here,wedeterminewhetherthispredictedmomentumdependence ofψDMcanaccountfortherelativelyunusualpropertiesofCrater II, giventhat itsrelatively widecore and low-velocity dispersion relative to the classical dSphs seems to support this possibility qualitatively.Similarly,ithasbeenproposedthatanultralightboson of m_ψc²= (0.6 − 1.4) ×10⁻²²eVaccounts for the largesize and modest velocity dispersion stars within Antlia II, consistent withbosonmassestimates formoremassivedwarf galaxieswith smallerdarkcores(Schiveetal.2014a)andplacesAntliaIIclose to the lower limitingJeans scale for galaxyformation permitted by theUncertainty Principleforthisverylightbosonmass. New spectroscopy data have revealed that Antlia II has a systematic

velocitygradientthatiscomparablewithitsvelocitydispersion(Ji etal.2021)indicatingthatithassufferedtidalelongationorpossibly rotating,atleastintheoutskirtswherethesignchangeofthevelocity gradientisapparent(Jietal.2021).ThetidalfieldofCraterIIisalso thoughttohavehadasignificanteffectgiventhesmallpericenterthat isnowestablishedforCraterIIinthelatest’Gaia’basedanalysis (Ji etal.2021),though itdoesnotshowanyclear evidencefora velocity gradientnorvisibleelongationalongitsorbit,unlike the caseofAntliaIIwheretidaleffectsappearmoreevident(Jietal.

2021).Moreover,therangeof core-to-halovariation predictedby thissimulationhasbeenshowntomatchwellthefamilyofprofiles ofdSphgalaxies,indicatingmostarestrippedatsomelevelrelative tothemoredistant‘isolated’dSphs(Pozoetal.2020).

SignificanttidalstrippingisnowfirmlyexpectedforCraterIIfrom itsGaia-basedorbit,whichisconfirmedtohavealargeellipticity, withacurrentradiusdistanceof117kpc(Torrealbaetal.2016),well withintheMilkyWayhalo(Sandersetal.2018),andapericenterof only18⁺₋¹⁴₁₀kpc(Fritzetal.2018;Battagliaetal.2021),sothatthat significanttidalstrippingisregardedtobelikely,perhapsresulting in 90percentreductioninitsmass,withthe possiblepresenceof visibletidaldistortion(Jietal.2021).

The‘ultralight’bosonmasssolutionexploredhere,withamassof 10⁻²²eV,isclaimedtounderestimatetheobservedamplitudeof theLyman-αpowerspectrumIrˇsiˇcetal.(2017)onsmallscales.This argumentreliesonanuncertainanalogywithprevious‘Warm’dark matterestimates, ratherthanemployingaself-consistenthydrody- namicalψDMsimulationthatarecomputationallytoodemanding currently.WealsoemphasizeempiricalevidencethatAGNactivity maysignificantlyboosttheforestpower-spectrumMadau&Haardt (2015),Padmanabhan&Loeb(2021),giventhedetectionsofdouble- peakedLyαemittersathighredshift,z6(Huetal.2016;Bosman etal.2020;Gronkeetal.2021)andthe wide‘gaps’in theforest at z> 5reported by (Beckeret al.2015),which imply sparsely distributedAGNs(Gangollieta.l2021)atsuchhighredshiftsmay significantlyboosttheforestvarianceinthepower-spectrumabove currentstandardCDM-basedpredictions.Theseobservationslend support to the proposal that AGN are responsible for the bulk of re-ionization (Madau & Haardt 2015), which would imply a different heatinghistory and less uniform re-ionization, affecting theinterpretationoftheforestpower-spectrum,especiallyonsmall scales.Moreover,Zhangetal.(2018)pointsoutthatthelower-mass limitsuggestedbyIrˇsiˇcetal.(2017)wascomputedwithouttaking intoaccountthesmall-scalequantumpressureintheψDMcontext thatsetsaJeansscaleinherenttoWaveDarkMatter,whichlimits theformationofsmall-scalestructure.

Theoutlineofthispaperisasfollows;first,wedescribetheradial structureandinternaldynamicsoftheψDMcoreinSection2for comparisonwiththedata(CheckFig. 1).Wethendescribethetidal effectsinSection3,predictedbyrecentψDMsimulations,anddis- cussthepossibleevolutionofCraterIIinSection4(CheckFigs2–6).

Finally,inSection5,wediscussourconclusionsregardingtheorigin ofCraterIIinthecontextofψDMwithsignificanttidalstripping.

2 C O M PA R I S O N O F WAV E DA R K M AT T E R P R E D I C T I O N S W I T H O B S E RVAT I O N S O F C R AT E R I I

Ultralight bosons,suchasAxions, provideanincreasinglyviable interpretation of dark matter as explored in Widrow & Kaiser (1993),Hu,Barkana&Gruzinov(2000),Arvanitakietal.(2010), Bozek etal.(2015),Schiveetal. (2014a),and Huietal. (2017).

Inthesimplestcase,withoutanyself-interactions,thebosonmass

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Figure2. EvolutionofthedensityandvelocitydispersionprofilespredictedforCraterII.Left-handpanel:possibleevolutionofCraterII‘sstellarprofileina ψDMcontext,withthemeasureprofileshowninorange,whereastheredandthegreenrepresentpreviouslessstrippedprofileswith50percentand75percent moremass.Noticehowtheprofilebecomes‘stretched’withabroadercore(markedbytheverticallines)andabiggerdensitygapbetweenthecoreandthe halo(largerC−H).Rightpanel:evolutionofthevelocitydispersionprofileduetotidalstrippingcorrespondingtothesameepochsastheleft-handpanel.

Thepeakofthedispersionmovestolargerradiusasstrippingincreases,followingtheexpansionofthecore,andthedistinctioninvelocitybetweenthecore andthehalodiminishes.

istheonlyfreeparameter,whichifsufficientlylightmeansthede- Broglie wavelengthexceeds the mean-freepath,set by the mean cosmologicaldensityofdarkmatter,andthussatisfiesthe ground state condition fora Bose-Einstein condensate. In this case, the density field is simply described by one coupled Schroedinger- Poissonequation,whichinco-movingcoordinatesreads:

i ∂

∂τ + ∇² 2 −aV

ψ=0, (1)

∇²V =4π

|ψ|²−1

. (2)

Here,ψisthewavefunction,Visthegravitationalpotential,andais thecosmologicalscalefactor.Thesystemisnormalizedtothetime- scaledτ=χ^1/2a⁻²dt,andtothelength-scaleξ=χ^1/4(mB/)^1/2x,and χ= ³₂H₀²₀where₀isthecurrentdensityparameterWidrow&

Kaiser(1993;Check Schiveetal.(2014a) tounderstand howthe simulationsarenormalizedtothecosmologicaldensitywhereξ is used).

Recentlyithasprovenpossiblewithadvanced GPUcomputing to make the first cosmological simulations that solve the above equations,(Schiveetal.2014a;Schwabeetal.2016;Moczetal.

2017;May&Springel2021),demonstratingthatlarge-scalestructure evolvesintoapatternoffilamentsandvoidsthatisindistinguishable fromCDM(Schiveetal.2014a),asexpectedforthisnon-relativitc formofdarkmatter.However,the virializedhaloesformedinthe ψDMsimulationsareverydifferentfromCDM,displayingaperva- siveselfinterferingwavestructure,withasolitoniccore,representing the ground state that naturally explain the dark matter cores of dSphs (Schiveet al. 2014b). Surrounding the solitoncore, there isanextendedhalowitha‘granular’textureonthede-Brogliescale, dueto interferenceofexcitedstates,butwhichwhenazimuthally averagedfollowscloselytheNavarro-Frank-White(NFW)density profile(Navarro,Frenk&White1996;Woo&Chiueh2009;Schive etal.2014a,b).Thisgranularinterferencestructurewithinthehalois predictedtoproducenoticeablelensingfluxanomalies(Chanetal.

2020;Hui et al.2021) that are pervasive and hence, statistically unliketherelativelyraresubhalostructureofCDM.

Thefittingformulaforthedensityprofileofthesolitoniccoreina ψDMhaloisobtainedfromcosmologicalsimulations(Schiveetal.

2014a,b):

ρc(r)∼ 1.9a⁻¹(m_ψ/10⁻²³eV)⁻²(r_c/kpc)⁻⁴

[1+9.1×10⁻²(r/r_c)²]⁸ Mpc⁻³, (3)

wherethevaluesoftheconstantsare:c₁=1.9,c₂=10⁻²³,andc₃= 9.1×10⁻²;mψisthebosonmassandrcisthesolitoniccoreradius.

Thelatterscaleswiththeproductofthegalaxymassandbosonmass, obeyingthefollowingthescalingrelationthathasbeenderivedfrom oursimulationsSchiveetal.(2014b):

r_c=1.6 10⁻²²

m_ψeV

a¹^/² ζ(z)

ζ(0) ₋₁_/₆

M_H 10⁹M

₋₁_/₃

kpc, (4)

wherea=1/(1+z).Beyondthesoliton,atradiilargerthanatransi- tionscale(r_t),thesimulationsalsorevealthehaloisapproximately NFWinform,presumablyreflectingthe non-relativisticnatureof condensates beyond the deBroglie scale, and thereforethe total densityprofilecanbewrittenas:

ρDM(r)=

ρ_c(r) if r<r_t,

ρ₀ r rs

1 + _r^r_s2 otherwise, (5)

Moreexplicitly,thescaleradiusofthesolitonicsolution,which representsthegroundstateoftheSchrodinger-Poissonequation,is relatedtosizetothehalothroughtheuncertaintyprinciple.From cosmological simulations, the latteris found to holdnon-locally, relatingalocalpropertywithaglobalone(formoredetailswerefer toSchiveetal.(2014b)).

The classical dwarf galaxies are known to be dominated by DM,and sothestarsaretreatedastracerparticlesGregoryetal.

(2019),McConnachie&Irwin(2006),McConnachieetal.(2006), and Kang & Ricotti (2019) moving in the gravitationalpotential generatedbyDMhalodensitydistribution.

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Crater II tidal waves 2627

Inthiscontext,thecorrespondingvelocitydispersionprofilecan bepredictedbysolvingthesphericallysymmetricJeansequation:

d(ρ_∗(r)σ_r²(r))

dr =−ρ∗(r)GM_DM(r)

r² −2βρ_∗(r)σ_r²(r)

r , (6)

where MDM(r) is the mass DM halo obtained by integrating the spherically symmetric density profile in equation (5), β is the anisotropy parameter [see Binney & Tremaine 2008 Binney &

Tremaine(2008),equation(4.61)],and ρ_∗(r)isthe stellardensity distributiondefinedbythesolitonicwavedarkmatterprofile:

ρ_∗(r)=

ρ1∗(r) if r<rt,

ρ₀₂_∗ r rs∗

1 + _r^r_s∗2 otherwise, (7)

where,

ρ₁_∗(r)= ρ₀_∗

[1+9.1×10⁻²(r/r_c)²]⁸N_∗kpc⁻³, (8) Here,rs∗isthe3Dscaleradiusofthestellarhalocorresponding toρ₀_∗thecentralstellardensity,ρ₀₂_∗isthenormalizationofρ₀_∗at thetransitionradiusandthetransitionradius,rt,isthepointwhere the solitonstructure ends,and the halobegins atthe juncture of thecore andhaloprofiles.Theequivalence ofequation(7)& (8) withequation(5)&(3),respectively.Thisbehaviourwehavenow establishedholdsgenerallyforthewell-studieddSphgalaxiesthat allfitwellthesolitoncoreprofileandwithextendedhaloesPozo etal.(2020).Thisbehaviourmaybetakentoimplysimplythatthe starstracethedarkmatter,asmaybeexpectedapproximatelyindark matterdominatedgalaxiesofspheroidalmorphology.Notealsothat thesolitonformiscloselysimilartothestandardlyadoptedPlummer formwidelyusedtomodeldSphgalaxies,whereitfitswelltoseveral timesthecoreradiusbutfallswellshortoftheextendedstellarhaloes nowcommonlyfound(Torrealbaetal.2019;Pozoetal.2020;Chiti etal.2021;Collinsetal.2021)

Finally,thepredictedvelocitydispersionprofilecanbeprojected alongthelineofsighttocomparewiththeobservations,aspresented inFig.4:

σ_los²(R)= 2 _∗(R)

∞ R

1−βR²

r²

σ_r²(r)ρ_∗(r)

(r²−R²)¹^/²rdr, (9) where,

_∗(R)=2

∞ R

ρ_∗(r)(r²−R²)⁻¹^/²rdr. (10)

Wenowapplytheabovetothenewlymeasureddispersionprofile of Crater II dwarf galaxy, discovered by Torrealba et al. (2016) andsomeidealψDMcases.Thisunusualgalaxywasidentifiedin imagingdataoftheVSTATLASsurveyandseemedtobelocatedat 120kpcfromthesun.Designedtosearchforextendedlow-surface- brightnessemission(Torrealbaetal.2016).ThegalaxyCraterIIis oneofthelargestofthelow-massdwarfsorbitingtheMilkyWay, witha half-lightradius ∼1.08kpc, andalso hasarelatively low surfacebrightnessdwarfs,similartocasesofTucII,TucIV,andUMa II.Itsstellarvelocitydispersionprofilehasrecentlybeenmeasured withdeepspectroscopybyCaldwelletal.(2017)andfoundtobe unusuallylowwithasurprisinglylowmeanvalueofonly2.7kms⁻¹ tracedto over 1kpc shownin Fig.1, whereweseethe data are consistentwiththecharacteristicψDMσform,whichpeaksatthe solitonradiusanddeclinesintothehalo(seeFig.1).

3 T I DA L E F F E C T S

Belowwedescribedifferenteffectsofthetidalstrippingincluding themassloss,coreenlargement,andthecore-halodensitytransition.

WefirstlycalculatetherelationbetweenM_H(halomass)andM_c (coremass)usingequation(3)untilthecoreradius,asindicatedin equation(5).Afterthat,thecoremass-lossrateisinferredforeach orbitwiththefollowingformula(Duetal.2018):

d(M_c(t))/dt=−2×M_c(t)e^a[³^μ(^M²^γ^c⁽^t))^]²⁺^b[³^μ(^M²^γ^c⁽^t))^]⁺^c, (11) Withthebest-fittingparametersvalues:(a,b,c)≡(5.8979410⁻⁵,

−8.7273310⁻²,1.6774),γ,theeffectofthecentrifugalforceowing tosynchronousrotationoftheorbitingsatellite,assumingittobea rigidbody,fixedto3/2forasoliton(Duetal.2018)andμ,thedensity ratiobetweenthecentraldensityofthesolitonρcandtheaverage densityofthehostwithintheorbitalradiusρ_host,μ≡ρ_c/ρ_host(Du etal.2018).Nevertheless,μmustberecalculatedacrosstheorbits duetothecoredensitylossalongeachorbit:

μ=4.38×10¹⁰m⁶_ψM_c⁴d³m⁻_host¹ , (12) ForψDM,thecoreandhaloarecoupled,withthegroundstate soliton surroundedby the haloes of excited states. A core–halo relationship has been established in the simulations, with more massive solitons formed in moremassive haloes that are denser becauseofthehighermomentum.Masslossbytidalforcesreadily stripsthetenuoushaloand,inturn,isexpectedtoaffectthesolitonvia thecore–halorelation,butinasmallerproportion(Duetal.2018).

Previously, wehave found anapproximate proportionalitybe- tweenr_candtheobservedhalf-lightradius,r_h,forlocaldSphgalaxies inPozoetal.(2020),ofabout1.3.Thisrh/rcratiohasalsobeen pointed out by Lazaret al. (2020), Schive etal. (2014a, b).So, computingthemasslossduetotidalstrippingusingequation(11),we continuouslyincrementr_cbyupdatingthemasslossinequation(3).

Finally,weusethe aboveratiobetweenr_cand r_htoincrementr_h acrosstheorbits(seeright-handpanelofFig.5).Theenlargement ofr_t,[extractedfromthesimulationsmadebySchiveetal.(2020), where it isdefined in whichratioshould the transitionradiusof aMilkyway’ssatellitedwarfgalaxyincreaseduetoMilkyWay’s tidalforces: ‘Thehalosurroundingthecentralsolitonisfoundto be vulnerableto tidaldisruption; the density atr > r_tdecreases by morethan anorder of magnitudeafter∼2 Gyr’]willexplain the observedC−H[densitydropbetweenthecoreand thehalo, _C₋_H=logρ_C/ρ_H,whereρ_Cistheasymptoticcentralcorestellar density and ρH is the stellar densityatthe transitionradius (rt)]

changesinthehaloofthegalaxies.

4 M O D E L R E S U LT S

4.1 CraterII

Here,wecomparethemeasuredvelocitydispersionandthestellar profileofCraterIIwithψDM.Wefitthedatawiththe following free parameters;the coreradiusr_cofthe solitonprofile givenby equation(8),thetransitionradiusr_tbetweenthecoreandthehalo, the central3D stellardensityρ₀_∗, andthe 3D scaleradiusofthe stellarhalors∗describingthescaleradiusoftheNFW-likehalothat weinferfromfittingtotheouterstellarprofilebeyondthetransition radius.Notethatthebosonmassisfixed,withavalueof1.5×10⁻²² eV,consistentwithourpreviousdynamicalworkondwarfgalaxies inthecontextofψDMSchiveetal.(2014a),Schiveetal.(2014b), Broadhurstetal.(2020),andPozoetal.(2020).

Ingeneratingourmodelprofiles,wesolvethesphericallysym- metric Jeansequation,describedabove, equation(6),subjectto a totalmassforCraterIIof2.93⁺₋²₁^._.⁹⁹₄₄×10⁸M,whichweobtainfrom fittingthecoreradiussize[equation(8)]inthestellarprofiledatawith

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Figure1. StellarandvelocitydispersionprofilesofCraterIIcomparedwithψDM.Upperpanel:thedatapointsarethestarcountsofCraterIIrebinnedfrom Torrealbaetal.(2016),afterapplyingbackgroundsubtractionbasedontheirasymptoticlimit[redcurveinfig.5ofTorrealbaetal.(2016)],comparedtothe ψDMprofileshownasthegreenshadedarea,representingthe2σ rangeoftheposteriordistributionofprofiles,includingthesolitoncoreandtheouterhalo approximatebytheNFWform.Theapparentexcessat∼3kpcseemstobeaproductofpossiblecontaminationbybackgroundgalaxies(Torrealbaetal.2016).

AstandardPlummerprofileisalsoshown,whichisverysimilartothesolitonformbutunderpredictstheobservationsoutsidethecore.Theorangevertical shadedareaindictsthetransitionradiusoftheψDMcore.Middlepanel:thisshowsthesamecomparisonastheupperpanel,butonalinearscale,sotheextent ofthehalocanbeappreciatedbetter.Lowerpanel:thedatapointsarethevelocitydispersionmeasurementsfromCaldwelletal.(2017),whicharecompared tothepredictedvelocitydispersionforCraterIIcorrespondingtothestellarprofilefitstothestarcountsintheupperandcentralpanelsforthe2σrangeofthe MCMCfits(seeFig.6fordetails).Theverticalgreyshadedareaindicatesthestellarcoreradiusanduncertaintyfromtheabove-stellarprofile.

Table1. ObservationsandψDMprofilefits.Column1:dwarfgalaxyname;Column2:coreradiusrc;Column3:transition radiusrt;Column4:dynamicalmassM(r<rh);Column5:observeddynamicalmassM(r<rh),obs;Column6:haloMass Mh;Column7:observedstellarage;Column8:observedmeanstellarmetallicity;Column9:referencesoftheobserved data.

Galaxy rc rt M(r<rh) M(r<rh),obs Mh ∗age,obs [Fe/H],obs Refs (kpc) (kpc) (10⁶M) (10⁶M) (10⁸M) Gyr

CraterII 0.71⁺₋₀⁰^._.⁰⁹₀₈ 1.68⁺₋₀⁰^._.²⁵₂₃ 7.17⁺₋₄¹²_.₅₆^.⁸³ 4.4⁺₋₀¹^._.²₉ 2.93⁺₋₁²^._.⁹⁹₄₄ ∼10 −1.98⁺₋₀⁰^._.¹₁ Caldwelletal.

(2017)

afixedbosonmassof1.5×10⁻²²eV,whichisconsistentwiththe dynamicalmassestimatedbyCaldwelletal.[2017;seeTable(1)].

Thetransitionradius,rt,isexpectedtobetwoorthreetimeslarger thanthecoreradiusinsimulationsofψDMSchiveetal.(2014a,b), whichweshowbelowisconsistentwithrt∼2.4rcthatwederive hereforCraterII.

Theresultsare listedin Table(1),andFig.(1)shows theself- consistency of these data in both stellar and kinematic profiles.

Thetransitionradius,r_t,atwhichthe profilechanges frombeing dominated by the halo rather than the soliton is marked with a vertical orangeline in Fig. (1). The green and purple shadowed areasrepresentthe2σrange,respectively.Thecomparisonshownin Fig.(1)betweenthe modelsandthe data representaconsistency check, where we simply employ the Jeans equations in making

predictionsforthe velocitydispersionprofile[purplemodelband in lowerpanelof Fig.(1)]wheninputtingthesetof core+halo profilesthatacceptablyfitthestellarprofiledata[greenmodelband Fig.(1)]andsubjecttoaconstraintonthetotalgalaxymasssetby themeanlevelofvelocitydispersionofabout3kms⁻¹.Note,the limitedprecisionofthevelocitydispersiondatadoesnotyetwarrant aclassiccombinedJeansanalysis.

Fig. (2) shows our predictedevolution of the stellarprofile of Crater IIas well as the changein the velocitydispersion profile for twochoices ofgalaxy mass and spanninga massloss of up to 50 percent. As a consequence of continued core mass loss (Duetal.2018),awideningofthecore isinduced,describedby equation(4).Itisimportanttopointouthowbothdensitiesofcore andhaloseemtodecrease,withthehalochangesmorestronglyover

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Crater II tidal waves 2629

time,in goodagreementwiththehalo’sgreaterweaknessagainst tidal forces (Schive et al. 2020). This tidally induced mass loss results in a reduction of the velocity dispersion while widening thecoresothatthedispersion profilehasalesspronounced peak that shifts the largerradius, reflecting the wideningcore as tidal strippingproceedsasseeninFig.5fortheparametersofCraterII.

Thisisinlinewiththe simulationsofFu,Simon&Alarc´onJara (2019) where prolongedtidalstripping shouldproduce adrop in themeanvelocitydispersioninconjunctionwithahalf-lightradius increasing(Sandersetal.2018;Fattahietal.2019;Torrealbaetal.

2019).

Weconcludethattheextendedcoresizeofrc0.71kpcofCrater IIisistheproductofitslowhalomass(10⁸)andsignificanttidal strippingofthehalohaveoccurred, whichisnaturalinthe ψDM context,whileCDMstrugglestoexplaintheobservedcombination of low-velocity dispersion and largeradius (Fattahi etal. 2019).

Moreover,Fig.1clearlyshowhowacuspyNFWprofileisunableto explainthestellardensityandkinematicbehaviourinthecore,under theassumptionthatstarstracethedarkmatter.Wealsonotethatwe havenotadoptedthecommonlyusedPlummerprofileforthestellar profile,preferringinsteadthesolitonformthatfitswellthestellar profileofCraterII(equation(7))derivedinSection2.

4.2 Generalcase

Here,wepredictthetidalevolutionofdwarfgalaxyprofilesinthe contextofψDM.Fig.(3)showshowthesolitoncoremassshould decreasewithtimeoverseveralGyrs,accordingtoequation(11).The solitonprofileremainsunchangeduntilitbecomesstripped,thenthe corerelaxesintoasofterprofileduringthisprocess,(seeFigs(3)and (4).Thisright-handpanelofFig.(3)showsthe solitonbecoming wider,indicated bythedifferencebetweenthickanddashed lines ofthesamecolour,duetothehalomasslossdescribedbyequation (4)andatthesametimetheamplitudeofthedensitygapandthe transition radiuscan be seen to increase as the halo is stripped.

SimilarbehaviourisnoticeableintherecentψDMsimulationsof (Schiveetal.2020),wheretidalstrippinghasbeenapproximatedfor theorbitingdwarfEridanusII.

Fig.(4)showsillustrativevelocitydispersionprofilesforarange ofψDMmassprofileshighlightingthetransitionfromthesoliton core to the outerNFW-likeouterprofile (Schiveetal. 2014a, b; Vicens,Salvado&Miralda-Escudero2018).Thevelocitydispersion profilesare listedinthe right-upperpaneland coverone orderof magnitudein thetotalmassstartingfrom5×10⁹to5×10¹⁰M. Solidanddashedlinesdifferentiatetheoriginalidealisolatedprofile fromthestrippedoneforeachsystem.IntermsoftheJeans-based calculation of the dispersion profile, the choice of β isnot very important,affecting the velocity dispersion well within the core, where it rises (orfall) sharply for a positive(or negative) value ofbetaandremainsflatifisothermal.Thedataappeartofavoura mildlynegativevalueforβasthedispersionoftheinnermostbin islowerthanthe mean,thoughquiteuncertain,as canbeseenin Fig.1(lowerpanel),consistentwithouradoptedβ=−0.5andthis issimilartothevaluechosenbyCaldwelletal.(2017)formodelling dwarfspheroidals,tocountertherisingdispersionprofilethatwould otherwise result from the ‘cusp’ of an NFW profile. The main influenceonthevelocitydispersionprofileisfromthepresenceofthe solitoncore,ratherthanthechoiceofβ,becausethesolitonisdense relativetothehalo(byafactorofabout30),generatingapeaked formataboutthecoreradius,ascanbeseeninFig.4andalsoin Fig.1.

5 D I S C U S S I O N A N D C O N C L U S I O N S

TheexistenceofCraterII,withitsunusuallylargesizeandrelatively low-velocity dispersion,hasbeen asurpriseas it strainsthe very definitionofa‘dwarf’galaxywithitslargesizeandalsobecauseitis insignificantlygreatertensionwithCDMthanthedSphgalaxies (Amorisco2019;Borukhovetskayaetal.2021),forwhichsizeable darkcoresareoftenclaimed,atoddswiththestandardpredictionfor collisionlessCDMparticles(orblackholes)thatlow-massgalaxies shouldbesmallanddenseratherthanlargeanddiffuselikeCraterII.

Instead, in this paper, we have shown that Crater II can be readilyunderstoodinthecontextofdarkmatterasaBose-Einstein condensatebycomparisonwiththeprofilesofgalaxiesgeneratedin ψDMsimulations(Schiveetal.2020),especiallyiftidalstripping isincluded.OneofthemostdistinguishingfeaturesofψDMisthe prominentsolitoncore.Thisisquiteunliketheusualsmoothcores generally exploredin othermodels, wherethe density turnsover towardsthecentrebecomingconstant,butinstead,thesolitoniccore ofψDMisprominent,raisedindensityabovethesurroundinghalo byafactorof30,representingthegroundstate.Furthermore,the solitondensityprofileischaracterizedonlybyaradiussetbythede Brogliewavelength,withthesameshape,irrespectiveofbosonmass, scalingonlywiththemomentumthatsetsthedeBrogliewavelength, sothatthesolitonissmallerinradiusandmoremassiveforhigher massandmoreconcentratedgalaxies.

Itshouldbeemphasizedthatthesimulationsalsomakeclearthat thereisamarkedtransitionbetweenthecoreandthehalo,andwe canseethatCraterIIdoespossessawell-definedcorewithsucha visibletransitioninitsstellarprofileataradiusof0.7kpc,shown inFig.1,andthisisdespitetherelativelylowsurfacebrightnessof CraterII,forwhichthestarcountsaremuchlowerthantypicalwell- studieddSphgalaxies.Theexistenceofthiscoreisalsosupported bythevelocitydispersionprofileofCraterII,whichwehaveshown isconsistentwithbeingpeakedataboutthestellarcoreradius,aswe haveshownispredictedforψDMinFigs3and4.

DeeperimagingandspectroscopyforCraterIIcanhelpconsid- erablyinclarifyingthelevelofcorrespondencebetweenthestellar profileandinternaldynamics,inparticular,itwillbeveryhelpfulto seewhetherthevelocitydispersionfallsinthehaloregionbeyond the current limit of r < 1.2 kpc covered by existing dynamical measurements,wherewepredictthedispersionto besignificantly lowerthaninthecorefortheψDMprofile,andquiteunliketherising profiles predictedby CDMforCraterII,whichmustbeassumed to be hostedby arelatively massivegalaxyof lowconcentration in the contextof CDM(Amorisco2019;Borukhovetskayaetal.

2021).

Wehavealsopointedoutthattheobservedstellarprofilebehaviour of CraterII iscontinuous withthe distinctivecore-halostructure thatappearstobeageneralfeatureoftheclassicaldSphgalaxies, establishedinourearlierworkPozoetal.(2020),wherewefound thatessentiallyallthewell-studieddSphgalaxieshaveaprominent stellarcorethataccuratelymatchestheuniquesolitonform,andalso thatthevelocitydispersionprofilesofthesedSphgalaxiesgenerally peaknearthestellarcoreradiusandarelowerinthehalo.However, despite this qualitative similaritybetweenCraterIIand the dSph class, thereis a clear differencein whichthe dSphs are about a factorthreesmaller,withameancoreradiusof0.25Kpc,compared to 0.7kpc forCraterII, andalsointermsofthe characteristic velocitydispersionthatisaboutthreetimesgreaterforthedSphs, withameanlevelof8−12kms⁻¹comparedtoonly2.7kms⁻¹for CraterIIsignificantlyhigherthanCraterII,andalsoofcourse,asthe name‘feeblegiant’suggests,CraterIIisrelativelymoreextended

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Figure3. TheevolutionofFourtypicaldSphψDMdensityprofilespredictedfororbitswithintheMilkyWay,withaconstantdensityratioof;μ=50.

Left-handpanel:coremasslossofthefourprofiles.Theverticaldashedlinemarksfourorbits.Thesmallpanelshowsindetailtheprofilesaftereachhaslost 50percentofthetotalmasstosteadytidalstrippingafterthefirstfourorbits.Right-handpanel:tidalevolutionoftheψDMdensityprofiles.Thethicklines representtheoriginalprofilesbeforethestrippingprocess,whilethedashedlinesrepresenttheirsituationafterfourorbits.Noticehowinallthecasesthecore becomesextendedduetolosingmassasthetransitionradiusincreases.

Figure 4. Velocity dispersion profile evolution for two tidally evolved profilesofdifferentgalaxymassspanningtheclassicaldwarfrange,asin Fig.3.Noticehowthecoreismoreevidentinthedispersionprofileforthe moreconcentrated,massivesoliton,indicatedbytheorangeprofile,compared tothesoftercore-halotransitionforthelessmassivedwarfindicatedbythe redprofile.

andofunusuallylowsurfacebrightnessthantypicaldSphgalaxies (Torrealbaetal.2016).

OuranalysisofCraterIIhasexaminedthepossibilitythattidal stripping in the context of WaveDM may accountfor the rather extremepropertiesofthisdwarfgalaxy,withitsrelativelylargesize andlowvelocitydispersion.Insupportofthis,weshowthestellar profileoftheCraterII(upperpanelFig.1)iswellfittedbyasoliton coreplusashallowNFWhaloandthatthiscombinedprofile,which is genericto WaveDM, is consistentwith the extended, shallow, velocitydispersionprofileofCraterII,(lowerpanelFig.1)wherethe lowmeanlevelofvelocitydispersionof3kms⁻¹correspondstoa totalgalaxymassofabout3×10⁸M.Theconsistencywefindhere mayindicatethatthestellarcoreandDMcorearesimilarinscale, thoughthisisbynomeansconclusiveatthecurrentlimitedprecision

ofthedispersionprofileandreliestosomeextentthelevelofvelocity anisotropyassumed,butmayimplythestarsbehaveessentiallyas tracerparticleswithinthedarkmatter-dominatedpotential.Detailed hydrodynamicalsimulationsofgasandstarformationwithindwarf galaxyhaloeswillberequiredforamoredefinitiveexplorationofthis relationshipbetweenstarsandDM,whichmustincludetherelaxation effects understood to be significant in randomly deflecting stars orbitingthroughthedeBrogliescaledensityfluctuationspredicted forWaveDMhaloes(Schiveetal.2014a)andproposedaspossible explanationfortheincreasingscaleheightofdiscstarsintheMilky WaywithstellaragebyChurch,Mocz&Ostriker(2019)andBar- Or, Fouvry & Tremaine (2019). For now, we content ourselves withthelargelyqualitativeconclusionthattidalstrippingofaDM- dominateddwarfgalaxyinthecontextofWaveDMcanplausibly resultinthe unusualpropertiesofCraterII, withanexpansionof itssolitoncorein responseto strippingofitsDMhalo,aneffect that follows fundamentally from the Uncertainty Principle for WaveDM.

Wehavebeenabletounderstandthesedifferencesinthecontext ofψDM,asthe possibleconsequenceoftidalstripping.Itisnow understoodthat CraterII hasasmallpericenterwithin the Milky Way, sothat tidal strippingshould besignificant and by this we haveshownaninterpretationofCraterIIasastrippeddSphgalaxy.

Thispossibilityfollowsdirectlyfromconsideringhowthe soliton expands as the halo mass is stripped away, as proposed by Du etal. (2018) thatisimplied bythe existenceof arelativelyclear relationshipestablishedintheψDMsimulationsbetweenthemass ofagalaxyand thesolitoncore,whichmayacttogetherwiththe strictinverserelationshiprequiredforasoliton(bytheUncertainty Principle), suchthatas galaxymassisreducedbytidalstripping, the momentum associated with the soliton ground state is also lowerandhencethesolitonexpandsasthedeBroglieWavelength is larger following the inverse soliton mass–radius relation, with areduce solitondensity andhence lowervelocity dispersion.We may conclude that in order for Crater II to have originated as a typicaldSph, its core hasexpanded by approximatelya factor of 2–3,and hencethe coremasswastwo–threetimeshigher and henceinitially,thetotalmasswouldhavebeen10−30×larger,

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Crater II tidal waves 2631

Figure5. Evolutionofmassandrhradiusduetotidalstripping.ThethicksolidanddashedlinesarewithaMilkywayhostof5×10¹²Mand10¹²M, respectively.Left-handpanel:PredictedtotalmasslossevolutionforCraterIIintheψDMcontext.Theorangehorizontallinerepresentsthebest-fittingmass ofCraterII(seeFig.1andTable1).ThesolidanddashedpurplehorizontallinesindicatesthemaximumandminimumallowedmassesofCraterIIfromour analysis,correspondinglimitingpurplecontoursofFig.1.Right-handpanel:predictedevolutionoftherhradiusgrowthforCraterII,inaψDMcontext.The orangehorizontallinerepresentCraterII’sactualrh.

Figure6. CraterII:correlateddistributionsofthefreeparameters.Ascanbeseen,thecoreradiusandtransitionradiusarewell-defineddespitetheflatinput priors,indicatingareliableresult.Thecontoursrepresentthe68percent,95percent,and99percentconfidencelevels.Thebest-fittingparametervaluesare themedians(witherrors),representedbythedashedblackones,andtabulatedinTable1.

assumingthecore-halomassscalingrelationscalingisfollowed,i.e.

m_sol∝ m¹_gal^/³.Wecancometothesamequantitativeconclusionby comparingthe velocity dispersion 3kms⁻¹of CraterII, which also differs by a factor of 3 with the typical8−10kms⁻¹ peak dispersionforthedSph’s.Thisagreementisquitecompellingforthe

ψDMinterpretationasthisfactordifferencewouldnotbeexpected, whereas forψDMit isarequirementastheuncertainty principle dictates thatthe solitonobeys r_solσ_sol= h/8πm_ψ, (taking2r_solas the width of the soliton wave packet) providing anapproximate bosonmassof2.51⁺₋₀⁰^._.₅₁⁶⁸×10⁻²²eV,andbecauseσ_solandr_solvary