Dark matter substructure and the “missing satellite problem”

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Dark matter

substructure and the

“missing satellite problem”

Galaxy Formation & Evolution

25 May 2007

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Basic problem

• Galaxy formation models based on CDM

always predict too many galaxies at the faint end of the luminosity function compared

with reality

• White & Rees (1978) predict faint end of LF to increase as -1.75 (-1.67 with current estimates of power spectrum), much

steeper than observed LF (-1 to -1.25)

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Missing satellites: Semi- analytic models

• Early semi-analytic galaxy formation models

(SAMs) -- based on the ideas of White & Rees -- also predict too many dwarf galaxies

• Cole (1991)

• White & Frenk (1991)

1991ApJ...367...45C 1991ApJ...379...52W 1991ApJ...379...52W

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Missing satellites: N- body simulations

• Although White & Rees (1978) claim that all substructure in DM halos is erased through dynamical friction, this was conclusively shown to be false by Ghigna et al. (1998)

• White & Rees ran into the

“overmerging” problem of N-body simulations: course numerical

resolution leads to (unrealistic) 2- body relaxation

• Note that early works (including Ghigna et al.) focus on clusters of galaxies

We have analysed both simulations and present results for each run, except when this would lead to duplication of plots or text without any gain in information. In these cases, we follow the policy of showing results for RUN2 when haloes are used as tracers, and RUN1 when we also require information on their internal structure.

This is because RUN2 has better spatial resolution but the same mass resolution as RUN1, although the softening length used in RUN1 will help suppress two-body relaxation effects and yield less noisy profiles of small haloes. (With hindsight, this caution appears to be superfluous.)

3 C L U S T E R S U B S T R U C T U R E A N D H A L O I D E N T I F I C AT I O N

3.1 The density distribution in the cluster

In past work, haloes in dissipationless N-body simulations have usually spontaneously dissolved when entering clusters. Two physi- cal effects conspire with the finite numerical resolution to erase dark matter haloes in clusters (Moore, Katz & Lake 1996). Halos are heated by cluster tides and halo–halo encounters, thus losing mass as they move into the potential well. (In the following, ‘tidal disruption’ refers to the sum of these effects, unless we excplicitly state otherwise.) When the halo radius approaches !3 times the

‘core radius’ (owing to either a density plateau or gravitational softening), the halo dissolves. Hence, it takes very high resolution to retain dark matter substructures at a distance 100–200 kpc from the centre of a cluster. Our numerical parameters were chosen so that haloes would survive at these scales. The wealth of haloes retained in our simulated cluster is visible in Fig. 1

The upper panel is a map of the density distribution in a box of size R200 (see Section 3.2 for a precise definition), centred on the cluster and projected on to a plane. Each particle is plotted using a grey-scale according to the logarithm of the local density [defined using an SPH smoothing kernel over 64 particles in a code called

SMOOTH(Stadel & Quinn 1997, http ref: http://www-hpcc.astro.wa- shington.edu/tools)]. Only regions with density contrast d > 30 are shown. The cluster boundaries, set at R200, correspond to the con tours of the central bright region. Much of the mass inside R200lies in the dark matter haloes that we will analyse here. Their projected distribution is shown as a ‘circle plot’ in the lower panel of Fig. 1.

The radius of each circle is the halo radius (Section 3.3) in units of R200. Note that haloes of similar central densities (similar brightness in the density map) may have largely different radii depending on their distances from the centre of the cluster. It is remarkable that substructure haloes cover such a large fraction of the projected cluster area. Comparing the two panels of Fig. 1, we note that the density map gives an excessive impression of overmerging within the central parts of the cluster with respect to the actual projected distribution of the haloes.

3.2 Cluster properties and evolution

We define the cluster centre as the position of its most bound particle. This particle is within a softening length of the centre of the most massive halo found bySKID (see Section 3.3). The density profile of the RUN1 cluster calculated in spherical shells is shown in the upper panel of Fig. 2 (the solid line is for z ¼ 0 and dotted line for z ¼ 0:5). The cluster forms at z ! 0:8 from the mergers of many haloes along a filamentary structure and at z ¼ 0:5 it has not yet virialized since it still has quite a lumpy structure, but the global density profile is roughly similar to that measured 5 Gyr later at

z ¼0. (We shall compare properties of the substructure identified at both epochs.)

The lower panel of this figure shows the circular velocity profile VcðRÞ " ½GMðRÞ=Rÿ1=2, where MðRÞ is the mass within R. The virial radius of the cluster is defined as the distance R200 for which the average density enclosed, ¯rCðR200Þ, is 200 times the cosmic density, rcr; we obtain R200 ¼1:95 Mpc at z ¼ 0 and 1:2 Mpc at z ¼ 0:5.

The cluster is not spherical and has axial ratios that are roughly 2:1:1. (In the following we will always use units of kpc and km s¹1 for lengths and velocities, unless we explicitly state otherwise.)

Fig. 3 shows the growth of the cluster mass with redshift.

Defining the formation redshift of the cluster as that where it has accreted half of its final mass, zform! 0:8 for our cluster. This is slightly earlier than expected for an average cluster of this mass from the Press–Schechter theory (Press & Schechter 1974; Lacey &

148 S. Ghigna et al.

!1998 RAS, MNRAS 300, 146–162 Figure 1. Density map (upper panel) and circle plot of the halo radii within the virial radius of the cluster (taken here as the length unit) at z ¼ 0. (The density map is the projection of the mass in a box and contains a few haloes at the periphery of the clsuter that do not appear in the circle plot.)

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• Klypin et al. (1999) were the first to compare N-body

simulations directly with the satellites of the MW and M31:

too many little subhalos!

• by a factor of >10x

• Note that this was also found by Moore et al. (1999), who used

the (nearly) scale-free nature of the CDM power spectrum to

“scale” their cluster simulations to MW-sized halos

• the ΛCDM power spectrum is not quite scale-free, so some dependence on halo mass is

seen (Gao et al. 1995)

FIG. 4.ÈProperties of satellite systems within 200 h~1 kpc from the host halo. T op: The three-dimensional rms velocity dispersion of satellites vs. the maximum circular velocity of the central halo. Solid and open circles denote "CDM and CDM halos, respectively. The solid line is the line of equal satellite rms velocity dispersion and the circular velocity of the host halo. Middle: The number of satellites with circular velocity larger than 10 km s~1 vs. circular velocity of the host halo. The solid line shows a rough approximation presented in the legend. Bottom: The cumulative circular VDF of satellites. Solid triangles show average VDF of MW and Andromeda satellites. Open circles present results for the CDM simula- tion, while the solid curve represents the average VDF of satellites in the

"CDM simulation for halos shown in the upper panels. To indicate the statistics, the scale on the right y-axis shows the total number of satellite halos in the "CDM simulation. Note that while the numbers of massive satellites ([50 km s~1) agree reasonably well with the observed number of satellites in the Local Group, models predict about 5 times more lower mass satellites withVcirc\10È30km s~1.

FIG. 5.ÈSame as in Fig. 4, but for satellites within 400h~1 kpc from the center of a host halo. In the bottom panel we also show the cumulative velocity function for the Ðeld halos (halos outside of 400 h~1 kpc spheres around seven massive halos), arbitrarily scaled up by a factor of 75. The di†erence at large circular velocitiesVcirc[50 km s~1 is not statistically signiÐcant. Comparison between these two curves indicates that the veloc- ity functions of isolated and satellite halos are very similar. As for the satellites within the central 200 h~1 kpc (Fig. 4), the number of satellites in the models and in the Local Group agrees reasonably well for massive satellites with Vcirc[50 km s~1 but disagrees by a factor of 10 for low- mass satellites withVcirc\10È30km s~1.

TABLE 3

SATELLITES IN "CDM MODEL INSIDE R \ 200/400 h~1 kpc FROM CENTRAL HALO

Halo(km s~1)Vcirc Halo Mass(h~1M_) Number of Satellites Fraction of Mass in Satellites (km s~1)Vrms (km s~1)Vrotation

140.5 . . . 2.93] 1011 9/15 0.053/0.112 99.4/94.4 28.6/15.0

278.2 . . . 3.90] 1012 39/94 0.041/0.049 334.9/287.6 29.8/11.8

205.2 . . . 1.22] 1012 27/44 0.025/0.051 191.7/168.0 20.0/11.3

175.2 . . . 6.26] 1011 5/10 0.105/0.135 129.1/120.5 41.5/45.2

259.5 . . . 2.74] 1012 24/52 0.017/0.029 305.0/257.3 97.1/16.8

302.3 . . . 5.12] 1012 37/105 0.055/0.112 394.6/331.6 39.4/15.7

198.9 . . . 1.33] 1012 24/58 0.048/0.049 206.1/169.3 17.7/12.1

169.8 . . . 7.91] 1011 17/26 0.053/0.067 162.8/156.0 9.3/5.0

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The “Via Lactea” simulation of Diemand et al. (2007).

Still not fully converged in center: two-body relaxation still

destroying too many subhalos

4 Dark matter substructure in the MW

Fig. 2.—Projected dark matter density-squared map of our simulated Milky Way-size halo (“Via Lactea”) at the present epoch. The image covers an area of 800 × 600 kpc, and the projection goes through a 600 kpc-deep cuboid containing a total of 110 million particles.

The logarithmic color scale covers 20 decades in density-square.

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The biggest subhalos have their own subhalos!

Diemand, Kuhlen, & Madau 5

Fig. 3.—Projected dark matter density-square map of the four most massive subhalos within the simulated Milky Way host at the present epoch. Sub-substructure is clearly visible. Only dark matter particles within the tidal radius rt are used for the projections. Clockwise from top left: (Msub, rt, rVmax) = (9.8 × 109M!, 40.1 kpc, 7.6 kpc), (3.7 × 109M!, 33.4 kpc, 4.0 kpc), (3.0 × 109M!, 28.0 kpc, 4.9 kpc), and (2.4 × 109 M!, 14.7 kpc, 6.1 kpc). The mean subhalo densities within the tidal radius (in units of the cosmic background dark matter density) are 1002, 654, 904, and 4950, respectively. These values are related to the local matter density of the host (72, 46, 59 and 397 in the same units), and correlate only weakly with the subhalo distance from the Galactic center (345, 374, 280 and 185 kpc).

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• Subhalo mass function in

“Via Lactea” simulation still increasing at lowest masses

• roll off is from overmerging

• need to fix time steps in their N-body code!

• At lowest masses (lowest V max ) problem still very

acute in “Via Lactea”

simulation

Diemand, Kuhlen, & Madau 7

]

sub [M M

106 107 108 109 1010

) subN( > M

1 10 102

103

104

Fig. 5.— Cumulative subhalo mass functions within rvir (upper curves) and 0.1 rvir (lower curves). Solid crosses: Via Lactea run.

Dashed crosses: lower resolution run. Solid line: power-law fit, N (> Msub) = 0.0064 × (Msub/Mhalo)−1.

dance of subhalos does increase with resolution in the relevant Vmax range, but not dramatically, i.e. the to- tal abundance of satellite galaxies in these models seems not to be very affected by their lower numerical resolu- tion. But what would these two models predict about satellite galaxies within the newly resolved inner sub- halos? The Kravtsov et al. (2004) model allows galaxy formation only in subhalos that were relatively massive (> 109 M!) before they fell into the gravitational poten- tial of the host. In the Moore et al. (2006) scenario only the earliest forming halos above the atomic cooling mass at z > 12 become proto-galactic building blocks (many of these systems remain well below 109 M! at all times).

By z = 0 they have built up a halo of stars, globular clus- ters and a few surviving dwarfs. We followed the assem- bly history of the inner subhalos backwards in time and found that only two of them lie above the (time depen- dent) minimum mass from Kravtsov et al. (2004). The same two systems are also the only ones that form early enough to become luminous in the (Moore et al. 2006) scenario. In the inner 0.1 rvir both models thus predict two dwarf galaxies, in good agreement with the numbers observed around the Milky Way and M31, and leave the other three inner subhalos above 10 km s−1 dark. There- fore the new, missing inner satellites problem seems to be resolvable in the same way as the well known Local Group wide problem. The overabundance of inner sub- halos might not be a problem for CDM, but it leads to the new and interesting prediction of having a number of relatively large, dark CDM subhalos orbiting in the inner halo, i.e. in the same region where the Milky Way galaxy is located.

3.4. Mass fractions in substructure and gravitational lensing

Standard smooth gravitational lens models have difficulties in explaining the relative fluxes of multiply-imaged quasars (Mao & Schneider 1998;

Metcalf & Zhao 2002). The discrepancy between the predicted and observed flux ratios is commonly referred to as the “anomalous flux ratio problem”, and dark matter substructure within the lens halo is one of the leading interpretations of such anomaly (e.g.

Metcalf & Madau 2001; Chiba 2002; Dalal & Kochanek 2002; Bradac et al. 2004; Mao et al. 2004; Amara et al.

2006; Maccio et al. 2006). Rather than the total mass fraction, lensing observations are sensitive to the mass fraction in substructure projected through a cylinder of radius 5-10 kpc around the lens center. Figure 7 depicts the fraction of the host halo mass within a sphere of radius r that is bound up in substructure, fsub(< r), as well as the substructure mass fraction in cylindrical projection of radius R, fsub(< R), measured in our Via Lactea simulation. The radial distribution follows the subhalo number density profile given in Diemand et al.

(2004b), i.e. it is more extended than the overall mass distribution. In the vicinity of R = 10 kpc, the surface mass density for all subhalos with Msub < 109 M! can be approximated as fsub(< R) = 0.002 (R/10 kpc).

Our total projected surface densities are consistent with, but on the low side of, estimates from semiana- lytic models (Zentner & Bullock 2003), although we find significantly larger contributions from the smallest sub- halos at large projected radii. The total projected sur- face densities is lower than the few percent value that seems to be required to explain the anomalous flux ratios (Dalal & Kochanek 2002; Metcalf et al. 2004). Whether

8 Dark matter substructure in the MW

[km/s]

V

max

1 2 3 4 5 6 7 8 910 20 30 40 50

)

max

N( > V

1 10 10

2

10

3

[km/s]

V

max

1 2 3 4 5 6 7 8 910 20 30 40 50

)

max

N( > V

1 10 10

2

Fig. 6.— Cumulative peak circular velocity function for all subhalos within rvir (upper panel) and for the subpopulation within the inner 0.1 rvir (lower panel). Solid lines without error bars: Via Lactea run. Dashed lines: lower resolution run. Solid line: power-law fit N (> Vmax) = (1/48) × (Vmax/Vmax,host)−3. Solid lines with error bars: observed number of dwarf galaxy satellites around the Milky Way (as in Mateo (1998); Klypin et al. (1999), plus the recently discovered Ursa Major dwarf (Willman et al. 2005; Kleyna et al. 2005)), within rvir and within 0.1 rvir (only Sagittarius). The vertical error bars show the Poissonian

N scatter.

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• But more massive halos contain more

substructure (Gao et al.

2004)

• At fixed mass, amount of substructure

• decreases with halo concentration

• decreases with formation redshift

6 L. Gao, et al.

-5 -4 -3 -2 -1

0.1 1 10

-5 -4 -3 -2 -1

0.1 1 10

10 11 12 13

0.01

10 11 12 13

0.01

Figure 5.Mass functions at z = 0 for subhaloes within radius r200 of their parent haloes. In the top left-hand panel we plot differential subhalo abundance as a function of scaled subhalo mass, mn= msub/Mhalo, for three ranges of halo mass in our GIF2 simulation, for GA3n and for our 8 cluster resimulations. In the top right-hand panel, we plot the corresponding cumulative mass functions. In the bottom left-hand panel, we plot differential subhalo abundance normalized to the total mass of the parent haloes, !Mhalodn/dmsub".

The corresponding cumulative mass functions are shown in the bottom right-hand panel.

and 243 haloes, respectively. In this plot we also include subhalo abundance functions for GA3n and for our 8 cluster simulations. If halo populations of differing mass were just scaled copies of each other, these various abundance func- tions would all agree. In fact, however, the differential and cumulative normalized mass functions of Fig. 5 depend sys- tematically on halo mass. The subhalo abundance in high- mass haloes is clearly higher (at given scaled subhalo mass) than in low-mass haloes. The difference between the rich

cluster haloes and the galaxy halo GA3n is a factor of 2.

The cluster haloes also clearly have more abundant sub- haloes than the lowest mass haloes in our GIF2 simulation.

Our simulation data agree with semi-analytical modelling by Zentner & Bullock (2003). These authors argued that, on average, the subhalo mass fraction should increase with halo mass because high mass haloes were assembled more recently. A trend in this direction is also clearly present in in high resolution simulation data of Diemand et al. (2004),

8 L. Gao, et al.

Figure 8.The relation between subhalo abundance and the concentration and the formation redshift of haloes. The left-hand panel shows the number of subhaloes as a function of halo concentration, as measured by Vmax/V200, for our GIF2 and cluster simulations.

Only subhaloes containing more than 0.1 per cent of the mass of their parent are considered in compiling these statistics. The middle and right-hand panels show the same measure of subhalo abundance as a function of halo formation times defined as the redshifts when the most massive progenitor has 50 per cent and 25 per cent of the final mass respectively. Open hexagons are for halos in the mass range 3 × 1014h−1M"< Mhalo < 1015h−1M"; filled triangles are for halos with 1014h−1M"< Mhalo < 3 × 1014h−1M"; and open squares are for haloes with 3 × 1013h−1M"< Mhalo< 1014h−1M".

Fig. 7 shows the average mass fraction (within r200) in sub- haloes more massive than given msub for GIF2 and cluster haloes in our three ranges of halo mass. These curves show clear trends which can already be inferred from Fig. 5. The subhalo mass fractions appear to converge to well-defined values as the lower limit on subhalo mass is reduced, and the asymptotic value is larger for high-mass than for low- mass haloes. Convergence is a result of the effective slope of the differential abundance function being larger than −2, while the trend with halo mass results from the apparent universality of the abundance function at low masses (when normalized by halo mass) together with a dependence of the high-mass cut-off on halo mass.

The masses of individual subhaloes, and so the value of this asymptotic mass fraction, will depend systematically on the algorithm used to define the subhaloes. A variety of different subhalo identification schemes have been used in published studies and undoubtedly account in part for the wide range of subhalo mass fractions quoted. Notice also that since most of the subhalo mass is in the biggest objects, there is a large halo-to-halo variation (well over a factor of 2) in the overall subhalo mass fraction. We show this scatter through the error bars on selected points in the curve for the most massive haloes in Fig. 7. These give the rms scatter of the individual values for the 15 clusters averaged together to make this curve.

4.4 Dependence of subhalo populations on halo concentration and formation time

As demonstrated in Fig. 5, subhaloes tend to be more abun- dant in more massive haloes. In this section, we show that strong trends are also apparent with halo concentration and with halo formation time. Such systematics are not surpris- ing since Navarro, Frenk & White (1996, 1997) showed that more massive haloes form later and have lower concentra- tions. They demonstrated that the density profiles of CDM haloes are well described by a simple fitting function with two parameters, ρsand rs. Here rsis a characteristic radius where the logarithmic density profile slope is −2, and ρs is the mass density at rs. They also showed that these two quantities are strongly correlated, implying a relation be- tween concentration parameter c = r200/rs and halo mass.

More massive haloes are less concentrated. They argued that this is because more massive haloes typically form later.

They also showed that at given mass, haloes which form earlier have higher concentrations, a result which has been confirmed by subsequent studies (Wechsler et al. 2000; Bul- lock et al. 2001; Zhao et al. 2003a, 2003b). This suggests that haloes of similar concentration or formation time should have similar formation histories and so similar numbers of subhaloes.

In the left-hand panel of Fig. 8 we show the num- ber of subhaloes as a function of the concentration of the host, as measured by Vmax/V200. (Using this measure of halo concentration avoids fitting a model to our numerical data). For this comparison, we count only subhaloes with msub/Mhalo > 0.001. This ensures that our results are free

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“Anomalous” flux ratios in gravitational lenses

• In a gravitationally-lensed QSO, the ratios of multiple-image fluxes should be

constrained such that the sum of the fluxes of the outer images should equal the flux of the inner image -- but this is violated in

several lenses

Could be due to substructure in the lens

(Mao & Schneider 1998; Dalal & Kochanek

2002)

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• However, it now seems clear that roughly half of the lenses with “anomalous” flux ratios have small, optically-identified

companions that were not included in the initial models (e.g., McKean et al. 2007)

• ...so the evidence for CDM substructure

from gravitational lenses is still open for

debate...

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Basic problem

• CDM-like power spectra --- mass

fluctuations decreasing with increasing mass, with no low-mass cutoff --- always overproduce the number of small halos

• This is why both the (E)PS and N-body models show the “missing satellite

problem”

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Solutions?

Reionization?

Efstathiou (1992) suggested that strong UV flux from reionization would heat the gas in

small halos at high redshift to the point that it wouldn’t be able to cool and form stars,

thus preventing the formation of small dwarf

galaxies

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Reionization or merging?

• Kauffmann, White &

Guiderdoni (1993) suggested that either drastic merging of small galaxies (small halos) or drastic suppression of

cooling in small halos might suppress dwarf formation

• both quite severe

1993MNRAS.264..201K

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Hidden as HVCs?

• Blitz et al. (1999) suggested that the High Velocity Clouds around the MW might contain a

significant amount of dark matter

• Assuming that

HVCs are at

large distances

from MW

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Klypin et al. (1999) find that the number of HVCs is similar to the number of DM subhalos in their

simulation

• Unfortunately, it now appears that most HVCs lie fairly close to the MW, so cannot be massive

enough (e.g., van Woerden et al. 2006)

still some uncertainty about the compact HVCs,

though

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Reionization + tidal stripping?

• Bullock, Kravtsov &

Weinberg (2000) resurrected the

Efstathiou (1992) idea, combined it with semi- analytic models and

showed that perhaps reionization could

suppress dwarf formation...

• when also considering dynamical friction and tidal stripping

No. 2, 2000 REIONIZATION AND GALACTIC SATELLITES 519

function measured in the cosmological N-body simulation of KKVP99. The upper dashed lines show the velocity func- tion if the e†ects of dynamical friction and/or tidal dis- ruption are ignored. The analytic model reproduces the N-body results remarkably well. We have not tuned any parameters to obtain this agreement, although we noted above that plausible changes in the assumed initial circular radius Rc could change the analytic prediction by D50%.

The good agreement suggests that our analytic model cap- tures the essential physics underlying the N-body results.

An interesting feature of the model is that the subhalos surviving at z \ 0 are only a small fraction of the halos actually accreted, most of which are destroyed by tidal dis- ruption. We discuss implications of this satellite destruction in ° 4.

2.2. Modeling Observable Satellites

The second step in our model is to determine which of the surviving halos at z \ 0 will host observable satellite gal- axies. The key assumption is that, after the reionization redshift We adopt a threshold of zre, gas accretion is suppressed in halos with km s ~1, based on the vc\

vT. vT\30

results of Thoul & Weinberg (1996), who showed that galaxy formation is suppressed in the presence of a pho- toionizing background for objects smaller than D30 km s ~1. This threshold was shown to be insensitive to the assumed spectral index and amplitude of the ionizing back- ground (a similar result was found by Quinn et al. 1996).

Shapiro et al. (1997; see also Shapiro & Raga 2000) and Barkana & Loeb (1999) have suggested that very low mass systems ( vc[10 km s ~1) could lose the gas they have already accreted after reionization occurs, but we do not consider this possibility here.

The calculation of the halo velocity function in ° 2.1 is approximate but straightforward, and we have checked its validity by comparing it to N-body simulations. Determin- ing which of these halos are luminous enough to represent known dwarf satellites requires more uncertain assump- tions about gas cooling and star formation. We adopt a simple model that has two free parameters: the reionization redshift zre and the fraction f \ M(zre)/Ma of a subhaloÏs mass that must be in place by zre in order for the halo to host an observable galaxy.

The value of zre is constrained to zreZ5 by observations of high- z quasars (e.g., Songaila et al. 1999) and to zre[50 by measurements of small-angle cosmic microwave back- ground (CMB) anisotropies, assuming typical ranges for the cosmological parameters (e.g., Griffiths, Barbosa, & Liddle 1999). The value of f is constrained by the requirement that observable halos have mass-to-light ratios in the range of observed dwarf satellites. For the subset of (dwarf irregular) satellite galaxies with well-determined masses, the mass-to- light ratios span the range M/L V^5È30 (Mateo 1998);

dwarf spheroidals have similar M/L V but with a broader range and larger observational uncertainties. We can esti- mate M/L V for model galaxies by assuming that they accrete a baryon mass fMa()b/)m) before zre and convert that accreted gas with efficiency v * into a stellar population with mass-to-light ratio A M M * / L V, obtaining

L V B \ f ~1 A )m

)b BA M *

L V B v *~1 F o . (3) The factor Fo is the fraction of the haloÏs virial mass Ma that lies within its Ðnal optical radius (which may itself be

a†ected by tidal truncation). For the M/L V values quoted above, the optical radius is typically D2 kpc (Mateo 1998), and representative mass proÐles of surviving halos imply kpc) ^ 0.5; however, this factor must be considered Fo(2 uncertain at the factor of 2 level. The value of v * is also uncertain because of the uncertain inÑuence of supernova feedback, but by deÐnition v * ¹ 1. Adopting a value typical for galactic disk stars (Binney & Mer- (M * / L V)^0.7

riÐeld 1998), ( from Burles & Tytler 1998), )m/)b)^7 (based on )m\0.3, h \ 0.7, and and

)bh2^0.02 v * \ 0.5,

we obtain Matching the mass-to- Fo\0.5, ( M/L V)^5f ~1.

light ratios of typical dwarf satellite galaxies then implies f D 0.3. With the uncertainties described above, a range f D 0.1È0.8 is plausible, and the range in observed M/L V could reÑect in large part the variations in f from galaxy to galaxy. Values of f [ 0.1 would imply excessive mass-to- light ratios, unless the factor Fo can be much smaller than we have assumed.

In sum, the two parameters that determine the fraction of surviving halos that are observable are zre and f, with plaus- ible values in the range zreD5È50 and f D 0.1È0.8. For a given subhalo of mass Ma and accretion redshift za, we use equation (2.26) of LC93 to probabilistically determine the redshift zf when the main progenitor of the subhalo was Ðrst more massive than Mf\fMa. We associate the subhalo with an observable galactic satellite only if zfºzre.

3 . RESULTS

Figure 2 shows results of our model for the speciÐc choices of zre\8 and f \ 0.3. The thin solid line is the velocity function of all surviving subhalos at z \ 0, repro- duced from Figure 1. The thick line and the shaded region

FIG. 2.ÈCumulative velocity function of all dark matter subhalos sur- viving at z \ 0 (thin solid line) and ““ observable ÏÏ halos ( zf[zre) (thick solid line with shading) for the speciÐc choice of zre\8 and f \ 0.3. The velocity function represents the average over 300 merger histories for halos of mass h ~1 The error bars and shading show the

Mvir(z\0)\1.1]1012 M

_ .

dispersion measured from di†erent merger histories. The observed velocity

function of satellite galaxies around the Milky Way and M31 is shown by

triangles.

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• Kravtsov, Gnedin & Klypin

(2004) extended this analysis using high-resolution N-body models and found that the

major effect is due to tidal stripping:

• halos that form dwarf

galaxies at z=0 are those that were much more massive at z re but were stripped by the host halo

• because they were massive early, they could form stars before reionization

radius. In this case, we set rt¼ Rvir. Throughout this paper, we denote the minimum of the virial mass and mass within rt

simply as M. For each halo we also construct the circular velocity profile Vc(r)¼ GM(<r)=r½ #1=2and compute the max- imum circular velocity profile Vm.

Figure 2 shows the particle distribution in the halo G1 at z¼ 0 along with the halos (circles) identified by the halo finder. The particles are color coded on a gray scale according to the logarithm of their density to enhance visibility of sub- structure clumps. The radius of the largest circle indicates the actual virial radius, Rvir, of the host halo (Rvir¼ 298 h$1kpc);

the radii of the other halos are the minimum of the truncation radius rtand Rvir. The figure demonstrates that the algorithm is efficient in identifying the substructure down to small masses.

4. CONSTRUCTING TRAJECTORIES

The halo finder described above was run at the 96 saved epochs between z¼ 10 and 0 with a typical spacing of

%(1 2) ; 108 yr between outputs. For each epoch, the halo

finder produced a halo catalog with positions, velocities, radii rh¼ min(rt; rvir), masses m(<rh), maximum of the circular velocity profile Vm, and the radius at which the maximum occurs, rmax. In addition, for each halo we save indices of all gravitationally bound DM particles located within rh.

This information is used to identify the progenitors of halos at successive epochs. Specifically, for a current epoch zi, starting at z¼ 0, we search progenitors for each halo at sev- eral previous epochs zi j as follows. First, we select a given fraction, fbd, of the most bound particles of the halos at the epochs of consideration. We then compare the fraction of these particles that is common to all pairs of halos at suc- cessive epochs and assume that the halo with the highest common fraction is the progenitor. The trajectories used in this study were constructed using fbd¼ 0:25. As the halo catalogs may miss some halos, especially near the completeness limit of the simulation, if the progenitor is not found at the previous epoch, we need to search at the earlier epoch, etc. In particular, if a halo is located within the search radius rsr of some larger

Fig. 3.—Three examples of the evolution of satellites of a MW-sized host (different columns). Top, Proper distance between the satellite and the center of the host halo as a function of time; middle, tidal force experienced by the object, calculated directly from gravitational potential, shown by the solid line. Dotted line shows the equivalent tidal force from the host halo with the host density profile approximated by an NFW model. Dashed line shows the contributions of all neighboring halos, including the host, with the density profiles of halos approximated by an NFW model with rmaxand Vmas measured in the simulation. See Appendix A for details on the tidal force calculation. Bottom, Maximum circular velocity Vm(solid lines) and bound mass m(<rt) (dashed lines) as a function of time. The three objects show different types of evolution: dramatic early stripping with a relatively quiescent evolution afterward (left), continuous dramatic tidal stripping (middle), and weak stripping and quiescent evolution (right).

DWARFS AND MISSING SATELLITES 485

No. 2, 2004

We distribute the gas on a spherically symmetric grid of 50 radial shells, according to the surface density of an exponential disk: !

g

(r) ¼ !

0

exp ( "r=r

d

). We use the observed Schmidt law of star formation to estimate the SFR: ˙ !

#

¼ 2:5 ; 10

"4

( !

g

=1 M

$

pc

"2

)

1:4

M

$

kpc

"2

yr

"1

. Only the shells above the threshold !

g

> !

th

% 5 M

$

pc

"2

form stars (Kennicutt 1998).

3. The scale length of the disk is determined by its an- gular momentum. For a rotationally supported disk it is approximately r

d

¼ 2

"1=2

k r

vir

. The value of the angular momentum parameter is drawn randomly from the proba- bility distribution,

p( k)dk ¼ 1 ffiffiffiffiffiffi p 2!

"

k

exp

"

" ln k= ¯k 2"

k2

# d k

k ; ð3Þ

with ¯ k ¼ 0:045 and "

k

¼ 0:56, according to the latest mea- surement by Vitvitska et al. (2002). This is a key assumption of the semianalytic models of galaxy formation.

However, small halos at high redshift could cool by atomic hydrogen only to about 10

4

K. If their virial temperature is only slightly above that equilibrium temperature, the gas would not be able to dissipate enough to reach a rotationally supported state. Instead, its distribution would be more ex- tended, which can have important implications for the star formation with a density threshold !

th

. This effect is partic- ularly important for dwarf halos.

We model the effect of inefficient dissipation by adopt- ing the expansion factor that depends on the ratio of the virial temperature to the equilibrium temperature 10

4

K. The gas would reach a Boltzmann distribution with the density M =r

3

/ exp (""=kT), where " is the potential energy. Using

the maximum circular velocity instead of the temperature and ignoring the slow variation of the potential, we can express the scale length of the gas as r

d

/ exp(c(V

4

=V

m

)

2

), where c is a normalization factor and V

4

¼ 16:7 km s

"1

is the virial ve- locity corresponding to T

vir

¼ 10

4

K. We find that c ¼ 10 is a best fit to the abundance and radial distribution of the LG galaxies (see x 6.2). This scaling also provides a good de- scription of the extent of the gas within halos in the cosmo- logical galaxy formation simulation described in Kravtsov &

Gnedin (2003). Thus, we set the size of the gaseous disk at each time step to be

r

d

¼ 2

"1=2

kr

vir

e

10ðV4=VmÞ2

: ð4Þ Of course, r

d

is not allowed to exceed the tidal radius of the halo, r

t

. The gas in large halos with V

m

3 V

4

can cool effi- ciently and reach rotational support, but for small halos with V

m

k V

4

the extended distribution reduces the central con- centration of the gas and hinders star formation.

4. Strong tidal forces, such as in the interacting or merging galaxies, may lead to a burst of star formation throughout the dwarf galaxy. The association of starbursts with strong peaks of the tidal force is motivated by theoretical models (Mayer et al. 2001a) and, to a certain extent, by observations (Zaritsky

& Harris 2004). The latter suggest that the tidally triggered star formation in the SMC can be accurately modeled as an instantaneous burst of star formation. Zaritsky & Harris (2004) find the best fit to their data when the SFR varies as r

"4:6

with the distance to the Galaxy. The tidal interaction parameter, I

tid

(see eq. [A7]), which reflects the integrated effect of a single tidal shock, is the most natural candidate for the parameterization of the tidally triggered SFR. Ignoring the adiabatic correction, it varies with the distance to the perturber approximately as I

tid

/ r

"4

(but see the discussion in x 6.2).

Fig. 8.—Fraction of satellites within a certain distance from the center of their host galaxy. The solid lines show distributions of the#CDM satellites in the three galactic halos, while the connected stars show the distribution of dwarf galaxies around the MW. The figure shows that radial distribution of observed satellites is more compact than that of the overall population of dark matter satellites. The dashed lines show distributions for the luminous satel- lites in our model (x 6). The population of luminous satellites is the same in this and previous figures.

Fig. 7.—Cumulative velocity function of the dark matter satellites in the three galactic halos (solid lines) compared with the average CVF of dwarf galaxies around the MW and Andromeda galaxies (stars). For the objects in simulations, Vcirc is the maximum circular velocity, while for the LG galaxies it is the circular velocity measured either from the rotation curve or from the line-of-sight velocity dispersion, assuming isotropic velocities. Both observed and simulated objects are selected within the radius of 200 h"1 kpc from the center of their host. The dashed lines show the velocity function for the luminous satellites in our model described inx 6. The minimum stellar mass of the luminous satellites for the three hosts ranges from (105 to (106 M$, comparable to the observed range.

KRAVTSOV, GNEDIN, & KLYPIN

490 Vol. 609

(20)

• Moreover, this idea also predicts (roughly) the correct spatial

distribution of (visible) dwarf satellites

• More concentrated than DM subhalos

We distribute the gas on a spherically symmetric grid of 50 radial shells, according to the surface density of an exponential disk: ! g (r) ¼ ! 0 exp ( "r=r d ). We use the observed Schmidt law of star formation to estimate the SFR: ˙ ! # ¼ 2:5 ; 10 "4 (! g =1 M $ pc "2 ) 1:4 M $ kpc "2 yr "1 . Only the shells above the threshold ! g > ! th % 5 M $ pc "2 form stars (Kennicutt 1998).

3. The scale length of the disk is determined by its an- gular momentum. For a rotationally supported disk it is approximately r d ¼ 2 "1=2 k r vir . The value of the angular momentum parameter is drawn randomly from the proba- bility distribution,

p( k)dk ¼ 1 ffiffiffiffiffiffi p 2!

" k exp

"

" ln k= ¯k 2" k 2

# d k

k ; ð3Þ

with ¯ k ¼ 0:045 and " k ¼ 0:56, according to the latest mea- surement by Vitvitska et al. (2002). This is a key assumption of the semianalytic models of galaxy formation.

However, small halos at high redshift could cool by atomic hydrogen only to about 10 4 K. If their virial temperature is only slightly above that equilibrium temperature, the gas would not be able to dissipate enough to reach a rotationally supported state. Instead, its distribution would be more ex- tended, which can have important implications for the star formation with a density threshold ! th . This effect is partic- ularly important for dwarf halos.

We model the effect of inefficient dissipation by adopt- ing the expansion factor that depends on the ratio of the virial temperature to the equilibrium temperature 10 4 K. The gas would reach a Boltzmann distribution with the density M =r 3 / exp (""=kT), where " is the potential energy. Using

the maximum circular velocity instead of the temperature and ignoring the slow variation of the potential, we can express the scale length of the gas as r d / exp(c(V 4 =V m ) 2 ), where c is a normalization factor and V 4 ¼ 16:7 km s "1 is the virial ve- locity corresponding to T vir ¼ 10 4 K. We find that c ¼ 10 is a best fit to the abundance and radial distribution of the LG galaxies (see x 6.2). This scaling also provides a good de- scription of the extent of the gas within halos in the cosmo- logical galaxy formation simulation described in Kravtsov &

Gnedin (2003). Thus, we set the size of the gaseous disk at each time step to be

r d ¼ 2 "1=2 kr vir e 10 ðV

4

=V

m

Þ

2

: ð4Þ Of course, r d is not allowed to exceed the tidal radius of the halo, r t . The gas in large halos with V m 3 V 4 can cool effi- ciently and reach rotational support, but for small halos with V m k V 4 the extended distribution reduces the central con- centration of the gas and hinders star formation.

4. Strong tidal forces, such as in the interacting or merging galaxies, may lead to a burst of star formation throughout the dwarf galaxy. The association of starbursts with strong peaks of the tidal force is motivated by theoretical models (Mayer et al. 2001a) and, to a certain extent, by observations (Zaritsky

& Harris 2004). The latter suggest that the tidally triggered star formation in the SMC can be accurately modeled as an instantaneous burst of star formation. Zaritsky & Harris (2004) find the best fit to their data when the SFR varies as r "4:6 with the distance to the Galaxy. The tidal interaction parameter, I tid (see eq. [A7]), which reflects the integrated effect of a single tidal shock, is the most natural candidate for the parameterization of the tidally triggered SFR. Ignoring the adiabatic correction, it varies with the distance to the perturber approximately as I tid / r "4 (but see the discussion in x 6.2).

Fig. 8.—Fraction of satellites within a certain distance from the center of their host galaxy. The solid lines show distributions of the #CDM satellites in the three galactic halos, while the connected stars show the distribution of dwarf galaxies around the MW. The figure shows that radial distribution of observed satellites is more compact than that of the overall population of dark matter satellites. The dashed lines show distributions for the luminous satel- lites in our model ( x 6). The population of luminous satellites is the same in this and previous figures.

Fig. 7.—Cumulative velocity function of the dark matter satellites in the three galactic halos (solid lines) compared with the average CVF of dwarf galaxies around the MW and Andromeda galaxies (stars). For the objects in simulations, V

circ

is the maximum circular velocity, while for the LG galaxies it is the circular velocity measured either from the rotation curve or from the line-of-sight velocity dispersion, assuming isotropic velocities. Both observed and simulated objects are selected within the radius of 200 h

"1

kpc from the center of their host. The dashed lines show the velocity function for the luminous satellites in our model described in x 6. The minimum stellar mass of the luminous satellites for the three hosts ranges from (10

5

to (10

6

M

$

, comparable to the observed range.

KRAVTSOV, GNEDIN, & KLYPIN

490 Vol. 609

(21)

• Moore et al. (2006)

show that this is due to the fact that the

surviving (luminous)

satellites correspond to some of the highest

peaks in the original density field

• see also Gao et al.

(2004)

Globular clusters and satellite galaxies 565

Figure 1. The high-redshift and present-day mass distribution in a region that forms a single galaxy in a hierarchical CDM Universe. The upper panel shows the density distribution at a redshift z = 12 from a region that will form a single galaxy at z = 0 (lower panel). The blue–pink colour scale shows the density of dark matter whilst the green regions show the particles from protogalaxies with virial temperature above 10

4

K that have collapsed at this epoch. These peaks have masses in the range 10

8

–10

10

M!. The lower panel shows same mass distribution at z = 0. Most of the rare peaks are located towards the centre of the galaxy today. The squares in both panels indicate those first objects that survive the merging process and can be associated with the visible satellite galaxies today orbiting within the final galactic mass halo. Most of the subhaloes stay dark since they collapse later after reionization has increased the Jeans mass.

is rapid and local metal enrichment occurs from stellar evolution.

Metal-poor Population II stars form in large numbers in haloes above M

H

≈ 10

8

[(1 + z)/10]

−3/2

M! (virial temperature 10

4

K), where gas can cool efficiently and fragment via excitation of hydrogen Lyα. At z > 12, these correspond to >2.5σ peaks of the initial Gaussian overdensity field: most of this material ends up within the inner few kpc of the Galaxy. Within the ≈1 Mpc turn-around region, a few hundred such protogalaxies are assembling their stellar systems (Kravtsov & Gnedin 2005). Typically 95 per cent of these first structures merge together within a time-scale of

a few Gyr, creating the inner Galactic dark halo and its associated old stellar population.

With an efficiency of turning baryons into stars and globular clus- ters of the order of f

= 10 per cent, we successfully reproduce the total luminosity of the old halo population and the old dwarf spheroidal satellites. The fraction of baryons in dark matter haloes above the atomic cooling mass at z = 12 exceeds f

c

= 1 per cent. A normal stellar population with a Salpeter-type initial mass function emits about 4000 hydrogen-ionizing photons per stellar baryon. A star formation efficiency of 10 per cent therefore implies the emis- sion of 4 000 × f

× f

c

∼ a few Lyman-continuum photons per baryon in the Universe. This may be enough to photoionize and drive to a higher adiabatic vast portion of the intergalactic medium, thereby quenching gas accretion and star formation in nearby low- mass haloes.

3 C O N N E C T I O N TO G L O BU L A R C L U S T E R S A N D H A L O S TA R S

The globular clusters that were once within the merging protogalax- ies are so dense that they survive intact and will orbit freely within the Galaxy. The surviving protogalaxies may be the precursors of the old satellite galaxies, some of which host old globular clusters such as Fornax, whose morphology and stellar populations are de- termined by ongoing gravitational and hydrodynamical interactions with the Milky Way (e.g. Mayer et al. 2005).

Recent papers have attempted to address the origin of the spatial distribution of globular clusters (e.g. Parmentier & Grebel 2005;

Parmentier & Gilmore 2005). Most compelling for this model and one of the key results in this paper is that we naturally reproduce the spatial clustering of each of these old components of the galaxy. The radial distribution of material that formed from >2.5σ peaks at z >

12 now falls off as ρ( r) ∝ r

−3.5

within the Galactic halo – just as the observed old halo stars and metal-poor globular clusters (cf. Fig. 2).

10–2 10–1 100

10–2 10–1 100 101 102 103 104 105 106

r/rvir ρ(r) / ρ crit.

all

2.5σ

3.5σ

stellar halo ~ r–3.5 globular clusters dwarf galaxies

Figure 2. The radial distribution of old stellar systems compared with rare peaks within a z = 0 $CDM galaxy. The thick blue curve is the total mass distribution today. The labelled green curves show the present-day distribu- tion of material that collapsed into 1, 2, 2.5, 3 and 3.5σ peaks at a redshift z = 12. The circles show the observed spatial distribution of the Milky Way’s old metal-poor globular cluster system. The dashed line indicates a power law ρ( r) ∝ r

−3.5

which represents the old halo stellar population. The squares show the radial distribution of surviving 2.5σ peaks which are slightly more extended than the overall Navarro–Frenk–White-like mass distribution, in good agreement with the observed spatial distribution of the Milky Way’s satellites.

(C

2006 The Authors. Journal compilation

(C

2006 RAS, MNRAS 368, 563–570

Figure

Updating...

References

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