Small-scale structure of dark matter and microlensing

30 Pages·0.58 MB·English

by  A. V. Gurevich

Physics±Uspekhi40(9)869±898(1997) #1997UspekhiFizicheskikhNauk,RussianAcademyofSciences
REVIEWSOFTOPICALPROBLEMS PACSnumbers:95.35.+d,98.35.±a
Small-scale structure of dark matter and microlensing
A V Gurevich, K P Zybin, V A Sirota
Contents
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Abstract. It has been revealed using microlensing that a con- pactandnon-compactobjectsisconsideredindetail.Theresults
t
s siderablepart,possiblymorethanhalf,ofthedarkmatterinthe ofmicrolensingobservationsaredescribedandcomparedwith
a
halo of our Galaxy consists of objects with a mass spectrum theory.Possibleastrophysicalmanifestationsofthepresenceof
rangingfrom0.05to0.8ofthesolarmass.Whatisthenatureof small-scalestructurearepointedout.Thefieldisbeingexten-
these objects? There exist two hypotheses. According to one, sivelystudiedandisoffundamentalinterestforcosmologyand
theseareJupitertypeplanetsorsmallstars(brownandwhite astrophysics.
dwarfs)consistingofnormalbaryonicmatter.Accordingtothe
other, these are non-compact objects, i.e., small-scale forma-
1. Introduction
tionsinnon-baryonicdarkmatter.Here,atheoryisproposed
describing the possibility of the existence of non-compact ob- Observations show that the greater part (over 90%) of the
jects inthe halo of our Galaxy, their structure and formation matterintheUniverseismadeupofinvisible(dark)matter
fromnon-baryonicmatter.Thetheoryofmicrolensingoncom- [1], and it remains unknown what particles it consists of.
From analysis of the processes of primary nucleosynthesis
and galaxy formation it follows, however, that dark-matter
AVGurevich,KPZybin,VASirota PNLebedevPhysicsInstitute,
particles must be of non-baryonic origin and must interact
RussianAcademyofSciences,
Leninski|¯ prosp.53,117924Moscow,Russia rather weakly both among themselves and with baryonic
Tel.[phone] matter.Theseparticlesaretypicallyassumedtobeeitherthe
E-mail:[email protected],[email protected], hypothetical heavy particles predicted by supersymmetry
[email protected] theory (customarily referred to as neutralinos), or light
axions, or cosmic strings. They all may constitute the so-
Received8April1997
calledcolddarkmatter(CDM)[2,3].
UspekhiFizicheskikhNauk167(9)913±943(1997)
TranslatedbyMVTsaplina;editedbyLVSemenova Non-dissipative dark matter plays a decisive role in the
formation of the large-scale structure of the Universe, i.e.,
galaxies, clusters of galaxies, and superclusters. At the
870 AVGurevich,KPZybin,VASirota Physics±Uspekhi40(9)
nonlinear stage of development of initially small perturba- However, the number of dark objects thus revealed
tions there occurs a gravitational compression and a sub- appearedtobeverylarge.Adetailedanalysiscarriedoutin
sequent kinetic mixing of the dark matter leading to the recentpapers(seeRef.[16])demonstratesthattheobserved
formation of stationary self-trapped objects with a singular objectsmakeupperhapsover50%ofthetotalmassofdark
densitydistributionratthecenter[4,5] matterinthehaloofourGalaxy.So,wefindacontradiction
with the conventional ideas of the non-baryonic nature of
ÿa
r(r)=Kr ; a(cid:25)1:8: (1)
dark matter. Furthermore, with such an interpretation one
encounterssomedifficultiesinwhatconcernstheconsistency
HereKisaparameterdependingontheinstantoftimewhen between microlensing data and direct optical observations
the object was formed. It is of importance that the form of fromtheHubbletelescope[18].
initial perturbation does not practically affect the scaling In this connection it was hypothesized in Ref. [12] that
parametera. microlensinginthehalorevealsnotJupitertypeobjectsand
The dissipative baryonic matter, which makes up but a coldstars,butobjectsofsmall-scalehierarchicaldark-matter
smallfractionofthetotalmassofthematter,islocatedinthe structuresconsistingofnon-baryonicmatter.Theseobjects,
center, i.e., in the peak of density, where it forms galaxies, however,arenon-compactandhavesizesoftheorderofthe
whilethenon-dissipativedarkmatter,distributedaccording Einsteinradius,characterizingmicrolensing,ormayevenbe
to the law (1), forms a giant galactic halo. This theory is severaltimesgreater.Forthisreason,thetheoryofmicrolen-
confirmedbytheobservationaldata[6±8]. sing by non-compact dark-matter objects was developed in
Owing to gravitational forces, galaxies unite to form Ref. [19] (see also Ref. [20]). A detailed comparison of this
clusterswhich, in turn,formlargeAbell clusters and super- theory with the available observational data gives a good
clusters thus showing up hierarchical clumping. The domi- agreement[19,13].Itshouldneverthelessbeemphasizedthat
nant role in this process is also played by dark matter in thesameobservationaldataareinagreementwiththetheory
which, irrespective of the object's size, the distribution (1) of microlensing by compact objects. The difference in the
inevitably sets in. This property of hierarchical clumping is results of these theories now lies within the experimental
clearlypronouncedinthepaircorrelationfunctionofgalaxies error. Therefore, the question of the presence of both non-
andotherobjects.Atthispoint,thetheory[9]alsoagreeswell compact and compact objects in the halo of our Galaxy
withtheobservationaldata[1,10,11]. remainsopen.
We have up to now considered clumping of large-scale Ouraimhereistogiveareviewofthepresentstateofthis
objects. What happens to objects that are smaller than problem. We should stress that if in comparison with the
galaxies? How far can the hierarchical structure extend theorysomenon-compactobjectsarediscoveredasaresultof
towardssmallscales?Ithadnormallybeenassumedthatthe improvedprecisionandprocessingofmicrolensingobserva-
minimal objects formed of dark matter were halos of small tion,thiswillsimultaneouslymeanadirectdiscoveryofnon-
7 8
galaxieswithamassof(10 ÿ10 )M(cid:12).However,theideasof baryonic CDM for the reason that no baryonic objects of
possible scales of minimal objects have lately changed suchamass,scaleandluminositymayexist.
radically. In the papers by the present authors [12, 13] it is
shown thatdarkmattercan formvery smallgravitationally
2. Minimal dark-matter objects
boundobjects.Themassoftheseobjectsisdeterminedbythe
structureofthespectrumofinitialfluctuations.Thestructure
of the spectrum on small scales is now unavailable. On the 2.1 Primary spectrum cut-off
basisofthehypothesis[12,13]concerningthecharacterofthe Dark-matterobjectsareformedupondevelopmentofJeans
initialspectruminthisregion,itwasdemonstratedthatinthe instability:asaresultofgravitationalattraction,smallinitial
dark matter on these scales a hierarchical structure fairly perturbations of a uniformly distributed matter increase.
similartothelarge-scalestructuremaydevelop,extendingto Crucial for the formation of such a structure is the form of
quitealargeamountofobjectswithmassesaslargeasthatof theinitialdensityfluctuations
theSun.
The question, however, is whether such dark-matter d(x;t)=r(x;t)ÿr0(t) ; (2)
objects can exist now, and if they can, in what form. How r0(t)
longdotheyliveandhowaretheydistributedinthehaloof
the Galaxy? The analysis of these questions shows [13] that where r(x;t) is the local density of matter and r0(t) is the
thelifetimeofsmall-scaleobjectsismuchlongerthantheage mean density of matter. The initial fluctuations di0(cid:12)(x) a(cid:12)r2e
oftheUniverse.Theyaredistributedinthehaloaccordingto customarily represented by the Fourier spectrum (cid:12)di(k)(cid:12) ,
thelaw(1).Thus,thetheorypointsoutthepossibilityofthe where
existence of a large amount of small-scale non-baryonic …+1
objects in the Galactic halo with a fairly broad mass di(k)= di0(x)exp(ikx)dx: (3)
spectrum. ÿ1
Manifestationsofdarkmatteronlargescalesuptonow The form of the spectrum of initial fluctuations is
havebeenregisteredonlyinobservationsofthedynamicsand determinedbyphysicalprocessesintheearlyUniverse.The
gravitational lensing of large-scale structures. Experiments spectrum
onmicrolensingwhichrevealedthatasubstantialamountof (cid:12) (cid:12)2
invisibleobjectsofmassM(cid:24)(0:05ÿ0:8)M(cid:12)[14±16]existin (cid:12)di(k)(cid:12) /k (4)
thehaloofourGalaxyarethereforeafundamentalnewstep.
Theseare usually supposed tobe brown or white dwarfs or isusedconventionally,suggestedin1965byYaBZel'dovich
planets (`Jupiters') consisting of ordinary baryonic matter [21] and confirmed by recent observations [22] over large
[15,17]. scales.
September,1997 Small-scalestructureofdarkmatterandmicrolensing 871
The spectrum (4) is cut off for low values of the wave 2.2 Linear growth of fluctuations
ÿ1
number k on the scale of the horizon radius Rh . For high Thesmall-scaledark-matterstructure,asforthelarge-scale,
valuesofthewavenumberk!1,thecut-offoccursduring appearsowingtothegrowthofsmallfluctuationsuponthe
decoupling,i.e.,intheperiodwhentheparticlesofthedark developmentofJeansinstability.Animportantpeculiarityof
matter escape thermal equilibrium. Note that one typically this process is that at the radiation-dominated stage of
considersbothHDMandCDM(oracombination).HDMis Universe expansion, the increase of fluctuations is slow.
understoodasweaklyinteractingparticlesthatleavethermo- Their comparatively rapid growth starts just after equili-
dynamic equilibrium when they are still relativistic (for briumhassetinandtheexpansionisalreadydeterminedby
example, light neutrinos). CDM is thought of as consisting matterratherthanradiation[24].
2
ofratherheavyparticlesofmassmx410 eVwhosestrong We shall discuss the dynamics of the growth of dark-
interactionstopsassoonasthetemperatureintheexpanding matter fluctuations. For simplicity we shall assume the
2
Universe lowers to T(cid:25)mxc . The concentration of such parameter O to be equal to unity, that is, the density of
particles may continue to decrease for some time at the mattertobeequaltothecriticaldensity.Inthiscase,onscales
expense of annihilation. In both cases, the spectrum varies lessthanthehorizon,onecanusetheNewtonianapproxima-
onsmallscalesduringfreezing-outofdark-matterparticles. tion in the analysis of fluctuation dynamics. The equations
Afreekineticspreadoftheseparticlesresultsinasmearingof describing the linear growth of density fluctuations d(x;t)
thefluctuationswhosescaleislessthanthehorizonradiusat takeasimpleform[1]
themomenttx,whenthetemperatureis 2 (cid:18) (cid:19)2 (cid:18) (cid:19)
d d 2 da dd 3 E 1 da 1 1 E
2 + = d; = + : (8)
T(tx)=mxc : (5) dt2 a dt dt 8 a3 a dt 4 a4 a3
tx
It is the horizon radius Rh at this instant of time that Herea(t)isthescalefactordeterminingtheexpansionofthe
determinesthecharacteristicscaleofthecut-off[23]: Universe.TheparameterEdescribesthematter-to-radiation
1 density ratio rx=rg. Before the moment teq of equilibrium
kmax (cid:25) tx : (6) onset,whena<aeqand
Rh
On large scales k5kmax, the fluctuation spectrum (3), (4) a(teq)=aeq =1 ; E=rx ; (9)
remains practically unaltered, whereas on small scales E rg
k>kmaxthefreespreadofparticlescausesastrongdamping
offluctuations,whichleadstoasharpfallofthespectrumon the dominating particles are photons [the first summand in
ÿ1
thescalesl<kmax.Thecalculatedvariationsofthespectrum Eqn (8) for a(t)] and after the equilibrium onset, i.e. for
(4)duetothisprocessarepresentedinFig.1[23].Theposition t>teq, nonrelativistic X-particles are predominant. At the
ofthepeakisdeterminedbyrelations(5)and(6).Thepeakis radiation-dominated stage, that is, in the period when
fairlybroad: radiation is predominant, the scale factor a(t) increases as
p(cid:129)(cid:129)
t,whileattheduststage,thatis,intheperiodwhenmatteris
0:5kmax <k<2kmax: (7) 2=3
predominant, it increases as t [1]. Since the density of
ÿ3 ÿ4
Afterthispeak,asharpfallofthespectrumisobserved.The matterisrx /a andtheradiationdensityisrg /a ,then
ÿ1
maximum in the spectrum and the steep fall when k>kmax rx=rg /a/(1+z) . As the present-day value of rx=rg is
areindicativeofthefactthatintherealizationofthespectrum known,itiseasytofindthedensityofmatterandtheredshift
ÿ1
therearenoperturbationswithscalesbelowlmin (cid:25)(2kmax) . attheinstantoftimeteq:
Consequently,thesmallestobjectsformedofdarkmatterwill
ÿ16 ÿ3
justhavescalelmin. rx (cid:25)rg (cid:25)req (cid:25)3(cid:2)10 gcm ;
4
teq =t(zeq); zeq '3(cid:2)10 : (10)
d
Inequations(8),thenormalizationofthescalefactora(t)is
1
chosenso that we havea=1 atthe instantof time tx when
dark-matterparticlesescapethermalequilibrium[thisinstant
0 0
oftimeisdefinedbyEqn(5)].AccordinglyE=rx=rg,where
2 0 0
rx and rg are the respective densities of X-particles and
photons at the instant tx. Since the temperature is
2
T(cid:24)10 eVatthemomentofequilibriumteq,itfollowsthat
2
iftheparticlemassismx410 eV,thentx5teq.Thismeans
that in CDM the moment tx is attributed to the radiation-
dominatedstageandE51.
In equation (8) for d one can get rid of the time by
introducingthevariable
p(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)
m= 1+aE:
1 10 Indeed,expressingdtfromthesecondequationofthesystem
k=kmax (8)intermsofda,wederiveauniversalequationford:
(cid:20) (cid:21)
Figure1.Zel'dovich±Harrisonspectrumtransformationduetofreeflight d 2 dd
ofdark-matterparticles[23]:(1)statisticalweightg=30,(2)g=200(16). (1ÿm ) +6d=0: (11)
dm dm
872 AVGurevich,KPZybin,VASirota Physics±Uspekhi40(9)
ÿ9
The solution to this equation is the second Legendre densityratio(rg=rb (cid:25)10 ):
polynomial:
ÿ8 mb
p(cid:25)10 ;
d 2 mx
=3m ÿ1=3Ea(t)+2: (12)
d0 wherembandrbarethemassandthedensityofbaryons.
Thisimpliesthatifdarkmatterconsistsofheavyparticles
Itshouldbeemphasizedthatdisnormalizedheretoasmall mx51GeV,themassoftheminimalobjectsisverysmall:
quantity d0 which is the initial value of fluctuations at the (cid:18) (cid:19)ÿ3
moment tx. Expression (12) describes the linear growth of ÿ6 ÿ7 mx
Mx (cid:24)(10 ÿ10 )M(cid:12) : (17)
fluctuations both at the radiation-dominated and the dust 1GeV
stages.Weseethat,duringtheradiationdominatedstage,the
fluctuationsgrowslowly,increasingonlyby2.5timesupuntil Theestimate(6),(17)givesthespectrumcut-offscaleandthe
the moment z=zeq, i.e. a=aeq. After this, a rapid growth mass of a possible minimal object for the assumption that
begins boththethermodynamicdark-matterparticledecouplingand
thecessationofinteractionoftheseparticleswithotherstakes
d /3Ea/t2=3: (13) placeatthetimeinstanttx(5).Ifthedecouplingandespecially
d0 interactioncessationtakeupmoretime(whichisthecase,for
example, for supersymmetric particles Ð neutralinos), the
2.3 Mass of the minimal dark-matter objects cut-offscaleandthemassoftheminimalobjectmayincrease
The linear growth of the fluctuations stops as soon as the substantially.
magnitude of perturbations becomes comparable with the Anotherpossibilityfortheappearanceofsmall-massnon-
mean density of matter d(cid:24)1. After this, the stage of baryonicstructuresisduetoaxions.Thenonlineareffectsof
gravitational compression and kinetic mixing sets in and evolutionoftheaxionfieldintheearlyUniversewhichmay
leads to the formation of stable spherically symmetric leadtotheformationofgravitationallyboundmini-clusters
objects in the center of which a singular density distribu- were investigated in Ref. [26]. These mini-clusters have a
ÿ12 10
tion (1) is observed. This process was considered in detail characteristicmassM(cid:24)10 M(cid:12)andsizeR(cid:24)10 cm.
in the previous review by the present authors [5]. As a
result, a hierarchical structure of gravitationally bounded 2.4 Scale of the minimal objects
objects develops from the primary spectrum of fluctua- Aswehaveseenabove,themassoftheminimalobjectsdoes
tions. not depend on the spectrum of initial perturbations but is
Itisofimportancethatthisstructureislimitedfrombelow determinedonlybythedark-matterparticlemass.Thescale
bythemassMminofminimalobjects: oftheminimalobjectsisdetermined,onthecontrary,bythe
amplitude of the fluctuation spectrum d. Indeed, the mean
3
Mmin (cid:25)rx(tx)ax: (14) density of matter r0(t) in the Universe decreases with time.
Becausethemassofanobjectisfixed[seeEqns(15),(16)],its
Here rx(tx) is the dark-matter density at the moment of sizeisnaturallydeterminedbythedark-matterdensityatthe
decoupling,whentheparticlesmx (5)aresingledout,andax instant of time tc when it was formed. The quantity tc is
is the scale of the horizon at the same instant of time, specifiedbytheconditionoftransitiontothenonlinearstage
tx
ax =Rh.Theinstanttx inEqn(14)dependsonthemassof of compression in the region of the spectrum maximum dm
dark-matter particles. Consequently, it is the mass mx that wheretheminimalobjectsaregenerated:
exertsacrucialinfluenceonMmin.InthecaseofHDM,when
2 dm(tc)(cid:24)1: (18)
mx510 eV,wehave,asshowninRef.[23],
Thus,thescaleRxofaminimalobjectisdefinedas
3
mpl
Mmin (cid:25) m2x : (15) Rmin (cid:25)(cid:20)Mmin(cid:21)1=3; (19)
r0(tc)
1=2
Herempl =(hc=G) isthePlanckmass.Onecanseethatin
thecaseunderconsiderationthemassMmin isrelatedtothe wherer0(tc)isthedark-matterdensityatthemomentofits
massofdark-matterparticlesinaone-to-onemannerandis formationtc.
rather large for HDM. For example, for the hypothetical Within the period when radiation prevails, fluctuations
small-mass neutrinos with mx =10 eV (which were consid- increase very slowly (see Section 2.2). Since the initial
17
ered in Ref. [2]), the mass is Mx (cid:25)10 M(cid:12), which corre- fluctuations d0 are not large, relation (18) can be fulfilled
spondstothemassoflargeclusters. only provided that tc >teq, where teq is the moment of
2
For the case of CDM for mx410 eV, relation (15) transition from the radiation-dominated to the dust stage,
changes(seeRefs[25,13])tobecome which is determined by condition (9). The red shift zc that
3 (cid:18) (cid:19)2=3 correspondstothemomenttcisdefinedherebyrelations(12)
mpl nx g and(18).Theyimplythat
Mmin =p m2x ; p=ng 4 : (16) 3 zeq+1
zc+1= ; d0 <1: (20)
2 1=d0ÿ1
Here nx and ng are the respective concentrations of CDM
particlesandphotons,gisastatisticalweightoftheorderof SincethedensityintheUniversedecreasesrapidlyas
2(cid:2)102.ThefactorpinEqn(16)isduetoannihilationofa (cid:18) (cid:19)3
1+z
majorityofthematterintheperiodafterdecoupling.Itcanbe r=req ; (21)
estimated from the present-day relic photon-to-baryon 1+zeq
September,1997 Small-scalestructureofdarkmatterandmicrolensing 873
ÿ16 ÿ3
wherereq (cid:25)3(cid:2)10 gcm isthedensityofmatteratthe It is the fluctuation F(k) that determines sequence of
moment of equilibrium, it follows that the scale of the appearance of objects on different scales (l(cid:24)1=k) in the
minimalobjectsincreasesactivelyasd0decreases: hierarchical CDM structure: the first to appear are objects
(cid:26) (cid:27)1=3 (cid:18) (cid:19) withmaximumF(k)values,andsubsequentlyappearother,
Mmin 2 1 largerobjects.
Rm =K2 (cid:2) (cid:3) ÿ1 : (22)
4p= (3ÿa)req 3 d0 The form of the function F(k) for the Zel'dovich±
Harrison spectrum (4) is shown in Fig. 2 (curve 1). As is
ThisexpressionisspecifiedforthescaleRm,namely,itnow clear from Eqns (24) and (25), for k>keq it gradually
allowsforthefactorK2 (cid:25)0:1ÿ0:3ofnonlinearcompression transforms to a plain form. This picture is on the whole
inthecourseoftheobject'sformation(seeSection4.1)and confirmed by contemporary observational data over large
thedensitydistribution(1)insidetheobject. scales (l(cid:24)500ÿ1000 Mpc) [22]. Recent temperature mea-
ÿ3
Forexample,ford0 (cid:25)10 thecorrespondingredshiftat surementsintheregionofthefluctuationmaximum,thatis,
ÿ3
which the object was formed (21) is zc (cid:25)1:5(cid:2)10 zeq, and on scales l(cid:24)30ÿ50 Mpc, as well as on smaller scales are
ÿ6
the scale of the object of mass Mm =10 M(cid:12) is indicative of possible deviations from the Zel'dovich±
15
Rm (cid:25)5(cid:2)10 cm.Ifd0 (cid:25)0:4,thenzc (cid:25)zeq andthescaleof Harrison spectrum, which in the simplest form are reduced
13
thesameobjectismuchsmaller:Rm (cid:25)10 cm.Wealsonote toavariationoftheexponentofthespectrum[27]:
(th0:a1tÿth0:e5)sMca(cid:12)le, ofinobwjechtischwitwhea cahrearaicnteterrisetsitcedmaslasteMr,m =is (cid:12)(cid:12)di(k)(cid:12)(cid:12)2 /kn; n(cid:25)1:2ÿ1:4: (26)
18 ÿ3
(0:6ÿ3)(cid:2)10 cmwhend0 =10 and
Suchapower-lawspectrumisalsopresentedinFig.2(curve
14
Rm (cid:25)(3ÿ15)(cid:2)10 cm (23) 2). Other notable deviations of the spectrum of initial
fluctuations from the Zel'dovich±Harrison spectrum are
whend0 =0:4. revealed in observations of distribution of the density of
matteronscalesof30±50Mpc[3].
It is, however, of importance that between the regions
3. Hierarchical structure 25 26
wherethespectrumhasalreadybeenmeasuredl510 ÿ10
Up to now we have only discussed minimal dark-matter cm and the region of minimum-scale objects
ÿ6 13 15
objects. They are first to appear. The presence of a wide M(cid:24)(10 ÿ1)M(cid:12), that is, l(cid:24)10 ÿ10 cm, which is of
spectrumofperturbationsleadstothesubsequentgeneration particular interest for us, there is a huge region with a
of objects on various scales. Large-scale objects can trap spectrum whose state ispracticallyunknown. Note that the
smaller-scale ones and be themselves trapped by still larger function F(k) is strictly flat in the region k>keq for the
objects.Exactlythistypeofhierarchicalstructureisobserved Zel'dovich±Harrisonspectrumonly.Itmaybeassumed,for
in large-scale formations in the Universe, such as galaxies, example,thatforkwhicharesufficientlylargebutbelowthe
clustersofgalaxies,largeAbellclusters,superclusters,andso cut-offregionk5kmax,thespectralfunctionF(k)hasanew
on.Itisquitenaturaltoexpectthatananalogoushierarchical region where it ascends. Such a non-standard spectrum is
structureoccursonsmallscalesaswell. showninFig.2(curve3).
The sequence of objects formation on different scales is Inthefirstmodel,curve1(letuscallitstandard),avery
determinedbythefluctuationspectrum.Goingoverfromthe broadspectrumofobjects,fromminimalbodiestogalaxies,
Fourier transform and back to x-space corresponds to appearalmostsimultaneouslyinthehierarchicalstructure.In
3
integration over d k and is, therefore, equivalent to multi- the second model (which we refer to as power-law), first to
3
plicationofthefluctuationspectrumbyk .So,insteadofd(k) developarethesmall-scaleobjects,andsubsequentlyappear
[seeEqn(4)]itisreasonabletoconsiderthespectralfunction structures of much larger scales. This difference in the
F0(k)=k3(cid:12)(cid:12)di(k)(cid:12)(cid:12)2: (24) cehveanragcrteearteorfionbtjheectthfiorrdm(antoionn-stoannddairfdfe)rmenotdseclawlehserbeecsommalels-
scale structures develop at a moment close to teq and large-
Animportanttransformationofthefluctuationspectrum scalestructuresdevelopmuchlater.Itshouldbestressedthat
isconnectedwithpeculiaritiesofCDM.Thepointisthatin
thecaseofCDMthemomentofparticledecouplingandthe
logF
momentwhenthedark-matterdensitybeginstoprevaildiffer
appreciablyinenergies(and,accordingly,intimeorredshifts log0:4
z). As shown above [see Eqns (8), (12)], at the radiation- 3
dominated stage in dark matter inside the horizon Rh,
fluctuations increase very slowly; they are almost `frozen'. 2
Outside the horizon,i.e. for k4keq =1=Rh, where fluctua- 1 1
tions are small, they increase gradually in proportion with
time[1]:
dr
/t:
r
keq ks logkÿ1
Owingtothisprocess,fluctuationsonsmallscalesareleveled,
andthespectralfunctionistransformedasfollows[1,2]:
Figure 2. Form of the spectral function F(k): (1) for the Zel'dovich±
F0(k) Harrisonspectrum,(2)forapower-lawspectrum,(3)foranon-standard
F(k)= : (25)
1+(k=keq)4 spectrum.
874 AVGurevich,KPZybin,VASirota Physics±Uspekhi40(9)
the indicated differences in the course of object formation The total number of objects Nf can be evaluated from the
may have an appreciable effect on the conditions of their relation
furtherexistence,whichdeterminestoalargedegreetheform
Mf
ofthesmall-scalestructurenowadays. Nf =f ; (33)
Mi
3.1 Clustering where f is the fraction composed of objects Mi in the total
Theprobabilitythatasmall-scaleobjectwillbetrappedbya massoftheobjectMf.
large-scaleone,providedthatthedifferenceinsizeisnotvery
big, does not generally prove to be high. In the case of a 3.2 The lifetime of entrapped objects
power-law spectrum, this probability is determined by the We note first of all that objects of a hierarchical structure
clusteringparametere,thatis,theprobabilitythatanobject undergoalmostnochangesfromthemomentwhentheyare
of scale Rx appears inside another object of size Rf. An formed till the moment when they are trapped by larger
estimateoftheclusteringparameterehastheform[5]: objects. Indeed, during this period they have very small
(cid:18) (cid:19)3=2 peculiar velocities because the relative velocities at the
Rf 6 moment of formation are practically zero and the distance
e=4pbln ; where b<0:016 : (27)
Rx m+5 betweentheobjectsincreasesowingtothegeneralcosmolo-
gical expansion. The relative velocity and the possibility of
Herem=nÿ4istheexponentofthespectralfunction.This collisionofobjectsareexclusivelyduetotheirentrapmentin
deviationoftheexponentofthespectrumfromthevalue(26) thegravitationalfieldoflarger-scaleformations.So,wemay
is due to the spectral function transformation (25). Expres- speak only of the lifetime of objects trapped in larger-scale
sion (27) gives a theoretical estimate `from above' for the structures.
clusteringparameter[28].Theestimationoftheparameterb Entrapped objects may experience interactions. Elastic
obtainedfromcomparisonwithobservationsforalarge-scale collisions between objects proceeding without energy
structureyields[29] exchange may affect only the distribution (1) of matter in
(cid:18) (cid:19)3=2 them,whereasitisinelasticprocessesthatareresponsiblefor
6 ÿ3 destruction.Theprincipalmechanismofdestructionistidal
b=b0 ; b0 (cid:25)(3ÿ6)(cid:2)10 : (28)
m+5 interaction,occurringwhenobjectsflybyoneanotherorby
starsatclosedistances.Forthecaseofordinarystars,where
Asuccessiveentrapmentofsmallerobjectsbyobjectsof energy dissipation takes place inside a star, the role of tidal
increasingly large scales presents a picture of hierarchical interactionisdescribedindetailinRef.[30].Inourcase,too,
clumping.Therelationbetweenthesizeandthemassinthe thephysicsofthedestructionprocessisrelatedprimarilyto
caseofhierarchicalclumpingisgivenby[12] tidal forces because these forces are responsible for the
(cid:18) (cid:19)(m+5)=6 increase of kinetic energy of particles inside objects, and if
Rx Mx thisenergybecomesequaltothebindingenergyoftheobject,
= : (29)
Rf Mf weobserveadestruction.Letusinvestigatethisprocessmore
closely.
Proceeding from relations (27)±(29) one can determine Supposeanon-compactobjectofmassMx andscaleRx
the mass ratio of larger Mf and smaller Mi objects to fit a interactswithanotherobjectflyingbyatthedistanceRora
probability of the order of unity that the larger object will massivebody(star)ofmassM.ThevelocityVoftheirrelative
entrapthesmallerone.Assuminge(cid:25)1informula(27)and motion greatly exceeds the pa(cid:2)rticle velocity vx in(cid:3)side the
1=2
allowingforEqns(28)and(29),wefind indicated non-compact object vx (cid:24)(GMx=Rx) . There-
(cid:20) 1=2(cid:21) fore,inthefirstapproximationtheparticlescanbeconsidered
Mf (m+5) tobeatrest.
=exp : (30)
Mi 4p61=2b0 As a result of the interaction, the object Mx as a whole
gainsadditionalvelocity,whichmeansanelasticscatteringof
So,fortheZel'dovich±Harrisonspectrum(n=1)wehave objects, and moreover the particles inside it gain the
additional velocity of relative motion. This is the tidal
Mf =1:9(cid:2)103 for b0 =6(cid:2)10ÿ3; interaction. The force of tidal interaction Ft is connected
Mi with the finite dimension of the non-compact body and is
Mf 6 ÿ3 determinedbythedifferenceofforcesactingonitsparticles:
=3:6(cid:2)10 for b0 =3(cid:2)10 (31)
Mi 2GMMx
Ft = R3 Rx:
andforasharplyincreasingspectrum(n=5)
Theshiftoftheparticlesundertheactionofthisforceis
Mf 5 ÿ3
Mi =3:7(cid:2)10 for b0 =6(cid:2)10 ; Ft 2 GM 2 R
Mf 11 ÿ3 Dx=2Mx(Dt) = R3 Rx(Dt) ; Dt=V :
=1:4(cid:2)10 for b0 =3(cid:2)10 :
Mi
Accordingly,theenergydissipatedinoneeventis
According to the theory, all the entrapped objects will be 2 2 2
2G M MxRx
distributedinsideth(cid:18)ela(cid:19)rgÿeaobjectaccordingtothelaw(1): DE=FtDx= R4V2 :
3ÿa r
n(r)=4pR3f Nf Rf : (32) Ttfh=e1=t(impRe2nVof), wthheerefrneeis tphaethnumbebtewredeennsictyololifsioobnjsectiss.
September,1997 Small-scalestructureofdarkmatterandmicrolensing 875
IntroducingtheeffectivecollisionfrequencynE andthetidal tionallyboundandcanexistinsidealarge-scaleobject.Itis
destructioncross-sectionsEaccordingtotherelations the Hubble expansion and the sequence of appearance of
(cid:28) (cid:29) objectsÐfirstsmall-scale,andthenincreasinglylarge-scale
dE E
= =nEE; nE =nVsE (34) Ð that enabled the formation of the hierarchical structure.
dt tf Thesequenceofappearanceofobjects,aswasshownabove,
isspecifiedbytheformofthespectralfunctionF(k)(Fig.2):
(Eistheabsolutevalueofthetotalenergy),weeventuallyfind small-scaleobjectsappearearlierthanlarge-scaleonesifthe
(cid:18) (cid:19)2 2 2 functionF(k)growsmonotonicallywithincreasingk.Forthe
2 M Rx vx standardZel'dovich±Harrisonspectrumthisisnotthecase
sE =pRx Mx R2 V2 ; R5Rx; (35) because the function F(k) over a broad range of scales for
k>keqisnearlyconstant(Fig.2,curve1).Inthiscase,objects
wherevx isthemeanvelocityofdark-matterparticlesinside of all sizes for k>keq appear almost simultaneously. But
theobjects, given this, smaller objects inside larger ones are in no way
distinguished in density. This means that they cannot
2 2
Mxvx GMx originate through a regular process, and if they are distin-
=E= :
2 Rx guishedowingtofluctuations,theirdensitydiffersverylittle
from the mean density of the surrounding matter, and they
FromEqn(35)itcanbeseenthatthemaximumcross-section aredestroyedrapidly.
isachievedforR(cid:25)Rx.Letusnowtakeaccountofthefact From this it follows that in the case of a standard
that the velocity vx is determined by the gravitational spectrum (curve 1 in Fig. 2) the hierarchical structure does
potential of a considered object Mx, while the velocity V is not at all depend on the spectrum cut-off at small scales. It
determined by the gravitational potential of a much larger- begins in the region where a deviation appears in the flat
ÿ1
scale formation in which it is trapped. Consequently, as spectral function F(k), that is, for k4keq. The scale keq is
2 2
indicatedabove,wealwayshaveV 4vx.Hence,thecross- smallerthanthehorizonradiusbyapproximatelyanorderof
sectionoftidaldestructionofobjectsbyoneanotherisalways magnitude at the moment of equilibrium onset. It corre-
rather small as compared to their geometric size. Thus, the sponds in mass to small galaxies. Consequently, in the case
lifetimeoftheobjectsissubstantiallylargerthanthetimeof ofastandardspectrumthehierarchicalstructurebeginsinthe
theirfreepathbetweenstraightforwardhead-oncollisions. region of small galaxies and stretches towards larger scales.
AsisclearfromEqns(34),thelifetimeofobjectsofmass Thisagreeswellwiththeobservedlarge-scalestructureofthe
Mx trappedinalarge-scaleobjectofmassMf,i.e.,thetidal Universe.
destructiontimetf,isdeterminedbytherelation Thepictureisapproximatelythesameforthepower-law
spectrumwhichdiffersfromtheZel'dovich±Harrisonspec-
1
tf = : (36) trum (n=1:4, curve 2 in Fig. 2). Although the hierarchical
n(r)sEVf structuremayinthiscasestartdevelopingearly,thesurvival
small-scaleobjectstosurviveishamperedbecauseofthesmall
Here,theconcentrationoftrappedobjectsn(r)dependenton difference in density. A sufficiently long-lived structure
ÿ1
thedistancertothecenterofthelarge-scaleobjectisgivenby occurs on a scale larger than keq when the slope of the
relations(32)and(33)andthevelocityVfisequalto spectral function F(k) changes sharply. This again leads
r(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129) mainly to the possibility of the appearance of a full
Mf hierarchical structure in the Universe beginning with small
Vf = 2G : (37)
Rf galaxies.
Thesituationisessentiallydifferentinthecaseofthenon-
Inexpression(35),forthecross-sectiononecanputM=Mx, standard function F(k) represented in Fig. 2 (curve 3). The
R=Rx.Proceedingfromrelations(35)±(37),wefind hierarchicalstructuredevelopsherefirstintheregionofsmall
2 r(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:18) (cid:19)a scales,fork>ks,butitbreakssharplyfork4ksandappears
tf = 4Rf Rf r : (38) again only on galactic scales k4keq. Small-scale objects
(3ÿa)Rxf 2GMf Rf k5ks originateearlyandhave,therefore,averyhighdark-
matterdensityingalactic-scaleobjectsthatarisemuchlater.
Herefistheportionofthemassofthelarge-scaleobjectMf Under such conditions small-scale dark-matter objects with
contained in the small-scale entrapped objects Mx. For k5ks arestronglypronounced.Aswillbeseenbelow,they
example,thetotalnumberofentrappedobjectsofmassMx mayexistintheUniverseforaratherlongtime.
isNx =fMf=Mx.Itcanbeseenthatthelifetimeofentrapped
objects decreases strongly with decreasing r, that is, 3.4 Small-scale structure Ð the main hypothesis
approachingthecenter.Onaverage,itincreasesproportion- Sufficientlyreliabledataonthespectrumofinitialperturba-
ally tothe object size ratio Rf=Rx as wellas with increasing tionsintheUniversearenowavailableonlyforlargescales
characteristictimeofoscillationsinthelarge-scaleobjectand l>1ÿ10 Mpc. As to small scales l<1ÿ10 Mpc, the
withdecreasingparameterf. information is next to null. The basic hypothesis that we
suggestedinRef.[13]readsthatforsmallscalesthespectral
3.3 Destruction of objects in the hierarchical structure function exhibits a non-standard nature as shown in Fig. 2
Theoriginationofthehierarchicalstructureiscloselyrelated (curve3)andforthisreasonthesmall-scalestructuredevelops
totheHubbleexpansionoftheUniverse.Indeed,becauseof intheUniverse.Ontheassumptionthatitisthisstructurethat
ÿ2
the expansion, the mean density of matter r0 /t falls isobservedinthemicrolensingonobjectsinthehaloofthe
rapidly.Asaresult,thoseobjectsthatappearedearlierhave Galaxy one may conclude that objects of mass 0:1ÿ0:5M(cid:12)
a higher density. Therefore, they are more strongly gravita- correspondtothepointofasharpsmall-scale(ks)spectrum
876 AVGurevich,KPZybin,VASirota Physics±Uspekhi40(9)
cut-off.Furthermore,theamplitudeofthespectralfunction Usingnowthevaluesoftheparameters[6,13]
intheplateau(orpeak)regionis
12
Mh (cid:25)2(cid:2)10 M(cid:12); Rh (cid:25)200kpc;
Fm =dim (cid:25)0:3ÿ0:4; (39) 14
Mx (cid:25)0:5M(cid:12); Rx (cid:25)4(cid:2)10 cm; f(cid:25)0:5;
which exceeds the value of the constant F0 in the standard wefind
spectrumbyovertwoordersofmagnitude. (cid:18) (cid:19)3=2(cid:18) 14 (cid:19)
On this basis we obtain that the small-scale hierarchical 8 Rh 4(cid:2)10 cm
t0 =7(cid:2)10 t0
structureofdark-matterobjectsextendstotheregion 200kpc Rx
s(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)
Mx (cid:24)(0:01ÿ1)M(cid:12): (40) 2(cid:2)1012M(cid:12)(cid:18) r (cid:19)1:8
(cid:2) : (45)
Mh 200kpc
Since the initial amplitude is high (39), it follows that as is
impliedbyEqn(20),thefluctuationsreachanonlinearstage
17
andformgravitationallycompressedobjectswithinaperiod Heret0 (cid:25)3(cid:2)10 sisthelifetimeoftheUniverse.Fromthis
oftimeclosetothemomentofequilibriumonsetteq.Thescale one can see that the lifetime of objects in the halo is many
oftheappearingobjects(22)and(23)is ordersofmagnitudelargerthanthelifetimeoftheUniverse.
Approaching the central part of the halo, t0 decreases
14 15
Rx (cid:24)(10 ÿ10 )cm: (41) substantially, but here, too, below scales of the order of
10kpcitremainsmuchgreaterthant0.
This scale grows with increasing mass Mx in proportion to In the galactic region, one should also involve the
1=3
Mx orsomewhatfaster,thegrowthratedependingonthe interactionwithstars.Makinguseofrelations(35),(36),we
slopeofthespectralfunction. arriveatthelifetime
We shall emphasize that the principal hypothesis of the (cid:18) (cid:19)(cid:18) (cid:19)
14
characterofthespectrumofinitialperturbationshasnotyet 3 4(cid:2)10 cm V
beencosmologicallygrounded. ts (cid:25)4(cid:2)10 t0 Rx 300kmsÿ1
(cid:20) ÿ3(cid:21)(cid:18) (cid:19)(cid:18) (cid:19)2
3.5 Lifetime of small-scale objects in the Galaxy 1pc Mx M(cid:12)
(cid:2) ; (46)
The small-scale structure with the characteristic mass spec- Ns(r) 0:5M(cid:12) Ms
trum (40) and scales (41) develops, under assumption (39),
overtheperiodclosetothemomentofequilibriumonsetteq. where Ns(r) is the density of the stars and Ms is their
Atthismoment,theamplitudeoffluctuationsonlargescales characteristic mass. We can see that outside the central
ÿ3
is only F0 (cid:24)10 . The large-scale structure, therefore, regionr>rGc,whererGc (cid:24)0:1ÿ1kpc,thisquantityremains
developsmuchlater,whenz<z0 (z0410). As aresult, the largerthanthelifetimet0.
meandark-matterdensityinlarge-scalestructures,including Thus, dark-matter objects may exist at the present time
10
thehaloofourGalaxy,ismany((cid:24)10 )ordersofmagnitude notonlyinthehalo,butinthegreaterpartoftheGalaxyas
lowerthanthedensityinsmall-scaleobjects.Thesmall-scale well. It is in the region r<rGc close to the center that they
objects are thus strongly distinguished dense formations in mustalreadybedestroyedasaresultoftidalinteractionwith
theGalactichalo. stars.
They,naturally,interactamongthemselvesaswellaswith The interaction of dark-matter objects with gas may be
starsandgalacticgasandcan,inprinciple,decayasaresultof quite diversified. First of all, since they form before
thisinteraction.Weshalldeterminetheirlifetime.Tothisend, recombination, it follows that after recombination and gas
we shall first estimate the total number of objects in the coolingtotemperatures
Galactichalo
mpMxG 3
T<T0; T0 (cid:25) (cid:25)2:6(cid:2)10 K; (47)
NxMx =fMh: (42) Rx
wheremp istheprotonmass,gascondensation startsinthe
Herefisthefractionofdarkmattercontainedinsmall-scale dark-matterobjects,whichleadstotheformationofbaryonic
structures, Mh is the total mass of the dark matter in the cores.Thisprocessmayproceedevennow.Then,obviously,a
halo, Mx is the characteristic mass of an object. The substantial difference must appear in scale and, perhaps, in
distribution of dark-matter objects obeys the fundamental thecompositionand structure ofbaryoniccoresinthehalo
law (1). Consequently, the density of these objects in the andtheGalaxy,wherethegasdensityishigh.Baryonicbodies
halo is intheGalaxymaynotonlybemuchgreaterthaninthehalo,
(cid:18) (cid:19)ÿa but they may also have an appreciable effect on the
3ÿa ÿ3 r distributionofthedarkmattersurroundingthecore.Specific
nx(r)= NxRh ; (43)
4p Rh featuresofthestructureofdark-matterobjectswithbaryonic
coreswillbediscussedatlengthinthesectiontofollow.
whereRhisthesizeofthehalo. Notethatbeingbaregravitationalbodiesonwhichstar-
Allowing for the tidal interaction of the dark-matter constituting gas is condensed, the dark-matter objects
objects, their lifetime is determined by expression (38). existing in the Galaxy may also have a strong effect on the
SubstitutingEqns(42)and(43),weobtain starformationprocess.Onbirthandignitionofastar,dark-
2 r(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:18) (cid:19)a matterparticlesmayeitherleaveitorstayinside(depending
t0 = 4Rh Rh r : (44) on their mass mx) producing a noticeable effect on its
(3ÿa)Rxf 2GMh Rh burning.ThispointwillalsobediscussedbrieflyinSection4.
September,1997 Small-scalestructureofdarkmatterandmicrolensing 877
Concludingthissectionweemphasizethatthesmall-scale (0) 2
runi = r0: (52)
structures considered within the proposed theory are essen- 5
tiallynewobjectsinthestructureofmatterintheUniverse.
(1)
Theyareformedbynon-interactingdark-matterparticlesand Auniformballofradiusr1anddensityrunicorrespondstothe
existowingtogravitationalforcesonly,i.e.,theparticlesare singulardensitydistribution(49).FromEqns(49)weobtain
entrappedbythegravitationalfieldcreatedbytheseobjects.
4 (1) 28p ÿ12=7
An important feature of the objects is, as follows from pruni = Kr1 : (53)
3 9
Eqns (40) and (41), that their mean density is
ÿ11 ÿ13 ÿ3
r/10 ÿ10 gcm .Inviewofthis,inspiteofasimilar Furthermore, in view of mass conservation we have the
mass,thevolumetheyoccupyismuchlargerthanistypical relation
forbaryonicbodiesÐtheirsizeexceedsthatofacorrespond-
(0) 3 (1) 3
ingcompactbaryonicbodybyapproximatelyfourordersof runir0 =runir1: (54)
magnitude. For this reason we henceforth refer to them as
non-compact objects (NO). It is noteworthy that, having a From Eqns (49), (52), and (53) we find the coefficients of
similarmass,abaryonicgasoccupiesamuchlarger(by7±9 compressionindensityKrandinradiusKr:
ordersofmagnitude)volumeintheGalaxythanNO.Thus,
(1) (cid:18) (cid:19)2(cid:18) (cid:19)7=3
onecanstatethatintheirgeneralpropertiesÐmass,scale, runi 40 5 r1 ÿ1=3
and luminosity Ð NO are specific objects of non-baryonic Kr =r(u0n)i ' 9p 2 ; Kr =r0 =Kr ; (55)
matter.Weaklyluminousbaryonicobjectsofthesamemass
andsizedonotexistinnature. whichgivesKr '20andKr '0:3.Letusemphasizethatthe
coefficientsofcompression(55)donotdependonaparticular
unifiedformtowhichboththedistributionsarebrought.For
4. Structure of non-compact objects
example,theresultwillbethesameifwebringthedistribution
2 2
(49) to the form (48) assuming r(r)=r1(1ÿr =r1) and
4.1 Coefficient of nonlinear compression determinetheparametersr1andr1usingmassconservation.
In the course of gravitational collapse leading to the Wecanseethatthenonlineargravitationalcompressionis
formation of gravitationally bound objects of non-interact- sufficientlylargeevenbeforethefirstsingularity.Theanalysis
ingdark-matterparticles,nonlinearcompressionoccurs.Let carriedoutinRef.[32]showsthatinthecourseofsubsequent
us determine the parameters of this compression. Gravita- mixing and the onset of the stationary state it does not
tionallycompressedobjectsareformedintheneighborhood generally increase. This remains valid if all the particles are
oflocalmaxima oftheinitialdensity. Suppose atthe initial kept in the entrapment region. However, depending on the
momentthedensitydistributionnearthismaximumr=0has initial conditions, some particles may actually leave the
theform entrapment region, and as a result of conservation of the
(cid:18) (cid:19) totalenergythisprocessinevitablyleadstostrengtheningthe
2
r
object'scompression.
r=r0 1ÿ 2 ; r<r0: (48)
r0
4.2 Density distribution in non-compact objects
Forsimplicity,wehaveassumedheretheinitialdistribution The distribution of density in NO depends mainly on the
tobesymmetric(intheneighborhoodofathree-dimensional development of Jeans instability at the nonlinear stage, but
maximum of any form the results are quite similar [5, 31]). thelinearstageoffluctuationgrowthalsoexertsaconsider-
The evolution of the nonlinear process of gravitational ableinfluenceonthisprocess.So,thescalinglaw(1),(49)is
compressionoftheinitialcluster(48)leadstotheappearance characterizedbytwofactors,namely,theslowlineargrowth
p(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)
inatimeoftheorderoftheJeanstime(tJ 'p 3=8r)ofa ofinitialdensityfluctuationsandtherapidcompressionwith
singularpointinthecenterwithadensitydistribution[4] subsequentkineticmixingatthenonlinearstage.Inthecase
(cid:18) (cid:19)6=7 oflargescales,theinitialdensityfluctuationsdi0[seeEqns(2)
ÿ12=7 3 40 12=7 and (3)] are always small, and therefore the unstable mode
r(r)=Kr ; K= r0r0 : (49)
7 9p increases for a substantially long time before the nonlinear
collapsesetsin.Withinthistimeinterval,thestablemodesare
Let us assume the distribution (49) to be restricted to the damped strongly. In the course of nonlinear collapse,
radius r1. For r>r1 we put r=0. The forms of the however, one of the damping modes starts increasing and
distributions (48) and (49) differ strongly. Under these doessoveryrapidly,sothatintheneighborhoodofr=0it
conditions, in order to find the value of the effective outruns the increasing mode [33]. As a result, in a small
parameterofcompressionitisreasonabletobring boththe neighborhoodofr=0thelaw(1),(49)iscutoffonthescalerc
distributionstoaunifiedform.Itisnaturaltochooseassuch [5],where
aformauniformballwithaconstantdensity
3
rc =di0Rx: (56)
r=runi: (50)
Totheinitialdistribution(48),weaddauniformballofthe HereRxisthecharacteristicscaleofanobject.Themagnitude
samemass ofinitialdensityfluctuationsdi0onlargescalesisrathersmall,
ÿ3
M= 8 pr0r30 =4 pr(u0n)ir30 (51) tdoi0v'ery10sma.lTlshciaslmese:afrnosmthEaqtnth(e56sc)aitlifnogllloawws(t4h9a)tirscf(cid:24)ulf1il0lÿed9Ruxp.
15 3
Thus,itsactualcut-offisnaturallycausedbytheinfluenceof
(0)
andradiusr0.Thenthedensityruniisrelatedtotheparameter otherprocesses,forinstance,bytheactionofbaryonicmatter
r0asfollows ortheappearanceofagiantblackhole[34].
878 AVGurevich,KPZybin,VASirota Physics±Uspekhi40(9)
For small-scale objects the situation is essentially differ- From relations (60) and (62) it follows that in the case
ent. As shown above [see Eqn (39)], the initial density Mb (cid:24)0:05Mx the region of appreciable influence of the
fluctuations on small scales, where their effective growth baryonic-object potential is of the order of
begins, are sufficiently high: di0 (cid:24)0:3ÿ0:5for t=teq. Then rb (cid:25)(0:05ÿ0:1)Rx. For smaller values of Mb the region of
fromEqn(56)itfollowsthatthedistribution(1),(49)iscutoff influenceofthepotentialcbisevensmaller.
ratherearly,namelywhen We shall assume the time of oscillations of dark-matter
particlestrappedinthepotential(62)tobemuchsmallerthan
rc (cid:24)(0:05ÿ0:1)Rx: (57)
thecharacteristictimeofbaryoncoolingandbaryonicbody
Asaresult,theparticledensityintheCDMofasmall-scale formationinthecenterofNO.Fortheobjectsofinterest,(40)
NOcanapproximatelyberepresentedas and (41), the time of oscillations is t0 (cid:24)10 years, so this
8 conditionisalwayswellfulfilled.Inthiscase,toconsiderthe
>><r0(cid:18); (cid:19)ÿa 0<r<rc; variationofthedistributionfunctionofdark-matterparticles
r= r0 r ; rc <r<Rx; (58) under the influence of baryons, one can use the adiabatic
>>: rc approximation. The initial adiabatic invariant Ii is deter-
0; r>Rx; minedbytherelation
wherethescalercisdeterminedaccordingtoEqn(57)andthe …rmaxr(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)m(cid:129)(cid:129)(cid:129)(cid:129)2(cid:129)(cid:129)
densityr0isrelatedtotheobject'smassMxandthescaleRxas Ii = rmin Eÿcd(r)ÿ2r2 dr; Ii =Ii(E;m): (63)
r0 '3ÿa MxRxaÿ3rÿca: (59) HereEistheenergyofaparticlemovinginthepotentialcd,
4p normalizedtothemassmx;mistheangularmomentumofthe
particle; rmin, rmax are reflection points defined as points at
4.3 Influence of baryonic component which the integrand vanishes. The initial distribution func-
on the structure of non-compact objects tionis
Up to now we have ignored the influence of the small
baryoniccomponentonthestructureofdark-matterinNO. f=f0(Ii)=f0(E;m): (64)
However,astheycooldown,baryonstendtothebottomof
thepotentialwellcreatedbythedarkmatterandcondenseat After the baryonic body formation, the adiabatic invar-
thecenterformingacompactbaryonicobjectofmassMb.It iantIboftheparticleisalreadydescribedbyanotherequation
createsanadditionalpotential …rmaxr(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)G(cid:129)(cid:129)(cid:129)(cid:129)M(cid:129)(cid:129)(cid:129)(cid:129)b(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)m(cid:129)(cid:129)(cid:129)(cid:129)2(cid:129)(cid:129)
cb =ÿGMr b : Ib = rmin Eÿc(r)+ r ÿ2r2 dr; (65)
The scale of influence of this potential, rb, depends on the whereGMb=risthebaryonicbodypotentialandc(r)isthe
massofabaryonicobjectandcanreadilybeestimatedonthe potentialcreatedbythenon-baryonicmatter.Inviewofthe
basisoftherelation factthattheadiabaticinvariantispreservedinaslowprocess,
I=const=Ib,therelation
cb(rb)=cd(rb); (60) (cid:12)
(cid:12)
where cd(r) is the potential created by the non-baryonic f(I)=f0(Ii)(cid:12)Ii=Ib (66)
component with the density distribution rd(r) (58), (59). It
canbefoundthroughasolutionofthePoissonequation holds.Heref0(Ii)istheinitialdistributionfunction,butgiven
1 d (cid:18) 2 dcd(cid:19) this, the function Ib(E;m) (65) already differs from Ii(E;m)
r2 dr r dr =4pGrd(r) (61) (63).Expressions(63)±(66)completelydescribethedeforma-
tion of the distribution function f(E;m) of dark-matter
with the boundary condition cd !0 as r!1; it has the particlesundertheinfluenceofthebaryonicbodypotential.
form The distributions of the potential and particle density are
8 ((cid:20) (cid:18) (cid:19)2 (cid:21) describedherebytheequations
>>>>>>>>>>>>>>>>>>>>>>>>><GRMx(cid:2)(cid:2)x(cid:18)(cid:20)1Rr3ÿcx(cid:19)ÿ6a2(cid:18)aÿarÿcrrc(cid:19)233ÿÿÿa+aa(cid:21))ÿ2a1((23; ÿÿaa)) r<rc; Dr(cr)==42ppGrm2rx;…01 dm2…c0+m2=(2r2)p(cid:129)E(cid:129)(cid:129)(cid:129)(cid:129)ÿ(cid:129)(cid:129)(cid:129)(cid:129)f(cid:129)c(cid:129)((cid:129)E(cid:129)(cid:129)ÿ(cid:129)(cid:129);(cid:129)(cid:129)m(cid:129)m(cid:129)(cid:129))(cid:129)2(cid:129)(cid:129)=(cid:129)(cid:129)(cid:129)((cid:129)2(cid:129)(cid:129)(cid:129)r(cid:129)(cid:129)2(cid:129)(cid:129))(cid:129) d(E67:)
cd =>>>>>>>>>>>>>>>>>>>>>>>>>:ÿÿGGÿMMr32xxÿÿ+; aaG3R+MxRRxr(cid:20)xx(cid:21)2(cid:20)ÿ11ÿa(cid:18)a3R(cid:18)rxRr(cid:19)cx2(cid:19)ÿa3ÿa(cid:21)ÿ1; Rrcx44rr4; Rx; trpn((ehS6q(oa7eeruttc))ueTap,tnrttioooaoitaoitlgnarledneyelnest,tht,4cieeowe.ars4nrlmsh)aewcintincnhit(dhtthreeih)aedd,tmlEelheaytcnqaeenassnrdnNssimsteoytoO(inhn6folre3eisftn)tdhtbre±haiuaayse(rcr6kdb.tt6su-ihom)msTr.edtearhaWy,ilbte.blto-aeueWmnrprtsyeiehaeodorsasnetnsshlunlhifocrsusabbiutltnarballycedrtoftsyididoissoroiynosstnntthslivrdcfaaie(beotnEpufEbta;ehtolqmnityeodhnssd)nyeees,,
r
Mb=Mx !0, when the perturbation of the potential c
a=1:8: (62) created by non-baryonic matter can be neglected to a first