Abstract
We use the Millennium Simulation series to investigate the relation between the accretion history and mass profile of cold dark matter (CDM) haloes. We find that the mean inner density within the scale radius, r−2 (where the halo density profile has isothermal slope), is directly proportional to the critical density of the Universe at the time when the virial mass of the main progenitor equals the mass enclosed within r−2. Scaled to these characteristic values of mass and density, the average mass accretion history, expressed in terms of the critical density of the Universe, M(ρcrit(z)), resembles that of the enclosed density profile, M(〈ρ〉), at z = 0. Both follow closely the Navarro, Frenk & White (NFW) profile, which suggests that the similarity of halo mass profiles originates from the mass-independence of halo accretion histories. Support for this interpretation is provided by outlier haloes whose accretion histories deviate from the NFW shape; their mass profiles show correlated deviations from NFW and are better approximated by Einasto profiles. Fitting both M(〈ρ〉) and M(ρcrit) with either NFW or Einasto profiles yield concentration and shape parameters that are correlated, confirming and extending earlier work that has linked the concentration of a halo with its accretion history. These correlations also confirm that halo structure is insensitive to initial conditions: only haloes whose accretion histories differ greatly from the NFW shape show notable deviations from NFW in their mass profiles. As a result, the NFW profile provides acceptable fits to hot dark matter haloes, which do not form hierarchically, and for fluctuation power spectra other than CDM. Our findings, however, predict a subtle but systematic dependence of mass profile shape on accretion history which, if confirmed, would provide strong support for the link between accretion history and halo structure we propose here.
INTRODUCTION
Numerical simulations have shown that the equilibrium structure of cold dark matter (CDM) haloes is approximately self-similar. Spherically averaged density profiles, in particular, are well approximated by scaling a simple formula proposed by Navarro, Frenk & White (1995, 1996, NFW). The NFW profile has a fixed shape, and is characterized by a logarithmic slope that steepens gradually from the centre outwards. As such, it may be fully specified by just two parameters, usually chosen to be either the virial radius and a characteristic density or, equivalently, the halo virial mass and a concentration parameter. (See Section 3.1 for a summary of relevant formulae and definitions.)
The gently varying slope of the NFW profile confounded early theoretical expectations, which had envisioned a simple power-law behaviour (e.g. Fillmore & Goldreich 1984; Hoffman & Shaham 1985; Quinn, Salmon & Zurek 1986; Crone, Evrard & Richstone 1994), and has motivated a number of proposals to explain its origin (see Frenk & White 2012 for a recent review). Most attempts have taken as guidance the secondary infall model first proposed by Gunn & Gott (1972), complemented by various conjectures about the role of mergers (e.g. Salvador-Sole, Solanes & Manrique 1998), dynamical friction (e.g. Nusser & Sheth 1999), angular momentum (e.g. Williams, Babul & Dalcanton 2004) or adiabatic invariants (e.g. Avila-Reese, Firmani & Hernández 1998; Dalal, Lithwick & Kuhlen 2010), or else have argued that entropy generation during virialization might be behind the halo structural similarity (see, e.g. Taylor & Navarro 2001; Pontzen & Governato 2013).
No general consensus has yet emerged, however, reflecting the difficulty that all of these models face when trying to explain why the same NFW profile seems to fit the structure of haloes formed through hierarchical clustering regardless of power spectrum (Navarro, Frenk & White 1997), as well as that of hot dark matter haloes or of systems formed through monolithic collapse (e.g. Huss, Jain & Steinmetz 1999; Wang & White 2009).
In addition, none of these models provides a thorough explanation for the redshift-dependent correlations between mass and concentration seen in simulations, their scatter, or their dependence on cosmological parameters and power spectra. Halo concentration, which depends only weakly on mass, was originally linked to halo collapse time (Navarro et al. 1997), but attempts to reproduce the simulation results with simple prescriptions based on that proposal have met with limited success (Bullock et al. 2001; Eke, Navarro & Steinmetz 2001; Neto et al. 2007; Gao et al. 2008; Macciò, Dutton & van den Bosch 2008).
Better results have been obtained with empirical models that relate concentration to halo mass accretion history (MAH) and, in particular, to the time when the main halo progenitor switches from a period of ‘fast growth’ to one of ‘slow growth’ (Wechsler et al. 2002; Zhao et al. 2003; Lu et al. 2006). The success of these models is not, however, unqualified. Zhao et al. (2009), for example, argue that halo concentration is determined at the time when the main progenitor first reaches 4 per cent of the final mass, but there seems to be no natural justification for why concentration should be related to this particular, rather arbitrary time of a halo's assembly history.
Further complicating matters, there is now convincing evidence that a number of haloes have density profiles that deviate slightly, but significantly, from the NFW profile (Navarro et al. 2004). Accounting for these deviations requires the introduction of an additional shape parameter, thus, breaking the structural similarity of CDM haloes. One parametrization that results in excellent fits is the Einasto profile, where the logarithmic slope is a simple power law of radius, d ln ρ/d ln r ∝ (r/r−2)α: the shape parameter, α, is the exponent of the power law. This finding has now been verified by additional work (Merritt et al. 2005, 2006; Gao et al. 2008; Hayashi & White 2008; Stadel et al. 2009; Navarro et al. 2010; Ludlow et al. 2011) but there is no clear understanding of what breaks the similarity or what determines the value of α for a particular halo.
We explore these issues here using a large ensemble of haloes selected from the three Millennium Simulations, MS-I (Springel et al. 2005), MS-II (Boylan-Kolchin et al. 2009) and MS-XXL (Angulo et al. 2012), collectively referred to hereafter as MS. These are amongst the largest cosmological N-body simulations available, and provide us with thousands of well-resolved haloes spanning more than four decades in mass. Merger trees are available for all these simulations, making them an ideal data set to explore the relation between accretion history and mass profiles. In addition, the numerical homogeneity and sheer size of the volumes surveyed by the MS allow us to combine large numbers of haloes with similar properties to smooth out statistical fluctuations and idiosyncrasies of individual systems that might obscure the general trends. Our analysis reveals a subtle but well-defined relation between mass profile and accretion history that offers valuable new clues to the origin of the structure of CDM haloes.
The plan of this paper is as follows. We describe the simulations in Section 2 and the analysis procedure in Section 3. We present our main results in Section 4 and summarize our main conclusions in Section 5.
NUMERICAL SIMULATIONS
Our analysis focuses on dark matter haloes identified in the three Millennium Simulations. We provide here a brief summary of these simulations and of their associated halo catalogues. We refer the reader to the original papers for extensive details on each of the MS runs.
The Millennium Simulation suite
All MS runs adopt a flat, Wilkinson Microwave Anisotropy Probe one-normalized Lambda cold dark matter (LCDM) cosmology with the following cosmological parameters: Ωm = 0.25, ΩΛ = 1 − Ωm = 0.75, h = 0.73, n = 1 and σ8 = 0.9. Here Ωi is the present-day contribution of component i to the total matter energy density in units of the critical density for closure, ρcrit; σ8 is the rms mass fluctuation in 8 h−1Mpc spheres, linearly extrapolated to z = 0; n is the spectral index of primordial density fluctuations and h is the Hubble parameter. In addition to using the same cosmological parameters, the MS runs also adopted the same sequence of outputs in order to facilitate comparisons between the runs.
MS-II follows the dark matter distribution using 21603 particles of mass mp = 6.89 × 106 h− 1 M⊙ in a 100 h−1Mpc periodic box. MS-I has the same total particle number, but follows the evolution of structure in a comoving box of 500 h−1 Mpc on a side; each particle in MS-I is thus 125 times more massive than in MS-II, or mp = 8.61 × 108 h− 1 M⊙. MS-XXL is the largest of the three simulations in both box size and particle number; it follows 67203 particles of mass mp = 6.17 × 109 h− 1 M⊙ in a 3 h−1Gpc box.
Halo catalogues
A friends-of-friends (FOF) group finder (Davis et al. 1985) was run on the fly for each simulation output using a linking length of b = 0.2 times the mean interparticle separation and a minimum particle number Nmin = 20. The subhalo finder subfind, (Springel et al. 2001) was then run to identify self-bound substructure within each FOF halo.
subfind dissects each FOF halo into one dominant structure (the main halo) and a number of subhaloes that trace the self-bound remnants of accreted systems. We will focus our analysis on main haloes identified at z = 0 that contain at least N200 = 5000 particles within their virial radius.1
Since dark matter haloes are dynamical systems, transients induced by mergers or ongoing accretion can lead to rapid fluctuations in the structure of a halo that are poorly captured with simple fitting formulae. We therefore impose three criteria to flag systems that are clearly out of equilibrium. We consider a halo to be dynamically ‘relaxed’ if it simultaneously satisfies all three of the following conditions (Neto et al. 2007): (i) fsub < 0.1, (ii) doff < 0.07 and (iii) 2T/|U| < 1.35. Here, fsub is the fraction of the halo's virial mass contributed by subhaloes, |$d_{\rm off}=|{\bf r}_{\rm p}-{\bf r}_{\rm CM}|/r_{200}$| is the distance between the halo barycenter and the location of its potential minimum, expressed in units of r200 and 2T/|U| is the virial ratio of kinetic to potential energies, measured in the halo rest frame. None of our conclusions is heavily affected by these restrictions. Unrelaxed systems make up only 20 per cent of all haloes with virial mass of the order of 1012 M⊙ and 25 per cent of ∼1013 M⊙ haloes. Only at very large halo masses, such as cluster-sized ∼1014 M⊙ systems, the unrelaxed fraction exceeds 50 per cent. We refer the reader to Neto et al. (2007) for further discussion of these criteria, and to Ludlow et al. (2012) for a discussion of how the inclusion of out-of-equilibrium systems may impact the mass–concentration relation at large halo masses.
ANALYSIS
Fitting formulae
We note that, for given concentration, an Einasto profile with α ≈ 0.18 resembles closely an NFW profile over roughly two decades in radius or enclosed mass. Profiles with other values of α deviate systematically from the NFW shape (see, e.g. Navarro et al. 2004, 2010).
Profile fitting
Our analysis deals primarily with the spherically averaged density profiles of relaxed CDM haloes identified at z = 0 in each MS. We construct radial profiles using 32 concentric bins, equally spaced in log r, spanning the radial range −2.5 ≤ log r/r200 ≤ 0.
In practice, the parameters ρ−2 and r−2 can be expressed in a variety of equivalent forms, such as virial mass and concentration (M200,c), or the magnitude and location of the peak in the circular velocity curve (Vmax, rmax). In order to ease comparisons with previous work, we characterize the dark matter halo mass profile in terms of its virial mass M0 = M200(z = 0), its concentration c = r200/r−2, and its Einasto ‘shape’ parameter, α. The Einasto profile provides an excellent description of the density profile of relaxed MS haloes: the median value of ψ is just |$0.073^{+0.014}_{-0.011}$|, where the range represents the 25th and 75th percentiles.
An analogous procedure is used when NFW fits need to be performed; in this case, the two parameters estimated by the fit can also be expressed as the virial mass and concentration.
The fits are carried out over a radial range rmin < r < r200. The fitting procedure yields robust estimates for ρ−2, r−2 and α, provided rmin is chosen to be the minimum of either rconv or 0.05 × r200. Here, rconv is the convergence radius defined by Power et al. (2003), where circular velocity profiles converge to better than ∼10 per cent.
Mass profiles and accretion histories
The left-hand panel of Fig. 1 illustrates the role of c and α in describing the density profile. This figure shows the density profile of MS-II haloes selected in a narrow range of mass, 1.24 < log M200/1010 h−1 M⊙ < 1.54. (Densities are weighted by r2 in order to enhance the dynamic range of the plot.) Each profile corresponds to different systems, grouped by concentration: the green squares track the median2 profile of haloes with average concentration for that mass; the blue circles and red triangles correspond to haloes with concentration ∼50 per cent higher and lower than the average, respectively (see boxes in the top panel of Fig. 2).
Halo density profiles and accretion histories. Left-hand panel: median density profiles of MS-II relaxed haloes in the mass range 1.24 < log M200/(1010 h−1 M⊙) < 1.54 (corresponding to particle numbers in the range 2.5 × 104 < N200 < 5 × 104), selected according to their concentration (see boxes in the top panel of Fig. 2). Densities are shown scaled to ρ0, the critical density at z = 0, and weighted by r2 in order to enhance the dynamic range of the plot. Radii are scaled to the virial radius, r200. The best-fitting Einasto profiles are shown by the thin solid curves, with parameters listed in the legend. Dot–dashed curves indicate NFW profiles (whose shape is fixed in these units) matched at the scale radius, r−2, where the r2ρ profiles peak. Arrows indicate the half-mass radius, r1/2. Right-hand panel: median MAHs of the same set of haloes chosen for the left-hand panel. Halo accretion history is defined as the evolution of the mass of the main progenitor, expressed in units of the mass of the halo at z = 0. The heavy circles indicate the redshift, z−2, when the progenitor's mass equals the mass, M−2, enclosed within the scale radius at z = 0. The starred symbols indicate the half-mass formation redshift.
Mass dependence of the best-fitting Einasto parameters for all haloes in our sample at z = 0. Only relaxed haloes with more than 5000 particles within the virial radius are considered. The top and bottom panels show, respectively, the concentration, c = r200/r−2, and shape parameter, α, as a function of halo virial mass. Individual points are coloured according to the third parameter (see colour bar on the right of each panel). The connected symbols trace the median values for each Millennium Simulation (see legend in the top panel); thin solid lines delineate the 25 to 75 percentile range. The dashed curves indicate the fitting formulae proposed by Gao et al. (2008). For clarity only 10 000 haloes per simulation are shown in this figure. Haloes shown in grey are systems where the best-fitting scale radius is smaller than the convergence radius; these fits are deemed unreliable and the corresponding haloes are not included in the analysis. The grey vertical bars highlight three different mass bins used to explore parameter variations at fixed halo mass (see Sections 4.3 and 4.4). The small boxes indicate haloes in each of those bins with average, higher-than-average and lower-than-average values of α (bottom panel) or of the concentration (top panel).
In the scaled units of Fig. 1 the scale radius, r−2, signals the location of the maximum of each curve, and different concentrations show as shifts in the position of the maxima, which are indicated by large filled circles. In addition to their different concentrations, the profiles differ as well in α, which increases with decreasing concentration (see legends in Fig. 1). Arrows indicate the half-mass radius of each profile. Dot–dashed curves show NFW profiles (whose shape is fixed in this plot) with the same concentration as the best Einasto fit (solid lines). The density profile curves more gently than NFW for α ≲ 0.18 and less gradually than NFW for α ≳ 0.18, respectively.
The (median) MAHs corresponding to the same sets of haloes are shown in the right-hand panel of Fig. 1. We define the MAH of a halo as the evolution of the virial mass of the main progenitor,3 usually expressed as a function of the scalefactor a = 1/(1 + z), and normalized to the present-day value, M0 = M200(z = 0). As expected, more concentrated haloes accrete a larger fraction of their final mass earlier on. The filled stars indicate the ‘half-mass formation redshift’, z1/2, whereas the filled circles indicate z−2, the redshift when the mass of the main progenitor first reaches M−2, the mass enclosed within r−2 at z = 0.
RESULTS
The mass–concentration–shape relations
The top panel of Fig. 2 shows the mass–concentration relation for our sample of relaxed haloes at z = 0. Concentrations are estimated from Einasto fits, and are colour coded by the shape parameter, α, as indicated by the colour bar. The open symbols track the median concentrations as a function of mass. The thin solid lines trace the 25th and 75th percentiles of the scatter at fixed mass. Different symbols are used for the different MS runs, as specified in the legend. Note the excellent agreement in the overlapping mass range of each simulation, which indicates that our fitting procedure is robust to the effects of numerical resolution.
The bottom panel of Fig. 2 shows the mass–α relation, coloured this time by concentration. The trend is again consistent with earlier work; the median values of α are fairly insensitive to halo mass, except at the highest masses, where it increases slightly. The mass–concentration–shape trends are consistent with earlier work; for example, the dashed lines correspond to the fitting formulae proposed by Gao et al. (2008) and reproduce the overall trends very well.
Fig. 2 illustrates an interesting point already hinted at in Fig. 1: the shape parameter seems to correlate with concentration at given mass. Interestingly, haloes of average concentration have approximately the same shape parameter (α ≈ 0.18, i.e. quite similar to NFW), regardless of mass. Haloes with higher-than-average concentration have smaller values of α and vice versa. This suggests that the same mechanism responsible, at given mass, for deviations in concentration from the mean might also be behind the different mass profile shapes at z = 0 parametrized by α. We explore this possibility next.
Characteristic densities and assembly times
As pointed out by Navarro et al. (1997) and confirmed by subsequent work (see, e.g. Jing 2000), the scatter in concentration is closely related to the accretion history of a halo: the earlier (later) a halo is assembled the higher (lower) its concentration.
This is clear from the assembly histories shown in Fig. 1, which illustrate as well that defining ‘formation time’ in a way that correlates strongly and unequivocally with concentration is not straightforward. For example, the often-used half-mass formation redshift, z1/2, varies only weakly with c, making it an unreliable proxy for concentration (Neto et al. 2007). An ideal definition of formation time would result in a natural correspondence between the characteristic density of a halo at z = 0 and the density of the Universe at the time of its assembly.
We explore two possibilities in Fig 3. Here, we show the mean density enclosed within various characteristic radii at z = 0 versus the critical density of the Universe at the time when the main progenitor mass equals the mass enclosed within the same radii.
Relation between mass profiles at z = 0 and accretion histories for relaxed, well-resolved haloes (N200 > 2.5 × 104) in our sample. Individual haloes are coloured by mass, according to the colour bar at the top of the plot. Left-hand panels: mean enclosed densities within the radii, r1/4, r1/2 and r3/4, containing, respectively, 25, 50 and 75 per cent of the virial mass, shown as a function of the (critical) density of the Universe at the time when the progenitor's virial mass equals the mass enclosed within each of those radii at z = 0. These densities are correlated, as expected if denser haloes collapse earlier. However, the dependence varies with radius and is generally quite weak. This explains, for example, why measures of halo density (such as the concentration) correlate only poorly with the half-mass formation time. Medians, quartiles and 10/90 percentiles are indicated by the box-and-whisker symbols. Right-hand panels: same as the left-hand panels, but for radii equal to half, one and two times the scale radius, r−2. The dotted line indicates direct proportionality, scaled vertically to best fit the data of each panel (fit parameters given in the legends). The excellent agreement between this simple scaling and the data implies that, expressed in units of the scale radius, the shape of the mass profile of a halo is intimately related to that of the accretion history of its main progenitor.
The left-hand panels correspond to radii enclosing 1/4, 1/2 and 3/4 of the virial mass of the halo. The dots indicate individual haloes coloured by halo mass, as shown in the colour bar at the top. Boxes and whiskers trace the 10th, 25th, 75th and 90th percentiles in bins of ρcrit. Note the tight but rather weak (and non-linear) correlation between densities at these radii. This confirms our earlier statement that ‘half-mass’ formation times are unreliable indicators of halo characteristic density: haloes with very different z1/2 may nevertheless have similar concentrations.
The right-hand panels of Fig. 3 show the same density correlations, but measured at various multiples of r−2, the scale radius of the mass profile at z = 0. The middle panel shows that the mean density within r−2, |$\langle \rho _{-2} \rangle =M_{-2}/(4\pi/3)r_{-2}^3$| is directly proportional to the critical density of the Universe at the time when the virial mass of the main progenitor equals M−2. Intriguingly, this is also true at r−2/2 (top-right panel) and at 2 × r−2 (bottom-right panel), although with different proportionality constants (listed in the figure legends).
This means that there is an intimate relation between the mass profile of a halo and the shape of its MAH, in the sense that, once the scale radius is specified, the MAH can be reconstructed from the mass profile, and vice versa. Since mass profiles are nearly self-similar when scaled to r−2, this implies that accretion histories must also be approximately self-similar when scaled appropriately. The MAH self-similarity has been previously discussed by van den Bosch (2002), but its relation to the shape of the mass profile, as highlighted here, has so far not been recognized.
NFW accretion histories and mass profiles
We explore further the relation between MAH and mass profile by casting both in a way that simplifies their comparison, i.e. in terms of mass versus density. In the case of the mass profile, this is just the enclosed mass–mean inner density relation, M(〈ρ〉) (see Section 3.1). For the MAH, this reduces to expressing the virial mass of the main progenitor in terms of the critical density, rather than the redshift, M(ρcrit(z)). In what follows, we shall scale all masses to the virial mass of the halo at z = 0, M0; ρcrit(z) to the value at present, ρ0; and 〈ρ〉 to 200 ρ0.
The top-left panel of Fig. 4 shows, in these scaled units, the average M(〈ρ〉) profile for haloes in three different narrow mass bins (indicated by the grey vertical bars in the bottom panel of Fig. 2). These mean profiles are computed by averaging halo masses, for given 〈ρ〉, after scaling all individual haloes as indicated above. As expected, each profile is well fitted by an NFW profile where the concentration increases gradually with decreasing mass. The heavy symbols on each profile indicate the value of M−2 and 〈ρ−2〉. The top-right panel shows the same data, but scaled to these characteristic masses and densities. Clearly, the three profiles follow closely the same NFW shape, which is fixed in these units.
Average mass profiles at z = 0 and accretion histories for haloes in three different mass bins (see shaded regions in the bottom panel of Fig. 2). Top left: average mass profiles of all haloes in each bin, plotted as enclosed mass (in units of M200), versus inner density (in units of 200 × the critical density). The dashed lines are best-fitting NFW profiles, which have a single adjustable parameter, the concentration, c = r200/r−2. The heavy filled symbols indicate the enclosed mass, M−2, and density, 〈ρ−2〉, at the scale radius of each profile. Residuals from the best fits are shown in the bottom inset. Top right: same as top-left panel, but scaled to the enclosed mass, M−2, and overdensity, 〈ρ−2〉, at the scale radius. Scaled in this manner, halo mass profiles all look alike and are very well approximated by an NFW profile (dashed curve). Bottom left: average accretion histories of the same haloes shown in the top panels. The plots show the growth of the virial mass of the main progenitor, normalized to the final mass at z = 0, as a function of time, expressed in terms of the critical density of the Universe at each redshift. The dashed curves are not fits to the data. Rather, they indicate accretion histories parametrized, as in the top panel, by an NFW profile in this M–ρ plane. The single adjustable parameter to these profiles is fully specified by the heavy filled symbols, which indicate M−2, chosen to match that of the mass profiles (top-left panel) and by ρcrit(z−2), computed as 776 〈ρ−2〉 following the correlation shown in the middle panel of Fig. 3. The light-coloured heavy symbols indicate the scale mass and density of the predicted NFW profile; the dark filled symbols mark the location of the halo characteristic mass and the corresponding formation time. Bottom right: same accretion histories as in the bottom-left panel, but scaled to the characteristic values of the MAH: M−2 and 〈ρ−2〉 (the light heavy symbols in the bottom-left panel). Note the remarkable similarity in the shape of the halo mass profiles at z = 0 and that of the accretion histories of their main progenitors.
The corresponding MAHs, computed as above by averaging accretion histories of scaled individual haloes, are shown in the bottom-left panel of Fig. 4. The heavy symbols on each profile again indicate the value of M−2 (as in the above panel), as well as ρcrit(z−2) = 776 〈ρ−2〉, computed using the relation shown in the middle-right panel of Fig. 3.
In these scaled units, a single point can be used to specify the ‘concentration’ of an NFW profile, which is shown by the dashed curves. Interestingly, these provide excellent descriptions of the MAHs: rescaled to their own characteristic density and mass they all look alike and also follow closely the NFW shape (bottom-right panel of Fig. 4). The MAHs and mass profiles of CDM haloes are not only nearly self-similar: they both have similar shapes that may be approximated very well by the NFW profile.
Relation between concentration parameters obtained from NFW fits to the average accretion histories and mass profiles shown in Fig. 4. The dashed curve indicates the expected concentration–concentration dependence given the correlations shown in the middle-right panel of Fig. 3, assuming an NFW profile. The dotted line shows the best fit obtained using equation (6); the parameters of the fit are provided in Table 1. Note that the relation is rather shallow, indicating that even haloes whose accretion histories differ greatly may have similar concentrations, a result consistent with the weak mass–concentration dependence reported in earlier work.
Parameters obtained for best fits of equation (6) to the concentration–concentration relations for NFW profiles and for Einasto profiles with several values of the shape parameter α. For all cases provided, fits are accurate to better than ≲3 per cent over the range −0.5 ≤ log c [MAH] ≤ 1.5.
| NFW | ||||
|---|---|---|---|---|
| a1 | a2 | a3 | ||
| 2.521 | 0.729 | 0.988 | ||
| Einasto | ||||
| α | a1 | a2 | a3 | |
| 0.10 | 4.124 | 0.849 | 0.833 | |
| 0.15 | 3.365 | 0.692 | 0.899 | |
| 0.20 | 2.946 | 0.614 | 0.953 | |
| 0.25 | 2.697 | 0.557 | 1.003 | |
| 0.30 | 2.504 | 0.530 | 1.042 | |
| 0.35 | 2.322 | 0.528 | 1.068 | |
| 0.40 | 2.154 | 0.543 | 1.084 |
| NFW | ||||
|---|---|---|---|---|
| a1 | a2 | a3 | ||
| 2.521 | 0.729 | 0.988 | ||
| Einasto | ||||
| α | a1 | a2 | a3 | |
| 0.10 | 4.124 | 0.849 | 0.833 | |
| 0.15 | 3.365 | 0.692 | 0.899 | |
| 0.20 | 2.946 | 0.614 | 0.953 | |
| 0.25 | 2.697 | 0.557 | 1.003 | |
| 0.30 | 2.504 | 0.530 | 1.042 | |
| 0.35 | 2.322 | 0.528 | 1.068 | |
| 0.40 | 2.154 | 0.543 | 1.084 |
Parameters obtained for best fits of equation (6) to the concentration–concentration relations for NFW profiles and for Einasto profiles with several values of the shape parameter α. For all cases provided, fits are accurate to better than ≲3 per cent over the range −0.5 ≤ log c [MAH] ≤ 1.5.
| NFW | ||||
|---|---|---|---|---|
| a1 | a2 | a3 | ||
| 2.521 | 0.729 | 0.988 | ||
| Einasto | ||||
| α | a1 | a2 | a3 | |
| 0.10 | 4.124 | 0.849 | 0.833 | |
| 0.15 | 3.365 | 0.692 | 0.899 | |
| 0.20 | 2.946 | 0.614 | 0.953 | |
| 0.25 | 2.697 | 0.557 | 1.003 | |
| 0.30 | 2.504 | 0.530 | 1.042 | |
| 0.35 | 2.322 | 0.528 | 1.068 | |
| 0.40 | 2.154 | 0.543 | 1.084 |
| NFW | ||||
|---|---|---|---|---|
| a1 | a2 | a3 | ||
| 2.521 | 0.729 | 0.988 | ||
| Einasto | ||||
| α | a1 | a2 | a3 | |
| 0.10 | 4.124 | 0.849 | 0.833 | |
| 0.15 | 3.365 | 0.692 | 0.899 | |
| 0.20 | 2.946 | 0.614 | 0.953 | |
| 0.25 | 2.697 | 0.557 | 1.003 | |
| 0.30 | 2.504 | 0.530 | 1.042 | |
| 0.35 | 2.322 | 0.528 | 1.068 | |
| 0.40 | 2.154 | 0.543 | 1.084 |
Note that this is consistent with earlier claims that halo concentration is linked to the time when halo growth switches from a fast- to a slow-accretion phase (Wechsler et al. 2002; Zhao et al. 2003). In our interpretation, since both the MAH and the mass profile follow the same NFW shape the scale radius of one tracks that of the other: the ‘curvature’ of the MAH is therefore reflected in that of the mass profile. Note as well that the relation shown in Fig. 5 is rather weak; in other words, even large changes in the MAH map on to a small range of concentrations in the mass profiles. This is at the root of the weak correlation between concentration and virial mass reported in earlier work (see, e.g. Neto et al. 2007).
Einasto accretion histories and mass profiles
The striking similarity between the shapes of the MAH and mass profile discussed above suggests an explanation for why haloes that are outliers in the mass–concentration relation tend to have mass profiles that differ more significantly from NFW (i.e. they have α parameters that differ from 0.18, see Fig. 2). In this interpretation, outliers in M200–c have MAH shapes that differ systematically from the mean, NFW-like shape.
In order to test this, we may use the Einasto formula to fit both MAH and mass profiles. Fig. 6 shows Einasto M–ρ profiles for various values of α, and compares them with an NFW profile of the same concentration. (This figure uses the same scalings as Fig. 4.) As stated earlier, over the range of mass and density plotted here the NFW profile is essentially indistinguishable from an α = 0.18 Einasto profile, but systematic deviations become apparent for other values of α. Interpreting Fig. 6 as a MAH, we see that α > 0.18 corresponds to haloes that are assembled more rapidly than expected from the NFW shape. The opposite holds for α < 0.18. This behaviour is clearly seen in the residuals from NFW, which are shown in the bottom inset of the figure.
MAHs, M(ρcrit), corresponding to Einasto profiles, compared with NFW. Note that NFW resembles closely an Einasto profile with α ∼ 0.18 or so. Larger or smaller values of α correspond to haloes that have been assembled more or less rapidly than the NFW curve, respectively. Residuals from NFW are shown in the bottom panel.
The top panels of Fig. 7 show the average M(〈ρ〉) profiles of haloes in three different narrow mass bins chosen to have different values of α. These are haloes whose Einasto parameters fall in the boxes drawn in the bottom panel of Fig. 2. The best-fitting NFW profile for each mass bin (as in Fig. 4) is indicated by a dashed curve in each panel. Deviations from the NFW curve are shown in the residuals panels. As expected, the residuals have different shapes depending on the value of their shape parameter α.
As Fig. 4 but for haloes with higher-than-average (blue), average (green), or lower-than-average (red) values of the Einasto parameter α (see boxes in the bottom panel of Fig. 2). Left-, middle and right-hand panels correspond to each of the three mass bins, as indicated in the legends. Top panels: average mass profiles compared with the best-fitting NFW profile for all haloes of the same mass (see top-left panel of Fig. 4). Residuals from that profile are shown at the bottom of each panel. Note the similarity between the residual curves of similar colour at all masses. Different values of α imply different profile shapes, and deviate systematically from NFW. Bottom panels: average MAHs corresponding to the same haloes as in the top panels. The dashed curves indicate the average ‘NFW accretion histories’ for each mass bin, as shown in the bottom-left panel of Fig. 4. Residuals from this average history are shown in the bottom inset of each panel. Note the similarity between the shapes of the residual curves of similar colours in all panels. This indicates that the MAH is intimately linked to the mass profile at z = 0. Haloes that, at z = 0, have mass profiles that deviate from NFW in a particular way have accretion histories that deviate from the NFW shape in a similar way.
The bottom panels of Fig. 7 show the corresponding average MAHs and compare them with the mean predicted MAHs shown in the bottom panels of Fig. 4. The latter are the NFW MAHs that result from the 〈ρ−2〉–ρcrit(z−2) correlation shown in Fig. 3. The residuals from this predicted MAH are clearly similar in shape to those in the top panels: in other words, on average, haloes whose mass profiles deviate from NFW have MAHs that deviate from the NFW shape in a similar way.
Quantitatively, this implies that the best-fitting Einasto parameters of both MAHs and mass profiles must be correlated. Since we expect the correlations to be weak (see, e.g. Fig. 5) we group haloes by mass, concentration and shape parameters (as measured from their mass profiles) before fitting Einasto profiles to their corresponding average MAHs. To prevent possible biases induced by numerical-resolution effects the grouping is such that we retain only well-resolved haloes with similar numbers of particles, 25 000 < N200 < 50 000. Statistical fluctuations are reduced by averaging over groups of at least 25 haloes, using a grid in the c–α plane with a mesh of width δ log c = 0.079 and δα = 0.026.
The results are shown by the heavy symbols in Fig. 8. The left-hand panel shows that the MAH shape parameter is clearly correlated with the shape parameter of the mass profile. Symbols of different colours are used for haloes in each of the three mass bins, which correspond to different MS. Note that they all delineate the same trend, despite the fact that they span a range of roughly four decades in virial mass. Note as well that parameters corresponding to individual haloes (shown as the grey dots in the figure) correlate less well, and that the scatter is larger for the MS-XXL haloes. This is because individual MAHs are often quite complex, especially when major mergers are involved or haloes are recently assembled and still unrelaxed (even though they pass the relaxation criteria set out in Section 2.2), as is the case for many MS-XXL systems (Ludlow et al. 2012). These MAHs cannot be well approximated by the Einasto shape, thus, hindering the interpretation of their fit parameters. We therefore focus the discussion on the parameters fitted to the averaged profiles, shown with the heavy symbols in Fig. 8.
Concentrations and shape parameters of Einasto profiles fitted to either accretion histories or mass profiles at z = 0. The heavy symbols correspond to well-resolved haloes grouped according to the c and α parameters of their mass profile (see details in the text). The grey dots correspond to individual haloes in the same three mass bins chosen in Fig. 7. The left-hand panel shows that the shape of the MAH and that of the mass profile are correlated. The panel on the right is analogous to Fig. 5 and shows that the same applies to the concentrations. In this case, the relation depends on the value of α, as shown by the coloured lines labelled in the legend. The heavy symbols are of the same type as in the left-hand panel, but coloured by α (see inset). Note that the correlations are relatively shallow, implying that even large departures from NFW-like MAHs lead only to minor deviations from NFW in the mass profiles.
The relation between shape parameters is quite weak (the 1:1 relation is indicated by the dotted line), implying that large variations in MAH shape map on to a narrower range of mass profile shapes. As a result, even haloes that assemble early and over a very short period of time, such as those whose MAH is characterized by α ∼ 0.5 (see Fig. 6), end up with α ∼ 0.25 mass profiles that differ only slightly from NFW. This is consistent with earlier findings that haloes assembled monolithically and without protracted accretion, such as those formed in hot dark matter universes, are nevertheless well approximated by NFW profiles (e.g. Huss et al. 1999; Wang & White 2009).
It also explains why the NFW profile fits rather well haloes formed in hierarchical scenarios other than LCDM. For example, the accretion histories of haloes formed in scale-free scenarios characterized by a power-law spectrum of density fluctuations, P(k) ∝ kn, depends on n, but rather weakly. For given n, the MAH shapes are also, on average, independent of halo mass. Fitting Einasto profiles to accretion histories taken from Zhao et al. (2009), we find that haloes that form from white noise spectra (n = 0) have MAHs well described by α ∼ 0.1. For n = −2, on the other hand, the average MAH shape is roughly α ∼ 0.2. These different MAHs result in only a subtle change in mass profile (see left-hand panel of Fig. 8), which would have been undetectable at the numerical resolution probed by earlier work. It may, however, be behind the claim by Knollmann, Power & Knebe (2008) that the inner slope of the density profile varies systematically with the spectral index n.
The right-hand panel of Fig. 8 shows, on the other hand, the correlation between the best-fitting Einasto concentration parameters of the MAH and mass profiles, for the same set of haloes shown in the left-hand panel of the same figure. This is analogous to Fig. 5, but for Einasto, rather than NFW, concentrations. (The NFW c–c correlation is shown by a dashed line.) Because of the extra parameter, the relation between Einasto concentrations depends on α, and is indicated by the coloured lines in the figure for three values of α. The symbol types in both panels are the same, but are coloured by α in the right-hand panel (see colour bar inset). Note that, for given α, the c–c relation for MS haloes follows closely the expected correlations. As for the NFW profile, the concentration–concentration relation for Einasto profiles can also be approximated by equation (6). In Table 1, we provide the best-fitting parameters obtained by fitting equation (6) to these relations for several different values of α.
As in the left-hand panel, the grey dots in the right-hand panel of Fig. 8 correspond to fits to individual halo MAH and mass profiles. These clearly follow the same trend as the averaged profiles but with larger scatter. In particular, complexities in the MAH caused by major mergers result at times in extreme values for the ‘concentration’ measured from accretion histories. These affect in particular large mass haloes that have recently been assembled. As discussed by Ludlow et al. (2012), many such haloes pass the relaxation criteria but are actually out of equilibrium and in a particular phase of their virialization process. These haloes are actually responsible for most of the outlier points in Fig. 8. Although a detailed investigation of the accuracy with which our model can predict the concentrations of individual haloes from their MAHs is beyond the scope of this paper, we plan to return to this subject in forthcoming work.
We conclude that there is strong evidence for a link between the concentration and shape of the mass profile of haloes and their accretion histories. The correlations are well defined but weak, in the sense that even MAHs whose shapes deviate substantially from the mean lead to haloes that depart only subtly from the average, NFW-like mass profile. This is probably due to the virialization process, where non-linear effects lead to a substantial but incomplete erasure of memory of the initial conditions from the equilibrium structure of a halo. The structural similarity of CDM haloes thus seems to arise from the mass-independence of MAH shapes aided by the homogenizing effect of halo virialization.
SUMMARY AND CONCLUSIONS
We have examined the mass profile and accretion histories of equilibrium CDM haloes identified in the Millennium Simulation series. As reported in earlier work, halo mass profiles are well approximated by NFW profiles which, at given virial mass, are characterized by a single parameter, the concentration, c = r200/r−2. Although in general deviations from the NFW profile are small, improved fits may be achieved using Einasto profiles characterized, at given virial mass, by the concentration and an extra shape parameter, α.
Our main finding is that these parameters are strongly linked to the accretion history of a halo. The mean density within the scale radius, r−2, is directly proportional to the critical density of the Universe at the time when the main progenitor's mass equals that within r−2. Scaled to these characteristic values of mass and density the shape of the MAH, expressed as M(ρcrit(z)), is, on average, independent of halo mass. Furthermore, this shape is nearly identical to that of the enclosed mass–mean inner density profile (M(〈ρ〉)) of the halo at z = 0, which can be well approximated by the NFW profile.
This result suggests that the structural similarity of haloes of different mass is related to the fact that their accretion histories are independent of mass. It also clarifies how the accretion history determines the concentration of a halo; since accretion history and mass profile follow the same NFW shape, there is a unique correspondence between the ‘concentration’ parameters of either one, as shown in Fig. 5.
This conclusion is strengthened by the finding that haloes whose mass profiles deviate from NFW and are better approximated by Einasto profiles have MAHs that deviate from the NFW shape in a similar way. This suggests that the extra shape parameter of the Einasto profile arises because some haloes have accretion history shapes that differ from NFW. Indeed, fitting Einasto profiles to both M(〈ρ〉) and M(ρcrit(z)) yields correlated concentration and shape parameters. The correlations are clear but weak, implying that only haloes whose accretion histories deviate strongly from the NFW shape would have mass profiles that deviate notably from the average, NFW-like shape. We ascribe this result to the convergent effects of virialization, which partially erase the memory of initial conditions from the halo structure.
Our results suggest that the density profiles of haloes formed in hierarchical scenarios other than CDM or monolithically, as in a hot dark matter Universe, are not truly self-similar. The deviations from similarity, however, are subtle, and a dedicated program of high-resolution numerical simulations is needed in order to validate this prediction.
We may use these findings to predict the dependence of halo concentration on mass and redshift, as well as the influence of varying the spectral index or the cosmological parameters, provided that realistic accretion histories are available, either through direct numerical simulation or through well-tested semi-analytic modelling (e.g. Monaco et al. 2002; van den Bosch 2002; Zhao et al. 2009). Such studies would help to reveal any shortcomings in our interpretation and should shed further light on the mechanisms responsible for CDM halo structure. We plan to address these issues in a forthcoming paper.
Finally, our findings provide some endorsement to the many previous studies that have sought a link between the final structure of a halo and its evolutionary history (e.g. Bullock et al. 2001; Wechsler et al. 2002; Alvarez, Ahn & Shapiro 2003; Zhao et al. 2003, 2009; Tasitsiomi et al. 2004; Lu et al. 2006; Wong & Taylor 2012) but still fall short of providing a full account of what determines the structure of dark matter haloes. Hopefully, the link with the accretion history we describe here will help to guide future theoretical work in order to unravel the mechanism at the root of the remarkable structural similarity of dark matter haloes.
We would like to thank Gerard Lemson for useful discussions and the Virgo Consortium for access to the MS data. We would also like to thank the anonymous referee for a prompt report that helped improve our paper. ADL acknowledges financial support from the SFB (956) from the Deutsche Forschungsgemeinschaft. REA is supported by Advanced Grant 246797 GALFORMOD from the European Research Council. MB-K acknowledges support from the Southern California Center for Galaxy Evolution, a multicampus research programme funded by the University of California Office of Research. VS acknowledges partial support by TR33, ‘The Dark Universe’, of the Deutsche Forschungsgemeinschaft.
We define the virial radius, r200, of a halo as the radius of a sphere centred at the potential minimum that encloses a mean density of 200 × ρcrit. We identify all virial quantities (i.e. measured within r200) with a ‘200’ subscript. Note that all particles are used to compute r200, not just those bound to the main halo.
Median profiles are computed at each radius after scaling all individual profiles as in Fig. 1.
The main progenitor of a given dark matter halo is found by tracing backwards in time the most massive halo along the main branch of its merger tree.
