Flat Central Density Profile and Constant DM Surface Density in Galaxies from Scalar Field Dark Matter

0.31 MB·English

by  Victor H. Robles

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed17January2012 (MNLATEXstylefilev2.2)
Flat Central Density Profile and Constant DM Surface
Density in Galaxies from Scalar Field Dark Matter
Victor H. Robles⋆ and T. Matos†
2
1 Departamento de Física,Centro de Investigación y de Estudios Avanzados del IPN, AP14-740, 0700 D.F., México
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2

n
17January2012
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ABSTRACT
1
Thescalarfielddarkmatter(SFDM)modelproposesthatgalaxiesformbycondensa-

] tionofascalarfield(SF)veryearlyintheuniverseformingBose-EinsteinCondensates
O (BEC) drops, i.e., in this model haloes of galaxies are gigantic drops of SF. Here big
C structures form like in the LCDM model, by hierarchy,thus all the predictions of the
LCDM model at big scales are reproduced by SFDM. This model predicts that all
.
h galaxiesmust be very similar and exist for bigger redshifts than in the LCDM model.
p In this workwe show thatBEC dark matter haloes fit high-resolutionrotationcurves
-
o of a sample of thirteen low surface brightness galaxies. We compare our fits to those
r obtainedusingaNavarro-Frenk-WhiteandPseudo-Isothermal(PI)profilesandfound
t abetteragreementwiththeSFDMandPIprofiles.Themeanvalueofthelogarithmic
s
a inner density slopes is α = −0.27 ± 0.18. As a second result we find a natural way
[ to define the core radius with the advantage of being model-independent. Using this

new definition in the BEC density profile we find that the recent observation of the
1
v constant dark matter central surface density can be reproduced. We conclude that in
2 light of the difficulties that the standard model is currently facing the SFDM model
3 can be a worthy alternative to keep exploring further.
0
3 Key words: cosmology:darkmatter–cosmology:observations–galaxies:fundamen-
. tal parameters
1
0
2
1
: 1 INTRODUCTION For instance, in the ΛCDM simulations the halos present
v
rising densities towards the central region behaving as
Xi In the Λ Cold Dark Matter (ΛCDM or LCDM) model, ρ ∼ r−1 well within 1 kpc (Navarro et al. 2010). On the
known as the standard model of cosmology, the formation
r other hand various observations suggest that the rotation
a of structures in the universe is through a hierarchical pro-
curves are more consistent with a constant central den-
cess ofgrowth of structures,meaningthat small structures,
sity(Kuzio deNaray et al.2008;deBlok2010),thisismost
like small haloes of galaxies merge to form bigger ones, like
commonlyknownasthecusp/coreproblem(deBlok 2010).
galaxyclustersandsuperclustershaloes.Inthispicture,the
Studyingawiderangeofgalaxiesofdifferentmorpholo-
universe contains around 96 per cent of an unknown form
gies and with magnitudes in the interval −22 6 M 6 −8
of energythat isusually called darkmatter(DM)and dark B
Donato et al. (2009) fit their rotation curves (RC) using a
energy. The ΛCDM model can successfully describe cosmo-
Burkert profile for theDM (Burkert 1995) and found that
logical observations such as the large scale distribution of
galaxies,thetemperaturevariationsinthecosmicmicrowave log(µ0/M⊙pc−2)=2.15±0.2 (1)
backgroundradiationandtherecentaccelerationoftheuni-
remains approximately constant, where
verse(Coles2005;Lahav & Liddle2010;Peebles et al.2009;
Guo et al. 2011).
µ0 =ρ0r0 (2)
However, recent observations in far and nearby galax-
ies have shown that the model faces serious conflicts when withρ0 thecentralDMdensityandr0 thecoreradius.Sim-
trying to explain the galaxy formation at small scales ilar results where found in Kormendy & Freeman (2004);
(see Robles & Matos 2011 for a review; Friedmann 2011). Spanoet al.(2008).Exploringfurthertheconstantvalueof
µ0 for the DM, Gentile et al. (2009) found that within r0
the DM central surface density in terms of the mass inside
⋆ E-mail:vrobles@fis.cinvestav.mx it (M<r0) is < Σ >0,DM= M<r0/πr02 ≈ 72+−4227M⊙pc−2,the
† E-mail:tmatos@fis.cinvestav.mx gravitational acceleration due to DM felt by a test particle
(cid:13)c 0000RAS
2 Victor H. Robles and T. Matos
at theradius r0 was found to be (iii) Withthismass,thecriticalmassofcollapseforareal
gDM(r0)=Gπ<Σ>0,DM=3.2+−11..82×10−9cms−2, (3) hscaalolaersfi(eAldlcuisbijeursrte1e0t12alM. 2⊙0,0i2.)e.., the one observed in galaxy
additionallytheyreportedtheacceleration duetothelumi-
(iv) The scalar field has a natural cut off, thus the
nous matterat r0 to begbar(r0)=5.7+−32..88×10−10cms−2. substructure in clusters of galaxies is avoided naturally.
IntheΛCDMmodelthegalaxies haveevolvedthrough With a scalar field mass of m ∼ 10−22eV the amount
φ
numerousmergersandgrownindifferentenvironments,the
of substructure is compatible with the observed one
star formation and basic properties of the galaxies are not
(Matos & Ureña2001; Suárez& Matos 2011).
expected to be a common factor among them. Therefore
(v) WeexpectthatSFDMformsgalaxiesearlierthanthe
explaining both the constancy of µ0 and the core in the cold dark matter model, because theyform BECs at acrit-
centralregionsofgalaxiesseemsveryunlikelyinthismodel.
ical temperature T >> TeV. So if SFDM is right, we have
c
These problems can be used to test alternative DM
to see big galaxies at big redshifts with similar features.
models. There are other models that do not include DM
(vi) And recently it has been demonstrated that SFDM
but instead modify the Newtonian force law F= ma, one
haloes maintain satellite galaxies going around big galaxies
of them is called MOND and was proposed by Milgrom
for enough time to explain the existence of old stars in the
(Milgrom 2010; Sanders 2009), in this modification New- satellites, provided that the mass of the SF is just m ∼
φ
ton’slawreadsF=maµ(a/a0),wherethefixedacceleration 10−22eV(Lora et al. 2011).
scale a0 divides the Newtonian and MONDian regimes, for
x << 1 µ(x) = x we have the MONDian regime and for
The idea was first considered by Sin (1994);
x >> 1 we recover the usual Newtonian acceleration, the
Ji S. U.& Sin S.J. (1994) and independently intro-
valueof a0∼1.2×10−10m/s2. ducedbyGuzmán & Matos(2000).IntheBECmodel,DM
Lately, thescalar field dark matter(SFDM) model has
halos can bedescribed in thenon-relativistic regime, where
received much attention. When the scalar field contains a
DM halos can beseen as a Newtonian gas. If we consider a
self-interaction this model is also called the Bose-Einstein
SFself-interaction,weneedtoaddaquartictermtotheSF
Condensate(BEC)darkmattermodel,bothnamesareused
potential,inthiscasetheequationofstateoftheSFisthat
in the literature and are interchangeable.The main idea is
of a polytope of index n=1 (Suárez & Matos 2011; Harko
simple (Guzmán & Matos 2000). The SFDM model pro-
2011a). Different issues of BEC DM halos and the cosmo-
poses that galaxies form by condensation of a scalar field
logical behavior of the BEC model have been studied in
(SF)withanultra-lightmassoftheorderofm ∼10−22eV.
φ Colpi et al. (1986); Gleiser et al. (1988); Bohmer & Harko
Therefore, when we mention the SFDM or BEC model in
(2007); Harko (2011a); Matos & Guzmán (2001); Chavanis
this paper we are describing a scalar field that conden-
(2011).
satessomehowandbecomesBoseEinsteinCondensatedark
It is a fact that any model trying to become a serious
matter. From this mass it follows that the critical temper-
alternative to ΛCDM has to succeed not only in reproduc-
ature of condensation T ∼ 1/m5/3 ∼TeV is very high,
c φ ing observations in which the standard model fails but also
thus, they form Bose-Einstein Condensates (BEC) drops
hastokeepthesolid descriptionatlargescale.Forthisrea-
very early in the universe. It has been proposed that these
son, our aim in this work is to test the BEC model with
drops are the haloes of galaxies (see Matos & Ureña 2001),
thetwo observations mentioned above, the cusp/core prob-
i.e.,thathaloesaregiganticdropsofSF.Ontheotherside,
lem (Bohmer & Harko 2007) and the constant DM central
big structures form like in the LCDM model, by hierarchy
surface density. In order to do this, we used the Thomas-
(Matos & Ureña2001;Suárez& Matos2011),thus,allpre-
Fermi approximation and a static BEC DM halo to fit ro-
dictionsoftheLCDMmodelatbigscalesarereproducedby
tation curvesof a set of galaxies. However, so far therewas
SFDM. In other words, in the SFDM model the haloes of
no comparison between the density profile and the data, in
galaxies do not form hierarchically, they from at the same
this work we fill this blank by fitting rotation curves of 13
timeandinthesamewaywhentheuniversereachesthecrit-
high resolution low surface brightness (LSB) galaxies and
icaltemperatureofcondensationoftheSF,inasimilarway
additionally compare the fits to two characteristic density
as water dropsform in theclouds. From thisit follows that
profiles 1) the cuspy Navarro-Frenk-White (NFW) profile
allgalaxiesmustbeverysimilarbecausetheyformedinthe
that results from N-body simulations using ΛCDM and 2)
samemannerandatthesamemoment.Therefore,fromthis
thePseudoIsothermal(PI)coreprofile.Thecomparisonal-
modelwehavetoexpectthatthereexistwellformedgalaxy
low ustoshow that themodelis in general agreement with
haloes at bigger redshifts than in theLCDM model. Inthis
the data and with a core in the central region. For our sec-
modelthescalarparticleswiththissmallmassaresuchthat
ondresultwefoundthatthemeaningofacoreissomewhat
their wave properties avoid the cusp and reduce the high
ambiguous. In order to clarify the meaning and unify the
number of small satellites (Hu,Barkana& Gruzinov 2000)
description, we propose a new definition for the core and
which is anotherproblem that is still present in theΛCDM
core radius that allow us to decide when a density profile
model(see Robles & Matos 2011 for a review;Klypin et al.
is cusp or core. Using thisdefinition in theBEC model dis-
1999).Summarizing,itisremarkablethatwithonlyonefree
cussed above we find that the BEC model can reproduced
parameter,theultra-lightscalarfieldmass(m ∼10−22eV),
φ the constant value of µ0 and as a crosscheck we used the
theSFDM model fits:
PIprofileandfindourresultstobeinverygood agreement
(i) The evolution of the cosmological densities withobservations.Thisarguesinfavorofthemodelandour
(Matos,Vazquez & Magaña 2009). definition.
(ii) The rotation curves of big galaxies (Harko 2011a; In section 2, we describe the density profiles we use to
Bernal et al. 2008) and LSB galaxies. fittherotation curvesandwegiveourdefinitionofthecore
(cid:13)c 0000RAS,MNRAS000,000–000
FlatCentral DensityProfile andConstantDMSurface Densityin GalaxiesfromScalarFieldDarkMatter 3
and core radius. In section 3, we fit the galaxy data and Additionaly, the logarithmic slope of the rotation curve is
obtain thefittingparameters andthecoreradiusforthePI defined(Harko2011b) by
andBECdensityprofiles.Insection4,wediscussourresults
d(log V)
and in section 5 we give our conclusions. β= (12)
d(log r)
from (9) we get
2 DARK MATTER HALO DENSITY 1 (πr/R)2
PROFILES β=− 1− . (13)
2 1−(πr/R)cot(πr/R)
(cid:20) (cid:21)
Inthissection weprovidethedarkmatterprofilesthat will
be used in the analysis. In the last part of the section we
2.2 Pseudo Isothermal profile
brieflydescribe theusualmeaning of thecore and establish
a new definition for it. All the empirical core profiles that exist in the literature
fit two parameters, a scale radius and a scale density. A
characteristic profile of this typeis
2.1 BEC profile
ρPI
The case in which the dark matter is in the form of a ρ = 0 , (14)
PI 1 + (r/R )2
staticBose-EinsteincondensateandthenumberofDMpar- c
ticles in the ground state is very large was considered in this is the PI profile (Begeman et al. 1991). Here R is the
c
Bohmer & Harko (2007). Following this paper and assum- scale radius and ρPI is the central density. The rotation
0
ing the Thomas-Fermi approximation (Dalfovo et al. 1999) curveis
which neglects the anisotropic pressure terms that are rel-
evant only in the boundary of the condensate, the system V(r) = 4πGρPIR2 1− Rc arctan r . (15)
of equations describing the static BEC in a gravitational PI s 0 c(cid:18) r (cid:18)Rc(cid:19)(cid:19)
potential V is given by
ρ
∇p =−ρ∇V (4) 2.3 Navarro-Frenk-White profile
m
(cid:18) (cid:19) The NFW profile emerges from numerical simulations
∇2V =4πGρ, (5) that use only CDM and are based on the ΛCDM model
(Dubinskiet al. 1991; Navarro et al. 1996, 1997). In addi-
with thefollowing equation of state tion to this, we have chosen this profile because it is repre-
p(ρ)=U0ρ2, (6) sentative of what is called the cuspy behavior (α ≈ −1) in
the center of galaxies due to DM. The NFW density pro-
whereU0 = 2πm~32a, ρis themass densityof thestaticBEC firelesp(eNctaivvaerlyrobeyt al. 1997) and the rotation curve are given
configuration and p is the pressure, as we are considering
ρ
zero temperature p is not a thermal pressure but instead it ρ (r)= i (16)
is producedbythestrong repulsiveinteraction between the NFW (r/Rs)(1 + r/Rs)2
groundstatebosons.Assumingsphericalsymmetryandde-
notingRastheradiusatwhichthepressureanddensityare 1 r r/R
z2e0r0o7,)the density profile takes the form (Bohmer & Harko VNFW(r)=p4πGρiRs3sr(cid:20)ln(cid:18)1 + Rs(cid:19)− 1 + r/sR(s1(cid:21)7,)
ρB(r)=ρB0 sink(rkr) (7) ρthieisharelolacteodllatposetdheanddenRsis2tyisoaf cthhaeraucntievreirssteicartadthiues.moment
where k = Gm3/~2a=π/R and ρB =ρ (0) is the BEC
0 B
central density, m is the mass of the DM particle and a is 2.4 Meaning of the core radius and cusp/core
p
thescattering lenght. The mass at theradius r is given by discrepancy
4πρB sin(kr)
m(r)= 0 r −cos(kr) , (8) Inthelargescalesimulationsthatusecollissionlesscolddark
k2 (cid:18) kr (cid:19) matter the inner region of DM halos show a density distri-
from here the tangential velocity V of a test particle at a bution described by a power law ρ ∼ rα with α ≈ −1,
B
distance r, is such behavour is what is now called a cusp. On the other
hand, observations mainly in dwarf and LSB galaxies seem
4πGρB sin(kr)
V2(r)= 0 −cos(kr) . (9) to prefer a central density going as ρ ∼ r0. This discrep-
B k2 kr
(cid:18) (cid:19) ancy between observation and theCDM model receives the
The logaritmic slope of a density profile is defined as name of cusp/core problem. Among the empirical profiles
d(log ρ) most frequently used to describe the constant density be-
α= (10) havior in these galaxies are the PI (Begeman et al. 1991),
d(log r)
the isothermal (Athanassoula et al. 1987) and the Burkert
using (7) in (10) it is obtained (Harko2011b)
profile(Burkert1995).Eventhoughtheirbehaviorissimilar
πr πr in the central region and is specified by the central density
α(r)=− 1− cot . (11)
R R fitting parameter, their second parameter called the core
(cid:20) (cid:18) (cid:19)(cid:21)
(cid:13)c 0000RAS,MNRAS000,000–000
4 Victor H. Robles and T. Matos
radius does not represent the same idea. For instance, in
F568-3 F568-3 DM only
the PI profile (eq.(14)) we see that the core radius will be
the distance in which the density is half the central den- 100 100
wsithye.nFρobrurtkh(eRBbuurrkk)er=t ρpbruorfikl/e4thanedcoforer arandiisuosthRecbrumrkalwpirlolfiblee m/s) 6800 6800
c 0 k
(I) (Spanoet al. 2008) ρI(RcI)=ρI0/23/2. Hence, we see an V ( 40 40
20 20
ambiguity in the meaning of the core radius, they get the
0 0
samenamebuttheinterpretation dependson theprofile.If 0 2 4 6 8 10 0 2 4 6 8 10
wewanttocomparethecentraldensityofLSBgalaxieswith r (kpc) r (kpc)
that of NFW, it usually suffice to have a qualitative com-
parison, so far this is what we have been doing by fitting F583-1 F583-1 DM only
empirical profiles. However, high resolution rotation curves 100 100
demandamorequantitativecomparison.Indeed,ifwewant 80 80
s) 60 60
to test models by fitting RCs we have to know the specific m/
k 40 40
meaning and size of the core, then we will be able to tell if V (
20 20
amodelisconsistent with acuspornot bymakingadirect
0 0
comparison with thedata.
0 2 4 6 8 10 0 2 4 6 8 10
Forthisreasonweaskourselvesthequestion:Whatand
r (kpc) r (kpc)
where is the core? To solve the previous ambiguity and to
unify the concept for future comparisons, we found that a Figure 1.Contribution ofthe baryons tothe rotation curve for
gooddefinitionforthecoreisaregionwherethedensitypro- F568-3andF583-1.Wedenoteobserveddatabyblackdotswith
file presents logarithmic slopes α>−1 and the core radius error bars, dark matter with blue asterisks, the disk with cyan
will be the radius at which the core begins, that is to say, squares and thethe gas withmagenta squared boxes. Wefitthe
forradiussmallerthanthecoreradiuswewillhaveα>−1, figures on the leftasummingthe minimundiskhypothesis while
this means that its valuer′ is determined bythe equation fortheonesontherightwesubstractinquadrature thebaryons
andfitonlythedarkmatter.Wenoticethatthebarioniccompo-
α(r′)=−1 (18) nentisnotdominantintheouterregionsandthatthedifference
inthefitsisbarelynoticeable.
The advantages of this definition are that the interpreta-
tion is independent of the profile chosen (also notice that
it applies to the total density profile and is not restricted
1
to that of DM) and in virtue of the same definition we can 0.75 0.99
directly tell if a DM model profile is cored or cuspy. With
0.5 0.96
ournewdefinitionthespecificdistanceatwhichthecorera- β (r) 0.25 β (r) 0.93
dius occurs still depends on the profile chosen but now the
0
physicalinterpretationisonlyone.Inthefollowingwhenwe 0.9
-0.25
refertoboththecoreandcoreradiusweadopttheprevious 0 2 4 6 8 10 0 1 2 3
interpretation. r (kpc) r (kpc)
Applying the definition to (7) we get the core radius
for theBEC profile R and for comparison we use (14) be- Figure 2. Logarithmic slope of the rotation curve (β) for three
B
values of R. From left to right R = 6,8,10. The plots on the
causeinturnsoutthattheparameterR correspondstothe
c
leftshowacommonbehavioralwaysreachingzerobeforeR.The
core radius as defined above. Finally, fitting the NFW pro-
figure on the right shows the region where a linear behavior in
file provides a direct comparison between a cusp and core
theRCspeedisstillvalid,the(green)horizontallineboundsthis
and hence tothe cusp/core problem.
regionandtheradiusinwhichthisboundaryisreachedwasfound
tober≈0.31Rineachcurve.
3 FITS AND DATA
baryonsisagoodhypothesis,weincludeinFig.1tworepre-
Weseefrom(7)thattheBECmodelsatisfiesρ∼r0nearthe sentativeexamples (F568-3 and F583-1). For thesegalaxies
origin,butapriorithisdoesnotimplyconsistencywithob- weplotthecontributionofthegas,diskandthedarkmatter
servedRCs.Therefore,wefittheprofilesinsection2tothir- separately.Wedidthefittingfirstconsideringthetotalcon-
teenhighresolutionobservedRCsofasampleofLSBgalax- tribution and then using only DM (marked as 14 and 15 in
ies. TheRCswere takenfrom asubsampleofde Blok et al. Table1and2).Wefoundnosubstantialdifferenceinourval-
(2001), we chose galaxies that haveat least 3 values within ues,whichcanbeseenfromourresultsinTable1and2.As
∼1 kpc, not presenting bulbs and the quality in the RC in theothergalaxies belong tothecategory of DMdominated
Hα is good as defined in McGaugh S. S.et al. (2001). The galaxiesasotherauthorshaveshown(Kuzio deNaray et al.
RCsinthisworkomitgalaxiespresentinghighasymmetries 2008; de Blok et al. 2001), neglecting baryons in ouranaly-
andincludedintheerrorbarsareexperimentalerrorsinthe ses will not modify substantially ourresults.
velocity measurement, inclination and small asymmetries. Asthedifferencebetweenacoreandacuspismostno-
BecausetheDMisthedominantmass componentforthese tableonlyfordatavaluesinside1kpcandgiventhatinthe
galaxies we adopt the minimum disk hypothesis which ne- interval ∼ 1 to 10 kpc the slopes of core and cusp profiles
glects baryon contribution to the observed RC. In order to are very similar, which can lead to the wrong conclusion
showthatinLSBanddwarfgalaxiesneglectingtheeffectof thatcuspyhalosareconsistentwithobservations,wedeter-
(cid:13)c 0000RAS,MNRAS000,000–000
FlatCentral DensityProfile andConstantDMSurface Densityin GalaxiesfromScalarFieldDarkMatter 5
100
30 40
80
m/s) 20 m/s) 30 m/s) 60
V (k V (k 20 V (k 40
10
10 ESO1870510 20 ESO3020120
ESO1200211
0 0 0
0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10
r (kpc) r (kpc) r (kpc)
-1 -0.5
-1
ESO1200211
ESO1870510
-1.5 -1.5 -1 ESO3020120
ρ) ρ) ρ) -1.5
g( -2 g( -2 g(
o o o
L L L
-2
-2.5 -2.5
-2.5
-3 -3
-1.2 -0.9 -0.6 -0.3 0 0.3 0.6 -0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 -0.6 -0.3 0 0.3 0.6 0.9
Log r (kpc) Log r(kpc) Log r (kpc)
70 120
60 100
100
50 80
m/s) 40 m/s) 80 m/s) 60
V (k 30 V (k 4600 V (k 40
20
10 ESO3050090 20 ESO4880049 20 F568-3
0 0 0
0 1 2 3 4 5 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10
r (kpc) r (kpc) r (kpc)
0.5 -1
ESO3050090 0 ESO4880049 F568-3
-1.5 -0.5 -1.5
ρ) ρ) -1 ρ)
g( -2 g( g( -2
o o -1.5 o
L L L
-2 -2.5
-2.5 -2.5
-3 -3
-0.4 -0.2 0 0.2 0.4 0.6 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 -0.2 0 0.2 0.4 0.6 0.8 1
Log r (kpc) Log r (kpc) Log r (kpc)
100
40 80
80
30 60
s) 60 s) s)
m/ m/ m/
V (k 40 V (k 20 V (k 40
20 F583-1 10 U4115 20 U11557
0 0 0
0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7
r (kpc) r (kpc) r (kpc)
-0.5 0
F583-1 U4115 -1
-1 U11557
-0.5
ρ) -1.5 ρ) ρ) -1.5
g( g( g(
o o o
L L L
-2 -1 -2
-2.5
-2.5
-0.9 -0.6 -0.3 0 0.3 0.6 0.9 -1.2 -0.8 -0.4 0 -0.6-0.4-0.2 0 0.2 0.4 0.6 0.8
Log r (kpc) Log r (kpc) Log r (kpc)
(cid:13)c 0000RAS,MNRAS000,000–000
6 Victor H. Robles and T. Matos
40
140 245
30 120 210
s) s) 100 s) 175
m/ 20 m/ 80 m/ 140
k k k
V ( V ( 60 V ( 105
10 U11583 40 70
20 U11616 35 U11748
0 0 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6
r (kpc) r (kpc) r (kpc)
-0.6 -0.5
-0.8 U11583 U11616 0 U11748
-1 -1
-0.5
ρg() -1.2 ρg() ρg()
Lo -1.4 Lo -1.5 Lo -1
-1.6
-1.5
-1.8 -2
-2 -2
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.4 -0.2 0 0.2 0.4 0.6
Log r (kpc) Log r (kpc) Log r (kpc)
-0.5
160
140 U11819
-1
120
V (km/s) 1 680000 ρLog() -1.5
-2
40
20 U11819
-2.5
0
0 2 4 6 8 0 0.2 0.4 0.6 0.8
r (kpc) Log r (kpc)
Figure 3. Observed LSB galaxy rotation curves and density profiles with the best halo fits. Below each RC is its density profile
alongwiththefits. Shownare thePI(green dashed-line); theBEC(red solidline)andNFW(blackdobble-dotted line)DMhalo
profiles,theobservationaldataisdrawnwitherrorbars.ThegrayarrowdenotesthebestfittothedatawithinR1 andthevertical
arrows denotethePI(blue)andBEC(magenta) coreradius.
mined the logarithmic slope and the uncertainty following an upperbound for the region in which the linear behavior
deBlok et al.(2001)withthedifferencethatwefitonlythe (V ∼ r) remains valid, we obtain an upper bound radius
B
datawithin1kpcandthatthereisnoneedofan uncertain of r ≈ 0.31R. This means that for values of r 6 0.31R we
“break radius”. expect V ∼ r. The latter can be used as a test to the
B
In Table 1 we list the fitting parameters of the profiles BEC model by fitting theRCs and veryfingthis solid-body
of section 2, we also include the values of the logarithmic behaviorwithinthementionedregion.ThefitsoftheRCsin
slope and its uncertainty, the value R1 denotes the nearest Fig. 3provethat thesolid-body likebehaviorcharacterized
radius to1kpcwhere a datapoint is given.Weobtain αby by a linear increase of the velocity in the central region is
fitting values inside R1, and we also report the core radius satisfied by the BEC model, in fact, it is more consistent
for the BEC profile R in order to compare it with R . In with thecore PI and BEC profiles than thecuspy NFW.
B c
Table 2 we report both the value of eq. (3) for the BEC If we now turn to the density profiles, our fits within
profileandthelogarithmofeq.(2)forPIandBECprofiles. R1 give an average value of α=−0.27 ± 0.18 consistent
In Fig. 3, we show our fits to the RC data and the with those obtained in deBlok et al. (2001) α=−0.2 ± 0.2
density profiles, also shown are the core radius in the BEC and with α=−0.29 ± 0.07 reported by Oh Se-Heon et al.
(magenta) and in PI (blue) profiles. The gray arrow is the (2011) analyzing 7 THINGS dwarf galaxies. The case of
fitthatdeterminesα,thesizeofthearrowdenotesthefitted ESO1870510 might be considered to be consistent with
region and is boundedby R1. NFWprofile,howeverit istheinnermostvaluethatconsid-
erably decreases α, being an irregular galaxy more central
data near the innermost region is required to discard the
possibilityofanyviolenteventthatmighthavecausedsuch
4 DISCUSSION
a slope value.
In Fig. 2 we plot β using three different values of R. From The density profiles corresponding to the RCs fits are
this figure we see a common behavior. We have that β is also shown in Fig.3 for each galaxy. We see that the BEC
a decreasing function of r and is zero before R, which tell fits slightly deviates for the farthest data points as a re-
us that (9) always reaches a maximun before R. We also sult of the finite size of the radius R that is fixed by the
notice that there is a region in which V ∼ r, we can take samedata.Thisdiscrepancyisduetothefactthatthehalo
B
this region to be when 0.9 6 β 6 1. If we use β = 0.9 as might be more extended than the value R. As a matter of
(cid:13)c 0000RAS,MNRAS000,000–000
FlatCentral DensityProfile andConstantDMSurface Densityin GalaxiesfromScalarFieldDarkMatter 7
Table 1.Best-fitparameters oftheprofiles insection2.
(a)
Label Galaxy ρP0I Rc χ2PI ρB0 R RB χ2B
(M⊙pc−3) (kpc) (M⊙pc−3) (kpc) (kpc)
1 ESO1200211 0.0464 0.57 1.45 0.0138 2.92 1.46 0.44
2 ESO1870510 0.0548 0.96 0.63 0.0329 2.93 1.465 0.2
3 ESO3020120 0.0598 1.89 0.03 0.0229 8.95 4.425 0.92
4 ESO3050090 0.0276 2.09 0.04 0.0217 4.81 3.04 0.1
5 ESO4880049 0.1035 1.62 0.99 0.0549 5.36 2.68 0.79
6 U4115 0.1514 0.93 0.10 0.1438 1.52 0.76 0.15
7 U11557 0.0156 5.37 0.08 0.0143 9.38 4.69 0.07
8 U11611 0.2065 1.46 0.14 0.0771 6.51 3.225 5.24
9 U11748 1.678 0.98 1.43 0.4205 5.33 2.665 4.37
10 U11819 0.0869 3.03 0.20 0.0539 8.86 4.43 1.09
11 U11583 0.119 0.64 0.12 0.0953 1.43 0.715 0.84
12 F568-3 0.0361 3.01 0.53 0.0248 8.78 4.39 0.11
13 F583-1 0.0317 2.6 0.54 0.019 8.53 4.26 0.48
14 F568-3(DM) 0.0264 3.44 0.84 0.0202 8.96 4.48 0.25
15 F583-1(DM) 0.0279 2.79 0.58 0.0177 8.66 4.33 0.37
(b)
Label Galaxy ρi Rs χ2NFW α ∆α R1
(×10−3M⊙pc−3) (kpc) (kpc)
1 ESO1200211 2.45 5.7 0.24 -0.04 0.53 0.95
2 ESO1870510 0.761 31.82 0.05 -1.09 0.76 1.13
3 ESO3020120 2.65 19.72 0.32 -0.2 0.16 1.4
4 ESO3050090 0.0328 705.67 0.22 -0.7 0.03 1.1
5 ESO4880049 1.42 52.27 0.16 -0.09 0.06 0.96
6 U4115 0.139 341.74 0.78 -0.27 0.24 0.89
7 U11557 0.0108 2849.65 1.43 0.2 0.31 0.9
8 U11611 11.59 14 1.77 -0.15 0.31 1.1
9 U11748 204.58 5.53 3.41 -0.38 0.11 1.1
10 U11819 1.19 101.24 1.07 -0.64 0.13 2.5
11 U11583 0.136 238.568 0.81 -0.2 0.24 0.95
12 F568-3 0.378 120.78 2.36 0.03 0.02 1.9
13 F583-1 0.345 102.349 0.55 -0.03 0.04 1.1
14 F568-3(DM) 0.0715 515.68 2.87 0.34 0.27 1.9
15 F583-1(DM) 0.329 102.349 0.58 0.02 0.07 1.1
afact,themoreextendedthe“flat” outerregion intheRCs in the solution of system (4) and (5). Nevertheless, it can
the more conspicuous the discrepancy. The main reason of beshown (Guzmán & Ureña-López2003)that asystemsof
thiscomesfromFig.2whereweinferthattherotationcurve many nodes is unstable, therefore so far no final conclusion
speed always presents a maximum value followed by a con- has been reached.
tinuos decrease, which means that to avoid the mentioned
When comparing the BEC and PI core radius we find
discrepancy we need that the BEC rotation curve profile a general difference of ∼ 2 kpc, the core size in the PI pro-
remains approximately constant after its maximum. From
file is approximately 2 kpc smaller than the BEC core size,
Fig.1 we see that the total rotation curve is dominated by
but the PI central density is larger. In U4115, U11557 and
thedarkmattercontribution,speciallyintheouterregions.
U11583bothprofilesareverysimilarwhichresultsinasim-
Hence,unlessthebaryonsbecomethedominantcomponent
ilar core and central density values, this can also be taken
intheouterregions,which doesnotseem tobeobserved,it
as a consistency check for our core definition.
isunlikelythataddingthebarioniccontributiontotheRCs
Comparing thevaluesof R inTable1wedidnotfind
in ourgalaxies will solve thediscrepancy. B
a tendency to a common value. Assuming that the core ra-
dius determines the transition where the DM distribution
Some solutions to keep the BEC rotation curve con-
changes from the outer region to theinner constant central
stant after its maximum include finite temperature correc-
density,thelackofauniquevaluemeansthatthereisnota
tions to (14) (Harko& Madarassy 2011), this suffice to al-
common radius at which this transition takes place.
leviate the latter problem in LSB galaxies and dwarfs but
not for bigger galaxies. Other authors proposed including ForoursecondtestweuseR tocalculate(2).Wehave
B
vortexlattices(Rindler-Daller & Shapiro2011;Zinner2011) alreadyseenthatR andR aregenerallydifferentandR
c B B
andaddingmorenodes(Sin1994;Ji S.U. & Sin S. J.1994) isnotafitparameter. Henceapriori R isnotexpectedto
B
(cid:13)c 0000RAS,MNRAS000,000–000
8 Victor H. Robles and T. Matos
Table 2.Derivedquantities fromtheparameters inTable1.
Label Galaxy logµPI a logµB gB
0 0 DM
(×10−9cms−2)
1 ESO1200211 1.42±0.13 1.60±0.50 0.908
2 ESO1870510 1.72±0.78 1.68±0.58 2.15
3 ESO3020120 2.05±0.52 2.00±0.51 2.17
4 ESO3050090 1.76±0.59 1.81±0.60 4.58
5 ESO4880049 2.22±0.11 2.16±1.16 6.63
6 U4115 2.14±0.40 2.03±0.23 4.93
7 U11557 1.92±0.31 1.82±0.54 3.03
8 U11611 2.97±0.13 2.40±0.42 11.3
9 U11748 3.21±1.98 3.05±0.78 50.4
10 U11819 2.42±0.30 2.37±0.25 10.8
11 U11583 1.88±0.25 1.83±0.53 3.08
12 F568-3 2.03±0.35 2.03±0.53 4.91
13 F583-1 1.91±0.23 1.91±0.65 3.66
14 F568-3(DM) 1.95±0.51 1.95±0.68 4.09
15 F583-1(DM) 1.89±0.13 1.88±0.59 3.45
a BothµP0I andµB0 unitsareM⊙/pc2
6 wasused asacrosscheck andwith (1)in which amuchbig-
ger sample was used. The agreement has shown that the
5 BEC
BEC model is capable of reproducing the constancy of the
4 valueµ0,somethingthatbecauseofthecuspynatureisnot
B)0 possible in theNFW profile.
µg( 3 In Fig. 4 we plot the above values for each galaxy. We
o
L 2 definetheDMcentralsurfacedensity(mentioned inthein-
troduction) for theBEC profileby
1
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13
N <Σ>B =M /πr2, (21)
0,DM <RB B
6
5 PI
PI)0 4 Twahbelree2Mw<eRBseeisthobattafionredU1fr1o7m48(8th)eevvaalluuaetleodgµaBt,RPIB.isFcroonm-
µg( 3 siderably above the rest and with the largest 0uncertainty.
o
L 2 Forthisreason, in thefollowing analysis weomit both,this
valueandthesmallestonethatcorrespondstoESO1200211.
1
Doing this we get an average value of eq. (19) given by
0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 << Σ >B0,DM>≈ 191.35 M⊙pc−2, and for the acceleration
feltbyatestparticlelocatedinR duetoDMonlywehave
N B
g (R )≈ 5.2 ×10−9cms−2 broadly consistent with (3).
DM B
Figure4.Plotoflog(cid:0)µB0/M⊙pc−2(cid:1)andlog(cid:0)µP0I/M⊙pc−2(cid:1)for The fact that all galaxies present approximately the
eachgalaxy.Ndenotes thegalaxyaccordingtoTable1.Herewe same order of magnitude in g (R ) might suggest that
DM B
observethatthesevaluesremainapproximatelyconstantinboth R represents more than a transition towards a constant
B
profiles, this also serves as a crosscheck for our definition of RB density, it can give us information about the close relation
in the BEC profile. The green dashed-line represents the mean
betweenDMandthebaryons.Moreover,inviewofthelack
valuesgivenin(19)and(20).
of a unique core radius, we can interpret the transition in
DM distribution as an effect of crossing a certain accelera-
correlate with ρB. However, with the values of Table 1 we tionscaleinsteadofaradiallengthscale.Suchinterpretation
0
obtain remindsusthatgiveninMONDbutwiththebigdifference
that the acceleration scale found is for DM and is not a
log(µB0/M⊙pc−2)=logρB0RB =2.05±0.56 (19) postulate of the model.
log(µP0I/M⊙pc−2)=logρP0IRc =2.08±0.46 (20) whetThoerdaenteramccienleerwathiiocnhsincatelerporretaatleionngtchaussceaslethweetwrainllsinteioend
for the average values in the BEC and PI profiles respec- tostudythepropertiesoflargersamplesofgalaxiesobserved
tively.Weseetheexcellentagreementof(19)with(20)that with thenew telescopes.
(cid:13)c 0000RAS,MNRAS000,000–000
FlatCentral DensityProfile andConstantDMSurface Densityin GalaxiesfromScalarFieldDarkMatter 9
5 CONCLUSIONS de Blok W.J.G., 2010, Adv. Astron., 2010, Article ID
789293, 14 pages.
In this paper we find that the BEC model gives a constant
de Blok W. J. G. et al., 2001, ApJ, 552, L23
density profile that is consistent with RCs of dark matter
Donato et al., 2009, MNRAS,397, 1169
dominated galaxies. The profile is as good as one of the
DubinskiJ., Carlberg R.G., 1991, ApJ, 378, 496
most frequently used empirical core profiles but with the
Friedmann D. E., 2011, arXiv:0912.1668v4
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