Structure of Dark Matter Halos From Hierarchical Clustering. III. Shallowing of The Inner C

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What is dark matter?
About 65 years ago, the first time that evidence of the existence of dark matter. At that time, Fulizizha Popovich found a large cluster of galaxies in the galaxy has a very high velocity, unless the quality of galaxy clusters is based on the number of calculations in which stars are more than 100 times the value, or cluster of galaxies can not bound lives of these galaxies. After decades of observation and analysis confirmed this. Although the nature of dark matter is still unknown, but by 80 years, accounting for about 20% of the energy density of dark matter to be widely accepted.
What is dark matter?
now we know that dark matter has become an important part of the universe. The total mass of dark matter is ordinary matter, 6.3 times the energy density in the universe, accounting for 1 / 4, but also important is that dark matter dominated the formation of cosmic structures. Now the nature of dark matter remains a mystery, but assuming it is a weak interaction of subatomic particles, then the resulting large-scale structure of the universe is consistent with the observations. Recently, however, the structure of galaxies and galaxy subanalysis shows that this assumption and the difference between observations, which at the same time provide a variety of possible dark matter theory was useless. Small-scale structure through the density, distribution, evolution and its environment studies can distinguish between these potential dark matter model for the nature of dark matter to bring a new dawn.

Dark Matter Halos around Galaxies

Dark Matter Halos around GalaxiesP.SalucciSISSA,via Beirut2-4,I-34013Trieste,ItalyM.PersicOsservatorio Astronomico di Trieste,via Beirut2-4,I-34013Trieste, ItalyAbstract.We present evidence that all galaxies,of any Hubble type and lumi-nosity,bear the kinematical signature of a mass component distributed diﬀerently from the luminous matter.We review and/or derive the DM halo properties of galaxies of diﬀerent morphologies:spirals,LSBs,ellip-ticals,dwarf irregulars and dwarf spheroidals.We show that the halo density proﬁleM h(x)=M h(1)(1+a2)x3x2+a2(with x≡R/R opt),across both the Hubble and luminosity sequences, matches all the available data that include,for ellipticals:properties of the X-ray emitting gas and the kinematics of planetary nebulae,stars, and HI disks;for spirals,LSBs and dIrr’s:stellar and HI rotation curves; and,ﬁnally,for dSph’s the motions of individual stars.The dark+luminous mass structure is obtained:(a)in spirals,LSBs, and dIrr’s by modelling the extraordinary properties of the Universal Rotation Curve(URC),to which all these types conform(i.e.the URC luminosity dependence and the smallness of its rms scatter and cosmic variance);(b)in ellipticals and dSph’s,by modelling the coadded mass proﬁles(or the M/L ratios)in terms of a luminous spheroid and the above-speciﬁed dark halo.A main feature of galactic structure is that the dark and visible matter are well mixed already in the luminous region.The transition between the inner,star-dominated regions and the outer,halo-dominated region,moves progressively inwards with decreasing luminosity,to the extent that very-low-L stellar systems(disks or spheroids)are not self-gravitating,while in high-L systems the dark matter becomes a main mass component only beyond the optical edge.A halo core radius,comparable to the optical radius,is detected at all luminosities and for all morphologies.The luminous mass fraction varies with luminosity in a fashion common to all galaxy types:it is comparable with the cosmological baryon fraction at L>L∗but it decreases by more than a factor102at L<<L∗.1LSBsFigure1.The loci populated by the various families of galaxies inthe central brightness vs.luminosity plane.For each Hubble type,the central halo density increases with de-creasing luminosity:sequences of denser stellar systems(dwarfs,ellipti-cals,HSBs,LSBs in decreasing order)correspond in turn to sequences of denser halos.Then,the dark halo structure of galaxiesﬁts into a well ordered pattern underlying a uniﬁed picture for the mass distribution of galaxies across the Hubble sequence.1.IntroductionOver scales∼1kpc up to the Hubble radius,the dynamics of cosmological systems is inﬂuenced,and often dominated,by non-radiating matter which re-veals itself only through a gravitational interaction with the luminous matter. As the observational evidence has been accumulating,it has become apparent that understanding the nature,history and structural properties of this dark component,is the focal point of Cosmology at the end of the millennium.In particular,the halos of dark matter,detected around galaxies,have driven the dissipative infall of baryons that,modulo a variety of initial conditions,has built the bulge/disk/spheroid systems we observe today.It is remarkable that the∼1011galaxies present within the Hubble radius can be classiﬁed in a very small number of types:ellipticals,spirals,low-surface-brightness(LSB)galaxies, dwarf spirals and dwarf irregulars.The main characterizing property of these families is their position in theµ0,M B(central surface brightness,magnitude)2plane(see Fig.1).In this plane spiral galaxies lie at the center and show a very small range(∼0.5mag)inµ0;ellipticals are very bright systems,and span only a factor10in luminosity,that however well correlates with central brightness; LSB galaxies are the counterpart of spirals at low surface brightness;and,ﬁnally, dwarfs are very-low-luminosity spheroids or disks which barely join the faintest normal systems and span the largest interval inµ0.Cosmologically,all galaxy types are equally important for at least two reasons:(a)those types having a lower average luminosity are however much more numerous and hence can store a signiﬁcant amount of baryons,and(b)the properties that characterize and diﬀerentiate the various Hubble types,i.e.the angular momentum content and the stellar populations,are intimately related to the process of galaxy formation.A systematic presence of dark matter wasﬁrst found in spirals,speciﬁcally from the non-keplerian shape of their rotation curves(Rubin et al.1980;Bosma 1981),and in dSph’s from their very high tidal M/L ratios(Faber&Lin1983). Later,dark matter has been sistematically found also around dwarf spirals and LSB galaxies(Romanishin et al.1982).For ellipticals,the situation is less clear:certainly at least some(if not all)show evidence of a massive dark halo in which the luminous spheroid is embedded(e.g.Fabricant&Gorenstein1983). Therefore,the claim for the ubiquitous presence of dark halos around galaxies may be observationally supported.Theoretically,this claim is the natural outcome of the bottom-up cosmo-logical scenarios in which galaxies form inside dark matter halos(probably with a universal density proﬁle;e.g.,Frenk et al.1988and Navarro et al.1996). We point out that this prediction has so far been tested only in spirals where reliable DM proﬁles have been obtained(Persic,Salucci&Stel1996;hereafter PSS96).Along the Hubble sequence,the systematic presence of dark matter in galaxies and its relation with the luminous matter has so far been poorly known. However,in the past year or so a number of observational breakthroughs(some of which presented at this conference)have allowed us to obtain the gravita-tional potential in numerous galaxies of diﬀerent Hubble types(including also LSB galaxies).The time is now ripe for attempting to derive the general mass proﬁle of dark matter halos,as a function of galaxy luminosity and morphology.In this paper(which is also a review describing much recent work),we will try to answer, for theﬁrst time,a simple(cosmological)question:Given a galaxy of Hubble type T and luminosity L,which halo is it embedded in?The aim of this article is then to derive/review,by means of proper mass modelling,the mass distribution in galaxies of diﬀerent luminosities and Hubble Types and toﬁt all of the various pieces into one uniﬁed scheme of galaxy structure.Notice that the theoretical implications of the results presented here will be discussed elsewhere.In detail,the plan of the paper is as follows:in section2we review the properties of DM halos in spiral galaxies;in section 3we work out the DM mass distribution in LSBs and perform a comparative analysis with that in spirals;in section4we derive/review the halo properties of elliptical galaxies;in sections5and6we derive the properties of dwarf galaxies, irregulars and spheroidals.Finally,in section7we propose a uniﬁed scheme for the DM halos around galaxies and their interaction with the luminous matter.(A value of H0=75km s−1Mpc−1is assumed throughout.)3Figure2.The RC slope at R opt vs luminosity(left)and V opt(right)for the1100RCs of Persic&Salucci1995and PSS96.2.Spiral GalaxiesThe luminous(∼stellar)matter in spiral galaxies is distributed in two com-ponents:a concentrated,spheroidal bulge,with projected density distribution approximately described byI(R)=I0e−7.67(R/r e)1/4(1) (with r e being the half-light radius;but see Broeils&Courteau1997),and an extended thin disk with surface luminosity distribution very well described(see Fig.3)by:I(R)=I0e−R/R D(2) (with R D being the disk scale-length;Freeman1970).Let us take R opt as the ra-dius encircling83%of the integrated light:for an exponential disk R opt=3.2R D is the limit of the stellar disk.The relative importance of the two luminous com-ponents deﬁnes the Hubble sequence of spirals,going from the bulge-dominated Sa galaxies to the progressively more disk-dominated Sb/Sc/Sd galaxies.We recall that the spiral arms are non-axisymmetric density perturbations,traced by newly-formed bright stars or HII regions,which are conspicuous in the light distribution and perturb the circular velocityﬁeld through small-amplitude sinu-soidal disturbances(i.e.,wiggles in the rotation curves),but they are immaterial to the axisymmetric gravitational potential.This digression is to recall thatﬁt-ting these features with a mass model is a mistake!The rotation curves of spiral galaxies do not show any keplerian fall-oﬀat outer radii(e.g.:Rubin et al.1980;Bosma1981).Moreover,their shapes at R>(1−2)R D are inconsistent with the light distribution,so unveiling the presence of a DM component.PSS96,analyzing approximately1100RCs,about 100of which extended out to∼<2R opt,found that the luminosity speciﬁes the entire axisymmetric rotationﬁeld of spiral galaxies.At any chosen normalized4Figure3.Averaged spirals I-light proﬁles at diﬀerent luminosities.Each L bin includes hundreths of galaxiesradius x≡R/R opt,both the RC amplitude and the local slope strongly correlate with the galaxy luminosity(in particular,for x=1see Fig.2;for outer radii see PSS96,Salucci&Frenk1989,and Casertano&van Gorkom1991).Remarkably, the rms scatter around such relationships is much smaller than the variation of slopes among galaxies(see PSS96).This has led to the concept of the Universal Rotation Curve(URC)of spiral galaxies(PSS96and Persic&Salucci1991; see Fig.5).The rotation velocity of a galaxy of luminosity L/L∗at a radius x≡R/R opt is well described by:V URC(x)=V(R opt)0.72+0.44logLL∗1.97x1.22(x2+0.782)1.43+ +0.28−0.44logLL∗1+2.25LL∗0.4x2x2+2.25(LL∗)0.41/2km s−1.(3) (with log L∗/L =10.4in the B-band).Remarkably,spirals show a very small cosmic variance around the URC.In80%of the cases the diﬀerence between the individual RCs and the URC is smaller than the observational errors,and in most of the remaining cases it is due to a bulge not considered in eq.(3) (Hendry et al.1997;PSS96).This result has been conﬁrmed by a Principal Component Analysis study of URC(Rhee1996;Rhee&van Albada1997): they found that the twoﬁrst components alone account for∼90%of the total variance of the RC shapes.Thus,spirals sweep a narrow locus in the RC-proﬁle/amplitude/luminosity space.The luminosity dependence of the URC strongly contrasts with the self-similarity of the luminosity distribution of stellar disks(Fig.3):the luminosity proﬁles L(x)∝xx I(x)dx do not depend on luminosity.This reﬂects the5Figure4.Coadded rotation curves(ﬁlled circles with error bars) repruduced by URC(solid line)Also shown the separate dark/luminous contributions(dotted line:disk;dashed line:halo.)150300Figure5.The URC surface.6discrepancy between the distribution of light and that of the gravitating mass. Noticeably,this discrepancy increases with radius x and with decreasing galaxy luminosity L.The URC can beﬁtted by a combination of two components:(a) an exponential thin disk,approximated for0.04R opt<R≤2R opt asV2d(x)=V2(R opt)β1.97x1.22(x+0.78),(4)and(b)a spherical halo represented byV2h(x)=V2(R opt)(1−β)(1+a2)x2(x2+a2),(5)M h(x)=G−1V2(1)R opt(1−β)(1+a2)x3(x+a),with x≡R/R opt being the normalized galactocentric radius,β≡V2d(R opt)/V2opt the disk mass fraction at R opt,and a the halo core radius(in units of R opt).The disk+haloﬁts to the URC are extremely good(ﬁtting errors are within1%on average)at all luminosities(see Fig.4)whenβ=0.72+0.44logLL∗,(6)a=1.5LL∗1/5.(7)Thus we detect,for the DM component,a central constant-density region of size∼R opt,slightly increasing with luminosity.The transition between the inner,luminous-matter-dominated regime and the outer,DM-dominated regime occurs well inside the optical radius:typically at r<<R opt in low-luminosity galaxies,and farther out,closer to∼R opt,at high luminosities(see Fig.4).Thus, the ordinary and dark matter are well mixed in the very stellar regions of spirals.The total halo mass can be evaluated by extrapolating the halo out to the radius,R200,encompassing a mean overdensity of200.Weﬁnd:R200=250LL∗0.2kpc(8)M200∼2×1012LL∗0.5M ,(9)in good agreement with results derived from satellite and pair kinematics(Charl-ton&Salpeter1991;Zaritsky et al.1993).Notice thatM200 L B 75L BL∗−0.5(10)(in solar units).This implies that the halo mass function is not parallel to the observed galaxy luminosity function(Ashman,Salucci&Persic1993).7The luminosity dependence of the disk mass fraction can be interpreted in terms of a mass-dependent eﬃciency in transforming the primordial gas fraction into stars.In fact,the total(i.e.,computed at R200)luminous mass fraction of a galaxy of luminosity L is:M M200 0.05LL∗0.8.(11)This suggests that only the brightest objects reach a value comparable with the primordial valueΩBBN∼<0.10,while in low-L galaxies only a small fraction of their original baryon content has been turned into stars.The luminosity dependence of the DM fraction found by Persic&Salucci(1988,1990)has been conﬁrmed by direct mass modelling(e.g.:Broeils1992;Broeils&Courteau1997; Sincotte&Carignan1997;see also Ashman1992).On scales(0.2−1)R200,the halos have mostly the same structure,with a density proﬁle very similar at all masses and an amplitude that scales onlyvery weakly with mass,like V200∝M0.15200.This explains why the kinematics ofsatellite galaxies orbiting around spirals show velocities(relative to the primary) uncorrelated with the primary’s luminosity(Zaritsky1997).At very inner radii, however,the self-similarity of the proﬁle breaks down,the core radius becoming smaller for decreasing M200according to:core radiusR200=0.075M20010M0.6.(12)The central density scales with mass as:ρh(0)=6.3×104ρcM2001012M−1.3(13)(ρc is the critical density of the universe).The regularities of the luminous-to-dark mass structure can be represented as a curve in the space deﬁned by dark-to-luminous mass ratio,(halo core radius)-to-(optical radius)ratio,central halo density(see PSS96).This curve is a structural counterpart of the URC and represents the only locus of the mani-fold where spiral galaxies can be found.The main properties of dark matter in spiral galaxies can then be summarized as follows:substantial amounts of dark matter are detected in the optical regions of all spirals,starting at smaller radii for lower luminosities;the dark and the luminous matter are well mixed;the structure of the halos is universal:it involves a core radius compara-ble in size with the optical radius and a central density scaling inversely with luminosity;the ratio between luminous and dark matter is a function of luminosity(or mass)among galaxies.At a given(normalized)radius,this ratio increses with increasing luminosity:the global visible-to-dark mass ratio spans the range be-tween∼ΩBBN at very high luminosities and∼10−4at L<<L∗.The discovery of these features,describing the various stages of the pro-cesses leading to present-day galaxies,supersedes the observationally disproved paradigms of”ﬂat rotation curves”and”cosmic conspiracy”.8Figure 6.Coadded rotation curve of the sample of LSB galaxies with V opt ∼(70±30)km s −1.The solid line represents the V URC of spirals of similar V opt .3.Low-Surface-Brightness GalaxiesThe central surface luminosity of spirals is normally about constant,at µ0(B )=21.65±0.30.Recently,however,a large population of disk systems with a signif-icantly lower surface brigthness (µ0(B )=24−25)has been detected (Schombert et al.1992;Driver et al.1994;Morshidi et al.1997;see also Disney &Phillips 1983).In these systems the light distribution follows that of an exponential thin disk (McGough &Bothun 1994),with total magnitude ∼1.5mag fainter than that of normal (HSB)spirals.At the faint end of the LSB luminosity distribu-tion (M B ∼−16),the galaxies have very extended HI disks whose kinematics yields the mass structure (de Blok,McGaugh &van der Hulst 1996).In detail,de Blok et al.have published the HI rotation curves of 19low-L LSBs,having M B ∼−17mag,V max ∼70km s −1,and (remarkably)R opt ∼(8−10)kpc,roughly independent of luminosity.These galaxies are the counterpart of the faintest HSB spirals.Notice that,although the maximum velocities are similar,the optical sizes of LSBs are ∼3times larger than those of HSBs.The objects in de Blok et al.(1996)are all within a small range of magni-tudes and optical velocities,M B =−17±0.5,V opt ∼(70±30)km s −1.From these data we construct,as we did for spirals,the coadded rotation curve V (R R opt ,70).Notice that,unlike for spirals,rotation curves of LSBs with higher V opt are not available:so we can compare LSB and spiral RCs only at their lowest velocities,shown for spirals in the top left panel in Fig.4and for LSBs in Fig.6(points).In detail,in the latter ﬁgure we plot V (R/R opt ),the coadded RC obtained from the de Blok et al.data,together with the spiral V URC for the same range of maximum velocities (50–100km s −1)(see PSS96for all of the details).The9Figure 7.The radius–luminosity relation (left)and the Tully-Fisher relation (right)for the Matthews sample of LSBs.Dashed curve:ﬁt to the data.Solid curve:predictions of a mass model identical to that of spirals.Dotted line:prediction of a mass model model with constant (M/L )agreement is striking:no diﬀerence can be detected between the LSB and HSB rotation curves,which coincide within the observational uncertainties.Notice that the LSB and HSB rotation curves are identical when the radial coordinate is normalized to the disk length-scale (essential procedure for determining the DM distribution).The fact that,when expressed in physical radii,LSB rotation curves rise to their maximum more gently than HSB RCs (see Fig.3of de Blok &McGough 1997),depends exclusively on the larger LSB disk scalelengths.For LSBs we adopt a mass model including,as for spirals,(i)an exponential thin disk and (ii)a dark halo of mass proﬁle given by eq.(5).The LSB coadded RC is extremely well ﬁtted by eqs.(4)and (5)with the ‘spiral’values β=0.1±0.04and a 0.75.These parameters imply that LSB galaxies are completely dominated by a dark halo with a large ∼(5−6)kpc core radius.Higher-luminosity (up to M B ∼−22)LSBs do exist (e.g.,Sprayberry et al.1993):their structure can be tentatively investigated by means of their Tully-Fisher relationship.Observationally,these objects show a good correlation between luminosity and the corrected linewidth w 0,i (Matthews et al.1997):log w 0,i =2.41+0.69log L V L ∗V +0.16log 2 L V L ∗V ,(14)where L ∗V corresponds to M ∗V =−20.87(see Fig.7).The quantity w 0,i is,as inspirals,a good measure of the circular velocity at ∼R opt :thenw 20,i R opt G [M h (R opt )+M bar (R opt )],(15)where M bar =M D +M HI .For LSB spirals we take the same dark-to-visible mass ratios as for HSB spirals,M h (R opt )M bar (R opt ) 9×(L B 0.04L ∗)−1(see conclusions of PSS96),and we assume M bar (R opt )∝L k B .Finally,by using the LSB luminosity-radius relation,R opt =12(L/L ∗)0.25kpc (see Fig.7),we are able to reproduce10Figure8.LSB(M/L) (left:solid curve;the dotted line refers tospirals)and central overdensity(right)as a function of luminosity.the observed M V–log w0,i relation:in Fig.7we show the excellent agreement between the observed linewidths and those predicted by our mass model when k=1.8and the LSBs stay on the‘spiral’β–L and a–L relationships.At∼L∗,the LSB stellar M/L ratios are similar to those of HSBs,but they decline steeply with decreasing luminosity(see Fig.8,left panel).This result is in good agreement with the(M/L) ratios obtained by applying the maximum-disk hypothesis to solve for the galactic structure:in this case log(M/L) increases from∼−0.3to0.5across the entire LSB luminosity(velocity)range(de Blok &McGaugh1997).The central overdensities,3/(4πGρc)(1−β)(V opt/R opt)2(1+a2)/a2,are shown in Fig.8(right panel)as a function of luminosity.They are smaller and less dependent on galaxy luminosity than those in normal spirals,in agreement with theﬁndings by de Blok&McGough(1997).With the caveat of the relative smallness of the present sample,we claim that LSB galaxies are indistinguishable from HSBs in the(β,a)space describing the coupling between dark and visible matter.To end this section,we comment on the argument,raised sometimes,accord-ing to which the discovery of a large number of LSBs would make theﬁndings on DM in spirals(e.g.PSS96)irrelevant,in that normal spirals would represent an unfair,biased sample of galaxies.There is no doubt that the existence of a population of LSB galaxies,contiguous to the HSBs,aﬀects the interpretation, and even the physical meanings,of luminosity functions and number counts. However,it is exactly the characteristic diﬀerence between the two families that answers the point.Even prior to any mass modelling,we can argue that in the Universe there are about1011disk systems obeying the“Freeman law”:µ0=21.5±0.5independent of galaxy luminosity.It is obviously necessary to know the DM properties of this family,independently of the existence of another family of disk systems not obeying such a“law”,and maybe not even observ-able.Of course,investigating the DM properties of LSBs is equally important and crucial.Furthermore,the argument raised above appears even more im-material after the analysis of the LSB rotation curves.In fact,as far as the structural parameters are concerned,HSBs and LSBs are indistinguishable.In11Figure9.The mass as a function of radius for the ellipticals of oursubsamples,with luminosities<log L/L∗>=−0.4and<log L/L∗=0.4(left and right,respectively).other words,LSBs follow the spiral V URC(R/R opt),but strongly deviate with respect to the spirals’luminosity–(disk length-scale)relation.4.Elliptical GalaxiesElliptical galaxies are pressure-supported,triaxial stellar systems whose orbital structure may depend on their angular momentum content,degrees of triaxial-ity and velocity dispersion anisotropy.As is well known,the derivation of the mass distribution from stellar motions is not straightforward as it is from ro-tation curves in disk systems,because the kinematics of the former is strongly aﬀected by geometry,rotation,and anisotropies.However,in addition to stellar kinematics,there are a number of mass tracers(X-ray emitting halos,planetary nebulae,ionized and neutral disks)which crucially help to probe the gravita-tional potential out to external radii(see Danziger1997).The luminosity proﬁleρ (r)of E’s is obtained from the observed projected surface density,which follows the de Vaucouleurs proﬁle[see eq.(1)],by assuming an intrinsic shape and deprojecting.A very good approximation forρ (r)is theHernquist proﬁle:ρ (r)=M2π1y(y+c)(16)(where y=1.81r/r e and c=1).As the Hernquist proﬁle is no longer a good ﬁt for R>>r e,from actual surface-photometry proﬁles we have evaluated thatR opt 2r e,where r e=6(L/L∗)0.7kpc(from Djorgovski&Davis1987).All ellipticals belong,within a very small cosmic scatter(<12%),to a relation ofthe type:r e∝σA0I B e,(17) where r e is the half-light radius,I e is the mean surface brightness within r e,and σ0is the observed(projected)central velocity dispersion.In the logarithmic space,this corresponds to a plane,the Fundamental Plane of ellipticals(Djor-1200.51 1.52012300.511.520123Figure 10.Coadded mass proﬁle of ellipticals,ﬁlled cirles,with thebest ﬁt mass model (solid line)govski &Davis 1987;Dressler et al.1987),that constrains the properties of the DM distribution.Observations indicate A =1.23(1.66)and B =−0.82(−0.75)in the V -band and K -band,respectively (e.g.,Djorgovski &Santiago 1993).Assuming a constant M/L and structural homology,the virial theorem predicts A =2,B =−1.The simplest explanation of the departure of A from the virial expectation involves a systematic variation of M/L with L ,which also accounts for the wavelength dependence of A (Djorgovski &Santiago 1993).The departure of B from the virial expectation is likely to be due to the breakdown of the homology of the luminosity structure (see Caon,Capaccioli &D’Onofrio 1993;Graham &Colless 1997).Assuming spherical symmetry and isotropic stellar motions (so M 3.4G −1σ20r e and L =2πI e r 2e ),the FP implies that M/L |r e ,inclusive of dark matter,is “low”:4−9(Lanzoni 1994;Bender,Burstein &Faber 1992),and roughly consistent with the values predicted by the stellar population models (Tinsley 1981).No large amounts of DM are needed on these scales,as it also emerges from mass models of individual ellipticals,obtained by analyzing the line proﬁles of the l.o.s velocity dispersion (van der Marel &Franx 1993),which show that the DM fraction inside r e is substantially less than 50%(i.e.M/L B ∼<10;see van der Marel 1991,1994;Rix et al.1997;Saglia et al.1997a,b;Carollo &Danziger 1994;Carollo et al.1995),as in spirals.We now investigate in some detail the eﬀects of DM on the Fundamental Plane.For this purpose,let us describe,for mathematical simplicity,the dark halo by a Hernquist proﬁle [eq.(16)]with a lower mass concentation than the luminous spheroid:c =2(see Lanzoni 1994).We recall that dark halos are likely to have an innermost constant-density region not described by eq.(16):this,13however,has no great relevance here,in that most of the dark mass is located outside the core where eq.(16)is likely to hold.Without loss of generality,we consider isotropic models (Lanzoni 1994):the radial dispersion velocity σr is related,through the Jeans equation,to the mass distribution by:σ2r (r )=G M dark (r )+M (r )r d log ρd log r −1.(18)The projected velocity dispersion is then σP (R )=2Σ (R ) ∞R ρ (r )σ2r (r )√2−R 2dr .This equation shows that,at any radius (including r =0),the measured projected ve-locity dispersion σP (R )depends on the distributions of both dark and luminous matter out to r ∼(2−3)r e .Conversely,in spirals the circular velocity V (R )depends essentially only on the mass inside R ,namely on just the luminous mass when R →0.If M 200is the total galaxy mass,we getσP (0)=3.4GM r e M 20010M 2/5.(19)The mass dependence of σP (0),combined with the thinness of the FP (i.e.:δR e /R e ∼<0.12),constrains the scatter that would arise,according to eq.(19),from random variations of the total amount of DM mass in galaxies with the same luminous mass.From eq.(17)we get δσ/σ∼<0.121.4,while eq.(19)implies dσ/σ=5/2δM/M .This means that,over 2orders of magnitude in M ,any random variation of the dark mass must be less than 20%(Lanzoni 1994;Renzini &Ciotti 1993;Djorgovski &Davis 1987).From eq.(17)and since M ∝L 1.2,we ﬁnally get M M 200∝σ5/2,so that M 200∝L 0.5−0.6,as in spirals (PSS96).Let us notice that the these well-established constraints on the amount of darkmatter in ellipticals,due to the existence of the Fundamental Plane,are however at strong variance with the extremely high values of the central M/L ratios,15−30,found in some objects,as a result of the dynamical models of Bertin et al.(1992)and Danziger (1997).We can determine the parameters of the dark and visible matter distribu-tion by means of a variety of tracers of the ellipticals’gravitational ﬁeld (see the review by Danziger 1997).This provides the (dark +luminous)mass dis-tribution for 12galaxies,with magnitudes ranging between −20<M B <−23.In order to investigate the luminosity dependence of the mass distribution,we divide the sample into 2subsamples,with average values <log L/L ∗>=−0.4and <log L/L ∗>=0.4respectively.In Fig.9we plot,as a function of R/r e ,the normalized mass proﬁle of ellipticals,M (R )/M (r e ),for each galaxy;and in Fig.10the coadded mass distributions for the high-L and the low-L subsamples that can be very well reproduced (solid lines)by a two-component mass model which includes a luminous Hernquist spheroid and a DM halo given by eq.(5).The resulting ﬁt is excellent (see Fig.10)when β,the luminous mass fraction inside R opt 2r e ,scales with luminosity as in disk systems,and the halo core radius a (expressed in units of R opt )scales asa =0.8L L ∗ 0.15.(20)14。

Investigating the Nature of Dark Matter

Investigating the Nature of DarkMatterThe phrase “dark matter” has become a buzzword in modern astrophysics as well as popular culture, and yet we still know very little about what dark matter really is. It is a mysterious substance that makes up 27% of the universe and that cannot be observed directly, but can only be inferred from the gravitational effects it has on visible matter. Therefore, dark matter is a topic of intense research and debate in the scientific community. In this article, we will explore the key aspects of dark matter and the different ways scientists are working to uncover its nature.What is Dark Matter?As mentioned, dark matter is a substance that does not emit, absorb or reflect light, hence its name. It does not interact strongly with electromagnetic forces, but it does with gravity, which is why its presence can be inferred from the gravitational effects it has on visible matter. One of the most well-known examples of this is the rotation curve of spiral galaxies. According to the laws of classical mechanics, the velocity of stars and gas in a galaxy should decrease as one moves away from the center, as the gravitational attraction of the visible matter decreases. However, observations have shown that the velocity remains constant or even increases, suggesting that there is an invisible mass that is causing this anomaly. This invisible mass is the dark matter.Another piece of evidence for the existence of dark matter is the distribution of matter in the universe as revealed by the cosmic microwave background radiation, which is the afterglow of the Big Bang. The pattern of temperature fluctuations in this radiation shows that the matter in the universe is not distributed evenly, but is rather clumped up in large structures such as galaxies and clusters of galaxies. However, this clumping up cannot be explained solely by the gravitational influence of visible matter; there must be an additional source of gravity, i.e. dark matter, to explain the observed distribution.Moreover, measurements of the large-scale structure of the universe, such as the distribution of galaxies and galaxy clusters, also point to the existence of dark matter.What is Dark Matter Made of?Despite its importance in shaping the structure of the universe, the identity of dark matter remains unknown. There are several hypotheses about what dark matter might be made of, but none of them has been conclusively proven yet. One popular hypothesis is that dark matter is composed of weakly interacting massive particles (WIMPs), which are hypothetical particles that would interact with normal matter only through the weak nuclear force and gravity. The idea is that WIMPs were produced in the early universe when it was hot and dense, and have been moving around freely ever since. If they collide with normal matter, they would transfer some of their energy and momentum, producing detectable signals. In fact, several experiments have been designed to detect WIMP interactions, such as the Large Underground Xenon (LUX) experiment and the Super Cryogenic Dark Matter Search (SuperCDMS).Another hypothesis is that dark matter is made of axions, which are theoretical particles that were originally proposed to explain a different problem in physics, the strong CP problem. The idea is that axions would be very light and weakly interacting, making them difficult to detect, but would still affect the motion of galaxies and other cosmic structures. The Axion Dark Matter eXperiment (ADMX) is currently searching for evidence of axions in a laboratory at the University of Washington.A third hypothesis is that dark matter is composed of primordial black holes, which are black holes that were formed by the collapse of a density fluctuation in the early universe. The idea is that these black holes could have a mass range that would make them more likely to be dark matter, and that their interactions with normal matter could produce observable effects. However, this hypothesis is less favored by most researchers, as the formation and stability of such black holes would require very specific conditions.ConclusionDespite decades of research, the nature of dark matter remains one of the most intriguing and elusive topics in astrophysics. It remains a theoretical construct that cannot be directly observed, but its effects on the motion and structure of the cosmos are undeniable. Researchers are continuing to study dark matter using a variety of tools and techniques, from telescopes that measure gravitational lensing to underground experiments that look for WIMP interactions. The hope is that someday we will finally be able to unravel the mystery of what dark matter is made of, and in doing so, gain a better understanding of the universe and our place in it.。

暗物质与暗能量

暗物质与暗能量什么是暗物质暗物质（Dark Matter）是一种比电子和光子还要小的物质，不带电荷，不与电子发生干扰，能够穿越电磁波和引力场，是宇宙的重要组成部分。

暗物质的密度非常小，但是数量非常庞大。

自从牛顿发现了万有引力定律以来, 人们就一直尝试用引力理论来解释各种天体的运动规律, 在这个过程中, “暗物质”的概念很早就已经形成了。

现代意义下的暗物质概念是瑞士天文学家家弗里兹·兹威基（Fritz Zwicky）早在1933 年研究后发星系团中星系运动的速度弥散时就提出来了。

他根据所测得的星系速度弥散并应用维理定理得到了后发星系团的质光比, 发现其比太阳的质光比要大400 倍左右。

1934 年，他在研究星系团中星系的轨道速度时，为了解释“缺失的物质”问题而正式提出了暗物质的概念．但当时并没引起太多的关注，直到40 年后，人们在研究星系中恒星的运动时遇到类似的困难: 人们发现如果仅考虑可见( 发光) 物体彼此之间的相互吸引力，那么各式各样的发光天体( 包括恒星、恒星团、气状星云，或整个星系) 运动的速度要比人们预想的快一些。

暗物质存在最直接的证据来自于漩涡星系旋转曲线的测量。

通常测量的旋转曲线在距离星系中心很远的地方会变平, 并且一直延伸到可见的星系盘边缘以外很远的地方都不会下降。

如果没有暗物质存在, 很容易得到在距离很远的地方旋转速度会随距离下降: v(r)= GM(r)! r ∝1!r因此, 平坦的旋转曲线就意味着星系中包含了更多的物质。

2003 年，Wilkinson 微波背景各向异性探测( WMAP) 、Sloan数字巡天( SDSS) 和最近的超新星( SN) 等天文观测以其对宇宙学参数的精确测量，进一步有力地证实了暗物质的存在．这在人类探索宇宙奥秘和物质基本结构的道路上无疑是一个光辉的成就．最新数据显示，在宇宙能量构成中，暗能量占72%，暗物质占23%，重子类物质只占了5%左右．暗物质的探测暗物质的探测可以分为如下3 种方法。

MACHOs in dark matter haloes

a r X i v :a s t r o -p h /0602394v 1 17 F eb 2006Mon.Not.R.Astron.Soc.000,000–000(0000)Printed 5th February 2008(MN L A T E X style ﬁle v2.2)MACHOs in dark matter haloesJanne Holopainen 1,Chris Flynn 1,Alexander Knebe 2,Stuart P.Gill 3,Brad K.Gibson 41Tuorla Observatory,V¨a is¨a l¨a ntie 20,Piikki¨o ,FIN-21500,Finland2AstrophysikalischesInstitut Potsdam,An der Sternwarte 16,14482Potsdam,Germany3Columbia University,Department of Astronomy,550West 120th Street,New York,NY 10027,USA 4Centre for Astrophysics,University of Central Lancashire,Preston,PR12HE,United Kingdom5th February 2008ABSTRACTUsing eight dark matter haloes extracted from fully-self consistent cosmological N -body simulations,we perform microlensing experiments.A hypothetical observer is placed at a distance of 8.5kpc from the centre of the halo measuring optical depths,event durations and event rates towards the direction of the Large Magellanic Cloud.We simulate 1600microlensing experiments for each halo.Assuming that the whole halo consists of MACHOs,f =1.0,and a single MACHO mass is m M =1.0M ⊙,thesimulations yield mean values of τ=4.7+5.0−2.2×10−7and Γ=1.6+1.3−0.6×10−6events star −1yr −1.We ﬁnd that triaxiality and substructure can have major eﬀects on the measured values so that τand Γvalues of up to three times the mean can be found.If we ﬁt our values of τand Γto the MACHO collaboration observations (Alcock et al.2000),we ﬁnd f =0.23+0.15−0.13and m M =0.44+0.24−0.16.Five out of the eight haloes under investigation produce f and m M values mainly concentrated within these bounds.Key words:gravitational lensing –Galaxy:structure –dark matter –methods:N -body simulations1INTRODUCTIONExperiments such as MACHO (Alcock et al.2000)and POINT-AGAPE (Calchi Novati et al.2005)have detected microlensing events towards the Large Magellanic Cloud (LMC)and the M31,supporting the view that some fraction of the Milky Way dark halo may be comprised of massive as-tronomical compact halo objects (MACHOs).The fraction of the dark halo mass in MACHOs,as well as the nature of these lensing objects and their individual masses can be constrained to some extent by combining observations of microlensing events with analytical models of the dark halo (e.g.Kerins 1998;Alcock et al.2000;Cardone et al.2001).However,the real dark halo might not be as simple as in an-alytical models,which typically have spherical or azimuthal symmetry and,in particular,smoothly distributed matter.Cosmological simulations indicate that dark matter haloes are neither isotropic nor homogeneous but are rather triax-ial (e.g.Warren et al.1992)and contain a notable amount of substructure (e.g.Klypin et al.1999).Even for the Milky Way it is still unclear if the enveloping dark matter halo has spherical or triaxial morphology:a detailed investigation of the tidal tail of the Sgr dwarf galaxy supports the notion of a nearly spherical Galactic potential (Ibata et al.2001;Majewski et al.2003)whereas that the data are also claimed to be consistent with a prolate or oblate halo (Helmi 2004).The shape of the halo has an eﬀect on the predicted numberof microlenses along diﬀerent lines-of-sight,for comparison with the results of microlensing experiments.In this paper,we examine the eﬀects of dark halo mor-phology and dark halo clumpiness on microlensing surveys conducted by hypothetical observers located in eight N -body dark matter haloes formed fully self-consistently in cosmological simulations of the concordance ΛCDM model.The eight haloes are scaled to approximately match the Milky Way’s dark halo in mass and rotation curve prop-erties,and hypothetical observers are placed on the surface of a “Solar sphere”(i.e.8.5kpc from the dark halo’s centre)from where they conduct microlensing experiments along lines-of-sight simulating the Sun’s line-of-sight to the LMC.The eﬀects on microlensing of triaxiality and clumping in the haloes are examined.The paper expands upon work in a study by Widrow &Dubinski (1998)although we take a slightly dif-ferent point of view.Instead of studying the errors caused by diﬀerent analytical models ﬁtted to an N -body halo,we analyse the microlensing survey properties of the simulated haloes directly.That is,we do not try to gauge the credibil-ity of analytical microlensing descriptions but rather use our self-consistent haloes for the inverse problem in microlensing (Cardone et al.2001).In other words,the free parameters (the MACHO mass and the fraction of matter in MACHOs)c0000RAS2Holopainen et al.are determined from the observations via the optical depth, event duration and event rate predictions of the models.This not only allows us to make predictions for these parameters based upon fully self-consistent halo models but also to in-vestigate the importance of shape and substructure content of the haloes.The paper is structured as follows.We describe the N-body haloes and their preparation in Section2,the sim-ulation details are covered in Section3,equations for the observables are derived in Section4,results are given in Section5,some discussion is presented in Section6,ﬁnal conclusions are in Section7.2THE HALOES2.1Cosmological simulation detailsOur analysis is based on a suite of eight high-resolution N-body simulations(Gill,Knebe&Gibson2004a)carried out using the publicly available adaptive mesh reﬁnement code MLAPM(Knebe,Green&Binney2001)in a standard ΛCDM cosmology(Ω0=0.3,Ωλ=0.7,Ωb h2=0.04,h= 0.7,σ8=0.9).Each run focuses on the formation and evo-lution of galaxy cluster sized object containing of order one million collisionless,dark matter particles,with mass reso-lution1.6×108h−1M⊙and force resolution∼2h−1kpc(of order0.05%of the host’s virial radius).The simulations have suﬃcient resolution to follow the orbits of satellites within the very central regions of the host potential( 5–10%of the virial radius)and the time resolution to resolve the satel-lite orbits with good accuracy(snapshots are stored with a temporal spacing of∆t≈170Myr).Such temporal resolu-tion provides of order10-20timesteps per orbit per satellite galaxy,thus allowing these simulations to be used in a pre-vious paper to accurately measure the orbital parameters of each individual satellite galaxy(Gill et al.2004b).The clusters were chosen to sample a variety of en-vironments.We deﬁne the virial radius R vir as the point where the density of the host(measured in terms of the cosmological background densityρb)drops below the virial overdensity∆vir=340.This choice for∆vir is based upon the dissipationless spherical top-hat collapse model and is a function of both cosmological model and time.We fur-ther applied a lower mass cut for all the satellite galax-ies of1.6×1010h−1M⊙(100particles).Further speciﬁc details of the host haloes,such as masses,density pro-ﬁles,triaxialities,environment and merger histories,can be found in Gill,Knebe&Gibson(2004a)and Gill et al. (2004b).Table1gives a summary of a number of rele-vant global properties of our halo sample.There is a promi-nent spread not only in the number of satellites with mass M sat>1.6×1010h−1M⊙but also in age,reﬂecting the dif-ferent dynamical states of the systems under consideration.2.2DownscalingAs we intend to perform microlensing experiments directly comparable to the results of the MACHO collaboration we need to scale our haloes to the size of the Milky Way.For this purpose we follow Helmi,White&Springel(2003)and Table1.Summary of the eight host dark matter haloes.The superscript cl indicates that the values are for the unscaled clus-ters.Column2shows the virial radius,R vir;Column3the virial mass,M vir;Column4the redshift of formation,z form;Column5 the age in Gyr;and theﬁnal column an estimate of the number of satellites(subhaloes)in each halo,N sat.apply an adjustment to the length scale,requiring that den-sities remain unchanged.Hence the scaling relations are:r=γr cl(1) v=γv cl(2) m=γ3m cl(3) t=t cl(4)where r is any distance,v is velocity,m is mass,t is time and the superscript cl refers to the unscaled value.Because the haloes are downscaled,γwill always be in the range γ∈[0,1].Toﬁnd a suitableγ,the maximum circular velocity of each halo is required to be220km s−1.The resulting(scaled) rotation curves for all eight haloes are shown in Figure1. Thisﬁgure highlights that Halo#8is a special system—its evolution is dominated by the interaction of three merging haloes(Gill et al.2004b).We do not consider Halo#8to be an acceptable model of the Milky Way,but we include it into the analysis as an extreme case.One might argue that our(initially)cluster sized ob-jects should not be used as models for the Milky Way for they formed in a diﬀerent kind of environment with less time to settle to(dynamical)equilibrium(i.e.the oldest of our systems is8.3Gyr vs.∼12Gyr for the Milky Way). However,Helmi,White&Springel(2003)showed that more than90%of the total mass in the central region of a realis-tic Milky Way model was in place about1.5Gyr after the formation of the object.Our study focuses exclusively on this central region(i.e.the inner15%in radius)boosting our conﬁdence that our haloes do serve as credible models of the Milky Way for our purposes.In Table2we list the scaled radii and masses,as well as the scaling factorγ,for each halo.Note that we have adopted a Hubble constant of h=0.7,which applies throughout the paper.The circular velocity(rotation)curves for the scaled haloes are shown in Figure1and demonstrate how the scaled mass M(<r)is accumulated out to400kpc.Two vertical lines mark the position of observers at the Solar circle and the distance of the LMC from the halo centre; within this region,we note that the mass proﬁles are similarc 0000RAS,MNRAS000,000–000MACHOs in dark matter haloes3Table 2.Properties of the haloes after downscaling to Milky Way size.Column 2shows the scale factor for each halo,γ;Column 3shows the scaled virial radius,R vir ;Column 4the scaled virial mass,M vir ;and the ﬁnal column the scaled particle mass,m p .20406080100120140160180200220240110100V c(k m s -1)r (kpc)Halo 1Halo 2Halo 3Halo 4Halo 5Halo 6Halo 7Halo 8Figure 1.Circular velocity curves of the downscaled haloes(V c =4Holopainen et al.Table3.Substructure within the inner70kpc of the(down-scaled)haloes.Column1shows the name of the halo;Column2 the distance from the centre of the subhalo to the centre of the host,l sat;Column3the truncation radius of the subhalo(within the truncation radius,the density of the subhalo is larger than the background density of the host halo),r sat;Column4the number of bound particles in the subhalo,N sat;and theﬁnal column the mass of the subhalo,M sat.ﬁxed at89.1◦(the original angle between( r⊙, r LMC− r⊙)in Galactocentric coordinates).This conﬁguration(illustrated in Figure3)gives1600individual sightlines,and microlens-ing cones,which sample diﬀerent sets of particles within a halo.The choice for the number of sightlines is a compro-mise between computation time and sampling density.As seen in Figure3,1600sightlines sample the volume quite suﬃciently,even when the sightlines are drawn as lines in-stead of actual cones.The microlensing cone itself is deﬁned by the observer and the source.Figure2illustrates that we treat the source (i.e.LMC)as a circular disk with diameter d S.The parti-cles inside the cone are considered to be gravitational lenses. However,this kind of setup can lead to a handful of parti-cles(those closest to the observer)contributing most to the optical depth.This leads to high sampling noise,and is es-sentially due to the limited resolution of the cosmological simulation(see Figure4).It follows that the particle distri-bution has to be smoothed in some manner to suppress this noise;we describe this and the related issue of the MACHO masses in the next section.3.2Mass resolution vs.MACHO massIn order to compute microlensing optical depths and other properties from the particles in the simulations,we need to get from a mass scale of order106M⊙,i.e.that of the particles in the simulation,down to the mass scale of the MACHOS,of order1M⊙;i.e.we account for the fact that a typical particle in the simulations represents of order106 MACHOs.To overcome the limitation imposed by the mass reso-lution of the simulation we recall that individual particles inFigure2.The microlensing sightline is deﬁned by the locations of the observer and the source and the cone by the sightline and the source diameter.We use l S=50.1kpc and d S=4.52kpc. The particles inside the cone are treated as gravitational lenses. Typically∼100particles from the cosmological simulation are found inside a cone.This number is increased to∼10000by breaking them up into subparticles according to the recipe out-lined in Section3.2.the simulation are not treated asδ-functions but have aﬁ-nite size.The extent of a particle(i.e.its size)is determined —in our case—by the spacing of the grid,as we are us-ing an adaptive mesh reﬁnement code(i.e.MLAPM).In MLAPM the mass of each particle is assigned to the grid via the so-called triangular-shaped cloud(TSC)mass-assignment scheme(Hockney&Eastwood1981)which spreads every in-dividual particle mass over the host and surrounding33−1 cells.The corresponding particle shape(in1D)reads as fol-lows:S(x)= 1−|x|MACHOs in dark matter haloes5Figure3.Sightlines used in the microlensing simulation.The image shows all1600sightlines of the40observers.The observers are located on a sphere with a radius of r⊙=8.5kpc.The sphere can be seen as the“dark ball”in the centre of the sightlines.The radius of the sphere which holds the sources is49.5kpc.Every observer has40sources(LMCs)at a distance of50.1kpc.The sightlines are not uniformly distributed because there is a limited number of observers and the angle( r obs, r LMC− r obs)is kept ﬁxed at89.1◦.This is the original angle between( r⊙, r LMC− r⊙) in Galactocentric coordinates.the optical depth measurements obtained when the particles within the cones have been resampled to“subparticles”.In the un-resampled case,the handful of particles which hap-pen to reside close to the observer are found to dominate the calculation of the optical depth and other quantities of interest such as event duration and event rate.Variations in the optical depth from sightline to sightline can vary by up to a factor of three in the un-resampled sample cones,sim-ply because a handful of particles dominate the microlens-ing.Convergence experiments have shown that breaking the original simulation particles down to100subparticles is suf-ﬁcient to reduce the noise from this source to be negligible; if we were to use more subparticles,there is no further gain in accuracy.The subparticles introduced have masses of order104 M⊙,so we are unfortunately still orders of magnitudes away from actual MACHO masses.In the formulae introduced later(Section4),each subparticle is expressed as a sin-gular concentration of MACHOs.That is,a subparticle is treated as a set of MACHOs which have the same posi-tion and velocity as the subparticle itself,and for conve-nience in the simulations the MACHO mass is chosen to be m M=1.0M⊙(although this is later relaxed by scaling).Fig-ure6demonstrates the hierarchy of particles starting from the cosmological simulation particle down to the individual MACHOs.123456789100102030405060708090100110120130τcone1−7Cone indexOriginal particles in useTSC subparticles in useparison between the use of cosmological simula-tion particles and triangular-shaped clouds containing100sub-particles.For the illustration,we distributed132observers on a perimeter of a disk with a radius of8.5kpc,located on the xy-plane.The volumes of consecutive cones overlap by half.Without any resolution enhancements the variation in optical depth from cone to cone can be as large as a factor of three!This is due to the insuﬃcient mass resolution of the cosmological simulation. We reached the Poissonian noise level,in which cone-to-cone vari-ations are<20%,with100subparticles,by experimenting.The large scale trend is due to triaxiality of the halo.4THE MICROLENSING EXPERIMENTS4.1DeﬁnitionsBefore going to the microlensing equations,we deﬁne what we mean by certain terms:A microlensing source is a circular region,oriented perpendicularly to the line-of-sight of the observer,in which uniformly distributed background source stars are located. Because of the statistical nature of the source model,the number of background stars aﬀects only the microlensing event rate.The region with a diameter d S and a distance l S,is shown in Figure2.A source star is a star located somewhere in the disk of the microlensing source.A microlens is also a circular region,oriented perpen-dicularly to the line-of-sight of the observer and centered on a dark particle.The region has a radius r E,which de-pends on the location and mass of the particle.Lenses are always inside a microlensing cone between the source and the observer.A microlensing event is a detectable ampliﬁcation of a source star caused by a microlens.Detectable means that the light of a source star is ampliﬁed by a factor larger than1.34.This occurs when the sightline to a source star passesa particle within the lens radius,also known as the Einstein radius.4.2Optical depthThe Einstein radius r E of a gravitational lens is deﬁned as r2E=4Gml−1l−16Holopainen etal.Figure 5.For this illustrative example of the subparticle clouds,we placed 222cosmological simulation particles onto the xy -plane and broke each particle down to 10,000subparticles.To get the subparticle positions,we used the triangular-shaped cloud den-sity proﬁle identically to the actual cosmological simulation.The subparticles form a cube (of a size of the corresponding MLAPM grid cell)around the position of the cosmological simulation par-ticle.In the real microlensing simulation,the whole volume under investigation is ﬁlled with intersecting cubes and the void areas between the cubes seen here are not present.Figure 6.Splitting the cosmological simulation particle to 100subparticles and a subparticle to N M MACHOs.In the ﬁrst phase,we break the cosmological simulation particle to 100subparticles.The resulting subparticle represents an order of N M ∼104MA-CHOs with a mass of 1M ⊙.The subparticle is treated as a MA-CHO concentration where the represented MACHOs all have a single location (l M ≡l sub ),mass (m M ≡m sub /N M )and velocity (v M ≡v sub ).where m is the mass of the lens,l is the distance to the lens,l S is the distance to the source and A =4Gl sub−1ωS,(8)where ωis the solid Einstein angle of the lens and ωS is the solid angle of the source.When the solid angles are small,this can be approximated to τ≃r Er S2=Am Θ−21l S,(9)where r S is the radius of the source and Θ−2=(l S /r S )2.Note that Θis approximately the half opening angle of the cone.The optical depth of a subparticle with a mass m sub is deﬁned as τsub =A Θ−2m sub1l S.(10)We take this value to be the total optical depth from all the ∼104MACHOs the subparticle represents.Optical depth is additive,as long as the lenses do not overlap or cover the whole source,and so the total optical depth in a microlensing cone which contains N subparticles is τcone =N iτ(i )sub=A Θ−2m subN i1l S(11)From Equation 11it follows that the optical depth in a cone depends mainly on the distances of the subparticles re-lated to the observer.The closer the particles are the larger is the optical depth.Section 3.2covered the details of par-ticle breaking,which also has a large eﬀect on τcone .Note that τcone does not depend on the MACHO mass m M .4.3Event durationThe event duration of a lens describes the typical duration of the amplifying event the lens would produce.The detected event durations in the MACHO experiment are the order of 100days.Event duration depends on the tangential velocity with which a lens would seem to pass a source star and on the Einstein radius of the lens.The equation for an individual subparticle ist sub =πv sub ,(12)where r sub is the lens radius of the subparticle and v sub the apparent tangential velocity diﬀerence between the sub-particle and the source,respective to the observer.c0000RAS,MNRAS 000,000–000MACHOs in dark matter haloes7 The termπ2m1/2subl subl sub−12m1/2Ml subl sub−1t M=N Mτsub/N Mˆtsub,(15)whereΓM=τMπA1/2Θ−2m−1/2Mm subv subl sub−1πA1/2Θ−2m−1/2Mm subNi v(i)sub l(i)sub−18Holopainen et al.Table4.The mean optical depths and event rates,with their standard deviations,over the1600cones for each halo.See Figure 9for the corresponding(diﬀerential)distribution functions—the numbers represented here are the integrated values of those func-tions.We also list the modes of the distributions:ˆt exp columns give the most probable event duration for the respective distri-bution.Error limits show the variations between cones(i.e.mi-crolensing experiments)in a halo.mean4.7+5.0−2.1861.6+1.3−0.668MACHOs in dark matter haloes90 50 100 150 200 250 300 350 400 450 500 5101520253035404550a p p a r e n t t a n g e n t i a l v e l o c i t y (k m s -1)distance from observer (kpc)Figure 8.An example which shows the anticorrelation between apparent tangential velocities and locations of subparticles inside a microlensing cone.This anticorrelation explains why triaxiality does not show up in Γas clearly as in τ.The size range of the triangular-shaped clouds also shows up clearly in the plot.is done for all cones,we bin the observables and calculate the upper and lower fractiles so that they hold 65%of the values.Especially the optical depth fractiles reveal clear tri-axiality signal.In the ﬁrst panel in Figure 10,the mean optical depth is largest when the sources are close to the z -axis (i.e.the major triaxial axis),as one would expect.For haloes #1,#2and #7,the optical depth on opposite sides of the xy -plane diﬀers somewhat,demonstrating that matter is not neces-sarily distributed with azimuthal symmetry in the simulated (or real)haloes.There are high mean optical depth values near the pos-itive y -axis for Halo #1in the second panel.This anomaly is due to the subhalo discussed in previous Section 5.2.Now we can see that the subhalo is located close to the positive y -axis.This is conﬁrmed by the true location of the subhalo,which is found to be l sat =(−3.37,19.8,1.36)kpc.The lowest panel is diﬀerent from the two previous ones because we are measuring tri axiality.The optical depth val-ues from sources near the x -axis seem to be the lowest amongst all panels.These low values (τcone ∼4×10−7)can also be seen in the second panel as the lower fractiles at α∼90◦.Thus,by these “observations”we can verify that the x -axis is the minor axis.In the lowest panel,the fractiles bend upwards at α∼90◦because the sources near both the z -and y -axis produce larger values than the sources near the x -axis.Triaxiality can not be seen as clearly in Figure 11as in Figure 10because Γis aﬀected by the lens velocities whereas τis not.Apparently,the velocities do not carry enough infor-mation about the triaxiality and the correlation between the source location and the observables are somewhat “washed out”.This is conﬁrmed in Figure 8where we show that there is virtually no correlation between the apparent tangential velocity and the location of the subparticles inside a cone.Nevertheless,the most striking features,i.e.the z -axis ma-jority and the subhalo in Halo #1,are still clearly visible in Γ.For example,a suitable subhalo can produce roughly twice as many events as a similar area without one,based on the second panel in Figure 11.5.4Estimating MACHO mass and halo mass fraction in MACHOsThe MACHO collaboration used several analytical halo models to estimate the typical,individual MACHO mass,m M ,and the mass fraction of MACHOs in the halo,f ,re-sponsible for the lensing events observed.For example,oneof their models ﬁts m M =0.6+0.28−0.20M ⊙and f =0.21+0.10−0.07.We calculate our own estimates for m M and f to see how our N -body halo models compare to the analytical ones in making these predictions.As the observational constraint,we use the blending corrected event durations chosen by the MACHO collab-oration’s criterion “A”,ˆtst (A ),from Table 8in Alcock et al.(2000).These values are not corrected for the observa-tional eﬃciency function E (ˆt)nor have they been reduced for the events caused by the known stellar foreground pop-ulations,given in Table 12in Alcock et al.(2000).We adopt N exp (A )=2.67for known population events and ar-rive at the observational values of τobs =3.38×10−8and Γobs =1.69×10−7events star −1yr −1.We then compare these values with our simulation produced values,τcone and Γcone ,to get m M and f .The simulation values are correctedwith the eﬃciency function E (ˆt)prior to calculating the pre-dicted values.The calculations are based on the following equations.First,we assume that f ∝τ,(18)which simply describes that that the probability of ob-serving an event is proportional to the number of lenses in the whole halo.From Equation 17,and from the fact that the number of observed events also follows the total number of lenses,we get 1Γ∝fm −1/2M.(19)These two equations permit us to estimate m M and f .From Equation 18,we get f obsτcone.(20)For the microlensing simulations we assumed f cone =1.0and thus,f =f obs =τobsm cone M=f obsΓobs2,(22)where f cone =1.0and m cone M=1.0M ⊙.Thus,we get the equation for the MACHO mass,m M =m obs M =f obsΓcone10Holopainen et al.Table 5.Same as Table 4buthereweusedtheMACHOcol-laboration’s eﬃciency function A.These numbers should corre-spond to the E (ˆt)–uncorrected observations if the MACHO frac-tion would satisfy f =1.0and the MACHO mass m M =1.0M ⊙.The distributions are not shown because they are essentially sim-ilar to Figure 9except for the smaller integrated values (given in Tables 5and 6).mean1.7+1.9−0.8900.6+0.5−0.272Haloτcone ˆt exp Γcone ˆtexp 10−7days10−6daysevents star −1yr −1#12.7+2.9−1.3840.9+0.8−0.466#22.9+3.6−1.3841.0+1.0−0.466#32.1+1.9−0.9840.7+0.5−0.364#42.4+2.7−1.2860.8+0.7−0.374#52.0+1.6−1.1940.6+0.4−0.369#62.4+2.5−0.9990.8+0.6−0.371#72.3+3.2−1.2890.7+0.8−0.361#81.3+1.0−0.51040.4+0.3−0.181educated guesses about the shape in form of some given pa-rameters (e.g.triaxiality,density proﬁle,etc.).Widrow &Dubinski (1998)used a so called microlens-ing tube to get better number statistics for their event rates,and as they note,this “distorts the geometry of a realistic microlensing experiment”.However,our solution (the intro-duction of subparticles in accordance to the TSC mass as-sigment scheme)to the insuﬃcient mass resolution of the cosmological simulation preserves the geometry.The downsides of using N -body haloes are:(i)We as-sume that the spatial and velocity distribution of MACHOs follows dark matter.(ii)The clusters we use are not as old as the Milky Way.Our models are based upon pure dark matter simula-tions which may not be appropriate if a signiﬁcant fraction of the dark matter is composed of MACHOs,since they are composed of baryons,and baryonic physics has been ex-plicitly ignored!We thus implicitly assume that the spa-tial and velocity distribution of MACHOs follows the re-spective distributions of the underlying dark matter.Based upon these assumptions dark satellites comprised of MA-CHOs would be detectable through excess optical depth values and event duration anomalies in microlensing exper-iments.Simulations suggest that the majority of these dark satellites can be located as far as 400kpc from the halo cen-tre.Thus,multiple background sources at distances over 400kpc would be needed to detect possible dark subhaloes in a Milky Way sized dark halo.For example,POINT-AGAPE (Belokurov et al.2005,Calchi Novati et al.2005)is a sur-vey that is in principle able to detect even a dark MACHO satellite,on top of “free”MACHOs in the dark halo,because its source,M31,is distant enough.From the results,we can see that Halo #8is too young to be used as a Milky Way dark halo model,but all the other haloes seem to behave well.The fact that Halo #8consists of three merging smaller haloes explains the peculiar values it produces.The other haloes are not experiencing any violent dynamical changes —a requirement we would expect a model of the Milky Way dark halo to fulﬁll.Obviously,our MACHO mass function is a δ-function.A more complex function would force us to calculate event durations and rates for individual MACHOs instead of sub-c0000RAS,MNRAS 000,000–000。

The physics of dark matter and its properties

The physics of dark matter and itsproperties在宇宙中，黑暗物质是一个神秘而令人着迷的存在。

尽管其所占比例高达总质量的大约85%，但我们几乎对其一无所知。

我们观测到的宇宙中的星系、星际云气、恒星以及行星均只占宇宙总质量的不到15%。

那么黑暗物质是什么？为何数量如此之多？如何探测？黑暗物质的物理特性相对较为简单，但要找到黑暗物质所组成的微观粒子仍是一项困难的挑战。

黑暗物质公认的物理模型主要包括冷暗物质和热暗物质两种类型。

冷暗物质的作用类似于大自然的“胶水”，在宇宙演化中形成纤细且复杂的结构。

冷暗物质的平均速度下降至较低水平，因此形成了绕着银河中心旋转的「暗物质圆盘」，这一圆盘横跨整个银盘，在银盘的周围包覆着暗物质球。

如此巨大且稳定的团簇引领我们去研究它的内在机制。

相对于冷暗物质的热暗物质因为它们的速度过快，因此不会集聚到足以支撑宇宙中的结构。

热暗物质可以解释一些复杂的宇宙学问题，比如银河争夺中的“卫星猜想”，同样也可以用于解释太阳系及行星诸如其轨道的形成等问题。

黑暗物质的物理性质实际上是和常见物质相似的。

这意味着，黑暗物质可能是由一种或几种微观粒子组成的，可以通过我们目前的粒子加速器来探寻其细节。

在加速器的实验中，会以极高的速度让微观粒子不断碰撞，通过这种碰撞来获得新物种。

正是通过这种途径，科学家们成功发现了爱因斯坦所预测的带电粒子中的一种——希格斯玻色子。

而它也许并不是我们探测黑暗物质的“氢原子”，但其精纯的性质、深厚的知识体系和先进的实验技术为黑暗物质研究者提供了宝贵的指导。

黑暗物质是如此神秘，以至于对于它的探测及研究，就像是一个寻找到一位布加迪超跑物主的游戏。

有着更快更准确的探测技术，才能有着可能发现黑暗物质。

DOMINO的黑暗物质观测是对现有探测方式的重大突破，不仅如此，有关黑暗物质的实验裂变也已成为目前物理学中最火热的研究领域之一。

总的来说，黑暗物质的探寻和研究是相当具有挑战性的，要想达到理想的研究结果，需要一个有才华的科研团队，也需要更加精密的实验方法和仪器来进行探测，一次次的试验和尝试，才有可能让我们找到这个宇宙深处的谜团。

The structure of dark matter halos. Observation versus theory

tion. All halo density proﬁles can be ﬁt accurately by the simple formula,
ρ(r)
=
ρ¯ (r/rs)(1 +
r/rs)2
(1)
where ρ¯ and rs are two free parameters. It is very interesting that Navarro et al 12,14 ﬁnd a strong correlation between ρ¯ and rs. Dark matter halos seem
observations. Early cosmological calculations did not have enough resolution
in order to resolve the density structure of DM halos in detail. Recent highresolution simulations 11,12,14 however have shown that in the inner and outer regions dark matter halos depart signiﬁcantly from an r−2 power-law distribu-
Unfortunately it is diﬃcult to observationally verify these numerical results as galaxies are in general gravitationally dominated by their visible baryonic components in the inner regions, while in the outer regions there is not enough visible material in order to measure accurately a rotation curve. In the inner region the inferred DM proﬁles will depend strongly on how much baryonic mass is subtracted, which in turn depends on the assumed baryonic mass-tolight ratio. The situation becomes even more complicated by the fact that a dominating baryonic component will gravitationally aﬀect and change the cold dark matter density proﬁle 13.

SUSY and Dark Matter Constraints from the LHC

Furthermore, in order to be consistent with the observed structure of the universe, WMAP favours cold dark matter, or matter comprised of particles that are non-relativistic when galaxy formation starts. This naturally leads one to conclude that some sort of weakly interacting massive particle, or WIMP, is providing this dark contribution to our universe, and it is here that exciting astrophysics suddenly evolves into challenging and captivating particle physics! The Large Hadron Collider (LHC) at CERN in Geneva will start taking data next year and the purpose of this talk is to look at how much the LHC will be able to say about the dark matter problem, from the direct production and observation of WIMP candidates. Given the
In SUSY theories, all existing particles of the standard model have partners with opposite spin statistics called sparticles. Furthermore, one can impose a symmetry called R-parity under which the standard model particles are even whilst the SUSY particles are odd. This has two important phenomenological consequences: 1. We will pair produce sparticles at the LHC. 2. The lightest sparticle (LSP) is absolutely stable. Thus, the LSP is a natural WIMP candidate, and any consideration of SUSY models is highly relevant to the search for dark matter. It is noted that in diﬀerent regions of the SUSY parameter space, one obtains diﬀerent LSP’s, with possible options including the gluino, sneutrino, gravitino and the lightest neutralino. The last particle in this list is an admixture of the superpartners of the neutral SM gauge bosons, and remains the subject of the majority of studies. Thus, it is the only SUSY candidate to be considered from now on. 2.2 Reconstructing SUSY Models at the LHC

Grand Unification, Dark Matter, Baryon Asymmetry, and the Small Scale Structure of the Univ

a r X i v :h e p -p h /0503112v 1 11 M a r 2005Grand Uniﬁcation,Dark Matter,Baryon Asymmetry,and the Small Scale Structureof the UniverseRyuichiro Kitano and Ian LowSchool of Natural Sciences,Institute for Advanced Study,Princeton,NJ 08540We consider a minimal grand uniﬁed model where the dark matter arises from non-thermal decays of a messenger particle in the TeV range.The messenger particle compensates for the baryon asymmetry in the standard model and gives similar number densities to both the baryon and the dark matter.The non-thermal dark matter,if massive in the GeV range,could have a free-streaming scale in the order of 0.1Mpc and potentially resolve the discrepancies between observations and the ΛCDM model on the small scale structure of the Universe.Moreover,a GeV scale dark matter naturally leads to the observed puzzling proximity of baryonic and dark matter densities.Uniﬁcation of gauge couplings is achieved by choosing a “Higgsino”messenger.INTRODUCTIONThe standard model of particle physics is believed to be incomplete.For decades the strongest arguments are based more on aesthetic reasonings than on empirical evidence.One example is the ﬁne-tuning in the mass of the scalar Higgs requires new physics at around 1TeV to stabilize the electroweak scale,for which the benchmark solution is weak scale supersymmetry (SUSY).Another example is the gauge coupling uniﬁcation which suggests a grand uniﬁed theory (GUT)at high energy scale [1],assuming a desert between the GUT scale and the electroweak scale where all the particles come in complete multiplets of the GUT group.It turned out that in the minimally supersymmetric standard model (MSSM)[2],gauge couplings unify to a much better precision than in the standard model,which adds to the attractiveness of the weak scale SUSY.On the empirical side the situation has dramatically improved over the last few years due to insights from precision cosmological observations,including the exis-tence of dark matter,the acceleration of the cosmic expansion,the baryon-asymmetric Universe,and a nearly scale-invariant density ﬂuctuations,none of which can be explained within the standard model.(A minimal model addressing all these issues has been proposed in [3].)Emerging from the observations is a description of the Universe based on the ΛCDM model [4]:a tiny cosmological constant plus cold dark matter.Neither the MSSM nor the ΛCDM model is perfect,however.In the MSSM the most notable problems are the SUSY ﬂavor problem,new ﬂavor violations from various superpartners,and the non-observation of a light Higgs as well as any light sparticles,which implies ﬁne-tuning at a few percents level [5].These problems prompted the proposal [6]of giving up SUSY as a solution to the hierarchy problem.But if gauge coupling uniﬁcation (and the neutralino dark matter)is the main motivation for SUSY,there is a much simpler model,the standard model with “Higgsinos”,which is just as good [7].On the other hand,there has been evidencesuggesting,albeit not yet conclusively,inconsistencies between observations and numerical simulations of the ΛCDM model on the galactic and sub-galactic scales [8];it seems that ΛCDM model predicts too much power on the small scales.One way out is to introduce warm dark matter (WDM)[9,10]which has a small free-streaming scale λFS 0.1Mpc,hence suppressing density ﬂuctuations on scales smaller than λFS .In this letter,using uniﬁcation instead of naturalness as the main incentive,we show that the three otherwise independent aspects:gauge coupling uniﬁcation,the puzzling proximity of ΩDM and Ωb ,and the small free-streaming scale,can all be intertwined in a non-trivial way.It is based on the model discussed in [11]where the dark matter arises from late-time decays of a heavy messenger particle compensating for the baryon asymmetry in the standard model.First,gauge coupling uniﬁcation suggests the existence of “Higgsinos,”which is the messenger particle,at the TeV scale and sets the GUT scale to be ∼1014GeV [7].Next cosmology constrains the decay temperature of the “Higgsinos”to be ∼10MeV and determines the scale of the higher dimensional operators responsible for the Higgsino decay to be 1014−15GeV [11],a most natural value in the GUT picture.Then the measured ratio of ΩDM /Ωb can be used to ﬁx the dark matter mass to be ∼1GeV,which turns out to come with the necessary free-streaming scale to suppress small scale structures of the Universe.Alternatively,the need for a sub Mpc free-streaming scale can be used to argue for a GeV scale dark matter,with which the ΩDM is naturally close to the Ωb .THE MODELThe basic idea in [11]is that the dark matter S is produced non-thermally from the decay of a heavy mes-senger particle X ,which carries the baryon number and compensates for the baryon asymmetry in the Universe.Both S and X ,which we call the dark sector,are assumed to be charged under a Z 2symmetry,the T -parity,while2 the whole standard model is T-even.The dark matter Sis then a stable particle being the lightest T-odd particle(LTP).At the time of baryogenesis,we assume that theB−L number is distributed between the T-even andT-odd sectors,resulting in the following relation:n SMB−L=−n X B−L=−q B−L(n X−n¯X),(1)where q B−L is the B−L charge of the messenger X,andn SM B−L and n X B−L are the B−L number densities in thestandard model and the dark sector,respectively.On the other hand,since both X and¯X eventually decay into the LTP,the dark matter candidate S,its number density is given by the total number of X and¯X particlesn DM=n tot X≡n X+n¯X,(2)which is independent of the n SMB−L in Eq.(1)and wouldsuggest there is no connection between the baryonic and dark matter densities,unless n X≫n¯X∼0or the other way around.This implies the lifetime of X should be long enough so that it does not decay until after most of the¯X particles annihilate with X,which will be the case if there is no relevant or marginal operator contributing to the decay of X.Then at temperature T<m X,where m X is the mass of the messenger,particle X starts to annihilate with its anti-particle¯X through gauge interactions and we are left with an abundance of X.Consequently,n SMB−L=−n X B−L≃−q B−L n DM.(3) This is a very general framework.In[11]we worked out the cosmological constraints,as well as an example and the associated collider phenomenology.The model we are interested here,is that with a singlet scalar dark matter S and a“Higgsino”messenger X.Recently it is pointed out[7]that standard model plus Higgsinos achieves gauge coupling uniﬁcation at around1014GeV with an accuracy similar to that of the MSSM.At two-loop level,for m X=1TeV this minimal model predicts smaller values ofαs than measured by O(5%),whereas larger values are predicted in the MSSM.A GUT scale in the order of1014GeV is too small for a conventional GUT model due to the constraint from proton decay. One possibility,as discussed in[7],is to embed the setup in extra-dimensional or deconstructed models to lower the string scale down to1014GeV or so.This has the bonus of resolving the triplet-doublet splitting problem in conventional GUT models.A crucial departure from the model in[7]arises in that there the dark matter is a mixture of the neutral component of the Higgsinos and an extra singlet fermion,which is the usual WIMP scenario, whereas in our model the dark matter comes from non-thermal decays of Higgsinos and has a mass which will be determined to be in the GeV range.The scenario proceeds in three stages.In theﬁrst stage,baryogenesis is achieved by the out-of-equilibrium,CP-violating decay of a T-odd scalar particle P into ¯ℓi+X andℓi+¯X,whereℓi is the lepton doublet in the i th generation in the standard model.CP violation enters through the phases in the Yukawa couplings of the P withℓi and X if there are more than one P’s.So at one loop one obtains an asymmetry proportional to the imaginary parts of the Yukawa couplings of the P’s. After the P’s drop out of the thermal equilibrium,we are left with an asymmetry in lepton number in the standard model.Note that this mechanism is reminiscent of the leptogenesis[12],in which P is a heavy neutrino and X is the ordinary scalar Higgs.The eﬀective B−L number of X in Eq.(1)is determined to be q B−L=−1.The second stage is the complete annihilation of the Higgsino with its anti-particles.This constrains the mass of the messenger particle m X so that n X≫n¯X∼0. The thermally averaged cross section is estimated to be g4/(256πm2X)for s-wave annihilations into SU(2)gauge bosons which gives an upper bound on m X:m X≪150TeV 1GeV20,(5)where T d is the temperature at which X decays.The long lifetime can be realized by assuming that X only decays through a dimensionﬁve operatorO decay=11TeV 1/2≪M 1014GeVm X3 of the Universe is restricted to bem X≤m P T R 1TeV MΩb =3.16m DM4These discrepancies do not diminish the tremendous successes of the ΛCDM model on larger scales,however,if these problems are real,they present a great opportunity.The simplest known mechanism for smoothing out small scale structures is the Landau damping.In this regard a popular candidate is the WMD,hotdark mattercooleddown,which has a free-streaming scale [10]λFS =0.2(ΩWDM h 2)1keV−4a (t )dt ≃t EQv (t )3TeVGeVT dMpc ,where v (t )and a (t )are the physical velocity and the FRW scale factor,respectively,of the NTDM at time t ,and t EQ denotes the time for matter-radiation equality.Therefore,for a prototypical m X /m S /T d =3TeV/1GeV/10MeV scenario that is well-motivated from previ-ous discussions,the NTDM is able to produce enough power on the small scales to be consistent with the Lyman αforest data and potentially resolve the dis-crepancies mentioned above.There are also constraints coming from studies of phase space density [18],which are weaker than those coming from Lyman αforest [19].On the other hand,the NTDM is not exactly the same as the WDM,since they still have diﬀerent momentum distributions.As an example,power spectrum of non-thermal production of neutralinos by the decay of topo-logical defects was considered in [19]and found to be diﬀerent from that of a 1keV WDM for k >5h Mpc −1.SUMMARYUsing gauge coupling uniﬁcation and cosmology as the main hints for physics beyond the standard model,we have considered a minimal GUT model which attributes a common origin to the baryon asymmetry and the dark matter,giving similar number densities to both the dark matter and the baryon.The dark matter,a singlet scalar,is produced by non-thermal decays of a messenger particle,the Higgsinos,whose existence implies the uniﬁcation at ∼1014GeV.There are checkson the the model from several orthogonal directions.Cosmological bounds point to a Higgsino in the TeV range and a mass scale consistent with the GUT.The ratio ΩDM /Ωb can be used to determine the mass of the dark matter to be GeV,which in turn gives a sub Mpc free-streaming scale consistent with observations and,at the same time,has the potential of resolving the ΛCDM crisis by reducing the power spectrum on small scales.Cosmology may be the best arena to test this model.The Higgsinos,on the other hand,might be too heavy to be discovered in the coming collider experiments.Acknowledgments We beneﬁted from conversations with Neal Dalal and Carlos Pe˜n a-Garay.This work is supported by funds from the IAS and in part by the DOE grant number DE-FG02-90ER40542.[1]H.Georgi and S.L.Glashow,Phys.Rev.Lett.32,438(1974);H.Georgi,H.R.Quinn and S.Weinberg,Phys.Rev.Lett.33,451(1974).[2]S.Dimopoulos and H.Georgi,Nucl.Phys.B 193,150(1981);S.Dimopoulos,S.Raby and F.Wilczek,Phys.Rev.D 24,1681(1981).[3]H.Davoudiasl,R.Kitano,T.Li and H.Murayama,arXiv:hep-ph/0405097.[4]See,for example,D.N.Spergel et al.Astrophys.J.Suppl.148,175(2003)[arXiv:astro-ph/0302209].[5]L.Giusti,A.Romanino and A.Strumia,Nucl.Phys.B550,3(1999)[arXiv:hep-ph/9811386].[6]N.Arkani-Hamed and S.Dimopoulos,arXiv:hep-th/0405159.[7]N.Arkani-Hamed,S.Dimopoulos and S.Kachru,arXiv:hep-th/0501082.[8]For a review,see J.P.Ostriker and P.J.Steinhardt,Science 300(2003)1909[arXiv:astro-ph/0306402].[9]S.Colombi,S.Dodelson and L.M.Widrow,Astrophys.J.458,1(1996)[arXiv:astro-ph/9505029].[10]P.Bode,J.P.Ostriker and N.Turok,Astrophys.J.556,93(2001)[arXiv:astro-ph/0010389].[11]R.Kitano and I.Low,Phys.Rev.D 71,023510(2005)[arXiv:hep-ph/0411133].[12]M.Fukugita and T.Yanagida,Phys.Lett.B 174,45(1986).[13]J.A.Harvey and M.S.Turner,Phys.Rev.D 42,3344(1990).[14]E.W.Kolb and M.S.Turner,“The Early Universe,”Redwood City,USA:Addison-Wesley (1990).[15]V.K.Narayanan,D.N.Spergel,R.Dave and C.P.Ma,arXiv:astro-ph/0005095.[16]M.Viel,J.Lesgourgues,M.G.Haehnelt,S.Matarreseand A.Riotto,arXiv:astro-ph/0501562.[17]P.Colin,V.Avila-Reese and O.Valenzuela,arXiv:astro-ph/0009317;V.Avila-Reese,P.Colin,O.Valenzuela,E.D’Onghia and C.Firmani,arXiv:astro-ph/0010525.[18]C.J.Hogan and J.J.Dalcanton,Phys.Rev.D 62,063511(2000)[arXiv:astro-ph/0002330].[19]W.B.Lin,D.H.Huang,X.Zhang and R.H.Bran-denberger,Phys.Rev.Lett.86,954(2001)[arXiv:astro-ph/0009003].。

Extended Press-Schechter theory and the density profiles of dark matter haloes

a r X i v :a s t r o -p h /0305300v 1 16 M a y 2003Mon.Not.R.Astron.Soc.000,1–9(2003)Printed 2February 2008(MN L A T E X style ﬁle v2.2)Extended Press-Schechter theory and the density proﬁlesof dark matter haloesNicos Hiotelis ⋆†1st Experimental Lyceum of Athens,Ipitou 15,Plaka,10557,Athens,Greece,E-mail:hiotelis@ipta.demokritos.grAccepted ...............Received ................;in original form ...........ABSTRACTAn inside-out model for the formation of haloes in a hierarchical clustering scenario is studied.The method combines the picture of the spherical infall model and a mod-iﬁcation of the extended Press-Schechter theory.The mass accretion rate of a halo is deﬁned to be the rate of its mass increase due to minor mergers.The accreted mass is deposited at the outer shells without changing the density proﬁle of the halo inside its current virial radius.We applied the method to a ﬂat ΛCDM Universe.The resulting density proﬁles are compared to analytical models proposed in the literature,and a very good agreement is found.A trend is found of the inner density proﬁle becoming steeper for larger halo mass,that also results from recent N-body simulations.Addi-tionally,present-day concentrations as well as their time evolution are derived and it is shown that they reproduce the results of large cosmological N-body simulations.Key words:cosmology:theory –dark matter –galaxies:haloes –structure –for-mation1INTRODUCTIONNumerical studies (Quinn,Salmon &Zurek (1986);Frenk et al.(1988);Dubinski &Galberg (1991);Crone,Evrard &Richstone (1994);Navarro,Frenk &White (1997),here-after NFW;Cole &Lacey (1996);Huss,Jain &Steinmetz (1999);Fukushige &Makino (1997);Moore et al.(1998),hereafter MGQSL;Jing &Suto (2000),hereafter JS:Hern-quist (1990),hereafter H90,Kravtsov at al.(1998),Klypin at al.(2001))show that the density proﬁles of dark matter haloes are ﬁtted by models of the formρf (r )=ρcd r=λ+µν(r/r s )µ2Nicos Hiotelisfor stydying the formation of structures.Recently,such a modiﬁed PS approximation was combined with a spherical infall model picture of formation by Manrique et al.(2003), MRSSS hereafter.Their results are in good agreement with those of N-body simulations.In this paper we use the formalism of MRSSS,with justiﬁed modiﬁcations,and the same model parameters as in BKPD.We compare the characteristics of the resulting structures with those in N-body results.In Section2, we discuss the modiﬁed PS theory and its application to the calculation of the density proﬁle.In Section3,the characteristics of the resulting dark matter haloes are presented.A discussion is given in Section4.2EXTENDED AND MODIFIEDPRESS-SCHECHTER THEORYOne of the major goals of the spherical infall model is the PS approximation.It states that the comoving density of haloes with mass in the range M,M+d M at time t is given by the relation:N(M,t)d M= πδc(t)M e−δc2(t)d M|d M(3)whereσ(M)is the present-day rms massﬂuctuation on co-moving scale containing mass M and is related to the power spectrum P by the following relationσ2(M)=23πρb0R3=Ωm0H20d t=H i g13f i∆i(7) andΩΛ,i is the initial values of the quantityΩΛ(a)=Λ/(3H2(a)).Eq.7is derived under the assumption that the initial velocity v i of the shell is v i=H i r i−v pec,i where the initial peculiar velocity,v pec,i,is given according to the linear theory by the relation v pec,i=170[1−Ωm,i(1+Ωm,i)](Lahav et al.(1991)).The radius of the maximum expansion is r ta=s ta r i,where s ta is the root of the equation g(s)=0that corresponds to zero velocity(d s/d t=0).The sphere reaches its turn-around radius at timet ta=12(s)d s(8) and then collapses at time t c=2t ta.The scale factor a of the Universe obeys the equation: d a2(a)(9) whereX(a)=1+Ωm,0(a−1−1)+ΩΛ,0(a2−1)(10) and the subscript0denotes the present-day values of the parameters.However,the time and the scale factor a are related by the equationt=12(u)d u(11) Setting t=t c in the above equation and solving for a one ﬁnds the scale factor a c at the epoch of collapse.If we call δi,c(t)the initial overdensity required for the spherical region to collapse at that time t and take into account the linear theory for the evolution of the matter density contrastδ=δρ/ρ,we haveδ∝1a a0X−3/2(u)d u=D(a),(12) (Peebles1980),thenδc(t)is givenδc(t)=δi,c(t)D(t0)D(t i)D(t0)D(t)(14) whereδcrit(t)is the linear extrapolation of the initial over-density up to the time t of its collapse.In an Einstein-de Sitter universe(Ωm=1,ΩΛ=0)this value is independent on the time of collapse and isδcrit≈1.686.In other cos-mologies it has a weak dependence on the time of collapse (e.g.Eke,Cole&Frenk(1996)).In aﬂat universe it can beapproximated by the formulaδcrit(t)≈1.686Ω0.0055m,0(t).The PS mass function agrees relatively well with the results of N-body simulations(e.g.Efstathiou,Frenk& White(1985),Efstathiou&Rees(1988);White,Efstathiou &Frenk(1993),Lacey&Cole(1994);Gelb&Bertschinger (1994);Bond&Myers(1996))while it deviates in detail at both the high and low masses.Recent improvements(Sheth &Tormen(1999);Sheth,Mo&Tormen(2001),see also Jenkins et al.(2001))allow a better approximation involving some more parameters.The application of the above approx-imation to the model studied in this paper is a subject of future research.Lacey&Cole(1993)extended the PS theory using the idea of a Brownian random walk,and were able to calcu-late analytically tractable expressions for the mass function,c 2003RAS,MNRAS000,1–9Extended Press-Schechter theory and the density proﬁles of dark matter haloes3merger rates,and other properties.They show that the in-stantaneous transition rate at t from haloes with mass M to haloes with mass between M ′,M ′+d M ′is given byr (M →M ′,t )d M ′=2d t1d M ′1−σ2(M ′)21σ2(M )d M ′(15)This provides the fraction of the total number of haloeswith mass M at t ,which give rise per unit time to haloes with mass in the range M ′,M ′+d M ′through instantaneous mergers of any amplitude.An interesting modiﬁcation of the extended PS theory is the distinction between minor and major mergers (Manrique &Salvador-Sol´e (1996);Kitayama &Suto (1996);Salvador-Sol´e et al.(1998);Percival,Miller &Peacock (2000),Cohn et al.(2001)).Mergers that produce a fractional increase below a given threshold ∆m are regarded as minor.This kind of mergers corresponds to an accretion.Consequently,the rate at which haloes increase their mass due to minor mergers is the in-stantaneous mass accretion rate and is given by the relationr a m (M,t )=M (1+∆m )M(M ′−M )r (M →M ′,t )d M ′(16)Thus the rate of the increase of halo’s mass due to the ac-cretion is d M (t )ρb (a )=1a i3(1+∆i )(18)where c f is the collapse factor of the sphere deﬁned as the ratio of its ﬁnal radius to its turnaround -hav et al.(1991)applied the virial theorem to the viri-alized ﬁnal sphere assuming a ﬂat overdensity and found the collapse factor to be c f ≈(1−n/2)/(2−n/2)wheren =(Λr 3ta )/(3GM ).For an Einstein-de Sitter Universe ∆vir (a )≈18π2at any time.For ﬂat models with cosmolog-ical constant,signiﬁcantly good analytical approximations of ∆vir exist.Bryan &Norman (1998)proposed for ∆vir the following approximation∆vir (a )≈(18π2−82x −39x 2)/Ωm (a )(19)where x ≡1−Ωm (a ).MRSSS considered the following picture of the forma-tion of a halo:At time t i an halo of virial mass M i and virial radius R i is formed and at later times it accretes mass ac-cording to the Eq.(17).Assuming that the accreted mass is deposited in an outer spherical shell without changing the density proﬁle inside its current radius,then M vir (t )−M i =R vir (t )R i4πr 2ρ(r )d r(20)The current radius R vir contains a mass with mean den-sity ∆vir (a )times the mean density of the Universe ρb (t ).Therefore,R vir (t )=3M vir (t )r a m [M vir (t ),t ]d[ln[ρb (t )∆vir (t )]]8πG∆vir (a )×1+M vir (a )a −d ln[∆vir (a )]d a=H −10X −1/2(a )r am [M vir (t ),t ].(24)Integrating Eq.17and using Eqs.21and 23,we obtain the growth of virial mass and virial radius and,in a parametric form,the density proﬁle of haloes.3DENSITY PROFILES OF DARK MATTERHALOESThe results described in this section are derived for a ﬂat universe with Ωm,0=0.3and ΩΛ,0=0.7.We used two forms of power spectrum.The ﬁrst one -named spect1-is the one proposed by Efstathiou,Bond &White (1992).It is based on the results of the COBE DMR experiment and is given by the relation:P spect 1(k )=Bk[1+a 1k 1/2+a 2k +a 3k 3/2+a 4k 2]b(26)The values for the parameters are:n =1,a 1=−1.5598,a 2=47.986,a 3=117.77,a 4=321.92and b =1.8606.We used the top-hat window function that has a Fourier transform given by:ˆW(kR )=3(sin(kR )−kR cos(kR ))4Nicos Hiotelis-4-22log (M/M unit )0.51l o g (M /M u n i t )σFigure 1.rms mass ﬂuctuation as a function of mass.Solid line:for spect1,dotted line:for spect2are shown.It must be noted that we use a system of units with M unit =1012h −1M ⊙,R unit =h −1Mpc and t unit =1.515×107h −1years.In this system of units H 0/H unit =1.5276.3.1Present day structuresIn the approximation used in this paper,for given models of the Universe and power spectrum there is only one free pa-rameter,that is the value of the threshold ∆m (see Eq.16).We found that the resulting density proﬁles are sensitive to the value of ∆m .As an example,the density proﬁles of two systems with the same present-day mass 1012h −1M ⊙and diﬀerent values of ∆m are plotted in Fig.2.Both den-sity proﬁles are derived from spect2.The solid line -shown in Fig.2-corresponds to the system derived for ∆m =0.21while the dotted line to the system for ∆m =0.5.The den-sity proﬁle for smaller ∆m is steeper at both the inner and the outer regions.Additionally,for diﬀerent ∆m the concen-trations of the haloes are diﬀerent too (a detailed description of the way the concentration is calculated is given below).The system that results for ∆m =0.21has c vir =15.2,while the one for ∆m =0.5has c vir =8.7.In order to calculate the density proﬁles (that will be presented below),we used as a basic criterion the concentrations of the present-day struc-tures.In fact,we have chosen the values of ∆m =0.23and ∆m =0.21for spect1and spect2respectively,because the concentrations resulting from these values are close to the results of the toy-model of BKPD.This model is constructed by BKPD to reproduce the results of their N-body simula-tions and it is able to give the concentration c vir of a virial mass M vir at any scale factor a .First,the scale factor a c at the epoch of collapse is calculated,solving the following equation-2.5-2-1.5-1log R1234l o g ρFigure 2.For spect2:Density proﬁles for two haloes having the same mass 1012h −1M ⊙.Solid line:∆m =0.21.Dotted line:∆m =0.5σ[M ∗(a c )]=σ[F M vir (a )](28)where F =0.01and M ∗is the typical collapsing mass.Then,the concentration is calculated using the formulac vir ,BKPD [M vir (a ),a ]=Kaλ+µν−n1/µr s (31)This formula gives the radius r n at which the logarithmic slope equals to n .According to the model presented in this paper,haloes grow inside-out.Thus,the value of c vir repre-sents the way of halo growth.In Fig.3,the concentration isc2003RAS,MNRAS 000,1–9Extended Press-Schechter theory and the density proﬁles of dark matter haloes510101102M vir /M unit681012141618c Figure 3.Concentration as a function of present-day virial mass.From the top,the ﬁrst pair of curves are for spect1and the second for spect2.Solid curves:our results.Dotted curves:BKPD toy-model results.plotted as a function of the present-day virial mass.From the top of the ﬁgure,the ﬁrst pair of curves (solid and dot-ted)correspond to spect1and the second pair to spect2.Solid curves show our results while dotted curves depict the results of the toy-model of BKPD.A very good agreement between the values of the concentration is shown.In partic-ular,concentrations resulting from spect2are in agreement with those obtained for the model of BKPD for the whole range of mass presented.On the other hand,small diﬀer-ences appear for very small and very large masses in the case of spect1.In Fig.4we present the density proﬁles of the re-sulting structures with present-day masses in the range of 0.2×1011h −1M ⊙to 8×1014h −1M ⊙.The left-hand side ﬁg-ures (a1,b1,c1,d1,e1)have been produced using spect1,while the right-hand ones using spect2.Figures (a1)and (a2)correspond to mass 0.2×1011h −1M ⊙,(b1)and (b2)have mass 1012h −1M ⊙,(c1)and (c2)to mass 1013h −1M ⊙,(d1)and (d2)to mass 1014h −1M ⊙and (e1)and (e2)cor-respond to mass 8×1014h −1M ⊙.Solid lines represent the resulting density proﬁles while dotted lines are the ﬁts using the general formula of Eq.1.It is shown that the ﬁts using the general formula of Eq.1are exact.We also ﬁt every halo density proﬁle using the analytical models that have been proposed in the literature (H90,NFW,MGQSL,JS)and are described in Section 1.The best ﬁt of these models to our resulting proﬁles is shown in Fig.4(circles).This best ﬁt is found by the minimizing procedure described above,for λ,µand νconstants and equal to the proposed values,while ρc and r s are the only ﬁtting parameters.Best ﬁt for the result-ing density proﬁle in (a1)is the H90model,in (a2)and (b2)the NFW model,in (b1),(c1)and (c2)the MGQSL model24l o g ()ρJS(d1)5JS(d2)-1.5-0.5log (R /R vir )024JS(e1)-1.5-0.5log (R /R vir )24JS(e2)-1.5-0.5024MGQSL(c2)24MGQSL(c1)024l o g ()MGQSLρ(b1)24NFW(b2)24H90(a1)24NFW(a2)Figure 4.Density proﬁles as a function of radius.Solid curves:re-sulting density proﬁles.Dotted curves:ﬁts of the resulting densityproﬁles using the formula of Eq.1.Circles:best bit to our results using models proposed in the literature (H90,NFW,MGQSL,JS).Left-hand side:spect1.Right-hand side:spect2and in (d1),(d2),(e1)and (e2)the JS model.Additionally,haloes of diﬀerent mass are ﬁtted well by diﬀerent analytical models.This is due to the diﬀerent inner and outer slopes of the density proﬁles.Inner slope,(deﬁned as that at ra-dial distance r =10−2R vir ),is an increasing function of the virial mass of the halo.For example,in the case of spect2the inner slope varies from 1.43for M =1012h −1M ⊙to 1.65for M =8×1014h −1M ⊙.Additionally,outer slope -at r =R vir -is a decreasing function of the virial mass and it varies from 3.67to 2.64for the above range of masses.Although density proﬁles resulting in simulations seem to be similar,systematic trends that relate them with the power spectrum have been reported.For example,Subra-manian,Cen &Ostriker (2000)found in the results of their N-body simulations the following:for power spectra of the form P (k )∝k n the density proﬁles have steeper cores for larger n .Therefore,a dependence of the density proﬁle on the power spectrum is expected.This dependence is shown in our results comparing the proﬁles of haloes with the samec2003RAS,MNRAS 000,1–96Nicos Hiotelis200400600800log (M vir /M unit )11.5λFigure 5.Exponent λas a function of present-day virial mass for both power spectra.It is shown that λis an increasing function of the virial mass.Solid curve:spect1.Dashed curve:spect2present day mass.It should be noted that the method stud-ied in this manuscript is applicable for the era of slow accre-tion when the infalling matter is in the form of small haloes that have mass less than ∆m times the mass of the parent halo.This kind of accretion occurs at the late stages of for-mation and thus determines the proﬁle of the outer regions of the halo under study.However,the values of the inner slopes may be questionable.Real haloes have followed dif-ferent mass growth histories and thus their properties show a signiﬁcant scatter about a mean value.Unfortunately,the method studied in this manuscript results one proﬁle for a halo of given mass.Thus,its purpose is just to approximate the mean density proﬁle of a large number of mass growth histories.Since the mass growth history resulting from the method is in good agreement with the mean growth his-tory resulting from N-body simulation -as it will be shown below-then the values of the inner slopes could be close to the ones of N-body simulations.A Monte Carlo analytic ap-proach based on the construction of a large number of mass accretion histories is under study.This study could answer to some of the above problems.In Fig.5the exponent λis plotted,that gives the asymptotic slope at R →0,derived by the general ﬁt as a function of present-day virial mass for both power spec-tra.It is shown that the exponent λis an increasing function of virial mass.This trend of the inner density proﬁle is also found in the results of recent N-body simulations (Ricotti (2002)).3.2Time evolutionIn Fig.6we plot mass growth curves.The curves show M vir (a )as a function of a in a logarithmic slope.The solidlines show our resulting structures and the dotted lines show the mass growth curves of the model proposed by Wech-sler et al.(2002).The curves of the left panel correspond to spect1while those of the right panel to spect2.From the top to the bottom,the curves correspond to masses 2×1011h −1M ⊙,1012h −1M ⊙,1013h −1M ⊙,1014h −1M ⊙and 8×1014h −1M ⊙respectively.It is obvious that massive haloes show substantial increase of their mass up to late times while the growth curves of less massive haloes tend to ﬂatten out earlier.This behaviour of mass growth curves characterizes the hierarchical clustering scenario where small haloes are formed earlier than more massive ones.Addition-ally,it helps to deﬁne the term ”formation time”by a mea-surable way.Wechsler et al.(2002)deﬁne as formation scale factor ˜a c the scale factor when the logarithmic slope of mass growth,(d lnM (a )/d lna ),falls below some speciﬁed value,S .They use the value S =2.It should be noted that this def-inition of formation scale factor diﬀers from a c ,deﬁned by BKPD,since a c is the value of the scale factor at the epoch the typical collapsing mass is F times the virial mass of the halo.We found that the values of ˜a c and a c for F =0.01and S =2are diﬀerent.This is also noticed in Wechsler et al.(2002)since they state that ˜a c and a c have similar values for S =2but for F =0.015.However,the use of the value F =0.015in the toy-model of BKPD changes the resulting concentrations and so our basic criterion for the choice of the threshold ∆m is not satisﬁed.Therefore,it is preferable to choose a diﬀerent value of S for the deﬁnition of ˜a c ,that of S =1.5.In Fig.6,the dotted lines show the mass growth curves of the model proposed by Wechsler et al.(2002).In this model the mass growth is calculated using the relation:M vir (a )=M vir,0exp[−˜a c S (1/a −1)](32)where M vir,0is the present-day virial mass and the formation scale factor ˜a c is deﬁned by the condition d lnM (a )/d lna =S with S =1.5.In Fig.6,a very satis-factory agreement is shown,particularly for the less massive haloes.We have to note that our model haloes grow inside-out.Therefore,in early enough times -when the slope of the density is smaller that 2all the way from the centre up to the current radius-it is meaningless to deﬁne c vir .Once the building of the halo has proceeded beyond the point with slope 2,the evolution of c vir is due to the growth of the virial radius and is given by c vir [M (a ),a ]=c vir (M 0)R vir [M (a )]Extended Press-Schechter theory and the density proﬁles of dark matter haloes 7-1-0.8-0.6-0.4-0.20log(a)-1-0.8-0.6-0.4-0.2l o g (M v i r /M 0)-1-0.8-0.6-0.4-0.20log(a)Figure 6.Mass growth curves as a function of scale factor a .Left panel,right panel:spect1,spect2respectively.From top to the bottom the curves correspond to masses 0.2×1011h −1M ⊙,1012h −1M ⊙,1013h −1M ⊙,1014h −1M ⊙and 8×1014h −1M ⊙respectively.0.50.60.70.80.91a024*********.50.60.70.80.91a2468101214161820c [M (a )]Figure 7.Concentration as a function of the scale factor a .Left panel:spect1.Right panel:spect2.Solid lines:our results.Dotted lines:the results of toy-model of BKPD.From top to the bottom the lines correspond to masses 0.2×1011h −1M ⊙,1012h −1M ⊙,1013h −1M ⊙,1014h −1M ⊙and 8×1014h −1M ⊙respectively.struction of analytical models requires a number of crucial assumptions.The model studied in this paper was proposed by MRSSS and assumes that(i)The rate of mass accretion is deﬁned by the rate ofminor mergers(ii)Haloes grow inside-out.The accreted mass is de-posited at the outer shells without changing the density pro-ﬁle of the halo inside its current virial radiusc2003RAS,MNRAS 000,1–98Nicos HiotelisTheﬁrst assumption indicates that structures presented in this paper formed by a gentle accretion of mass.The phys-ical process implied by the second assumption is that the infalling matter does not penetrate the current virial radius. This process requires an amount of non-radial motion.This amount has to be large enough so that the pericenter of the accreted mass is larger than the current virial radius.It should be noted that a density proﬁle that results from a radial collapse has inner slope steeper than2.It is the pres-ence of non-radial motion during the collapse that leads to inner slopes shallower than2.(e.g.Nusser(2001),Hiotelis (2002),Subramanian,Cen&Ostriker(2000)).Non-radial motions are always present in the structures formed in N-body simulations.Despite the above assumptions,the results of the model studied in this paper are in good agreement with the results of N-body simulations.The summary of these results is as follows:(i)Density proﬁles of haloes are close to the analytical models proposed in the literature as goodﬁts to the results of N-body simulations.A trend of the inner slope of the density proﬁle as an increasing function of the mass of the halo is also found,in agreement with recent results of N-body simulations.(ii)Concentration is a decreasing function of virial mass. Its values are in agreement with the results of numerical methods.(iii)Massive haloes increase their mass substantially up to late times.Growth curves of less massive haloes tend to ﬂatten out earlier.The concentrations of less massive haloes evolve more rapidly while those of more massive haloes evolve slowly.Taking into account the number of assumptions and approx-imations used to build the model presented in this paper,we can conclude that the agreement with the results of N-body simulations is very good.Consequently,this model provides a very promising method to deal with the process of struc-ture formation.Further improvements to this model could help to understand better the physical picture during this process.5ACKNOWLEDGEMENTSI would like to thank the Empirikion Foundation for itsﬁ-nancial support.REFERENCESAvila-Reese V.,Firmani C.,Hern´a dez X.,1998,ApJ,505, 37Bond J.R.,Cole S.,Efstathiou G.,Kaiser N.,1991,ApJ, 379,440Bond J.R.,Myers S.,1996,ApJS,103,41Bower R.J.,1991,MNRAS,248,332Bryan G.,Norman M.,1998,ApJ,495,80.Bullock J.S.,Kolatt A.,Primack J.R.,Dekel A.,2001,MN-RAS,321,559(BKPD)Cohn J.D.,Bagla J.S.,White M.,2001MNRAS,325,1053 Cole S.,Lacey C.,1996,MNRAS,281,716Crone M.M.,Evrard A.E.,Richstone D.O.,1994,ApJ,434, 402Dubinski J.,Calberg R.,1991,ApJ,378,496.Efstathiou G.,Frenk C.S.,White S.D.M.,1985,ApJ,292, 371Efstathiou G.,Rees M.,1988,MNRAS,230,5 Efstathiou G.,Bond J.R.,White S.D.M.,1992,MNRAS, 258,1Eke V.R,Cole S.,Frenk C.S,1996,MNRAS,282,263 Frenk C.S,White S.D.M.,Davis M.,Efstathiou G.,1988, ApJ,327,507Fukushige T.,Makino J.,1997,ApJ,477,L9Gelb J.,Bertschinger E.,1994,ApJ,436,467Henriksen R.N.,Widrow L.M.,1999,MNRAS,302,321 Hernquist L.,1990,ApJ,356,359Hiotelis N.,2002,A&A,382,84Huss A.,Jain B.,Steinmetz M.,1999,MNRAS,308,1011 Jenkins A.,Frenk C.S.,White S.D.M.,Colberg J.M.,Cole S.,Evrard A.E.,Couchman H.M.P.,Yoshida N.,2001,MN-RAS,321,372Jing Y.P.,Suto Y.,2000,ApJ,529,L69(JS)Kitayama T.,Suto Y.,1996,MNRAS,280,638Kravtsov A.V.,Klypin A.A.,Bullock J.S.,Primack J.R., 1998,ApJ,502,48Klypin A.A.,Kravtsov A.V.,Bullock J.S.,Primack J.R., 2001,ApJ,554,903Kull A.,1999,ApJ,516,L5Lacey C.,Cole S.,1993,MNRAS,262,627Lacey C.,Cole S.,1994,MNRAS,271,676Lahav O.,Lilje P.B.,Primack J.R.,Rees M.J.,1991,MN-RAS,251,128Lokas E.L.,2000,MNRAS,311,423Manrique A.,Salvador-Sol´e E.,1996,ApJ,467,504 Manrique A.,Raig A.,Salvador-Sol´e E.,Sanchis T.,Solanes J.M.,2003,preprint(astro-ph/0304378)(MRSSS)Moore B.,Governato F.,Quinn T.,Stadel ke G.,1998, ApJ,499,L5(MGQSL)Navarro J.F.,Frenk C.S.White S.D.M.,1997,ApJ,490, 493(NFW)Nusser A.,Sheth R.,1999,MNRAS,303,685Nusser A.,2001,MNRAS,325,1397Peebles P.J.E.,1980,The Large-Scale Structure of the Uni-verse,Princeton Univ.Press,Princeton,NJPercival W.J.,Miller L.,Peacock J.A.,2000,MNRAS,318, 273Press W.H.,Schechter P.,1974,ApJ,187,425.Quinn P.J.,Salmon J.K.,Zurek W.H.,1986,Nature,322, 329Raig A.,Gonz´a lez-Casado G.,Salvador-Sol´e,1998,ApJ, 508,L129Ricotti M.,2002,preprint(astro-ph/0212146)Salvador-Sol´e E.,Solanes J.M.,Manrique A.,1998,ApJ, 499,542Sheth R.K.,Tormen G.,1999,MNRAS,308,119Sheth R.K.,Mo H.J.,Tormen G.,2001,MNRAS,323,1 Smith C.C.,Klypin A.,Gross M.A.K.,Primack J.R., Holtzman J.,1998,MNRAS,297,910Subramanian K.,Cen R.Y.,Ostriker J.P.,2000,ApJ,538, 528Syer D.,White S.D.M.,1988,MNRAS,293,337 Wechsler R.H.,Bullock J.S.,Primack J.R.,Kratsov A.V., Dekel A.,2002,ApJ,568,52c 2003RAS,MNRAS000,1–9Extended Press-Schechter theory and the density proﬁles of dark matter haloes9 White S.D.M.,Efstathiou G.,Frenk C.,1993,MNRAS,262,1023This paper has been typeset from a T E X/L A T E Xﬁle preparedby the author.c 2003RAS,MNRAS000,1–9。

Galaxy Formation and Dark Matter

a r X i v :a s t r o -p h /0603209v 1 8 M a r 20062Joseph Silkmodel is the spectrum of primordial densityﬂuctuations,measured in the lin-ear regime via the temperature anisotropies of the CMB.This provides the ini-tial conditions for large-scale structure and galaxy formation via gravitational instability once the universe is matter-dominated.Dark matter consequently provides the gravitational potential wells within which galaxies formed.The dark matter and galaxy formation paradigms are inextricably interdependent. Unfortunately we have not yet identiﬁed a dark matter candidate,nor do we yet understand the fundamental aspects of galaxy formation.Nevertheless, cosmologists have not been deterred,and have even been encouraged to de-velop novel probes and theories that seek to advance our understanding of these forefront issues.Progress has been made on the baryonic dark matter front.Only about half of the baryons initially present in galaxies,or more precisely,on the comoving scales over which galaxies formed,are directly observed.We cannot predict with any certainty the mass fraction in dark baryons.Yet there are excellent candidates for the dark baryons,both compact and especially diﬀuse.In contrast,we have at least one elegant and moderately compelling theory of particle physics,SUSY,that predicts the observed fraction of nonbaryonic dark matter.Unfortunately,we have no idea yet as to whether the required stable supersymmetric particles actually exist.In this review,I willﬁrst describe the increasingly standard precision model of cosmology that enables us to provide an inventory of cosmic baryons.I summarise the current situation with regard to possible baryonic dark mat-ter.I discuss how nonbaryonic matter has been successfully used to provide an infrastructure for galaxy formation,and review the astrophysical issues, primarily centering on star formation and feedback.I conclude with the out-look for future progress.for nonbaryonic dark matter detection and galaxy formation.2Precision cosmologyModern cosmology has emphatically laid down a challenge to theorists.A combination of new experiments has unambiguously measured the key pa-rameters of our cosmological model that describes the universe.These include the temperatureﬂuctuations in the cosmic microwave background,the large galaxy redshift surveys,gravitational shear distortions of distant galaxies by lensing,the studies of the intergalactic medium via the distribution of ab-sorbing neutral clouds along diﬀerent lines of sight and the use of distant Type Ia supernovae as standard candles.Cosmologists now debate the er-ror bars of the standard model parameters.The ingredients of the standard model in eﬀect deﬁne the model.These most crucially are the Friedmann-Robertson-Walker metric and the Friedmann-Lemaitre equations,and the contents of the universe:baryons,neutrinos,photons,baryons,dark mat-ter and dark energy.On these constituents is superimposed a distribution ofGalaxy Formation and Dark Matter3 primordial adiabatic density(scalar)ﬂuctuations characterised by a power spectrum of speciﬁed amplitude and spectral index.In addition,there may be a primordial gravity wave tensor mode ofﬂuctuations.The number of free parameters in the standard model is14,of which the most signiﬁcant are: H0,Ωb,Ωm,ΩΛ,Ωγ,Ων,σ8,n s,r,n T,andτ.One can also add an equation of state for dark energy parameter,w=−pΛ/ρΛ,in eﬀect really a function of redshift,and a rolling scalar(and possibly tensor)index,dn s/dlnk.No single observational set constrains all,or even most,of these parame-ters.There are well-known degeneracies,most notably betweenΩΛandΩm,σ8 andτ,andσ8andΩm.However use of multiple data sets helps to break these degeneracies.For example,CMB anisotropiesﬁx the combinationΩm+ΩΛif a Hubble constant prior is adopted,as well asΩb h2andΩm h2,and SNIa constrain the(approximate)combinationΩm−ΩΛ.Both weak lensing andpeculiar velocity surveys specify the productΩ−0.6m σ8.Lyman alpha forest sur-veys extend the latter measurement to Mpc comoving scales,probing the cur-rently nonlinear regime.Finally,baryon oscillations are providing a measure ofΩm/Ωb,independently of the CMB.Interpretation in terms of a standard model(Friedmann-Lemaitre plus adiabaticﬂuctuations)yields the concor-dance model with remarkably small error bars[1].Theﬂatness of space is measured to beΩtotal=1.02±0.02.Dark energy in the form of a cosmological constant dominates the universe,withΩΛ= 0.72±0.02.The dark energy equation of state is indistinguishable from that of a cosmological constant,with w≡pΛ/ρΛc2=−0.99±0.1,this uncertainty holding to z∼0.5.Even at z∼1,the claimed uncertainty around w=−1 is only20percent.Non-baryonic dark matter dominates over baryons with Ωm=0.27±0.02andΩb=0.044±0.004.Most of the baryons are non-luminous,sinceΩ∗≈0.005.The spectrum of primordial densityﬂuctuations is unambiguously mea-sured both in the CMB and in the large-scale galaxy distribution from deep redshift surveys,and found to be approximately scale-invariant,with scalar index n s=0.98±0.02.One can also constrain a possible relic gravitational wave background,a key prediction of inﬂationary cosmology,by the tensor mode limit on relic gravitational waves:T/S<0.36.It has been argued that a fundamental test of inﬂation requires sensitivity at a level T/S>∼0.01[2]. Neutrinos are known to have mass as a consequence of atmospheric(ντ,νµ) and solar(νµ,νe)oscillations,with a deduced mass in excess of0.001eV for the lightest neutrino.From the power spectrum of the densityﬂuctuations, the inferred mass limit(on the sum of the3neutrino masses)isΣmν<0.4eV.However one note of caution should be added.These tight error bars all depend on adoption of simple priors.If these are extended,to allow,for ex-ample,for an admixture of generic primordial isocurvatureﬂuctuations,the error bars on many of these parameters increase dramatically,by up to an order of magnitude.Clearly,the devil is in the observational details.Popular models of inﬂation predict that n≈0.97.Space is expected to be very close toﬂat,withΩ=4Joseph Silk1+O(10−5).The numbers of rare massive objects at high redshift is speciﬁed by the theory of gaussian randomﬁelds applied to the primordial linear density ﬂuctuations.The universe as viewed in the CMB should be isotropic.Any deviations from these predictions would be immensely exciting.Suppose deviations were to be found.This would allow all sorts of pos-sible extensions to the standard model of cosmology.One might consider the signatures of string relics of superstrings or transplanckian features in δT/T|k[3].Large-scale cosmology might be aﬀected by compact topology or global anisotropy with observable signatures in CMB temperature and polar-isation maps[4].The initial conditions might involve primordial nongaussian-ity.Anthropically constrained landscape scenarios of the metauniverse prefer a slightly open universe[5].Some of these features,and others,could be a consequence of compactiﬁcation from higher dimensions.3The global baryon inventoryThere are several independent approaches to obtaining the baryon abun-dance in the universe.At z∼109,primordial nucleosynthesis of the light elements yieldsΩb=0.04±0.004.At the epoch of matter-radiation decou-pling,z∼1000,the ratios of odd and even CMB acoustic peak heights set Ωb=0.044±0.003.At more recent epochs,Lyman alpha forest modelling of the intergalactic medium at z∼3as viewed in absorption along diﬀer-ent lines of sight towards high redshift quasars at z∼3yieldsΩb≈0.04. At the present epoch,on very large scales,of order10Mpc comoving linear regime equivalent,the intracluster baryon fraction measured via x-ray obser-vations of massive galaxy clusters provides a baryon fraction of15%.This translates intoΩb≈0.04.In summary,we infer thatΩb=0.04±0.005and Ωb/Ωm=0.15±0.02.One’s immediate impression is that,at least until very recently,most of the baryons in the universe today are not accounted for.The reasoning is as follows.The luminous content in the form of stars sums toΩb≈0.004or10% in spheroids,andΩb≈0.002or5%in disks.There is also hot intracluster gas amounting toΩb≈0.002or5%.Current epoch observations of the cold/warm photo-ionised IGM via the nearby Lyman alpha/beta forest at104−105K as well as CIII(at z∼0)yield a much larger baryonic reservoir of gas,Ωb≈0.012 or30%.This gas is metal-poor,with an abundance of about10%solar[6].So far,we have only accounted for50%of current epoch baryons.The probable breakthrough,however,has come with recent detections of the warm-hot intergalactic medium at T<∼105−106K at z∼0,observed in OVI absorption in the UV and especially via x-ray absorption via OVII and OVIII hydrogen-like transitions towards low redshift luminous AGN.Some-thing likeΩb≈0.012or30%of the primordial baryon fraction appears to be in this form,enriched(in oxygen,at least)to about10%of the solar value[7]. We now have>∼80%of the baryons accounted for today.The total baryonGalaxy Formation and Dark Matter5 content sums toΩb=0.032±0.005.Given the measurement uncertainties, this would seem to remove any strong case for more exotic forms of dark baryons being present.However,the situation is not so simple.The Andromeda Galaxy and our own galaxy are especially well-studied regions,where dark matter and baryons can be probed in detail.In the Milky Way Galaxy,the virial mass out to100 kpc is M virial≈1012M⊙,whereas the baryonic mass,mostly in stars,is M∗≈6−8×1010M⊙.The inferred baryon fraction is at most8%[8].Similar statements may be made for massive elliptical galaxies[9].These in fact are upper limits as the dark mass estimate is a lower bound.I infer that globally,there is no problem.Nevertheless the outstanding question is:where are the galactic baryons?Most of the baryons are globally accounted for.But this is not the case for our own galaxy and most likely for all comparable galaxies.We cannot account for a mass in baryons comparable to that in stars.It is possible that up to10%of all the baryons may be dark, and that the dark baryons are comparable in mass to the galactic stars.4The“missing”baryonsThere are several possibilities for the“missing”baryons.Perhaps they never were present in the protogalaxy.Or they are in the outer galaxy.Or,ﬁnally, they may have been ejected.Theﬁrst of these options seems very unlikely(although we return below to a variant on this).Consider the second option.The most likely candidates for dark baryons are massive baryonic objects or MACHOs.These are con-strained by several gravitational microlensing experiments.The allowed mass range is between10−8and10M⊙,and the best current limit on the MACHO abundance is<∼20%of the dark halo mass.In fact,one experiment,that of the MACHO Collaboration,claims a detection from some20events seen towards the LMC,most of which cannot be accounted for by star-star mi-crolensing.The observed range of ampliﬁcation time-scales speciﬁes the mass of the lensing objects.The preferred MACHO mass is around∼0.5M⊙.This mass favours an interpretation in terms of old halo white dwarfs. Main sequence stars in this mass range can be excluded.Current searches for halo high velocity old white dwarfs utilise the predicted colours and proper motions as a discriminant fromﬁeld dwarfs,and set a limit of<∼4%of the dark halo mass on a possible old white dwarf component in the halo[10].How-ever even if this limit were to apply,an extreme star formation history and protogalactic IMF would be required.Observations at high redshift both of star-forming galaxies and of the diﬀuse extragalactic light background,com-bined with chemical evolution and SNIa constraints,make such an hypothesis extremely implausible.If the empirical mass range constraint is relaxed,theory does not exclude either primordial brown dwarfs(0.01−0.1M⊙),primordial black holes(mass6Joseph Silk>∼10−16M⊙)or even cold dense H2clumps<∼1M⊙.The latter have been invoked in the Milky Way halo in order to account for extreme halo scattering events[11]or unidentiﬁed submillimetre sources[12].However these possibili-ties seem to be truly acts of the last resort in the absence of any more physical explanation.There is indeed another possibility that seems far less ad hoc.The nearby intergalactic medium is enriched to about10%of the solar metallicity,and contains of order50%of the baryons in photo-ionised and collisionally ionised phases.This strongly suggests that ejection from galaxies via early winds must have occurred,and moreover would inevitably have expelled a substantial fraction of the baryons along with the heavy elements.Supporting evidence comes from x-ray observations of nearby galaxy groups,which demonstrate that many of these are baryonically closed systems,containing their prescribed allotment of baryons.There are candidates for young galaxies undergoing extensive mass loss via winds.These are the Lyman break galaxies at z∼2−4.Observations of spectral line displacements of the interstellar gas relative to the stellar component as well as of line widths are indicative of early winds from L∗galaxies[13].Studies of nearby starburst galaxies,essentially lower luminosity counterparts of the distant LBGs,show that the gas outﬂow rate in winds is of order the star formation rate.The intracluster medium to z∼1is enriched to about a third of the solar metallicity,again suggestive of massive early winds, in this case from early-type galaxies.Hence the“missing”baryons could be in the IGM,with about as much mass ejected in baryons as in stars remaining.The ejection hypothesis however has to confront a theoretical diﬃculty. Winds from L∗galaxies cannot be reproduced by hydrodynamical simulations of forming galaxies[14].The momentum source for gas expulsion appeals to supernovae.SN feedback works for dwarf galaxies and can explain the observed outﬂows in these systems.However an alternative feedback source is needed for massive galaxies.This most likely is associated with AGN,and the ubiquitous presence of central supermassive black holes in galaxy spheroids.First,however,I address a more pressing and not unrelated problem, namely given that90percent of the matter in the universe is nonbaryonic and cold,how well does CDM fare in confronting galaxy formation models? 5Large-scale structure and cold dark matter:the issues The cold dark matter hypothesis has had some remarkable successes in con-fronting observations of the large-scale structure of the universe.These have stemmed from predictions,now veriﬁed,of the amplitude of the temperature ﬂuctuations in the cosmic microwave background that are directly associated with the seeds of structure formation.The initial conditions for gravitational instability to operate in the expanding universe were measured.The forma-tion of galaxies and galaxy clusters was explained,as was theﬁlamentary na-Galaxy Formation and Dark Matter7 ture of the large-scale structure of the galaxy distribution.Nor was only the amplitude conﬁrmed as a prerequisite for structure formation.The Harrison-Zeldovich-Peebles ansatz of an initially scale-invariantﬂuctuation spectrum, later motivated by inﬂationary cosmology,has now been conﬁrmed over scales from0.1to10000Mpc,via a combination of CMB,large-scale galaxy distri-bution and IGM measurements.Despite these stunning successes,diﬃculties remain in reconciling the-ory with observations.These centre on two aspects:the uncertainties in star formation physics that render any deﬁnitive predictions of observed galaxy properties unreliable,and the detailed nature of the dark matter distribution on small scales,where the simulations are also incomplete.The former issues include such observables as the galaxy luminosity func-tion,disk sizes and mass-to-light ratios,and the presence of old,red massive galaxies at high redshift.These diﬃculties in the confrontation of galaxy for-mation theory and observational data are plausibly resolved by improving the prescriptions for star formation and feedback,although there are as yet no deﬁnitive answers.The latter issues require high resolution dark matter sim-ulations combined with hydrodynamic simulations of the baryons including star formation and feedback.I will focusﬁrst on the dark matter conundrums,and in particular on the challenges posed by theoretical predictions of dark matter clumpiness,cuspi-ness and concentration.Implementation of numerical simulations of dark halos of galaxies in the context of hierarchical galaxy formation yields repeatable and reliable results at resolutions of up to∼105M⊙in M∗halos.It is clear that the simulations predict an order of magnitude or more dwarf galaxy halos than are observed as dwarf galaxies.It is more controversial but probably true that the dark halos of dwarf galaxies and of barred galaxies do not have the ∼r−1central cusps predicted by high resolution simulations.The dark matter concentration parameter,deﬁned by the ratio of r200,approximately the virial scale,to the scale length,within which the cusp proﬁle is found,measures the cosmological density at virialisation,and hence should be substantially lower for late-forming galaxy clusters than for galaxies.This may not be the case in the best-studied examples of massive gravitationally lensed clusters,cf.[15]. There are also examples of early-forming massive clusters[16]6Resurrection via astrophysicsThere are at least two viewpoints about resolving the dark matter issues, involving either fundamental physics or astrophysics.Tinkering with funda-mental physics,in essence,opens up a Pandora’s box of phenomenology.It seems to me that one shouldﬁrst take the more conservative approach of examining the impact of astrophysics on the dark matter distribution before advocating more fundamental changes.Of course if one could learn about fun-damental physics,such as a new theory of gravity or higher dimensional dark8Joseph Silkmatter relics from dark matter modelling,this would represent an unprece-dented and unique breakthrough.But the prospect of such revelations may be premature.Astrophysical resolution involves two complementary approaches.One in-corporates star and AGN feedback in the dense baryonic core that forms by gas dissipation.Massive gas outﬂows can eﬀectively weaken the dark matter gravity,at least in the central cusp.These may include stellar feedback driving massive winds via supernovae augmented by a top-heavy IMF and/or by hy-pernovae,or the impact of supermassive black hole-driven outﬂows.Another mechanism that shows some promise in terms of generating an isothermal dark matter core is dynamical feedback,via a central massive rotating gas bar.Such bars may form generically and dissolve rapidly,but their dynamical impact on the dark matter has not yet been fully evaluated[17,18,19].All of these are radical procedures,but some are more radical than others. To proceed,one has to better understand when and how galaxies formed. Fundamental questions in galaxy formation theory still remain unresolved. Why do massive galaxies assemble early?And how can their stars form rapidly, as inferred from theα/F e abundance ratios?Where are the baryons today? And if,as observations suggest,they are in the intergalactic medium,including both the photo-ionised Lymanαforest and the collisionally ionised warm-hot intergalactic medium(WHIM),how and when is the intergalactic medium (IGM)enriched to0.1of the solar value?Can the galaxy luminosity function be reconciled with the dark matter halo mass function?Does the predicted dark matter concentration allow a simultaneous explanation of both the Tully-Fisher relation,the fundamental plane and the galaxy luminosity function? And for that matter,is the dark matter distribution consistent with barred galaxy and low surface brightness dwarf galaxy rotation curves?The observational data that motivates many of these questions can be traced back to the colour constraints on the interpretation of galaxy spectral energy distributions by population synthesis modelling[20,21].The galaxy distribution is bimodal in colour,and this can be seen very clearly in studying galaxy clusters.The presence of a red envelope in distant clusters of galaxies testiﬁes to the early formation of massive ellipticals.A major recent break-through has been the realisation from UV observations with GALEX that many ellipticals,despite being red,have an ongoing trickle of star formation. Mostﬁeld galaxies and those on the outskirts of clusters are blue,and are actively forming stars.The general conclusion is that there must be two modes of global star formation:quiescent and starburst.The ineﬃcient,long-lived,disk mode is motivated by cold gas accretion and global disk instability.The low eﬃciency is due to negative feedback.The disk mode is relatively quiescent and contin-ues to form stars for a Hubble time.The violent starburst mode is necessarily eﬃcient as inferred from the[α/F e]clock.It is motivated by mergers,in-cluding observations and simulations,as well as by CDM theory.The highGalaxy Formation and Dark Matter9 eﬃciency is presumably due to positive feedback,but it is not clear how the feedback is provided.7What determines the mass of a galaxy?The luminosity function of galaxies describes the stellar mass function of galaxies.It is biased by star formation in the B(blue)band but is a good tracer in the near-infrared(K)band.It is sensitive to the halo mass,at least for spiral galaxies,as demonstrated by rotation curves.There is a charac-teristic luminosity,and hence a characteristic stellar mass,associated with galaxies:L∗≈3×1010L⊙and M∗≈1011M⊙.The luminosity function de-clines exponentially at L>L∗.This is most likely a manifestation of strong feedback.Considerﬁrst the mass-scale of a galaxy.There is no diﬀerence in dark matter properties between galaxy,group or cluster scales,but there is a very distinct diﬀerence in baryonic appearance.Speciﬁcally,the baryons are mostly in stars below a galaxy mass scale of M∗and mostly in hot gas for systems much more massive than M∗,such as galaxy groups[22]and clusters.A sim-ple explanation comes from considerations of gas cooling and star formation eﬃciency.It does not matter whether the gas infall initially is cold or whether it virialises during infall.The gas generically will be clumpy,and cloud colli-sions will be at the virial velocity.In order for the gas to form stars eﬃciently, a necessary condition is that the cooling time of the shocked gas be less than a dynamical time,or t cool<∼t dyn.The inferred upper limit on the stellar mass,for stars to form within a dynamical time in a halo of baryon fraction f b and mean densityρh,can be written asM∗=Aβm2βp G−(3+β)/2(t cool/t dyn)βf1−βb ρ(β−1)/2 h,where the cooling rate has been taken to beΛ=Av2−3/βs,withβ≈1being appropriate for metal-free cooling in the temperature range105−106K.This yields a characteristic mass M∗/m p≈0.1α3α−2g(m p/m e)(t cool/t dyn)≈1068, whereαg=Gm2p/e2.This is comparable to the stellar mass associated with the characteristic scale in the Schechterﬁt to the luminosity function,and also the scale at which galaxy scaling relations change slope.However there is no reason to believe that the dynamical time argument gives as sharp a feature as is observed in the decline of the galaxy luminosity function to high luminosities.Additional physics is needed.8Outﬂows from disksIn the quiescent mode,the clumpy nature of accretion suggests that ministar-bursts might occur.In fact,what is more pertinent is the runaway nature of10Joseph Silksupernova feedback in a cold gas-rich disk.Initially,exploding stars compress cold gas and stimulate more star formation.Negative feedback is eventually guaranteed in part as the cold gas supply is exhausted and also as the cold gas is ejected in plumes and fountains from the disk,subsequently to cool and fall back.Global simulations have inadequate dynamical range to follow the mul-tiphase interstellar medium,supernova heating and star formation.The fol-lowing toy model provides an analytical description of disk star formation.I assume that self-regulation applies to the hot gasﬁlling factor1−e−Q,where Q is the porosity and is deﬁned by(SN bubble rate)×(maximum bubble4-volume)∝(star formation rate)× turbulent pressure−1.4 .One can now write the star formation rate as[23]αS×rotation rate×gas densitywithαS≡Q×ǫ.Hereǫ=(σgas/σf)2.7,where theﬁducial velocity dispersion σf≈20kms−1 E SN/1051ergs 0.6(200M⊙/m SN)0.4.Here m SN is the mass in stars formed per supernova and E SN is the initial kinetic energy in the supernova explosion.The star formation eﬃciency Qǫis0.02 σgas400kms−1 m SN E SN .The observed mean value is0.017[24].Also,the analytic expression derived for the star formation rate agrees with that found in3-D multiphase simulations [25].In fact,the observed distribution of young stars in merging galaxies can-not beﬁt by modelling the star formation rate with a Schmidt-Kennicutt law, but requires the incorporation of a turbulence-like term[26],as incorporated in this simple model.To extract the wind,one might expect that the outﬂow rate equals the product of the star formation rate,the hot gas volumeﬁlling factor,and the mass loading factor(f L).This reduces to∼Q2ǫ˙M∗,or˙M outflow≈f Lα2Sǫ−1M gasΩ.If Q is of order50%,then the outﬂow rate is of order the star formation rate,but this evidently only is the case for dwarf galaxies.Once ǫ≫1,the wind is suppressed.This begs the question of how massive disks such as our own and M31 have depleted their initial baryon content by of order50percent.One cannot appeal to protospheroid outﬂows initiated by AGN(see below)to resolve this issue.Presumably baryon depletion in late-type massive disks(with small spheroids)must have occurred during the disk assembly phase.A collection of gas-rich dwarfs most likely assembled into a current epoch massive disk, and outﬂows from the dwarfs could plausibly have expelled of order half of the baryons into the Local Group or even beyond.However weak lensing studiesﬁnd that the typical late-type galaxy in a cluster environment appears to have utilised its full complement of baryons over a Hubble time[27],whereas an early-type galaxy may indeed have expelled about half of its baryons into the intracluster medium.9Outﬂows from protospheroidsGalaxy spheroids formed early.The inferred high eﬃciency of star formation on a short time-scale,as inferred from theα/F e enhancement,is suggestive of a feedback mechanism distinct from,and much more eﬃcient than,super-novae.The preferred context for such a mechanism is that of ultraluminous star-bursts.Major mergers between galaxies produce extreme gas concentrations that provide an environment for the formation of supermassive black holes. The observed correlation between SMBH mass and the spheroid velocity dis-persion suggests contemporaneous SMBH growth and coupled formation of the oldest galactic stars.The spheroid stars are old and formed when the galaxy formed.Hence the SMBHs,which account via the empirical correla-tion for approximately0.001of the spheroid mass,must have formed in the protogalaxy more or less contemporaneously with the spheroid.Supermassive black hole growth is certainly favoured in the gas-rich protogalactic environ-ment.Another clue is that both SMBHs,as viewed in AGN and quasars,and massive galaxy spheroids formed anti-hierarchically at a similar epoch,peak-ing at z∼2.Massive systems form before less massive systems.This could be a consequence of the same feedback mechanism,which necessarily must be positive in order to favour the massive systems.Supernova feedback is negative and is most eﬀective in low mass systems.SMBH outﬂows provide an intriguing possibility for positive feedback that merits further exploration. What is lacking for the moment is quantitative evidence for the frequency with which AGN activity is associated with ultraluminous infrared galaxies. Nevertheless,AGN feedback seems to provide the most promising direction for progress.A speciﬁc mechanism for positive feedback appeals to SMBH-induced out-ﬂows interacting with the clumpy protogalactic medium.Twin jets are accel-erated from the vicinity of the SMBH along the minor axis of the accretion disk.These jets are the fundamental power source for the high non-thermal lu-minosities and the huge turbulent velocities measured in the nuclear emission line regions in active galactic nuclei and quasars.The jets drive hot spots at a velocity of order0.1c that impact the protogalactic gas.In a cloudy medium, the jets are frustrated and generate turbulence.The jets are surrounded by hot cocoons that engulf and overpressure ambient protogalactic clouds[28]. These clouds collapse and form stars.The speed of the cocoon as it overtakes the ambient gas clouds greatly exceeds the local gravitational velocity.In this。

The Science of Dark Matter and Its Discovery

The Science of Dark Matter and ItsDiscoveryIntroductionDark matter is an elusive substance that makes up about 27% of the universe. It neither emits nor absorbs light, making it invisible to telescopes. Scientists have been studying dark matter for decades, and its discovery is considered one of the greatest mysteries of modern physics. In this article, we will delve into the science of dark matter, its properties, and the research that has led to its discovery.What is Dark Matter?Dark matter is a hypothetical substance that does not interact with light or any other form of electromagnetic radiation. It is invisible to telescopes, but its presence is inferred from its gravitational effects on objects that emit light. Dark matter is thought to be five times more abundant than visible matter, which is what stars, planets, and galaxies are made of.The Properties of Dark MatterAlthough scientists have yet to observe dark matter directly, they have been able to infer its properties from its gravitational effects. Dark matter is thought to be cold, meaning that its particles move relatively slowly. It is also believed to be non-interacting, meaning that it does not interact with other particles except through the force of gravity.Dark matter is widely thought to be made up of weakly interacting massive particles (WIMPs), which are particles that interact with each other only through the weak force and gravity. Other proposed candidates for dark matter particles include axions and sterile neutrinos, but these have not been observed directly.The Search for Dark MatterThe search for dark matter has been ongoing for several decades. One of the most promising methods for detecting dark matter involves looking for the energetic particles that result from the annihilation of dark matter particles. This method is called indirect detection and involves searching for gamma rays, neutrinos, or cosmic rays that are produced by the decay or annihilation of dark matter particles.Another way to detect dark matter is through the direct detection method, which involves looking for the recoil of atomic nuclei in a detector after they have been struck by dark matter particles. This method requires a sophisticated detector that can detect even the slightest signal. Several experiments are currently underway to detect dark matter particles using these methods.Discovery of Dark MatterThe discovery of dark matter can be traced back to the 1930s when Swiss astronomer Fritz Zwicky observed that the visible matter in the Coma cluster of galaxies was not enough to hold the cluster together. He hypothesized the presence of invisible matter that was holding the cluster together, which he called dark matter.Over the years, other scientists have provided evidence for the existence of dark matter. In the 1970s, Vera Rubin and Kent Ford studied the rotation curves of galaxies and found that the observed mass could not account for the observed rotation speeds. They concluded that there must be more mass in the form of dark matter that was holding the galaxies together.More recently, the European Space Agency’s Planck satellite produced a detailed map of the cosmic microwave background radiation, which is thought to be leftover radiation from the Big Bang. The map provided strong evidence for the existence of dark matter and its abundance in the universe.ConclusionThe discovery of dark matter is one of the most exciting and challenging areas of modern physics. Scientists continue to search for dark matter using a variety of methods, including indirect and direct detection. Although dark matter has yet to be observeddirectly, its presence and properties can be inferred from its gravitational effects on visible matter. As we continue to unravel the mysteries of dark matter, we are sure to gain new insights and a deeper understanding of the universe we inhabit.。

The Mass of Virialized Dark Matter Haloes

The structure of haloes in phase-space, already discussed in Busha et al. (2005), is as follows: In the inner parts the average radial velocities are zero. Around the virial radius there are some signs of infall or outﬂow, which strongly correlate with the mass scale (Prada et al. 2006). At large distances, the Hubble ﬂow is the most remarkable feature. This structure shows a substantial diﬀerence in dark matter haloes associated with diﬀerent masses (see left panel in Figure 1). Although the transition from the inner region to the Hubble ﬂow shows monotonically increasing radial velocity in the low-mass and the galacticsize halo, the situation is diﬀerent for clusters where there is a signiﬁcant tail in the velocity distribution with large negative velocities. This infall is altering the equilibrium of the particle distribution in the halo, and hence that region is no longer virialized.

Dogs that Don't Bark (The Tale of Baryonic Dark Matter in Galaxies)

a rXiv:as tr o-ph/21132v113Nov22February 2,20081:31WSPC/Trim Size:9in x 6in for Proceedings review5DOGS THAT DON’T BARK?(THE TALE OF BARYONIC DARK MATTER IN GALAXIES)N.W.EVANS Institute of Astronomy,Madingley Rd,Cambridge,CB21ST,England E-mail:nwe@ This article reviews the nature and distribution of baryonic dark matter in galax-ies,with a particular emphasis on the Milky Way.The microlensing experiments towards the Large Magellanic Clouds,the Andromeda Galaxy and the bulge pro-vide evidence on the characteristic mass and abundance of baryonic dark matter,as do direct searches for local counterparts of dark halo populations.1.Introduction Fifteen years or so ago,it was commonly argued;“If we want to believe the observations rather than our prejudice,we should take as our best bet that dark haloes are baryonic.”1Such a viewpoint is not often heard today.This change-of-mind has been enforced upon us largely by the microlensing experiments.Particle dark matter diﬀers from (most types of)baryonic dark matter in that it does not produce microlensing events.The familiarparade of baryonic candidates has now been whittled down,and perhaps only one remains as a possible substantial contributor to the dark matter in the Galaxy’s halo.This review assesses the distribution of missing matter in the Galaxy (Section 2),the likely baryonic dark matter suspects (Section3),the evidence from microlensing (Section 4)and from the halo white dwarf searches (Section 5).2.Missing Mass in the Galaxy2.1.The Inner PartsIt is now clear 2that there is little dark matter in the inner parts of big galaxies like the Milky Way.Here,the mass budget is dominated by the baryons in the luminous disk and the bulge.1February2,20081:31WSPC/Trim Size:9in x6in for Proceedings review5 2Figure1.Left:Likelihood contours for the total mass M of the Milky Way halo(inunits of1011M⊙)and the velocity anisotropyβ.Results including(solid curves)andexcluding(dotted curves)Leo I are shown.Contours are at heights of0.32,0.1,0.045and0.01of peak height and the most likely values are indicated by plus signs.Right:Likelihood contours for the total mass M of the M31halo(in units of1011M⊙)and thevelocity anisotropy radius r a.[From Wilkinson et al.2001]There are three strong pieces of evidence.First,models of the Milky Way in which the dark halo makes little contribution within the central∼5kpc are already strongly supported by simulations of the gasﬂow in theGalactic bar3.To reproduce the terminal velocities of the HI gas,the barand disk must provide almost all of the gravity forceﬁeld within the innerfew kpc.Second,bars in galaxy models having halos of moderate or highcentral density all experience strong drag from dynamical friction.The barin the Milky Way is able to maintain its observed high pattern speed only ifthe halo has a central density low enough for the disk to provide most of thecentral attraction in the inner Galaxy4.Third,for the Milky Way,thereare extremely high microlensing optical depths towards Baade’s Windowin the bulge.Almost all the matter in the inner parts of the Galaxy mustbe capable of causing microlensing(and hence probably baryonic)5.Thecentral∼5kpc of the Milky Way contain little particle dark matter.2.2.The Outer PartsThe total mass of the Milky Way galaxy is not known very well.This is be-cause the gas rotation curve cannot be traced beyond∼20kpc,leaving onlydistant globular clusters and satellite galaxies as tracers of the dark matterpotential.The dataset of positions and radial velocities(sometimes propermotions as well)of∼20satellite galaxies and distant globular clusters isFebruary2,20081:31WSPC/Trim Size:9in x6in for Proceedings review53 sparse.Thus,most investigators6have chosen to make strong assump-tions about the underlying halo model,using Bayesian likelihood methodsto estimate the total mass and the eccentricity of the orbits.Typical recentresults are shown in Figure1.The solid(dotted)contours in the left panelof Figure1show the likelihood including(excluding)one of the most dis-tant and troublesome of the satellite galaxies(Leo I)from the dataset.Themost likely total mass of the Milky Way galaxy is∼2×1012M⊙includingLeo I and9.1×1011M⊙excluding Leo I.For comparison,the right panel ofFigure1shows likelihood contours for the M31halo,using the dataset ofprojected positions and velocities of globular clusters and satellite galaxies.The most likely mass of M31’s halo is∼1×1012M⊙.Given the large un-certainties in the estimates,a reasonable conclusion is that both the MilkyWay galaxy and M31have equally massive dark haloes.The total mass indark matter is about ten times greater than the total mass in stars.Theouter parts of both the Milky Way galaxy and M31are overwhelminglydominated by dark matter.2.3.The Solar NeighbourhoodFor all direct detection experiments,the crucial question is:how much darkmatter is there in the solar neighbourhood?By analyzing the line-of-sightvelocities and distances of K dwarf stars seen towards the south Galacticpole,Kuijken&Gilmore7showed that at the solar radius there is∼71±6M⊙pc−2of material within1.1kpc of the Galactic plane.Measurementsof the proper motions and parallaxes of stars that lie within200pc of theSun have yielded estimates of the local density of all matter8.For example,Cr´e z´e et al.found(76±15)mM⊙pc−2;Pham found(111±10)mM⊙pc−2;Holmberg&Flynn found(102±6)mM⊙pc−2.By counting disk M dwarfs in Hubble Space Telescopeﬁelds,the vertical proﬁle of these objects is known to be well modelled by9ν(z)=0.435sech2(z/270pc)+0.565exp(−|z|/440pc).(1) The eﬀective thickness of the disk’s stellar mass is51ˆz≡February2,20081:31WSPC/Trim Size:9in x6in for Proceedings review54have that stars contribute26.9M⊙pc−2to the71±6M⊙pc−2of matterthat lies within1.1kpc of the plane.Gas(primarily hydrogen and helium)contributes13.7M⊙pc−2.Thus,∼41M⊙pc−2of the mass within1.1kpcof the plane can be accounted for by stars and gas,and the remaining∼30M⊙pc−2should be contributed by dark matter5.The overall erroron this last number could easily be as large as15M⊙pc−2each way.3.The Usual SuspectsThis Section lines up the baryonic dark matter suspects,which could makeup some of the copious amounts of missing matter in the Galaxy.3.1.Red and Brown DwarfsRed dwarfs(M dwarfs)have masses between0.5M⊙and0.08M⊙.Theyshine due to hydrogen burning in their cores.Judging from local samples,red dwarfs are about4times more common than all other stars combined.About80%of all the stars in the solar neighbourhood are red dwarfs11.The local number density of red dwarfs12as reckoned from surveys suchas the8-parsec sample is∼0.07per cubic pc.Brown dwarfs are objects lighter than∼0.08M⊙.They are too light to ignite hydrogen.They are brightest when born and then continuouslycool and dim.Since1997,near-infrared surveys(DENIS and2MASS)havebeen steadfastly uncovering brown dwarfs13.There are over∼100goodcandidates now(as well as two new spectral classes,L and T dwarfs).Thelocal number density of brown dwarfs14is very uncertain but it may be ashigh as0.1per cubic pc.In which case,the total number of brown dwarfsmay exceed the total number of stars in the Galaxy.Both red and brown dwarfs are seemingly very common in the Galactic disk(and probably the bulge and spheroid too).But,it is now clear thatmost of the missing mass in the Galactic halo cannot be ascribed to eitherred or brown dwarfs.Red dwarfs are ruled out because they are not seen in suﬃcient abun-dance in long exposures of high Galactic latitudeﬁelds using the HubbleSpace Telescope Wide Field Camera.More speciﬁcally,less than1%of themass of the halo can be in the form of red dwarfs15.Brown dwarfs areruled out because they produce microlensing events towards the MagellanicClouds with typical timescales∼15days.This is much shorter than thetimescales of the observed events,which are∼40days.Let us recollectthat the only parameter in a microlensing event providing any physical in-February2,20081:31WSPC/Trim Size:9in x6in for Proceedings review55 formation is the timescale.This encodes the mass with the velocities anddistances of both the source star and the microlens.Hence,the masses ofthe microlenses cannot be deduced on an event by event basis,but typicalmasses can be deduced using models of the Galactic halo.To minimisethe mass,the transverse velocity of the microlens with respect to the lineof sight must be reduced.In the outer halo,radial anisotropy is best fordoing this;closer to the Solar circle,tangential anisotropy is best.Byusing a constraint on the total kinetic energy of the lensing population,the microlens mass can be minimised over all orientations of the velocitydispersion tensor.This minimum mass is∼>0.1M⊙,which lies above thehydrogen-burning limit16.So,the microlenses cannot be brown dwarfs.3.2.White and Beige DwarfsWhite dwarfs are objects with mass∼0.5M⊙,the remnants of stars withmasses in the range1-8M⊙.The local number density of white dwarfs17is0.005per cubic pc.This is reckoned from samples believed complete to13pc.If so,then white dwarfs are about100times rarer than red dwarfsand brown dwarfs.For many years,white dwarfs were regarded as very improbable can-didates for the dark matter in galactic haloes.The main problem is thatthe progenitor stars are likeﬁlthy furnaces,disgorging metals into the ISM.Carbon,nitrogen,helium and deuterium are seriously overproduced,asjudged by the present abundances of stars in the Galactic halo18.Evenif all the ejecta of a population of white dwarfs are removed by Galacticwinds,the mass budget is enormous,exceeding that of the entire LocalGroup.It needs a contrived IMF so as to avoid leaving large numbers ofvisible main sequence precursors still burning today in the halo.Theseproblems19remain largely unsolved.But,the microlensing results(withtheir preferred typical mass of the microlenses of∼0.5M⊙)have sparked alot of activity in the area of white dwarf searches–without success so far.Beige dwarfs have masses up to∼0.2M⊙.These objects are supposed to form by slow accretion of gas onto planets or brown dwarfs.Providedthe accretion energy is radiated away,the temperature in the core neverrises high enough to ignite hydrogen20.As beige dwarfs are envisagedas primordial objects rather than the end-points of stellar evolution,thisingeniously circumvents the problem of pollution by metals.Unhappily,the most recent calculations suggest that the accretion rate needs to be∼0.1M⊙Gyr−1–too slow to allow their manufacture in this Universe.February2,20081:31WSPC/Trim Size:9in x6in for Proceedings review563.3.Neutron Stars and Black HolesNeutron stars are the remnants of stars with initial masses in the range8-20M⊙,while black holes are the remnants of stars larger than20M⊙.However,neutron stars and black hole remnants less than∼105M⊙cannotmake up the bulk of the dark matter as their precursors generate unaccept-able metal production or background light21.Stars larger than∼105M⊙collapse directly to black holes without excessive nucleosynthetic or background light production.They cause mi-crolensing events with timescales∼>50yr,which are too long to be de-tectable by current surveys.There are some noticeable dynamical eﬀects.For example,stellar encounters with such black holes produce a power-lawtail in the energy distribution.Accordingly,Lacey&Ostriker’s original pa-per22correctly predicted the existence of the(then unknown)thick disk.Hence,supermassive black holes remain genuine suspects.4.The Evidence from MicrolensingMicrolensing towards the Large Magellanic Cloud and the Andromedagalaxy provides direct evidence on the fraction of dark matter in haloesthat is baryonic.Microlensing towards the bulge provide indirect evidenceon the structure of the Galactic dark halo.4.1.Microlensing towards the Magellanic CloudsThe original motivation23of the microlensing experiments was to detect theeﬀects of baryonic dark objects in the Galactic halo on background starsin the nearby satellite galaxies,the Large and Small Magellanic Cloud.From5.7years of data,the MACHO collaboration24found between13to17microlensing events towards the Large Magellanic Cloud(LMC)andreckoned that the optical depth(or probability of microlensing)wasτ∼×10−7.They argued that,interpreted as a dark halo population,the1.2+0.4−0.3most likely mass of the microlenses is between0.15and0.9M⊙,seeminglyimplicating white dwarfs.The total mass in the objects out to50kpc is∼9+4−3×1010M⊙.This is∼<20%of the mass of the halo.In stark contrast,after8years of monitoring the Magellanic Clouds,the EROS collaboration25secured just a“meagre crop of three microlensing candidates towards theLMC”.EROS monitor a wider solid angle of less crowdedﬁelds in theLMC.So,blending and contamination by lenses in the LMC itself(so-called“self-lensing”)are much less important.EROS do not report theirFebruary2,20081:31WSPC/Trim Size:9in x6in for Proceedings review57 results in terms of optical depth,but their experiment seemingly implies alower value than that preferred by MACHO.One possibility is that the lenses lie in or close to the Large Magellanic Cloud or in some intervening population,rather than being true denizensof the dark halo.A number of ingenious suggestions26have been made–the LMC disk,tidal debris,the warped Milky Way disk,an interveningsatellite galaxy.For one reason or another,none of these ideas have gaineda concensus,although some contain ingredients of merit.What we do knowfor sure is that some of the lenses do not lie in the dark halo.There are nowfour exotic events27(two binary caustic crossing events,one long timescaleevent with no detectable parallax,one xallarap event)for which the locationof the event can be more or less inferred.In all four cases,the lens almostcertainly lies in the Magellanic Clouds.Most recently of all,there has beenthe direct imaging of one the microlenses by Alcock et al.28,revealing itto be a nearby low-mass star in the disk of the Milky Way.Atﬁrst glance,all this seems strong evidence that most of the lenses do not lie in the darkhalo;however,there are biases in exotic events that favour the discoveryof events in which the lens and source are close together.In other words,there is no compelling evidence either for or against a Galactic halo originof the microlenses.They may equally well lie in the dark halo or they maylie in the Magellanic Clouds or in intervening populations.Aﬁnal possibility that deserves serious consideration is that some mi-crolensing events may have been misidentiﬁed.For example,there areexpected to be∼20supernovae in background galaxies behind the LMCand brighter than MACHO’s limiting magnitude during the experiment’slifetime;this number may be larger by at least a factor of two,depending onthe supernova contribution from faint galaxies.So,supernova contamina-tion is a serious problem24.The bumps do no repeat and rise up from aﬂatbaseline.They diﬀer from microlensing curves in that they are asymmetric,but such asymmetry may not be obvious in noisy or sparsely sampled data.In fact,MACHO’s data are taken at a site where the median seeing is∼2.1arcsec so the quality of the data is sometimes poor.An interesting recentbreakthrough by Belokurov and co-workers29has been the development ofneural networks to identify microlensing events in massive variability sur-veys.This replaces judgements made by human experts with judgementsbased on strict statistical criteria.This technique has thus far been appliedonly towards the Galactic bulge,but it already hints that some events mayhave been misclassiﬁed.An analysis of the events towards the LMC can beexpected from this group soon.February2,20081:31WSPC/Trim Size:9in x6in for Proceedings review58Table1.Parameters for the4POINT-AGAPE candidates.Here,∆Ris the magnitude(Johnson/Cousins)of the maximum sourceﬂux varia-tion,t E is the Einstein timescale,t1/2is the full-width half-maximumand A max is the maximum ampliﬁcation.All these events have veryhigh ampliﬁcation and short full-width half-maximum timescale.[FromPaulin-Henriksson et al.2002].reference t1/2(days)A maxPA-99-N1 1.917.54+1.33−1.15PA-99-N225.013.33+0.75−0.67PA-00-S3 2.318.88+8.15−5.89PA-00-S4 2.1211+16456−120Figure2.The location of4microlensing events detected by POINT-AGAPE towardsM31.Also marked are the twoﬁelds that straddle the north and south of M31.[FromPaulin-Henriksson et al.2002]4.2.Microlensing towards M31Microlensing experiments towards M31have the potential to clarify theambiguous results towards the LMC.This is because M31is highly inclined(i∼77◦).Lines of sight to disk stars in the north or near side of M31areshorter than those to the south or far side.Microlensing by a spheroidaldark halo will have a characteristic signature with an excess of events on thefar side of the M31disk30.This signal is absent when the microlenses liein the stellar disk or bulge of M31.A number of groups31are now carryingout large-scale surveys of M31to look for this near-far disk asymmetry.InFebruary2,20081:31WSPC/Trim Size:9in x6in for Proceedings review59 M31,the individual stars are not resolved,so that theﬂux on the detectorelements(pixels or superpixels)is monitored.Novel techniques have beendeveloped to monitorﬂux changes of unresolved stars in the face of seeingvariations32.Recently,the POINT-AGAPE collaboration33,34has reported results from two years of data taken with the Wide Field Camera on the2.5mIsaac Newton Telescope.Theﬁelds are shown in Figure2;they are∼0.3deg2and located north and south of the centre of M31.Two years of dataare not yet suﬃcient to look for any gradient signal,but they are enoughto identify a sample of some convincing high signal-to-noise candidates.Table1lists the characteristics of four such events,designated PA-99-N1,PA-99-N2,PA-00-S1and PA-00-S4.Here,N(or S)tells us whether theevent occurs in the northern or southernﬁeld,while99or(00)tells uswhether the event peaks in1999(or2000).The events are selected on thebasis of a set of severe selection criteria.There are∼350candidates witha single,substantial,symmetric bump which is a goodﬁt to the standardPaczy´n ski form.Many of these are variable stars and a longer baselineis needed to provide discrimination.Accordingly,Paulin-Henriksson et al.insist upon a short full-width half-maximum timescale(t1/2<25days)andaﬂux variation exceeding theﬂux of a21st magnitude star(∆R<21).Therationale for this is that microlensing is the only astrophysical process thatcan cause such hugeﬂuctuations on such very short timescales.This cutleaves eight microlensing candidates35,of which the four listed in Table1are the most convincing.These early results are tantalizing.Microlensing events in the inner few arcminutes are overwhelmingly due to stellar lenses in the bulge36.Hence,PA-99-N1and PA-00-S3are likely to be caused by low mass stars in thebulge.PA-00-S4lies about22′from the centre of M31,but it is only3′fromthe centre of the foreground dwarf elliptical galaxy M32.Although the lenscould be a dark object in M31’s halo,the closeness to M32suggests thata stellar lens in M32is more probable33.The fourth event PA-99-N2liesfar out in the M31disk and there are no obvious concentrations of stellarlenses along the line of sight.Atﬁrst glance,this looks a good candidatefor a lens in the dark halo of either of our Galaxy or M31.However,theself-lensing optical depth(that is,the probability that an M31disk star islensed by another M31disk star)is34τdisk≈4πGΣdisk h sec2i100M⊙pc−2hFebruary2,20081:31WSPC/Trim Size:9in x6in for Proceedings review510whereΣdisk is the disk column density,h is its exponential scale height andwe have normalised the formula to likely values.In fact,τdisk is comparableto the halo optical depth at this location for a20%baryon fraction.How-ever,the disk self-lensing hypothesis makes a reasonably model-independentprediction about the event timescales,namelyt E ≈116Σdisk≈100days M100M⊙pc−2 −1/2(4)which is in good agreement with the t E of∼90days for PA-99-N2.These arguments are teasingly suggestive rather than conclusive.Cer-tainly,the notion that dark halo lenses are responsible for all of these events is not especially favoured,although it is cannot be rejected right now33.In the cases where halo lenses may be responsible(PA-99-N2,PA-00-S4and perhaps PA-99-N1),stellar lensing is equally likely.Had one or several of these events been projected against the far side of the M31disk,well away from the M31bulge and from M32,then halo lensing would have been the likely culprit.4.3.Microlensing towards the BulgeLines of sight towards the bulge do not probe the halo dark matter directly, as almost all the lensing events are probably caused by low mass stars in the disk and the bulge.Rather surprisingly,however,we do learn something concerning the structure of the Galaxy’s dark halo from these experiments.Table2shows the measurements of the optical depth to microlensing to the red clumps stars in the bulge37.Red clump stars are bright stars that are known to reside in the bulge.The experiments have remained very consistent with an optical depth ofτ∼>3.0×10−6.Figure3shows contours of optical depth in three barred models of the inner Galaxy38. All three models have been derived from the same dataset,namely the infrared surface photometry measured by the DIRBE instrument on the COBE satellite,but made diﬀerent corrections for absorption and emission by dust.Models such as those based on constant mass-to-light deprojections of the infrared photometry39are not able to reproduce these high optical depths.Freudenreich’s model40does come closer(as Figure3shows), although it too has some diﬃculties with the highest values,such as the most recent results from Sumi et al.(the MOA collaboration).A crucial diﬀerence between baryonic and particle matter is that the former can cause microlensing events,while the latter cannot.To get these high optical depths,almost all the matter permitted by the rotation curveFebruary2,20081:31WSPC/Trim Size:9in x6in for Proceedings review511 Table2.The microlensing optical depth recorded by various ex-perimental groups towards locations in the Galactic bulge.Collaboration Optical DepthUdalski et al.(1994)∼3.3×10−6Alcock et al.(1995)∼3.25×10−6Alcock et al.(1997)3.9+1.8×10−6−1.2×10−6Alcock et al.(2000)3.23+0.52−0.50Popowski et al.(2000)2.0±0.4×10−6×10−6Sumi et al.(2002)3.40+0.94−0.73Figure3.Contours of microlensing optical depth to the red clump giants(in unitsof10−6)in the three Galaxy models,excluding(full lines)and including(dotted lines)spirality.The optical depths reported by Alcock et al.(2000)and Popowski et al.(2000)are shown in boxes.Light(or dark)gray boxes correspond to EROS(or OGLE)ﬁelds.[From Evans&Belokurov2002]must be baryonic within the inner∼5kpc.Figure4shows aﬁt to thetangent-velocity data(short dashed line)originally derived by Binney etal.41.Any model must lie below this curve.The dotted curve shows thecontribution to the rotation curve from a bulge and disk judged to reproducean optical depth of2×10−6towards Baade’s Window(itself a conservativevalue,lower than most of the measurements in Table2).The long-dashedcurve shows the contribution of the dark halo to the rotation curve usingthe local column densities of dark matter derived in Section2.3,assumingthe cusped Navarro-Frenk-White(NFW)model42currently favoured bycosmological simulations.The total rotation curve always lies above thedata.The high optical depths to microlensing seen towards the bulge areenough to rule out cusped dark halo models like Navarro-Frenk-White forthe Milky Way5.February 2,20081:31WSPC/Trim Size:9in x 6in for Proceedings review51202468050100150200250R (in kpc)NFW 024********150200250R (in kpc)NFWFigure 4.The panels show the circular-speed curves generated by the gas disk together with enough stars to yield τ=2×10−6(dotted curves)and by NFW haloes (long-dashed curves).The combined rotation curve is shown as a solid line and must lie below the ﬁt to the tangent-velocity data (short-dashed line).The panels diﬀer in the local dark matter column density [From Binney &Evans 2001].5.The Evidence from White Dwarf SurveysThere have been a number of searches for local examples of high velocity,very cool white dwarfs that might be representatives of the halo population causing the microlensing towards the LMC.First,Ibata et al.43claimed the detection of 5faint objects with sig-niﬁcant proper motion in the Hubble Deep Field.They argued that the observations were consistent with old white dwarfs with hydrogen atmo-spheres.They claimed that this provided a local mass density which,if extrapolated,was suﬃcient to account for the microlensing results.Strictly speaking,Ibata et al.found 5faint objects whose light centroids shifted between the ﬁrst and second epoch of exposures (separated by 2years).For point sources,such centroid shifts might be indicative of proper motions;however,they also can arise easily enough for extended or variable sources.Richer 44withdrew the results after an analysis of the third epoch data,taken 5years after the original Hubble Deep Field.None of the 5objects possessed a statistically signiﬁcant proper motion.Whatever the objects are,they are not moving and so certainly not high velocity white dwarfs.Second,Oppenheimer et al.45also claimed “direct detection of galactichalo dark matter”.They identiﬁed candidates with sub-luminosity andFebruary2,20081:31WSPC/Trim Size:9in x6in for Proceedings review513 with high intrinsic proper motions from SuperCOSMOS Sky Survey Plates.They followed this up with spectroscopy to discover38cool white dwarfs.Oppenheimer et al.derive U and V velocities for each target by settingW=0,where(U,V,W)are the components towards the Galactic Centre,in the direction of Galactic rotation and perpendicular to the Galactic planerespectively.Systems with[U2+(V+35)2]1/2>94kms−1are identiﬁed asmembers of the halo.They derive a halo white dwarf density of∼>2.2×10−4stars per cubic pc.This is a factor of10higher than the expected density ofwhite dwarfs in the stellar halo.However,this result has been contested byReid et al.46,who question the validity of the kinematic discriminant.Forexample,there is a signiﬁcant excess of prograde rotators in Oppenheimeret al.’s sample:34out of38.This is exactly the behaviour expected ifa substantial number of the white dwarfs are drawn from the thick diskrather than the halo.Most true halo populations are only weakly rotating.Despite false alarms,such halo white dwarf surveys are clearly worth pursuing.The discovery of a local counterpart to the putative microlensingpopulation would be a substantial breakthrough.6.ConclusionsEvidence from dynamics and particularly microlensing has made manybaryonic dark matter candidates unlikely as components of galaxy haloes.The constraints on stellar baryonic dark matter are especially harsh,withbrown,red,beige and white dwarfs ruled out as dominant contributors.Only at the very high mass end(supermassive black holes)do possibilitiesremain for building the Galactic halo entirely from dark baryonic objects.It is curious that none of the microlensing events towards the Magel-lanic Clouds or Andromeda can be ascribed to lenses in the dark halo withsurety.Some of the events have been almost certainly identiﬁed with stellarpopulations.This includes the exotic lenses towards the Magellanic Cloudsand some of the events towards M31.This need not have been the case.Unambiguous halo candidates could have been found–for example,binarycaustic crossing events implicating a halo lens in the experiments towardsthe Magellanic Clouds or short timescale events far out in the M31disk.Similarly,local searches could have identiﬁed a convincing counterpart toany halo baryonic dark matter population–but did not!The dogs could have barked three times in the night47(during the MACHO experiment,in the POINT-AGAPE datasets,in the white dwarfsearches).Each time,the dogs stayed silent.。

黑暗物质高中英语

黑暗物质高中英语Dark Matter: An Enigma in the CosmosThe universe is a vast and mysterious expanse, filled with countless celestial bodies, each with its own unique story to tell. Among the many wonders that captivate the scientific community, one stands out as a true enigma: dark matter. This elusive and perplexing phenomenon has been the subject of intense research and speculation for decades, as scientists strive to unravel its secrets and understand its role in shaping the very fabric of the universe.At its core, dark matter is a hypothetical form of matter that cannot be directly observed, yet its existence is inferred from its gravitational effects on the visible matter and the structure of the universe. Unlike the familiar matter that makes up the stars, planets, and the very world we inhabit, dark matter does not interact with electromagnetic radiation, rendering it invisible to our telescopes and other instruments. It is this very property that has made it so challenging to study and understand.The first inklings of dark matter's existence can be traced back to the early 20th century, when Swiss astronomer Fritz Zwicky observed themotion of galaxies within the Coma Cluster. He noticed that the galaxies were moving at much higher speeds than expected, suggesting the presence of a significant amount of unseen mass holding the cluster together. This observation laid the foundation for the concept of dark matter, though it would take decades before the scientific community fully embraced the idea.As our understanding of the universe has evolved, the evidence for dark matter has become increasingly compelling. Observations of the cosmic microwave background radiation, the leftover glow from the Big Bang, have revealed intricate patterns that can only be explained by the presence of a substantial amount of dark matter. Additionally, gravitational lensing, the bending of light by massive objects, has provided further confirmation of dark matter's existence, as the observed lensing effects are far greater than can be accounted for by visible matter alone.Despite the overwhelming evidence, the nature of dark matter remains elusive. Scientists have proposed numerous theories to explain its composition, ranging from exotic subatomic particles to modifications of our understanding of gravity. The search for dark matter particles, such as weakly interacting massive particles (WIMPs) and axions, has become a major focus of particle physics research, with scientists around the world employing sophisticated detectors and experiments in an attempt to directly observe these elusiveentities.One of the most intriguing aspects of dark matter is its role in the formation and evolution of the universe. Cosmological models suggest that dark matter was a crucial component in the early stages of the universe, providing the gravitational scaffolding upon which the first structures, such as galaxies and galaxy clusters, were able to form. Without the stabilizing influence of dark matter, the universe as we know it may have never come into being.As researchers continue to delve deeper into the mysteries of dark matter, they are also confronted with the broader implications of its existence. The discovery of dark matter has challenged our understanding of the fundamental laws of physics, forcing us to re-examine our theories and potentially pave the way for revolutionary new insights. The search for dark matter has also sparked the imagination of the public, captivating the minds of scientists and science enthusiasts alike, as they collectively strive to unravel one of the greatest unsolved puzzles of our time.In the pursuit of understanding dark matter, scientists have employed a wide range of cutting-edge technologies and techniques. From the construction of massive underground detectors designedto capture the faint signals of dark matter particles, to the development of sophisticated computer simulations that model thebehavior of dark matter on cosmic scales, the quest to unlock the secrets of this elusive substance has driven the field of astrophysics and cosmology to new frontiers.As the search for dark matter continues, the scientific community remains hopeful that a breakthrough is on the horizon. The potential rewards of such a discovery are immense, as it could not only shed light on the nature of the universe but also lead to a deeper understanding of the fundamental forces that govern our reality. Whether the answer lies in the detection of exotic particles, the refinement of our theories of gravity, or some entirely unexpected revelation, the pursuit of dark matter remains one of the most captivating and high-stakes endeavors in the realm of scientific exploration.In the end, the mystery of dark matter serves as a poignant reminder of the limitless boundaries of human knowledge and the boundless potential of the universe to surprise and inspire us. As we continue to push the boundaries of our understanding, we are reminded that the greatest discoveries often lie in the shadows, waiting to be uncovered by the curious and the courageous. The pursuit of dark matter, therefore, is not just a quest for scientific knowledge, but a testament to the human spirit's unwavering determination to unravel the deepest secrets of the cosmos.。

The structure of matter and its properties

The structure of matter and itspropertiesMatter is what makes up everything in the universe, from the smallest particles to the largest stars. It is the physical substance that occupies space and has mass. Matter is made up of atoms, which are the building blocks of elements. Atoms, in turn, are made up of subatomic particles such as protons, neutrons, and electrons.The structure of matter is an important concept in physics and chemistry. Understanding the structure of matter helps us understand its properties and behavior. The structure of matter can be studied at different levels, ranging from subatomic particles to the macroscopic scale.At the subatomic level, matter is composed of particles such as quarks, leptons, and bosons. These particles interact with one another through fundamental forces such as the strong force, weak force, electromagnetic force, and gravitational force. The strong force holds quarks together to form protons and neutrons, while the weak force is responsible for the decay of subatomic particles.Moving up to the atomic scale, matter is composed of atoms, which are made up of a nucleus (containing protons and neutrons) surrounded by electrons. The number of protons in the nucleus determines the element to which the atom belongs. Atoms can also have different isotopes, which have the same number of protons but different numbers of neutrons in the nucleus.The properties of matter depend on its structure. For example, different elements have different chemical and physical properties because of the differences in their atomic structure. The way in which atoms are arranged in a material also has an impact on its properties. For example, diamond and graphite are both made up of carbon atoms, but they have different structures and properties. Diamond is hard and transparent, while graphite is soft and opaque.The properties of matter can be classified as physical or chemical. Physical properties include characteristics such as density, melting point, boiling point, and conductivity. Chemical properties, on the other hand, refer to the behavior of matter in the presence of other substances, such as its ability to react with other chemicals.The behavior of matter can also be described in terms of its states. Matter can exist as a solid, liquid, or gas, depending on the temperature and pressure. These states are characterized by differences in the arrangement and motion of atoms or molecules. For example, in a solid, the atoms or molecules are tightly packed and vibrate in place, while in a gas, they are widely spaced and move freely.The nature of matter is also related to its energy content. Matter can have different forms of energy, including kinetic energy (energy of motion), potential energy (energy of position), and thermal energy (energy related to temperature). These different forms of energy can be exchanged between matter and its surroundings, leading to changes in the matter's properties.In conclusion, the structure of matter and its properties are closely intertwined. Understanding the structure of matter allows us to predict and explain its behavior in different circumstances. Matter's properties depend on its structure, energy content, and the environment in which it is located. This knowledge is fundamental to many areas of science, including physics, chemistry, and materials science.。

Dark Matter Halo Structure in CDM Hydrodynamical Simulations

arXiv:as tr o-ph/981116v113Jan1998Mon.Not.R.Astron.Soc.000,000–000(0000)Printed 1February 2008(MN L A T E X style ﬁle v1.4)Dark Matter Halo Structure in CDM Hydrodynamical SimulationsP.B.Tissera12,R.Dom´ınguez-Tenreiro 11PostalAddress:Departamento de F´ısica Te´o rica C-XI,Universidad Aut´o noma de Madrid,Cantoblanco 28049,Madrid,Spain 2Imperial College of Science,Technology and Medicine,Blackett Laboratory,Prince Consort Road,London,United Kingdom E-mail:rosa@astrohp.ft.uam.es1February 2008ABSTRACT We have carried out a comparative analysis of the properties of dark mat-ter halos in N-body and hydrodynamical simulations.We analyze their density proﬁles,shapes and kinematical properties with the aim of assessing the eﬀects that hydrodynamical processes might produce on the evolution of the dark matter component.The simulations performed allow us to reproduce dark matter halos with high resolution,although the range of circular velocities is limited.We ﬁnd that for halos with circular velocities of [150−200]km s −1at the virial radius,the presence of baryons aﬀects the evolution of the dark mat-ter component in the central region modifying the density proﬁles,shapes and velocity dispersions.We also analyze the rotation velocity curves of disk-like structures and compare them with observational results.Key words:Galaxies:evolution –Galaxies:formation –Hydrodynamics –Methods:numericalc 0000RAS21INTRODUCTIONThe importance of understanding the formation and evolution of dark matter halos is based upon the hypothesis that they contain relevant information about cosmological parameters and the power spectrum of initial densityﬂuctuations.Theoretical works based mainly on the second infall model,ﬁrst studied by Gunn&Gott(1972),have predicted diﬀerent behavior for the density proﬁles of dark matter halos.These models simpliﬁed the problem of the collapse of the structure by assuming spherical symmetry and purely radial motions. Fillmore&Goldreich(1984)and Bertschinger(1985)found self-similarity solutions for the secondary infall collapse.These models predict the formation of virialized objects with dark matter density proﬁles that are approximately isothermal.Moutarde et al.(1995)show that this result is not restricted to the fully spherically symmetric case.Hoﬀman&Shaham(1985) and Hoﬀman(1988)extended the analysis to models with diﬀerent density parameters(Ω) and power spectra(P(k)∝k n,−3<n<4)concluding that density proﬁles of dark matter halos should steepen for larger values of n and lowerΩ.However,because of the oversimpliﬁed hypothesis assumed,these models fail to describe consistently the collapse and evolution of the structure in a hierarchical clustering scenario where the structure is formed through the collapse and aggregation of substructure.Numerical simulations are well suited to follow the hierarchical built-up of halos througthout their linear and non-linear regime(Frenk et al. 1985,1988,Quinn et al.1986,Efstathiou et al.1988,Zurek et al.1988,Dubinski&Carlberg 1991,Warren et al.1992).During the last years there have been numerous works on the properties of dark matter halos and their implications for explaining some observational facts such as the rotation velocity curves of disk galaxies,cores of dwarf galaxies,gravitational lenses of background galaxies by galaxy clusters,etc.These techniques have resulted in a very useful tool limited mainly by numerical resolution.Several works have been carried out recently which have taken advantage of the last improvements in computational facilites and numerical techniques in order to perform high resolution numerical simulations.In particular,several works have been dedicated to the study of dark halo density proﬁles in N-body numerical simulations where only the gravitationalﬁeld is integrated.The most recent works by Navarro et al.(1995,hereafter NFW,1996a,b),Cole&Lacey(1997), Tormen et al.(1997),have shown that halos in purely dynamical simulations in scale-free and Cold Dark Matter(CDM)universes in low-density,ﬂat and open cosmogonies,can be ﬁtted by the following curve:x−αρ(r)∝Properties of Galactic Objects3 matter density proﬁles are always well-described by equation(1)and that only the values of its parameters show some dependence with the spectrum or cosmology.In particular,it has been asserted in their works that the only primordial information that dark matter proﬁles remember is their time of collapse.So far all recent works have dealt with dark matter halos in purely gravitational scenarios, describing its evolution and properties as a function of mass and initial conditions.However, halos contain a fraction of baryonic matter that,although in a smaller proportion,may inﬂuence the evolution of the dark matter.There have been several attempts to address the eﬀects of dissipation on the evolution of the dark matter component in galactic halos using diﬀerent approaches to model it(Blumenthal et al.1986,Flores et al.1993,Flores& Primack1994,Dubinski1994).Blumenthal et al.(1986)used analytic models and dissipative particle collisions in N-body simulations to study the response of the dark matter halo to the presence of baryons,with the aim at understanding the origin of rotation curves of spiral galaxies.Subsequent works(Flores et al.1993,Flores&Primack1994)used their results to deep on the analysis of dark matter structure in halos.Dubinski(1994)analyzed the eﬀects of dissipation on the dark matter halo by introducing an analytical potential well representing disk-like and spheroidal structures in its central region.This author found that when the baryonic component is taken into account,the overall potential well is modiﬁed in the central region leading to a change in the density proﬁles and shapes,in agreement with the result of Blumenthal et al.(1986).Evrard et al.(1994)analyzed the formation of galaxies in cosmological hydrodynamical simulations,ﬁnding that halos are more spherical than in purely dynamical ones.Regarding clusters of galaxies,there have been several numerical works on the analysis of their density proﬁles.Observational estimates of gas and dark matter density proﬁles in X-ray clusters show controversial results since gas density proﬁles seem to be shallower (ρ∝r−1)than those of the dark matter component,deduced by gravitational lensing tech-niques.Navarro et al.(1995)analyzed hydrodynamical simulations of galactic clusters and found that they could beﬁtted by a shallower power law:ρ∝r−1,in the central region.On the other hand,Aninos&Norman(1996)obtained a diﬀerent result from high resolution simulations of galaxy clusters using a two-level grid code.Theyﬁnd that both gas and dark matter density proﬁles follow a unique power lawρ∝r−9/4.They pointed out the presence of a non-convergent gas density core even for their highest resolution run,in disagreement with Navarro et al.(1995)results.It is an uncomfortable situation that numerical studies of dark matter halos of cluster of galaxies using diﬀerent techniques yield diﬀerent results.Nu-merical resolution problems and artefacts are still unavoidable limitations,together with our ignorance of the detailed physics involved in the formation of the structure in the Universe.From an analytical point of view,Chieze et al.(1997)and Teyssier et al.(1997)studied the adiabatic collapse of spherical symmetric perturbations of gas and dark matter.They found a segregation between gas and dark matter components.The dark matter density pro-ﬁles was found to be steeper than that of the gas.According to these authors,the main eﬀect of the joint evolution of their adiabatic purely spherical collapse is the compression of the gas core by the dark matter.But they found no changes in the dark matter density proﬁles. Their hypothesis of geometry and non-treatment of cooling eﬀects may result in diﬀerences in theﬁnal properties of the gas and dark matter distributions when compared to the results of cosmological hydrodynamical simulations which also include cooling mechanisms.Studying halos which contain both components,baryonic and dark matter,with numer-ical techniques is still a quite diﬃcult task to accomplish,principally because of the high resolution required by the dissipative component,but also because of the complexity of the diﬀerent processes related to the physics of baryons such as star formation,supernovae,etc.c 0000RAS,MNRAS000,000–0004The aim of this paper is to present a comparative analysis of galactic halos formed in simu-lations with and without hydrodynamical eﬀects,and to assess any changes in the properties of the dark matter arising as a consequence of the presence of the dissipative component. Our results attempt to describe the eﬀects of the joint evolution of the dark matter and baryons,principally in the central region of galactic-like halos.They do not intend to be conclusive because of our numerical resolution limits.We also attempt to estimate the eﬀects of star formation history on the properties of the objects at z=0.This paper is organized as follows.Section2describes the simulations.Section3deals with the analysis of dark matter halos and disk-like objects properties,and discusses the results.Section4outlines the conclusions.2SIMULATIONSThe initial distributions of positions and velocities of particles in all simulations have been set using COSMICS(kindly provided by E.Bertschinger).They are consistent with standard CDM cosmologies:Ω=1,Ωb=0.1,Λ=0,b≡1/σ8=2.5or1.7whereΩandΩb are the cos-mological and the baryonic density parameter,respectively,Λis the cosmological constant and b,the bias parameter.We have adopted a Hubble constant of H o=100h kms−1Mpc−1 with h=0.5.We studied two sets of simulations with diﬀerent initial conditions(set I and II).The simulations in each set share the same initial conditions but have diﬀerent hydro-dynamical parameters.In all simulations we followed the evolution of262144particles in a periodic box of10Mpc.All dark,gas and star particles have the same mass,2.6×108M⊙. The initial distribution of gas particles has been choosen randomly from the total parti-cle distribution provided by COSMICS.The hydrodynamical ones(simulations I.2,I.3and II.2)were performed using a Smooth Particle Hydrodynamical(SPH)code(see Tissera et al.1997for details of the SPH implementation)coupled to the high resolution adapta-tive particle-particle-particle-mesh code(AP3M),kindly provided by Thomas&Couchman (1992)(simulations I.2and I.3correponds to simulations6and3in Tissera et al.1997, Table I).For comparison,the same initial conditions were run taking into account only gravitational eﬀects using the AP3M(simulations I.1and II.1).The gravitational softening used wasǫg=5kpc.Unfortunately,this SPH code does not allow us to use individual time steps.The integrations were carried out with1000steps of∆t=1.3×107yrs.All simulations were integrated only gravitationally from z>50to z=10,and from there the SPH forces were also taken into account(Evrard et al.1994)up to z=0.Table I lists their main characteristics.Hydrodynamical runs also include a star formation algorithm(Tissera et al.1997)which allows to transform cold gas in high density regions into stars.The parametersηandρstar in Table I are related to the star formation scheme:ηis the star formation eﬃciency and ρstar is a critical density above which a cold gas particle in a convergentﬂow is transformed into stars.The value ofρstar is estimated by requiring the cooling time of the gas cloud represented by a particle to be smaller than its dynamical time.The actual value ofρstar≈7×10−26g/cm3used here is a minimun limit for the gas density to fulﬁll this requirement (Navarro&White1993).Because of the high eﬃciency of the radiative cooling and the fact that no heating sources have been included,the star formation process is very eﬃcient (a lower cutoﬀof T=104K has been used for the temperatures).This leads to an early transformation of the gas into stars,and prevents the formation of disk-like structures.As a consequence,in simulations I.3and II.2all objects are spheroids.In simulation I.2a smaller value ofηwas used.This allowed the gas to settle into disk-like structures,although thec 0000RAS,MNRAS000,000–000Properties of Galactic Objects5 star formation history of these objects was artiﬁcially delayed to low redshift.Objects in set II are dynamically more evolved than those in set I because of their higherσ8value.By the same reason,objects in run II.2have suﬀered from an earlier transformation of the gas into stars than those in run I.3,even if they have the same values of theηandρstar parameters.The simulations performed allow us to study the properties of a set of typically resembling galactic-like objects with very diﬀerent formation history and of their dark matter halos. Some of them have undergone multiple mergers(Tissera et al.1996)while others have experienced a more quiescent evolution.So diﬀerences on the properties of halos which may arise as a result of a diﬀerent evolutionary path may be assessed.However,we do not have a large range of circular velocities to study the dependence of the results on the mass.3ANALYSISAs mentioned in the introduction,there have been several analytical and numerical works on the properties of the dark matter structure.Most of them were carried out using N-body simulations which provide the correct gravitational evolution of the dark matter.Neverthe-less,serveral authors(eg.Blumenthal et al.1986,Carlberg et al.1986,Flores&Primack 1994,Dubinski1994)have pointed out that the presence of baryonic matter might aﬀect the evolution of the dark component by modifying the potential well in the central region, and producing a change in their shapes and density proﬁles.However,there have not been exhaustive studies of halos in SPH simulations.Hence,it is very interesting to carry out a comparative analysis of the properties of dark halos in purely dynamical and hydrody-namical simulations.We run sets of simulations sharing the same initial conditions but with and without hydrodynamical eﬀects.A direct comparison between the shapes and density proﬁles and an assessment of the possible eﬀects caused by dissipational mechanisms can be then carried out.In this work,we analyzed only the more massive objects whose halos and the galaxy-like objects they host(GLOs,see below)are represented by a fairly large number of particles.We have chosen halos in hydrodynamical runs which contain more than300baryonic particles at their virial radius(r200).Hence we describe the properties of the dark halos of20galaxy-like objects in hydrodynamical simulations and12in purely gravitational ones.Halos have been identiﬁed at their virial radius according to the discussion of White&Frenk(1991).The virial radius is the radius where the density contrast of theﬂuctuation reaches a value ofδρ/ρ≈200.In the hydrodynamical case,baryons are included so the virial radius corresponds to that of the total mass distribution(baryons and dark matter).In simulations of set II,we have discarded some objects that belong to groups and therefore,share a common dark halo.However,in set I we keep halo7because it hosts only two main baryonic clumps which assembled at very low redshift.The halo mass center is deﬁned by an interative method which searches for the mass center of concentric spheres of decreasing radius.The process is halted when the number of particles in the sphere equals a certainﬁxed number.This algorithm has been proved to converge quickly.The baryonic clumps within r200of a halo are isolated using a friends-of-friends algorithm with a linking length of the order of10% the mean particle separation.We will refer to them as galaxy-like objects.It has to be noted that the center of mass of the halos may not coincide with the center of mass of the principal galaxy it hosts.This is due to the presence of satellites within its virial radius.The diﬀerence between both mass centres may be of order of3−6kpc.Table II summarizes the principal characteristics of halos and the GLOs they host(spheroids(S),disks(D)and pairs(SP or DP)).c 0000RAS,MNRAS000,000–0006b/a 0.60.70.80.91 1.10.50.60.70.80.91Figure 1.3.1ShapesIn order to analyze the shapes of dark matter halos,we follow a procedure based upon the discussion presented by Curir et al.(1993,and references therein).Basically,we calculated a sequence of concentric ellipsoids containing fractions of the total mass of the objects.We identify inside this region a sequence of 10concentric ellipsoids containing fractions of 10%to 100%of the total mass:r =[x 2+y 2(c/a )2]1/2(2)The semiaxes of the triaxial ellipoids (a >b >c )are calculated iteratively :bI 11)1/2,c I 11)1/2(3)where I 11>I 22>I 33are the eigenvalues of the tensor of inertia (I jk =i x j i x k i a=c aand cProperties of Galactic Objects 7r(kpc)01002003000.50.60.70.80.91r (kpc)01002003000.50.60.70.80.91Figure 2.An inspection of the diﬀerent ellipsoids deﬁned according to equation 2shows the de-pendence of shapes on radius.From r =100kpc to r 200the shapes are quite stable.So we calculated the axis for a sequence of 10ellipsoids containing fractions from 10%to 100%of the dark mass within r =100kpc.In Fig.2a,we can see the ratios b/a (open)and c/a (solid)for halos containing a disk-like (pentagons)and a spheroidal (triangles)structure (both objects are in simulation I.2).All objects tend to be more prolate in the central region becoming more oblate-like with increasing radius.If we compare the distribution of shapes of the halos with disks and spheroids,we ﬁnd that halos with disks tend to be slightly more prolate in the central regionsIn Fig.2b,we compare the shapes of a halo containing a disk in I.2(pentagons)and its counterpart in I.3,where all objects are spheroids (triangles).A weak trend to more oblate structures is found in halos hosting spheroids.In this case,the ratio c/a (solid triangles)is smaller than c/a for halos with disks (solid pentagons),except in the central region where the relation is the inverse.Although our sample is quite small to draw a conclusion from it,the general trend agrees with Dubinski’s work who found that adding a spheroidal potential well at the center of a halo produced a stronger eﬀect on its shape than placing a disk-like potential.3.2Kinematical PropertiesWe calculated the velocity dispersion tensor in spherical coordinates in the reference system of each halo:σ2k (r i )=18where¯V k (r i)=1σ2r.Fig.3a showsβdark for halo4in I.2and halo2in I.3(heavy lines),and their gravitational counterparts(light lines)as examples. All halos are nearly isotropic at the center but show an increasing anisotropy as a function of the distance to its mass center,reaching an average maximun ofβdark≈0.3−0.5.So the outer regions of the halos are more dominated by radial dispersions.There are no signiﬁcant systematic diﬀerences among the values ofβdark of halos in diﬀerent simulations.However, small diﬀerences may be covered up by the noise originated by the discreteness in the number of particles.In order to have an estimation of the strength of the signal over this noise,we measuredβdark for a simulated dark matter halo of10000particles consistent with a King model(King1966).The rms dispersion found was∆rms≈0.11.Hence,the anisotropy of the velocity distribution estimated for the simulated halos can be regarded as a real signal. On the other hand,the statistical signiﬁcant features individualized in the graphics of Fig.3a can be matched with the presence of satellites or substructures within r200.The positions of these substructures change within simulations mainly due to modiﬁcations in the collapse times of satellites originated by hydrodynamical eﬀects(Navarro et al.1995).By contrast,a diﬀerence is found among their total velocity dispersions.Fig.3b shows the velocity dispersions of halos1,2,3,4in simulations I.1(light lines)and I.2(heavy lines). It can be seen from thisﬁgure thatσof halos in hydrodynamical simulations increases by a percentage of approximately59%between r=100kpc and r=10kpc.In the purely gravitational simulations,however,the percentage is signiﬁcantly smaller,10%,remaining practicallyﬂat over the whole range of radii.This eﬀect is clearly related to the presence of a baryonic core in the central region of the halo which modiﬁes the overall potential well.We have also compared the velocity dispersion proﬁles of halos in simulations I.2and I.3. As mentioned above,these simulations diﬀer from each other only on their star formation eﬃciency which has led to the formation of spheroidal objects in I.3,while,conversely,I.2has also disk-like structures.We found that the velocity dispersion of a halo hosting a disk and the same halo with a spheroid in its central region,does not show any signiﬁcant diﬀerence in our simulations.3.3Halo Density ProﬁlesWe calculated the density proﬁles of dark matter halos for simulations in sets I and II.The density proﬁles have been deﬁned by binning the particles in spherical shells centred at the halo mass center.We work with weighted proﬁles using shells containing aﬁxed number of particles(30or50).We also estimated the density proﬁles by binning in the logarithm of the radius.Both procedures yield the same results.We have preferred to carry out the analysisc 0000RAS,MNRAS000,000–000Properties of Galactic Objects9Figure3.using the weighted proﬁles since the other one gives too much weight to the central region where the resolution is lower(r<2ǫg).Weﬁrst analyzed the density proﬁles of dark matter halos in purely dynamical runs (I.1and II.1).We found that these halosﬁt the NFW proﬁle remarkably well.So their density proﬁles steep up as r−1in the central region as predicted by NFW and Hern.We also observed that more massive halos are less concentrated than smaller ones.The concentration parameters c NF W found for our halos are consistent with the values published by NFW (Table II).Fig.4shows four halo density proﬁles in a N-body run(I.1)and the corresponding ﬁtting using NFW model(solid and dashed lines respectively).Following the same procedure,we calculated the density proﬁles of dark matter halos in the hydrodynamical runs(simulations I.2,I.3and II.2).These halos show some diﬀerences as can be seen in Fig.5,where the density proﬁles of halos1and3in simulations I.1and I.2have been plotted.The solid line represents the density proﬁles in the hydrodynamical run and the dashed lines the corresponding proﬁles in the purely dynamical one.Since the simulations share the same initial conditions for both,velocities and positions,and gravitational parameters(softening,force errors),the diﬀerences found among the density proﬁles may be directly related with hydrodynamical eﬀects.Theseﬁgures show clearly how the density changes because of the presence of a baryonic core at the center of the halo.In Fig.6,we plot the density proﬁles of halos1,2,3,4in simulation I.2(solid lines).We also include the bestﬁt obtained using NFW proﬁles(dot-long dashed lines).The slope of the density proﬁles of these halos is steeper in the central c 0000RAS,MNRAS000,000–00010Figure4.region,so the proﬁles deﬁned by Navarro et al.(1995)are too curved toﬁt them⋆.Our proﬁles seem to follow a simple power law over a larger range of radius than NFW proﬁles. The radius at which NFW proﬁles depart from our simulated proﬁles correlates with the radius of the galaxy-like object in the halo(we deﬁne r b as the radius at which the baryonic and dark mattter density proﬁles intersect each other.See section3.3.1for details).It has to be stressed that the dynamical resolution of our halos is comparable with NFW objects for the same range of circular velocities.We use a standard algorithm toﬁt to these proﬁles a power law in the central region (ρ∝rα),looking for the value ofαthat gives the bestﬁt.Values ranging from-1.9to -2where found.Hence,we decided to keep the functional form deﬁned by equation1,but increasing the slope of the power law for small radius:ρ(r)=A c3c2T D(1+c T D)(7)whereρdark=M200dark/v200.The mean dark matter density at r200,ρdark,can be rewritten⋆The parameterδc has been redeﬁned for consistency in a similar way as in equation6:δc=c3M dark200ρ−1critFigure5.Figure6.(1−f b)200where f b is the fraction of baryons within r200.This fraction is almost constant(≈Ωb)for all halos.The proﬁle deﬁned by equation(6)wasﬁtted to all halos in the hydrodynamical simulations.In Fig.6,we also plot the bestﬁt for halos1,2,3, and4using equation(6)(dotted lines).The parameters obtained are summarized in Table II.As can be seen from this table,c T D values are smaller than c NF W ones,implying larger values of r s for halos in hydrodynamical runs.This can be understood taking into account that,in equation(6),r s is the radius that sets the transition fromρ(r)∝r−4for larger radius toρ(r)∝r−2for the intermediate region,while,in NFW,r s delimites the change in the slope fromρ(r)∝r−3toρ(r)∝r−1in the central region.Fig.7shows the correlation found between the logarithm of c T D and M200for the den-sity proﬁles of halos in simulations I.1(open pentagons),I.2(solid squares)and I.3(open triangles).The zero point of the relation for objects in I.2and I.3has been redeﬁned by adding an arbitrary constant to the logarithm of c T D in order to place them in the same region of the diagram as the logarithm of c NF W.Thisﬁgure shows that larger masses are c 0000RAS,MNRAS000,000–000Figure7.less concentrated,as pointed out by NFW for purely dynamical simulations.We also include the theoretical predictions for c NF W from the collapse time assuming NFW density proﬁles (we used an algorithm kindly provided by these authors to estimate the collapse time of the halos).With thisﬁgure we intend only to compare the slope of the relations for the hydro-dynamical and N-body runs.The slope of the correlation for halos in I.2and I.3seems to be consistent with the relation obtained from the purely dynamical simulations.Note,however, that the range of masses is small to get a general trend.We did notﬁnd a diﬀerence between halos hosting a disk or a spheroid.In any case,a large sample with higher hydrodynamical resolution and dynamical range is needed to have a more robust statistical signal.We measured the ratio M sph darkFigure8.the presence of baryons does change the matter distribution and this is clearly observed in our simulations,although the eﬀect may be not as strong as measured here.We will come back to this point in the next sections.3.3.1Baryonic Density ProﬁlesWe construct the density proﬁles for the baryonic matter belonging to each halo listed in Table II by following the same procedure described in section3.3for the dark matter component.The density proﬁles of baryons are not as accurate as the dark matter ones since the number of particles used to solve the gaseous component is lower than the one used for the dark matter.As an illustration of the general behaviour of the baryonic component,Fig.9shows the density proﬁles for baryons and dark matter for two halos in set I.These two proﬁles intersect each other at a certain radius,r b,which can be taken as a measure of the physical size of the central galaxy-like object.The same is true for the other GLOs listed in Table II. The r b values range from14kpc to30kpc,but they are diﬃcult to be properly determined because of the noisy character of the baryonic density proﬁles.Noteﬁrst that the baryonic density proﬁles(dot-long dashed and dotted lines)are steeper than the dark matter ones(solid lines).The slope for radius greater than r b isρbar(r)∝r−3. Secondly,that the baryonic dark matter proﬁles in the central region depend on the star formation parameters used.In simulation I.2(dotted lines),baryons seem to have collapsed more dramatically than in I.3.In this last run(dot-long dashed lines),the baryonic density proﬁles follow the dark matter for r<r b,changing its slope toρbar(r)∝r−2.This is consistent with the fact that the gas has been more eﬃciently transformed into stars and that stars behave in a similar fashion as dark matter particles since they are only aﬀected by gravitational forces.On the other hand,in I.2the higher concentration of baryons in the central region is the consequence of a higher eﬃciency in the cooling of the gas particles. Hence,the history of star formation is relevant to theﬁnal distribution of baryons in the central region.However,from our simulations we cannot detect a clear change in the dark matter density proﬁles(solid lines)as a result of the diﬀerence in the baryonic distributions.c 0000RAS,MNRAS000,000–000。

Detailed theoretical predictions of the outskirts of dark matter halos

D RAFT VERSION F EBRUARY 5, 2008
A Preprint typeset using L TEX style emulateapj v. 10/09/06
DETAILED THEORETICAL PREDICTIONS FOR THE OUTSKIRTS OF DARK MATTER HALOS
1. INTRODUCTION
The study of the density proﬁle of cold dark matter halos beyond the virial radius is a subject of considerable relevance. From an observational point of view, knowledge of the shape of the density proﬁle far beyond the virial radius is essential for an appropriate interpretation of gravitational lensing phenomena (e.g. Smith et al. 2001; Guzik & Seljak 2002; Hoekstra et al. 2004; Sheldon et al. 2004), the pattern of Lyman alpha absorption around virialized systems (e.g. Barkana 2004; Bajtlik, Duncan & Ostriker 1988) as well as the motion of satellite galaxies as a test for dark matter distribution at large radii (Zaritsky & White 1994, Zaritsky et al. 1997; Prada et al. 2003, Brainerd 2004; Conroy et al. 2004). From the theoretical point of view, the study of the properties of dark matter halos at several virial radius in cosmological simulations provides an excellent benchmark for developing and testing the basic theoretical framework which will be decisive for a full understanding of the physical origin and formation of the ΛCDM halos. Understanding halo properties involves a set of theoretical considerations. First, we have the issue of choosing the correct initial density proﬁle. Also, there is the question of which processes are relevant to the gravitational evolution of the initial proﬁle: is only the spherical collapse what matters or is thich radius can we use the standard spherical collapse model without shell crossing? are highly asymmetrical processes, like merging, relevant? In order to answer these questions it is very convenient to focus ﬁrst on those properties of the halos which involve the fewest theoretical uncertainties. The dark matter density proﬁles at several virial radius

Dark matter and structure formation a review

a r X i v :0801.1091v 1 [a s t r o -p h ] 7 J a n 2008Dark matter,density perturbations and structure formationA.Del Popolo 1,2,31Bo ˘g azi ¸c i University,Physics Department,80815Bebek,Istanbul,Turkey2Dipartimento di Matematica,Universit`a Statale di Bergamo,via dei Caniana,2,24127,Bergamo,ITALY 3Istanbul Technical University,Ayazaga Campus,Faculty of Science and Letters,34469Maslak/ISTANBUL,TurkeyAbstract —-This paper provides a review of the variants of dark matter which are thought to be fundamental components of the universe and their role in origin and evolution of structures and some new original results concerning improvements to the spherical collapse model.In particular,I show how the spherical collapse model is modiﬁed when we take into account dynamical friction and tidal torques.1.INTRODUCTIONThe origin and evolution of large scale structure is today the outstanding problem in cosmology.This is the most fundamental question we can ask about the universe whose solution should help us to better understand problems as the epoch of galaxy formation,the clustering in the galaxy distribution,the amplitude and form of anisotropies in the microwave background radiation.Several has been the ap-proaches and models trying to attack and solve this problem:no one has given a ﬁnal answer.The leading idea of all structure formation theories is that structures was born from small perturbations in the other-wise uniform distribution of matter in the early Universe,which is supposed to be,in great part,dark (matter not detectable through light emission).With the term Dark Matter cosmologists indicate an hypothetic material component of the universe which does not emit directly electromagnetic radiation (unless it decays in particles having this property ([1],but also see [2])).Dark matter,cannot be revealed directly,but nevertheless it is necessary to postulate its existence in order to explain the discrepancies between the observed dynamical proper-ties of galaxies and clusters of galaxies and the theoretical predictions based upon models of these objects assuming that the only matter present is the visible one.If in the space were present a diﬀused material component having gravitational mass,but unable to emit electromagnetic radiation in signiﬁcative quantity,this discrepancy could be eliminated ([3]).The study of Dark Matter has as its ﬁnality the explanation of formation of galaxies and in general of cosmic structures.For this reason,in the last decades,the origin of cosmic structures has been “framed”in models in which Dark Matter constitutes the skeleton of cosmic structures and supply the most part of the mass of which the same is made.There are essentially two ways in which matter in the universe can be revealed:by means of radiation,by itself emitted,or by means of its gravitational interaction with baryonic matter which gives rise to cosmic structures.Electromagnetic radiation permits to reveal only baryonic matter.In the second case,we can only tell that we are in presence of matter that interacts by means of gravitation with the luminous mass in the universe.The original hypotheses on Dark Matter go back to measures performed by Oort ([4])of the surface density of matter in the galactic disk,which was obtained through the study of the stars motion in direction orthogonal to the galactic plane.The result obtained by Oort,which was after him named “Oort Limit”,gave a value of ρ=0.15M 0pc −3for the mass density,and a mass,in the region studied,superior to that present in stars.Nowadays,we know that the quoted discrepancy is due to the presence of HI in the solarclusters (a Cluster)and the total mass contained in galaxies of the same clusters.These and other researches from the thirties to now,have conﬁrmed that a great part of the mass in the universe does not emit radiation that can be directly observed.1.1Determination of Ωand Dark MatterThe simplest cosmological model that describes,in a suf-ﬁcient coherent manner,the evolution of the universe,from 10−2s after the initial singularity to now,is the so called Standard Cosmological Model (or Hot Big Bang model).It is based upon the Friedmann-Robertson-Walker (FRW)met-ric,which is given by:ds 2=c 2dt 2−a (t )2dr 22g ik R =−8πGa 2˙a 2+k3ρ(4)2¨a a 2+k2 the components of the today universe are galaxies.If weassume that galaxies motion satisfy Weyl([9])postulate,the velocity vector of a galaxy is given by u i=(1,0,0,0),and then the system behaves as a system made of dust forwhich we have p=0.Only two of the three Friedmannequations are independent,because theﬁrst connectsdensity,ρto the expansion parameter a(t).The characterof the solutions of these equations depends on the valueof the curvature parameter,k,which is also determinedby the initial conditions by means of Eq. 3.The solutionto the equations now written shows that ifρis largerthanρc=3H2ρc .In this case,theconditionΩ=1corresponds to k=0,Ω<1corresponds to k=−1,andΩ>1corresponds to k=1.1The value ofΩcan be calculated in several ways.The most common methods are the dynamical methods,in which the eﬀects of gravity are used,and kinematics methods sensible to the evolution of the scale factor and to the space-time geometry.The results obtained forΩwith these diﬀerent methods are summarized in the following.Dynamical methods:(a)Rotation curves:The contribution of spiral galaxies to the density in the universe is calculated by using their rotation curves and the third Kepler ing the last it is possible to obtain the mass of a spiral galaxy from the equation:M(r)=v2r/G(6) where v is the velocity of a test particle at a distance r from the center and M(r)is the mass internal to the circular orbit of the particle.In order to determine the mass M is necessary to have knowledge of the term v2in Eq.(6)and this can be done from the study of the rotation curves through the21cm line of HI.Rotation curves of galaxies are characterized by a peak reached at distances of some Kpcs and a behavior typicallyﬂat for the regions at distance larger than that of the peak.A peculiarity is that the expected Keplerian fall is not observed.This result is consistent with extended haloes containing masses till10times the galactic mass observed in the optical ([10]).The previous result is obtained assuming that the halo mass obtained with this method is distributed in a spherical region so that we can use Eq.(6)and that we neglect the tidal interaction with the neighboring galaxies which tend to produce an expansion of the halo.After M and the luminosity of a series of elliptical galaxies is determined,the contribution to the density of the universeL>ℓwhereℓis the luminosity per unit volume due to galaxies and can be obtained from the galactic luminosity functionφ(L)dL,which describes the number of galaxies per Mpc3and luminosity range L,L+dL.The value that is usually assumed forℓis ℓ=2.4h108L bo Mpc−3.The arguments used lead to a value ofΩg for the luminous parts of spiral galaxies ofΩg≤0.01, while for haloesΩh≥0.03−0.1.The result shows that the halo mass is noteworthy larger than the galactic mass observable in the optical([11]).(b)Virial theorem:In the case of non spiral galaxies and clusters,the mass can be obtained using the virial theorem2T+V=0,withT∼=3c≈Ω0.6λρ(9) ([13]).Then given the overdensityδρρcan be ob-tained from the overdensity of galaxiesδn gρ=δn g3 (d)Kinematic methods:These methods are based upon the use of relations be-tween physical quantities dependent on cosmological param-eters.An example of those relations is the relation luminos-ity distance-redshift:H0d L=z+1F the luminosity distance,L the absolute luminos-ity,and F theﬂux.By means of the relations luminosity-redshift,angle-redshift,number of objects-redshift,it is possible to determine the parameter of deceleration q0=−¨a0a0=100hkm/Mpcs are the scale factor and the Hubble constant,nowadays).At the same time q0is connected toΩby means of q0=Ω•Growth rate ofﬂparisons of presentday structure withﬂuctuations at the last scatteringof the cosmic microwave background(CMB)or withhigh redshift objects of the young universe.The methods and current estimates are summarized in Table3.The estimates based on virialized objects typi-cally yield low values ofΩm∼0.2−0.3.The global mea-sures,large-scale structure and cosmicﬂows typically indi-cate higher valuesofΩm∼0.4−1.Bahcall et al.([17]),showed that the evolution of the number density of rich clusters of galaxies breaks the degen-eracy betweenΩ(the mass density ratio of the universe)and σ8(the normalization of the power spectrum),σ8Ω0.5≃0.5, that follows from the observed present-day abundance of rich clusters.The evolution of high-mass(Coma-like)clus-ters is strong inΩ=1,low-σ8models(such as the standard biased CDM model withσ8≃0.5),where the number den-sity of clusters decreases by a factor of∼103from z=0 to z≃0.5;the same clusters show only mild evolution in low-Ω,high-σ8models,where the decrease is a factor of ∼10.This diagnostic provides a most powerful constraint onΩ.Using observations of clusters to z≃0.5−1,the authors found only mild evolution in the observed cluster abundance,andΩ=0.3±0.1andσ8=0.85±0.15(for Λ=0models;forΩ+Λ=1models,Ω=0.34±0.13). ferreira et al.([18]),proposed an alternative method to estimate v12directly from peculiar velocity samples,which contain redshift-independent distances as well as galaxy red-shifts.In contrast to other dynamical measures which de-termineβ≡Ω0.6σ8,this method can provide an estimate of Ω0.6σ28for a range ofσ8whereΩis the cosmological density parameter,whileσ8is the standard normalization for the power spectrum of densityﬂuctuations.Melchiorri([19]),used the angular power spectrum of the Cosmic Microwave Background,measured during the North American testﬂight of the BOOMERANG experiment,to constrain the geometry of the universe.Within the class of Cold Dark Matter models,theyﬁnd that the overall frac-tional energy density of the universe,Ω,is constrained to be0.85≤Ω≤1.25at the68%conﬁdence level. Branchini([20]),compared the density and velocityﬁelds as extracted from the Abell/ACO clusters to the corre-spondingﬁelds recovered by the POTENT method from the Mark III peculiar velocities of galaxies.Quantitative comparisons within a volume containing∼12independent samples yieldβc≡Ω0.6/b c=0.22±0.08,where b c is the cluster biasing parameter at15h−1Mpc.If b c∼4.5,as in-dicated by the cluster correlation function,their result is consistent withΩ∼1.(f)Inﬂation:It is widely supposed that the very early universe experi-enced an era of inﬂation(see[21],[22],[13]).By‘inﬂation’one means that the scale factor has positive acceleration,¨a>0,corresponding to repulsive gravity and3p<−ρ. During inﬂation aH=˙a is increasing,so that comoving scales are leaving the horizon(Hubble distance)rather than entering it,and it is supposed that at the beginning of in-ﬂation the observable universe was well within the horizon. The inﬂationary hypothesis is attractive because it holds out the possibility of calculating cosmological quantities, given the Lagrangian describing the fundamental interac-tions.The Standard Model,describing the interactions up to energies of order1T eV,is not viable in this context be-cause it does not permit inﬂation,but this should not be re-garded as a serious setback because it is universally agreed4mology.The nature of the required extension is not yet known,though it is conceivable that it could become known in the foreseeable future.But even without a speciﬁc model of the interactions(ie.,a speciﬁc Lagrangian),the inﬂation-ary hypothesis can still oﬀer guidance about what to expect in cosmology.More dramatically,one can turn around the theory-to-observation sequence,to rule out otherwise rea-sonable models.The importance of inﬂation is connected to:a)the origin of density perturbations,which could origi-nate during inﬂation as quantumﬂuctuations,which be-come classical as they leave the horizon and remain so on re-entry.The original quantumﬂuctuations are of exactly the same type as those of the electromagneticﬁeld,which give rise to the experimentally observed Casimir eﬀect. b)One of the most dramatic and simple eﬀects is that there is noﬁne-tuning of the initial value of the density parame-terΩ=8πρ/3m2P l H2.From the Friedmann equation,Ωis given byΩ−1=(K3An argument has been given forΩ0very close to1on the basis of eﬀects on the cmb anisotropy from regions far outside the observable simplest one([21])invokes a scalarﬁeld,termed the inﬂa-tonﬁeld.An alternative([23])is to invoke a modiﬁcation of Einstein gravity,and combinations of the two mecha-nisms have also been proposed.During inﬂation however, the proposed modiﬁcations of gravity can be abolished by redeﬁning the spacetime metric tensor,so that one recovers the scalarﬁeld case.We focus on it for the moment,but modiﬁed gravity models will be included later in our survey of speciﬁc models.In comoving coordinates a homogeneous scalarﬁeldφwith minimal coupling to gravity has the equation of motion¨φ+3H˙φ+V′(φ)=0(13) Its energy density and pressure areρ=V+12˙φ2(15)If such aﬁeld dominatesρand p,the inﬂationary condition 3p<ρis achieved provided that theﬁeld rolls suﬃciently slowly,˙φ2<V(16)Practically all of the usually considered models of inﬂation satisfy three conditions.First,the motion of theﬁeld is overdamped,so that the‘force’V′balances the‘friction term’3H˙φ,˙φ≃−116π V′38π8πV′′5 in which they are satisﬁed and we are adopting that nomen-clature here.Practically all of the usually considered modelsof inﬂation satisfy the slow-roll conditions more or less well.It should be noted that theﬁrst slow-roll condition is ona quite diﬀerent footing from the other two,being a state-ment about the solution of theﬁeld equation as opposed toa statement about the potential that deﬁnes this equation.What we are saying is that in the usually considered modelsone can show that theﬁrst condition is an attractor solu-tion,in a regime typically characterized by the other twoconditions,and that moreover reasonable initial conditionsonφwill ensure that this solution is achieved well beforethe observable universe leaves the horizon.It is importantto remember that there are strong observational limits forthe parameters previously introduced(e.g.ǫ,η).For ex-ample[27]studied the possible contribution of a stochasticgravitational wave background to the anisotropy of the cos-mic microwave background in cold and mixed dark matter(CDM and MDM)models.This contribution was testedagainst detections of CMB anisotropy at large and inter-mediate angular scales.The bestﬁt parameters(i.e.thosewhich maximize the likelihood)are(with95%conﬁdence)n S=1.23+0.17−0.15andR(n S)=C T2π2f(n S)=2.4+3.4−2.2(23)wheref(n S)=Γ(3−n S)Γ(3+n SΓ2(4−n S2)(24)The previous constraintﬁxes the value ofǫas well that ofη2η=n s−1+2ǫ(25) Theyﬁnd that by including the possibility of such back-ground in CMB data analysis it can drastically alter the conclusion on the remaining cosmological parameters.More stringent constraints on some of the previous parameters are given in section1.12.(h)Conclusions:We have seen the possible values ofΩusing diﬀerent meth-ods.We have to add that Cosmologists are“attracted”by a value ofΩ0=1.This value ofΩis requested by inﬂa-tionary theory.The previous data lead us to the following hypotheses:i)Ω0<0.12;in this case one can suppose that the uni-verse is fundamentally made of baryonic matter(black holes; Jupiters;white dwarfs).ii)Ω0>0.12;in this case in order to have aﬂat universe,it is necessary a non-baryonic component.Ωb=1is excluded by several reasons(see[28],[13].The remaining possibilities are:1)existence of a smooth component withΩ=0.8.The test of a smooth component can be done with kinematic methods.2)Existence of a cosmological term,absolutely smooth to whom correspond an energy densityρvac=Λsthe number of particles per unit comoving volume and we remember that n is the number density of species and s the entropy density, we obtain a contribution of the species to the actual density of the universe asΩh2=0.28Y(T f)(m6limits([13]).The solution to the problem was proposed by Peccei-Quinn in1977([36])in terms of a spontaneous sym-metry breaking scheme.To this symmetry breaking should be associated a Nambu-Goldstone boson:the axion.Theaxion mass ranges between10−12ev-1Gev.In cosmology there are two ranges of interest:10−6ev≤m a≤10−3ev; 3ev≤m a≤8ev.Axion production in the quoted range can originate due to a series of astrophysical processes([13])and several are the ways these particles can be detected. Nevertheless the eﬀort of researchers expecially in USA, Japan and Italy,axions remain hypothetical particles. They are in any case the most important CDM candidates.In the following,I am going to speak about the basic ideas of structure formation.I shall write about density perturbations,their spectrum and evolution,about correla-tion functions and their time evolution,etc.1.3Origin of structuresObserving our universe,we notice a clear evidence of in-homogeneity when we consider small scales(Mpcs).In clus-ters density reaches values of103times larger than the av-erage density,and in galaxies it has values105larger than the average density([13]).If we consider scales larger than 102Mpcs universe appears isoptric as it is observed in the radio-galaxies counts,in CMBR,in the X background([11]). The isotropy at the decoupling time,t dec,at which matter and radiation decoupled,universe was very homogeneous, as showed by the simple relation:δρT(28)([13])4.The diﬀerence between the actual universe and that at decoupling is evident.The transformation between a highly homogeneous universe,at early times,to an highly local non homogeneous one,can be explained supposing that at t dec were present small inhomogeneities which grow up because of the gravitational instability mechanism([37]). Events leading to structure formation can be enumerated as follows:(a)Origin of quantumﬂuctuations at Planck epoch.(b)Fluctuations enter the horizon and they grow linearly till recombination.(c)Perturbations grow up in a diﬀerent way for HDM and CDM in the post-recombination phase,till they reach the non-linear phase.(d)Collapse and structure formation.Before t dec inhomogeneities in baryonic components could not grow because photons and baryons were strictly cou-pled.This problem was not present for the CDM compo-nent.Then CDM perturbations started to grow up before those in the baryonic component when universe was mat-ter dominated.The epoch t eq≈4.4∗1010(Ω0h2)−2sec,at which matter and radiation density are almost equal,can be considered as the epoch at which structures started to form.The study of structure formation is fundamentally an initial value problem.Data necessary for starting this study are:1)Value ofΩ0.In CDM models the value chosen for this parameter is1,in conformity with inﬂationary theory pre-dictions.2)The values ofΩi for the diﬀerent components in the uni-verse.For example in the case of baryons,nucleosynthesis gives us the limit0.014≤Ωb≤0.15whileΩW IMP S≈0.9.3)The perturbation spectrum and the nature of pertur-bations(adiabatic or isocurvature).The spectrum gener-ally used is that of Harrison-Zeldovich:P(k)=Ak n with n=1.The perturbation more used are adiabatic or curva-ture.This choice is dictated from the comparison between theory and observations of CMBR anisotropy.1.4The spectrum of density perturbationIn order to study the distribution of matter density in the universe it is generally assumed that this distribution is given by the superposition of plane waves independently evolving,at least until they are in the linear regime(this means till the overdensityδ=ρ−ρthe average density in the volume and withρ(r)the density in r,it is possible to deﬁne the density contrast as:δ(r)=ρ(r)−l(and similar conditions for the other components)and for the periodicity conditionδ(x,y,L)=δ(x,y,0)(and similar conditions for the other components). Fourier coeﬃcientsδk are complex quantities given by:δk=1σ2)(32)([28]).The quantityσthat is present in Eq.(32)is the variance of the densityﬁeld and is deﬁned as:σ2=<δ2>= k<|δk|2>=1(2π)3 P(k)d3k=17It can be deﬁned as the joint probability of ﬁnding an over-density δin two distinct points of space:ξ(r ,t )=<δ(r ,t )δ(r +x ,t )>(35)([38]),where averages are averageson anensemble obtainedfromseveralrealizations ofuniverse.Correlation functioncan be expressed as the joint probability of ﬁnding a galaxy in a volume δV 1and another in a volume δV 2separated by a distance r 12:δ2P =n 2V [1+ξ(r 12)]δV 1δV 2(36)where n V is the average number of galaxies per unit volume.The concept of correlation function,given in this terms,can be enlarged to the case of three or more points.Correlation functions have a fundamental role in the study of clustering of matter.If we want to use this function for a complete description of clustering,one needs to know the correlation functions of order larger than two ([39]).By means of correlation functions it is possible to study the evolution of clustering.The correlation functions are,in fact,connected one another by means of an inﬁnite system of equations obtained from moments of Boltzmann equa-tion which constitutes the BBGKY (Bogolyubov-Green-Kirkwood-Yvon)hierarchy ([40]).This hierarchy can be transformed into a closed system of equation using closure conditions.Solving the system one gets information on cor-relation functions.In order to show the relation between perturbation spec-trum and two-points correlation function,we introduce inEq.(35),Eq.(30),recalling that δ∗k =δ(−k )and taking the limit V u →∞,the average in the Eq.(35)can be expressed in terms of the integral:ξ(r )=12π2k 2P (k )sin (kr )b (t p )2T 2(k ;t f )P (k ;t p )(39)where b(t)is the law of grow of perturbations,in the linear regime.In the case of CDM models the transfer function is:T (k )= 1+ ak +(bk )1.5+(ck )2 ν−1S=3ρr−δρmT−δρm8The distribution function f that appears in the previous equations cannot be obtained from observations.It is possi-ble to measure moments of f (density,average velocity,etc.).We want now to obtain the evolution equations for δ.Forthis reason,we start integrating Eq.(44)on p and after using Eq.(43),we get:a 3ρb∂δa 2▽p fd 3p =0(45)If we deﬁne velocity as:v =pfd 3p(46)and introduce it in Eq.(45)we get:ρb ∂δa▽(ρv )=0(47)We can now multiply Eq.(44)for p and integrate it on the momentum:∂ma 2∂βp αp βfd 3p +a 3ρ(x ,t )φ,α=0(48)this last in Eq.(45)leaves us with:∂2δa ∂δa2▽[(1+δ)▽φ]+1ma 2fd 3p(50)the equation for the evolution of overdensity becomes:∂2δa ∂δa 2▽[(1+δ)▽φ]+1∂t 2+2˙a ∂t=4πGρb δ(52)This equation in an Einstein-de Sitter universe (Ω=1,Λ=0)has the solutions:δ+=A +(x )t2a 2=82(1−Ω0)(cosh η−1)(55)t (η)=Ω0a 2=823t.(57)Before concluding this section,we want to ﬁnd an ex-pression for the velocity ﬁeld in the linear ingthe equation of motion p =ma 2˙x ,d pdt+v ˙a a=Gρb ad 3xδ(x ,t )x −x ′4π∂|x ′−x |(59)([38]).This solution is valid just as that for δin the linear regime.At time t =t 0this regime is valid on scales larger than 8h −1Mpc .1.7Non-linear phaseLinear evolution is valid only if δ<<1or similarly,if the mass variance,σ,is much less than unity.When this condi-tion is no longer veriﬁed (e.g.,if we consider scales smaller than 8h −1Mpc),it is necessary to develope a non-linear theory.In regions smaller than 8h −1Mpc galaxies are not a Poisson distribution but they tend to cluster.If one wants to study the properties of galactic structures or clusters of galaxies,it is necessary to introduce a non-linear theory of clustering.A theory of this last item is too complicated to be developed in a purely theoretical fashion.The problem can be faced assuming certain approximations that simpli-ﬁes it ([47])or as often it is done,by using N-Body simu-lations of the interesting system.The approximations are often used to furnish the initial data to simulations.In the simulations,a large number of particles are randomly dis-tributed in a sphere,in the points of a cubic grid,in order to eliminate small scale noise.The initial spectrum is ob-tained perturbing the initial positions by means of a super-position of plane waves having random distributed phases and wave vector ([48]).Obviously,the universe is considered in expansion (or comoving coordinates are used),and then the equation of motion of particles are numerically solved.For what concerns the analytical approximations one of the most used is that of [47].This gives a solution to the prob-lem of the grow of perturbations in an universe with p =0not only in the linear regime but even in the mildly non-linear regime.In this approximation,one supposes to have particles with initial position given in Lagrangian coordi-nates q .The positions of particles,at a given time t,are given by:9 where x indicates the Eulerian coordinates,p(q)describesthe initial densityﬂuctuations and b(t)describes their growin the linear phase and it satisﬁes the equation:d2bdtdadt =dbρ ∂q jρ δjk+b(t)∂p k3p(q)= k i k3exp(i kq)(67)([28]),that leads us back to the linear theory.In other words,Ze‘ldovich approximation is able to reproduce the linear theory,and is also able to give a good approximationin regions withδρρ>>ing the expression for p(q),theJacobian in Eq.(64)is a real matrix and symmetric that can be diagonalized.With this p(q)the perturbed density can be written as:ρ(q,t)=(1−b(t)λ1(q))(1−b(t)λ2(q))(1−b(t)λ3(q))(68) whereλ1,λ2,λ3are the three eigenvalues of the Jacobian, describing the expansion and contraction of mass along the principal axes.From the structure of the last equation,we notice that in regions of high density Eq.(68)becomes inﬁnite and the structure of collapse in a pancake,in a ﬁlamentary structure or in a node,according to values of eigenvalues.Some N-body simulations([49])tried to ver-ify the prediction of Ze‘ldovich approximation,using initial conditions generated using a spectrum with a cut-oﬀat low frequencies.The results showed a good agreement between theory and simulations,for the initial phases of the evolu-tion(a(t)=3.6).Going on,the approximation is no more valid starting from the time of shell-crossing.After shell-crossing,particles does not oscillate any longer around the structure but they pass through it making it vanish.This problem has been partly solved supposing that particles, before reaching the singularity they sticks the one on the other,due to a dissipative term that simulates gravity and then collects on the forming structure.This model is known as“adesion-model”([50]).Summarizing,Zel’dovich approximation gives a description of the transition between linear and non-linear phase.It 1.8Quasi-linear regimeWe have seen in the previous section that in the case of regions of dimension smaller than8h−1Mpc,the linear theory is no more a good approximation and a new theory is needed or N-body simulations.Non-linear theory is able to calculate quantities as the formation redshift of a given class of objects as galaxies and clusters,the number of bound objects having masses larger than a given one,the average virial velocity and the correlation function.It is possible to get an estimate of the given quantities as that of other not cited,using an intermediate theory between the linear and non-linear theory:the quasi-linear theory. This last is obtained adding to the linear theory a model of gravitational collapse,just as the spherical collapse model.Important results that the theory gives is the bottom-up formation of structures(in the CDM model). Other important results are obtained if we identify density peaks in linear regime with sites of structure formation. Two important papers in the development of this theory are[51]and that of[52].This last paper is an application of the ideas of the quasi-linear theory to the CDM model. The principles of this approach are the following:•Regions of mass larger than M that collapsed can be identiﬁed with regions where the density contrast evolved according to linear regime,δ(M,x),has a value larger than a threshold,δc.•After collapse regions does not fragment.The major drawbacks of the theory,as described in[52]are fundamentally the fact that the estimates that can be ob-tained by means of this theory depends on the threshold δc,on the ratio between theﬁltering mass and that of ob-jects and from other parameters.Nevertheless,this theory has helped cosmologists in obtaining estimate of important quantities as those previously quoted,and at same time give evidences that leads to exclude very low values for spectrum normalization.1.9Spherical CollapseSpherical symmetry is one of the few cases in which grav-itational collapse can be solved exactly([53];[38]).In fact, as a consequence of Birkhoﬀ’s theorem,a spherical pertur-bation evolves as a FRW Universe with density equal to the mean density inside the perturbation.The simplest spherical perturbation is the top-hat one, i.e.a constant overdensityδinside a sphere of radius R; to avoid a feedback reaction on the background model,the overdensity has to be surrounded by a spherical underdense shell,such to make the total perturbation vanish.The evo-lution of the radius of the perturbation is then given by a Friedmann equation.The evolution of a spherical perturbation depends only on its initial overdensity.In an Einstein-de Sitter background, any spherical overdensity reaches a singularity(collapse)at aﬁnal time:t c=3π3δ(t i) −3/2t i.(69) By that time its linear density contrast reaches the value:3/2。

The mysteries of the universe Dark matter

The mysteries of the universe DarkmatterDark matter is one of the greatest mysteries of the universe. Despite its elusive nature, scientists have been able to gather some information about it through various observations and experiments. In this essay, we will explore the concept of dark matter, its significance in the universe, and the ongoing efforts to understand it. First and foremost, what is dark matter? Dark matter is a hypothetical form of matter that is thought to make up approximately 27% of the universe's total mass and energy. Unlike ordinary matter, which is composed of atoms and molecules, dark matter is believed to be non-baryonic, meaning it does not consist of the same particles as normal matter. This makes it extremely difficult to detect, as it does not interact with light or other forms of electromagnetic radiation. The significance of dark matter lies in its gravitational effects on visible matter. Through its gravitational pull, dark matter is responsible for the formation and structure of galaxies and galaxy clusters. Without the presence of dark matter, the universe would look very different, with galaxies failing to hold together and maintain their shape. Therefore, understanding dark matter is crucial to our comprehension of the universe as a whole. Despite its importance, the nature of dark matter remains largely unknown. Scientists have proposed various theories and conducted numerous experiments in an attempt to detect and understand dark matter, but so far, it has eluded direct observation. One of the leading candidates for dark matter is weakly interacting massive particles (WIMPs), which are predicted by certain particle physics theories. However, efforts to detect WIMPs directly have thus far been unsuccessful. In addition to direct detection experiments, scientists also study dark matter through its indirect effects on visible matter and radiation. For example, the gravitational lensing of light by dark matter has been observed, providing indirect evidence for its existence. Furthermore, the cosmic microwave background radiation, which is the afterglow of the Big Bang, contains subtle imprints of dark matter's influence on the early universe. In conclusion, dark matter is a perplexing and enigmatic component of the universe. Its existence isinferred from its gravitational effects, but its true nature and composition remain elusive. Scientists continue to investigate dark matter through a variety of methods, hoping to unlock the secrets of this mysterious substance. Understanding dark matter is not only essential for advancing our knowledge of the cosmos but also for unraveling the fundamental workings of the universe.。

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a r X i v :a s t r o -p h /0306203v 1 10 J u n 2003Structure of Dark Matter Halos From Hierarchical Clustering.III.Shallowing of The Inner CuspToshiyuki FukushigeDepartment of General Systems Studies,College of Arts and Sciences,University of Tokyo,3-8-1Komaba,Meguro-ku,Tokyo 153,JapanAtsushi KawaiFaculty of Human and Social Studies,Saitama Institue of Technology,1690Fusaiji,Okabe,Ohsato,Saitama 369-0293,JapanJunichiro MakinoDepartment of Astronomy,School of Sciences,University of Tokyo,7-3-1Hongo,Bunkyo-ku,Tokyo 117,JapanABSTRACTWe investigate the structure of the dark matter halo formed in the colddark matter scenarios by N-body simulations with parallel treecode on GRAPE cluster systems.We simulated 8halos with the mass of 4.4×1014M ⊙to1.6×1015M ⊙in the SCDM and LCDM model using up to 30million particles.With the resolution of our simulations,the density proﬁle is reliable down to 0.2percent of the virial radius.Our results show that the slope of inner cusp within 1percent virial radius is shallower than −1.5,and the radius wherethe shallowing starts exhibits run-to-run variation,which means the innermost proﬁle is not universal.Subject headings:cosmology:theory —dark matter —galaxies:clusters:general —methods:N-body simulations 1.IntroductionSince the ”ﬁnding”of the universal proﬁle by Navarro,Frenk,and White (1996,1997,hereafter NFW),the structure of dark matter halos formed through dissipationlesshierarchical clustering from cosmological initial setting has been explored by many researchers.NFW performed a number of N-body simulations of the halo formation using 10-20k particles and found that the proﬁle of dark matter halo could beﬁtted to a simple formula(hereafter,NFW proﬁle)ρ0ρ=.(2)(r/r0)1.5[1+(r/r0)1.5]Fukushige and Makino(2001,Paper I;2003,Paper II)performed two series of N-body simulations,and found that the halos have density cusps proportional to r−1.5,independent of the halo mass and cosmological models.On the other hands,other researchers obtained the slope of inner cusp shallower than −1.5and close to that in the NFW proﬁle.Jing and Suto(2000,2002)performed a series of N-body simulations and concluded that the power of the cusp depends on mass.It varies from−1.5for galaxy mass halo to−1.1for cluster mass halo.Klypin et al.(2001)obtained the slope at the center that could be approximated by r−1.5.They,however,argued that the NFWﬁt is still good up to their resolution limit.Power et al.(2003)simulated an LCDM galaxy-sized halo with3M particles and claimed that their circular velocity proﬁle obtained is in better agreement with the NFW proﬁle than with the M99proﬁle.The purpose of this paper is to explore the inner structure of the dark matter halo by means of N-body simulations with about10times higher mass resolution than that of previous simulations.We simulated the formation of8cluster-sized halos in the SCDM and LCDM models using parallel Barnes-Hut treecode on parallel GRAPE cluster.The structure of this paper is as follows.In section2,we describe the model of ourN-body simulation.In section3,we present the results of simulation.Section4is forconclusion and discussion.2.Simulation MethodWe consider two cosmological models,SCDM model(Ω0=1.0,h=0.5,σ8=0.6)andLCDM model(Ω0=0.3,λ0=0.7,h=0.7,σ8=1.0).Here,Ω0is the density parameter,λ0is the dimensionless cosmological constant,and H0=100h km/s·Mpc−1at the presentepoch.The amplitudes of the power spectrum in CDM models are normalized using thetop-hatﬁltered mass variance at8h−1Mpc according to the cluster abundance(Kitayama&Suto1997).We simulate the formation of the dark matter halos using the”re-simulation”method,which has been a standard method for the simulation of halo formation since NFW(1996).The procedure for setting the initial condition of halos are the same as that used in PaperII.Weﬁrst performed large scale cosmological simulations with3.7×106particles ina sphere of300h Mpc comoving radius.We regard spherical overdensity regions aroundlocal potential minima within r v as candidate halos.We deﬁne the radius r v such thatthe spherical overdensity inside is178Ω0.30times the critical density for SCDM and178Ω0.40times for LCDM model(Eke,Cole,Frenk1998).We selected8halos from the catalog of candidate halos.The selected halos aresummarized in Table1.We selected the three most massive halos and one halo randomlyfrom halo candidates lying within200h Mpc from the center in both models(so that theexternal tidalﬁeld can be included).We express a region within5r v from the center ofthe halo at z=0in the cosmological simulation with larger number of particles.We placeparticles whose mass is as same as that in the cosmological simulation in a sphere of∼100h Mpc comoving radius surrounding the high resolution region,in order to express theexternal tidalﬁeld.The total number of particles,N,is listed in Table1.The generationof initial densityﬂuctuation were done on HITACHI SR8000(1node)in InformationTechnology Center,University of Tokyo using COSMICS2package(Bertschinger2001).We integrate the system directly in the physical coordinates for both the cosmologicaland halo simulations.We used a leap-ﬂog integrator with shared and constant timestep.The step size for the cosmological simulation is∆t/(t H−t i)=1/1024and that for the halo simulation is1/4096.Here,t H is the Hubble time and t i is the time at which the simulationstarts.The gravitational softening is constant in the physical coordinates and the lengthεgrav is5kpc for the cosmological simulation,and1kpc for Runs S1,S2,L1,and L2,and 2kpc for other runs of halo simulations.The force calculation is done with the parallel Barnes-Hut tree code on GRAPE clusters (Kawai,Makino2003)1.GRAPE is a special-purpose computer designed to accelerateN-body simulations.The parallelization scheme we used are basically the same as Warren &Salmon’s(1993)Hashed Oct-Tree algorithm,except that we incorporated Barnes’(1990) modiﬁed algorithm.The modiﬁcation is necessary in order to make GRAPE work eﬃciently (Makino1991).We use only the dipole expansion and the opening parameterθ=0.4for the cosmological simulation andθ=0.5for the halo simulation.For high-resolution halo simulations,we used both a parallel GRAPE-5cluster at University of Tokyo and a parallel MDGRAPE-2cluster at RIKEN.The parallel GRAPE-5 cluster consists of8host computers(Pentium4/1.9GHz,i845)each of which has one GRAPE-5(Kawai et al.2000)board.The parallel MDGRAPE-2cluster consists of8host computers(Pentium4/2.2GHz,i850)each of which has one MDGRAPE-2(Susukita et al. 2003)board.For cosmological simulations we used one board GRAPE-5.The simulation presented below needs,for example in Run S2,∼300seconds per timestep,and thus one run(4096timesteps)is completed in350hours(wallclock time)with the GRAPE-5cluster.3.Results3.1.SnapshotsFigure1shows the particle distribution for Run S2at z=0.58and0.The length of the side for each panel is6.67Mpc.For these plots,we shifted the origin of coordinates to the position of the potential minimum.In Table1,we summarized the radius r v,the mass M v,and the number of particles N v within r v at z=0.3.2.Density ProﬁleFigures3and4show the density proﬁles of all runs at z=0for SCDM and LCDM models,respectively.The exception is Run L4,for which we plot the density proﬁle atz=0.06because the merging process occurs just near the center of halos at z=0.The position of the center of the halo was determined using the potential minimum and thedensity was averaged over each spherical shell whose width is log10(∆r)=0.0172.For the illustrative purpose,the densities are shifted vertically.We plot the densities by the thick(colored magenta in online edition)lines only if the criteria for two-body relaxation introduced in Paper I,t rel(r)/t>3,is satisﬁed,where t rel(r) is the local two-body relaxation time given by0.065v3t rel=3.3.1.Two-body relaxationWe test the reliability of the criterion(3)using the simulations of the same initial condition as used in Run S1but with several diﬀerent values for the total number of particles(N).Except for N,we used the same simulation parameters as in Run S1.Figure 6show the cumulative mass M r(r)within the radii,0.1,0.03,0.01,0.005,and0.003Mpc, as a function of time,for three simulations with29(Run S1),14and1million particles within r v.Figure7shows theﬁnal density proﬁles for three simulations.The vertical bars indicates the reliability limit obtained by the criterion(3).In Figure6,we can see that the cumulative mass evolution obtained in the simulations with29and14millions particle are in good agreement for r>0.01Mpc.This agreement indicates that our criterion(0.009Mpc for14millions particle run)gives a good reliability limit.In Figure7we can also see that the densityρobtained in the simulations with29 and14millions particle are in good agreement outside the reliability limit of14millions particle run(0.009Mpc).The agreement of the averaged density is somewhat worse than that of the density.This is because the averaged density is integrated quantity.Any error of the density inside the sphere of radius r aﬀects the average density at radius r.Recently,Power et al.(2003)proposed another reliability criterion for the two-body relaxation,given byt rel(r)8ln N(r) ρaveat t/(t H−t i)=0.78125.We plot the proﬁle at this time since,in the simulation with ∆t/(t H−t i)=1/2048,merging process occurs near the center of halos at around z=0.In theseﬁgures we can see that larger stepsize makes the central proﬁle shallower.Thedensity proﬁle outside of0.007Mpc converges even by adapting1/2048.Therefore,we can conclude that the stepsize of∆t/(t H−t i)=1/4096did not introduce any numerical artifact.Power et al.(2003)investigated inﬂuences of the large stepsize on the proﬁle,andshowed that the inﬂuence depends also on the softening length.They found that if potential√softening length is larger than an optimal length,ε≃4r v/=15 ∆tt c(r v)3.5.EvolutionFigures12and13show the growth of the density proﬁle for all runs.The virial radii and the masses within the virial radius at the redshift plotted are summarized in Table3. Weﬁt these proﬁles to the M99proﬁle.Theﬁtting procedure is as same as that for Figure 3.The scale radii r0obtained by theﬁtting are summarized in Table3.At the inner region(r<0.03Mpc),we can see the density keeps almost unchanged from relatively higher redshift for all runs.This fact also can be seen in the evolution of the cumulative mass shown in Figure6.This means that the density at the inner region is determined by that of the smaller halo that collapsed at higher redshift.The density proﬁle of the outer region is formed as the halo grows and shows universality.Moreover,the agreement with the M99proﬁle at higher redshift is very well down to the radius at which the cusp shallowing can be seen at z=0,independent of the cosmological model we simulated in this paper.Figure14shows the relation between the scale radius r0and densityρ0obtained by theﬁtting.We can see clearly an evolutionary pass along a line,ρ0∝r−1.50,also independent of the cosmological model.3.6.Diﬀerent FittingIn section3.2,we see that the agreement with the M99proﬁle is not good at the inner region(r<0.02Mpc),and also that with the NFW proﬁle is worse in the whole range of proﬁle in section3.4.Therefore,it is worthwhile toﬁt the results to other proﬁles.Here, we try toﬁt the results to two diﬀerent proﬁles.Firstly,weﬁt the results to a proﬁle that has an inner cusp shallower than that of the M99proﬁle and steeper than the NFW proﬁle[ﬁtting(1)],given asρ0ρn1=(7)C0(r/r c)β[1+(r/r c)]1.5−β[1+(r/r0)1.5]where1/C0=(r0/r c)β[1+r0/r c]1.5−β,(8) r c is another scale radius.Although this proﬁle includes more parameters toﬁt,it is based on the observation that two diﬀerent mechanisms might be working in the growth of the halo as suggested by the analyses in section3.5.In Figure15,weﬁt the density proﬁles to the proﬁle given by equation(7).Here,for simplicity,we setβ=0for all runs and,therefore,the equation(7)becomesρ0ρn2=Although we found interesting features in the inner structure of dark matter halo by new simulations with much higher resolution,we could not achieve theﬁnal understanding of the structure.One question remained is whether the CDM halo has aﬂat core or not. Another question is whether the same shallowing can be seen in the halo of galaxy or dwarf galaxy size.The origin of the inner structure is also still unclear.In order to answer these questions,we are now planning to perform larger simulations using new GRAPE cluster system.We are grateful to Yasushi Suto and Atsushi Taruya for many helpful discussions.We gratefully acknowledge the use of the initial condition generator in the publicly available code COSMICS2developed by E.Bertschinger.We would like to thank all people who have made contribution to the development of the MDGRAPE-2system at RIKEN.A part of numerical computations were carried out on the GRAPE system at ADAC(the Astronomical Data Analysis Center)of the National Astronomical Observatory,Japan. This research was partially supported by the Research for the Future Program(JSPS-RFTP 97P01102)and by the Grants-in-Aid(14740127and13440058)of Japan Society for the Promotion of Science.Part of this work is carried out while A.K.is a special postdoctal researcher of RIKEN.REFERENCESBarnes,J.E.1990,p.Phys.,87,161Barnes,J.E.,&Hut,P.1986,Nature,824,446Bertschinger,E.,2001,ApJS,137,1Eke,V.R.,Cole,S.,&Frenk C.S.1996,MNRAS,282,263Fukushige,T.,&Makino,J.1997,ApJ,477,L9Fukushige,T.,&Makino,J.2001,ApJ,557,533(Paper I)Fukushige,T.,&Makino,J.2003,ApJ,588,674(Paper II)Ghigna,S.,Moore,B.,Governato,F.,Lake,G.,Quinn,T.,&Stadel,J.2000,ApJ,544,616 Jing,Y.P.,&Suto,Y.2000,ApJ,529,L69Jing,Y.P.,&Suto,Y.2002,ApJ,574,538Kawai,A.,Fukushige,T.,Makino,J.,&Taiji,M.2000,PASJ,52,659Kawai,A.,Makino,J.,proc.of IAU Symposium208,2003,in pressKitayama,T.,&Suto,Y.1997,ApJ,490,557Klypin,A.,Kravtson,A.V,,Bullock,J.S.,&Primack,J.R.2001,ApJ,554,903 Makino,J.1991,PASJ,43,621Moore,B.,Governato,F.,Quinn T.,Statal,J.,&Lake,G.1998,ApJ,499,L5Moore,B.,Quinn T.,Governato,F.,Statal,J.,&Lake,G.1999,MNRAS,310,1147 Navarro,J.F.,Frenk,C.S.,&White,S.D.M.,1996,ApJ,462,563Navarro,J.F.,Frenk,C.S.,&White,S.D.M.,1997,ApJ,490,493Power,C.,Navarro,J.F.,Jenkins,A.,Frenk,C.S.,White,S.D.M.,Springel,V.,Stadal, J.,&Quinn,T.,2003,MNRAS,338,14Susukita,R.,Ebisuzaki,T.,Elmegreen,B.G.,Furusawa,H.,Kato,K.,Kawai,A., Kobayashi,Y.,Koishi,T.,McNiven,G.D.,Narumi,T.,Yasuoka,K.2003,submitted to p.Phys.Warren,M.S.,Salmon,J.K.,proc.of Supercompuing’93,1993,(IEEE Comp.Soc.),12Table1:Run PropertiesModel Run N(×106)m(107M⊙)1+z i1.58×1015 3.0829.21.21×10152.8431.21.21×10152.84 4.54.47×1014 2.03 6.9LCDM L162.8 3.8051.1 L259.9 2.6752.7L316.79.0147.5L413.5 5.6749.4 r0(Mpc)r0(Mpc)r0(Mpc)r c(Mpc)forρM99forρNFW forρn1forρn2S1 3.05S2 2.82S3 2.82S4 1.97L1 2.40L2 2.13L3 2.13L4 1.82Table3:Parameters at higher redshift Run zS1 3.30.651.6×10140.250.151.6×10130.0561.2S3 1.2S4 1.2L1 4.50.421.2×10140.200.0867.3×10120.051.9L3 1.9L4 1.9Fig.1.—Snapplots from Run S2at z=0.58.The length of the side is equal to6.67Mpc.Fig.2.—Same as Figure1,but at z=0.Fig. 3.—Density proﬁle of the halos for all runs of the SCDM model at z=0.Only the densities plotted in the thick lines(colored magenta in online edition)satisfy criterion(3) in section3.2at r<r v.The labels indicate the run name.The proﬁles except for Run S1 are vertically shifted downward by1,2,3dex for Runs S2,S3,and S4,respectively.The vertical bar above the proﬁles indicate0.01r v.The solid curves(colored blue)indicate thedensity proﬁle given by equation(2)(M99proﬁle).Fig.4.—Same as Figure3,but for the LCDM model.Fig.5.—Density proﬁles for all runs scaled by r0andρ0(Table2).The solid(colored bluein online edition)curves indicates the density proﬁle given by equation(2)(M99proﬁle).Fig.6.—Cumulative mass within radii,0.1,0.03,0.01,0.005,and0.003Mpc,as a function of time,for three simulations with29(Run S1,thick lines),14(intermediate thick,colored blue in online edition)and1(thin,colored red)million particles.Fig.7.—Density proﬁles for three simulations with29(Run S1,thick lines),14(intermediate thick,colored blue in online edition)and1(thin,colored red)million particles.The vertical bars indicate the reliability limits obtained by criterion(3)in section3.2.Fig.8.—Same as Figure 6,but for for three simulations with ∆t/(t H −t i )=1/4096(Run S1,thick lines),1/2048(intermediate thick,colored blue in online edition)and 1/1024(thin,colored red).Fig.9.—Density proﬁles for three simulations with ∆t/(t H −t i )=1/4096(Run S1,thick lines),1/2048(intermediate thick,colored blue in online edition)and 1/1024(colored red).The vertical bars indicate the reliability limits obtained by criterion (3)in section 3.2.Fig.10.—Density proﬁles for all runs.The solid curves(colored blue in online edition)indicate the density proﬁle given by equation(1)(NFW proﬁle).Fig.11.—Residuals(ρ−ρM99)/ρM99and(ρ−ρNFW)/ρNFW as a function of radius.Fig.12.—Evolution of density proﬁle for all runs of the SCDM model.The numbers near proﬁles indicate the redshift.The proﬁles at z=0are plotted by the thin lines.Only the densities plotted in the thick(colored magenta in online edition)lines satisfy criterion(3) in section3.2at r<r v.The solid curves(colored blue)indicate the density proﬁle given byequation(2)(M99proﬁle).Fig.13.—Same as Figure12,but for the LCDM model.Fig.14.—The scale densityρ0as a function of the scale radius r0(Table3)at the redshiftplotted in Figures12and13.The dashed line indicatesρ0∝r−1.50.Fig.15.—Density proﬁles for all runs at the inner region.The solid curves indicate the density proﬁles given by equations(6)(colored green in online edition)and(7)(coloredblue).Fig.16.—Residuals(ρ−ρn1)/ρn1and(ρ−ρn2)/ρn2as a function of radius.。