Rotation Curves of High-Resolution LSB and SPARC Galaxies in Wave (Fuzzy) and Multistate (Ultra-light Boson) Scalar Field Dark Matter

4.7 MB·

by  Tula Bernal

MNRAS000,1–17(2016) Preprint5thJanuary2017 CompiledusingMNRASLATEXstylefilev3.0
Rotation Curves of High-Resolution LSB and SPARC Galaxies in
Wave (Fuzzy) and Multistate (Ultra-light Boson) Scalar Field Dark
Matter
T. Bernal,1(cid:63) L. M. Fernández-Hernández,1† T. Matos2,3‡ and M. A. Rodríguez-Meza1,3§
1DepartamentodeFísica,InstitutoNacionaldeInvestigacionesNucleares,AP18-1027,CiudaddeMéxico11801,México
2DepartamentodeFísica,CentrodeInvestigaciónydeEstudiosAvanzadosdelIPN,AP14-740,CiudaddeMéxico07000,México
3PartoftheInstitutoAvanzadodeCosmología(IAC)Collaboration
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5thJanuary2017
2

n
a ABSTRACT
J ColdDarkMatter(CDM)hasshowntobeanexcellentcandidateforthedarkmatteroftheuni-

4 verseatlargescales,howeveritpresentsseveraldifficultieswithobservationsatthegalactic
level.Ontheotherside,theScalarFieldDarkMatter(SFDM),alsocalledFuzzy,Wave,Bose-

] EinsteinCondensateorUltra-lightAxionDM,isidenticaltoCDMatcosmologicalscalesbut
A
different at the galactic ones. Because of its quantum nature, the SFDM forms haloes with
G
core density profiles; it has a natural cut-off in its matter power spectrum, thus it fits well
. the amount of satellite galaxies in the Milky Way neighbourhood, and predicts well-formed
h
galaxiesathighredshifts.Intherecentyearsastronomershavemeasuredtherotationcurves
p
andtheamountofluminousmatterandgasingalaxieswithgreataccuracy.Inthisworkwe
-
o reproduce the rotation curves of high-resolution LSB and SPARC galaxies with two differ-
r ent SFDM profiles: (1) The soliton+NFW profile in the Wave DM (ψDM) model, arising
t
s empirically from cosmological simulations of real, non-interacting SF at zero temperature,
a
and (2) the multistate SFDM profile, an exact solution to the Einstein-Klein-Gordon evolu-
[
tion eqs. for a SF perturbation, taking into account the self-interaction and temperature of

1 the real SF, introducing several quantum states as a realistic model for a SFDM halo. From
v thefitswiththesoliton+NFWprofile,withoutassuminganycosmologicalrestrictiononthe
2 boson mass m , we obtained 0.264 < m /(10−23eV/c2) < 30.0 and for the core radius
ψ ψ
1 0.311 < r /kpc < 4.90.AdditionallyweshowthemultistateSFDMmodelfitstheobserva-
c
9
tionsbetterthantheempiricalsoliton+NFWprofileinaverysimpleway,evenatthecentres
0
of the galaxies, and it reproduces naturally the wiggles present in some galaxies, being a
0
theoreticallymotivatedframeworkalternativetotheψDMprofile.
.
1
0 Keywords: galaxies:haloes,galaxies:structure,(cosmology:)darkmatter
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1
:
v
i 1 INTRODUCTION energycontentoftheuniverse,settingthecontributionofdarkmat-
X
terto∼26%,meanwhilethebaryonicmatterisonly∼5%andthe
r Darkmatter(DM)waspostulatedfirstinordertoexplaintherota-
a cosmologicalconstantΛordarkenergyis∼69%(PlanckCollab-
tioncurvesofdiskgalaxiesandtheobservedvelocitydispersionsin
orationetal.2016).
galaxyclusters,aswellastheobservationalmass-to-lightratiosin
galaxiesandclustersofgalaxies(Zwicky1933,1937;Smith1936; ThemostaccepteddarkmattermodelistheColdDarkMat-
Rubinetal.1980;Rubin1983).Later,itsnecessitywasevidentto ter (CDM), which is very successful at reproducing the observa-
explainthegravitationallenses,thestructureformationintheearly tionsatcosmologicalscales;however,atthegalacticlevelitfaces
universe,theacousticbaryonicoscillations,thepowerspectrumof some problems (see e.g. Weinberg et al. 2015). One of the diffi-
galaxies,amongotherastrophysicalandcosmologicalphenomena cultiesistheso-called“cusp/core”problem,sincefromCDMN-
(seee.g.Bertoneetal.2005;Bennettetal.2013).Thespacemis- body simulations the DM haloes are assembled with the “cuspy”
sionPlanckhasobtainedthemostprecisemapofthetotalmatter- Navarro-Frenk-Whitedensityprofile(Navarroetal.1997),which
isproportionalto1/ratsmallradiiandto1/r3atlargedistances,
meanwhile many observations suggest a constant central density
(cid:63) E-mail:tbernal@fis.cinvestav.mx
or“core”profile,e.g.inrotationcurvesofgalaxies(Mooreetal.
† E-mail:lfernandez@fis.cinvestav.mx
‡ E-mail:tmatos@fis.cinvestav.mx 1999;deBloketal.2001a;McGaughetal.2007,2016)anddwarf
§ Email:[email protected] spheroidal(dSph)galaxies(Klypinetal.1999;Kroupaetal.2010;
(cid:13)c 2016TheAuthors
2 T.Bernaletal.
Boylan-Kolchinetal.2011;Walker2013;Pawlowskietal.2014). with the characteristic ultra-light SF mass, the bosons condense
AnotherissueistheCDMpredictionoftoomanysatellitehaloes veryearlyintheuniverseatcriticalcondensationtemperaturesof
aroundbiggalaxiesliketheMilkyWay(Sawalaetal.2016)which TeV(Matos&Ureña-López2001),makingupBose-EinsteinCon-
havenotbeenobserved.Aside,itfailstoreproducethephase-space densates(BEC)interpretedasthedarkmatterhaloes.Oneimport-
distribution of satellites around the Milky Way and Andromeda antpropertyoftheSFDMisthat,atcosmologicalscales,itbehaves
galaxies (Pawlowski et al. 2012; Ibata et al. 2013, 2014) and the asdustandreproducesthesameobservationsaswellasCDM:the
internal dynamics in tidal dwarf galaxies (Gentile et al. 2007b; cosmic microwave background and the mass power spectrum at
Kroupa 2012). Finally, another issue might lie in the early form- large scales (Rodríguez-Montoya et al. 2010; Hlozek et al. 2015;
ationofbiggalaxies,sincetheCDMmodelpredictsbiggalaxies Schiveetal.2016).Recently,Schiveetal.(014a,b)haverunhigh-
wereformedhierarchicallyfromhaloeslessmassivethan1012M , resolution cosmological simulations of Scalar Field (Wave) DM
(cid:12)
as is the typical case, but there are some recent observations of andreproducedthesameresults.
massivegalaxiesatveryhighredshifts(5 (cid:54) z < 6)(Caputietal. Inaseriesofpapers(Matos&Ureña-López2007;Bernaletal.
2015). 2008;Robles&Matos2012)itwasshownthattheSFDMforms
Such problems at the galactic scales might be solved by in- corehaloes(seealsoHarko2011).FurtherfeaturesoftheSFDM
troducing the baryonic physics to the simulations, which can be wereanalysed,forexample,thegravitationallensingwasstudied
relevantatthecentresofthegalaxiesandgalaxyclustersthrough byNúñezetal.(2010);Robles&Matos(2013a);theψ4-SFpoten-
theinclusion,forexample,ofstarformation,supernovaexplosions, tialwasstudiedbyMatos&Suarez(2011);multistatesolutionsfor
stellarwinds,activegalacticnuclei,etc.(seee.g.Governatoetal. theSF,i.e.bosonsinexcitedstatesoutsidethegroundenergylevel,
2010;Teyssieretal.2013;Pontzen&Governato2012;Madauetal. havebeenconsidered(seee.g.Bernaletal.2010;Robles&Matos
2014;DiCintioetal.2014;Pontzen&Governato2014).Froma 2013b;Martinez-Medinaetal.2015a;Bernaletal.2016).Numer-
phenomenological pointof view,empirical CDMdensity profiles icalsimulationsofgalaxyformationwereperformedbyMartinez-
have been proposed in order to explain the observations, e.g. the Medina & Matos (2014); Martinez-Medina et al. (2015b), where
Burkert profile (Burkert 1996) and the generalised NFW profile thecharacteristicspiralarmsandbarsofadiskgalaxywereeasily
(Zhao 1996). Aside, there are some alternatives to CDM seeking generated.Itwasalsoshownthatsatellitegalaxiesarestablearound
to solve the discrepancies at galactic scales without appealing to SFDMhaloes(Roblesetal.2015).Andastatisticalstudyofhigh-
baryonicprocesses,forinstance,WarmDarkMatter(Zavalaetal. resolutionrotationcurvesofspiralgalaxiesintheBEC-DMmodel
2009;Navarroetal.2010;Lovelletal.2012),Self-InteractingDark hasbeendeveloped(Fernández-Hernándezetal.2016).
Matter(Spergel&Steinhardt2000;Yoshidaetal.2000;Davéetal. Asmentionedbefore,theSFDMmodelhasmanyvariants,de-
2001;Elbertetal.2015)andScalarFieldDarkMatter(SFDM)(see pendingonthespecificcharacteristicstudiedbydifferentauthors:
Matos & Guzman 2000; Magaña et al. 2012; Suárez et al. 2014; theSFmightberealorcomplexandpossessaself-interaction,it
Matos&Robles2016;Marsh2016;Huietal.2016,andreferences mighttakeintoaccountalsothetemperatureoftheSF,byintrodu-
therein). cingasuitableSFpotential.ThemodelhasbeennamedasFuzzy
We are interested in the SFDM model, studied within some DM (FDM) (Hu et al. 2000), Wave DM (WDM or ψDM) (Bray
specialcasesandnameddifferentlydependingontheauthors.The 2012;Schiveetal.014a)orUltra-lightAxion(ULA)DM(Hlozek
motivation is the natural solution emerging from this model to et al. 2015); Bose-Einstein Condensate (BEC) DM (Boehmer &
theCDMproblemsmentionedbefore.TheSFDMmodelassumes Harko2007);MultistateSFDM(Robles&Matos2013b),etc.
dark matter is a spin−0 scalar field (SF) ψ, with a typical ultra- IntheBEC-DMmodel(Boehmer&Harko2007)itisassumed
lightmassm ∼10−23−10−22eV/c2,whichmightincludeself- a dominant self-interaction between the particles (Thomas-Fermi
ψ
interactions. The first time when this idea was mentioned was in limit);inthismodelallthebosonslieinthegroundstateatzero
Baldeschietal.(1983);sincethentheideawasrediscoveredseveral temperature;themeanbosonmassfromtheanalyticdensitypro-
timesusingdifferentnames(seee.g.Membradoetal.1989;Press file is m ∼ 10−6eV/c2. This model has been widely studied,
ψ
etal.1990;Sin1994;Ji&Sin1994;Lee&Koh1996;Matos& showing up sharp discrepancies in dwarf galaxies (Diez-Tejedor
Guzman2000;Sahni&Wang2000;Peebles2000;Goodman2000; etal.2014),diskgalaxies(Boehmer&Harko2007;Robles&Ma-
Matos&Ureña-López2000,2001;Huetal.2000;Wetterich2001; tos 2012; Fernández-Hernández et al. 2016) and galaxy clusters
Arbeyetal.2001;Boehmer&Harko2007;Matosetal.2009;Woo (Bernal et al. 2016), ruling out this approximation as a realistic
&Chiueh2009;Bray2010;Lundgrenetal.2010;Robles&Matos SFDMmodel.
2013b; Marsh et al. 2014; Schive et al. 014a) and more recently ThefirstmodelstudiedinthepresentworkistheWave,Fuzzy
by Hui et al. (2016). However, the first systematic study of this or Ultra-light Axion DM, which share the same characteristics:
idea started in 1998 by Guzman et al. (1999); Matos & Guzman aquadraticnon-thermal,non-interactingpotentialwithultra-light
(2000),showingthattheobservedrotationcurvesofdiskgalaxies massesm ∼10−23−10−21eV/c2.Fromthecosmologicalsim-
ψ
can be reproduced by the SFDM model, and the cosmology was ulationbySchiveetal.(014a,b),anempiricaldensityprofilewas
studiedforthefirsttimebyMatosetal.(2000);Matos&Ureña- obtainedfortheWaveDMmodel,composedbyacouplingofthe
López(2000).OthersystematicstudiesoftheSFDMmodelwere asymptoticNFWdeclinewithaninnersoliton-likeprofilefromthe
performed by Arbey et al. (2001); Arbey et al. (2002) and more SFDM(“soliton+NFW”profile)(Schiveetal.014a;Marsh&Pop
recentlybyMarshetal.(2014);Marsh(2016). 2015).
In the works by Matos et al. (2000); Matos & Ureña-López Secondly,weinvestigatethemultistateSFDM(Robles&Ma-
(2000), using the amount of satellite galaxies observed in the vi- tos 2013b), which considers the SF is thermal at the very early
cinity of the Milky Way, it was found that the mass of the scalar universe, interacting with the radiation and the rest of matter. By
fieldshouldbem ∼10−22eV/c2.Withthisultra-lightmass,Al- introducingaself-interacting,temperature-correctedSFpotential,
ψ
cubierreetal.(2002)foundthroughnumericalsimulationsthatthe theauthorsobtainedanexactanalyticsolutionfortheSFdensity,
gravitational collapse of a SF configuration forms stable objects includingthegroundandexcitedquantumstates.Thismodelhas
withmassesM oftheorderofagalaxyhalo:M ∼1012M .Also, beenprovedsuccessfulinfittingtheobservationsofdwarfspher-
(cid:12)
MNRAS000,1–17(2016)
RCsofHigh-ResolutionLSBandSPARCGalaxiesinWave(Fuzzy)andMultistateSFDM 3
oidalgalaxies(Martinez-Medinaetal.2015a),therotationcurves generalpotentialisgivenby
ofdiskgalaxies(Robles&Matos2013b),thestronggravitational
1 1
lensing(Robles&Matos2013a)andthedynamicalmassesfrom V(ψ)= 2m2ψψ2+ 4λψ4, (2)
X-rayobservationsofclustersofgalaxies(Bernaletal.2016).
theso-called“double-well”potential,whichincludesthemassm
Inthisworkwereproducetheobservedrotationcurvesof18 ψ
of the SF and a self-interaction λ. For positive m2 the potential
high-resolutionlowsurfacebrightness(LSB)galaxies,whereonly ψ
has a single minimum at ψ = 0 and the potential is Z invari-
the DM is taken into account, 4 representative SPARC (Spitzer 2
ant;for−m2 thepotentialhastwominimaandtheZ symmetry
Photometry & Accurate Rotation Curves) galaxies and 2 other ψ 2
appearsspontaneouslybroken,beingofgreatinterestforphysical
galaxiesusedinRobles&Matos(2013b)(thelast6galaxieswith
situations(seeSubsection2.2).Arecentstudyoflargescalestruc-
high-resolutionphotometricinformation).Forthegalaxieswithba-
tureformationwiththelastpotentialhasbeendevelopedbySuárez
ryonic data, McGaugh et al. (2016); Lelli et al. (2016a) analysed
& Chavanis (2016). For this potential, in the hydrodynamic ap-
153galaxieswiththegasandstarsinformationfromtheSPARC
proach, it is possible to write an equation of state of the SF, res-
database(Lellietal.2016b),andfoundanempiricalradialacceler-
ultinginapolytropeofindexn=1.
ationrelationbetweentheobservedaccelerationfromtherotation
Inthecaseλ=0,thepotential(2)reducesto
curvesofthegalaxiesandtheaccelerationfromthebaryoniccom-
ponent,showingadeviationatthevalueg† = 1.2×10−10m/s2. 1
V(ψ)= m2ψ2, (3)
Thisresultcouldsuggestthatbaryonsarethesourceofthegravit- 2 ψ
ationalpotential,atleastatsmallradii.AsMcGaughetal.(2016)
whichhasbeenwidelyusedtomodeltheSFDMintheuniverse,as
pointedout,suchrelationcanbeexplainedastheendproductof
discussedinthenextSubsection.
galaxy formation processes (including the baryonic matter), new
DM physics or the result of a modified gravity law. In the CDM
paradigm, Ludlow et al. (2016) introduced different stellar and 2.1 Wave(Fuzzy)DarkMatter
AGNfeedbackprocessestoexplaintheobservedaccelerationrela-
The Fuzzy DM (FDM) (Hu et al. 2000), Wave DM (WDM or
tioninwell-resolvedgalaxiesfromtheEAGLEsimulation,show-
ψDM)(Bray2012;Schiveetal.014a)orUltra-lightAxion(ULA)
ingthattheempiricalrelationcanbe“accommodated”withinthe
DM(Hlozeketal.2015)(herewenameitasSchiveetal.(014a),
model. However, instead to look for complicated and diverse ba-
simply ψDM), considers the SFDM is described by the poten-
ryonicprocesses,inthisworkwetestthetwoSFDMmodelswith
tial(3),forultra-lightmassesm ∼10−23 −10−22eV andnull
theobservedrotationcurvesofsomerepresentativeSPARCgalax- ψ
self-interaction λ = 0. As showed by Matos & Ureña-López
ies,showingthatthemodelisconsistentwiththedata.
(2001); Schive et al. (014a); Ureña-López & González-Morales
The idea is to analyse statistically both approximations: the
(2016), this approximation presents a cut-off in the power spec-
studycangiveusaclueforwhetherthescalarfieldisthermalor
trum which suppresses the small scale structure formation below
not,oriftheself-interactionbetweenSFparticlescanbeimportant,
the de Broglie wavelength λ (corresponding to halo masses
assumingthescalarfieldisthedarkmatterofthegalaxies.Inorder deB
M<108M form ∼10−22eV),asaresultofthequantumprop-
todoso,thisworkisorganisedasfollows:InSection2webriefly (cid:12) ψ
erties of the model (Heisenberg uncertainty principle), solving in
explain the SFDM model and the two different approaches to be
this way the small scale structure overproduction in CDM. This
compared;inSection3wedescribethegalaxies’samplesusedto
approachassumestheSFDMisatzerotemperature,implyingall
fit the SFDM models; in Section 4 we present the results, and in
thebosonsareinthegroundstate,i.e.thelowestenergylevelwith
Section5wediscusstheresultsandpresentourconclusions.
no nodes. Also, Suarez & Matos (2011) showed that, within this
approach, the evolution of perturbations of the SFDM model is
identicaltoΛCDMandRodríguez-Montoyaetal.(2010)showed
2 SCALARFIELDDARKMATTER theSFDMmodelisconsistentwiththeacousticpeaksofthecos-
micmicrowavebackgroundforabosonmassm ∼10−22eV.
From particle physics motivation, spin-0 scalar fields are the ψ
Nevertheless,Lietal.(2014)foundthatthismodelisnotcon-
simplest bosonic particles, described by the Klein-Gordon (KG)
sistent with the Big Bang Nucleosynthesis (BBN) constraints for
equation:
thebosonmassm foranymodelwithλ = 0,at1σ-confidence
ψ
dV(ψ) level(CL),makingnecessarytointroduceaself-interactionintothe
(cid:3)ψ− =0, (1)
dψ SFpotential.Althoughtheirresultsallowtoalleviatesuchrestric-
tionsat2σ-CL,theirworkisamotivationtostudyaSFpotential
fortherealorcomplexscalarfieldψ andtheSFpotentialV(ψ);
includinganon-nullself-interaction(seethenextSubsection).
fromhereafterweusethenotation(cid:126) = 1 = c,forcthespeedof
Schiveetal.(014a,b)runahigh-resolutioncosmologicalsim-
lightand(cid:126)thereducedPlanck’sconstant.Thescalarfieldsarisein
ulation based on the dynamics of ψDM in the Newtonian limit,
cosmologytoexplaintheinflationaryepochintheearlyuniverse
governedbytheSchrödinger-Poisson(SP)system.Fromthesimu-
andasalternativestoexplaintheacceleratedexpansionoftheuni-
lation,Schiveetal.(014a)derivedanempiricaldensityprofilefor
verse;theycandescribealsocompactobjects,asbosonstarsand
the DM haloes, consisting of a soliton-like core in every system,
supermassiveblackholes.AsalternativetotheCDMparadigm,itis
prominentbeforeatransitionradius,embeddedinaNFWdensity
proposedthatDMhaloesareconstitutedbyultra-lightscalarfields,
halo,dominantatlargeradii.Themotivationofthisproposalisthe
forwhichthecompleteEinstein-Klein-Gordonequationsreduceto
expectedlossofphasecoherenceoftheψDMwavesatlargedis-
theSchrödinger-PoissonsystemintheNewtonianlimit.
tancesfromthecentreofdistribution,expectingatransitionfrom
Untilnow,thereisnotanagreementonthecorrectformofthe
thesolitontotheNFWprofile(Marsh&Pop2015).Suchprofile
potentialV(ψ)forDMapplicationsandsomeofthemhavebeen
canbeapproximatedby
proposedinordertoapplytheSFDMtheorytodifferentastronom-
ical and cosmological situations (see e.g. Suárez et al. 2014). A ρ (r)=Θ(r −r)ρ (r)+Θ(r−r )ρ (r), (4)
ψDM (cid:15) sol (cid:15) NFW
MNRAS000,1–17(2016)
4 T.Bernaletal.
where Θ is a step function, r is the transition radius where the freeparameters,thuswefitonlym ,r andr .Thecompleteex-
(cid:15) ψ c (cid:15)
densitychangesfromthesolitonprofile(Schiveetal.014a): pressionsarewritteninAppendixA.
ρ
ρ (r)= c , (5)
sol [1+0.091(r/rc)2]8 2.2 MultistateScalarFieldDarkMatter
totheNFWprofile(Navarroetal.1997): AsastepforwardonthestudyoftheSFDMmodel,Robles&Ma-
ρ tos(2013b)consideredthegeneraldouble-wellpotential(2),with
ρNFW(r)= (r/r )(1+s r/r )2. (6) ano-nullself-interactionterm,plusone-loopfinite-temperatureT
s s
corrections(Kolb&Turner1994):
Inequation(5),ρc :=1.9(mψ/10−23eV)−2(rc/kpc)−4M(cid:12)pc−3 1 1 1 π2
isthecentralsolitondensity,m thebosonmassandr thehalf- V(ψ)=− m2ψ2+ λψ4+ λψ2T2− T4, (7)
ψ c 2 ψ 4 8 90
lightradiusofthesoliton-likeregion.Inequation(6),ρ isrelated
s
in units k = 1 for the Boltzmann constant and where the Z
to the density of the universe at the moment the halo collapsed B 2
symmetry appears spontaneously broken (with the term −m2).
and r is a scale radius. The total density profile (4) has 5 free ψ
s
Suchself-interacting,finite-temperaturecorrectedpotentialismo-
parameters (m , r , r , ρ , r ), which are reduced to 4 asking
ψ c (cid:15) s s
tivated by the theoretical expectation of modern particle physics
forcontinuityofthefunctionatthetransitionradiusr andupto
(cid:15)
that,athightemperatures,symmetriesspontaneouslybrokentoday
3askingfordifferentiabilityatthesamepoint(seeAppendixA).
wererestored.Symmetrybreakingstudiesareofgreatinterestfor
Schiveetal.(014a)foundthatthetransitionradiusr is,ingeneral,
(cid:15)
diverse physical situations, assuming the universe has underwent
r >3r .
(cid:15) c
phasetransitionsduringitsevolution(seee.g.Kolb&Turner1994,
Thereareseveralconstraintsonthebosonmassm fromdif-
ψ
andreferencestherein),asforexample,theinflationaryera.With
ferent observations: From the study of the effect of tidal forces
thismotivationinmind,Robles&Matos(2013b)considerthatat
on the cold clumps in Ursa Minor, Lora et al. (2012) found
m ∼(0.3−1)×10−22eV,andinSextants,Lora&Magaña(2014) theveryearlyuniverse,theinitialSFfluctuationsfrominflationin-
ψ
foundm ∼(0.12−8)×10−22eV;fromCMBandgalaxycluster- teractwiththerestofmatterandradiationatveryhightemperatures
ψ
ingdata,Hlozeketal.(2015)constrainedthemassm >10−24eV; T (cid:29)Tc,forTcthecriticaltemperaturewhereψ=0isaminimum
ψ
ofthepotentialandthesymmetryisrestored.Inthiscase,theSFis
from the high-redshift galaxy luminosity function, Schive et al.
(2016)obtainedm >1.2×10−22eV;fromLyman-αobservations, embedded in a thermal bath at finite temperature T at very early
Sarkaretal.(2016)ψderivedm >10−23eV.Fromultra-faintdwarf times1. As the universe expands, the temperature decreases and
ψ
eventually the SF decouples from the rest of matter, evolving in-
spheroidalgalaxies(dSphs)galaxies,Calabrese&Spergel(2016)
derivedm ∼(3.7−5.6)×10−22eV. dependently. As the potential (7) depends on the 4th-power with
ψ
respecttoT,asthetemperaturecontinuesdecreasing,theSFgoes
For the classical dSph galaxies in the Milky Way, Marsh &
throughaZ spontaneoussymmetrybreaking(SSB),whichturns
Pop (2015) constrained m using the soliton+NFW density pro- 2
ψ
a minimum of the potential to a maximum, which increases the
file (4), through stellar populations’ observations in Fornax and
Sculptor and found m <1.1 × 10−22eV. More recently, Chen amplitudeoftheinitialSFperturbationsformingtheinitialgalaxy
ψ
haloes(seealsoMatos&Rodríguez-Meza2014,forthestudyof
et al. (2016) applied the Jeans analysis to the kinematic data of
theSSBofachargedcomplexSF).
theeightdSphstoconstrainm ;forallthedSphs,exceptforFor-
ψ
For cosmological applications, Robles & Matos (2013b) as-
nax, as the transition radii are outer the half-light radii r , they
c
sumedtheSSBatT=T takesplaceintheradiationdominatedera.
tookintoaccountthesolitoncoreprofile(5)only,derivingamass c
m = 1.18×10−22eV. For Fornax, they applied the complete TheyfoundthatatthatmomenttheSFfluctuationscanstartgrow-
ψ
densityprofile(4)andfoundalargerm =1.79×10−22eVand ing in the linear regime for T<Tc, until they reach a new stable
ψ
minimum. Then obtained the perturbed Einstein-Klein-Gordon
smallerr ,withrespecttothesoliton-onlymodelresults.Mostre-
c
evolution equations for the SF perturbation, δψ, in a Friedmann-
cently, González-Morales et al. (2016) constrained m with the
ψ
Lemaître-Robertson-Walker background spacetime. Under these
completesoliton+NFWprofile(4),usingkinematicmockdataof
assumptions,thegalactichaloescouldhavebeenformedalmostat
FornaxandSculptor,whichincludethestellarcomponentsofthe
thesametimeoftheSSBandwithsimilarmasses,M ∼1012M
galaxies, finding core radii r > 1.5 kpc and r > 1.2 kpc, re- (cid:12)
spectively,andm <4×10−2c3eV.Suchresultsacreintensionwith (formψ ∼ 10−22eV),whichisthetypicalmassofagalaxylike
ψ
ours.Lateron,suchhaloescanenterinthenon-linearregime,mer-
the estimates from the cold clumps in Ursa Minor and Sextants
ging and constituting larger structures, hierarchically, just like in
(Loraetal.2012;Lora&Magaña2014),thehigh-redshiftlumin-
theCDMmodel.
osityfunction(Schiveetal.2016)andtheultra-faintdSphsstudies
Under the linear approximation to describe the evolution of
(Calabrese&Spergel2016).González-Moralesetal.(2016)sup-
agalaxyhalo,intheNewtonianregime,theexact solutiontothe
pose that such small boson mass might be the result of baryonic
perturbation equations for the temperature-corrected potential (7)
feedbackprocessespresentinthedSphgalaxies.
isfoundas(Robles&Matos2013b)
For the LSB and SPARC galaxies analysed in the present
work,weassumethesoliton+NFWprofile(4)fortheψDMmodel (cid:20)sin(k r)(cid:21)2
ρj (r)=ρj j ; (8)
andexpectthetransitionradiusr(cid:15) iswellinsidetheradiusofthe SFDM 0 (kjr)
lastobservedpoint(spanningarangeR ∼1−30kpc,forthe
max thusthefinite-temperaturecorrectedpotential(7)impliestheexist-
maximumradiusofluminousmatter),thusweapplythecomplete
enceofdifferentexcitationstatesj assolutiontotheSFperturba-
density profile. Additionally, we impose continuity and differen-
tionequation,i.e.thebosonsarethermallydistributedintheground
tiability conditions to the total profile at the transition radius r ,
(cid:15)
assumingasmoothtransitionbetweenthesolitonandtheexternal
NFWhalo.Withthesetwoadditionalconditions,itispossibleto 1 Inequation(7),theterm∝ ψ2T2appearsduetotheinteractionofthe
writetheNFWparametersrsandρsinfunctionoftheotherthree SFwiththethermalbathandtheterm∝T4isduetothethermalbathonly.
MNRAS000,1–17(2016)
RCsofHigh-ResolutionLSBandSPARCGalaxiesinWave(Fuzzy)andMultistateSFDM 5
andexcitedstatesathigherenergylevels,andthegeneralsolution (2013b) (NGC 1003, data from SPARC; NGC 1560, data from
allowsaconfigurationoutofthefullycondensatesystem.Inthelast de Blok et al. (2001b); NGC 6946, data from McGaugh (2005)).
equation,j =1,2,3,...isthenumberoftheexcitedstaterequired InTable2weshowthecharacteristicsofthesegalaxies.
tofitthemassdistribution,ρj = ρj (0)isthecentraldensity The SPARC database contains the near-infrared photometry
0 SFDM
andtheradiusRoftheSFconfigurationisfixedthroughthecon- (tracing the stellar mass distribution) and high-resolution HI/Hα
ditionρj (R) = 0,i.e.k R = jπ.Thissolutionisnaturally rotationcurves(tracingthegravitationalpotentialouttolargeradii)
SFDM j
core.Themassdistributionisgivenby of175diskgalaxies,derivingwithhighresolutionthestellarand
(cid:20) (cid:21) gas components of the galaxies (Lelli et al. 2016b). The sample
4πGρ r sin(2k r)
Mj (r)= 0 1− j . (9) represents a wide range in morphological types, stellar masses,
SFDM k2 2 (2k r)
j j surface brightnesses and gas fractions, and the mass models are
Now,asthesolutiontotheSFperturbationequationislinear, reconstructedfromtheobserveddistributionsofstarsandgasfor
itispossibletohavecombinationsofexcitedstates;thismeansthat differentcharacteristicradiiandvaluesofthestellarmass-to-light
thetotaldensityρ canbewrittenasthesumofthedensities ratio.Thedatabaseincludeslowsurfacebrightness(LSB)galaxies,
SFDM
inthedifferentexcitedstates(Robles&Matos2013b): in major part gas-dominated dwarf galaxies; high surface bright-
ness(HSB)galaxies,inmajorpartbulge-dominated spiralgalax-
ρ (r)=(cid:88)ρj(cid:20)sin(jπr/R)(cid:21)2. (10) ies;andintermediatesurfacebrightness(ISB)galaxies,whichare
SFDM 0 (jπr/R) disk-dominatedsystems.
j
Usually,inLSBgalaxiesthedarkmattercomponentisdom-
ItisworthnotingthattheradiusRinthelastequationisthesame
inantevenatsmallradii,whilstinHSBgalaxiesthecontributionof
foralltheexcitedstatespresentintheconfigurationanditisdefined
thestellarandgascomponentstothetotalmassisimportantatthe
astheradiusoftheSFDMhalo.Suchprofilehasatleasttwofree
centralregions.However,McGaughetal.(2016)foundthat,even
parametersforonestateandeveryadditionalexcitedstateaddsone
when the dark matter is dominant at the inner regions for many
parameter, ρj. The disadvantage of this profile is that the phys-
0 galaxies, the observed acceleration in 153 galaxies of all types
ical parameters of the SF, m and λ, are degenerated with other
ψ (LSB,ISBandHSBgalaxies)fromtheSPARCdatabasestrongly
quantities,likethecriticaltemperatureT attheSSB,thetemperat-
c correlates with the acceleration from the baryonic matter, show-
ureofthehaloattheformationtime,etc.However,thismodelhas ingamassdiscrepancyatthevalueg†=1.2×10−10m/s2.There-
beenprovedsuccessfulinfittingtheobservationsofdwarfspher-
fore,inthiswork,weinvestigatetheconsistencyofthetwoSFDM
oidalgalaxies(Martinez-Medinaetal.2015a),therotationcurves
modelswiththeobservationalrotationcurvesoffourrepresentative
ofgalaxies(Robles&Matos2013b),thedynamicalmassesfrom
SPARCgalaxies,includingthebaryonicinformation.
X-rayobservationsofclustersofgalaxies(Bernaletal.2016),pre-
dictingthesizeoftheEinsteinradiusoflensedgalaxiesbystrong
gravitationallensing(Robles&Matos2013a)andthesurvivability
3.3 Statisticalcalibrationmethod
ofsatelliteSFDMhaloesorbitingaroundaMilkyWay-likegalaxy
(Roblesetal.2015). Thegalaxyrotationcurveisgivenby
(cid:114)
GM (r)
V(r)= T , (11)
r
3 GALAXIES’SAMPLES
where M = M +M +M +M , for M the
3.1 High-ResolutionLSBGalaxies T bulge disk gas DM DM
dark matter contribution depending on the density profile used
We are using two sets of galaxy rotation curves according to the (soliton+NFWormultistateSFDM).InthecaseoftheLSBgalax-
availability of photometric data. For the low surface brightness ies(Subsection3.1),MT = MDM,andforthegalaxieswithpho-
(LSB)galaxies,inwhichthedarkmatteristhedominantcompon- tometricinformation(Subsection3.2)weconsiderthedifferentba-
ent,weareusingtheobservedrotationcurvesof18high-resolution ryoniccomponents(bulge,disk-starsandgas),ifgiven.
LSB galaxies reported in de Blok et al. (2001a). In this work, In order to constrain the free parameters of every SFDM
theauthorsconsideredthevisibledatacontribution,classifyingthe model,weusedtheMarkovChainMonteCarlo(MCMC)method
galaxiesaccordingwiththeavailabilityofphotometricdataandus- (Gamerman1997),throughamaximisationofthelikelihoodfunc-
ingthreemodels:theminimumdiskmodel,wherethedarkmatter tionL(p)givenby
istheprincipalcomponentinthehaloes,makingzeroallthevisible
1 (cid:18) ∆TC−1∆(cid:19)
components for the galaxies with photometry, a constant M/L∗ L(p)= exp − , (12)
(2π)N/2|C|1/2 2
andthemaximumdiskmodelswhereonlythegalaxieswithphoto-
metrydataareconsidered. wherepisthevectorofparameters,Nthenumberofobservational
Giventhatweareinterestedintestingdarkmattermodelsthe points for each galaxy, ∆ = V (r )−V (r ,p), for V
obs i model i obs
set without photometric data will be used. This set is in Table 1, theobservationalcircularvelocityattheradiusr andV the
i model
wherewedescribesomeimportantcharacteristicsofeachgalaxy. derived total velocity for a given SFDM model computed in the
samepositionwhereV wasmeasured,andCadiagonalmatrix.
obs
We sample the parameter space from uniform prior ranges
3.2 NGCgalaxieswithphotometricdata
withtwoMarkovchainsandtestedtheconvergenceofthefitwith
Inthisworkweselected,asrepresentativesystemsofthesecond theGelman-Rubinconvergencecriterion(R−1 < 0.1)(Gelman
setofgalaxieswithphotometricdata,thethreesamplegalaxiesin &Rubin1992).Thefittingparametersand1σ and2σ confidence
McGaughetal.(2016)(NGC7814,6503,3741)fromtheSPARC rangesarecomputedfromtheMarkovchainswith30%asburn-in.
(SpitzerPhotometry&AccurateRotationCurves)database(Lelli Simultaneously, we used the data analysis software ROOT
et al. 2016b), and the sample galaxies used in Robles & Matos (Brun & Rademakers 1997). The method minimises the χ2 er-
red
MNRAS000,1–17(2016)
6 T.Bernaletal.
High-resolutionLSBgalaxies
Galaxy Morphology Mabs Rmax Vmax D
Name (mag) (kpc) (km/s) (Mpc)
(1) (2) (3) (4) (5) (6)
ESO-LV014-0040 Spiral -21.6 29.2 263 212
ESO-LV084-0411 Edge-on -18.1 8.9 61 80
ESO-LV120-0211 FuzzyMagellanicbar -15.6 3.5 25 15
ESO-LV187-0510 Irregularspiral,flocculent -16.5 3.0 40 18
ESO-LV206-0140 Spiral -19.2 11.6 118 60
ESO-LV302-0120 Spiral,hintofbar? -19.1 11.0 86 69
ESO-LV305-0090 Barredspiral -17.3 4.8 54 11
ESO-LV425-0180 Spiral -20.5 14.4 145 86
ESO-LV488-0490 InclinedMagellanicbar -16.8 6.0 97 22
F730-V1 Spiral ... 11.9 145 144
UGC4115 Im -12.4 1.0 40 3.2
UGC11454 Fuzzyspiral,smallcore −18.6a 11.9 152 91
UGC11557 SBdms -20.0 6.2 95 22
UGC11583 dI −14.0a 1.5 36 5
UGC11616 Sc −20.3a 9.6 143 73
UGC11648 I −21.0a 12.7 145 48
UGC11748 Sbc −22.9a 21.0 242 73
UGC11819 dG −20.3a 11.7 153 60
Table1.InthisTableweshowthecharacteristicsofthe18high-resolutionLSBgalaxies(deBloketal.2001a)usedinthepresentwork.Thecolumnsread:
(1)Thenameofthegalaxy,(2)Morphology,(3)Absolutemagnitude(aisZwickymagnitude17),(4)Maximumobservationalradius,(5)Maximumvelocity
oftherotationcurveand(6)Distancetothegalaxy.
Galaxieswithphotometricdata withthedensityprofile(10),alsoseparatingthefitswiththehigh-
Galaxy Morphology Mabs Rmax Vmax D resolution LSB galaxies (Subsection 4.2.1) and the galaxies with
Name (mag) (kpc) (km/s) (Mpc) photometricdata(Subsection4.2.2).
(1) (2) (3) (4) (5) (6)
NGC7814a Sabl -20.15d 5.03 218.9 14.40 4.1 Soliton+NFWfits
NGC6503a Scd -17.7e 3.04 116.3 6.26
NGC3741a Im -13.13f 5.14 50.1 3.21 4.1.1 High-resolutionLSBgalaxies
NGC1003a SA(s)cd -19.2 31.3 115 11.8
Table3showstheresultsforthe18high-resolutionLSBgalaxies
NGC1560b Sd -15.9 8.3 78 3.0
NGC6946c SABcd — — 224.3 10.1 reported in de Blok et al. (2001a), for the ψDM model with the
soliton+NFWdensityprofile(4).AsexplainedinAppendixA,we
restrictedthedensityprofileaskingforcontinuityanddifferentiab-
Table2.ThesameasTable1forthegalaxieswithphotometricdata. ilityofthefunction(4)atthetransitionradiusr ,asasmoothtrans-
(cid:15)
aDatafromtheSPARCdatabase(Lellietal.2016b).
itionbetweenthesolitonandNFWregions.Withtheserestrictions
bDatafromdeBloketal.(2001b).
wehavethreefreeparameters:thecentraldensityρ ,thesoliton-
cDatafromMcGaugh(2005). c
likecoreradiusr ,andthetransitionradiusr .Wereportthe1σ
dMonachesietal.(2016). c (cid:15)
eKodaetal.(2015). errorsfromtheMCMCmethodused,alsothereducedχ2red errors
fGentileetal.(2007a);Duttaetal.(2009). foreachgalaxy.Fromρc andrc weobtainedthebosonmassmψ,
andwithr wecomputedtheNFWparameters,r andρ (seeAp-
(cid:15) s s
pendixA),andadditionally,derivethevirialradiusr andcon-
200
rorstoobtainthebestfitfromtheobservations: centrationparameterc:=r /r ,2showninthesameTable.
200 s
Figure1showstheresultingplotsforthedensityprofile(4)for
χ2red =(cid:88)N (cid:18)Vobs(ri)−σVmodel(ri,p)(cid:19)2 , (13) thefittingparametersshowninTable3;thetransitionradiusr(cid:15) is
i=1 i theverticallineshownforallthegalaxies.Theobservationalerror
barsarebigforsomegalaxiesinthesampleand,ingeneral,thefits
whereσ istheerrorinthemeasurementofV (r ).
i obs i aregood,exceptattheouterregionofUGC11748.Figure2shows
the 1σ and 2σ contours for the parameters m and r from the
ψ c
MCMCanalysis,andthetransitionradiusr isthescattercoloured
(cid:15)
4 RESULTS plot.
FromTable3itisnoticeablethat,ingeneral,r isoftheorder
InthisSectionweshowtheresultsofthefitswiththetwoSFDM c
models. In Subsection 4.1 we present the results for the ψDM
model with the soliton+NFW density profile (4), separately for 2 r200 istheradiuswherethedensityis200timesthecriticaldensityof
the high-resolution LSB galaxies (Subsection 4.1.1) and for the theuniverse,anddefinesthehaloradius.IntheCDMframework,concen-
galaxieswithphotometricinformation(Subsection4.1.2).InSub- trationsarestronglycorrelatedwiththehaloformationepoch,andmustbe
section 4.2 we show the results for the multistate SFDM model inagreementwiththeN-bodysimulations.
MNRAS000,1–17(2016)
RCsofHigh-ResolutionLSBandSPARCGalaxiesinWave(Fuzzy)andMultistateSFDM 7
EESSOO 00114400004400 EESSOO 00884400441111 EESSOO 11220000221111
80 35
300
70 30
250 60 25
V [km/s] 125000 V [km/s] 4500 V [km/s] 1250
30 10
100
20 5
50 10 0
00 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 9 -50 0.5 1 1.5 2 2.5 3 3.5
r [kpc] r [kpc] r [kpc]
EESSOO 11887700551100 EESSOO 22006600114400 EESSOO 33002200112200
45 140 100
40 120
35 80
100
30
m/s] 25 m/s] 80 m/s] 60
V [k 20 V [k 60 V [k 40
15
40 20
10
5 20 0
0 0.5 1 1.5 2 2.5 3 00 2 4 6 8 10 12 0 2 4 6 8 10 12
r [kpc] r [kpc] r [kpc]
EESSOO 33005500009900 EESSOO 44225500118800 EESSOO 44888800004499
70 180 120
60 160 100
50 140
V [km/s] 3400 V [km/s] 11028000 V [km/s] 6800
20 60 40
10 40 20
0 20 0
-100 1 2 3 4 5 00 2 4 6 8 10 12 14 0 1 2 3 4 5 6
r [kpc] r [kpc] r [kpc]
FF 773300__vv11 UU 44111155 UU 1111445544
160 160
140 40 140
120 120
30
m/s] 100 m/s] m/s] 100
V [k 80 V [k 20 V [k 80
60 60
10
40 40
20 0 20
00 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 00 2 4 6 8 10 12
r [kpc] r [kpc] r [kpc]
UU 1111555577 UU 1111558833 UU 1111661166
160
40
80 140
30 120
V [km/s] 4600 V [km/s] 20 V [km/s] 10800
60
10
20 40
0 20
0
0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 00 2 4 6 8 10
r [kpc] r [kpc] r [kpc]
UU 1111664488 UU 1111774488 UU 1111881199
160
140 250 160
140
120
200 120
V [km/s] 10800 V [km/s] 150 V [km/s] 10800
60 100 60
40 40
50
20 20
00 2 4 6 8 10 12 00 2 4 6 8 10 12 14 16 18 20 22 00 2 4 6 8 10 12
r [kpc] r [kpc] r [kpc]
Figure1.Bestfitsforthe18high-resolutionLSBgalaxieswithoutphotometryfortheψDMmodelusingthesoliton+NFWprofile,correspondingtothefitting
parametersshowninTable3.Theverticallinescorrespondtothetransitionradiir(cid:15).ItisworthnotingthatinUGC11648r(cid:15) isnegligible,inthiscasethe
soliton-likecontributiontothetotalprofileisnotrelevant.
MNRAS000,1–17(2016)
8 T.Bernaletal.
High-resolutionLSBgalaxies
Soliton+NFWprofile
FITTINGPARAMETERS
Galaxy ρc(M(cid:12)/pc3) rc(kpc) r(cid:15)(kpc) mψ(10−23eV) c r200(kpc) χ2red
ESO-LV014-0040 0.222+0.044 3.12±0.76 3.23±0.92 0.352+0.063 21.029 254.131 0.658
−0.091 −0.16
ESO-LV084-0411 0.00588+0.00068 4.90+1.5 4.0+1.1 0.986+0.011 — — 0.092
−0.0015 −0.99 −1.6 −0.58
ESO-LV120-0211 0.0266+0.0072 1.01+0.17 1.44+0.35 9.6+1.6 44.524 21.505 0.074
−0.0091 −0.27 −0.78 −3.9
ESO-LV187-0510 0.0466+0.0092 1.20+0.27 1.46+0.36 5.28+0.87 21.118 37.729 0.101
−0.014 −0.34 −0.84 −2.7
ESO-LV206-0140 0.172+0.031 1.65+0.31 1.80+0.39 1.34+0.25 23.063 105.844 0.133
−0.048 −0.36 −0.56 −0.49
ESO-LV302-0120 0.0423+0.0068 2.66±0.57 3.28+0.89 1.09+0.14 21.556 78.971 0.032
−0.012 −1.4 −0.48
ESO-LV305-0090 0.0265+0.0044 2.26+0.68 2.30+0.77 2.589+0.020 — — 0.076
−0.0099 −0.89 −1.4 −1.9
ESO-LV425-0180 0.0404+0.0067 3.7±1.1 3.42+1.2 0.653+0.053 6.139 251.269 0.509
−0.022 −0.83 −0.35
ESO-LV488-0490 0.089+0.014 1.98±0.45 2.24+0.52 1.37+0.22 20.398 90.886 0.131
−0.025 −1.0 −0.62
F730-V1 0.181+0.035 1.84+0.34 1.95+0.39 1.08+0.22 21.181 129.491 0.329
−0.050 −0.46 −0.67 −0.44
UGC4115 0.152+0.020 1.06+0.14 1.037+0.075 6.5+1.1 14.418 91.143 0.132
−0.043 −0.66 −0.79 −5.8
UGC11454 0.157+0.030 1.96+0.39 1.96+0.44 1.0+0.21 16.037 156.477 0.364
−0.040 −0.44 −0.55 −0.38
UGC11557 0.0191+0.0018 3.7+1.4 2.77+0.94 1.152−0.072 — — 0.12
−0.0062 −1.2 −1.1 −0.83
UGC11583 0.095+0.022 0.86+0.11 1.45+0.25 7.2+1.9 — — 0.097
−0.026 −0.26 −1.0 −3.0
UGC11616 0.163+0.019 1.93±0.28 2.10+0.38 0.97+0.13 22.378 122.007 0.189
−0.032 −0.49 −0.27
UGC11648 2.26±0.25 0.0858+0.010 0.0737+0.0089 128+20 8.375 183.7 1.097
−0.0093 −0.0080 −30
UGC11748 0.97+0.13 1.70+0.23 2.14±0.35 0.502+0.056 77.024 159.495 1.978
−0.23 −0.21 −0.098
UGC11819 0.0710+0.0040 3.92+0.51 4.9±1.1 0.351+0.017 27.976 139.349 0.28
−0.0096 −0.26 −0.077
Table3.InthisTableweshowtheresultingparametersρc,rcandr(cid:15)fortheψDMmodelwiththesoliton+NFWdensityprofile(4),andtheresultingboson
massmψ,allthequantities±1σerrorsfromtheMCMCmethodused.Wealsoshowtheresultingr200fromtheNFWhaloradius,concentrationparameter
candχ2 errorsforthe18high-resolutionLSBgalaxiesindeBloketal.(2001a).
red
ofthetransitionradiusr .Thismeansthatthesolitoncontribution resultingbosonmassm ,thetransitionradiusr andthereduced
(cid:15) ψ (cid:15)
isoverlappedwiththeNFWhalo(asweaskforasmoothtransition χ2 errorsfromthefittingmethod.TheNFWparametersr and
red s
betweenbothprofiles),andfromthebestfits,thevaluesofr do ρ wereobtainedfromthelastthreefittingparameters,andwere-
(cid:15) s
notcorrespondtor >3r (Schiveetal.014a). ported in the same Table the corresponding virial radii r and
(cid:15) c 200
Moreover,fromFigure1itiseasytoseethatthesolitoncon- concentrationparametersc:=r /r .
200 s
tribution to the total density profile is negligible for UGC11648, In Figure 3 we show the plots for the corresponding fitting
meanwhileforESO4250180andF730-V1thesolitoncontribution parameters presented in Table 4; the vertical lines show the cor-
spansonlyoneobservationalpoint.Forthelastthreegalaxiesthe respondingtransitionradiusr foreachgalaxy.Aswenoticefrom
(cid:15)
maincontributiontothetotalvelocitycomesfromtheNFWpro- Table4andFigure3,thesoliton+NFWprofilefitstheobservations
file, and for UGC11648 is not necessary to add the soliton part. ofNGC7814,6503and6946onlyoncethebaryoniccontribution
Thus,toobtainthevalueofthebosonmasswediscardedthegalaxy is taken into account, except some points inside 5 kpc for NGC
UGC11648,sincefromthenegligiblesolitoncontributioninclud- 6503.ForNGC1003andNGC6946,thetransitionradiusr isvery
(cid:15)
ingm ,thevalueobtained(m ∼10−21 eV)isnotreliable.For smallwithrespecttothemaximumradiusR ,i.e.inthesecases
ψ ψ max
thelast15galaxies,thesolitoncontributionbecomesimportantin the whole rotation curve is better explained with the NFW pro-
order to fit the whole rotation curves. From the 17 galaxies (ex- fileandthesolitonpartdoesnotintroduceanyimprovement.For
ceptfromUGC11648)thevaluesobtainedforthebosonmassare NGC3741,whichisagas-dominateddwarfgalaxy,theNFWpro-
in the range 0.351 < m /10−23eV < 9.6, and core radius in filefitsthedataverywell,bothwithandwithoutphotometry.And
ψ
0.86<r/kpc<4.9. forNGC1003thefitisbetterincludingthebaryons,butaswenote,
evenwhenthefitisbetter,theNFWprofilecannotexplaintheos-
cillationsfromthehigh-resolutionobservations(seealsoRobles&
4.1.2 NGCgalaxieswithphotometricdata Matos2013b);inthiscasethemultistateSFDMprofileisrelevant
(seeSubsection4.2).Finally,NGC1560istheonlygalaxywhich
InTable4weshowthefittingparametersforthethreerepresent-
remainswithalmostthesamecontributionfromtheψDMmodel,
ativegalaxiesinMcGaughetal.(2016)(NGC7814,6503,3741)
withoutandwiththephotometricinformation.Theresultingranges
fromtheSPARCDatabase(Lellietal.2016b)andthethreesample
fromtheDM+baryonsfitsforthebosonmassandcoreradiusare:
galaxies analysed in Robles & Matos (2013b) (NGC 1003, data
0.264 < m /10−23eV < 30.0 and 0.311 < r/kpc < 4.84,
fromSPARC;NGC1560,datafromdeBloketal.(2001b);NGC ψ
respectively.
6946, data from McGaugh (2005)), for the ψDM model with the
soliton+NFW density profile (4). In the top panels we show the
resultsforthegalaxieswithoutthephotometricinformationandon
4.2 MultistateSFDMfits
thebottompanelstheresultsincludingthebulge,diskand/orgas
profiles, to analyse the effect of the baryons on the DM profile. InthisSubsectionwepresenttheresultsforthemultistateSFDM
We report the central density ρ and the core radius r , with the model.Itisworthmentioningthatwithahaloformedofoneex-
c c
MNRAS000,1–17(2016)
RCsofHigh-ResolutionLSBandSPARCGalaxiesinWave(Fuzzy)andMultistateSFDM 9
Figure2.Posteriordistributionsfortheparametersmψ(10−23eV)andrc(kpc),withthetransitionradiusr(cid:15)(kpc)asthecolouredscatterplot,forthehigh-
resolutionLSBgalaxiesintheψDMmodel,correspondingtothefittingparametersshowninTable3.Thecontoursarethe1σand2σconfidenceregions.
TheplotswereobtainedwiththeGetDist0.2.6Package.
citedstateonly(twofreeparameters)itispossibletoreproducethe (the halo radius). We also computed the mass in the higher en-
rotationcurvesofsomegalaxiesinthesample,however,insome ergy level M and in the lower state M , as well as the mass
j i
cases,thenumberoftheexcitedstateisveryhighandasaresult, ratio η = M /M . Ureña-López & Bernal (2010) showed it
j i
the halo radius R is much greater than the maximum observable is possible to have stable multistate configurations for values of
radiusR ;inthesecases,wepreferredlowerexcitationstates. η = M /M (cid:46) 1.3withrespecttothegroundstateM ,i.e.the
max j 1 1
Additionally,itisawell-knownresultthatitisnotpossibletorepro- multistatehaloesdonotdecaytothegroundstate.Therefore,from
ducethewholerotationcurveofbiggalaxieswiththegroundstate ourresults,wemightarguethatthefinalconfigurationsarestable
only(Boehmer&Harko2007;Robles&Matos2012;Fernández- wherethegroundstateappearsasadominantcomponent.However,
Hernández et al. 2016), so we do not include these results in the thereare8galaxiesinwhichthelowerdominantstateisi=2,3or
article. eveni=6;inthesesystemsthefinaldistributionindifferentexcit-
Thereportedcombinationsofexcitedstatesaretheonesthat ationlevelsmightbethefinalproductofinteractionswiththeba-
bestreproducethedata.Itisimportanttonotethatinsomecases, ryonicmatter.Furthernumericalsimulationsareneededtoinvest-
the resulting configurations do not display the ground state as a igatethestabilityofmultistateconfigurationswithoutadominant
dominantcomponent.Thismightbetheresultofthedifferentform- groundstateandtheinfluenceofthebaryonicmatterintheSFDM
ationhistoriesforeverygalaxy,includinginteractionswiththeba- halo.
ryonicmatterresultingindiversemultistateconfigurations.
With the haloes formed by two states only we found a very
goodagreementwiththeobservationalvelocitiesforallthegalax-
4.2.1 High-resolutionLSBgalaxies
ies.Figure4showstheresultingbest-fitprofilesforthesegalaxies.
In Table 5 we show the results for the 18 high-resolution LSB Forsomegalaxies,forexample,F730-V1andUGC11748,thefits
galaxies from de Blok et al. (2001a), for the multistate SFDM mightbeimprovedbyintroducingathirdexcitedstate;howeverit
model(10)withtwoexcitedstates.Wereportthefittingparamet- isnotnecessarytoincludeadditionaldegreesoffreedomandwe
ers:ρi andρj (thecentraldensitiesforthestatesiandj)andR donotreportthesefitsinthework.
0 0
MNRAS000,1–17(2016)
10 T.Bernaletal.
NGCgalaxieswithphotometricdata
DM-onlyfits
Soliton+NFWprofile
Galaxy ρc(M(cid:12)/pc3) rc(kpc) r(cid:15)(kpc) mψ(10−23eV) c r200(kpc) χ2red
NGC7814 47.3±4.8 0.0872+0.010 0.0774+0.0096 27.0+2.8 69.425 147.32 5.541
−0.0067 −0.0063 −5.5
NGC6503 14.3+1.9 0.0464+0.0036 0.0402+0.0032 171±20 28.59 98.725 6.38
−1.3 −0.0028 −0.0025
NGC3741 1.33+0.13 0.0367±0.0050 0.0315±0.0043 932+200 6.032 72.329 0.646
−0.20 −300
NGC1003 0.90+0.52 0.154+0.046 0.133+0.040 69+70 8.778 124.246 4.864
−0.24 −0.065 −0.056 −30
NGC1560 0.0563+0.0064 1.39±0.18 1.31±0.20 3.16+0.50 7.983 95.974 0.395
−0.0088 −0.78
NGC6946 0.0514+0.0032 4.92+0.36 6.28±0.70 0.254+0.019 27.294 150.211 1.616
−0.0047 −0.30 −0.029
DM+baryonsfits
Soliton+NFWprofile
Galaxy ρc(M(cid:12)/pc3) rc(kpc) r(cid:15)(kpc) mψ(10−23eV) c r200(kpc) χ2red
NGC7814 0.0557+0.0066 4.84+0.80 5.6±1.2 0.264+0.028 18.492 177.278 0.86
−0.014 −0.66 −0.070
NGC6503 0.0459+0.0041 2.81+0.29 3.07+0.38 0.831+0.072 13.809 106.975 1.547
−0.0068 −0.24 −0.34 −0.13
NGC3741 0.0633+0.0082 0.68+0.21 0.60+0.18 18.0+1.4 4.687 93.217 2.115
−0.029 −0.27 −0.25 −12
NGC1003 0.231+0.061 0.311+0.033 0.269+0.029 30.0+8 5.062 135.755 2.93
−0.032 −0.052 −0.045 −4
NGC1560 0.0411+0.0042 1.46±0.20 1.38+0.20 3.35+0.47 7.113 86.89 0.234
−0.0065 −0.23 −0.83
NGC6946 0.27+0.16 0.62+0.33 0.55+0.30 8.7+7.9 9.376 143.84 5.34
−0.12 −0.27 −0.24 −4.9
Table4.InthisTableweshowthefittingparametersρc,rcandr(cid:15),theresultingbosonmassmψ,theconcentrationparameterscandr200radiifromthe
NFWprofile,andtheχ2 errorsforthesoliton+NFWprofile,fortheNGCgalaxieswithphotometricinformation.InthetoppanelweshowtheDM-onlyfits
red
andinthebottompanelthefitstakingintoaccountthebaryoniccontribution.
High-resolutionLSBgalaxies
MultistateSFDMwithtwostates
FITTINGPARAMETERS
Galaxy i,j R(kpc) ρi0(M(cid:12)/pc3) ρj0(M(cid:12)/pc3) Mi(1010M(cid:12)) Mj(1010M(cid:12)) η χ2red
ESO-LV014-0040 2,7 46.93±2.071 0.031±0.0052 0.2±0.025 26.5219 15.6727 0.59 0.1262
ESO-LV084-0411 1,6 19.7±8.85 0.0039±0.0012 0.0033±0.0024 0.7595 0.0214 0.03 0.0029
ESO-LV120-0211 3,8 15.56±2.919 0.0035±0.0024 0.029±0.0098 0.0253 0.0264 1.04 0.0204
ESO-LV187-0510 2,7 8.404±1.512 0.024±0.0078 0.045±0.022 0.1014 0.0127 0.13 0.0029
ESO-LV206-0140 2,7 23.1±1.51 0.021±0.0037 0.17±0.02 2.0117 1.4056 0.70 0.2341
ESO-LV302-0120 1,2 12.02±2.235 0.0097±0.0078 0.028±0.0086 1.0633 0.771 0.73 0.0084
ESO-LV305-0090 1,5 7.699±1.987 0.016±0.0048 0.025±0.013 0.3412 0.0173 0.051 0.0078
ESO-LV425-0180 1,9 26.3±5.25 0.008±0.0026 0.19±0.099 5.4805 1.5405 0.28 0.0054
ESO-LV488-0490 3,8 23.8±2.954 0.035±0.0094 0.064±0.017 1.0278 0.2134 0.21 0.0054
F730-V1 1,4 14.8±1.235 0.02±0.0041 0.18±0.02 3.9812 1.7529 0.44 1.0227
UGC4115 1,5 2.25±0.978 0.13±0.037 0.031±0.086 0.0356 0.0004 0.011 0.0023
UGC11454 1,3 14.27±0.624 0.025±0.0026 0.1±0.0077 4.4521 1.7955 0.40 0.2939
UGC11557 1,5 23.31±14.06 0.01±0.0028 0.0079±0.0049 0.8712 0.0603 0.069 0.0392
UGC11583 1,4 6.413±2.922 0.015±0.031 0.087±0.024 0.0199 0.0227 1.14 0.0801
UGC11616 2,5 25.56±1.844 0.021±0.0041 0.11±0.0078 2.5003 1.7909 0.72 0.5551
UGC11648 1,7 19.36±1.131 0.012±0.0012 0.23±0.017 4.3513 1.4266 0.33 0.6469
UGC11748 6,9 41.12±0.8 0.27±0.032 0.33±0.051 16.8126 9.5328 0.57 1.6754
UGC11819 2,7 24.7±0.997 0.043±0.0034 0.062±0.013 5.2072 0.5528 0.11 0.0931
Table5.InthisTableweshowthefittingparametersR,ρi andρjfortwoexcitationstatesi,jinthemultistateSFDMmodel,forthe18high-resolutionLSB
0 0
galaxiesindeBloketal.(2001a).WereporttheresultingmassesMiandMj foreachstate,themassratioη = Mj/Miandχ2rederrorsfromthefitting
method.
4.2.2 NGCgalaxieswithphotometricdata NGC1560,datafromdeBloketal.(2001b);NGC6946,datafrom
McGaugh(2005)).Forthehaloesformedbytwostateswereport:
ρi,ρj andR,all±1σerrorsfromthefittingmethodused.Wealso
InTable6wepresenttheresultsforthemultistateSFDMmodel, 0 0
reportthetotalmassesM andM ,andtheresultingmassratios
for the three representative SPARC galaxies in McGaugh et al. i j
η = M /M . For the DM+baryons fits we obtain four galaxies
(2016)(NGC7814,6503,3741)andthethreesamplegalaxiesana- j i
withadominantgroundstateandtwowithexcitedstatesonly.In
lysedinRobles&Matos(2013b)(NGC1003,datafromSPARC;
MNRAS000,1–17(2016)