The Aquarius Project: the subhaloes of galactic haloes

Authors


E-mail: volker@mpa-garching.mpg.de

ABSTRACT

We have performed the largest ever particle simulation of a Milky Way sized dark matter halo, and present the most comprehensive convergence study for an individual dark matter halo carried out thus far. We have also simulated a sample of six ultrahighly resolved Milky Way sized haloes, allowing us to estimate the halo-to-halo scatter in substructure statistics. In our largest simulation, we resolve nearly 300 000 gravitationally bound subhaloes within the virialized region of the halo. Simulations of the same object differing in mass resolution by factors of up to 1800 accurately reproduce the largest subhaloes with the same mass, maximum circular velocity and position, and yield good convergence for the abundance and internal properties of dark matter substructures. We detect up to four generations of subhaloes within subhaloes, but contrary to recent claims, we find less substructure in subhaloes than in the main halo when regions of equal mean overdensity are compared. The overall substructure mass fraction is much lower in subhaloes than in the main halo. Extrapolating the main halo's subhalo mass spectrum down to an Earth mass, we predict the mass fraction in substructure to be well below 3 per cent within 100 kpc, and to be below 0.1 per cent within the solar circle. The inner density profiles of subhaloes show no sign of converging to a fixed asymptotic slope and are well fitted by gently curving profiles of Einasto form. The mean concentrations of isolated haloes are accurately described by the fitting formula of Neto et al. down to maximum circular velocities of 1.5 km s−1, an extrapolation over some five orders of magnitude in mass. However, at equal maximum circular velocity, subhaloes are more concentrated than field haloes, with a characteristic density that is typically ∼2.6 times larger and increases with decreasing distance from halo centre.

1 INTRODUCTION

A major puzzle in cosmology is that the main matter component in today's Universe appears to be in the form of a yet-undiscovered elementary particle whose contribution to the cosmic density is more than five times that of ordinary baryonic matter (e.g. Komatsu et al. 2008). This particle interacts extremely weakly with atoms and photons, so that gravity alone has affected its distribution since very early times. Recent observations have established a standard paradigm in which dark matter emerged from the early Universe with negligible thermal velocities and a Gaussian and scale-free distribution of density fluctuations. In this ‘cold dark matter’ (CDM) hypothesis, quantum fluctuations during a very early period of cosmic inflation determine the statistics of the dark matter distribution at early epochs when the Universe was almost uniform (Guth 1981; Hawking 1982; Starobinsky 1982; Bardeen, Steinhardt & Turner 1983). Galaxies form from these initial conditions through the condensation of gas at the centres of a hierarchically aggregating population of quasi-equilibrium dark matter haloes (White & Rees 1978; White & Frenk 1991).

When the effects of the baryons can be neglected, the non-linear growth of dark matter structure is a well-posed problem where both the initial conditions and the evolution equations are known. This is an N-body problem par excellence. The faithfulness of late-time predictions (which must be confronted directly with observation to test the paradigm) is limited purely by numerical technique and by the available computing resources.

Over the past two decades, numerical simulations have played a pivotal role in establishing the viability of the CDM paradigm (e.g. Davis et al. 1985; Frenk et al. 1988; Warren et al. 1992; Cen et al. 1994; Gelb & Bertschinger 1994; Hernquist et al. 1996; Jenkins et al. 2001; Wambsganss, Bode & Ostriker 2004). They have led to the discovery of a universal internal structure for dark matter haloes (Navarro, Frenk & White 1996, 1997, hereafter NFW), and they have supplied precise predictions for the expected large-scale structure of the Universe. The matter distribution on scales from ∼50 kpc to the size of the observable Universe and the galaxy population predicted by hierarchical CDM scenarios have been compared directly with a wide array of observations. A recent example is the ‘Millennium Run’ (Springel et al. 2005), still one of the largest cosmological simulations ever carried out, which followed the formation and evolution of over 10 million galaxies by post-processing the stored simulation outputs (Bower et al. 2006; Croton et al. 2006). So far, the ΛCDM paradigm has passed these tests successfully, particularly those that consider the large-scale matter distribution.

Given this success in reproducing the large-scale structure of the Universe, it is important to test CDM predictions also on smaller scales, not least because these are sensitive to the nature of the dark matter. Indeed, a number of serious challenges to the paradigm have emerged on the scale of individual galaxies and their central structure. The realization that CDM haloes have cuspy dark matter density profiles led to a fierce debate about whether these are consistent with the rotation curves observed for low surface brightness, apparently dark matter dominated galaxies (Flores & Primack 1994; Moore 1994; McGaugh & de Blok 1998; de Blok, McGaugh & Rubin 2001; Hayashi et al. 2004; Hayashi & Navarro 2006). The abundance of small dark matter subhaloes predicted within CDM haloes has also drawn much attention (Klypin et al. 1999; Moore et al. 1999a). The total number is much larger than the number of known satellite galaxies surrounding the Milky Way, even accounting for the many recently discovered faint systems. It is still unclear whether this reflects an absence of the predicted low-mass objects or merely the fact that no stars were able to form within them. The issue of dark matter substructure within haloes is made more urgent by the prospect of observing dark matter particles in the near future, either by their annihilation radiation (e.g. Bergström, Ullio & Buckley 1998), or through direct detection in experiments here on Earth (reviewed e.g. by Gaitskell 2004). For both, a precise and quantitative understanding of the small-scale dark matter distribution within our Galaxy is needed.

Our Aquarius Project addresses these questions by studying the highly non-linear structure of Galaxy-sized CDM haloes in unprecedented detail using state-of-the art numerical simulations. We are particularly interested in the inner regions of these haloes and of their substructures, where the density contrast exceeds 106 and the astrophysical consequences of the nature of dark matter may be most clearly apparent. Quantifying such structure reliably through simulation is an acute challenge to numerical technique. In order to address this challenge, we use a newly developed version of our parallel simulation code, gadget-3 (based on Springel, Yoshida & White 2001b; Springel 2005), which allows us to cover an unprecedented dynamic range at high numerical accuracy. We carefully validate our simulation techniques and establish their range of numerical convergence through systematic convergence studies, thereby building confidence in the reliability of our results.

In order to evaluate the scatter in structural properties between haloes, we have simulated six different systems at high resolution, each having between 160 and 224 million particles within r50, the radius with mean enclosed overdensity 50 times the critical value.1 Each of these simulations is better resolved than any previously published high-resolution halo simulation except the very recent ‘Via Lactea II’ run of Diemand et al. (2008) which has 470 million particles within r50. For one of our haloes, we have increased the resolution by a further factor of 8, pushing the particle number within r50 to 1.47 billion. The gravitational softening length of this largest run is just 20.5 pc. We collectively refer to this suite of simulations as the Aquarius Project.

In the present paper, we describe our simulation techniques and analyse a number of basic properties of our z= 0 haloes. We focus in particular on the abundance of substructures, their radial distribution, and their internal density profiles. We also give results for the concentration of substructures, and for the fraction of the mass they contain at different radii. In a companion paper (Springel et al. 2008), we study implications for the detectability of dark matter annihilation within the Milky Way's dark matter halo, and in Navarro et al. (2008) we study the structure of the central density cusps of the main haloes. Future papers will study the evolution of our haloes and their substructure.

This paper is organized as follows. In Section 2, we introduce our simulation set and describe our numerical techniques. The abundance of dark matter substructures and their radial distribution within our haloes are analysed in Section 3. Then, in Section 4, we turn to an analysis of the abundance of substructure within subhaloes. In Section 5, we consider the density profiles of subhaloes and their concentrations. Finally, we summarize our conclusions in Section 6.

2 SIMULATION SET AND NUMERICAL TECHNIQUES

All our simulations follow halo formation within a periodic cube of side 100 h−1  Mpc ≃ 137  Mpc in a cosmology with parameters Ωm= 0.25, ΩΛ= 0.75, σ8= 0.9, ns= 1 and Hubble constant H0=100 h km s−1 Mpc−1= 73 km s−1 Mpc−1. These cosmological parameters are the same as used in the Millennium Simulation project, and are consistent with the current set of cosmological constraints within their uncertainties, in particular those from the Wilkinson Microwave Analysis Probe 1- and 5-yr data analyses.

2.1 Setting the initial conditions

The linear power spectrum used for making the initial conditions is based on a transfer function made by cmbfast (v4.5.1; Seljak & Zaldarriaga 1996) with Ωbaryon= 0.045. The transfer function was evaluated at z= 0 where the CDM and baryon transfer functions are virtually identical for wavenumbers k up to log10(kh/Mpc) = 2.5. For higher wavenumbers the effects of pressure are important for the baryons and are reflected in a feature present also in the CDM transfer function. Given that our simulations model only a CDM component and cannot account for the separate evolution of the baryons, we have chosen to ignore this baryon induced feature for our CDM power spectrum and have instead created a smooth composite transfer function (explained in detail below) which lacks this feature. In practice, for the Aquarius simulations, the difference between using our composite transfer function and the cmbfast transfer function is relatively modest. The size of the difference can be gauged by considering the fractional change in the rms linear density fluctuations for a spherical top-hat filter enclosing a mass of mean density corresponding to 32 particles (our resolution limit in the subsequent analyses). For our highest resolution simulation the rms using our composite transfer function is less than 2 per cent above that using the cmbfast CDM transfer function. For all other simulations the difference is even smaller.

Our composite transfer function was formed as follows. It was set equal to the cmbfast transfer function for log10(kh/Mpc) < 1, and equal to an analytic form based on the CDM only transfer function of Bardeen et al. (1986, hereafter BBKS) for log10(kh/Mpc) > 2. Over the intermediate range the composite transfer function is given by the linear combination of (1−w) times the cmbfast and w times the BBKS transfer function, where the weighting function w changes smoothly from 0 to 1. By using a value of Γ= 0.16 for the shape parameter in the BBKS formula and scaling the overall amplitude it proved possible to match the amplitudes of the cmbfast and BBKS functions everywhere in the transition region, 1 < log10(kh/Mpc) < 2, to better than 0.5 per cent, with the result that the overall transfer function is very smooth.

We selected target haloes for resimulation from a parent simulation carried out at homogeneous resolution with 9003 particles in this same box. We targeted haloes of roughly Milky Way mass and without a massive close neighbour at z= 0. We also checked that semi-analytic modelling applied to our target haloes predicated them to host late-type galaxies. Otherwise our selection was random. New initial conditions for the selected objects were then constructed by identifying the Lagrangian region from which each halo formed. The high-resolution region, which has an ‘amoeba’-like shape, was defined as the union of a set of small identical cubes joined face to face covering the whole of the Lagrangian region and forming a simply connected volume. Within this region the mass distribution was represented by a much larger number of lower mass particles. On making the initial conditions, additional small-scale power was added to the high-resolution region as dictated by the higher local particle Nyquist frequency. More distant regions were sampled with progressively more massive particles, but retaining sufficient resolution to ensure an accurate representation of the tidal field at all times. The initial displacements were imprinted using the Zeldovich approximation, and a ‘glass-like’ uniform particle load (White 1996) was used within the high-resolution regions of all our initial conditions. We also invested particular care to guarantee that all our final haloes are unaffected by contamination by heavier boundary particles. In fact, all our haloes are free of any boundary particles within the radius r50, except for simulations Aq-E-2, Aq-F-2 and Aq-F-3, where 71, nine and three heavier particles are found within this radius, respectively, corresponding to a fraction ≲10−5 of the total mass within this radius. Typically, about 30 per cent of the high-resolution particles in the new initial conditions end up in the virialized region of the final halo.

In setting up these zoomed initial conditions, all power from the parent simulation is deleted beyond some wavenumber which is smaller than the Nyquist wavenumber of the parent simulation, but substantially larger than the fundamental wavenumber of the cube enclosing the high-resolution region in which we generate the additional high-frequency waves. The latter replace the waves deleted from the fluctuation field of the parent simulation and extend its power spectrum up to the Nyquist frequency corresponding to the mean interparticle separation in the high-resolution region. When we create a series of simulations of the same object at differing mass resolution (i.e. with different particle masses in the high-resolution region) we are careful to ensure that all the waves used to create a lower resolution simulation are present with identical amplitude and phase in all higher resolution simulations. This means that every object which forms in the lower resolution simulation should also be present with identical mass and position in its higher resolution counterparts. This allows us to make detailed convergence tests on the properties of every non-linear object in our simulations, not just on the main halo.

In Fig. 1, we show the power spectrum as measured from the initial conditions of our highest resolution resimulation, Aq-A-1 from Table 1. All the measurements are made using a 10003 Fourier transform using a cloud-in-cell assignment scheme and taking into account for each mode the expected smoothing effect of the assignment scheme. Measuring the power spectrum accurately however requires a further measure. Even in the absence of the imposed perturbations, the unperturbed particle distribution has some measurable power. This power in a discrete particle representation, unlike a truly uniform mass distribution, does not grow by gravitational instability. Over most scales this extra power is negligible compared to that introduced when the density perturbations are added by displacing the particles which we wish to measure. However, at scales approaching the particle Nyquist frequency the contribution from the unperturbed particle distribution starts to become significant compared to the imposed power. To allow for this effect, which would otherwise lead to an overestimate of the input power, the values for all the points plotted are computed by differencing the power of the perturbed and unperturbed particle positions.

Figure 1.

Measured power spectrum (dimensionless variance Δ2(k) ∼k3P(k) per natural log interval) in our highest resolution ‘zoom’ initial conditions, Aq-A-1, linearly extrapolated to the values expected at z= 0. The lower red points show the power spectrum measured in our homogeneously sampled parent simulation, shifted down by 1 dex for clarity. The upper red circles show our measurement for the zoom initial conditions for the whole box, while cyan and magenta circles show measurements from smaller boxes centred within the high-resolution region. The solid black lines show the linear theory input spectrum. The vertical dashed line on the right-hand side marks the Nyquist frequency of the high-resolution region, while the left-hand vertical line is the joining point between the long wavelength modes from the parent simulation and the high-frequency modes added to the high-resolution cube.

Table 1.  Basic parameters of the Aquarius simulations. We have simulated six different haloes, each at several different numerical resolutions. The leftmost column gives the simulation name, encoding the halo (A–F), and the resolution level (1–5). mp is the particle mass, ε is the Plummer-equivalent gravitational softening length, Nhr is the number of high-resolution particles, and Nlr the number of low-resolution particles filling the rest of the volume. M200 is the virial mass of the halo, defined as the mass enclosed in a sphere with mean density 200 times the critical value. r200 gives the corresponding virial radius. We also give the mass and radius for a sphere of overdensity 50 times the critical density, denoted as M50 and r50. Note that this radius encloses a mean density 200 times the background density; in some studies (e.g. Diemand et al. 2007a) M50 and r50 have been defined as virial mass and radius. Finally, N50 gives the number of simulation particles within r50.
Namemp(M⊙)ε(pc)NhrNlrM200(M⊙)r200(kpc)M50(M⊙)r50(kpc)N50
Aq-A-11.712 × 10320.54252 607 000144 979 1541.839 × 1012245.762.523 × 1012433.481473 568 512
Aq-A-21.370 × 10465.8531 570 00075 296 1701.842 × 1012245.882.524 × 1012433.52184 243 536
Aq-A-34.911 × 104120.5148 285 00020 035 2791.836 × 1012245.642.524 × 1012433.5051 391 468
Aq-A-43.929 × 105342.518 535 972634 7931.838 × 1012245.702.524 × 1012433.526424 399
Aq-A-53.143 × 106684.92316 893634 7931.853 × 1012246.372.541 × 1012434.50808 479
Aq-B-26.447 × 10365.8658 815 01080 487 5988.194 × 1011187.701.045 × 1012323.12162 084 992
Aq-B-42.242 × 105342.518 949 101648 8748.345 × 1011188.851.050 × 1012323.604683 037
Aq-C-21.399 × 10465.8612 602 79578 634 8541.774 × 1012242.822.248 × 1012417.09160 630 624
Aq-C-43.213 × 105342.526 679 146613 1411.793 × 1012243.682.285 × 1012419.367110 775
Aq-D-21.397 × 10465.8391 881 10279 615 2741.774 × 1012242.852.519 × 1012433.21180 230 512
Aq-D-42.677 × 105342.420 455 156625 2721.791 × 1012243.602.565 × 1012435.859579 672
Aq-E-29.593 × 10365.8465 905 91674 119 9961.185 × 1012212.281.548 × 1012368.30161 323 676
Aq-E-42.604 × 105342.517 159 996633 1061.208 × 1012213.631.558 × 1012369.145982 797
Aq-F-26.776 × 10365.8414 336 000712 8391.135 × 1012209.211.517 × 1012365.87223 901 216
Aq-F-32.287 × 104120.5122 766 400712 8391.101 × 1012207.151.494 × 1012363.9865 320 572

In Fig. 1 the lower sequence of red filled circles shows the measured power spectrum (linearly extrapolated to z= 0 and offset by exactly one dex for clarity) from the parent 9003 particle simulation, plotted down to the shortest wavelength included in the refined initial conditions. The two black curves show the theoretical linear power spectrum, discussed above, with the upper curve extrapolated to z= 0 and the lower curve shifted down from it by one dex. The power spectrum of the parent simulation is evidently a very good match to the theoretical power spectrum except at low k where close agreement is not expected because only a few modes contribute. The upper red circles show the power spectrum measured in the initial conditions of the zoom simulation (linearly extrapolated to z= 0). The most massive particles in the refined initial conditions have a mass 1/413 of the entire 100 h−1 Mpc box. As the Nyquist frequency for particles of this mass corresponds to log(kh/Mpc−1) = 0.11, only points to the left-hand side of this limit are plotted. The cyan circles show the power spectrum measured from a box of side length 6.4 h−1 Mpc centred on the high-resolution region. The masses of the particles in this region vary, with the most massive particles having a mass of ∼ 7.23 × 10−8 of the region within which high-frequency power is added. (The latter is cubic with a side length of 7.06 h−1 Mpc.) The Nyquist frequency corresponding to the most massive particles is here log(kh/Mpc−1) = 2.03, and the cyan points are shown only to the left-hand side of this limit.

Finally, the magenta circles show the average power spectrum measured from eight boxes of side length h−1 Mpc inside the central high-resolution region, where all the particles are of the same mass. The largest possible cube that can be extracted where all the particles have the same mass is about 2.4 h−1 Mpc on a side. A 10003 Fourier transform is not large enough to make an accurate measurement of the power spectrum over this size of cube because the particle and Fourier mesh Nyquist frequencies are very close. Instead we placed eight non-overlapping h−1 Mpc cubes inside this volume. Averaging the eight regions reduces the expected scatter at low wavenumber, but makes no significant difference compared to a single measurement at high wavenumbers. Again the points are only plotted up to their Nyquist frequency, which is marked by the rightmost vertical dashed line. The leftmost vertical dashed line marks the joining point in the high-resolution cube between the long-wavelength waves from the parent simulation and the high-frequency waves in the high-resolution cube. The joining point corresponds to about 15 waves across the high-resolution cube.

These measurements of the power spectrum of our initial conditions show that we clearly achieve an excellent match to the desired linear input spectrum over many decades in spatial scale. We stress that such tests of the initial conditions are essential, as their quality is obviously of paramount importance for the accuracy of the evolved simulations. Note that in the following, ‘high-resolution region’ refers to the amoeba-shaped region where all the particles have the same mass rather than to the region to which high-frequency waves are added (which is larger). Similarly, the low-resolution region is everything outside the amoeba.

2.2 The Aquarius simulation suite

In Table 1, we provide an overview of the basic numerical parameters of our simulations. This includes a symbolic simulation name, the particle mass in the high-resolution region, the gravitational softening length, the total particle numbers in the high- and low-resolution regions, as well as various characteristic masses and radii for the final haloes, and the corresponding particle numbers. Our naming convention is such that we use the tags ‘Aq-A’ to ‘Aq-F’ to refer to simulations of the six Aquarius haloes. An additional suffix ‘1’–‘5’ denotes the resolution level. ‘Aq-A-1’ is our highest resolution calculation with ∼1.5 billion halo particles. We have level 2 simulations of all six haloes, corresponding to 160–224 million particles per halo.

We kept the gravitational softening length fixed in comoving coordinates throughout the evolution of all our haloes. The dynamics is then governed by a Hamiltonian and the phase-space density of the discretized particle system should be strictly conserved as a function of time (Springel 2005), modulo the noise introduced by finite force and time integration errors. Time-stepping was carried out with a kick-drift-kick leap-frog integrator where the time-steps were based on the local gravitational acceleration, together with a conservatively chosen maximum allowed time-step for all particles.

We define the virial mass MΔ and viral radius rΔ as the mass and radius of a sphere that encloses a mean density Δρcrit, where ρcrit is the critical density. Different choices for Δ are used in the literature. The most common ones are (1) a fixed value of Δ= 200 as in NFW's original work, (2) a value of Δ∼ 178 Ω0.45m based on a generalization of the spherical top-hat collapse model to low-density cosmologies (Eke, Cole & Frenk 1996; Bryan & Norman 1998) or (3) a value of Δ= 200 Ωm(z), which corresponds to a fixed overdensity relative to the background density.

We will frequently give results for the radius according to convention (3), for which Δ= 200 Ωm= 50 at z= 0, simply because this yields the largest radius and hence the largest number of substructures, which improves statistics. The corresponding radius is designated as r50, while r200 refers to the radius that encloses an overdensity of 200 with respect to the critical density, as is customary in the literature (except for Diemand et al. 2007a, and collaborators, who use r200 to refer to a radius that encloses 200 times the mean density, which is equivalent to r50 in our notation).

2.3 Integration technique

All our simulations were started at redshift z= 127, and were evolved with a new parallel TreePM code gadget-3 written especially for the Aquarius Project. This code offers much better scalability to large numbers of compute cores as well as higher basic speed than its parent code gadget-2 (Springel 2005). The gravitational field on large scales is calculated with a particle-mesh (PM) algorithm, while the short-range forces are delivered by a tree-based hierarchical multipole expansion, such that a very accurate and fast gravitational solver results. The scheme combines the high spatial resolution and relative insensitivity to clustering of tree algorithms with the unmatched speed and accuracy of the PM method to calculate the long-range gravitational field. We note that achieving our force resolution with a single mesh in a standard PM approach would require a grid with (107)3 cells – storing such a mesh would require several million petabytes. This illustrates the enormous dynamic range we are aiming for with our simulations.

In fact, the numerical challenge of the calculations is substantial. One challenge simply reflects the large dynamic range involved: gravitational time-scales are inversely proportional to the square root of the density, so simulating a CDM halo means dealing with a system where different regions evolve on time-scales which may differ by factors of thousands. A code with spatially dependent, adaptive time-stepping is mandatory; otherwise the most rapidly evolving regions – which usually include only a tiny fraction of the mass – force time-steps so short that the calculation grinds to a halt. A second challenge stems from the highly clustered spatial distribution of matter and affects, in particular, the scalability of parallel algorithms. A CDM halo is a nearly monolithic, highly concentrated structure with a well-defined centre. There is no obvious geometrical decomposition which can separate it into the large number of computationally equivalent domains required for optimal exploitation of the many processors available in high-performance parallel architectures. In our highest resolution calculation, the clustering is so extreme that far more than a billion particles (amounting to about one third of all particles in the simulation) collect in a region that encompasses less than 10−8 of the simulated volume. In addition, gravity couples the dynamics of matter throughout the halo and beyond, requiring efficient communication between all parts of the simulated region.

It is clear that parallelization of such calculations for distributed memory machines is difficult, yet is mandatory to make them feasible on today's supercomputers. We have carried out our most expensive calculation, the Aq-A-1 run, on the Altix 4700 supercomputer of the Leibniz Computing Centre (LRZ) in Garching/Germany, using 1024 CPUs and about 3 TB of main memory. The calculation took more than 3.5 million CPU hours to carry out about 101 400 time-steps that involved 6.72 × 1013 force calculations in total. We have stored 128 simulation dumps for this calculation, amounting to a data volume of about 45 TB. The other simulations of the Aquarius Project were in part calculated on the LRZ system, and in part on other supercomputers across Europe. These were the COSMA computer at Durham University/UK, the Bluegene/L system STELLA of the LOFAR consortium in Groningen/Netherlands, and a Bluegene/P system of the Max Planck Computing Centre in Garching. For all these simulations we also stored at least 128 outputs, but for Aq-A-2 and Aq-A-4 we kept 1024 dumps, and for Aq-A-3 half this number. This provides exquisite time resolution for studies of the detailed formation history of haloes and the evolution of their substructure. In the present study, however, we focus on an analysis of the objects at z= 0.

2.4 A first view of the simulations

In Figs 2 and 3, we show images2 of the dark matter distribution in our six high-resolution haloes at redshift z= 0. The brightness of each pixel is proportional to the logarithm of the squared dark matter density projected along the line of sight,

image(1)

while the colour hue encodes the mean dark matter velocity dispersion, weighted as

image(2)

Here the local dark matter density ρ(r) and the local velocity dispersion σloc(r) of the particles are estimated with an SPH kernel interpolation scheme based on 64 neighbours. We use a two-dimensional colour table (see Fig. 2) in which the information about the local dark matter ‘temperature’ is orthogonal to the density information; conversion of the images into grey-scale eliminates the velocity information but leaves the density information intact, with the latter being proportional to the dark matter annihilation luminosity.

Figure 2.

The top left-hand panel shows the projected dark matter density at z= 0 in a slice of thickness 13.7 Mpc through the full box (137 Mpc on a side) of our 9003 parent simulation, centred on the ‘Aq-A’ halo that was selected for resimulation. The other five panels show this halo resimulated at different numerical resolutions. In these panels, all particles within a cubic box of side length 2.5 ×r50 centred on the halo are shown. The image brightness is proportional to the logarithm of the squared dark matter density S(x, y) projected along the line of sight, and the colour hue encodes the local velocity dispersion weighted by the squared density along the line of sight. We use a two-dimensional colour table (as shown on the left-hand side) to show both of these quantities simultaneously. The colour hue information is orthogonal to the brightness information; when converted to black and white, only the density information remains, with a one-dimensional grey-scale colour map as shown on the left-hand side. The circles mark r50.

Figure 3.

Projected dark matter density in our six different high-resolution haloes at z= 0, at the ‘2’ resolution level. In each panel, all particles within a cubic box of side length 2.5 ×r50 centred on the halo are shown, and the circles mark the radius r50. The image brightness is proportional to the logarithm of the squared dark matter density, and the colour hue encodes the local particle velocity dispersion, with the same colour map as in Fig. 2.

Looking at these images it is clear that our haloes are filled with a sea of dark matter substructures of many different sizes. Fig. 2 shows that these repeat closely, as they should, between simulations of the same object at different resolution, albeit at slightly different positions. As we will show later in more detail, there are even substructures inside subhaloes. In fact, up to four such generations are resolved by our highest resolution calculation, the Aq-A-1 version of the ‘A’ object. It is clear that an important task in analysing this complex phase-space structure lies in finding the gravitationally bound substructures that orbit within the virialized regions of the haloes.

We address this complex problem with our SUBFIND algorithm (Springel et al. 2001a) which finds substructures using a topological excursion set method. Based on local dark matter density estimates calculated with the SPH kernel interpolation approach for all high-resolution particles, we first identify a set of subhalo candidates, which are locally overdense structures found within a given input group of particles identified with a FOF (friends-of-friends) group finder (Davis et al. 1985). These are then subjected to a gravitational unbinding procedure that iteratively eliminates all unbound particles. Provided more than 20 bound particles remain, we record the particle group as a genuine subhalo in our group catalogue. For each subhalo, we calculate a number of physical properties, such as the maximum circular velocity, spin and velocity dispersion, and we store the particles in order of the gravitational binding energy, which is useful for tracking subhaloes between simulation outputs at different times. We have fully parallelized the SUBFIND and FOF algorithms for distributed memory systems and inclined them in our simulation code gadget-3. Thus group finding can be done on the fly during the simulation, if desired. This is often advantageous as these calculations are computationally quite intense and require equally large memory as the dynamical simulation code itself.

The density values used by SUBFIND were based on SPH density estimates ρi with smoothing lengths hi that satisfy the implicit equation (4π/3) h3iρi= 64 mp (Springel & Hernquist 2002), where mp is the particle mass and an effective neighbour number of 64 was adopted. In order to test the sensitivity of inferred dark matter annihilation rates to the dark matter density estimator (see Springel et al. 2008), we have also calculated densities based on a three-dimensional Voronoi tessellation of the simulation volume, where the density estimate was defined as the mass of a particle divided by its Voronoi volume. To construct the Voronoi tessellation, we have written a parallel code that can rapidly calculate the Delaunay triangulation, and from it its topological dual, the Voronoi tessellation.

As an illustration of the extreme dynamic range of our simulation set and the degree of numerical convergence between the different resolutions, we show in Fig. 4 the spherically averaged density profiles for the five simulations of the ‘Aq-A’ halo. In each case, we draw the measured density profile as a thick solid line down to the smallest radius where convergence is expected based on the criteria of Power et al. (2003), and we continue the measurements as thin lines for scales where the force law is unaffected by the gravitational softening. Power et al. (2003) have shown that convergence of the density profile at a given radius requires that the two-body relaxation time at this radius be larger than the Hubble time. This condition can be cast into the form

image(3)

where N(r) is the number of particles inside r, and inline image is the average enclosed density. Note that this form of the convergence criterion is in principle also applicable to dark matter subhaloes (see below), but in this regime it has not been empirically validated so far.

Figure 4.

Spherically averaged density profile of the Aq-A halo at z= 0, at different numerical resolutions. Each of the profiles is plotted as a thick line for radii that are expected to be converged according to the resolution criteria of Power et al. (2003). These work very well for our simulation set. We continue the measurements as thin solid lines down to , where ε is the Plummer-equivalent gravitational softening length in the notation of Springel et al. (2001b). The dotted vertical lines mark the scale 2.8ε, beyond which the gravitational force law is Newtonian. The mass resolution changes by a factor of 1835 from the lowest to the highest resolution simulation in this series. Excellent convergence is achieved over the entire radial range where it is expected.

We find that there is very good agreement between the densities and enclosed masses for all radii larger than the convergence radius estimated in this way. The quality of this convergence is impressively demonstrated by Fig. 5, where we show the local logarithmic slope of the density profile, for the radial range where convergence is expected according to the Power criterion. There are some large fluctuations of the local slope in the outer parts of the halo, caused by substructures, which are remarkably well reproduced at the different resolutions. In the more relaxed inner regions, the local logarithmic slope varies smoothly with radius. In particular, it becomes gradually shallower towards the centre, as suggested by Navarro et al. (2004). In fact, the local slope becomes clearly shallower than −1 at the innermost converged radius. This has important implications for the structure of the central cusp which is analysed in full detail in Navarro et al. (2008). For the rest of this paper, we focus on an analysis of the dark matter substructures.

Figure 5.

Local logarithmic slope of the density profiles as a function of radius for the Aq-A halo simulated at different numerical resolution. Only the radial region that should be converged according to the criteria of Power et al. (2003) is shown. Note that the large fluctuations in the outer parts are caused by substructures but nevertheless reproduce well between simulations. In this regime, we expect significant halo-to-halo scatter.

3 SUBSTRUCTURE ABUNDANCE AND SPATIAL DISTRIBUTION

In this section, we investigate the abundance of dark matter substructures as measured by the SUBFIND algorithm. All our substructures consist of particle groups that are gravitationally self-bound and are overdense with respect to the local background. Every simulation particle can be part only of one subhalo, but we are able to detect substructure within substructure (see below). We count substructures down to a minimum of 20 bound particles.

3.1 Subhalo counts and substructure mass fraction

In Fig. 6, we show the differential abundance of subhaloes by mass (i.e. the number of subhaloes per unit mass interval) in our ‘A’ halo within r50, and we compare results for simulations of the same object at different mass resolution. For masses above ∼ 5 × 108  M⊙, the number of subhaloes is small and large halo-to-halo scatter may be expected (see below). However, for lower masses a smooth mass spectrum is present that is well described by a power law over many orders of magnitude. Multiplication by M2sub compresses the vertical scale drastically, so that the slope of this power law and deviations from it can be better studied. This is shown in the bottom panel of Fig. 6. We see that resolution effects become noticeable as a reduction in the number of objects at masses below a few hundred particles, but for sufficiently well-resolved subhaloes, very good convergence is reached. There is good evidence from the fully converged part of the differential mass function that it exhibits a true power-law behaviour, and that the slope of this power law is shallower than −2, though not by much. Our results are best fitted by a power law d N/d MM−1.9, the same slope found by Gao et al. (2004), but significantly steeper than Helmi, White & Springel (2002) found for their rich cluster halo. The exact value obtained for the slope in a formal fit varies slightly between −1.87 and −1.93, depending on the mass range selected for the fit; the steepest value of −1.93 is obtained when the fit is restricted to the mass range 106– 107  M⊙ for the Aq-A-1 simulation.

Figure 6.

Differential subhalo abundance by mass in the ‘A’ halo within the radius r50. We show the count of subhaloes per logarithmic mass interval for different resolution simulations of the same halo. The bottom panel shows the same data but multiplied by a factor M2sub to compress the vertical dynamic range. The dashed lines in both panels show a power law d N/d MM−1.9. For each of the resolutions, the vertical dotted lines in the lower panel mark the masses of subhaloes that contain 100 particles.

The small tilt of the slope n=−1.9 away from −2 is quite important. For n=−2, the total predicted mass in substructures smaller than a given limit m0 would be logarithmically divergent when extrapolated to arbitrarily small masses. If realized, this might suggest that there is no smooth halo at all, and that ultimately all the mass is contained in subhaloes. However, even for the logarithmically divergent case the total mass in substructures does not become large enough for this to happen, because a sharp cut-off in the subhalo mass spectrum is expected at the thermal free-streaming limit of the dark matter. Depending on the specific particle physics model, this cut-off lies around an Earth mass, at ∼ 10−6  M⊙, but could be as low as 10−12  M⊙ in certain scenarios (Chen, Kamionkowski & Zhang 2001; Hofmann, Schwarz & Stöcker 2001; Green, Hofmann & Schwarz 2004; Profumo, Sigurdson & Kamionkowski 2006).

Our measured mass function for the ‘A’ halo is well approximated by

image(4)

with n=−1.9, and an amplitude of a0= 8.21 × 107/M50= 3.26 × 10−5  M−1 for a pivot point of m0= 10−5M50= 2.52 × 107 M⊙. This means that the expected total mass in all subhaloes less massive than our resolution limit mres is

image(5)

where mlim is the thermal dark matter limit. For mlim→ 0 and our nominal subhalo resolution limit of mres= 3.24 × 104  M⊙ in the Aq-A-1 simulation, this gives Mtot= 1.1 × 1011  M⊙, corresponding to about 4.5 per cent of the mass of the halo within r50. While non-negligible, this is considerably smaller than the total mass in the substructures that are already resolved by the simulation. The latter is 13.2 per cent of the mass within r50 for the Aq-A-1 simulation. We hence conclude that despite the very broad mass spectrum assumed in this extrapolation, the total mass in subhaloes is still dominated by the most massive substructures, and an upper limit for the total mass fraction in subhaloes is ∼18 per cent within r50 for the ‘Aq-A’ halo.

We caution, however, that the extrapolation to the thermal limit extends over 10 orders of magnitude! This is illustrated explicitly in Fig. 7, where we show the mass fraction in substructures above a given mass limit, combining the direct simulation results with the extrapolation above. We also include an alternative extrapolation in which a steeper slope of −2 is assumed. In this case, the total mass fraction in substructures would approximately double if the thermal limit lies around one Earth mass. If it is much smaller, say at mlim∼ 10−12  M⊙, the mass fraction in substructure could grow to ∼50 per cent within r50, still leaving room for a substantial smooth halo component. Note, however, that within 100 kpc even this extreme extrapolation results in a substructure mass fraction of only about 5 per cent. Most of the mass of the inner halo is smoothly distributed.

Figure 7.

Expected mass fraction in subhaloes as a function of the limiting mass Mlim, inside r50 (top curves) and inside 100 kpc (bottom curves). The solid thin lines show an extrapolation of the direct simulation result with an n=−1.9 power law for the differential subhalo mass function. In this case, the total substructure mass converges at the low-mass end. The dotted lines show the prediction for the logarithmically divergent case, n=−2. In this case, we would expect the mass in substructures down to an Earth mass (vertical dashed line) to be about twice what we can resolve directly. This mass is a reasonable estimate for the thermal free-streaming limit in many supersymmetric theories where the dark matter particle is a neutralino. However, the parameters of these theories are sufficiently uncertain that the thermal limit could lie as low as 10−12  M⊙. Even in this case, the lumpy component of the halo would still be subdominant within r50, and would be a small fraction of the total mass of 9.32 × 1011  M⊙ within 100 kpc.

Within r50 the mass fraction in resolved substructures varies around 11 per cent for our six simulations at resolution level 2, each of which has at least 160 million particles in this region. Table 2 lists these numbers, which are 12.2 per cent (Aq-A-2 simulation), 10.5 per cent (Aq-B-2), 7.2 per cent (Aq-C-2), 13.1 per cent (Aq-D-2), 10.8 per cent (Aq-E-2) and 13.4 per cent (Aq-F-2). This gives an average of 11.2 per cent within r50 down to the relevant subhalo mass resolution limit, ∼ 2 × 105  M⊙. This is similar to the substructure mass fractions found by earlier work on galaxy cluster haloes (e.g. Ghigna et al. 1998; Springel et al. 2001a; De Lucia et al. 2004) and Galaxy-sized haloes (Stoehr et al. 2003) once the different limiting radius (r200 instead of r50) is corrected for. However, it is larger than the 5.3 per cent inside r50 reported by Diemand et al. (2007a) for a Milky Way sized halo.

Table 2.  Basic structural properties of the main haloes in our various simulations. The leftmost column gives the simulation name, Vmax is the maximum circular velocity, rmax is the radius where this maximum is reached, δV gives the characteristic density contrast based on the peak of the circular velocity curve, while cNFW is the same value converted to an equivalent NFW concentration under the assumption that the halo is reasonably well fitted by an NFW profile. zform gives the formation redshift of the halo, defined as the earliest epoch at which the M200 mass of the main halo progenitor exceeds half its final value. Finally, Nsub gives the total number of subhaloes that we resolve inside r50, and fcumulsub is their total mass fraction relative to all the mass inside r50.
NameVmax (km s−1)rmax(kpc)δVc*NFWzformNsubfcumulsub (per cent)
Aq-A-1208.7528.352.035 × 10416.111.93297 79113.20
Aq-A-2208.4928.142.060 × 10416.191.9345 02412.16
Aq-A-3209.2227.882.114 × 10416.351.9313 85411.34
Aq-A-4209.2428.202.067 × 10416.211.9319609.68
Aq-A-5209.1728.552.015 × 10416.041.932998.64
Aq-B-2157.6840.155.788 × 1039.721.3942 53710.54
Aq-B-4159.0344.314.834 × 1039.021.3916148.26
Aq-C-2222.4032.471.761 × 10415.212.2335 0227.17
Aq-C-4223.2033.631.654 × 10414.842.2319726.02
Aq-D-2203.2054.085.299 × 1039.371.5147 01413.06
Aq-D-4204.4755.765.046 × 1039.181.51311610.67
Aq-E-2179.0055.503.904 × 1038.262.2642 72510.75
Aq-E-4182.6854.594.202 × 1038.522.2620247.53
Aq-F-2169.0842.675.892 × 1039.790.5552 50313.39
Aq-F-3174.0543.765.937 × 1039.820.5512 9509.15

In Fig. 8, we compare the differential subhalo mass functions of these six haloes, counting the numbers of subhaloes as a function of their mass normalized to the M50 of their parent halo. Interestingly, this shows that at small subhalo masses the subhalo abundance per unit halo mass shows very little halo-to-halo scatter. In fact, the mean differential abundance is well fitted by equation (4) with the parameters given above, and the rms halo-to-halo scatter in the normalization is only ∼8 per cent.

Figure 8.

Differential subhalo abundance by mass for our six haloes simulated at resolution level 2. We here count all subhaloes inside r50 and plot their abundance as a function of Msub/M50. We see that at low mass the abundance of subhaloes is universal to good accuracy when their mass is expressed in units of the mass of their parent halo. The bottom panel shows the same data but multiplied by a factor M2sub to compress the vertical dynamic range. The dashed lines in both panels show the power law d N/d MM−1.9.

In Table 2, we also list a few other basic structural properties of our haloes, namely their maximum circular velocity Vmax, the radius rmax at which this velocity is attained, a simple measure for halo concentration, and the redshift at which the halo formed. One way to characterize the concentration of a halo is to express the mean overdensity within rmax in units of the critical density. This corresponds to the definition

image(6)

We can relate this quantity to a more familiar concentration measure based on the NFW density profile:

image(7)

where ρs and rs are a characteristic density and radius, respectively. Assuming this profile shape, the pair of values Vmax and rmax are sufficient to determine the density profile uniquely. The characteristic NFW overdensity δc is then

image(8)

which can be converted to the NFW concentration c through the equation

image(9)

We list the ‘NFW’ concentrations calculated in this way as c*NFW in Table 2. We note however that fits to the full density profile may yield slightly different results, since in this case the circular velocity curve of the fit will not necessarily peak exactly at r=rmax.

Defining a precise value for the total mass of a subhalo requires an operational definition of its ‘outer edge’. Different substructure detection algorithms define different effective boundaries and so produce systematically different total mass estimates. It may therefore be more robust to count subhaloes as a function of their peak circular velocity, which typically lies well within the object and so is insensitive to definitions of its edge. Note, however, that in small systems maximum circular velocity estimates can be more sensitive to numerical resolution effects than total mass estimates.

In Fig. 9, we show the cumulative abundance of subhaloes as a function of maximum circular velocity for our different resolution simulations of Aq-A. Again, there is good convergence. Indeed, at the massive end, the curves lie essentially on top of each other, showing that we are really seeing the same subhaloes, and that they are reproduced with the same maximum circular velocity in all the simulations. This suggests that we are also achieving good convergence for the internal structure of individual subhaloes, an issue that we will investigate further below.

Figure 9.

Cumulative subhalo abundance as a function of maximum subhalo circular velocity. The top panel shows the raw measurements from the simulations, while in the bottom panel, we have applied the correction of equation (10) to compensate approximately for the impact of the gravitational softening on Vmax. We show results for five simulations of the Aq-A halo carried out with differing mass resolution. The dashed line is the fitting function given for their own simulations by Reed et al. (2005), which also accurately matches the result for the ‘Via Lactea I’ simulation (Diemand et al. 2007a). This is clearly inconsistent with our own data.

However, it is worth noting that the individual measurements for the velocity functions peel away from their higher resolution counterparts comparatively early at low velocities, which suggests worse convergence than found for the subhalo mass functions at the low-mass end. This behaviour can be understood as an effect of the gravitational softening length ε, which lowers the maximum circular velocities of subhaloes for which rmax is not much larger than ε. To estimate the strength of this effect, we can imagine that the gravitational softening for an existing subhalo is adiabatically lowered from ε to zero. The angular momentum of individual particle orbits is then an adiabatic invariant. Assuming for simplicity that all particles are on circular orbits, and that the gravitational softening can be approximated as a Plummer force with softening length ε, the expected change of the maximum circular velocity is then

image(10)

In the lower panel of Fig. 9, we plot the cumulative velocity functions for these corrected maximum circular velocities. Clearly, the measurements line up more tightly down to lower Vmax, demonstrating explicitly that the convergence in the number of objects counted as a function of (corrected) circular velocity is in principle as good as that counted as a function of mass. Note that a similar correction can also be applied to the measured rmax values. However, for the remainder of this paper, we focus on the raw measurements from the simulations without applying a gravitational softening correction.

The dashed line in Fig. 9 shows the fit which Reed et al. (2005) quote for the subhalo abundance as a function of maximum circular velocity in their own simulations, N(>Vmax) = (1/48)(Vmax,sub/Vmax,host)−3. Diemand et al. (2007a) found this formula to fit the results from their own Via Lactea I simulation very well. Fig. 9 thus confirms the indication from subhalo mass fractions that our simulations show substantially more substructure than reported for Via Lactea I. This is particularly evident at lower subhalo masses which are unaffected by the small number effects which cause scatter in the abundance of massive subhaloes. With the help of J. Diemand and his collaborators, we have checked that this abundance difference is not a result of the different subhalo detection algorithms used in our two projects.

We do not think that this discrepancy can be explained by halo-to-halo scatter since it is much larger than the variation in substructure abundance among our own sample of haloes. This is demonstrated in Fig. 10, which shows the cumulative subhalo abundance distributions within r50 as a function of maximum subhalo circular velocity for all our resolution level 2 haloes. We plot subhalo count against subhalo maximum circular velocity normalized to V50, the circular velocity of the main halo at r50. Because the slope of the abundance curve is very close to −3, this is equivalent to plotting subhalo count normalized by the total parent halo mass within r50 (which is proportional to V350) against subhalo maximum circular velocity. There is remarkably little scatter between our simulations when normalized in this way; the rms scatter in amplitude in the power-law regime is around 10 per cent. The steep slope of the N(>Vmax) function together with the increase of concentration for smaller haloes (see section 5.2) means that the total dark matter annihilation luminosity of haloes is dominated by small substructures (e.g. Diemand et al. 2007a, 2008; Kuhlen, Diemand & Madau 2008; Springel et al. 2008). The figure also shows the substructure abundance reported for the Via Lactea I (dashed) and II (dotted) simulations (Diemand et al. 2007a, 2008), after rescaling to the normalization we prefer here.3 There is a difference of a factor of 3.1 between the mean abundance of small subhaloes in our simulations and in ‘Via Lactea I’. The Diemand et al. (2008) abundance for Via Lactea II differs substantially from that for Via Lactea I and is much closer to our results. Nevertheless, the abundance of small subhaloes in Via Lactea II is still 31 per cent lower than the mean for our set of six haloes, which is more than three times the rms scatter in abundance between our haloes.

Figure 10.

Cumulative subhalo abundance as a function of maximum subhalo circular velocity in units of the circular velocity of the main halo at r50. We show results for all six of our haloes at resolution level 2, and in addition we include our highest resolution result for the Aq-A-1 run. For comparison, we overplot fitting functions for the Via Lactea I and Via Lactea II simulations (Diemand et al. 2007a, 2008), appropriately rescaled from a normalization to Vmax,host to one by V50,host.

These results lead us to disagree with the assertion by Madau, Diemand & Kuhlen (2008) and Diemand et al. (2008) that differences of this magnitude lie within the halo-to-halo scatter. Instead, the substantial difference between ‘Via Lactea I’ and ‘Via Lactea II’ must have a systematic origin. We also think it unlikely that the higher abundance in our simulations reflects the small differences in the background cosmology assumed in the two projects, as suggested by Madau et al. (2008), even though this is a possibility we cannot exclude. For example, the Via Lactea simulations assumed a lower value for σ8 than we used, and we believe that lowering σ8 should result in slightly more substructure in objects of given mass, simply because these haloes then tend to form more recently which increases the number of surviving subhaloes within them (e.g. De Lucia et al. 2004). We have explicitly confirmed this effect by comparing the substructure abundances in the Millennium Simulation (with σ8= 0.9) with those in the simulations of Wang et al. (2008), which used the same cosmology except for taking σ8= 0.722. On the other hand, the different tilt assumed for the primordial power spectrum of the Via Lactea II simulation may have reduced the subhalo abundance and could perhaps be responsible for the difference (Zentner & Bullock 2003).

We note that the small halo-to-halo scatter in substructure abundance which we find also contradicts the recent suggestion by Ishiyama, Fukushige & Makino (2007) that the halo-to-halo variation in subhalo abundance could be very large, and that the apparent paucity of dwarfs surrounding the Milky Way might simply reflect the fact that our Galaxy happens to live in a low-density environment.

3.2 The spatial distribution of subhaloes

In Fig. 11, we show the radial distribution of subhaloes of different mass within our Aq-A-1 simulation. In the top panel, we plot the number density profile for different subhalo mass ranges, each normalized to the mean number density of subhaloes of this mass within r50. The number density of subhaloes increases towards halo centre, but much more slowly than the dark matter density, consistent with previous work (e.g. Ghigna et al. 1998; Diemand, Moore & Stadel 2004; Gao et al. 2004; Nagai & Kravtsov 2005; Diemand, Kuhlen & Madau 2007a). As a result, most subhaloes of a given mass are found in the outer parts of a halo, even though the number density of subhaloes is highest in the central regions. Another view of this behaviour is given in the bottom panel of Fig. 11, which histograms the abundance of subhaloes as a function of log radius so that the area under the curves is proportional to the total number of subhaloes. Clearly, the vast majority of subhaloes are found between ∼ 100  kpc and the outer radius of the halo.

Figure 11.

Subhalo number density profiles for different subhalo mass ranges in the Aq-A-1 simulation. In the top panel, the number density profiles for five logarithmic mass bins are shown, normalized to the mean number density within r50 (vertical dashed line). The profile shape appears independent of subhalo mass, and is well fitted by an Einasto profile with α= 0.678 and r−2= 199 kpc = 0.81r200. The bottom panel shows the number fraction of subhaloes per logarithmic interval in radius, on a linear–log plot. The area under the curves is proportional to subhalo number, showing that most subhaloes are found in the outermost parts of the halo.

Perhaps the most remarkable aspect of Fig. 11 is that there appears to be no trend in the shape of the number density profiles with subhalo mass. Previous work has already hinted at this behaviour (Diemand et al. 2004; Ludlow et al. 2008), which is here confirmed with much better statistics and over a much larger dynamic range in mass. We note that this disagrees with a tentative finding by De Lucia et al. (2004), who suggested that more massive substructures have a radial profile that is more strongly antibiased with respect to the mass than that of low-mass subhaloes. Like Ludlow et al. (2008) we find that the number density profile is well described by an Einasto profile (a fit to our measurements yields a shape parameter α= 0.678 and scale radius r−2=199 kpc = 0.81 r200). It is thus tempting to conjecture that this behaviour continues to (arbitrarily) small subhalo masses. If true, an interesting corollary is that there must be a smooth dark matter component which dominates the inner regions of haloes. Only the outer parts may have a substantial mass fraction in lumps (see also Fig. 7). This contrasts with previous speculations (Calcáneo-Roldán & Moore 2000; Moore et al. 2001) that all the mass of a halo may be bound in subhaloes.

Further light on this question is shed by Fig. 12, where we show the local mass fraction in subhaloes as a function of radius. In the top panel, we compare results for our six different haloes, with the radial coordinate normalized by r50. While there is some scatter between the different haloes, the general behaviour is rather similar and shows a rapid decline of the local mass fraction in substructures towards the inner parts of each halo. The mean of the six simulations (thick red line) is well fitted by a gently curving power law. It can be parametrized by

image(11)

with parameters α=−0.36, β= 0.87 and γ=−1.31. This fit is shown in the upper two panels of Fig. 12 as a thin black line. The middle panel is the same measurement, but for all the different resolution simulations of the Aq-A halo, while the bottom panel is the corresponding cumulative plot. These two panels give an impression of how well numerical convergence is achieved for this quantity.

Figure 12.

The mass fraction in subhaloes as a function of radius. In the top panel, we show results for the local mass fraction in substructures for our six different haloes, as a function of radius normalized by r50. The thick solid line shows the average of all the runs. In the middle panel, we consider the same quantity for the different resolution simulations of the Aq-A halo, while in the bottom panel we show the corresponding cumulative substructure fractions in the Aq-A halo. The solid line in the two upper panels is an empirical fit with a slowly running power-law index. The vertical dotted lines at 8 kpc in the middle and bottom panels mark the position of the solar circle; here the expected local mass fraction in subhaloes has dropped well below 10−3. The outer vertical dotted lines mark r50 for the Aq-A halo.

An interesting implication from Fig. 12 is an estimate of the fraction of the mass in substructures near the solar circle (marked by a vertical dashed line). At r= 8 kpc, the expected local mass fraction in substructure has dropped well below 10−3. This measurement appears converged, and accounting for unresolved substructure does not raise the fraction above 10−3 (compare Fig. 7). The dark matter distribution through which the Earth moves should therefore be mostly smooth, with only a very small contribution from gravitationally bound subhaloes.

4 SUBHALOES INSIDE SUBHALOES

In our simulations, we find several levels of substructure within substructure. Fig. 13 illustrates this by showing individually six of the largest Aq-A-1 subhaloes in enlarged frames. Clearly, all of these subhaloes have embedded substructures. Sometimes these second-generation subhaloes contain a further (third) level of substructure and, in a few cases, we even find a fourth generation of subhaloes embedded within these. An example is given in the bottom row of Fig. 13, which zooms recursively on regions of the subhalo labelled ‘f’ in the top left-hand panel. As shown in the bottom left-hand panel, subhalo ‘f’ has several components, each of which has identifiable subcomponents; we are able to identify up to four levels of this hierarchy of substructure in this system. We note that the hierarchy of nested structures is established directly by the recursive nature of the SUBFIND algorithm; at each level, a given substructure and its parent structure are surrounded by a common outer density contour that separates them from the next level in the hierarchy.

Figure 13.

Images of substructure within substructure. The top left-hand panel shows the dark matter distribution in a cubic region of side 2.5 ×r50 centred on the main halo in the Aq-A-1 simulation. The circles mark six subhaloes that are shown enlarged in the surrounding panels, and in the bottom left-hand panel, as indicated by the labels. All these first generation subhaloes contain other, smaller subhaloes which are clearly visible in the images. SUBFIND finds these second generation subhaloes and identifies them as daughter subhaloes of the larger subhaloes. If these (sub)subhaloes are large enough, they may contain a third generation of (sub)subhaloes, and sometimes even a fourth generation. The bottom panels show an example of such a situation. The subhalo shown on the bottom left-hand side contains another subhalo (circled) which is really made up of two main components and several smaller ones (bottom, second from left-hand side). The smaller of the two components is a third generation substructure (bottom, third from left-hand side) which itself contains three subhaloes which are thus fourth generation objects (bottom right-hand side).

It is important to quantify in detail the hierarchical nature of substructure, since this may have a number of consequences regarding indirect and direct dark matter search strategies. Recently, Shaw et al. (2007) suggested that the (sub)substructure distribution in subhaloes might be a scaled version of the substructure distribution in main haloes. This claim has been echoed by Diemand et al. (2008), who report roughly equal numbers of substructures inside radii enclosing a mean overdensity of 1000 times the cosmic average value (r250 in our notation) and centred at either subhaloes or the main halo. This result has been interpreted by Kuhlen et al. (2008) to imply that the (sub)subhalo abundance per unit mass of a subhalo should be roughly constant and equal to that of the main halo. This, however, seems unlikely because, as we have seen, local substructure abundance is a strong function of radius in main haloes, with most of the substructure found in the outer regions.

In this section, we present the first convergence studies ever attempted for (sub)substructure inside subhaloes in order to assess the alleged self-similarity of the substructure hierarchy. We begin by discussing a suitable definition for the outer edge of a subhalo, which allows us to measure the (sub)substructure mass fractions of subhaloes in a consistent manner. We then study the number and mass of subsubhaloes within that radius and compare them with the expectation from self-similarity. In order to compare with recent work by Diemand et al. (2008), we also carry out, for a few subhaloes, the same analysis within a radius of fixed overdensity, r250.

4.1 The radius of a subhalo

SUBFIND identifies substructures as locally overdense regions relative to the average background density. It is thus able to find substructures and assign masses to them without large biases throughout most of the halo. The procedure, however, may break down near the centre, where the average density of the main halo may overwhelm even the highest density peak of embedded substructures, leading to potential biases in the masses assigned to subhaloes by SUBFIND. It is therefore desirable to find a definition for the radius of a subhalo that is simple and physically meaningful and, at the same time, relatively insensitive to the accuracy with which SUBFIND assigns subhalo masses.

We have settled on the following operational procedure for determining a subhalo radius, rsub. Starting from the centre of a particular subhalo, we first measure spherically averaged density profiles for all the mass and for the gravitationally bound mass (as determined by SUBFIND). We then obtain a measure for the local density of the main halo at the position of the subhalo location by: (i) determining the radius r′ enclosing total mass equal to Msub; (ii) determining the amount of unbound mass M′ inside this radius (this is simply the difference between Msub and the bound mass within r′); (iii) defining ρloc=M′/(4πr′ 3/3). With this in hand, we operationally define the bounding radius rsub of the subhalo to be the radius at which the spherically averaged density profile of the bound mass has dropped below 0.02 ×ρloc.

The bound density profile drops very steeply near the edge of the subhalo, so changing the pre-factor 0.02 has only a minor influence on the radius determined in this way. We found that, with this choice, the edge of the subhalo is robustly determined right at the transition between the falling density profile of the subhalo, and the approximately constant local background density of the halo. Also, this radius tends to agree well with the minimum in the circular velocity curve constructed using all the mass around the subhaloes's centre. In Fig. 14, we show a typical example to illustrate this procedure. Note that the determination of ρloc, and hence the value of rsub obtained through the above procedure, is relatively insensitive to the precise value of Msub.

Figure 14.

An example of the determination of the subhalo radius rsub. The thick blue line shows the spherically averaged density profile of particles identified by SUBFIND as gravitationally bound to the subhalo, while the thin black line is the density profile of all particles. The purple line is the density profile of the difference, i.e. of all particles that are not bound to the subhalo. The dashed horizontal line is the mean local background density ρloc at the subhalo's location, estimated within the radius marked by the vertical dotted line, which encloses Msub in the profile of the total mass. The red vertical line marks our estimated radius rsub, at which point the bound density profile has dropped to 0.02 ×ρloc. The dashed blue vertical line shows the estimated tidal radius of the subhalo based on Msub.

In Fig. 15, we show how the subhalo radii determined in this way compare with tidal radii derived from the distances of subhaloes to halo centre, and their SUBFIND masses Msub. We calculate the tidal radius (Binney & Tremaine 1987; Tormen, Diaferio & Syer 1998) of a subhalo of mass Msub and distance r from the centre of the main halo as

image(12)

where M(<r) is the main halo mass within a sphere of radius r. Reassuringly, there is good agreement between rt and rsub in the mean, with some scatter. This gives us further confidence that our subhalo radii are physically meaningful, and that SUBFIND correctly identifies the self-bound regions of subhaloes.

Figure 15.

Comparison between estimated subhalo radii and tidal radii for the Aq-A-1 simulation. Only subhaloes with at least 104 particles have been included in this plot.

4.2 Matching individual subhaloes

In order to study convergence not only of the main halo but also of individual embedded subhaloes, we need an appropriate method to identify the same subhalo in simulations with different mass resolution. This is less straightforward than it may seem at first, since one cannot expect the subhalo to be at exactly the same position in different simulations. When the numerical resolution is changed, small phase offsets build up in the orbits of individual subhaloes, causing them to be at slightly different positions in different simulations, even though their histories and their internal structure may agree in detail (see e.g. Frenk et al. 1999).

One solution to this problem is to match subhaloes at very early times, by tracking the particles of a particular subhalo back to the (unperturbed) initial conditions. For each particle, we can then find the closest particle in another realization of the initial conditions created for the same halo but at different numerical resolution. These matched particles can then be tracked forward in time in the second simulation to see where they end up. This yields a set of possible subhalo matches in the second simulation, among which the one containing the largest number of matched particles is selected as the partner to the original subhalo. To increase robustness, one may require that the same match be obtained when the procedure is carried out in reverse, i.e. starting with the subhalo in the second simulation.

We have found this procedure to work quite robustly for our simulation set. To speed up the matching procedure in the unperturbed initial conditions, we have successfully applied the following trick. Our IDs are constructed as 63-bit Peano–Hilbert keys (Springel 2005), i.e. they correspond to positions along a space-filling fractal that tessellates our simulation volume with a fiducial grid of 221 cells per dimension. This corresponds to a comoving spatial resolution of around 65 pc, which is still considerably smaller than the mean particle spacing in the high-resolution region, even for the Aq-A-1 simulation. Exploiting the fact that positions that are close on the Peano–Hilbert curve are always close in three-dimensional space (the reverse is, however, not always true), we can accelerate the matching by finding the particle with the nearest Peano–Hilbert key in the second simulation. This always finds a particle that is very close, although it does not guarantee that it is the closest. This procedure turns out to be quite sufficient for the task at hand here.

In Fig. 16 we compare properties of subhaloes matched in this way in the Aq-A series of simulations. The two panels on the left-hand side show the masses and maximum circular velocities of subhaloes in the Aq-A-1, Aq-A-3 and Aq-A-4 simulations in units of the values measured for their counterparts in the Aq-A-2 simulation. Remarkably, the agreement is excellent with a surprisingly small scatter, and there is no obvious systematic offset between the different resolutions. Further analysis of these data is given in Springel et al. (2008), where we show that the maximum circular velocity of subhaloes can be trusted in the mean with an accuracy of 10 per cent down to Vmax∼ 1.5  km  s−1. Convergence for rmax is more difficult to achieve, but is still good in the mean down to rmax∼ 165  pc. In the right-hand panel of Fig. 16 we compare the projected spatial positions of matching subhaloes in the Aq-A-1, Aq-A-2, Aq-A-3 and Aq-A-4 simulations. While the agreement for the absolute coordinates is not perfect, especially for subhalo quartets close to the main halo's centre, matching subhaloes are generally found quite close together. The mean spatial offset is of the order of ∼ 30  kpc, which is much better than we have typically found in our older simulation work (Stoehr et al. 2003). This is a tribute to the improved integration accuracy in gadget-3, and to the high quality of our initial conditions.

Figure 16.

The two panels on the left-hand side compare the Msub and Vmax values of individual matched subhaloes in Aq-A-1, Aq-A-2, Aq-A-3 and Aq-A-4. In both panels, the measured values are ratioed against the value found in Aq-A-2, our second highest resolution simulation of the ‘A’ halo. In the panel on the right-hand side, we use small circles to indicate the projected positions of the matched subhaloes on the xz plane. The positions of each subhalo in the four different simulations are joined by thin lines. The mean positional off-set between matched subhaloes in Aq-A-4 and Aq-A-1 is ∼ 54  kpc. This shrinks to 28 and 26 kpc for matches of Aq-A-3 and Aq-A-2 to Aq-A-1, respectively. The inner and outer circles mark r200 and r50.

4.3 The abundance of substructures within subhaloes

Using our matched sets of subhaloes we are now ready to consider the convergence of the properties of substructure inside subhaloes. We begin by considering (sub)substructures within the subhalo radius, rsub, and extend the analysis later to regions of fixed overdensity, in order to compare with the results reported by Diemand et al. (2008).

We start by simply counting all subhaloes within rsub of a given subhalo's centre and comparing the count with the number expected from the assumption that subhaloes are simply scaled-down copies of the main halo. This expected number must be adjusted to take into account that, unlike the main halo, subhaloes have a different ‘edge’ (rsub), as well as comparatively poorer mass resolution. In practice, the self-similar expected number is computed by considering in the main halo only subhaloes with masses exceeding inline image, where 20 mp is our absolute SUBFIND mass limit for subhalo detection, Mmain is the main halo's mass within a radius that encloses the same overdensity as that of the subhalo within rsub, and inline image is the total mass within rsub of the subhalo's centre.

In Fig. 17 we compare the actual (sub)subhalo counts with the self-similar expectation, as a function of subhalo mass. Interestingly, we always find fewer (sub)subhaloes than expected from the self-similar hypothesis. The suppression in substructure abundance is not uniform; a few subhaloes have almost the full expected abundance, while others lie well below it. This is not entirely unexpected, and it is probably related to the time since each subhalo was accreted into the main halo and the extent to which it has lost mass to tides. Typically the number of (sub)subhaloes is a factor of 2 or 3 below that predicted by self-similarity.

Figure 17.

(Sub)substructure count within subhaloes as a function of their mass. The plot includes all subhaloes with more than 50 000 particles in the Aq-A-1 simulation. The actual (sub)subhalo counts for these subhaloes are shown as red dots and their average in logarithmically spaced mass bins is indicated by the purple line. Open circles show the (sub)subhalo count which these same subhaloes would have if they were scaled-down copies of the main halo, with the corresponding average indicated as a function of mass by the solid black line. When estimating these predictions, we correct for the lower effective resolution within subhaloes in comparison to the main halo by only counting main halo subhaloes above the appropriately scaled mass limit.

The mass fraction of subhaloes in the form of embedded substructure provides another way of quantifying (sub)substructure abundance. To estimate this we simply measure the total mass of all (sub)subhaloes within rsub and divide it by the total mass M(rsub) contained within this same radius. Because the minimum subhalo size identified by SUBFIND is always 20 particles regardless of the mass of the parent object (main halo or subhalo), we correct the measured cumulative substructure mass by extrapolating the mass function from 20mp down to 20mpM(rsub)/Mmain using a slope N(> m) ∝m−0.9. Note that Mmain is here the mass of the main halo measured within the radius that encloses an equal mean overdensity as the subhalo within rsub.

In Fig. 18, we compare the substructure mass fraction of subhaloes to those of the main halo. The latter is computed within the radius where the main halo density equals that of the subhalo [inline image]. As shown in Fig. 18, computed in this way the main halo prediction is a monotonically decreasing function of ρsub. Interestingly, this line forms an accurate upper bound to the measured substructure mass fractions of subhaloes. Close to the line, the self-similar expectation is fulfilled, but the fact that most subhaloes lie considerably below the line confirms that substructure is on average significantly underabundant in subhaloes when compared to the main halo.

Figure 18.

(Sub)substructure mass fraction in subhaloes as a function of their radius within the main halo, expressed as the mean enclosed density in units of the cosmic mean density. The innermost subhaloes are thus on the right-hand side. The plot includes all subhaloes with more than 50 000 particles in Aq-A-1 (shown as symbols). All (sub)subhaloes within the estimated subhalo radius rsub are counted, and their total mass is normalized by the total mass within rsub. To be able to compare all measured values of fsub on an equal footing, we add a correction to account for (sub)subhaloes below our resolution limit but above the mass which corresponds to the resolution limit in the main halo after scaling down to each subhalo. The solid red line gives the substructure mass fraction which the subhaloes would have if they were scaled down versions of the main halo, but with all material outside the tidal radius removed. The horizontal blue line is the substructure mass fraction for the main halo within r50.

For completeness, and in order to ease comparison with previous work, we show in Fig. 19 the cumulative subhalo abundance for 12 of our more massive subhaloes in the Aq-A halo. For the first time, we consider the convergence of the (sub)substructure mass function for individual subhaloes, based on the Aq-A-1, Aq-A-2 and Aq-A-3 simulations and our matched set of subhaloes. Here we count all substructures within a sphere (of radius r250) centred on the subhalo, and of mean overdensity 250 times the critical density (and so 1000 times the cosmic mean density, as chosen by Diemand et al. 2008). A further criterion for selecting the subhaloes shown in Fig. 19 was that their tidal radius rt should exceed r250, thus ensuring that the region where (sub)subhaloes are counted really lies inside the subhalo. Note that this criterion is actually rather restrictive, as it precludes subhaloes from study that are at radii r < r250= 258 kpc.

Figure 19.

Cumulative mass function of (sub)subhaloes within subhaloes. In each panel, we show results for one subhalo, and we compare results for different numerical resolutions, corresponding to simulations Aq-A-1, Aq-A-2 and Aq-A-3. Only substructures within a distance r250 of the subhalo centre are counted, and the substructure mass is normalized to the mass within this radius M250. The thin black power law shows the subhalo abundance of main haloes within a radius enclosing the same mean overdensity. The labels in each panel give the subhalo mass Msub as determined by SUBFIND, the values of M250 and r250, and the distance d of the subhalo from the centre of the main halo.

The (sub)subhaloes shown in Fig. 19 are counted as a function of their mass normalized by M250, the total mass of each subhalo within its own r250. When the substructure in main haloes is counted in this way, a near-universal cumulative subhalo mass function is found, as we show explicitly in Fig. 20 for our sample of six haloes simulated at resolution level 2. There is a well-defined mean relationship with little scatter. For reference, we include a power-law fit to this relation in the various panels of Fig. 19. Clearly, also in this case most subhaloes show a cumulative substructure mass function with similar slope but with a normalization that is typically substantially lower. Only a few subhaloes have a substructure abundance which approaches that predicted by scaling our results the main halo.

Figure 20.

Cumulative subhalo mass function within r250 for our six different haloes at level 2 resolution, as well as for our highest resolution run Aq-A-1. Subhalo masses are measured in units of M250 in each case. The dashed line is a fit to the mean mass function.

We may also compare the substructure abundance of subhaloes inside r250 with that of field haloes of equal M250, such that both are equally well resolved and biases due to different numerical resolutions are excluded. Fig. 21 shows our results for this comparison, both in terms of the count of all substructures down to the resolution limit, and by just counting subhaloes with a maximum circular velocity larger then 0.1 V250, which effectively measures the amplitude of the (sub)subhalo velocity function. We here used uncontaminated field haloes found in the high-resolution region around the main halo in Aq-A-1, and compared them to subhaloes in the outer parts of the main halo (with r > 258 kpc, such that their tidal radius is larger than r250). Again we find an offset of about a factor of 2 in the mean substructure abundance between field haloes and genuine subhaloes.

Figure 21.

Substructure count in subhaloes of Aq-A-1 within r250 (red filled circles) compared with field haloes (hollow black circles) inside r250, as a function of M250. The top panel shows the substructure count down to our resolution limit, while the bottom panel gives the count above a limiting maximum circular velocity equal to 0.1 V250, which effectively measures the amplitude of the subhalo velocity function. The solid lines give averages for independent logarithmic mass bins. We see that there is a systematic offset in the substructure abundance of field haloes and genuine subhaloes. The downturn of the count above 0.1 V250 for masses below ∼ 109  M⊙ is due to resolution limitations.

We conclude that the (sub)substructure abundance in subhaloes is not, in general, a scaled version of that in main haloes. Rather, the self-similar expectation provides an upper limit on the abundance of these second generation substructures; less (sub)substructure is typically found. This reflects the fact that the substructure abundance of a subhalo is not only diminished by tidal truncation once it falls into a larger structure; in addition, its retained substructures continue to lose mass to the main subhalo through tidal effects and, in strong contrast to the situation for main haloes, they are not continually replenished by infall of new subhaloes from the field. The substructure deficit in subhaloes when compared to main haloes at the same mean overdensity is expected to grow with time as they orbit within their main haloes. It will therefore be more marked in subhaloes in the inner halo, which are typically ‘older’.

Our results thus caution strongly against the assumption that subhaloes typically have mass fractions in substructure similar to the main halo (as suggested by e.g. Shaw et al. 2007; Kuhlen et al. 2008).

5 INTERNAL STRUCTURE OF SUBHALOES

In this section we study the structural properties of subhaloes, and we compare them to the properties of similar mass isolated haloes. We analyse how the density profiles of individual subhaloes converge as numerical resolution is increased, and we measure the concentration of subhaloes as a function of mass, circular velocity and radial distance. We then compare with the corresponding relations for field haloes.

5.1 Density profiles for subhaloes

The internal density structure of non-linear dark matter haloes is one of the most important predictions obtained from numerical simulations of the CDM paradigm. The density profile directly affects the rotation curves of galaxies, the gravitational lensing properties of dark matter haloes, and the X-ray luminosity and SZ signal of galaxy groups and clusters. The density profile of subhaloes also determines the kinematics of the stars in satellite galaxies, which are observationally accessible for the dwarf spheroidals around the Milky Way (e.g. Stoehr et al. 2002; Strigari et al. 2007a). Furthermore, the inner density profiles of haloes and subhaloes are critical for estimating the luminosity in dark matter annihilation radiation, in case such a decay channel exists.

About a decade ago, NFW were able to show that the spherically averaged density profiles of dark matter haloes have an approximately universal shape that is well described by a simple fitting formula

image(13)

which has become known as the NFW profile. In this double power law, the local logarithmic slope gradually changes from a value of −3 in the outer parts to an asymptotic slope of −1 in the inner parts. The spatial scale rs of this transition is treated as a fitting parameter and is often parametrized in terms of the concentration c=r200/rs of the halo, which is, in fact, simply a reparametrization of δc, the characteristic overdensity relative to the critical density: δc= (200/3) c3/[ ln (1 +c) −c/(1 +c)]. NFW showed the concentration to depend systematically on halo mass, a finding that can be interpreted as reflecting the density of the universe at the time of halo formation. A number of analytic fitting functions for this dependence have been proposed in the literature (NFW; Bullock et al. 2001; Eke, Navarro & Steinmetz 2001), but only recent large-volume simulations have been able to calibrate it reliably for rare objects like massive galaxy clusters (Gao et al. 2007; Neto et al. 2007).

Ever since the discovery of the NFW profile, the structure of the inner cusp has been the subject of much discussion and controversy. As computing power has increased, many groups have re-examined the value of the inner slope using ever bigger and better resolved simulations, but no consensus has yet emerged (Fukushige & Makino 1997; Moore et al. 1998, 1999b; Ghigna et al. 2000; Jing & Suto 2000; Fukushige & Makino 2001; Klypin et al. 2001; Jing & Suto 2002; Fukushige & Makino 2003; Power et al. 2003; Fukushige, Kawai & Makino 2004; Navarro et al. 2004; Diemand et al. 2005; Stoehr 2006; Knollmann, Power & Knebe 2008).

It has often been claimed that the inner cusps of haloes and subhaloes may have slopes less than −1, with some studies even proposing an asymptotic slope of −1.5 (Moore et al. 1999b; Fukushige & Makino 2001). For main haloes this proposition has been ruled out in recent years by newer generations of simulations. Nevertheless, the idea that the asymptotic slope is typically steeper than −1 (e.g. ∼ − 1.2) is still widespread and has been reiterated in recent papers, even though this is clearly inconsistent with e.g. Fig. 4 or the numerical data in Navarro et al. (2004).

With respect to the density profiles of subhaloes, the situation is even more unclear. So far few studies have examined this question directly. Stoehr (2006) found that the circular velocity curves of subhaloes are best fitted by a parabolic function relating log V to log r, implying that the density profiles become shallower in the centre than NFW. On the other hand, Diemand et al. (2008) recently argued that subhaloes have steep cusps with a mean asymptotic slope of −1.2.

We want to emphasize from the outset that the nature of halo and subhalo density profiles, becoming gradually and monotonically shallower towards the centre, makes it easy to arrive at the wrong conclusion for the structure of the inner cusp. Almost all numerical simulations to date have been able to produce demonstrably converged results for the density profile only in regions where the local slope is significantly steeper than −1. They have also all shown that the slope at the innermost measured point is significantly shallower than at radii a factor of a few further out. Thus, although no slope as shallow as −1 has been found, there is also no convincing evidence that the values measured are close to the asymptotic value, if one exists. Most claims of steep inner cusp slopes are simply based on the assertion that the slope measured at the innermost resolved point continues all the way to the centre.

Navarro et al. (2004) argued that the local logarithmic slope of halo profiles changes smoothly with radius and is poorly fitted by models like those of NFW or Moore that tend to an asymptotic value on small scales. They showed that in their simulation data the radial change of the local logarithmic slope can be well described by a power law in radius, of the form

image(14)

which corresponds to a density profile

image(15)

Here ρ−2 and r−2 are the density and radius at the point where the local slope is −2. This profile was first used by Einasto (1965) to describe the stellar halo of the Milky Way, so we refer to it as the Einasto profile. The introduction of a shape parameter, α may be expected, of course, to provide improved fits, but we note that fixing α∼ 0.16 gives a two-parameter function which still fits mean halo profiles much better than the NFW form over a wide range of halo masses (i.e. with maximum residuals of a few per cent rather than 10 per cent; Gao et al. 2007). Further evidence for a profile where local slope changes gradually has been presented by Stoehr et al. (2003); Graham et al. (2006); Stoehr (2006). For reference, we note that the enclosed mass for the Einasto profile is

image(16)

where γ(a, x) is the lower incomplete gamma function. For a value of α= 0.18 the radius where the maximum circular velocity peaks is given by rmax= 2.189 r−2, and the maximum circular velocity is related to the parameters of the profile by V2max= 11.19 G r2−2 ρ−2.

No published simulation to date has had enough dynamic range to measure the logarithmic slope of the density profile in the region where the Einasto model would predict it to be shallower than −1, so only indirect arguments could be advanced for this behaviour (Navarro et al. 2004). This situation has changed with the Aquarius Project, as can be seen from Fig. 4, and in Navarro et al. (2008) we provide a detailed analysis of this question. In the following, we focus on the density profiles of dark matter subhaloes, where the available particle number is, of course, much smaller. Our best resolved subhaloes in the Aq-A-1 simulation contain more than 10 million particles, allowing a relatively precise characterization of their density profiles. Until recently, such particle numbers represented the state of the art for simulations of main haloes.

In Fig. 22, we show spherically averaged density profiles for nine subhaloes within the Aq-A halo. For each we compare up to five different resolutions, covering a factor of ∼1835 in particle mass. The density profiles line up quite well outside their individual resolution limits, as predicted by the convergence criterion of Power et al. (2003) in the form given in equation (3). Individual profiles in the panels are plotted as thick solid lines at radii where convergence is expected according to this criterion, but they are extended inwards as thin lines to twice the gravitational softening length (the gravitational force is exactly Newtonian outside the radii marked by vertical dashed lines). These density profiles are based on particles that are gravitationally bound to the subhaloes, but for comparison we also show a profile for each subhalo that includes all the mass (i.e. including unbound particles; thick dashed lines). It is clear that the background density dominates beyond the ‘edge’ of each subhalo. It is therefore important that this region is excluded when fitting analytic model density profiles to the subhaloes.

Figure 22.

Subhalo density profiles for nine different subhaloes in the Aq-A halo, simulated with varying resolution. The profiles show the bound mass only and are drawn with thick lines for the radial range where convergence is expected, based on the criterion of Power et al. (2003). They are continued with thin lines down to the scale . Vertical dashed lines mark the radii where the force law becomes Newtonian (2.8ε). The dot–dashed purple line in each panel is the density profile of all the mass around the subhalo's centre (i.e. including unbound mass). The thin black line shows a fit with the Einasto profile. The labels in each panel give the maximum circular velocity, mass and distance d to halo centre for each subhalo. α is the shape parameter of the Einasto profile, which we here allowed to vary freely in our fits.

In making such fits, we restrict ourselves to the radial range between the convergence radius (equation 3) and the largest radius where the density of bound mass exceeds 80 per cent of the total mass density. The density profiles themselves are measured in a set of radial shells spaced equally in log r. To define the best fit, we minimize the sum of the squared differences in the log between measurement and model, i.e. we characterize the goodness of fit by a quantity

image(17)

where the sum extends over all bins i. We then minimize Q with respect to the parameters of the model profile. We have included such fits as thin solid lines in Fig. 22, based on the Einasto profile, allowing the third parameter α to vary as well. The resulting values of α and the maximum circular velocities of the subhaloes, as well as their mass and distance to the main halo's centre are shown as labels in the individual panels.

It is clear from Fig. 22 that the Einasto profile provides a good description of subhalo radial density profiles, but due to the large dynamic range on the vertical axis combined with the narrow radial range over which the density profile can be fitted, it is not clear in this representation whether the Einasto fit is significantly better than fits with other analytic functions, like the NFW or Moore profiles.

Further insight can be obtained by studying the local logarithmic slopes of the subhalo density profiles as a function of radius, which we show in Fig. 23, obtained by finite differencing of the measured density profiles. Again, we compare the differing resolutions available for Aq-A, and plot the results as thick lines for radii where we expect convergence according to Power et al. (2003), continuing them with thin lines towards smaller scales. The convergence criterion appears to work quite well and in most cases accurately delineates a limit beyond which the profiles suddenly start to become significantly flatter. At larger radii, the local slopes change continuously and smoothly with radius. For several subhaloes, we have direct evidence that for the local slope is significantly shallower than −1.5 in the innermost converged bin, thereby ruling out the Moore profile for at least some dark matter subhaloes. In one case, we find convergence to a slope which is clearly shallower than −1.2. As for main haloes, extrapolation of the shape of these curves to smaller radii suggests that profiles that will become significantly shallower before reaching an asymptotic inner slope, if one exists. From these results it seems very unlikely that typical dark matter subhaloes could have power-law cusps with slopes as steep as −1.2, as recently suggested by Diemand et al. (2008).

Figure 23.

Local logarithmic slope (obtained by numerical differentiation) of the measured density profile for nine different subhaloes, at varying numerical resolution. As in Fig. 22, the thick line style is used in regions where numerical convergence is expected. The thin solid line shows a fit with the Einasto profile. Slopes of −1.5 (corresponding to the Moore profile) and of −1.0 (the NFW profile) are marked with horizontal dashed lines.

Another way to arrive at a similar conclusion is not to consider the numerically differentiated density profile, but rather the maximum asymptotic inner slope

image(18)

which can be supported by the enclosed mass at a certain radius. This quantity was introduced by Navarro et al. (2004). It requires converged values for both the local density and the enclosed mass at each radius r. This is a more stringent convergence requirement than asking that the density alone be converged. Nevertheless, it can provide a powerful lower limit on the profile slope in the inner regions; there cannot possibly be a cusp steeper than ρ∝r−β since there is simply not enough mass enclosed to support it. In Fig. 24, we show β(r) as a function of radius for the same subhaloes as before, using the same approach to mark the Power et al. (2003) convergence radius. We see that this convergence criterion is not conservative enough in some of cases, where the enclosed mass is not fully converged for the last bin. The Power et al. (2003) criterion was actually designed for this quantity, but it has only been tested for main haloes, and it is not surprising that we find subhaloes to be somewhat more demanding. Nevertheless, this figure reinforces our earlier conclusion. For most of the subhaloes, a central dark matter cusp as steep as the Moore profile can be safely excluded, and in a few cases, the limit is shallower than ∼− 1.3. Again, the shape of β(r) suggests that limits are likely to tighten considerably once still smaller scales can be resolved.

Figure 24.

The maximum possible inner asymptotic slope of the density profiles of nine subhaloes in the Aq-A-1 simulation. The plot uses the same line styles as in Fig. 23. Rather than the local slope of the density profile, this plot shows the most negative slope that is consistent both with the locally enclosed mass, and the local spherically averaged density.

Finally, we would like to answer objectively the question whether the Einasto model fits subhalo profiles better than the NFW or Moore models; in other words, whether it produces smaller residuals overall. To test this question, we fix α for the Einasto profile at α= 0.18 so that there are only two free parameters left, as in the NFW and Moore profiles. (These are a characteristic overdensity and a radial scale.) Our results are insensitive to varying α in the range ∼ 0.16– 0.20. We estimate best fits for 526 subhalo profiles (considering all subhaloes in Aq-A-1 with more than 20 000 particles) by minimizing the quantity Q defined by equation (17) over the radial range between the Power convergence radius and an outer radius defined as above. In Fig. 25 we show the results. We plot the mean residual per bin with symbols giving results for the Einasto profile to illustrate the typical scatter. The solid coloured lines are means for the three different profile shapes, calculated for logarithmic bins of subhalo mass. We see that the Einasto profile consistently produces the lowest residuals, followed by the NFW profile, while the Moore profile is consistently the worst. The relatively small difference in the quality of the fit between the NFW and Moore profiles is due to the fact that the resolution limitations for the subhaloes restrict the fits to comparatively large radii where the two still have quite similar shape. There appears to be no systematic trend with subhalo mass.

Figure 25.

Quality of fits to subhalo density profiles, based on three different two-parameter models, an NFW profile, a Moore profile and an Einasto profile with α= 0.18. The circles show a measure for the mean deviation per bin, Q, for 526 subhaloes in the main halo of the Aq-A-1 simulation. The subhaloes considered contain between 20 000 and nearly ∼10 million particles. The lines in different colours show averages in logarithmic mass bins for each of the three profiles.

We conclude that the density profiles of subhaloes show similar behaviour to those of main haloes; the local logarithmic slope becomes gradually shallower with decreasing radius. There is no evidence that a fixed asymptotic power law has been reached at the innermost converged points. Inner cusps as steep as the Moore profile are excluded for most objects, and for some objects we can already exclude logarithmic slopes as steep as −1.3. We note that over the resolved range the Einasto density profile yields circular velocity profiles that are also well described by the parametrization of Stoehr (2006), but the latter will clearly become invalid at very small scales where it yields a density maximum at a finite radius.

5.2 The concentration of subhaloes

Because the density profiles of dark matter haloes are not pure power laws it is possible to assign them a characteristic density or ‘concentration’. Perhaps the simplest such measure is the overdensity (relative to critical) within the radius where the circular velocity curve peaks (see equation 6). In many studies it has been found that haloes of a given mass exhibit a well-defined characteristic concentration (NFW; Bullock et al. 2001; Eke et al. 2001; Gao et al. 2007; Neto et al. 2007, or in other words, that the radius rmax at which the circular velocity peaks is tightly correlated with the maximum circular velocity Vmax. Recently, Neto et al. (2007) have given an accurate fit to this relation for haloes with masses between about 1012 and 1015  M⊙, based on the good statistics provided by the Millennium Simulation.

In the top panel of Fig. 26, we show the relationship between rmax and Vmax as measured for main haloes in our Aq-A simulation. These are haloes that are outside of the main Milky Way sized halo, but are still contained in the high-resolution region. We make sure to include only haloes that are free of any contamination by boundary particles. Comparing the various resolutions available for the ‘A’ halo, it can be seen that the correlation can be trusted down to about Vmax∼ 1.5  km s−1 for our highest resolution calculation, the Aq-A-1 run. Remarkably, we find that the power-law relation of Neto et al. (2007) describes our measurements very accurately, despite the fact that this is an extrapolation by several orders of magnitude into a regime which was previously unconstrained by numerical data.

Figure 26.

Relation between rmax and Vmax for main haloes (top) and subhaloes (bottom) in the Aq-A series of simulations. We compare results for simulations of different resolution for this halo, and we use solid lines to mark the mean of log rmax in each bin. The dashed red lines enclose 68 per cent of the distribution for the Aq-A-1 simulation. The solid line is an extrapolation to smaller mass of the result of Neto et al. (2007) for the haloes of the Millennium Simulation, while the dotted power law in the lower panel is a fit to our results for subhaloes, lying a factor of 0.62 lower.

The bottom panel of Fig. 26 shows the equivalent measurements for subhaloes that are contained within r50 of the main halo. Clearly, these subhaloes are typically more concentrated than haloes of the same circular velocity in the field, as first found by Ghigna et al. (1998), Bullock et al. (2001). At equal Vmax, the rmax values of subhaloes are on average 62 per cent of those of field haloes, corresponding to a 2.6 times higher characteristic density. This can be understood as a result of tidal mass-loss. As shown by Diemand, Kuhlen & Madau (2007b) and Peñarrubia, Navarro & McConnachie (2008), stripping reduces both Vmax and rmax, but the reduction in rmax is larger, so that the concentration increases (see also Hayashi et al. 2003; Kazantzidis et al. 2004; Bullock & Johnston 2005). We note that this effect also increases the characteristic density and the dark matter annihilation luminosity of subhaloes relative to haloes in the field when they are compared at equal mass, contrary to the arguments of Strigari et al. (2007b).

The subhalo masses Msub are tightly correlated with Vmax as well, as shown in Fig. 27. However, the slope of this relation, MsubV3.5max, is somewhat steeper than expected for a self-similar scaling of subhalo structure with size. This is again a consequence of tidal mass-loss, which affects the mass of a subhalo more than its maximum circular velocity.

Figure 27.

Mean relation between subhalo mass as assigned by SUBFIND and maximum halo circular velocity. The dashed red lines enclose 68 per cent of the distribution around the mean (calculated as an average of log Msub) for the Aq-A-1 simulation. The dotted line is a power-law fit, Msub≃ 3.37 × 107(Vmax/10  km s−1)3.49, to the results of Aq-A-1.

Another interesting quantity to consider is the mean characteristic density contrast δV=V2max/(H0rmax)2 of subhaloes. In Fig. 28, we show the dependence of this measure of concentration on circular velocity and subhalo mass. Clearly, the concentration increases strongly with decreasing subhalo mass. Interestingly, this trend is equally strong for subhalo samples at different radii, but the absolute values of the concentrations are larger at smaller radii. This is illustrated in Fig. 28 which compares results for the inner halo (r < 50 kpc) and for a shell at large radii (r > 300 kpc) with results for the halo as a whole.

Figure 28.

Subhalo characteristic density (which is a measure of concentration) as a function of maximum circular velocity (left-hand panel) and subhalo mass (right-hand panel). We show results for all haloes in the r50 radius, as well as separately for the inner halo within 50 kpc, and for an outer shell between 300 kpc and r50. The solid line in the left-hand panel gives an extrapolation of the result which Neto et al. (2007) quote for haloes in the Millennium Simulation, while the dotted power law lies a factor of 2.6 higher and fits our results for the subhaloes within r50. The dotted line in the right-hand panel, δV≃ 5.80 × 108(Msub/108  M⊙)−0.18, is a fit to our results for all subhaloes within r50.

This radial trend is more directly displayed in Fig. 29, where we show the mean characteristic density contrast as a function of radius for samples selected above different lower cut-offs in circular velocity. In general, subhalo concentrations rise towards halo centres, as found by Diemand et al. (2007b, 2008). For comparison, we also show results for Via Lactea II, as recently published by Diemand et al. (2008) where a cut-off of 5 km s−1 was used. Interestingly, our subhaloes are substantially more concentrated than those in Via Lactea II for the same lower cut-off. The Via Lactea II subhaloes are actually slightly less concentrated than our subhaloes selected above 10 km s−1. However, the origin of this offset may well lie in differences in the adopted cosmologies (Macciò, Dutton & van den Bosch 2008).

Figure 29.

Subhalo concentration as a function of radius, for subhaloes with maximum circular velocity larger than 2.5, 5 or 10 km s−1. The dotted line is a fit to our result for the 5 km s−1 sample, which yields δV≃ 3.77 × 106  (r/kpc)−0.63. For comparison, we have also included the result quoted recently by Diemand et al. (2008) for the Via Lactea II simulation, which selected subhaloes with Vmax > 5 km s−1. Clearly, our subhaloes are more concentrated than theirs at the same circular velocity, a difference that may be due to differences in the adopted cosmologies (e.g. Macciò et al. 2008).

6 SUMMARY

In this paper, we have presented first results from the Aquarius Project, a Virgo Consortium4 programme to carry out high-resolution dark matter simulations of Milky Way sized haloes in the ΛCDM cosmology. This project seeks clues to the formation of galaxies and to the nature of the dark matter by designing strategies for exploring the formation of our Galaxy and its luminous and dark satellites, for searching for signals from dark matter annihilation, and for designing experiments for the direct detection of dark matter.

In our approach, we pay great attention to validating our numerical results to careful convergence studies. In addition, we explore possible uncertainties in predictions for the Milky Way resulting from the scatter in properties between otherwise similar haloes. Thus, we simulate not just one realization at ultrahigh resolution, but rather a sample of (currently) six different haloes. Our ambition is to redefine the state of the art in this field with respect to the accuracy of the cosmological N-body simulations, and the rigour with which quantitative statements about halo structure can be made.

Our new simulation code gadget-3, developed specifically for the Aquarius Project, is a highly efficient, massively parallel N-body code. It offers much better scalability to large numbers of compute cores and a higher basic speed than its parent code gadget-2 (Springel 2005). It is able to cope efficiently with the tight coupling of around 1.5 billion particles in a single non-linear object, split up across 1024 CPUs. Some of our simulations at resolution level 2 were run on an even larger number of compute cores, using up to 4096 cores of a Bluegene/P computer. Here we used a novel feature in gadget-3 that can exploit additional compute cores in shared-memory nodes by means of threads (based on the POSIX pthreads library) yielding a mix of distributed and shared memory parallelism. The ability to simulate this high degree of clustering and non-linearity on massively parallel architectures is a prerequisite for exploiting the power of upcoming petaflop computers for the next generation of high-precision simulations of cosmological structure formation.

The results presented above demonstrate that we have created a remarkably accurate set of simulations, reaching very good convergence for the dark matter density profile and the substructure mass function over the maximum range that could be expected. Even the location, mass and internal structure of individual large dark matter subhaloes reproduce well between simulations of differing resolution, a level of convergence which exceeds anything previously reported in the literature.

The abundance of dark matter subhaloes is remarkably uniform across our halo sample when normalized to parent halo mass, and when considering subhaloes sufficiently small that fluctuations due to counting statistics are unimportant. The differential subhalo mass function is tilted to a slope slightly shallower than the critical value −2, so that, even when extrapolated to arbitrarily small masses, the total mass fraction in substructures remains small, less than 3 per cent within 100 kpc of halo centre, and less than 20 per cent within r50∼ 400  kpc. Adopting the logarithmically divergent slope −2 (which our results appear to exclude) does not increase these mass fractions by more than a factor of 2 or 3 for lower mass limits in the range 10−6 to 10−12  M⊙, which plausibly correspond to the thermal free-streaming limit if the dark matter is the lightest sypersymmetric particle. The inner halo is dominated by a smoothly distributed dark matter component, not by substructure.

Independent of their present mass, substructures have a strong preference to be found in the outer regions of haloes. For example, we estimate that at most a fraction of 10−3 of the dark matter at the solar circle is in bound subclumps. The rest is smoothly distributed. Note, however, that this smooth component is expected to have a rich structure in velocity space, being composed of a large number (perhaps 105 or more) of cold streams (Helmi, White & Springel 2003; Vogelsberger et al. 2008).

Contrary to previous claims, we find that substructure in subhaloes is not a scaled-down version of substructure in main haloes. Subhaloes typically have less substructure than main haloes. This is due to two causes. Tidal stripping removes the outer substructure-rich parts of haloes when they fall into a larger system and become subhaloes. In addition, as the retained substructure ages it decreases in mass and number and is not replaced by the infall of new objects. As a result, the substructure mass fraction in subhaloes is often much smaller than in main haloes, particularly for subhaloes in the inner regions which are the most heavily stripped and also, typically, the ‘oldest’.

We have presented the first detailed convergence study of the shape of subhalo density profiles to be based on simulation sets where the same subhalo can be identified in simulations of differing mass resolution. We find that the inner regions of subhaloes, well inside their tidal truncation radii, can be well fitted by NFW or Einasto profiles. Einasto fits are typically preferred, even when the shape parameter α is fixed to a standard value, e.g. α= 0.18. We have also studied how the local logarithmic slope of the density profile varies with radius, finding profiles to become gradually shallower towards the centre with no sign of approaching an asymptotic power-law behaviour. This is very similar to the behaviour of the central cusp in isolated dark matter haloes (Navarro et al. 2004, 2008). We find many subhaloes for which the slope at the innermost converged point is substantially shallower than −1.5, and a few where it is shallower than −1.2. The Moore profile appears firmly excluded as a description of the inner regions of subhaloes. It should not be used when modelling the CDM annihilation signal, as in a number of recent papers (e.g. Baltz et al. 2008).

The concentration of subhaloes is higher than that of haloes of the same circular velocity or of the same mass in the field. This can be understood as a consequence of tidal truncation and mass-loss (Kazantzidis et al. 2004; Bullock & Johnston 2005; Peñarrubia et al. 2008) which lead to a larger reduction of rmax than of Vmax. Interestingly, we find that the relationship between rmax and Vmax for field haloes is very well fitted by the fitting function given by Neto et al. (2007) for the Millennium Simulation, even though this involves an extrapolation over many orders of magnitude towards lower mass. At the same maximum circular velocity, we find that the rmax values of subhaloes are, on average, only 62 per cent of those of field haloes.

We note that our results disagree with those of the recent Via Lactea I and II simulations (Diemand et al. 2007a, 2008; Kuhlen et al. 2008; Madau et al. 2008) on several important points. We find substantially more substructure than reported for the Via Lactea simulations, and the discrepancy with Via Lactea I is larger than the expected halo-to-halo scatter, based on our own simulation set. We also differ in our conclusions about the amount of (sub)substructure in subhaloes, which we demonstrate to be less than predicted by the hypothesis that subhaloes are tidally truncated, but otherwise scaled-down versions of field haloes. Our subhaloes are also more concentrated than those found in the Via Lactea II simulation, but this difference may simply be due to cosmology. Finally, we disagree with the claim of Diemand et al. (2008) that subhaloes have central power-law cusps with a mean slope of −1.2.

In future work, we will analyse the detailed formation history of the ‘Aquarius’ haloes and the evolution of their substructure. We will also build a new generation of semi-analytic models to follow the evolution of the baryonic component, and we will compare these with full hydrodynamical simulations of these same haloes that we have already begun to carry out. This should bring new insights into galaxy formation, and directly address possible small-scale challenges to the ΛCDM theory. The verdict about whether CDM works on such scales is still pending.

Footnotes

  • 1

    We use this unconventional outer radius for our haloes, rather than the standard r200, to facilitate comparison with Diemand, Kuhlen & Madau (2007a) and Diemand et al. (2008) who quote results for their Via Lactea simulations within r50 although they refer to this radius as ‘r200’.

  • 2

    Further images and videos of the formation process of the haloes are available at http://www.mpa-garching.mpg.de/aquarius.

  • 3

    Note that V50,host unambiguously characterizes the enclosed mass within r50, the region in which subhaloes are counted. This is not the case for Vmax,host, the velocity scale chosen by Diemand et al. (2008), because it is additionally affected by halo concentration.

  • 4

    The Virgo Consortium is an international collaboration of astronomers working on supercomputer simulations of cosmic structure formation, see http://www.virgo.dur.ac.uk.

ACKNOWLEDGMENTS

The simulations for the Aquarius Project were carried out at the Leibniz Computing Centre, Garching, Germany, at the Computing Centre of the Max-Planck-Society in Garching, at the Institute for Computational Cosmology in Durham, and on the ‘STELLA’ supercomputer of the LOFAR experiment at the University of Groningen. SDMW acknowledges the Aspen Centre for Physics for providing the perfect atmosphere for final editing of this paper.