Marriage à-la-MOND: Baryonic dark matter in galaxy clusters and the cooling flow puzzle

0.41 MB·English

by  Mordehai Milgrom

Marriage`a-la-MOND: Baryonicdarkmatteringalaxyclusters and
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MordehaiMilgrom

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Centerfor Astrophysics, WeizmannInstitute
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J

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Abstract

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h I start with a brief introduction to MOND phenomenology and its possible roots in cosmology–a notion that may
p
turnouttobethemostfarreachingaspectofMOND.NextIdiscusstheimplicationsofMONDforthedarkmatter
-
o (DM)doctrine:MOND’ssuccessesimplythatbaryonsdetermineeverything.ForDMthiswouldmeanthatthepuny
r tail of leftover baryons in galaxies wags the hefty DM dog. This has to occur in many intricate ways, and despite
st the haphazard construction history of galaxies–a very tall order. I then concentrate on galaxy clusters in light of
a MOND, which still requires some yet undetected cluster dark matter, presumably in some baryonic form (CBDM).
[ This CBDM might contribute to the heating of the x-ray emitting gas and thus alleviate the cooling-flow puzzle.

MOND, qua theory of dynamics, does not directly enter the microphysics of the gas; however, it does force a new
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v outlook on therole of DM in shaping thecluster gasdynamics: MONDtells usthat thecluster DM is not cold dark
3 matter, is not so abundant,and is not expected in galaxies; it is thusnot subject to constraints on baryonic DM in
0 galaxies. The mass in CBDM required in a whole cluster is, typically, similar to that in hot gas, but is rather more
2 centrallyconcentrated,totallydominatingthecore.TheCBDMcontributiontothebaryonbudgetintheuniverseis
4 thus small. Its properties, deduced for isolated clusters, are consistent with the observations of the “bullet cluster”.
2. Itskinetic-energyreservoir ismuchlargerthanthat ofthehot gasin thecore,and wouldsufficetokeepthegas hot
1 for many cooling times. Heating can be effected in various ways dependingon theexact nature of the CBDM, from
7 verymassive black holesto cool, compact gas clouds.
0
:
v
Keywords: GalaxyClusters, Cosmology, darkmatter
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1. introduction Binney 2005 for a discussion of its pros and cons).
Dark matter (DM), in its presently favoredvariety
Recentobservationsofthecoresofgalaxyclusters ofweaklyinteractingparticles,isnotdeemedafac-
have undermined the long standing “cooling flow” tor inthese considerations,as itdoes notaffectthe
paradigm.Theseobservationsdonotdetecttheex- gasdynamicsinthecore(butsee,e.g.,Chuzhoyand
pected telltale signs of cooling, and thus point to Nusser2006forpossibleeffectsofmorestronglyin-
some,stillmoot,mechanismthatheatsthecoregas teractingDM).
and keeps it at an elevated temperature despite its MONDwasintroducedasanalternativetoNew-
shortcoolingtime(forrecentreviewssee,e.g.,Bauer toniandynamicsthatseekstoexplainthedynamics
et al. 2005; Peterson and Fabian 2006; Sanderson ingalacticsystemswithoutDM.It,arguably,works
et al. 2006). Several heating mechanism have been verywellforgalaxies,galaxygroups,andsuperclus-
proposed. The most popular, at present, seems to ters,andhasmadeconsiderabletheoreticalheadway
beheatingbycentralAGNactivity(See,e.g.,Peter- (for reviews see, e.g., Sanders and McGaugh 2002,
son et al. 2003, Omma et al. 2004, and Nipoti and Scarpa 2006, Bekenstein 2006,Milgrom 2008).But
Preprintsubmitted toElsevier 2February2008
in galaxy clusters MOND does not completely ex- 2. Anoverview ofMONDphenomenology
plain away the mass discrepancy. This is particu-
larlysointheverycoresofclusterswherealargere- MONDcanbedescribedasamodificationofdy-
mainingdiscrepancyisfound.SoevenwithMOND namicsatlowaccelerations(Milgrom1983a).More
onehastoinvokeyetundetectedmatterpeculiarto precisely,the MOND paradigmrests onthe follow-
clusters: presumably in some baryonic form, here- ing premises: (i) There appears a new constant in
afterCBDM.Massiveneutrinoswithmassnearthe dynamics, a0, with the dimensions of acceleration.
present experimental upper limit of 2 ev were also (ii)Intheformallimita0 →0classical(pre-MOND)
proposed(Sanders2003,2007). dynamics is restored. (iii) For purely gravitational
MOND determines the overall potential field in systems,inthedeep-MONDlimit,a0 →∞,thelim-
the cluster, but does not directly affect the micro- iting equations can be brought to a form in which
physics of the cooling flow and core gas. It does, the constantsa0 andG,andallmassesintheprob-
however,shednew light onthe possible role of DM lem,mi,appearonlyasmiGa0(Milgrom2005).This
inclustergasdynamics:(i)Unlikethestandardview is a fiat that guarantees the required MOND phe-
accordingtowhichtheDMconstituentsareweakly nomenologyinpurelygravitationalsystems.
interacting particles, in the context of MOND the These requirements can be incorporated in vari-
clusterDMislikelytobebaryonic;so,itcanpossi- ous MOND theories. For example, in the nonrela-
blyinteractwiththegasinwaysrelevanttothecool- tivisticregimeonecanmodifythePoissonequation
ingpuzzle.(ii)Accordingto MOND,this CBDMis for the gravitationalfield(BekensteinandMilgrom
peculiartoclusters;so,itisnotsubjecttothestrin- 1984),oronecanmodifythekineticactionofparti-
gentconstraintswehaveonubiquitousbaryonicDM clesleadingtomodifiedinertia(Milgrom1994a).It
candidatesingalaxies(e.g.,CarrandSakellariadou followsfromthebasicassumptionsthatanyMOND
1999, Carr 2000, Kamaya and Silk 2002). (iii) Al- theory must be nonlinear even in the nonrelativis-
though in the inner parts of clusters, the required ticregime(seee.g.,Milgrom2008fordetails).From
amount of CBDM is large compared with that ob- the above MOND tenets (and the assumption that
served in stars and gas, the total amount required a0istheonlynewdimensionedconstant)alsofollow
for the cluster at large is comparable to the mass some important scaling laws (under which Ga0 is
of the hot gas. This renders the baryonic option invariant)forthedeepMONDlimitofpurelygravi-
more palatable. It also greatly relaxes constraints tationalsystemsinanyMONDtheory(seeMilgrom
on baryonic DM in clusters based on standard dy- 2008formoredetails):Thetheoryisinvariantunder
namics, which requires 5-10 times more DM. Such scalingoftheradiir→λr,andtimest→λt(masses
smallquantitiesofanewbaryoniccomponenteasily areunscaled).(Thisispartofthe conformalinvari-
satisfynucleosynthesisconstraints. ance, in space alone, of the deep MOND limit in a
I discuss the possibility that such a baryon com- particularformulationofMOND–Milgrom1997).
ponentcouldkeepthecoregasfromcoolingprecipi- As a consequences of this scaling invariance we
tously,thushelpingalleviatethe“coolingflow”puz- deduce that if r(t) is a trajectory of a body in a
zle.ThetotalkineticenergybudgetoftheCBDMis configuration of masses m at positions r (t), then
i i
known,andis sufficientto supply heatingformany ˆr(t) = λr(t/λ) is a trajectory for the configuration
Hubble times. The idea of heating the cluster gas wherem areatλr (t/λ),andthevelocitiesonthat
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withbaryonicDMisnotnew(e.g.Walker1999),but trajectoryarevˆ(t)=v(t/λ). There isanother scal-
as explained above, it takes quite a different shape ingproperty:ifwescaleallthemasses,alltrajectory
inthe contextofMOND. pathsremainthesamebutthebodytraversesthem
I start by reviewing in section 2 the basic phe- with allvelocitiesscalingasm1/4.
nomenology of MOND, and its possible origin in Ultimately, one would like to incorporate these
cosmology;its significance and implications for the principlesinarelativisticextensionofGeneralRel-
DM paradigmare discussed in section 3.In section ativity. The state of the art of this effort is the
4 I summarize what MOND tells us about DM in TeVeS theory and its derivatives (Bekenstein 2004,
clusters.Insection5IlistcandidatesfortheCBDM andBrunetonandEsposito-Farese2007).Ishallnot
anddiscussheatingmechanisms. discusstheorieshere;forarecentreviewseeBeken-
stein (2006). The theories proposed so far involve
afreefunctionthatessentiallyinterpolatesbetween
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the abovetwolimits.
Schematically, a0 plays a similar role to c in rel-
ativity, or to ~ in quantum theory. On one hand,
they all mark the boundaries between the classical
andthemodifiedregimes;soformallypushingthese
boundaries to the appropriate limits (c → ∞, ~ →
0,ora0 →0)one restoresthe correspondingclassi-
cal theory.On the other hand, these constants also
feature prominently in the physics of the modified
regime and appear in various phenomenologicalre-
lations. Examples of such relations for ~ are: the
Fig.1.Galaxymassvs.asymptotic velocity. Left:analogto
black body spectrum, the photoelectric effect, the thetraditionalTully-Fisherplotinvolvingstellarmassalone.
Hydrogen atomic spectrum, the quantum Hall ef- Right: gas mass is included. The solid line has the log-log
fect, etc.. For c we could mention the (relativistic) slope of 4, predicted by MOND and is not a fit (McGaugh
2005b).
Doppler effect, the mass-velocity relation, the life-
time-velocityrelation,andtheradiusofablackhole.
These relations, in themselves, are independent in dius where V2/R=a0.Ingalaxieswhosecentral
the sense that they do not follow from each other acceleration is below a0 (low surface brightness
and are made related only through the underlying galaxies) there is a discrepancy at all radii (Mil-
theory. The MOND paradigm similarly predicts a grom 1983b, for tests see Sanders and McGaugh
numberoflawsrelatingtogalacticmotions,someof 2002,McGaugh2006).
whicharequalitative,but manyofwhicharequan- 4. Foraconcentratedmass,M,wellwithinitstran-
titative and involvea0; they may be likened to Ke- sitionradius,rt ≡(MG/a0)1/2,rt playsaspecial
pler’s laws of planetary motions1. More details of role(somewhatakintothatofthe Schwarzschild
thesepredictionsandhowtheyfollowinMONDcan radiusinGeneralRelativity).Forexample,ashell
befoundinMilgrom(2008).Hereisapartiallistof of phantom DM may appear in the vicinity of r
t
suchlaws. (MilgromandSanders2007).
1. Therotationcurvesforanisolatedmassisasymp- 5. Isothermal spheres have mean surface densities
totically flat: V(r) →V∞ (Milgrom1983b).This Σ¯ . a0/G (Milgrom 1984) underlying the Fish
follows for all MOND theories from the above law for quasi-isothermal stellar systems (see dis-
mentionedscalingoflengthandtime. cussionin SandersandMcGaugh2002).
2. The mass-velocity relation (aka the baryonic TF 6. A mass-velocity-dispersionrelation σ4 ∼ MGa0
relation) between the total mass of a body and underlying the Faber-Jacksonrelationfor ellipti-
its V∞: V∞4 = MGa0 (Milgrom 1983b, see anal- calgalaxies(Milgrom1984,1994b).
ysis in McGaugh 2005a, and Fig. 1 here). This 7. Thereis adifferenceinthe stabilitypropertiesof
alsofollowsinallMONDtheoriesfromtheabove discs with mean surface density Σ¯ . a0/G and
mentioned scaling of orbital velocities with mass with a0/G. Σ¯ (Milgrom 1989a,Brada and Mil-
inthe deepMOND limit. grom1999b,TiretandCombes 2007).
3. Indiscgalaxieswithhighcentralaccelerationsthe 8. The excess acceleration that MOND produces
mass discrepancy appears always around the ra- overNewtoniandynamics,for a givenmass,can-
not much exceed a0 (Brada and Milgrom 1999a,
1 The nominal Kepler laws were all discovered before the as confirmed for a sample of disc galaxies by
underlyingtheoryofNewtoniandynamicswaspropounded. MilgromandSanders2005).
Fromthistheoryadditional“Kepler’slaws”canbededuced, 9. An external acceleration field, g , affects the in-
e
such as the dependence of the Kepler constant onthe mass
ternal dynamics of a system imbedded in it. For
of the central star. MOND was constructed by assuming
example, if the system’s intrinsic acceleration is
only one law–the asymptotic flatness of rotation curves–
based on rather anecdotal evidence existing at the time, smaller than ge , and both are smaller than a0,
and it was helped by a rough constraint from the Tully- the internal dynamics are quasi-Newtonian with
Fisherrelation.MONDthenelevatedthesetwointoabsolute an effective gravitational constant ≈ Ga0/ge
predictions(replacingtheoriginal,velocity-luminosityTully-
(Milgrom1983a,1986a,BekensteinandMilgrom
Fisherrelationbyonebetweenvelocityandtotalmass,and
1984, with applications in e.g., Brada and Mil-
predictingtheexactpowerinthisrelation,stillmootatthe
time). grom 2000a,b, Angus and McGaugh 2007, and
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Wu et al. 2007). Some aspects of this prediction stellar-mass-dominated NGC 2903. More examples
violatethestrongequivalenceprinciple,andthus are shown in Fig. 3. Not only are the full rotation
conflictwiththe predictions ofDM. curveswellfitwiththeabovesingleparameter,but
10. The thin lens approximation breaks down in ingasdominatedgalaxieseventhisparameterisal-
MOND, which also conflicts with the prediction most immaterial, and MOND then practically pre-
ofDM. dicts the rotationcurve.When the M/L parameter
11. Disc galaxies have a disc “DM” component in isimportant,itsvalues,obtainedwiththeMOND1-
additiontoaspheroidalone(Milgrom2001).See parameterfits,areingoodagreementwiththeoret-
confirming analysis in Milgrom(2001), Kalberla ical, population-synthesis estimates (e.g., Sanders
et al. (2007), and Sa´nchez-Salcedo, Saha, and andVerheijen1998).
Narayan (2007). This prediction conflicts with
the expectationfromCDM.
12. Negative density of “dark matter” is predicted
in some locations (Milgrom 1986b). Again this
conflicts withpredictionsofDM. 2.1. a0?
Thesepredictionsareallindependentinthesense
that one can construct galaxy models of baryons The constant a0 appears in several of the above
plusDMsuchthatanysubsetoftheserelationsare MOND laws of galactic motions, and its value
satisfiedbutnotanyoftheothers.Thus,inthecon- was initially determined, with consistent results,
text of DM they will each require its own separate by appealing to them (Milgrom 1983b). However,
explanation. In fact, as I indicated, some of these the best leverage on a0 comes now from rota-
haveimplications thatconflictwith the predictions tion curve analysis. For example, the value found
of CDM, and among them some that conflict with in Milgrom (1988) was a0 ≈ 1.3 × 10−8cms−2,
anyformofDM. and Begeman et al. (1991) found, with better
But,theflagshipofMONDphenomenologyisthe data, a0 ≈ 1.2 × 10−8cms−2 (both with H0 =
detailed prediction of the full rotationcurves of in- 75kms−1Mpc−1).ItwasnotedinMilgrom(1983a)
dividual disc galaxies from their observed baryonic that a0 ≈ cH0/2π, and because of the still mys-
mass alone. Its importance was evident from the terious cosmic coincidence (ΩΛ ∼ 1) we also have
outset(Milgrom1983b);but,testingthisprediction a0 ≈c(Λ/3)1/2/2π.
had had to await the advent of extended rotation In the context of quantum theory, the Planck
curves afforded by HI observations:Rotation curve lengthandthe Planckmass,constructedfrom~,G
analysis in MOND started only some five years af- andc,tellus wherewecanexpectcombinedeffects
teritappeared,withKent(1987)andtherectifying of strong gravity and quantum physics. Similarly,
sequel by Milgrom (1988). Some of the subsequent a0 defines a length scale, ℓ0 ≡ c2/a0 ≈ 1029 cm,
studies where by Begeman et al. (1991), Sanders and a mass, M0 ≡ c4/Ga0 ≈ 6 × 1023M⊙, that
(1996), Sanders and Verheijen (1998), de Blok and tell us where to expect MOND effects combined
McGaugh (1998),Bottema et al. (2002),Gentile et with strong gravity. In view of the above coinci-
al.(2004),CorbelliandSalucci(2007),Sandersand dences of a0 with cosmologicalaccelerationparam-
Noordermeer(2007),andBarnesetal.(2007).There eters, ℓ0 and M0 are of the order of the Hubble
arenowoftheorderof100galaxiesforwhichMOND radius ℓ0 ≈ 2πℓH, with ℓH ≡ cH0−1 ≈ c(Λ/3)−1/2,
has been tested in this way: one uses the observed and the cosmological mass parameter (“the mass
baryonic mass in a disc galaxy to predict the ob- within the horizon”), M0 ≈ 2πMU, with MU ≡
servedrotationcurveaccordingtoMOND(convert- c3G−1(Λ/3)−1/2 ≈ c3G−1H−1, respectively. This
0
ing stellar lightto mass using the M/L value–mass means that combined effects are expected only for
tolightratio–afree parameter).Itis,ofcourse,the theuniverseatlarge,andthattherearenostrongly
complete failure of this procedure with Newtonian MONDish blackholes.
dynamics that leads to DM in galaxies. MOND is Thesecoincidencesmaypointtoaconnectionbe-
performingextremelywellinthisregard.Ishowex- tween MOND and cosmology: either the same new
amples in Fig. 2 for three galaxies of rather differ- physics explain MOND and the “dark energy” ef-
entproperties:fromthelowmass,lowacceleration, fects, or MOND is an expressionin localphysics of
gas dominant NGC 1560, through the intermedi- whatever agent is responsible for the observed cos-
ate NGC 3657,to the highmass,highacceleration, mic acceleration(as embodied in Λ). If the param-
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Fig.2.TheobservedandMONDrotationcurves(insolidlines)forNGC3657(left),NGC1560(cnter),andNGC2903(right).
The first from Sanders (2006), the last two from Sanders and McGaugh (2002). Points are data, dashed and dotted lines for
the last two galaxies are the Newtonian curves calculated for the stars and gas alone; the reverse for the first (they add in
quadrature to givethe fullNewtonian curve).
Fig. 3. Additional MOND rotation curves from Sanders (1996) and de Blok & McGaugh (1998) (left) and from Sanders &
Verheijen(1998)(right).(MONDcurvesinsolid;stellardiscNewtoniancurvesindotted;gasindot-dash;andstellarbulgein
longdashed.)
eter ac ≡ MU/ℓ2H = cH0 ≈ c(Λ/3)1/2 is somehow close to its surface: There, a “constant of nature”–
felt by local physics, it may not be surprising that theGalileifree-fallacceleration,g =M⊕G/R⊕2,ap-
dynamicsisdifferentforaccelerationaboveandbe- pearsinthedynamics.Ifweknowtheescapespeed,
low this value. But why is it that cosmology enters ves = (2M⊕G/R⊕)1/2, or the orbital speed on a
localphysics throughits characteristicacceleration near earth orbit vorb = (M⊕G/R⊕)1/2–which can
andnot,forexample,throughitscharacteristicmass serveaspossibleanalogsofc–wewouldnotethere-
or length? I discussed possible reasons in Milgrom lation between this, g, and R⊕, which is essentially
(1999).Notethatasimilarthinghappensintheway the same as that between, c, a0, andℓH. We would
earth’s gravity enters the dynamics in experiments thenprobablydeducethattherelationisasignthat
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Galileanfreefallisaneffectivelawresultingfroma differsgreatlyfromMONDasregardstheoriginand
deeper theory,which,ofcourse,itis. nature of mass discrepancies they predict in galac-
Its relation with cosmological parameters raises tic systems. In MOND, these discrepancies are not
thepossibilitythata0 varieswithcosmictime(Mil- real; the full dynamics in a givensystem, including
grom1989b,2002).If it is alwaysrelatedto cH0 in theimaginarydiscrepancies,arepredicteduniquely
thesameway,orifitisrelatedtothedensityof“dark fromthe presentlyobserved(baryonic)massdistri-
energy”,andthischanges,thena0wouldfollowsuit. bution. In particular, the pattern of these discrep-
Itisalsopossible,however,thatΛisaconstantand ancies is predicted, and is observed, to follow the
soisa0.Thepossibilityofavariablea0opensupin- well defined relations discussed in section 2. In the
terestingpossibilities.Suchvariationsmay,inturn, language of DM one can say that MOND predicts
inducesecularevolutioningalacticsystems;forex- that the distribution of normal matter (baryons)
ample, the mass-velocity relations dictate that the fully determines all aspects of the DM distribution
velocitiesinasystemofagivenmassshoulddecrease ineachandeverygalacticsystem.Incontrast,inthe
with the cosmological decrease in a0 (like a10/4 in DMparadigmtheexpecteddistributionsofthetwo
thedeepMONDregime),andthiswouldbeaccom- components depend strongly on details of the par-
panied by changesin radius,since adiabatic invari- ticularhistoryofthesystem,sincethe twoaresub-
ants, such as possibly rv, have to stay fixed (Mil- jecttodifferentinfluences.Theformationprocessof
grom1989b).Suchvariationsina0 couldalsobe in a givengalactic systemand its ensuing unknowable
thebasisofthecosmologicalcoincidenceΩΛ ∼1via historyofmergers,cannibalism,gasaccretion,ejec-
anthropic considerations (Milgrom 1989b, Sanders tionofbaryonsby supernovaeand/orrampressure
2001).Also,the generalconnectionof MOND with stripping, etc., all go into determining the present
cosmology, and, in particular, the possibility of a0 amount,anddistribution,ofbaryonsandofDM.A
varying, may provide a mechanism for inducing a strong and direct evidence for such differentiation
localarrowoftimebythecosmologicalone.Forex- comes from the recent realization that the ratio of
ample,ifinanexpandinguniversea0decreases,this baryon mass to the DM mass required in galaxies
fact is felt by small local system, and can induce a is much smaller than the cosmic value, with which
preferreddirectionofevolutioninthem. protogalaxies should have started their way (by
Analyzingthe dataofGenzeletal.(2006)onthe an order of magnitude, typically). This evidence
rotation curve of a galaxy at z = 2.38, I find that comes, for example, from probing large galactic
theyareconsistentwithMONDwiththelocalvalue radiiwithweaklensing;e.g.byKleinheinrichetal.
of a0; but, the large error margins, and the fact (2004), Mandelbaum et al. (2005), and by Parker
that at the last measured point the galaxy is only et al. (2007) (see also McGaugh 2007 for evidence
marginallyMONDish, still allow appreciable varia- based on small radius data with CDM modeling).
tions ofa0 with red-shift. Evenfor galaxyclusters there is now some prelimi-
MOND, as it is formulated at present, does not, nary evidence that the observed baryon fraction is
in my opinion, provide a clear-cut tool for treating afewtensofpercentssmallerthanthecosmicvalue
cosmology (including the appearance of cosmolog- (Afshordietal.2007andreferencestherein).
ical DM). It is possible that the understanding of Another acute example of the large variety of
cosmology within MOND will come together with baryon vs. DM properties expected in the DM
the understanding of MOND within cosmology, in paradigm is brought into focus by the recent ob-
the sense alludedto above. servationoflargemass discrepancies inthree tidal-
debris dwarf galaxies (Bournaud et al. 2007). In
viewoftheirspecificformationscenario,hardlyany
3. Significance forthe DM paradigm CDM is predicted in them in the CDM paradigm.
This is very different from what is expected in pri-
Can the successes of MOND simply reflect the mordial dwarfs, and contrary to what is observed.
workings of DM, which are, somehow, summarized These dwarfs conform nicely to the predictions of
by a very simple unifying formula? Can the ubiq- MOND, which are based only on their presently
uitous appearance of the constant a0 in seemingly observed properties (Milgrom 2007, Gentile et al.
independentgalacticphenomenaemerge,somehow, 2007)
fromthe DMparadigm? How,then,canthe haphazardandsmallamount
The Newtonian-dynamics-with-DM paradigm ofleftoverbaryonsingalaxiesdeterminesomanyof
6
the propertiesofthe dominantDM halo,with such few hundred (M/L)⊙–were thought to represent
accuracy as evinced by MOND? I deem it quite mass discrepancies of order 100. The MOND cor-
inconceivable that DM will ever explain MOND rection I found in Milgrom (1983c) reduced the
and the relations it predicts for individual sys- M/L values substantially,but still lefta significant
tems. Achieving this within the complex scenario discrepancy (M/Lof a few to a few tens (M/L)⊙).
of galaxy formation in the DM paradigm would be I speculated there that the x-ray emission then
akintopredictingthedetailsofaplanetarysystem– knownto emanatefromclustersmightbespeakthe
planet masses, radii, and orbits–from knowledge of presenceoflargequantitiesofintra-clustergasthat
onlythe propertiesofthe centralstar. mayaccountformuchoftheremainingdiscrepancy.
Indeed,tomyknowledgenoneoftheMONDpre- This has been largely borne out by the identifica-
dictionslistedinsection2hasbeenshowntofollow tion and weighing of the hot, x-ray emitting gas,
intheDMparadigmevenasjuststatisticalcorrela- which has reduced the cluster mass discrepancy
tions, to say nothing of actual predictions for indi- (in standard physics and in MOND) by an order
vidual systems. Even the Tully-Fisher-like relation of magnitude. This, however, turned out to not
thathasbeenbruitedtofollowfromLCDMrequires quite suffice. Studies based on gas dynamics and
astrongassumptiononthepresentdaybaryonfrac- on lensing have helped determine the remaining
tioningalaxiesandonhowitvariesamonggalaxies discrepancy, e.g. by The and White (1988), Gerbal
(originallytheratiowasassumedtobeuniversaland et al. (1992), Sanders (1994,1999,2003),Aguirre et
equal to the cosmic value); but the DM paradigm al. (2001), Pointecouteau and Silk (2005), Angus
doesnotcomeclosetopredictingthisratio.Notonly et al. (2007b), and by Takahashi and Chiba 2007.
isn’tthe generalcorrelationpredicted,but the pro- In the framework of MOND we have to attribute
cessesthat causedonlya smallfractionofthe orig- the remaining discrepancy to yet undetected mat-
inalbaryonsto showup inpresentday galaxiesare ter. It is likely that this matter is baryonic in some
likelytohaveproducedalargescatterinthis ratio, form–and so I shall assume in this paper–although
andhenceinthepredictedbaryon-mass-velocityre- other possibilities have been raised, e.g., massive
lation,unlikewhatis observed. neutrinos (Sanders2003,2007).
Notealso,inthecontextofourverysubjecthere, The following rough picture emerges regarding
that recent comparisons of LCDM simulation with the distribution of CBDM: The observed accelera-
observationsofgalaxyclustersshowacleardiscrep- tions inside a few hundred kiloparsecs of the cen-
ancy: observed mass distributions in the cores are ter are of the order of or a few times larger than
significantlymorecentrallyconcentratedthanthose a0. Thus, MOND implies only a small correction
predicted by LCDM (Hennawi et al. 2007, Broad- there. So most of the discrepancy observed in the
hurst& Barkana2008). coremustbeduetoCBDM.Theratio,l,ofaccumu-
Torecapitulate,thefactthattheMONDlawsare lated CBDM to visible baryons there is around 10
satisfied in nature speaks against the Newtonian- to a few tens. The results of Sanders (1999) scaled
dynamics-plus-DMparadigmintwoways:First,be- to H0 = 75 kms−1Mpc−1 correspond roughly to
causeitsupportsMONDasacompetingparadigm. l = 2 at 1 Mpc. [Pointecouteau and Silk (2005)
Second, because in itself, and without reference to claim higher values. But after correcting for two
MOND, this fact directly disagreeswith the expec- oversights on their part their results are consistent
tationsfromtheDMparadigm:Auniqueconnection with those of Sanders: for H0 = 75 kms−1Mpc−1
between the baryons and DM that holds galaxy by theyusethesmallvalueofa0 appropriateforH0 =
galaxyfliesinthefaceofthehaphazardbaryon-DM 50kms−1Mpc−1,andtheytookthemeanradiusin
relationexpectedinthe DM paradigm. Sanderssampletoberathersmallerthanitis.]The
valueofldecreasecontinuouslywithradius,asseen,
for example, in the small-samplestudy of Angus et
4. Clusterdark matter inlightofMOND
al. (2007b).Since at the maximalradii in the anal-
ysesthegasmassisseentoincreasefasterthanthe
It was realized early on that MOND does not MONDdynamicalmass,wecanextrapolatetoeven
fully account for the mass discrepancy in galaxy higher radii and conclude that for the cluster as a
clusters. At the advent of MOND, the baryonic wholelisabout1orevensmaller.McGaugh(2007)
budgetofclusterswasthoughttoconsistofgalaxies reachesasimilarconclusionbasedonextrapolating
only: the large M/L values deduced for clusters–a the mass-velocityrelationfromgalaxiestoclusters.
7
ThecontributionoftherequiredCBDMtotheto- the CBDM, while this has not happened in spirals.
tal baryonic budget in the universe is rather small. If this is connectedwiththe appearanceofmuchof
FukugitaandPeebles(2004)estimatethetotalcon- the baryonic content as hot gas,it may also be im-
tributionofthehotgasinclusterstoΩtobeabout portantforellipticalgalaxiesandgalaxygroupsen-
0.002, some 5 percent of the nucleosynthesis value. shroudedin hotgas.
It follows from the above that the contribution of
the CBDMis similar. 5. Heating byCBDM
If the CBDM is made of compact macroscopic
objects–asismostlikely–thenlikethegalaxies,these
5.1. Candidates andcooling mechanisms
must have sloshed through the hot gas for many
dynamicaltimes; this is implied by the CBDM dis-
Like CDM and neutrinos, some forms of CBDM
tribution being still rather extended. So, when two
are inherently ineffective in heating the gas. This
clusters collide, the CBDM will follow the galaxies
wouldbethecase,e.g.,forbrowndwarfs,planetary
in going throughthe collisionzone, andwill not be
sizeobjects,ortoocompact,coolgasclouds.Onthe
greatlyaffectedeveninhead-oncollisionswherethe
otherhand,blackholesofdifferentmassescouldac-
gas components of the two clusters coalesce.Clowe
creteandradiate,oriftheyaremassiveenoughthey
etal.(2006)haverecentlyusedweaklensingtomap
could lose kinetic energy to the gas by dynamical
themassdistributionaroundapairofcollidingclus-
friction.CBDMintheformofcooldensegasclouds
tersandindeedfounddarkmassconcentrationsco-
couldconverttheirkineticenergytothermalenergy
incidentwiththegalaxyconcentrationstothesides
of the hot gas if they are largeand massive enough
ofthegasagglomerationnearthecenterofcollision
to collide at a sufficient rate (e.g., Walker 1999). I
(See, however,Mahdavi et al. 2007).Such observa-
shallconcentrateontheCBDMkineticenergyasthe
tions donotadda puzzle inthe contextofMOND;
sourceofheat:IftheobjectsmakinguptheCBDM
they are very much in line with what we already
movewiththeclustervirialspeed,whichisareason-
knowaboutMONDdynamicsofsingleclusters.An-
able assumption considering their distribution, the
gusetal.(2007a)showthiswithadetailedanalysis.
reservoirof their kinetic energy in the core alone is
Another result of weak lensing analysis claimed
tens of times larger than the thermal energy of the
tobeadirectevidenceforDMinclustersistheap-
gasthere.IftheCBDMbodieshavemoreradialor-
parent ring of DM (Jee et al. 2007) in the galaxy
bits thecorecantapevenalargerreservoir.Sucha
clusterCl0024+17.Itturnsout,however,thatthis
heatingsourceissmoothlydistributedinspaceand
can be explained as a MOND effect (Milgrom and
rathersteady withtime.The volume densityofthe
Sanders2007):theringmightbeadirect“image”of
heating ratescaleslike the productoftwodensities
the MOND transition region analogous to the GR
(CBDM-CBDM for cloud-cloud collisions, CBDM-
transition region as marked by the Schwarzschild
gasfordynamicalfriction)sothereisachancethat
horizon. The observations and the MOND predic-
the balance of the cooling rate will extend over a
tions are shown in Fig. 4. Note also in this connec-
range of radii. All these are considered to be expe-
tion that gravitational lensing in MOND is more
dient for solving the puzzle (see e.g.Petersonet al.
difficult to interpret since the thin-lens approxima-
2003).Also,the resulting steady state temperature
tion fails completely, because of the nonlinearity of
of the cooling gas has to be a fraction of the virial
MOND;so,the structurealongthe line ofsighten-
temperature andno overheatingcanoccur.
tersstrongly(law10above).
Inowdiscussbrieflyonepossiblemechanismwith
OnemaywonderwhythisCBDMappearsinclus-
the CBDMinthe formofcooldensegasclouds.
tersbutnotinspiralgalaxies.Theunswermayliein
the fact that the two types of objects differ greatly
in appearance, baryonic content, and probably in 5.2. Heatingbydense gas clouds
their formationprocess.Most ofthe observedmass
inclusters is inthe formofhot,x-rayemitting gas, Cold,densegascloudsasDMhavebeendiscussed
with only a small fraction in stars, while in spirals by Pfenniger et al. (1994), Walker and Wardle
the situation is the opposite. Because the required (1998),andinmanysucceedingpapersbythemand
total amount of CBDM is comparable with that of others (see, e.g., Kamaya and Silk 2002, and ref-
the hotgas,itis conceivablethatinclusters afrac- erences therein). Notably, Walker (1999) discussed
tionofthebaryonshavesomehowgoneintoforming collisionsbetweensuchcloudsasadominantsource
8
3 3
2.5 2.5
2 2
ΣΣ/01.5 ΣΣ/01.5
1 1
0.5 0.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4
R/rt(total) R/rt(total)
Fig. 4. On the left: the deduced surface density distribution around the cluster Cl 0024+17 using weak lensing (Jee et al.
2007).Thesmallbumpistheallegedring.TheothertwoframesshowtheMONDpredictionsforverysimplisticconfigurations
produced by masseswell contained withinthe radiusof the bumptaken fromMilgromand Sanders (2007b).
of the x-ray emission from galaxy clusters. Walker, the mass and size distribution of the clouds should
however, assumed that these clouds constitute the be a function of location, with much mixing on the
ubiquitous DM, and also that they are responsi- cluster crossingtime scale.Thereis alsoredistribu-
ble for the so called extreme scattering events in tion ofheatandgasonhydrodynamicaltime scale,
the Galaxy. This determines the cloud parameters withcloudsfromlargeradii(perhapsevenfromthe
(mass and radius) quite stringently. As explained very outskirts of the cluster) replenishing those at
in the introduction, we are here totally free from smallerradiiwhere destructionis moreeffective.
such constraints, and can reconsider the process in Togetaroughideaofthepropertiesoftheclouds
anew light. that are today most efficient in heating, I assume
Ablation of the clouds by the hot gas is also at that at present all clouds are identical in size and
work, and contributes to the heating. The rate of mass, and require that the rate of kinetic energy
heatingbyablation,orcollisions,isproportionalto transfer due to cloud collisions balances cooling lo-
the mass flux a cloud sees in his rest frame. Since cally.Ialsoassumethatmostoftheenergyreleased
themassfluxinCBDMisratherlargerthanthatin goes into heat, as opposed to being radiated away
hotgasintheclustercore(bytheratiooftheirden- (see below). We can write the ratio of the heating
sities),onewouldestimatetherelativecontribution rate,E˙ ,tothecoolingrate,E˙ inauniformspher-
H c
ofablationtobesmall.However,itcanbemadeim- ical volume,representingthe cluster core,ofradius
portantifthecrosssectionofthecloudstoablation R=100R2 kpc as:
is largerthanthat forcollisions;e.g.,ifclouds have
an extended magnetic field of such strength that it E˙ /E˙ ∼η(M /M )×
H c DM gas
candragthegasbuthardlyaffectthemorecompact
×(T /T −1)(τ /τ )(Σ /Σ ), (1)
cloudsthatflyby.InwhatfollowsI’llconsideronly DM gas cool cross DM cl
collisions. where MDM = 1013MDM,13M⊙ and TDM are the
The processes occurring today regarding the
totalmassandkinetictemperatureoftheCBDMin
cloud-gas system cannot really be understood in
the core volume, while M and T are those of
gas gas
disjunction from the full system history. Because,
thegas;τ andτ arethecoolingtimeandthe
cool cross
typically,thetotalCBDMmassiscomparablewith
crossing time (= R/V, V being the 1-D velocity),
thatinhotgasitispossiblethat,infact,muchofthe Σ = M /πR2 is the mean surface density of
DM DM
presentlyobservedhotgasoriginatedincoldclouds 2
the core CBDM, Σ = M /πr is that of a single
cl cl
that collided and converted their kinetic energy to
cloud(M beingitsmass,andritsradius).Theco-
cl
heat (as proposed by Walker 1999). The evolution
efficientηisageometricalfactortakingintoaccount
of the cloud population is thus also an important
aGaussianvelocitydistributionoftheclouds,angle
aspect. The life time of a cloud against destruction
ofimpact,andimpactparameterdistribution.Using
by collisionsdepends onits sizeandits location,as results from Walker (1999) I find η ≈ 1.2. We have
well as on properties of the cloud population. So,
direct knowledge of all the quantities appearing on
9
therighthandside,exceptforΣ ,whichcanthusbe The other means of direct detection is via mi-
cl
constrained by the requirement for heating to bal- crolensing of quasar images behind a cluster. Since
ance cooling at present.Substituting typical values thecloudsurfacedensityhastobemuchlargerthan
for cooling cluster cores: τ /τ ∼ 100; M /M ∼ 1grcm−2,theyarewellwithintheirEinsteinradius
c cr B g
50and(T /T )∼3 forlensingatcosmologicaldistances,andshouldbe
DM gas
Σcl ∼104ηΣDM ∼103ηMDM,13R2−2grcm−2. (2) estffeeicntidviescmsoicfrtohleencsleosu.dTs,hwehsikchyicsotvheerpfarocbtoarbioliftyEfionr-
Itis difficult toestimate the massandsize ofthe appreciablemicrolensingamplificationis,ofcourse,
clouds separately. These may depend on how the similartothestronglensingprobabilitybytheclus-
clouds formed, which of them have already disap- tercore,whichistypicallynotmuchlessthenunity.
pear,whatsupports them,etc.. Microlensing of quasar images have been identified
Thefirstroughequalityineq.(2)tellsusthatthe (Fohlmeister 2006a,b). The expected microlensing
total sky cover factor of the clouds in the core is duration by clouds depends on the masses of in-
∼10−4,whichmeansthatthe cloud-cloud-collision dividual clouds. Some microlensing constraints on
life time of a single cloud is ∼ 104 crossing times. CBDM masses are mentioned by Carr (2000), but,
ThisturnsouttobeoftheorderoftheHubbletime, asexplainedaboveshouldberelaxedfortheCBDM
which may not be a mere coincidence, but a result quantities impliedby MOND.
ofcloudevolutionthatcullsoutcloudswithshorter
lifetimes.Thesmallcloudskycoverfactorisalsoin
line with the observationsofthe bullet cluster as it
meansthattheCBDMmassesoftwocollidingclus-
AcknowledgementsIthankBobSandersforcom-
terswouldgothrougheachotherpracticallyintact.
ments on the manuscript. This research was sup-
The second rough equality in eq.(2) tells us that
portedbyacenterofexcellencegrantfromtheIsrael
the cloud’s optical depth is very large. In a colli-
Science Foundation
sion, much of the kinetic energy is converted into
thermal energy. Part of this is converted back into
hydrodynamic energy of the gas debris expanding
fromthecollisionregion;anotherpartescapesasra-
diation. The photon escape time is typically t ∼ References
es
rτ/c, and is at least a few times longer in our case Afshordi, N., Lin, Y. -T., Nagai, D., Sanderson,
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exp
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10