The Astrophysycal Journal, 647:000-000, 2007 September 20
A Preprint typeset using L TEX style emulateapj v. 10/09/06
THE NATURE OF DARK MATTER AND THE DENSITY PROFILE AND CENTRAL BEHAVIOR OF RELAXED HALOS
e Eduard Salvador-Sol?1 , Alberto Manrique1 , ? Guillermo Gonzalez-Casado2 , and Steen H. Hansen3
1 Departament d’Astronomia i Meteorologia, and Institut de Ci`ncies del Cosmos (UB/IEEC) e associated with the Instituto de Ciencias del Espacio (CSIC), Universitat de Barcelona, Spain 2 Departament de Matem`tica Aplicada II, Universitat Polt`cnica de Catalunya, Spain and a e 3 Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Denmark The Astrophysycal Journal, 647:000-000, 2007 September 20
arXiv:astro-ph/0701134v3 31 Jul 2007
ABSTRACT We show that the two basic assumptions of the model recently proposed by Manrique and coworkers for the universal density pro?le of cold dark matter (CDM) halos, namely that these objects grow inside out in periods of smooth accretion and that their mass pro?le and its radial derivatives are all continuous functions, are both well understood in terms of the very nature of CDM. Those two assumptions allow one to derive the typical density pro?le of halos of a given mass from the accretion rate characteristic of the particular cosmology. This pro?le was shown by Manrique and coworkers to recover the results of numerical simulations. In the present paper, we investigate its behavior beyond the ranges covered by present-day N -body simulations. We ?nd that the central asymptotic logarithmic slope depends crucially on the shape of the power spectrum of density perturbations: it is equal to a constant negative value for power-law spectra and has central cores for the standard CDM power spectrum. The predicted density pro?le in the CDM case is well ?tted by the 3D S?rsic pro?le e over at least 10 decades in halo mass. The values of the S?rsic parameters depend on the mass of e the structure considered. A practical procedure is provided that allows one to infer the typical values of the best NFW or S?rsic ?tting law parameters for halos of any mass and redshift in any given e standard CDM cosmology. Subject headings: cosmology: theory – dark matter – galaxies: halos
The universal shape of the spherically averaged density pro?le of relaxed dark halos in high-resolution N -body simulations is considered one of the major predictions of standard cold dark matter (CDM) cosmologies. Down to 1 % of the virial radius R, it is well ?tted by the so-called NFW pro?le (Navarro et al. 1996, 1997), ρ(r) =
3 ρc rs , r(rs + r)2
speci?ed by only one mass-dependent parameter, the scale radius rs or equivalently the concentration c ≡ rs /R. At radii smaller than 1 % of the virial radius, however, the behavior of the density pro?le is unknown. On the basis of recent numerical simulations, some authors advocate a central asymptotic slope signi?cantly steeper (Moore et al. 1998; Jing & Suto 2000) or shallower (Taylor & Navarro 2001; Ricotti 2003; Hansen & Stadel 2006) than that of the NFW law. Others suggest an ever decreasing absolute value of the logarithmic slope (Power et al. 2003; Navarro 2004; Reed et al. 2005), which might tend to zero as a power of the radius as in the three-dimensional (3D) S?rsic (1968) or Einasto e (Einasto & Haud 1989) laws (Merritt et al. 2005, 2006). This uncertainty is the consequence of the fact that the origin of such a universal pro?le is poorly understood. Two extreme points of view have been envisaged. In one of these, it would be caused by repeated signi?cant mergers (Syer & White 1998; Raig et al. 1998; Subramanian et al. 2000; Dekel et al. 2003), while
in the other it would be essentially the result of smooth accretion or secondary infall (Avila-Reese et al. 1998; Nusser & Sheth 1999; Del Popolo et al. 2000; Kull 1999; Manrique et al. 2003; Williams et al. 2004; Ascasibar et al. 2004). The fact that the M ? c relation at z = 0 is consistent (Salvador-Sol? et al. 1998, Wechsler et al. 2002; e Zhao et al. 2003b) with the idea that all halos emerge from major mergers with similar values of c, which then decreases according to the inside-out growth of halos during the subsequent accretion phase, seems to favor an important role of mergers. However, the purely accretiondriven scenario is at least as attractive, as the inside-out growth during accretion leads to a typical density pro?le that appears roughly to have the NFW shape with the correct M ? c relation in any epoch and cosmology analyzed (Manrique et al. 2003, hereafter M03). Certainly the e?ects of major mergers cannot be neglected in hierarchical cosmologies, so both major mergers and accretion should contribute in shaping relaxed halos. However, as noted by M03, if the density pro?le arising from a major merger were set by the boundary conditions imposed by current accretion, then the density pro?le of halos would appear to be independent of their past aggregation history, so halos could be assumed to grow by pure accretion without any loss of generality. All the correlations shown by relaxed halos in numerical simulations can be recovered under this point of view (Salvador-Sol? et al. 2005). Simultaneously, this would e explain why halos with very di?erent initial conditions and aggregation histories have similar density pro?les (Romano-Diaz et al. 2006).
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In the present paper, we show that the M03 model relies on two basic assumptions, namely, (1) that halos grow inside out in periods of smooth accretion, and (2) that the mass pro?le and all its derivatives are continuous functions. The former assumption is supported by the results of numerical simulations (Salvador-Sol? et al. 2005; e Lu et al. 2006; Romano-Diaz et al. 2006, 2007), while the second one is at least not in contradiction with them. In the present paper we show that both assumptions are in fact sound from a theoretical point of view, as they can be related to the very nature of CDM. This renders the predictions of the model beyond the range of current simulations worth examining in detail, as is done below. For simplicity, we consider spherical structures, which at best is an approximation to the triaxial structures observed in numerical simulations. Secondary infall is known to be in?uenced by deviations from spherical symmetry (Bond & Myers 1996; Zaroubi et al. 1996). Yet, in an accompanying paper (Gon?lez-Casado et al. 2007), a we show that this does not seem to a?ect the fundamental role of the two assumptions given above. Another simplifying assumption used here is the neglect of substructure. It has recently been shown in high-resolution simulations that about 57 % of the halo mass is collected in the previous major merger (Faltenbacher et al. 2005). However, substructures (and sub-substructures) contribute only about 5 % of the total mass in halos (Diemand et al. 2007) whose density pro?le is well described by equation (1). It seems therefore a good ?rst approximation to ignore substructure. The paper is organized as follows. In §2, we show how the M03 model emerges from the properties of standard CDM. The behavior of the predicted density pro?le at extremely small radii and for a wide range of halo masses is investigated in §3. Our results are summarized in §4.
2. CDM PROPERTIES AND HALO DENSITY PROFILE 2.1. Inside-out growth during accretion
Fig. 1.— M ? c relation at z = 0 predicted in the concordance model by the M03 model for ?m = 0.26 (solid line) compared to the empirical relation traced by those points, with minimal error bars obtained by Zhao et al. (2003a) from high-resolution simulations (triangles with error bars).
We now argue that halos grow inside out during accretion, as is indeed found in numerical simulations, because of some properties of CDM; in particular, its characteristic power spectrum, leading to a slow halo accretion rate. As shown in §2.2, this has important consequences for the inner structure of these objects. Schematically, one can distinguish between minor and major mergers. In minor mergers, the relative mass increase produced, ? ≡ ?M/M , is so small that the system is left essentially unaltered, whereas in major mergers ? is large enough to cause rearrangements. The smaller the value of ?, the most frequent the mergers (Lacey & Cole 1993). For this reason, although individual minor mergers do not a?ect the structure of halos, their added contribution yields a smooth secular mass increase, the so-called accretion, with apparent effects on the aggregation track M (t) of the halo. The ˙ accretion-scaled rate, M /M (t), is given by (Raig et al. 2001, hereafter RGS01)
contributing to accretion. In contrast, less frequent major mergers yield notable sudden mass increases or discontinuities in M (t). After undergoing a major merger (and virializing), halos evolve as relaxed systems until the next major merger. The fact that standard CDM is nondecaying and nonself-annihilating1 guarantees that the mass collected during such periods is conserved. This does not yet imply that halos grow inside out during accretion, because their mass distribution might still vary due to energy gains or losses or to the action of accretion itself. Even if each individual minor merger would leave the halo unchanged their collective action might alter these systems. However, standard CDM is also dissipationless and therefore halos cannot loose energy. Furthermore, under the assumption of spherical symmetry halos cannot su?er tidal torques from surrounding matter, and hence the surrounding matter is unable to alter the kinetic energy of the halo. Therefore, the only possibility for a timevarying inner mass distribution of accreting halos is that the accretion process causes it itself. This possibility, that the process of accretion itself could alter the internal mass distribution, would be realized only if the accretion time 1/Ra were smaller than the dynamical time τcr . On the contrary, if 1/Ra is substantially larger than τcr , the adiabatic invariance of the inner halo structure will be guaranteed, and the halo will evolve inside out. Thus, by requiring 1/Ra to be C times larger than τcr , we are led to the equation C 4π G?vir (t)?(t) ρ 3
= Ra (M, t) ,
Ra (M, t) =
d? ? Rm (M, t, ?) ,
where Rm (M, t, ?) is the usual Lacey-Cole (Lacey & Cole 1993) instantaneous merger rate (see eq. [A1]) and ?m is the maximum value of ? for mergers
which gives the upper mass Ma for inside-out growth at t. In equation (3), ρ is the mean cosmic density and ?
1 Both decay and annihilation rates are extremely small for realistic dark matter particle candidates.
DENSITY PROFILE AND CENTRAL BEHAVIOR OF HALOS
Fig. 2.— Predicted density pro?les (solid lines) compared to their ?ts to a NFW pro?le (dashed lines) for halo masses at z = 0, ranging from 1016 M⊙ to 1011 M⊙ (or, equivalently, from 103 to 10?2 times the current critical mass M?0 for collapse), in the concordance model. Upper subpanels: Pro?les. Lower subpanels: Residuals of the logarithmic ?ts. The halo mass and the corresponding best-?tting value of the NFW concentration parameter are quoted in each panel.
?vir (t) is the virialization density contrast given e.g. by Bryan & Norman (1998). Here Ma is indeed an upper limit because Ra (M, t) is an increasing function of t; see RGS01. In any CDM cosmology analyzed, equation (3) appears to have no solution for values of C signi?cantly larger than unity in the relevant redshift range. This means that accretion is always slow enough for halos to grow inside out, as required by the M03 model. As previously mentioned, the inside-out growth of halos in accreting periods is unambiguously con?rmed by the results of N -body simulations (Salvador-Sol? et al. e 2005; Lu et al. 2006; Romano-Diaz et al. 2006, 2007). It is also consistent with the fact that dark matter structures preserve the memory of initial conditions, in the sense that the most initially overdense regions end up being the central regions of the ?nal structures (Diemand et al. 2005), implying that the spatial positions of particles are not signi?cantly perturbed by merging/accretion during the assembly of the structures. Likewise, the energy of the individual particles in the ?nal structure (at z = 0) is very strongly correlated with their energies at much earlier times (z = 10; Dantas & Ramos 2006). This shows that particles even preserve the memory of the initial energies.
2.2. Smoothness of the mass pro?le
Contrarily to an ordinary ?uid, CDM is collisionless and free-streaming and, hence, cannot support discontinuities (shock fronts) in the spatial distribution of any of its macroscopic properties. As a consequence, all radial pro?les in relaxed halos are necessarily smooth. This holds in particular for the mass pro?le, M (r), and its radial derivatives,2 which has the following consequence. The inside-out growth of a halo during accretion (§2.1) implies that the mass pro?le M (r) built at that interval is the simple conversion, through the de?nition of the instantaneous virial radius R(t) = 3M (t) 4π?vir (t)?(t) ρ
of the associated mass aggregation track M (t). The smoothness condition implies that the old M (r) pro?le must match perfectly with the new part of the pro?le built during that time. Since minor mergers only cause tiny discontinuities, the new piece of the M (t) track that they produce is well approximated by a smooth
2 We are naturally referring to the theoretical mass pro?le, equal to the average pro?le over random realizations or over time realizations as long as the halo evolves inside-out.
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function, and, as the functions ρ(t) and ?vir (t) in equa? tion (4) are also smooth functions, the corresponding piece of the M (r) pro?le automatically ful?lls the right smoothness condition. Thus, the system can grow, during accretion, without the need to essentially rearrange its structure. Only when a halo undergoes a major merger and its M (t) track su?ers a marked discontinuity will the mass pro?le prior to the major merger no longer match the piece that begins to develop after it. Since the M (r) pro?le cannot have any discontinuities, the halo is then forced to rearrange its mass distribution (through violent relaxation) to ful?ll the required smooth condition. In other words, the fundamental assumption of the M03 model that the mass distribution of relaxed halos is determined by their current accretion rate (through dramatic rearrangements of the structure on the occasion of major mergers and very tiny and negligible ones during accretion periods), is simply the natural consequence of the slowly accreting, nondecaying, nonselfannihilating, dissipationless, and collisionless nature of standard CDM.
2.3. The M03 model
The mass pro?le of a speci?c halo with mass Mi at time ti accreting at a given rate during any arbitrarily small time interval ?t around ti is therefore simply the smooth extension inward of the small piece of pro?le being built during that interval.3 Unfortunately, the smooth extension of a small piece of a function is hard to ?nd in practice, so the mass pro?les of real individual halos can hardly be obtained in this way. There is one case, however, in which such a smooth extension can readily be achieved: that of halos with Mi at ti accreting at the typical cosmological rate Ra (M (t), t) during any arbitrarily small interval of time around ti . In this case, the (unique) smooth extension we are looking for necessarily coincides with the smooth function M (t), the solution of the di?erential equation ˙ M = Ra (M (t), t) (5) M (t) for the boundary condition M (ti ) = Mi , properly converted from t to r by means of equation (4). Once the typical M (r) pro?le is known, by di?erentiating it and taking into account equations (4) and (5), one is led to the typical density pro?le for halos with Mi at ti proposed by M03, ˙ M 1 = [?vir (t)?(t)δ(t)]t(r) ρ (6) ρ(r) = 2 ˙ 4πr R
Fig. 3.— Relations between the NFW rs and Ms parameters obtained in the concordance model [sghown with solid lines for masses up to 10M? (z) and dashed lines beyond it] from the ?ts of the theoretical density pro?les of halos with varying mass at the redshifts 0.0, 1.03, 2.08, and 3.94. The dot-dashed line gives the analytical ?t (see text) to the mean of the solid lines.
This dependence is, however, so convoluted that the density pro?le (eq. ) must be inferred numerically. Only its central asymptotic behavior can be derived analytically, as will be shown in the next section.
3. SOME CONSEQUENCES OF THE MODEL
d ln(?vir ρ) ? 1 . (7) Ra (M (t), t) dt From equations (6) and (7) we see that the shape of this pro?le is ultimately set by the CDM power spectrum of density perturbations in the cosmology considered through the merger rate Rm (M, t, ?) used to calculate the accretion rate Ra (M, t) (see eqs. [A1] and ). δ(t) = 1 ?
3 The derivatives at any order of a smooth function, which is known in some ?nite domain, are completely determined at any point of that domain. Thus, by taking the Taylor series expansion of the function in that point, one can extend it in a unique way outside the domain.
From equation (2) we see that the exact shape of the density pro?le in equation (6) depends on ?m . This parameter marks the e?ective transition between minor and major mergers, and it can be determined from the empirical M ? c relation at some given redshift. For each given ?m value, the density pro?les, down to R/100, predicted at z = 0 in the concordance model characterized by (?m , ?Λ , h, σ8 ) = (0.3, 0.7, 0.7, 0.9) for halos with di?erent masses have been ?tted to the NFW pro?le to ?nd the best-?t values of c. Then we searched for the value of ?m that minimizes the departure of the theoretical M ? c relations from the empirical one drawn from high-resolution simulations by Zhao et al. (2003a). As shown in Figure 1, ?m = 0.26 gives an excellent ?t over almost 4 decades in mass (8 × 1010 h?1 M⊙ < M < 4 × 1014 h?1 M⊙ ). In Figure 2, we plot, down to a radius equal to the current resolution radius of most numerical simulations, the density pro?les predicted in the concordance model for halo masses at z = 0 ranging from 1011 to 1016 M⊙ . They are all well ?tted to a NFW pro?le, although there is a tendency for the theoretical pro?les for M 1014 M⊙ to deviate from that shape and approach a power law with logarithmic slope intermediate between the NFW asymptotic values of ?1 and ?3. This tendency also makes c increase very rapidly at large masses where the ?t by the NFW law is no longer acceptable. This causes the M ? c relation to deviate from its regular trend at smaller values of M (see Fig. 2). Both e?ects, already reported in M03, were later observed in simulated halos (Zhao et al.
DENSITY PROFILE AND CENTRAL BEHAVIOR OF HALOS
Fig. 4.— Same as Figure 2, but for the S?rsic pro?le. All the pro?les are now drawn down to a 1 pc radius and cover a wider range of e halo masses: 1016 ? 106 M⊙ (or, equivalently, from 103 M?0 to 10?7 M?0 ).
2003a; Tasitsiomi et al. 2004). This is a clear indication that the NFW pro?le does not provide an optimal ?t for very massive structures. Of course, above 1014 M⊙ halos are hardly in virial equilibrium, so such a deviation has essentially no practical e?ects. As explained in Salvador-Sol? et al. (2005), another ine teresting consequence of the M03 model is that the M (t) tracks traced by accreting halos (hence, growing inside out) coincide with curves of constant rs - and Ms - values, with Ms de?ned as the mass interior to rs . Thus, the intersection of those accretion tracks at any arbitrary redshift sets the relation Ms (rs ) between such a couple of parameters, implying that the Ms (rs ) relation satis?ed by halos is time-invariant. In Figure 3, we show how the di?erent Ms (rs ) curves obtained by ?tting the density pro?les predicted at di?erent redshifts to a NFW law overlap. There is only some deviation at large masses, where the density pro?les are not correctly described by the NFW pro?le. Such a time-invariant Ms (rs ) relation is well ?tted, for rs in the range 10?4 Mpc < rs < 1 Mpc, by
Substituting equation (8) into the relation Ms ln 2 ? 0.5 = M g(c) which holds for NFW pro?les, one is led to 4π ?vir (t) ρ(t) ? 6ρ 2.37 × 10 ?0
M 13 M 10 ⊙
g(c) , c2.45
Ms = 141 1013 M⊙
where g(c) stands for ln(1 + c) ? c/(1 + c) and ρ0 is ? the current mean cosmic density. Equation (10) is an implicit equation for the concentration of halos with any given mass and redshift. What about the central behavior of the predicted density pro?le? According to equation (4), small radii correspond to small cosmic times. In this asymptotic regime, all Friedman cosmologies approach the Einstein?de Sitter model in which ?vir (t) is constant and ρ(t) is (in the ? matter-dominated era when halos form) proportional to t?2 . If the power spectrum of density perturbations were of the power-law form P (k) ∝ k j , with the index j satisfying 1 > j > ?3 to guarantee hierarchical clustering, then the universe would be self-similar. The mass accretion in equation (5) would take the asymptotic form: 2 M (t) ∝ t j+3 (see the Appendix). The fact that both ?vir (t)?(t) and M (t) would then be power laws has two ρ
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consequences. First, the time dependence of their respective logarithmic derivatives on the right-hand side of equation (7) cancel, which implies that ρ(t) is proportional to ?vir (t)?(t), and hence to ρ(t). Second, the ρ ? virial radius given by equation (4) is also a power law, R(t) ∝ [M (t)/ρ(t)]1/3 ∝ t2(j+4)/[3(j+3)] . From this we ? get t(r), and thereby one ?nds This central behavior, fully in agreement with the numerical pro?les obtained from power-law power spectra, is particularly robust, as it does not depend on ?m . Note that it coincides with the solution derived in self-similar cosmologies by Ho?man & Shaham (1985) assuming spherical collapse. It is worth noting, however, that in that derivation, such an asymptotic behavior is restricted to j > ?1 so as to warrant the required adiabatic invariance (Fillmore & Goldreich 1984), while, in the present derivation, there is no such a restriction as the central density pro?le is not assumed to be built by spherical infall, but results instead from smooth adaptation to the boundary condition imposed by current accretion. The CDM power spectrum is, of course, not a power law. However, in the limit of small masses involved in that asymptotic regime, it tends to a power law of index j = ?3. Thus, according to equation (11), we expect a vanishing central logarithmic slope of ρ(r) for the standard CDM case. This is con?rmed by the numerical pro?les obtained in this case: as one goes deeper and deeper into the halo center, they become increasingly shallower. What is still more remarkable is that down to a radius as small as 1 pc, the density pro?les appear to be well ?tted by the 3D S?rsic or Einasto law e ρ(r) = ρ0 exp ? r rn
ρ(r) ∝ r?
Fig. 5.— Mass dependence at z = 0 of the S?rsic law parameters e n and cn ?tting the predicted density pro?les of halos in the concordance model (solid lines) and their ?ts by the simple expressions given in the text (dashed lines).
to the relations plotted in Figure 5, which are well approximated by n(M ) = 4.32 + 7.5 × 10?7 ln ln[cn (M )] = 13.3 + 7.5 × 10?8 ln M M⊙ M M⊙
(15) , (16)
leading to the following combined relation cn (M ) = 5.84 × 10
over at least 10 decades in halo mass (see Fig. 4), from 106 M⊙ to 1016 M⊙ . We remind the reader that for power-law spectra, equation (6) leads to density pro?les with central cusps, so the S?rsic shape is not a general e consequence of the M03 model, but it is speci?c to the standard CDM power spectrum. In fact, from the reasoning above we see that what causes the zero central logarithmic slope in the CDM case is the fact that the logarithmic accretion rate d ln M (t)/d ln t = tRa (M (t), t) diverges in the limit of small values of t. This is in contrast to the general power-law case, where the accretion rate remains ?nite. Similar to the characteristic density ρc in the NFW pro?le (eq. ), the central density ρ0 entering the 3D S?rsic law (eq. ) can be written in terms of the mass e M and the values of the two (instead of one) remaining parameters, n and either rn or cn ≡ R/rn : M ρ0 = P ?1 (3n, c1/n ) , (13) n 3 4πnrn Γ(3n) where Γ is the usual gamma function and x 1 P (a, x) = dt e?t ta?1 (14) Γ(a) 0 is the incomplete or regularized one. At z = 0, the two free parameters n and cn depend on M according
These expressions can be used to infer the typical values of the S?rsic parameters for present-day halos with any e mass. It is worth mentioning that the predicted values of n are of the same order of magnitude as the ones obtained by Merritt et al. (2005) from simulated halos with masses ranging from dwarf galaxies to galaxy clusters (the values of cn were not presented in that work). To obtain more general values of these parameters for halos of any mass and redshift, we can proceed as in the NFW case above. For reasons identical to the ones leading to the time-invariant relation Ms (rs ), the relations ρ0 (rn ) and n(rn ) must be time-invariant. In Figure 6, we see how the corresponding curves obtained from the ?t to the S?rsic pro?le of the same density pro?les as e used in Figure 3 overlap, indeed, even better than the Ms (rs ) curves do, since there are no large deviations at large masses. These invariant relations are well ?tted, for x ≡ log(rn /Mpc) in the range ?25 < x < ?9, by ρ0 (rn ) = ρ0 exp (A) , ? A = ?2.48 ? 4.73x ?0.270x2 ? 6.24 × 10?3 x3 , (18)
DENSITY PROFILE AND CENTRAL BEHAVIOR OF HALOS
Fig. 6.— Same as Figure 3 but for the ρ0 (rn ) and n(rn ) relations among the best-?t S?rsic parameters. The respective analytical ?ts e (see text) to the mean curve are shown in dot-dashed lines. Here ρ0 is the current mean cosmic density. ?
n(rn ) = ?4.43 ? 1.43x
Replacing these expressions into equation (13), we can solve for rn and then use the equation (19) to ?nd the value of n. This provides a very concrete prediction that can be tested with numerical simulations. One can take the very strong correlation shown in Figure 6, which allows one to ?t the density pro?le of any dark matter structure with only two free parameters: e.g. n and ρ0 . With this value of n (purely from the shape of the density pro?le), one now gets a value for the mass (from Fig. 5). This mass can trivially be compared to the true virial mass (which is naturally known in the simulation), and hence one can con?rm or reject the prediction of the accretion-driven model.
?0.0313x2 ? 2.98 × 10?4 x3 .
shown to be in overall agreement with the results of numerical simulations. In this model, the density pro?le of relaxed halos permanently adapts to the pro?le currently building up through accretion and does not depend on their past aggregation history. As a consequence, the typical density pro?le of halos of a given mass at a given epoch is set by their time-evolving cosmology-dependent typical accretion rates. Although halos have been assumed to be spherically symmetric throughout the present paper, this is not crucial for the M03 model. As will be shown in a following paper (Gonz?lez-Casado et al. 2007), the results prea sented here also hold for more realistic triaxial rotating halos. Furthermore, an approach similar to the one followed here allows one to explain not only their mass distribution, but also others of their structural and kinematic properties, such as the radial dependence of angular momentum. According to the M03 model, the central asymptotic behavior of the halo density pro?le depends, through the typical accretion rate, on the power spectrum of density perturbations. The prediction made in the case of power-law spectra should be possible to check by means of numerical simulations, provided one concentrates on massive halos, as these reach the asymptotic regime at larger radii. In the case of the standard CDM power spectrum, the model predicts a vanishing central logarithmic slope. The way this asymptotic behavior is reached is surprisingly simple: down to a radius as small as 1 pc, the density pro?le is well ?tted by the 3D S?rsic or Einasto e pro?le over at least 10 decades in halo mass. Another consequence of the M03 model with useful practical applications is the existence of time-invariant relations among the NFW or 3D S?rsic law parameters e (Ms and rs in the former case and ρ0 , rn , and n in the latter) ?tting the halo density pro?les. A code is publicly available4 that computes such invariant relations for any desired standard CDM cosmology. Some of these consequences can be readily tested by numerical simulations or by (X-ray or strong-lensing) observations, which should allow one to con?rm or reject the prediction for the central behavior of the density pro?le of halos that is made by the M03 model.
We have shown how the basic properties of standard CDM can justify the M03 model, which was previously
This work was supported by the Spanish DGES grant AYA2006-15492-C03-03. We thank Donghai Zhao and co-workers for kindly providing their data.
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APPENDIX THE ACCRETION TRACK ASYMPTOTIC BEHAVIOR
The instantaneous rate of mergers undergone by halos with mass M at time t yielding a relative mass increase of ?, is (Lacey & Cole 1993) Rm (M, t, ?) = 2/π M dδc dσ(M ′ ) σ 2 (M ′ ) dt dM ′ 1? σ 2 (M ′ ) σ 2 (M )
2 σ 2 (M ′ ) δc (t) 1? 2 2σ 2 (M ′ ) σ (M )
where M ′ ≡ M (1 + ?) is the ?nal mass, δc (t) is the linearly extrapolated, cosmology-dependent, critical density contrast of density ?uctuations at t for collapse at the present time t0 , and σ(M ) ≡ σ(M, t0 ) is the rms density contrast of ?uctuations on scale M at t0 , related to the zeroth-order spectral moment. In a self-similar universe with spectral index j, one has δc (t) = δc0 t t0
σ(M ) = σ(M?0 )
? j+3 6
where δc0 and M?0 are the current values of δc (t) and M? (t), respectively, M? (t) being the critical mass for collapse at t, the solution of the implicit equation σ(M? , t) = δc0 . Therefore, equation (A1) takes the form Rm (M, t, ?) =
j+3 2 j + 3 ν(M, t) (1 + ?)2 3 ?1 ν 2 (M, t) (1 + ?) 3 ? 1 exp ? 3/2 j+3 π 9t 2 (1 + ?) 3 ? 1 j+3
ν(M, t) =
M = M? (t)
Substituting equation (A3) into equation (2) and changing ? by x = ν 2 [(1 + ?)(j+3)/3 ? 1], we obtain the following expression for the accretion rate Ra (M, t) = 3Aν 2 (j + 3)t
xm (M,t,?m )
1 + ν ?2 x
1 + ν ?2 x
exp (?x/2) x2
where, for simplicity, we have dropped the explicit dependence of ν and used the notation A = 2/π (j + 3)/9 and xm (M, t, ?m ) = ν 2 [(1 + ?m )(j+3)/3 ? 1]. In the di?erential equation (eq. ), the variable M in Ra is replaced by the mass accretion track M (t). As halos grow through both accretion and major mergers, the accretion tracks M (t) increase with increasing time less rapidly than do M? (t), tracing the typical mass evolution of halos in any self-similar universe. Thus, ν(t) ≡ ν(M (t), t) is a decreasing function of t (see eq. [A4]) and, in the small-t asymptotic regime, ν(t)?2 tends to zero. Taking the Taylor series expansion of [1 + ν ?2 (t)x]3/(j+3) inside the integral on the right-hand side of equation (A5), at leading order in ν ?1 (t), we obtain √ xm (t) 9 2πAt?1 j+3 exp (?x/2) 9At?1 √ erf ν(t) (1 + ?m ) 3 ? 1 , = (A5) dx Ra (M (t), t) = (j + 3)2 0 x (j + 3)2
DENSITY PROFILE AND CENTRAL BEHAVIOR OF HALOS
where erf(x) is the error function. Given the value of constant A, and given that, for ν(t) tending to in?nity, the error function approaches unity, we are led to 2 ?1 Ra (M (t), t) = t . (A6) j+3 Finally, integrating equation (5) for such an accretion rate, we are led to the following asymptotic behavior for accretion tracks: 2 (A7) M (t) ∝ t j+3 . Note how it compares with the time dependence of the critical mass in a self-similar universe: M? (t) ∝ t j+3 .