aa r X i v : . [ a s t r o - ph ] N ov A Model for Dark Matter Halos
F.D.A. Hartwick
Department of Physics and Astronomy,University of Victoria, Victoria, BC, Canada, V8W 3P6
ABSTRACT
A halo model is presented which possesses a constant phase space density(Q) core followed by a radial CDM-like power law decrease in Q. The motivationfor the core is the allowance for a possible primordial phase space density limitsuch as the Tremaine-Gunn upper bound. The space density profile derived fromthis model has a constant density core and falls off rapidly beyond. The newmodel is shown to improve the fits to the observations of LSB galaxy rotationcurves, naturally provides a model which has been shown to result in a lengtheneddynamical friction time scale for the Fornax dwarf spheroidal galaxy and predictsa flattening of the density profile within the Einstein radius of galaxy clusters. Aconstant gas entropy floor is predicted whose adiabatic constant provides a lowerlimit in accord with observed galaxy cluster values. While ‘observable-sized’cores are not seen in standard cold dark matter (CDM) simulations, phase spaceconsiderations suggest that they could appear in warm dark matter (WDM)cosmological simulations and in certain hierarchically consistent SuperWIMPscenarios.
Subject headings: cosmology: dark matter
1. Introduction
Simulations of cold dark matter (CDM) cosmology predict halos whose density profilesare generally well described by what has become known as an NFW profile (Navarro et al.1997). A defining characteristic is the presence of a density cusp at the center. Anotherimportant property of CDM halo profiles is that they exhibit a power law in the parameter ρ/σ which extends over two orders of magnitude in radius beyond the resolution limit ofthe simulations. This was first shown by Taylor & Navarro (2001) from modelling based onthe NFW density profile and later directly from CDM simulations by Dehnen & McLaughlin(2005). Nonparametric models of the density profiles of CDM halos as well as alternate 2 –parameterizations are given by Merritt et al. (2006). In a companion paper Graham et al.(2006) discuss the power law nature of the ρ/σ profile.The NFW profile has been successfully fit to many observations including those of dwarfspheroidal and low surface brightness (LSB) galaxies as well as galaxy clusters. However,not all LSB rotation curves can be fit (Hayashi et al. 2004), and some authors have reporteddensity profiles near the centers of galaxy clusters which are shallower than predicted byNFW (e.g. Sand et al. 2002, Broadhurst et al. 2005). Recently, Goerdt et al. (2006) haveargued that a constant density core is required in the Fornax dwarf in order to understandwhy its resident globular clusters have not disappeared due dynamical friction, as mightbe expected if its dark matter halo was of NFW form. Modelling dark matter halos withconstant density cores is not new but it is usually done by parameterizing the density profile(e.g. see Burkert, 1995 and references above). Here the goal is to follow the effects of a finiteprimordial phase space density upper limit. Thus the constant density core results from asolution of the Jeans equation with a parameterized phase space density profile.Simple analytical arguments suggest that the effects of a primordial phase space densitybound should be seen in present structures even after many mergings (e.g. Dalcanton &Hogan, 2001). In the absence of cosmological simulations which include such a primordialbound, we rely here on the good agreement of predictions from a simple model with obser-vations to argue that standard CDM simulations and hence the NFW profile may not begiving a complete picture.
2. The Model
We start with a CDM like power law in the quantity ρ/σ where ρ is the local spacedensity and σ is the local radial velocity dispersion. The above quantity is often looselyreferred to as the phase space density (as it is here) but it is actually a ‘pseudo’ phase spacedensity (e.g. see Dekel & Arad (2004) for a discussion of the true 6-D phase space density).This power law is maintained in the outer regions of the model but with a continued riseat sufficiently small radius, the phase space density is assumed to reach the Tremaine &Gunn (1979) limit in the absence of other lower and less fundamental limiting effects. Thisquantum statistical upper limit on the phase space density applies to thermal particles as wellas fermions. Hence, as long as the particles are not bosons we expect an eventual cap/corein the central phase space density. Here a simple model is proposed in order to mimic thelingering effects of a putative but as yet unknown primordial phase space density bound(Q p ≡ ¯ ρ/ ¯ σ where ¯ ρ is the mean density and ¯ σ is the one dimensional velocity dispersion).With a constant central phase space density ( Q o ) core of size r c , the following profile is 3 –defined: ρ/σ ≡ Q ( r ) = Q o (1 + ( r/r c ) α ) . /α (1)The choice of a model independent power law index of 1.92 (close to that found byTaylor & Navarro) comes from the work of Dehnen & McLaughlin (2005). All but one of themodels presented here were computed with the shape parameter α = 1 .
92. Lower values of α result in a more gradual transition to the outer power law and as will be shown producevery similar results.The above expression for Q(r) is substituted into the Jeans equation after eliminating σ , thereby allowing the determination of the density profile ρ . (Note the assumption ofspherical symmetry). d log ρd log r = − . GM r r (cid:18) Qρ (cid:19) / − . β + 0 . d log Qd log r (2)with dM r d log r = ln(10)4 πr ρ (3)where β is the anisotropy parameter (Binney & Tremaine, 1987) and is positive for a predomi-nantly radial anisotropy. The following dimensionless number involving the initial conditionswas found to lie between 2 and 3 for all models presented here. γ = 4 πGr c Q o ( σ o ) / = 4 πGr c Q / o ρ / o (4)Integrations can be carried out with β = 0, but for the above range of γ , the dispersion isfound to increase outwards around the point of gradient change in Q(r). As CDM simulationsindicate a small radial anisotropy in the outer parts of a halo, at each step in the integration,trial values of β are stepped through (in units of 0.001) in order to determine that value whichmakes the logarithmic gradient in dispersion have the shallowest negative value. In this waya β profile is obtained starting at zero in the center (actually 0.001 for computational reasons)remaining small throughout the core and usually ending up ∼ . − .
3. At some point furtherout the scheme tries to make β decrease but it is constrained to remain at its maximumvalue. The above value of β max is a rough average of the outermost values determined forCDM halos (Fig. 3 of Dehnen & McLaughlin (2005)). Unlike models computed without 4 –the above simple, well defined prescription, those here exhibit consistent scaling relations(see discussion below). Furthermore, the initial rise of β is very nearly a linear function ofthe logarithmic density gradient as advocated by Hansen & Moore (2006). Beyond β ∼ . β and we have elected to keep it constant in this region.The core can be considered isothermal (i.e. constant velocity dispersion and negligiblevelocity anisotropy) within the region where the logarithmic density gradient is greater than-0.1. This radius is ∼ . c .Depending on what observations are given (i.e. initial rotation curve slope, the locationof the bend in the rotation curve or its amplitude) determines which of the parameters ρ o , r c or Q o one chooses to fix initially. A model is constructed by integrating equations (2) and (3)while systematically varying the other two parameters until the logarithmic density gradientbecomes -4.000 at M vir . M vir and R vir are defined as the values of mass and radius wherethe mean density becomes 100 × ρ c (h = 0 . ρ o determined for a converged model corresponds to a particular value of β max . Remarkably,any other model with the same ρ o can then be obtained from the following scaling relations(i.e. Q o ∝ M − , Q o ∝ σ − o and Q o ∝ r − c ) and hence is a member of a one parameter family of models. Interestingly, these scaling relations are identical to those discussed byDalcanton & Hogan (2001) to describe the results of ‘gentle’ merging given that during amerger Q cannot increase. Based on this discussion, the inverse relation between Q o andM found here suggests that equation (1), for all of its simplicity, is consistent with a formof hierarchical structure formation. As discussed by the above authors, it is decidedly notcompatible with ‘phase packing’ where one expects more massive objects (with higher centralvelocity dispersions) to have smaller core radii.In what follows, models are specified by the four parameters Q o (M ⊙ pc − ( km sec − ) − ),r c (kpc), ρ o (M ⊙ pc − ) and α and are enclosed in brackets.It is useful to express the equivalent gas entropy in terms of Q. We do this by evaluatingthe adiabatic constant K = kT n − / e . With Q in the same units as above we obtain Approximate solutions which obviate the need for the trial and error procedure are given in the appendix. K = 8 . × − µµ / e ((3 − β ( r )) / Q / = K o (3 − β ( r )) / r/r c ) α ) . /α (5)and K o = 8 . × − µµ / e Q − / o kev cm − Note that the entropy of the gas is initially constant at K o and then increases as apower law with index 1.28 as long as β remains constant in the outer region (see Fig. 1).We emphasize that inherent in equation (5) is the assumption that the entropy of the gasis the same as the entropy of the dark matter and that as merging continues the increasein entropy is the same for both components. Thus this value of K must be a lower limit tothe actual gas entropy and as such it provides a floor on which gas physics processes (i.e.cooling, heating, astration etc.) can be played out.The characteristics of a representative model (in this case for the galaxy cluster A1689)are shown in Fig. 1.
3. Confronting the Model with Observations
As a test of the model we apply it to three regimes of total mass: LSB galaxies, clustersof galaxies and dwarf spheroidal galaxies.
Hayashi et al. (2004) have derived best fit rotation curves for a sample of LSB galaxiesusing the NFW density profile. They divided their fits into three categories. The first (Aclass) provided good fits to the observations. The second (B class) included galaxies whichcould not be satisfactorily fit with ΛCDM-compatible parameters. Galaxies in the thirdgroup (C class) have irregular rotation curves. Fig. 2 shows the fits of the circular velocity(GM r /r ) / ∼ ssm/data.Following Hayashi et al. we let the smallest uncertainty in velocity be ± kmsec − . Theabove model provides adequate fits to both class A and class B samples. Parameters for theclass A galaxies (3 . × − , . , . , .
92) for ESO2060140 and (1 . × − , . , . , . β max ’s ∼ .
3) than the B groupgalaxies (3 . × − , . , . × − , .
92) for ESO0840411 and (1 . × − , . , . × − , .
92) for UGC5750 with β max ’s 0.205 and 0.220 respectively. Generally, models withhigher values of ρ o have cores with relatively larger values of Q o . 6 – The derived behavior of the dark matter density profile in the inner parts of galaxy clus-ters is controversial in part because of ‘contamination’ by the baryonic component in additionto the observational resolution difficulties. Here we fit our model to a recent gravitationallensing study of the massive cluster A1689 by Broadhurst et al. (2005). Standard integrationof the density profile provides the run of projected mass with radius. Figure 1 shows a modelwith parameters (2 . × − , . , . , .
92) fit to the Broadhurst et al. data. The modelhas a virial mass of 1 . × M ⊙ . Comparing this figure to Fig. 3 of the Broadhurst et al.paper shows that unlike the best fit NFW profile this model exhibits the desired propertiesof more flattening towards the center and more steepening towards the outside. The reduced χ statistic for this particular model fit is χ red = 23 . /dof = 23 . /
12 = 1 . Recently Goerdt et al. (2006) and S´anchez-Salcedo et al. (2006) have argued that theFornax dwarf must contain a large core in order that its globular clusters are not drawninto the center by dynamical friction. In order to test the sensitivity of this process to coresize, two models were constructed. One has parameters (1 . × − , . , . , .
92) and β max = 0 .
294 while the other has (3 . × − , . , . × − , .
92) and β max = 0 . . × M ⊙ . (Note that the value of r c of 0.92 kpc providesa region within which the logarithmic density gradient is less than -0.1 of only ∼ . τ df ≈ × V / ( ln (Λ) M GC ρ ) yrs). Here, Vis the velocity of the cluster with assumed mass M GC = 2 × M ⊙ and lnΛ is the coulomblogarithm which we take here to be 5 for consistency with Goerdt et al. Given that withinthe radial distance ∼ . r c the density and velocity dispersion are essentially constant and β ∼
0, we replace V /ρ with 3 √ σ o / ρ o = 3 √ o and obtain τ df ≈ . × M GC Q o yrs (6)While the above estimate for τ df is no substitute for a Goerdt et al. type of analysis, itdoes allow an intercomparison among our cored models. 7 –The above model with the largest core has a dynamical friction time scale of 7 . × years which is nearly four times that of the cluster with the smaller but higher density core.As a further check, a model with the same mass and r c but different α was made withparameters (4 . × − , . , . × − , . c but only by ∼ α while keeping the other parameters the same lowers the central entropy slightly). Thissimple analysis is in accord with the conclusions of Goerdt et al. and S´anchez-Salcedo etal. that increasing the core size leads to an increase in dynamical friction time scale. Thisincreased time scale approaches a Hubble time and because our calculated value of Q o is not necessarily assumed to be a result of phase packing remains within the constraints imposedby the sophisticated dynamical model of Fornax by Strigari et al. (2006).An additional check on the model comes from the recent work of Gilmore et al. (2007)whose analysis of the light distribution and velocity dispersion profile of several local dwarfspheroidal galaxies shows that shallow (cored) central density profiles with mean densities of0 . M ⊙ pc − (identical to that of our model above with r c = 0 . kpc ) are most consistentwith the observations.
4. Discussion
A new model for dark matter halos has been proposed and is successfully applied toobservations of objects with masses ranging from ∼ to ∼ M ⊙ . Fig. 3 is a graphicalsummary of the results which shows the scaling relations discussed earlier. It is importantto note key differences in structure occur with different scaling normalizations. The filledsymbols are structures with relatively high values of ρ o ( ∼ . M ⊙ pc − ) while the opensymbols are structures with lower values of ρ o ( ∼ . M ⊙ pc − ) and lower β max ’s. Closeexamination of Fig. 3 reveals a real systematic shift between these two groups of objects.Objects with identical central densities would lie essentially dispersionless along a line of theindicated slope. Further, for two models with the same mass, the one with the higher Q o hasthe higher ρ o and smaller r c (i.e. quantitatively ∂ log Q o / ∂ log ρ o = 0 .
67 and ∂ log Q o / ∂ logr c = − .
57 for a fixed halo mass). While the density is not expected to increase duringmerging, Hernquist et al. (1993) propose a scheme whereby the density decreases while thedispersion remains constant. Dalcanton & Hogan interpret this as a result of more violentmerging so that different merging histories at earlier times could account for the variationsin central density seen now. More observations are required to determine if the apparentdichotomy in central density is real and if so what its origin is. For example, if we assume 8 –that the dichotomy extends to galaxy clusters, then a model with the same mass as A1689but with one tenth the central density will have its gas entropy floor raised by ∼ .
8. Such achange in central gas entropy floor is one characterization differentiating cooling flow clustersfrom non cooling flow clusters.The error bars in each panel of Fig. 3 were calculated by assuming an empirical ap-proximation to γ (equation (4)) in terms of the central density (i.e. γ ∼ . ρ − . o ) andvariables ρ o and r c were then treated as independent with estimated uncertainties of ± . ± .
10 respectively.Rotation curves derived from the model and the NFW profile can be very similar (e.g.the two A class galaxies in § o and the mass of dark matterhalos over a range from ∼ to ∼ M ⊙ . An observational challenge is to find the lowermass limit to objects with dark matter halos thus providing an estimate of the primordialvalue of Q (Q p ). Knowledge of Q p allows the determination of the mass of the dark matterparticle (assuming that the particles are thermal) since then Q is proportional to the fourthpower of the particle mass (e.g. equating the value of Q o found above for Fornax with Q p provides a lower limit on this mass of 431 ev). An additional constraint comes from ananalysis of the power spectrum of the Ly α forest. From this one can determine the freestreaming length ( λ fs ) of the dark matter particle. This quantity in turn is simply related(in the case of thermal particles) to the particle mass. A recent determination of a limit on λ fs by Seljak et al. (2006) implies a thermal dark matter particle mass limit of >
10 kev(i.e. Q p > ∼ M ⊙ ).An attractive alternative to the above ‘classical’ WDM scenario has been proposed byStrigari et al. (2007). If the particles are non-thermally produced by the decay of a super-symmetric particle for example and if they are born sufficiently late then the initial velocitiesof the resulting daughter particles can be sufficiently high to yield a free streaming lengthcomparable to that found from the Ly α forest analysis but with Q p orders of magnitudelower than above (i.e. Q p ∼ − − − ) and a correspondingly much higher dark halo masslimit. This picture has the additional feature that it is hierarchical in the conventional CDMsense since the parent particles are born cold and being bosons they are not subject to the 9 –ultimate phase space density restriction.Future results from experimental particle physics and even more sophisticated cosmo-logical simulations should lead to a fuller understanding of the dark matter problem and theviability of the model.The author wishes to thank Drs. Julio Navarro, Tony Burke and Andi Mahdavi for usefuldiscussions, Dr. Greg Poole for introducing me to x-ray observations of galaxy clusters andthe referee for a constructive report. REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
11 –AppendixAs described above, a converged model is obtained by the somewhat cumbersome trialand error method. Below we present some analytical approximations which will allow a moreefficient exploration of parameter space and which illustrate more explicitly the two (scaling)parameter nature of the model (i.e. fix ρ o and vary Q o ). They were obtained by ‘fitting’to the converged solutions above. Letting α = 1 .
92 in equation (1) of the text, the densityprofile is approximated by ρ ( r ) ∼ ρ o (1 + ( r/r c ) . )(1 + ( r/ ( δ ( ρ o ) r c )) . ) (1)where r c ∼ . × − ρ − . o Q − / o (2)and δ ( ρ o ) ∼ . log ρ o + 24 .
97 (3)The velocity dispersion becomes σ ( r ) ∼ ( ρ o /Q o ) / (1 + ( r/ ( δ ( ρ o ) r c )) . ) / (4)Finally the value of β max is obtained by determining the maximum associated with thesmallest r value (i.e. the first maximum) in the following expression β ( r ) ∼ − . GM r / ( σ r ) − . d log Q/d log r − (5 / d log σ /d log r (5)where M r comes from the integration of equation (3) in the text.With these approximations, values of r c , M , R , are determined to < < < β max are between 1 .
4% and 14 .
1% with the largest deviations occurringat the lowest values of ρ o . Beyond the maxima the run of β is not reliable. Circular velocitymaxima derived from the above expressions are within 5% of the model values. It shouldbe emphasized that the above expressions and bounds were determined from models withcentral densities 5 . × − ≤ ρ o ≤ . × − . 12 – -1 0 1 2 3 4-5-4-3-2-10 log r (kpc) 0 1 2 311.522.53 log r (kpc)1 1.5 2 2.5 378910 log r (kpc) -1 0 1 2 3 400.10.20.30.40.5 log r (kpc) Fig. 1.— Attributes of a solution to equations (1),(2),& (3) for the galaxy cluster A1689.Upper left panel: The run of density (solid) and velocity dispersion (dashed) versus radialdistance. Upper right panel: The gas entropy profile before astrophysical processes changeit. Lower left panel: Observational data from Fig. 3 of Broadhurst et al. 2005 with themodel projected mass density superposed. Lower right panel: The run of β with radialdistance derived as described in §
2. Log(r c ) for this model is 1.52. 13 – Fig. 2.— Model fits to LSB galaxy rotation curves. The black line is the fit of an empiricalfitting function with the same parameters given in Figs 7 & 8 of Hayashi et al. 2004. Thered curve is the model fit. Recall that NFW profiles could not be well fit to the two groupB galaxies. The reduced χ for all model fits is <
1. 14 –
Fig. 3.— The model parameters for the six systems discussed here. Open circle- Fornaxdwarf, open triangles-group B LSB galaxies, solid triangles-group A LSB galaxies and solidcircle-galaxy cluster A1689. Clockwise from the upper left shows log( Q o ) versus centralvelocity dispersion, core radius, virial radius and virial mass. The dashed lines are not fitsbut illustrate the scaling relations described in the text (i.e. Q o ∝ σ − o , ∝ r − c , ∝ r − vir and ∝ M − virvir