Dark Matter Halos. They will have a large impact on galaxies

Transcription

1 Dark Matter Halos Dark matter halos form through gravitational instability. Density perturbations grow linearly until they reach a critical density, after which they turn around from the expansion of the Universe and collapse to form virialized dark matter halos. These halos continue to grow in mass (and size), either by accreting material from their neighborhood or by merging with other halos. In this module we will talk about: -The mass function of dark matter halos (i.e. the number density of halos as a function of halo mass), its dependence on cosmology, and its evolution with redshift. -The mass distribution of the progenitors of individual halos. -The merger rate of dark matter halos. -Intrinsic properties of dark matter halos, such as density profile, shape and angular momentum. They will have a large impact on galaxies

2 Dark Matter Halos In the peak formalism, it is assumed that the material which will collapse to form non-linear objects (i.e. dark matter halos) of given mass can be identified in the initial density field by first smoothing it with a filter of the appropriate scale and then locating all peaks above some threshold. The properties of peaks in a Gaussian random field can be analysed in a mathematically rigorous way, but the formalism nevertheless has significant limitations: - it cannot be used to obtain the mass function of nonlinear objects (dark halos) - it does not provide a model for how dark matter halos grow with time Alternative formalism: Press-Schechter mass function, the mass distribution of progenitors, merger rate and clustering properties

3 Density Peaks In the linear regime the overdensity field grows as δ(x,t) D(t), with D(t) the linear growth rate: for a given cosmology [which governs D(t)] we can determine the properties of the density field at any given time from those at some early time ti. Can we extend this to the non-linear regime in order, for example, to predict the distribution and masses of collapsed objects from δ(x,ti) without following the non-linear dynamics in detail? An object of mass M forms from an overdense region with volume V = M/ρ in the initial density field. It is tempting to assume that such a region will correspond to a peak in the density field after smoothing with window of characteristic scale R M 1/3. This suggests that a study of the density peaks of the smoothed density field can shed some light on the number density and spatial distribution of collapsed dark matter halos.

4 Peak Number Density

5 Peak Number Density

6 Peak Number Density

7 Peak Shape

8 Peak Shape

9 Shortcomings of the Peak Formalism It is tempting to interpret the number density of peaks in terms of a number density of collapsed objects of mass M ρr 3. However, there is a serious problem with this identification, because a mass element which is associated with a peak of δ 1 (x) = δ s (x;r1) can also be associated with a peak of δ2(x) = δs(x;r2), where R2 > R1. Should such a mass element be considered part of an object of mass M1 or of mass M2? R2 R1 If δ2 < δ1 the mass element can be considered part of both. The overdensity will first reach (at time t1) the critical value for collapse on scale R1, and then (at time t2 > t1) on scale R2. M1 is the mass of a collapsed object at t1 which merges to form a bigger object of mass M2 at t2. In the opposite case, where δ2 > δ1, the mass element apparently can never be part of a collapsed object of mass M1 but rather must be incorporated directly into a larger system of mass M2. Such peaks of the field δ1 should therefore be excluded when calculating the number density of M1 objects. This difficulty is known as the cloud-in-cloud problem. What is required to predict the mass function of collapsed objects is a method to partition the linear density field δ (x) at some early time into a set of disjoint regions (patches) each of which will form a single collapsed object at some later time.

10 Press-Schechter Formalism Consider the overdensity field δ(x,t), which in the linear regime evolves as δ(x,t) = δ 0 (x)d(t), where δ 0 (x) is the overdensity field linearly extrapolated to the present time and D(t) is the linear growth rate normalized to unity at the present. According to the spherical collapse model described, regions with δ(x,t) > δ c 1.69, or equivalently, with: δ 0 (x) > δ c /D(t) δ c (t), will have collapsed to form virialized objects. In order to assign masses to these collapsed regions, Press & Schechter (1974) considered the smoothed density field where W(x;R) is a window function of characteristic radius R corresponding to a mass M = γfρr 3, with γf a filterdependent shape parameter (i.e. γf = 4π/3 for a spherical top-hat filter). The ansatz of the Press-Schechter (PS) formalism is that the probability that δ s > δ c (t) is the same as the fraction of mass elements that at time t are contained in halos with mass greater than M. The probability that an overdense region of radius R has collapsed gives the fraction of mass in objects above that mass

11 Press-Schechter Formalism If δ 0 (x) is a Gaussian random field then so is δ s (x), and the probability that δ s > δ c (t) is given by: where is the mass variance of the smoothed density field with P(k) the power spectrum of the density perturbations, and W (kr) the Fourier transform of W (x; R). From the PS ansatz, this probability equals F(> M), the mass fraction of collapsed objects with mass greater than M.

12 Press-Schechter Formalism Problem: As M 0, then σ(r) and P[> δc(t)] 1/2. This would suggest that only half of the mass in the universe is part of collapsed objects of any mass (in linear theory, only regions that are initially overdense can collapse). However, underdense regions can be enclosed within larger overdense regions (cloud-in-cloud problem). Press & Schechter (1974) argued, without a proper demonstration, that the material in initially underdense regions will eventually be accreted by the collapsed objects, doubling their masses without changing the shape of the mass function. Press & Schechter therefore introduced a fudge factor 2 and adopted F(> M) = 2P[> δc(t)]. This results in a number density of collapsed objects with masses in the range M M + dm given by: Halos with mass M can only form in significant number when σ(m) > δ c(t). If we define a characteristic mass M (t) by: σ(m ) = δc(t) = δc/d(t), then only halos with M <M can have formed in significant number at time t. Since, in hierarchical models, D(t) increases with t and σ(m) decreases with M, the characteristic mass increases with time. Thus, as time passes, more and more massive halos will start to form.

13 Excursion set derivation of PS - extended PS Adopting δc=δc(t), S σ 2 (M) where M=γfρR 3, since in hierarchical models S is a monotonically declining function of M, a larger value of S corresponds to a smaller mass. Each location x in the density field δ 0 (x) corresponds to a trajectory δs(s), which reflects the value of the density field at that location when smoothed with a filter of mass S. In the limit S 0, which corresponds to M, we have that δs = 0 for each x. Increasing S corresponds to decreasing the filter mass, and δs starts to wander away from zero. In the excursion set formalism one adopts the sharp k-space filter for which γ f = 6π 2. The smoothed field is then: where k c = 1/R is the size of the top-hat in k-space, and δ k,0 are the Fourier modes of δ 0 (x). The advantage of using this particular filter is that the change δ s corresponding to an increase from k c to k c + k c is a Gaussian random variable with variance Thus the distribution of δ s is independent of the value of δ s (x;k c ). When k c [or the associated S = σ 2 (k ) = σ 2 (M) with M = 6π 2 ρk 3 ] is increased, the value of δ at a given point x executes a Markovian random walk. Note that for any other filter, the trajectory δ s (S) will not be Markovian.

14 extended PS Consider a given mass scale M1, corresponding to S1, as indicated by the vertical line. According to the PS ansatz, the fraction of all trajectories having δs > δc at S1 is equal to the fraction of mass elements in collapsed objects with mass M > M1. For trajectory B, which has δs < δc at S1, this mass element would not be part of a collapsed object with M > M1 according to the PS ansatz. However, since it has δs>δc over the interval S2<S<S3, it should be part of a halo with M>M3>M1 according to the same PS ansatz. Clearly the PS ansatz is not self-consistent. The problem is that it fails to account for the mass elements with trajectories such as B. A correction for this can be made rather easily by realizing that the trajectory from (S2,δc) to (S1,Q1) is equally as likely as the trajectory B (indicated by the dashed line) obtained by mirroring B for S S2 in δs = δc.this implies that each trajectory such as B that is missed in counting based on the PS ansatz corresponds to an equally likely trajectory, B, that passes through a point with δs > δc at S1. The fraction of mass in halos with M > M1 is therefore given by twice the fraction of trajectories that crosses S1 at δs > δc.

15 Extended PS To derive HMF from excursion set formalism, start by deriving the fraction of trajectories that have their first upcrossing of the barrier δs = δc at S > S1. These trajectories are associated with mass elements in collapsed objects of masses M < M1. Clearly, such a trajectory must have δs(s1) < δc. However, this includes B trajectories, which have their first upcrossing of the barrier at some S < S1. In order to exclude such trajectories, we use the fact that B and B have equal probabilities. Indeed, all trajectories that pass through (S,δs) = (S1,Q1) but have δs > δc for some S < S1 have a mirror trajectory that passes through (S,δs) = (S1,Q2), where Q2 = Q1 +2(δc Q1) = 2δc Q1. Thus, the fraction of trajectories with a first upcrossing at S > S1 is given by: where ν1 δc/ S1 using, for a Gaussian random field. If S as M 0, every trajectory will cross the barrier δc at some point. Hence, each mass element in the Universe is expected to be in a collapsed object with some mass M > 0. Consequently, F(> M) = 1 F(< M) and we obtain the HMF: the latter gives the fraction of trajectories that have their first upcrossing in the range (S, S + ds). Substituting in the top equation yields exactly the PS mass function without having to include a fudge factor of 2.

16 Extended PS In deriving the HMF using the excursion set formalism, we have made the ansatz that: The fraction of trajectories with a first upcrossing on a mass scale (M, M + dm) is equal to the mass fraction of the Universe in collapsed objects with masses in this range. However, the following example shows that this cannot formally be true: Consider a uniform spherical region of mass M and radius R, with density equal to the critical density for collapse at some time t, embedded in a larger region with lower density. According to the spherical collapse model, all mass elements in this spherical region will be part of a collapsed halo of mass M at time t. Now consider an interior point x which is at a distance r from the center of the spherical region, and calculate the mean overdensities within spheres centered on this point. Clearly, the overdensity reaches the critical density only for spheres with radii smaller than R r. Thus, the mass element at x has its first upcrossing at a mass scale M(R r) 3 /R 3, which is smaller than M unless x is at the center of M. (DRAW) This simple example suggests that the mass associated with the first upcrossing of a mass element is only a lower limit on the actual mass of the collapsed object to which the mass element belongs. Thus, the ansatz cannot be correct for individual mass elements. Yet, the resulting mass function seems to be in good agreement with numerical simulations. This paradox is often interpreted as indicating that the excursion set formalism predicts, in a statistical sense, how much mass ends up in collapsed objects of a certain mass at a given time, but that it cannot be used to predict the halo mass in which a particular mass element ends up.

17 Tests of PS Many crude assumptions in Press-Schechter (PS) and extended PS (EPS) - important to test HMF with simulations, which follow the growth and collapse of structures directly by solving the equations of motion for dark matter particles. The PS mass function over-predicts the abundance of sub-m halos, and under-predicts that of halos with M > M. The PS mass function for ellipsoidal collapse, on the other hand, fits simulations much better. However, comparisons depend on halos identification in the simulations. For example, (FOF) algorithm defines halos as structures whose particles are separated by distances less than the linking length, b, times the mean interparticle distance. A heuristic argument, based on spherical collapse, suggests that one should use b 0.2, for which the mean overdensity of a halo is 180. Different values for b result in different mass functions. More stringent test to e-ps is to predict the formation sites and masses of individual halos, and compare with simulations. The excursion-set formalism is not expected to work on an object-by-object basis. However, the mass at first upcrossing of a mass element should give a lower limit to the mass of the collapsed object to which the mass element belongs. For an ellipsoidal collapse barrier, this prediction was found to be consistent with N-body simulations.

18 Merger Trees Consider a spherical region (a patch) of mass M 2, corresponding to a mass variance S 2 = σ 2 (M 2 ), with linear overdensity δ 2 δ c (t 2 ) = δ c /D(t 2 ) so that it forms a collapsed object at time t 2. We are interested in the fraction of M 2 that was in collapsed objects of a certain mass at an earlier time t 1 < t 2. To proceed, we adopt the same excursion-set approach as we did in deriving the Press-Schechter (PS) formula, and calculate the probability for a random walk originating at (S,δ s ) = (S 2,δ 2 ) to execute a first upcrossing of the barrier δ s = δ 1 δ c (t 1 ) at S = S 1, corresponding to mass scale M 1. This is exactly the same problem as before except for a translation of the origin in the (S,δ s ) plane. Hence, the probability we want is given by: which follows from Eq. (7.56) upon replacing δ c by δ 1 δ 2 and S by S 1 S 2. According to the interpretation of the excursion set, Eq. (7.80) gives the fraction of mass elements in (M 2,δ 2 ) patches that were in collapsed objects of mass M 1 at the earlier time t 1. Converting from mass weighting to number weighting, one obtains the average number of progenitors at t 1 in the mass interval (M 1,M 1 +dm 1 ) which by time t 2 have merged to form a halo of mass M 2 : Although Eq. (7.81) is in good agreement with simulations for small t = t 2 t 1, underestimates the mass fraction in high mass progenitors for large t.

19 Merger Trees The conditional mass function derived above can be used to describe the statistical properties of the merger histories of dark matter halos. The merger history of a given dark matter halo can be represented pictorially by a merger tree such as that shown in Fig In this figure, time increases from top to bottom, and the widths of the branches of the tree represent the masses of the individual progenitors. If we start from an early time, the mass which ends up in a halo at the present time t 0 (the trunk of the tree) is distributed over many small branches; pairs of small branches merge into bigger ones as time goes on.

20 Merger Trees In order to construct a merger tree of a halo, it is most convenient to start from the trunk and work upwards. Consider a halo with mass M at a time t (which can be the present time, or any other time). At a slightly earlier time t t, the progenitor distribution, in a statistical sense, is given by Eq. (7.81). This can be used to draw a set of progenitor halos, after which the procedure is repeated for each of these progenitors, thus advancing upwards into the merger tree. The construction of a set of progenitor masses for a given parent halo mass needs to obey two requirements: 1) the number distribution of progenitor masses needs to follow Eq. (7.81). 2) mass needs to be conserved, so that in each individual realization the sum of the progenitor masses is equal to the mass of the parent halo. In principle, this requirement for mass conservation implies that the probability for the mass of the n th progenitor needs to be conditional on the masses of the n 1 progenitor halos already drawn. Unfortunately, these conditional probability functions are not derivable from the EPS formalism, so additional assumptions have to be made. This has resulted in the development of various algorithms for merger trees, each with its own pros and cons (Kauffmann & White, 1993; Sheth & Lemson, 1999; Somerville & Kolatt, 1999; Cole et al., 2000, 2008; Zhang et al., 2008; Neistein & Dekel, 2008).

21 Binary Merger Trees One of the simplest merger trees is the so called binary tree, in which the assumption is made that each merger involves exactly two progenitors (e.g. Lacey & Cole, 1993; Cole et al., 2000). To obtain masses of the two progenitor halos of a halo of mass M at time t: First draw a value for S from the mass-weighted probability function: Here δ = δc(t t) δc(t) reflects the time step used in the merger tree. The progenitor mass, Mp, corresponding to S follows from S(Mp) = S(M)+ S, and the mass of the second progenitor is simply given by Mp = M Mp. Ensures mass conservation, but two important shortcomings: 1) makes the implicit assumption that the probability of having a progenitor of mass Mp is equal to that of having a progenitor of mass Mp = M Mp. However, in general the EPS progenitor mass distribution does not respect this symmetry, especially at earlier times. 2) the assumption that a timestep involves only a single merger between two progenitor halos is only correct in the limit t 0. In practice, however, one has to adopt finite timesteps, for which the assumption of binarity is not strictly valid.

22 KW merger trees The tree-building scheme of Kauffmann & White (1993) avoids the assumption of binary mergers and ensures that the EPS progenitor mass distributions of Eq. (7.80) are reproduced for all t 1 and t 2 and for all M 1 above some resolution limit. For each descendant halo mass M at time t n one makes a library of N real realisations of possible progenitor sets at the earlier time t n 1 by using Eq. (7.80) to calculate the total number of progenitors (over all realisations) with mass M p < M in each of a number of mass bins. Proceeding from most massive to least massive, these progenitors are then distributed at random among the N real descendants with probability proportional to the remaining unassigned mass of the descendant, if this exceeds the mass of the progenitor, and zero otherwise. Once all progenitors have been assigned, the remaining descendant mass is assumed to have been accreted smoothly in objects below the resolution limit. Randomly chosen progenitor sets from such one-step libraries can then be combined into trees giving the complete merger history of each final halo.

23 SK merger trees A third method for constructing halo merger trees is the N-branch scheme with accretion of Somerville & Kolatt (1999). For each progenitor, a value for S is drawn from Eq. (7.82), and its corresponding mass, M p, is determined. For each new progenitor one checks whether the sum of the progenitor masses drawn thus far exceeds the mass of the parent, M. If this is the case the progenitor is rejected and a new progenitor mass is drawn. Any progenitor with M p < M min is added to the mass component M acc that is considered to be accreted onto the parent in a smooth fashion. Here again M min reflects the mass resolution chosen for the merger tree. This procedure is repeated until the total mass left, M left = M M acc M p, is less than M min. This remaining mass is then assigned to M acc and one moves on to the next time step. Although this algorithm, by construction, ensures exact mass conservation, and yields conditional mass functions that are in fair agreement with direct predictions from EPS theory, it is not very rigorous. In particular, mass conservation is enforced by hand, by rejecting progenitor masses that overflow the mass budget. Consequently, the mass distribution of first-drawn progenitors differs from that for second-drawn progenitors, and so on. Somewhat fortunately, the sum of these distributions matches the distribution of Eq. (7.81) for all progenitors, but only if sufficiently small time steps δ are used. These three tree-building algorithms were compared to each other, to the scheme of Cole et al. (2000), and to several new schemes by Zhang et al. (2008). Of the older schemes, only that of Kauffmann & White (1993) was found to reproduce the EPS conditional probability distributions over wide ranges of mass and time. Still, this is not a sufficient condition for an algorithm

24 Main Progenitor The full merger history of any individual dark matter halo is a complex structure containing a lot of information. A useful subset of this information is provided by the main progenitor history, sometimes called the mass accretion history or mass assembly history, which restricts attention to the main trunk of the merger tree. This main trunk is defined by following the branching of a merger tree back in time, and selecting at each branching point the most massive (main) progenitor. Fig. 7.5 The left-hand panel of Fig. 7.5 shows 25 examples of main-progenitor histories as function of redshift, M main (z), normalized by the present-day halo mass, M 0. All histories correspond to a halo of M 0 = h 1 M in an EdS cosmology. Note that these main progenitor histories reveal a large amount of scatter. The right-hand panel of Fig. 7.5 shows the average main progenitor histories, M main (z)/m 0, obtained by averaging 1000 random realizations for a given M 0. Results are shown for five values of M 0, as indicated. As is evident, on average halos that are more massive assemble later. As we demonstrate below, this is a characteristic of hierarchical structure formation.

25 Halo Assembly and Formation Times The concept of the main progenitor allows one to define a characteristic time for the assembly of a dark matter halo. The assembly time, t a, of a halo of mass M 0 is the time when its main progenitor first reaches a mass M 0 /2, i.e. M main (t a ) = M 0 /2. The probability distribution of t a can be obtained as follows. From Eq. (7.80) the probability that a random mass element of a collapsed object of mass M 0 at time t 0 was part of a progenitor of mass M 1 at an earlier time t 1 < t 0 is f FU (S 1,δ 1 S 0,δ 0 ) ds 1 /dm 1 dm 1, and the average number of progenitors that each M 0 -halo has is given by Eq. (7.81). However, since each object can have at most one progenitor with mass M 0 /2 < M 1 < M 0, the probability that any particular object has a progenitor in this mass range, and thus an assembly time earlier than t a is given by: where ω ã = ω (t a )

26 Halo Assembly and Formation Times The advantage of using the variables S and ω is that P(> ω ã ) depends only very mildly on halo mass, M 0, and cosmology. For example, for a power-law fluctuation spectrum with spectral index n, we can write: Upon substituting Eq. (7.87) in Eq. (7.85) it is clear that P(> ω ã ) is independent of M 0 and t 0 and with only weak dependence on the spectral index n.

27 Halo Assembly and Formation Times The shaded histograms in Fig. 7.6 show the differential distributions, dp/dω ã, for dark matter halos in the mass range M min < M < M max in a flat ΛCDM cosmology with Ω m,0 = 0.3, n = 1 and σ 8 = 0.9 obtained from an N-body simulation. The solid curves show the corresponding predictions from the EPS formalism. Halos in N-body simulations assemble earlier than predicted by the EPS formalism when adopting the spherical collapse model (better for ellipsoidal) probability of formation vs formation time

28 Halo Assembly and Formation Times For an EdS cosmology, δ c (t ) 1 + z, where z is the redshift corresponding to time t. Using the definition of ω ã, we obtain a simple, approximate expression for the median value of the assembly redshift: Here z 0 is the redshift at which the halo of mass M 0 is identified, and ω ã,m 1 is the median value of ω ã. From this we can see that, in a model with n > 3, halos of a higher mass assemble later, which is an important characteristic of hierarchical structure formation. For a given power spectrum, the distribution of assembly times for halos with the same M/M only depends on cosmology through the linear growth rate D(t). Since D(t) has a weaker dependence on t in a cosmology with lower Ω m,0, halos of a given mass typically assemble earlier in a lower-density universe.

29 Halo merger rates From Eq. (7.80) we can obtain the reverse conditional probability: where S i = σ 2 (M i ) and δ i = δ c (t i ) with S 1 > S 2 and δ 1 > δ 2, and f FU (S,δ c ) is given by Eq. (7.56). According to the interpretation of the excursion set, this is the conditional probability that a halo of mass M 1 at time t 1 is incorporated into a halo with a mass between M 2 and M 2 + dm 2 (M 2 > M 1 ) at a later time t 2 > t 1. If we set M 2 = M 1 + M and t 2 = t 1 + t, then Eq. (7.92) gives the probability for the halo to gain a mass M by merging or accretion in the time interval t. Thus, the rate at which a halo with mass M transits to a halo with mass between M and M + M is given by: where S 1 =σ 2 (M)andS 2 =σ 2 (M+ M). In any finite time interval t, the change in mass, M, can be due to the cumulative effects of more than one merger. However, for an infinitesimal time interval dt, the transition from M 1 to M 2 is most likely due a single merger event. Thus Eq. (7.93) gives the merger rate of a halo with mass M at time t with another halo with mass M.

30 Fig.7.7 shows P( M M,t) as a function of log( M/M) for power-law spectra with spectral index n = 2 and 0 in an EdS universe. (i) Halo Assembly and Formation Times For M M, P is a power law of M: P ( M) 1/2. This asymptote can be obtained directly from Eq. (7.93). Thus the number of merger events is dominated by minor mergers (those with M M), whereas the mass accretion rate (i.e. the mass-weighted merger rate) is dominated by major mergers with M M. (ii) For M M, mergers with halos of similar or larger masses are rare, while such mergers are more frequent for M M. (iii) Since a spectrum with a more negative n has more power on large scales relative to that on small scales, major mergers (i.e. those with a large M/M) happen more frequently for a spectrum with a more negative spectral index. minor mergers major mergers

31 Internal Structure of Dark Matter Halos To a first approximation, we can model a dark matter halo as a spherical object. In this case, the internal mass distribution is fully described by a density profile, ρ(r). As discussed above, different halos have different formation histories, so we may expect a significant halo-to-halo variation in density profile. But dark matter halos are highly non-linear, and it may be that information regarding their formation histories has largely been erased by their non-linear collapse. In the latter case, density profiles may be more closely related to the violent relaxation process than to initial conditions. In the similarity model of spherical collapse, if we start with an initial perturbation δ i (r) r 3ε, then the final profile will be ρ(r) r γ 3, where γ = 1 for ε 2/3, and γ = 3/(1+3ε) for ε > 2/3. The typical density profile around each particle in a linear density field with scale-free power spectrum, P(k) k n, is δ i (r) ξ(r) r (n+3). The typical halo profile will then be: Over the range of interest, 3 < n < 0, this model thus predicts that virialized halos resemble isothermal spheres.

32 Isothermal The simplest plausible model is therefore to assume that dark matter halos are truncated singular isothermal spheres: We define the limiting radius of a dark matter halo, r h, to be the radius within which the mean matter density is where ρ is the mean matter density of the Universe at the time in question, and ρ crit is the corresponding critical density for closure. Delta is the density contrast. Under this assumption, we have where M h is the mass of the halo (i.e. the mass within r h ), V h is its circular velocity at r h, and H(t) is the Hubble constant at time t. It is common practice to refer to r h and V c as the virial radius and virial velocity. A physically motivated choice would be to take h = vir. However, since the criterion for virialization is not strict, other definitions are also in use in the literature. For example, some studies adopt ρ h = 200 ρ, so that h = 200, while others use ρ h = 200 ρ crit, so that h = 200/Ω m. Note that different definitions imply different relations among M h, r h and V h.

33 NFW The isothermal model is, at best, an approximation. Many effects may cause deviations from the profile predicted by the simple similarity model. (i) collapse may never reach an equilibrium state in the outer region of a dark matter halo, (ii) non-radial motion may be important (iii) mergers associated with the (hierarchical) formation of a halo may render the spherical-collapse model invalid. Can only be addressed numerically. Using high resolution N-body simulations of structure formation in a CDM cosmogony, Navarro et al. (1996) showed that the density profiles of the simulated dark matter halos are shallower than r 2 at small radii and steeper at large radii. Navarro, Frenk & White (NFW) profile: Here r s is a scale radius, and δ char is a characteristic overdensity. The logarithmic slope of the NFW profile changes gradually from 1 near the center to 3 at large radii, and only resembles that of an isothermal sphere at radii r r s. In a follow-up paper, Navarro et al. (1997) found Eq. (7.138) to be a good representation of the equilibrium density profiles of dark matter halos of all masses and in all CDM-like cosmogonies. Thus, halos formed by dissipationless hierarchical clustering seem to have a universal density profile. The enclosed mass of the NFW profile is given by is the halo concentration parameter.

34 Internal Structure of Dark Matter Halos Note that the total mass, M h, of the halo is given by Eq.(7.139) with x = 1, and that this mass depends on the definition chosen for r h (different definitions for the bounding radius of a halo are in common use - different mass and concentration). Using Eqs. (7.138) (7.140), we obtain a relation between characteristic overdensity and concentration parameter, -for a given cosmology, the NFW profile is completely characterized by its mass, M, and its concentration parameter, c, or equivalently by r s and δ char. Navarro et al. (1997) showed that characteristic overdensity is closely related to formation time. Halos which form earlier are more concentrated. For a definition of formation time similar to that of Eq.(7.91), they found that the natural relation δ char Ω m,0 (1 + z f ) 3 describes how the overdensity of halos varies with their formation redshift. Halos that have experienced a recent major merger typically have low concentrations, c 4, while halos which have experienced a longer phase of relatively quiescent growth have larger concentrations (Zhao et al., 2003a, 2008). Physically, the central structure of a dark matter halo seems to be established through violent relaxation during a phase of major mergers, which leads to a universal NFW profile with c 4. Later accretion does not add much material to its inner regions, thus increasing r h while leaving r s almost unchanged. More massive halos assemble later, and thus had their last major merger more recently - lower concentration.

35 Internal Structure of Dark Matter Halos Although the NFW profile is widely used, numerical simulations of high resolution (Navarro et al., 2004; Hayashi & White, 2008; Gao et al., 2008; Springel et al., 2008) have shown that density profiles of dark matter halos show small deviations from NFW form and are more accurately described by an Einasto (1965) profile, with r 2 the radius at which the logarithmic slope of the density distribution is equal to 2 and ρ 2 = ρ(r 2 ). The best-fit values for the index α typically span the range 0.12 < α < 0.25 and increase systematically with increasing mass. Note that the systematic variation of α with halo mass demonstrates a small but significant deviation of mean density profiles from any universal shape. A characteristic property of the Einasto profile is that its logarithmic slope is a power-law in radius: Thus, contrary to an NFW profile, which has a central r 1 cusp, the logarithmic slope of the Einasto profile continues to become shallower as r 0 (however it only becomes <-1 at 0.01r 2 ). Why near-universal profiles? Different initial conditions give similar results, so they must result from relaxation processes in very general circumstances. The formation of a dark matter halo is in general clumpy and chaotic meaning that violent relaxation must play an important role (however cold collapse from asymmetric initial conditions produces objects with near-nfw density profiles, so hierarchical growth of structure is apparently not required).

36 Halo Shapes The collapse of overdensities in the cosmic density field is generically aspherical so halos should not be spherical. Halo equidensity surfaces can be described by ellipsoids, each of which is characterized by the lengths of its axes (a 1 a 2 a 3 ). These axes can be used to specify the dimensionless shape parameters, and/or the triaxiality parameter: Note that oblate and prolate shapes correspond to T = 0 and T = 1, respectively. The majority of CDM halos in numerical simulations have 0.5 <T <0.85, and 0.5 <s <0.75. In particular, less massive halos are more spherical, and halos of a given mass become flatter with increasing redshift. Shape is tightly correlated with its merger history: halos that assembled earlier are more spherical. -Halos that experienced a recent major merger have a tendency to be close to prolate, with the major axis reflecting the direction along which the last merger event occurred. -There is a strong tendency for the minor axes of halos to lie perpendicular to large-scale filaments. This alignment is found to be stronger for more massive halos, and shows that the shapes of dark matter halos reflect the large-scale tidal field in which they are embedded.

37 Halo Substructure When a small halo merges with a significantly larger halo it becomes a sub-halo orbiting within the potential well of its host. As it orbits, it is subjected to strong tidal forces from the host, which cause it to lose mass. In addition, the orbit itself evolves, as the sub-halo is subjected to dynamical friction which causes it to lose energy and angular momentum to the dark matter particles of its host. Whether a sub-halo survives as a self-bound entity depends on its mass (relative to that of its host), its density profile (which is related to its formation redshift) and its orbit.

38 The mass loss of sub haloes can be written as: Halo Substructure where r tid is its instantaneous tidal radius. We assume the latter to be the radius where the original (unstripped) density of the subhalo equals the density of its host s halo at its current orbital radius, r orb : i.e. ρ sub (r tid ) = ρ host (r orb ). If we assume that both ρ sub (r) and ρ host (r) are singular isothermal spheres, then so that both m and r tid decrease in direct proportion to r orb. The evolution of r orb is governed by dynamical friction, and, assuming a circular orbit, is given by with V c the circular velocity of the host halo. To leading order, the Coulomb logarithm lnλ is just a function of the mass ratio m/m, which is a constant (equal to v c 2 /V c 2 where v c is the circular velocity of the subhalo) under the physically reasonable assumption that M should be taken as the mass interior to the satellite s orbit. In this very simple model the satellite s mass and the radius of its (circular) orbit thus decrease linearly with time: where m i and r orb,i are the initial values of m and r orb respectively.

39 Halo Substructure Since we expect lnλ to be of the order of a few and r orb,i r vir, this predicts that newly accreted subhalos should merge completely with their host after of order 0.1(V c /v c ) 3 0.1m i /M times the initial orbital periods, where M is the total mass of the host. Thus infalling subalos with mass greater than a few percent that of the main object are predicted to merge rapidly, whereas substantially lower mass objects will survive for long times with little stripping. High resolution numerical simulations show that while this conclusion is qualitatively correct, Eq. (7.155) is a poor description of the typical mass loss behavior (Diemand et al., 2007). Most accreted halos fall in on highly elongated orbits and lose a large fraction of their mass at first pericenter. -The most massive and lowest concentration objects continue to lose mass thereafter and are often completely disrupted -Less massive and more concentrated objects often stabilize on their new orbit at their reduced mass. Thereafter they evolve rather little either in mass or in orbit. Regardless of this, host halos assembling earlier will end up with less mass in subhalos, simply because there has been more time for mass loss to operate. In a CDM cosmogony, halos that are more massive assemble later, and are thus expected to have a larger subhalo mass fraction, f 0, than lower mass objects.

40 Angular Momentum Another important property of a dark matter halo is its angular momentum. As originally pointed by Hoyle (1949) and first demonstrated explicitly using nu- merical simulation by Efstathiou & Jones (1979), asymmetric collapse in an expanding universe produces objects with significant angular momentum. This is traditionally parametrized through a dimensionless spin parameter, where J, E and M are the total angular momentum, energy and mass of the halo, respectively. For an isolated system, all these quantities are conserved during dissipationless gravitational evolution, and so, therefore, is λ itself. The spin parameter thus defined is roughly the square root of the ratio between the rotational and the total energy of the system, and so characterizes the overall importance of angular momentum relative to random motion. The angular momentum growth stops once a proto-galaxy separates from the overall expansion and starts to collapse. As an approximation, the final angular momentum of the proto- galaxy may therefore be estimated as the value of J predicted at the time t f when D(t f )δ i = 1. According to the Zel dovich approximation, this is approximately the time when the object collapses to form a pancake. This angular momentum may not correspond to the final angular momentum of a dark matter halo, because a dark matter halo may acquire significant amounts of angular momentum during the late stages of non-linear collapse and due to mergers with other halos.