Scaling behaviour of a scalar field model of dark matter halos

0.15 MB·English

by  B. Fuchs

Mon.Not.R.Astron.Soc.000,1–4(2003) Printed2February2008 (MNLATEXstylefilev2.2)
Scaling behaviour of a scalar field model of
dark matter halos
B. Fuchs1⋆ and E. W. Mielke2†
1Astronomisches Rechen–Institut, M¨onchhofstr. 12–14, 69120 Heidelberg, Germany,
4 2Departamento de F´ısica, Universidad Aut´onoma Metropolitana–Iztapalapa, Apartado Postal 55-534, C.P. 09340, M´exico, D.F., M´exico
0
0
2
Accepted .Received;inoriginalform2003September

n
a
J ABSTRACT

7 Galactic dark matter is modelled by a scalar field. In particular, it is shown that
2 an analytically solvable toy model with a non–linear self–interaction potential U(Φ)

leads to dark halo models which havethe formof quasi–isothermalspheres.We argue
1
thatthesefitbettertheobservedrotationcurvesofgalaxiesthanthecentrallycusped
v
halos of standard cold dark matter. The scalar field model predicts a proportionality
5
between the central densities of the dark halos and the inverse of their core radii.
7
5 We test this prediction successfully against a set of rotation curves of low surface
1 brightness galaxies and nearby bright galaxies.
0
Key words: galaxies: kinematics and dynamics
4
0
/
h
p
- 1 INTRODUCTION which we investigate here. For this purpose we recapitulate
o
brieflythefieldtheoreticalapproachinthenextsectionand
r Theevidencefor darkmatteris overwhelming, although its
t discussthenthepredictedscalinglawusingdecompositions
s nature is not clear. Most promising seems at present the oftherotation curvesofaset of nearbybright andlow sur-
a concept of cold dark matter. However, this is flawed on
: face brightness galaxies.
v galactic scales. One of its major difficulties is that mod-
i els of cold dark matter halos of galaxies show inner density
X cusps, ρ r−α, with α in the range 0.5 to 1.5 (Navarro,
∝ 2 SCALAR FIELD WITH A Φ6
r Frenk & White 1997, hereafter referred to as NFW, Moore
a SELF–INTERACTION POTENTIAL
etal.1999),whicharenotconsistentwithobservedrotation
curves of galaxies. For this and other reasons alternative MSP chose ad hoc for the scalar field an analytically solv-
candidatesfordarkmatterhavebeenproposed,mostpromi- abletoymodel(Mielke1978)withaΦ6typeself–interaction
nently self–interacting cold dark mater (Spergel & Stein- potential,
hardt 2000). Others have been based, as for example by
U(Φ)=m2 Φ2 1 χΦ4 , χΦ4 61, (1)
Goodman (2000), on field theoretical concepts.In thesame | | | | − | | | |
vein, Schunck (1997, 1999), Mielke & Schunck (2002) and where m is a tin(cid:0)y ‘bare’ m(cid:1)ass of the scalar field and χ a
Mielkeet al. (2002, hereafter referred toas MSP) havepro- couplingconstant.Bothcharacterizethehypotheticalscalar
posed a primordial almost massless scalar field with a re- field and are thought of as constants of nature. The self–
pulsive Φ6 self–interaction as dark matter candidate. They interaction in the radial Klein–Gordon equation takes the
showed that the corresponding non–linear Klein–Gordon form dU(P)/dP2=m2 3m2χP4,whereP =Φeimt.Fora
equation permits a soliton solution, which is equivalent to sphericallysymmetricco−nfigurationthecorrespondingnon–
a Newtonian mass configuration with the density profile linear Klein–Gordon equation simplifies to an Emden type
of a quasi-isothermal sphere. Such density profiles of dark equation
halos with homogeneous cores fit observed rotation curves
of galaxies (de Blok, McGaugh, & Rubin 2001, de Blok & P′′+ 2P′+3χP5 =0, (2)
x
Bosma 2002, hereafter referred to as dBMGR&dBB) much
better than cusped density profiles. Besides this interesting familiar from thetheoryofgaseous spheres.Ithasthecom-
densityprofile,MSP’smodelpredictsascalinglawbetween pletely regular exact solution
the central densities of the dark halos and their core radii
P(r)= χ−1/4 A , (3)
± r1+A2x2
⋆ E-mail:[email protected] wherethedimensionless radial coordinate x=mr hasbeen
† E-mail:[email protected] introducedandA=√χP2(0)isrelatedtothecentralvalue.
2 B. Fuchs and E. W. Mielke
Thesolution dependsessentially onthenon–linearcoupling buthavealsoconstructeddynamicalmodelsofthegalaxies.
parameter χ, since the limit χ 0 would be singular. This The observed rotation curvesare modeled as
→
feature is rather characteristic for soliton solutions. In the
v2(R)=v2 (R)+v2 (R)+v2 (R)+v2 (R),(10)
following we restrict ourselves totherange A61 forwhich c c,bulge c,disc c,isgas c,halo
thepotential U(|Φ|) remains positive. where vc,bulge, vc,disc, vc,isgas, and vc,halo denote the contri-
Thecanonicalenergy–momentumtensorofarelativistic butions due to the bulge, the stellar disc, the interstellar
spherically symmetric scalar field is diagonal, i.e. Tµν(Φ)= gas, and the dark halo, respectively. The radial variations
diag(ρ,−pr, −p⊥,−p⊥) with of vc,bulge(R), vc,disc(R), and vc,isgas(R) were derived from
ρ = 1 m2P2+P′2+U , tlihgehtobrsaetriovsatwioenrse, wlehftileasthfereneoprmaraalmizaetteiornssobfyththeefimtsasosf–ttoh–e
2
(cid:0) (cid:1) mass models to the data. Fits of the form (10) to observed
pr = ρ U, (4)
p⊥ = pr−−P′2, rportoavtiidoenfoarreeancohtograiloauxsylyseavmerbailgmuooudse.lsT,hounse,wdiBthMzGerRo&bduBlgBe
wheretheprimeindicatesaradialderivative.Theform(4)is anddiscmass, onemodel with a‘reasonable’ mass–to–light
familiar from perfect fluids, except that the radial and tan- ratioofthebulgeandthedisc,andfinallya‘maximum–disc’
gentialpressuresgeneratedbythescalarfieldareingeneral model with bulgeand disc masses at themaximum allowed
different, i.e. pr =p⊥. The scalar field proposed here is not by thedata. Furthermore these authors try for each galaxy
interacting by se6lf–gravity but exerts a gravitational force. two types of dark halo models. One is the cusped NFW
From (4) MSP find in flat spacetime theenergy–density density law and the second is the quasi–isothermal sphere.
While varying the disc contribution to the observed rota-
ρ = m2 2P2+P′2 χP6 tion curve leads to fits of the same quality, dBMGR&dBB
2 − findthatthequasi–isothermalspheremodelsofthedarkha-
(cid:2) (cid:3)
Am2 A4x2 A2 los givesignificantly betterfitstothedatathan thecusped
= 1+ − . (5)
√χ(1+A2x2)(cid:20) 2(1+A2x2)2(cid:21) NFWdensitylaw.Thusthescalarfieldmodelpresentedhere
isinthisaspectevensuperiortothecolddarkmattermodel
TheleadingtermoftheNewtoniantypemassconcentration
in its present form.
(5)isexactlythedensitylawofthequasi–isothermalsphere
Pradaetal.(2003)haveattemptedusingSDSSdataon
ρ(r)≃ rc2ρ0+rc2r2 . (6) skaptcelslictaelegathlaexoieusteorfhisaololatmedasshodsitstgriablauxtiioensst.oTphreoybfienodnt1h0a0t
the line–of–sight velocity dispersions of the satellites follow
At large radii the density falls of like ρ r−2 which cor- closely the radially declining velocity dispersion profile of
∝
responds to an asymptotically flat rotation curve. Compar- halo particles in a NFW halo. This implies an outer mass
ing with the MSP model, the central density of the quasi– densitydistributionoftheformρ r−3whichisatvariance
isothermal sphere is given by ρ0 Am2/√χ and the core withthepredictionofthequasi–iso∝thermalsphere(6).Inthe
≃
radiusisrc 1/mA.Thisimpliesascalinglawforthedark cold dark matter model the system of satellite galaxies is
≃
halos of theform assembled during the same accretion processes as the dark
m 1 1 halo, and Prada et al. (2003) assume consistently for the
ρ0≃ √χrc ∝ rc , (7) satellites the same distribution function in phase space as
forthehaloparticles.Inthescalarfieldmodel,however,the
where A, which may varyfrom halo to halo, cancels out.
darkhaloprovidesforthebaryonicmattersimplyaNewto-
Finally we note that this non–linearly coupled scalar
nianforcefield.Thedistributionfunctionofsatellitegalaxies
field exerts theradial and tangential pressures
inphasespaceisthusnotspecifiedandcanbemodelledac-
pr = m22 χP6+P′2 = 2√χ(A13+mA22x2)2 ≃ A23√mχ2, cthoredqinugastio–istohtehoebrmsearlvastpihonerse, ehvaesnaifshthaellopwoetrenptriaolfitlerotuhgahnoaf
(cid:2) (cid:3) NFW halo.
p⊥ = m2 χP6 P′2 = A3m2(1−A2x2) A3m2,(8) Next we examine the predicted scaling relation (7). A
2 − 2√χ(1+A2x2)3 ≃ 2√χ
(cid:2) (cid:3) relation of this type was found empirically by Salucci &
respectively. Thus, at the center the pressure is isotropic, Burkert (2000), although this was based on the universal
pr(0)=p⊥(0), whereas asymptotically at infiniteradius rotation curve model of Persic, Salucci & Stel (1996) and
the, also empirically derived, halo density profile of Burk-
m2 ert (1995). This densitylaw resembles thequasi–isothermal
pr,−p⊥→ 2√χAx4 . (9) sphere in that it has also a homogeneous core. In Fig. 1
we show central densities versus the inverses of the core
radii of quasi–isothermal dark halo models constructed by
3 ASTRONOMICAL TESTS dBMGR&dBB assuming for the discs mass–to–light ra-
tios consistent with current population synthesis models.
We have tested the theoretical model predictions against There is despite some scatter a clear correlation between
rieostattaikoenncfurrovmeddaBtMaoGfRa&sedtBoBf.loTwhesuarufathceorbsrhigahvtenemsseagsaulraexd- ρ(10.4a6nd0r.c−551)loovge(rr−se1v).erTahluosrtdheersdaorfkmhaaglonimtuoddee,lldoagt(aρ0s)eem∝
± c
high–resolution rotation curves of in total 54 galaxies. For to confirm statistically the scaling relation (7). The scatter
abouthalfofthemsurfacephotometryisavailable.Forthese inthecorrelationdiagramisprobablyduetotheneardegen-
galaxies the authors do not provide only kinematical data, eracy of thefitsto theobserved rotation curves,even if the
Scaling behaviour of a scalar field model of dark matter halos 3
structure, one of us has pointed out, judging from the im-
plied internal dynamics of the galactic discs, that the discs
might be near to maximum (Fuchs 2002, 2003a, b). This
would imply for the discs of some galaxies mass–to–light
ratios which are significantly higher than in current pop-
ulation synthesis models of the discs of low surface bright-
nessgalaxies.Thesecanbemodified,though,toyieldhigher
mass–to–light ratios (Lee et al. , in preparation). Therefore
we show in Fig. 2 the central densities versus the inverses
of the core radii of the dark halo models, if they are con-
structedassuming‘maximumdiscs’.Althoughthedarkhalo
parametersareshiftedtoothervalues,thelinearcorrelation
persists.Withlog(ρ0)∝(1.08±0.39)log(rc−1)thedarkhalo
modelsfitnearlyideallytothepredictedscalingrelation(7).
Included into Fig. 2 are also ‘maximum disc’ dark halo pa-
rametersofnearbybrightgalaxies(Fuchs1999,2003a,Fried
& Fuchs, in preparation) which fit also well to the scaling
law. We conclude from this discussion that the astronom-
ical tests are rather encouraging for the scalar field model
of dark halos presented here. What is missing at present,
Figure 1. Central densities versus the inverses of the core radii is of course a cosmological setting of the scalar field model.
ofdarkhalosoflowsurfacebrightnessgalaxiesderivedfrommod- We hope to address this and, in particular, the question of
ellingtheir rotationcurves. Thehalomodelsareconstructed as- large scale structure formation, which is on the other hand
suming‘realistic’mass–to–lightratiosforthediscs.Thesolidline successfullydescribedbycolddarkmattercosmology,inthe
isthepredictedρ0∝rc−1 relation. future.
ACKNOWLEDGMENTS
We would like to thank the organizers of the Tenth Marcel
Grossmann Meeting, in particular Remo Ruffini, for pro-
viding us the opportunity to discuss these topics in Rio de
Janeiro in July 2003 and to meet again after our joint stu-
dent times in the mid 1970th at theUniversity of Kiel.
REFERENCES
Burkert A., 1995, ApJ447, L25
de Blok W.J.G. & Bosma A., 2002, AA 385, 816
de Blok W.J.G., McGaugh S.S. & Rubin V.C., 2001, AJ
122, 2396
FuchsB., 1999, ASPConf. Ser. 182, 365
Fuchs B., 2002, in: Dark matter in astro- and particle
physics, H. V. Klapdor-Kleingrothaus, and R. D. Viollier
(eds.), Springer, Berlin, 28
FuchsB.,2003a,inProceedingsoftheFourthInternational
Workshop on the Identification of Dark Matter, Neil J. C.
Figure 2. Same as Fig. 1, but the halo models are constructed
Spooner and Vitaly Kudryavtsev (eds.), World Scientific,
assuming‘maximumdiscs’.Opensymbols:lowsurfacebrightness
Singapore, 72
galaxies,filledsymbols:nearbybrightgalaxies
FuchsB., 2003b, ASS284, 719
Goodman J., 2000, NewA 5, 103
Mielke E.W., 1978, Phys.Rev.D 18, 4525
disccontributionstotherotationcurvesarefixed.Moreover Mielke E.W. & Schunck F. E., 2002, Phys. Rev. D 66,
practice shows that the radial exponential scale lengths of 023503-1
the discs cannot be determined more precisely than about Mielke E.W., Schunck, F.E. & Peralta H.H., 2002, Gen.
20%.Thishasaconsiderableeffectforthedisccontributions Rel. Grav. 34, 1919
to the rotation curves (cf. equation 10) and consequently Moore B., Quinn T., Governato F., Stadel J. & Lake G.,
for the dark halo models and might also explain some of 1999, MNRAS310, 1147
thescatter ofthecorrelation diagram in Fig. 1.Webelieve, NavarroJ.F., FrenkC.S. & WhiteS.D.M., 1997, ApJ490,
however, that this has not changed the general trend. Us- 493
ing arguments of the density wave theory of galactic spiral Persic M., Salucci P.& StelF., 1996, MNRAS 281, 27
4 B. Fuchs and E. W. Mielke
Prada F., Vitvitska M., Klypin A. et al., 2003, ApJ 598,
260
Salucci P.& Burkert A., 2000, ApJL 537, L9
SpergelD.N.&SteinhardtP.J.,2000, Phys.Rev.Lett.84,
3760
Schunck F.E., 1997, in Aspects of Dark Matter in Astro-
and Particle Physics, H.V. Klapdor-Kleingrothaus and
Y. Ramachers (eds.), World Scientific,Singapore, 403
Schunck F.E., 1999, in Proc. of the Eighth Marcel Gross-
manMeetingonGeneralRelativity,T.PiranandR.Ruffini
(eds.), World Scientific, Singapore, 1447
ThispaperhasbeentypesetfromaTEX/LATEXfileprepared
by theauthor.