# Dark Matter Halo Mass Profiles

Dan Coe11affiliation:  Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr, MS 169-327, Pasadena, CA 91109

###### Abstract

I provide notes on the NFW, Einasto, Sérsic, and other mass profiles which
provide good fits to simulated dark matter halos (§3). I summarize various
published $c(M)$ relations: halo concentration as a function of mass (§1). The
definition of the virial radius is discussed and relations are given to
convert $c_{vir}$, $M_{vir}$, and $r_{vir}$ between various defined values of
the halo overdensity (§2).

###### Subject headings:

cosmology: dark matter — galaxies: clusters: general — methods: data analysis
— gravitational lensing

††slugcomment: Draft

##  1\. Mass-Concentration Relations

The mass profiles of galaxy clusters appear to be more centrally concentrated
than realized in simulations (Broadhurst et al., 2008; Oguri et al., 2009;
Sereno et al., 2010). If true, this may be evidence for Early Dark Energy (see
e.g., Grossi & Springel, 2009). Or perhaps there is a less exciting
explanation (e.g., Barkana & Loeb, 2009; Lapi & Cavaliere, 2009; Meneghetti et
al., 2010). For more details, see my discussion in Coe et al. (2010). More
conclusive results are expected from the CLASH HST MCT project111Cluster
Lensing And Supernova survey with Hubble,
http://www.stsci.edu/$\sim$postman/CLASH/ and perhaps LoCuSS (Okabe et al.,
2009; Richard et al., 2009).

More massive halos generally have lower concentrations than less massive
halos. This is seen in both simulations and observations, though less clearly
so far in the latter (see below). More massive halos form later, resulting in
lower concentrations reflecting the lower background density at the time of
formation Navarro et al. (1996).

For a given radial mass profile (see §3), the concentration is defined as:

$c_{vir}=r_{vir}/r_{-2},$ (1)

a mishmash ratio of the virial radius $r_{vir}$ and the radius $r_{-2}$ at
which $\rho\propto r^{-2}$. For an NFW profile, $r_{-2}=r_{s}$. The definition
of the virial radius $r_{vir}$ is discussed at length in §2, but it is
typically approximated as the region within which there is an average
overdensity of a certain value ($\Delta_{c}\sim 100$ or 200) above
$\rho_{crit}$. For clarity, one may quote the exact value of $\Delta_{c}$
used: $c_{200}$, for example.

In principle concentrations could be derived using any radial fitting profile
(§3). However the choice does matter as the profiles behave differently
between $r_{-2}$ and $r_{vir}$. Concentrations derived from NFW and Einasto
fits to the same halos (Duffy et al., 2008) are compared in Fig. 8. Einasto
$c(M)$ relations have been derived for the Millennium simulation relaxed (Gao
et al., 2008) and all Hayashi & White (2008) halos. Below we focus on $c(M)$
relations derived from NFW fits.

###  1.1. Current $c(M)$ measurements from NFW profile fits

The current best estimates for $c(M,z)$ are probably those given by Duffy et
al. (2008) and Macciò et al. (2008). Their findings are similar. Both analyze
simulations which use the WMAP5 cosmology, resulting in $\sim 20\%$ lower
concentrations than WMAP1 (Table 8) as used in the Millennium simulation (Neto
et al., 2007), for example. Duffy et al. (2008) find that present-day ($z=0$)
halos follow the following mass-concentration relation:

$c_{200}\simeq 5.74\left(\frac{M_{200}}{2\times
10^{12}h^{-1}M_{\odot}}\right)^{-0.097}.$ (2)

They provide a separate relation for relaxed clusters which are more symmetric
and thus better fit by radial profiles such as NFW. These have $15-20\%$
higher concentrations (Fig. 7):

$c_{200}\simeq 6.67\left(\frac{M_{200}}{2\times
10^{12}h^{-1}M_{\odot}}\right)^{-0.092}.$ (3)

Intrinsic scatters are $\Delta\log_{10}(c_{200})\simeq 0.15$. These relations
are plotted in Fig. 2 along with corresponding relations from Macciò et al.
(2008).

Duffy et al. (2008) also supply fitted functions to halos spanning the
redshift range $z=0-2$. Full:

$c_{200}\simeq\frac{5.71}{(1+z)^{0.47}}\left(\frac{M_{200}}{2\times
10^{12}h^{-1}M_{\odot}}\right)^{-0.097}$ (4)

and relaxed:

$c_{200}\simeq\frac{6.71}{(1+z)^{0.44}}\left(\frac{M_{200}}{2\times
10^{12}h^{-1}M_{\odot}}\right)^{-0.092}.$ (5)

In their Table 1, they provide uncertainties for these fit parameters as well
as corresponding values for $c_{vir}$ and $M_{vir}$. These
$c_{200}(M_{200},z)$ relations are plotted in Figs. 3 & 4. Also plotted are
the corresponding $c_{vir}(M_{vir},z)$ relations provided by (Duffy et al.,
2008). In Fig. 4, we plot the Bullock et al. (2001) $c\propto(1+z)^{-1}$
scaling for comparison. Note that Duffy08 find weaker dependencies on
redshift: $c_{200}\propto(1+z)^{-0.45}$; $c_{vir}\propto(1+z)^{-0.70}$.

Halo concentrations are sensitive to cosmology. A higher $\sigma_{8}$ causes
halos to form earlier, resulting in higher concentrations. This was the case
in the Millennium simulations which used the WMAP 1-year cosmology, including
$\sigma_{8}=0.9$. This yields concentrations $\sim 20\%$ higher than found in
simulations which use WMAP5’s $\sigma_{8}=0.796$222This value is in excellent
agreement with the WMAP 7-year maximum likelihood value $\sigma_{8}=0.803$
(Komatsu et al., 2010). (Duffy et al., 2008). The effect of cosmology was
explored in more detail by Macciò et al. (2008). These effects are shown in
Fig. 5.

Various derived $c(M)$ relations (for $z=0$) are plotted in Fig. 6. In our
Tables 3, 4, and 5, we provide $c_{200}(M_{200})$, $c_{vir}(M_{vir})$, and
$c(M,z)$ relations, respectively, as derived by Duffy et al. (2008), Macciò et
al. (2008), Neto et al. (2007), Bullock et al. (2001), Hennawi et al. (2007),
and Gentile et al. (2007).333I’ll take a crack at completeness, and mention
other papers with $c(M)$ relations neglected here (for no particular reason):
Eke et al. (2001); Wechsler et al. (2002); Alam et al. (2002); Zhao et al.
(2003); Dolag et al. (2004); Kuhlen et al. (2005); Lu et al. (2006); Shaw et
al. (2006); Dutton et al. (2007); Gnedin et al. (2007). And I have probably
missed still others! The latter is the original NFW $c(M)$ prescription
updated to the WMAP3 cosmology.

The various simulations considered here are outlined in Table 7. The relevant
details of their adopted cosmologies ($\Omega_{m}$, $\sigma_{8}$) are given in
Table 8. We provide the range of halo masses produced in each simulation. The
dangers of extrapolating $c(M)$ relations beyond these ranges have been cited
by Zhao et al. (2003), for example.

Neto et al. (2007, their Fig. 5) find that 10,000 particles within the virial
radius are required to yield robust concentration measurements. They note that
using fewer particles introduces scatter but does not appear to introduce bias
in their concentration measurements.

Hennawi et al. (2007) measure significantly larger concentrations for galaxy
clusters in their simulations. Their cluster concentrations are $\sim 50\%$
and $\sim 80\%$ larger than found by Duffy et al. (2008) and Macciò et al.
(2008), respectively (Fig. 6, right). Their use of $\sigma_{8}=0.95$ probably
only results in concentrations inflated by $\sim 20\%$ compared to the WMAP5
$\sigma_{8}=0.796$ simulations. The remaining disagreement may be a result of
their halo density fitting procedure which they claim is better for comparison
with lensing measurements. Specifically, they assign large uncertainties to
radial bins with large subhalos. This may bias the fitted profiles to be low
at large radius (where large subhalos typically reside) resulting in higher
concentrations. These results may considerably ease tensions between observed
and simulated halo concentrations. The differences in fitting procedure should
be better studied and understood.

###  1.2. Care in citing concentration expectations

A concern often noted is that the concentration measured for A1689 (in every
study to date) is higher than that found in simulations for a halo of A1689’s
mass. The concentration found in simulations has been cited loosely as $c\sim
5$ or $c\sim 5.5$ using a relation given by Bullock et al. (2001):

$c_{vir}\simeq\frac{9}{1+z}\left(\frac{M_{vir}}{1.3\times
10^{13}h^{-1}M_{\odot}}\right)^{-0.13},$ (6)

But the expected concentration is actually lower ($c_{200}\sim 3.0$,
exacerbating the disagreement with observations) for four reasons:

  * •

$c_{200}<c_{vir}$

  * •

$z>0$

  * •

WMAP7 vs. WMAP1

  * •

$M_{vir}\approx 1.4\times 10^{15}M_{\odot}/h>10^{15}M_{\odot}$

We also note that Bullock et al. (2001) did not simulate halos as massive as
A1689, with their most massive halos weighing in at $M\sim
10^{14}M_{\odot}/h$.

###  1.3. The observed $c(M)$ relation

Based on a compilation of 62 published measurements (including 10 new
measurements) of halo $c_{vir}$ and $M_{vir}$, Comerford & Natarajan (2007)
find the following relation (with a large scatter):

$c_{vir}\simeq\frac{14.5\pm 6.4}{1+z}\left(\frac{M_{vir}}{1.3\times
10^{13}h^{-1}M_{\odot}}\right)^{-0.15\pm 0.13}.$ (7)

For clusters as massive as A1689, Comerford’s relation converges toward that
of Hennawi et al. (2007), the former being only slightly higher.

This and other observed $c(M)$ relations are shown in Fig. 6 and detailed in
Table 6. The Comerford & Natarajan (2007) compilation includes both lensing
and X-ray determinations of $c$ and $M$, including the X-ray samples presented
by Buote et al. (2007) and Schmidt & Allen (2007). Each of these papers
presented their own $c(M)$ relation. A recent $c(M)$ relation from weak
lensing of individual halos was presented by Okabe et al. (2009). And $c(M)$
derived from stacked weak lensing analyses were presented by Johnston et al.
(2007) and Mandelbaum et al. (2008). It seems apparent that one should study a
wide enough range of halo masses to obtain a confident $c(M)$ relation.

##  2\. Overdensity within the Virial Radius

Various conventions are used to define the virial mass and radius. We explain
and show how to convert between different definitions.

###  2.1. Overdensity Definitions

The virial radius $r_{vir}$ designates the edge of the halo. Within this
radius, objects are supposed to be “virialized”: gravitationally bound and
settled into regular orbits. Outside this radius, objects are not in orbit
although they may still be infalling. In practice, there is no sharp dividing
line the two regions. And even if there were, it would be extremely difficult
to discern observationally for a given massive body. Meanwhile, the objects we
study are not always virialized. In fact, galaxy clusters are the largest
bodies which have had time to virialize given the age of the universe. Thus
some of the clusters we observe have virialized just recently, but many are
still in the process of doing so.

Despite these complications, we can define a virial radius for a massive body
based on theory and simulations. Early theoretical work (Peebles, 1980)
predicted that a sphere of material will collapse if its density exceeds
$1.686(1+z)$ times that of the background. After it collapses and virializes,
the sphere will obtain an average density

$\Delta_{c}\approx 18\pi^{2}\approx 178$ (8)

times the critical density $\rho_{crit}(z)$ at that redshift, where

$\rho_{crit}=\frac{3H^{2}(z)}{8\pi G}.$ (9)

Cole & Lacey (1996) cited this as a theoretical result and then confirmed it
in simulations.444Cole & Lacey (1996) spoke of this factor as the overdensity
above $\rho_{m}=\Omega_{m}\rho_{crit}$ rather than above $\rho_{crit}$. But as
$\Omega_{m}=1$ in their simulations, the two densities were equal and thus
interchangeable.

Navarro et al. (1996) adopted the nice round number of $\Delta_{c}=200$, which
has been used commonly ever since to allow for easy comparison between papers.
But the $\Delta_{c}\approx 178$ result was obtained in an Einstein de-Sitter
cosmology of ($\Omega_{m}$, $\Omega_{\Lambda}$) = (1, 0). In the concordance
cosmology ($\Omega_{m}$, $\Omega_{\Lambda}$) = (0.3, 0.7), we find a much
lower value of $\Delta_{c}\approx 100$, as we describe next.

At least three different forms have been given for $\Delta_{c}$ as a function
of cosmology. For a flat universe ($\Omega_{m}+\Omega_{\Lambda}=1$), Bryan &
Norman (1998) give

$\Delta_{c}\approx 18\pi^{2}-82\Omega_{\Lambda}-39\Omega_{\Lambda}^{2}.$ (10)

An approximation to this is given as (Eke et al., 1998)

$\Delta_{c}\approx 178\Omega_{m}^{0.45}.$ (11)

And Nakamura & Suto (1997, their Eq. C19; see also , their Eq. A17) give

$\Delta_{c}\approx 18\pi^{2}(1+0.4093x^{2.7152})\Omega_{m}$ (12)

with $x\equiv(1/\Omega_{m,0}-1)^{1/3}(1+z)^{-1}$ and
$\Omega_{m}(z)=1/(1+x^{3})$. Given the current concordance cosmology with
$\Omega_{m}=0.3$, these different expressions yield
$\Delta_{c}=101.1,103.5,100.3$, respectively for a halo at $z=0$. Or given
$\Omega_{m}=0.25$, $\Delta_{c}=94.2,95.4,93.5$. We note $\Omega_{m}=0.25$ is
in better agreement with the WMAP 7-year value (Komatsu et al., 2010) and
$h=0.742\pm 0.036$ from Riess et al. (2009).

The overdensity is often quoted as a factor $\Delta_{vir}$ above the mean
background density $\rho_{m}=\Omega_{m}\rho_{crit}$:

$\rho_{halo}=\Delta_{c}\rho_{crit}=\Delta_{vir}\rho_{m}$ (13)

With $\Delta_{c}=\Delta_{vir}\Omega_{m}$, $\Delta_{c}=101.1$ corresponds to
$\Delta_{vir}=337$ for $\Omega_{m}=0.3$. This value is cited by e.g., Bullock
et al. (2001) and Graham et al. (2006). Using the Eke et al. (1998) relation
and the Spergel et al. (2003) first-year WMAP value of $\Omega_{m}=0.268$,
Merritt et al. (2006, among others) give $\Delta_{vir}=368$.

To facilitate comparison among current and future investigations, we propose
that a value of $\Delta_{c}=100$ be adopted for present-day halos. This
corresponds to $\Delta_{vir}=333$ given $\Omega_{m}=0.3$, or the nice round
number $\Delta_{vir}=400$ given $\Omega_{m}=0.25$. We also note that for
$\Omega_{m}=0.25$, the Bryan & Norman (1998) expression yields
$\Delta_{c}\approx 94$, $\Delta_{vir}\approx 376$.

While results from simulations are most often reported for present-day halos,
Nature provides us observers with images of clusters as they were in the past.
Thus in the expressions above, we should replace the present day values of
$\Omega_{m,0}$ and $\Omega_{\Lambda,0}$ (here “$0$” subscripts have been added
for clarity) with:

$\Omega_{m}(z)=\frac{1}{1+(\Omega_{\Lambda,0}/\Omega_{m,0})(1+z)^{-3}}$ (14)

and $\Omega_{\Lambda}(z)=1-\Omega_{m}(z)$. For the massive galaxy cluster
A1689 at $z=0.1862$ and adopting $\Omega_{m}=0.3$, the widely used Bryan &
Norman (1998) expression yields $\Delta_{c}=116.6$ and the Nakamura & Suto
(1997) expression yields $\Delta_{c}=115$. The latter was adopted by
Broadhurst et al. (2005a, b, private communication) so we adopt it as well for
consistency in Coe et al. (2010).

In Figs. 13, 14, and 15, we plot $\Omega_{m}(z)$, $\Delta_{c}(z)$, and
$\Delta_{vir}(z)$.

###  2.2. Conversion between overdensity values $\Delta_{c}$

If the mass profile is well described by an NFW profile, then it is
straightforward to convert $c_{vir}$, $r_{vir}$, and $M_{vir}$ between
different conventions of $\Delta_{c}$ (c.f., Fig. 8). Converting from
$c_{200}$ ($\Delta_{c}=200$) to $c_{115}$ ($\Delta_{c}=115$), for example,
simply involves finding that value of $c_{115}$ which yields the same value of
$\delta_{c}$ (Eq. 38) as did $c_{200}$. This can be accomplished by a simple
root finding program, but the relation is very linear as shown in Fig. 9. Here
we provide expressions

$\displaystyle c_{94}$ $\displaystyle\approx$ $\displaystyle
1.328c_{200}+0.272$ (15) $\displaystyle c_{100}$ $\displaystyle\approx$
$\displaystyle 1.298c_{200}+0.246$ (16) $\displaystyle c_{115}$
$\displaystyle\approx$ $\displaystyle 1.232c_{200}+0.189$ (17)

which are accurate to within 0.5% for $2<c_{200}<25$.

These factors may be generalized:

$c_{vir}\approx a\thinspace c_{200}+b$ (18) $\displaystyle a$
$\displaystyle\approx$ $\displaystyle-1.119\log_{10}\Delta_{c}+3.537$ (19)
$\displaystyle b$ $\displaystyle\approx$
$\displaystyle-0.967\log_{10}\Delta_{c}+2.181$ (20)

to yield $c_{vir}$ within 1% for $3<c_{200}$ and $70<\Delta_{c}<140$. These
factors are plotted in Fig. 10.

An alternate expression gives $c_{vir}$ as a function of $c_{200}$ and
$\Delta_{c}$:

$c_{vir}=c_{200}+c_{200}^{0.9}10^{p}$ (21) $p=-(8.683\times
10^{-5})\Delta_{c}^{1.82}$ (22)

to within 1% for $3<c_{200}<35$ and $85<\Delta_{c}<165$ ($z<1$ or so).

See also Hu & Kravtsov (2003, Appendix C).

###  2.3. Virial Mass

Virial mass (the mass within $r_{vir}$) is given by

$M_{vir}=\frac{4}{3}\pi
r_{vir}^{3}\Delta_{c}\rho_{crit}(z)=\frac{r_{vir}^{3}\Delta_{c}H^{2}(z)}{2G}$
(23)

In Fig. 12 we plot this simple relation between $r_{vir}$ and $M_{vir}$.

Quoted values for $M_{200}$ and $r_{200}$ can also be converted to $M_{vir}$
and $r_{vir}$ as a function of $c_{200}$ and $\Delta_{c}$ as plotted in Fig.
11. For a given NFW curve (with fixed $r_{s}$ and $\rho_{s}$), $r_{vir}$
simply scales with $c_{vir}$ since $r_{s}$ stays fixed. So
$r_{vir}/r_{200}=c_{vir}/c_{200}$. As for the $M_{vir}$ conversions, we solved
for those numerically.

##  3\. Mass Profiles

###  3.1. Double Power Laws

In dark matter simulations, galaxy and cluster halos Navarro et al. (1996,
1997) were all shown to have mass density profiles well approximated by the
NFW profile:

$\rho(r)=\frac{\rho_{s}}{(r/r_{s})(1+r/r_{s})^{2}}.$ (24)

This profile behaves as $\rho\propto r^{-1}$ in the core, $\rho\propto r^{-2}$
at $r=r_{s}$, and steepens to $\rho\propto r^{-3}$ in the outskirts. The two
fit parameters $\rho_{s}$ and $r_{s}$ were shown to be related and a function
of halo mass. This “universal” profile is still a good approximation to
today’s simulated halos. However the higher resolution does reveal subtle
differences.

Deviations were sought for using a generalized version of the NFW profile
(Hernquist, 1990, his Eq. 43; see also Zhao, 1996; Wyithe et al., 2001):

$\rho(r)=\frac{2^{(\beta-\gamma)/\alpha}~{}\rho_{s}}{(r/r_{s})^{\gamma}[1+(r/r_{s})^{\alpha}]^{(\beta-\gamma)/\alpha}}.$
(25)

This profile behaves as $\rho\propto r^{-\gamma}$ in the core, and
$\rho\propto r^{-\beta}$ in the outskirts. The rate of transition is governed
by $\alpha$. Where NFW found ($\alpha$, $\beta$, $\gamma$) = (1, 3, 1), Moore
et al. (1999) instead found best fits of (1.5, 3, 1.5). Importantly, the inner
profile was steeper: $\gamma=1.5$, $\rho\propto r^{-1.5}$. There were many
other attempts to accurately resolve and measure this inner slope, including
Diemand et al. (2005) who found $\rho\propto r^{-1.2}$.

The fully generalized form in Equation 25 proves a bit too general with large
degeneracies between the free parameters (Klypin et al., 2001). Thus, in their
efforts to determine the central slope $\gamma$, authors often use one of two
constrained versions of Equation 25, either a “generalized NFW” profile with
($\alpha$, $\beta$, $\gamma$) = (1, 3, $\gamma$):

$\rho(r)=\frac{2^{3-\gamma}\rho_{s}}{(r/r_{s})^{\gamma}[1+(r/r_{s})]^{(3-\gamma)}},$
(26)

or what we might call a “generalized Moore” profile555This latter form is also
often referred to as a “generalized NFW” profile, although strictly speaking
it can only exactly reproduce the Moore profile and not that of NFW. with
($\alpha$, $\beta$, $\gamma$) = ($3-\gamma$, 3, $\gamma$):

$\rho(r)=\frac{2\rho_{s}}{(r/r_{s})^{\gamma}[1+(r/r_{s})^{3-\gamma}]}.$ (27)

Meanwhile, Dehnen & McLaughlin (2005) found
($\alpha$, $\beta$, $\gamma$) = (4/9, 31/9, 7/9):

$\rho(r)=\frac{2\rho_{s}^{6}}{(r/r_{s})^{7/9}[1+(r/r_{s})^{4/9}]^{6}}.$ (28)

and when accounting for anisotropy, the more general
{$\alpha$, $\beta$, $\gamma$} = {$(3-\gamma)/5$, $(18-\gamma)/5$, $\gamma$}:

$\rho(r)=\frac{2\rho_{s}^{6}}{(r/r_{s})^{\gamma}[1+(r/r_{s})^{(3-\gamma)/5}]^{6}}.$
(29)

###  3.2. Continuously Varying Power Laws

The original NFW proponents proposed a new profile which gradually flattens
all the way toward the center (Navarro et al., 2004). This profile was found
(Navarro et al., 2004; Merritt et al., 2005, 2006) to yield better fits to a
wide range of simulated dark matter halos than did the generalized NFW profile
(Eq. 26), which has an equal number (3) of free parameters, including the
central slope. Inner slopes as steep as $\rho(r)\propto r^{-1.2}$ are clearly
ruled out by recent simulations (Navarro et al., 2010).

The new Navarro et al. (2004) fitting form was quickly recognized (Merritt et
al., 2005) as the Sérsic (1968) profile generally applied to fitting the light
distributions of elliptical galaxies. The implications are intriguing: that
the collapse of massive bodies, be they luminous or dark matter, may lead to
similar profiles.

However to be precise, Navarro et al. (2004) fit a Sérsic-like profile to 3-D
density distributions, where the Sérsic profile was fit to 2-D surface density
distributions (of light). Einasto (1965) was first to use such a density law
to describe a 3-D distribution, namely the spatial distribution of old stars
within the Milky Way.

Today we distinguish between the “Einasto” and “Sérsic” mass profiles. The
former is fit to 3-D mass density $\rho(r)$ while the latter is fit to 2-D
projected mass distributions $\Sigma(R)$. Projected and deprojected
approximations to the Einasto and Sérsic profiles, respectively, have also
been derived (see Table 1).

The Sérsic (1968) profile is given by:

$\Sigma(R)=\Sigma_{e}\exp\left\\{-b_{n}\left[\left(\frac{R}{R_{e}}\right)^{1/n}-1\right]\right\\},$
(30)

There are three free parameters: $\Sigma_{e}$, $R_{e}$, and $n$, with $b_{n}$
being a function of $n$ (given below) such that half the mass is contained
within $R_{e}$. Note that the total mass of a Sérsic profile is finite, unlike
that for an NFW profile. A 3-D deprojected approximation is given by Prugniel
& Simien (1997).

The Einasto mass profile is a similar function but of 3-D mass density
$\rho(r)$:

$\rho(r)=\rho_{-2}\exp\left(-\frac{2}{\alpha}\left[\left(\frac{r}{r_{-2}}\right)^{\alpha}-1\right]\right)$
(31)

where $\rho_{-2}$ and $r_{-2}$ are the density and radius at which
$\rho(r)\propto r^{-2}$. The concentration is defined as
$c_{vir}=r_{vir}/r_{-2}$. Navarro et al. (2010) found $\alpha\approx 0.17$ for
galaxy-sized halos in the Aquarius simulation. Gao et al. (2008) concur and
found $\alpha$ increases to $\sim 0.3$ for the most massive clusters in the
Millennium simulation. Duffy et al. (2008) reduce the Einasto profile to two
free parameter by using the “peak height” $\alpha(\nu)$ relation from Gao et
al. (2008). A 2-D projected approximation of Einasto is given by Dhar &
Williams (2010).

Merritt et al. (2005, 2006) experimented with both of these and other fits to
3-D mass density profiles of simulated halos. In the latter paper, they
compared the performance of various formulae fit to 10 simulated halos (6
cluster-sized and 4 galaxy-sized). Their results for 3-parameter fits are
reprinted here in Table 2. The four tested profiles yielded similar results.
Einasto performed a bit better across the board. Prugniel-Simien (the
deprojected Sérsic profile) performed a bit better for clusters. Dehnen-
McLaughlin performed a bit better for galaxies. And the generalized NFW
profile (Eq. 26) was not far behind.

Given the small number of halos tested, and the similarity of the
performances, all of these fitting formulae might still be considered
reasonable choices. However the Einasto profile has become especially popular
(e.g., Hayashi & White, 2008; Gao et al., 2008; Duffy et al., 2008; Navarro et
al., 2010).

Recently a new fitting formula was proposed by Stadel et al. (2009). It yields
superior fits to the two high resolution halos tested (Via Lactea 2 and
GHALO):

$\rho(r)=\rho_{0}e^{-\lambda\left[\ln(1+r/R_{\lambda})\right]^{2}}$ (32)

with $\lambda\simeq 0.1$.

Table 1 Sérsic-like profiles of 2-D and 3-D density 3-D $\rho(r)$ |  | 2-D $\Sigma(R)$
---|---|---
Einasto (1965) | $\underrightarrow{\rm projected}$ | Dhar & Williams (2010)
Prugniel & Simien (1997) | $\underleftarrow{\rm deprojected}$ | Sérsic (1968)
Table 2 3-parameter fits: deviations from halo profiles measured in Merritt et al. (2006, their Table 4) Model | 6 clusters | 4 galaxies
---|---|---
Einasto | 0.028 | 0.026
Prugniel-Simienaadeprojected Sérsic | 0.025 | 0.030
generalized NFW | 0.032 | 0.028
Dehnen-McLaughlin | 0.034 | 0.023

###  3.3. Power Law Behaviors

Power law slopes for the above fitting formulae as a function of $r$ are
plotted in Figs. 16, 17, 18. The slopes were calculated numerically as
$d\ln\rho/d\ln r$. For fun, we note this is equivalent to
$(d\rho/dr)(r/\rho)$. The radii $r$ are given in units of $r_{-2}$ at which
$\rho\propto r^{-2}$.

We see that in principle, we should be able to distinguish between these
various profiles in both observed and simulated halos given sufficient
resolution at a large enough range of radii. Such clear determinations have so
far eluded us.

In Fig. 19, we plot power law slopes for $\rho(r)$, $\kappa(R)$, and $M(<R)$
for the NFW profile.

###  3.4. Profile Details

Here we provide useful expressions derived from the NFW and Sérsic profiles.

####  3.4.1 NFW Profile

Simulated galaxy and cluster halos Navarro et al. (1996, 1997) were shown to
all have mass density profiles well approximated by the NFW profile:

$\rho(r)=\frac{\rho_{s}}{(r/r_{s})(1+r/r_{s})^{2}}.$ (33)

The two fit parameters $\rho_{s}$ and $r_{s}$ were shown to be related and a
function of halo mass, as we discuss below. But the parameter making all the
buzz these days is the central mass concentration:

$c_{vir}=r_{vir}/r_{s},$ (34)

where $r_{vir}$ is the virial radius of the mass halo. As discussed above, the
virial radius is estimated as that which contains an average density
$\Delta_{c}\rho_{crit}$, for a total virial mass of

$M_{vir}=\frac{4}{3}\pi\Delta_{c}\rho_{crit}r_{vir}^{3}.$ (35)

For an NFW halo, the mass within a sphere with radius $r=xr_{s}$ can be found
by simply integrating the NFW profile (Eq. 33):

$\displaystyle M(r)$ $\displaystyle=$ $\displaystyle 4\pi
r_{s}^{3}\int_{0}^{x}dx^{\prime}x^{\prime
2}\frac{\rho_{s}}{x^{\prime}(1+x^{\prime})^{2}}$ (36) $\displaystyle=$
$\displaystyle 4\pi\rho_{s}r_{s}^{3}\left(\ln(1+x)-\frac{x}{1+x}\right)$ (37)

Combining Eqs. 34, 35, and 37, we find that the concentration parameter $c$
can be obtained from the following expression, as given in Navarro et al.
(1996):

$\frac{\rho_{s}}{\rho_{crit}}\equiv\delta_{c}=\frac{\Delta_{c}}{3}~{}\frac{c^{3}}{\ln(1+c)-c/(1+c)}.$
(38)

To fit the NFW profile to our gravitational lensing mass maps which measure
projected surface density, we integrate the NFW profile along the line of
sight (e.g., Golse & Kneib, 2002) to find the projected surface density:

$\Sigma(R)=2\rho_{s}r_{s}F(X)$ (39)

with $R=Xr_{s}$ and

$\displaystyle
F(X)=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{X^{2}-1}\left(1-\frac{1}{\sqrt{1-X^{2}}}\cosh^{-1}\frac{1}{X}\right)&(X<1)\\\
\displaystyle\frac{1}{3}~{}~{}~{}(X=1)\\\
\displaystyle\frac{1}{X^{2}-1}\left(1-\frac{1}{\sqrt{X^{2}-1}}\cos^{-1}\frac{1}{X}\right)&(X>1)\end{array}\right.$
(43)

Integrating once more over the area within $R$, we find the total mass within
a cylinder of radius $R$

$M(R)=4\pi r_{s}^{3}\rho_{s}G(X)$ (44)

with

$\displaystyle
G(X)=\ln\frac{X}{2}+\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{\sqrt{1-X^{2}}}\cosh^{-1}\frac{1}{X}&(X<1)\vspace{0.1in}\\\
\displaystyle 1\vspace{0.1in}&(X=1)\\\
\displaystyle\frac{1}{\sqrt{X^{2}-1}}\cos^{-1}\frac{1}{X}&(X>1)\end{array}\right.$
(48)

This should not be confused with Eq. 37 which gives mass within a sphere of
radius $r$.

From this we can obtain the shear due to an NFW mass profile:
$\gamma(R)=\bar{\kappa}(R)-\kappa(R)$:

$\gamma(R)=2\kappa_{s}\left(\frac{2G(X)}{X^{2}}-F(X)\right).$ (49)

The quantity measured in weak lensing studies is the reduced shear:

$g=\frac{(D_{LS}/D_{S})\gamma}{1-(D_{LS}/D_{S})\kappa},$ (50)

where we have finally given the redshift dependence. (All previous expressions
were given for a fiducial lensed source at $z_{s}=\infty$.)

####  3.4.2 Sérsic Profile

We now give the Sérsic (1968) profile and quantities derived from it (e.g.,
Graham & Driver, 2005; Terzić & Graham, 2005). Note that the Sérsic profile is
commonly used to describe the (projected 2-D) light profiles of elliptical
galaxies. Here instead it will be discussed as describing projected mass
profiles.

The Sérsic (1968) profile is given by:

$\Sigma(R)=\Sigma_{e}\exp\left\\{-b_{n}\left[\left(\frac{R}{R_{e}}\right)^{1/n}-1\right]\right\\},$
(51)

There are three free parameters: $\Sigma_{e}$, $R_{e}$, and $n$, with $b_{n}$
being a function of $n$ (given below) such that half the mass is contained
within $R_{e}$. (Note that the total mass of a Sérsic profile is finite,
unlike that for an NFW profile.) The total projected mass within a radius $R$
is given as:

$M(R)=2\pi\Sigma_{e}R_{e}^{2}ne^{b_{n}}b_{n}^{-2n}\hat{\gamma}(2n,x)$ (52)

where

$x=b_{n}(R/R_{e})^{1/n},$ (53)

and $\hat{\gamma}(a,x)=\int_{0}^{x}dte^{-t}t^{a-1}$ is the incomplete gamma
function (with the “hat” used to distinguish $\hat{\gamma}$ from the lensing
shear $\gamma$). Thus to satisfy $M(R_{e})=\text@frac{1}{2}M(R=\infty)$,
$b_{n}$ must obey:

$\Gamma(2n)=2\hat{\gamma}(2n,b_{n})$ (54)

where $\Gamma(a)=\hat{\gamma}(a,\infty)$ is the complete gamma function. In
SciPy’s “special” package, we find a routine to quickly calculate $b_{n}={\tt
gammaincinv(2*n,0.5)}$. An approximation may also be used (Prugniel & Simien,
1997):

$b_{n}\approx 2n-1/3+0.009876/n$ (55)

Lensing properties of the Sérsic profile have been derived and explored in
Cardone (2004) and Elíasdóttir & Möller (2007). Of special interest here is
the weak shear $\gamma=\bar{\kappa}-\kappa$. The average $\kappa$ within $R$
can be derived straightforwardly from the above expression for $M(R)$:

$\bar{\kappa}(R)=\frac{M(R)}{\pi
R^{2}\Sigma_{crit}}=2\kappa_{e}ne^{b_{n}}x^{-2n}\hat{\gamma}(2n,x)$ (56)

where we have introduced $\kappa_{e}=\Sigma_{e}/\Sigma_{crit}$. Meanwhile,
$\kappa(R)=\Sigma(R)/\Sigma_{crit}$ can be rewritten as:

$\kappa(R)=\kappa_{e}e^{(b_{n}-x)}$ (57)

Thus we find $\gamma(R)=\bar{\kappa}(R)-\kappa(R)$:

$\gamma(R)=\kappa_{e}e^{b_{n}}\left(2nx^{-2n}\hat{\gamma}(2n,x)-e^{-x}\right),$
(58)

with the reduced shear given as $g=(\gamma D_{LS}/D_{S})/(1-\kappa
D_{LS}/D_{S})$.

There are fewer published fits of Sérsic profiles to simulated cluster halos.
We do note that Merritt et al. (2005) found $n=2.38\pm 0.25$ for their cluster
sample.

We thank Angelo Neto for useful conversations about the Millennium simulation
and their study of halo profiles. This work was carried out at Jet Propulsion
Laboratory, California Institute of Technology, under a contract with NASA.

Figure 1.— For fixed $\mathbf{M_{200}=10^{15}M_{\odot}}$, we show what an
“over-concentrated” looks like: $c_{200}=9$ versus the expected concentration
($c_{200}=3$). This is roughly the case for A1689 (Coe et al., 2010). Dots
mark $r_{s}$ and dashed lines mark $r_{vir}$ and $M_{vir}$. Thicker lines are
used for two decades in radius about $r_{s}$ to emphasize the shifting of the
curves.

Figure 2.— Expected NFW concentration $c_{200}$ and 1-$\sigma$ scatter as a
function of halo mass $M_{200}$ for all clusters (left) and relaxed clusters
(right).

Figure 3.— Expected NFW concentration $c(M)$ for $z=0$ halos and $c(M,z)$ for
$z=0,1,2$ halos from Duffy et al. (2008).

Figure 4.— Expected NFW concentrations as a function of redshift for various
mass halos from Duffy et al. (2008) (see Table 5). Also plotted is the
$c\propto(1+z)^{-1}$ slope expected from Bullock et al. (2001). The $x$ axes
are plotted on scales of $\log(1+z)$.

Figure 5.— The effect of cosmology. Left: the Millennium simulation (Neto07)
used WMAP1 with a higher $\sigma_{8}$ resulting in higher concentrations.
Right: 3 different simulations with different cosmologies used.

Figure 6.— Comparing several studies, including both simulated and observed NFW $c(M)$.  Figure 7.— Ratio of NFW $c_{200}$ for relaxed vs. all $z=0$ halos in various simulations.  Figure 8.— Expected concentrations derived from Einasto profiles compared to NFW profiles. Plotted are relations for relaxed halos from Duffy et al. (2008). The yellow NFW lines were plotted in Fig. 3.  Table 3 NFW $c_{200}(M_{200})$ fit parameters ($z=0$): $c_{200}=c_{0}(M_{200}/M_{0})^{-\alpha}\pm\Delta c_{200}$ Sample |  | Cosmology | $c_{0}$ | $M_{0}[M_{\odot}/h]$ | $\alpha$ | $\Delta\log_{10}c_{200}$
---|---|---|---|---|---|---
Duffy08 | all | WMAP5 | 5.74 | $2\times 10^{12}$ | 0.097 | 0.15
Duffy08 | relaxed | WMAP5 | 6.67 | $2\times 10^{12}$ | 0.092 | 0.15
Maccio08 | all | WMAP5 | 6.12 | $10^{12}$ | 0.110 | 0.130
Maccio08 | relaxed | WMAP5 | 6.76 | $10^{12}$ | 0.098 | 0.105
Maccio08 | all | WMAP3 | 5.24 | $10^{12}$ | 0.088 | 0.132
Maccio08 | relaxed | WMAP3 | 5.87 | $10^{12}$ | 0.083 | 0.109
Maccio08 | all | WMAP1 | 7.57 | $10^{12}$ | 0.119 | 0.129
Maccio08 | relaxed | WMAP1 | 8.26 | $10^{12}$ | 0.104 | 0.111
Neto07 | all | WMAP1 | 4.67 | $10^{14}$ | 0.11 | 0.094
Neto07 | relaxed | WMAP1 | 5.26 | $10^{14}$ | 0.10 | 0.061
Table 4 NFW $c_{vir}(M_{vir})$ fit parameters ($z=0$): $c_{vir}=c_{0}(M_{vir}/M_{0})^{-\alpha}\pm\Delta c_{vir}$ Sample |  | Cosmology | $c_{0}$ | $M_{0}[M_{\odot}/h]$ | $\alpha$ | $\Delta\log_{10}c_{vir}$
---|---|---|---|---|---|---
Duffy08 | all | WMAP5 | $7.96\pm 0.17$ | $2\times 10^{12}$ | $0.091\pm 0.007$ | $\cdots$
Duffy08 | relaxed | WMAP5 | $9.23\pm 0.15$ | $2\times 10^{12}$ | $0.089^{+0.010}_{-0.013}$ | $\cdots$
Maccio08 | all | WMAP5 | 8.41 | $10^{12}$ | 0.108 | $\cdots$
Maccio08 | relaxed | WMAP5 | 9.35 | $10^{12}$ | 0.094 | $\cdots$
Maccio08 | all | WMAP3 | 7.26 | $10^{12}$ | 0.086 | $\cdots$
Maccio08 | relaxed | WMAP3 | 8.22 | $10^{12}$ | 0.080 | $\cdots$
Maccio08 | all | WMAP1 | 10.26 | $10^{12}$ | 0.114 | $\cdots$
Maccio08 | relaxed | WMAP1 | 11.25 | $10^{12}$ | 0.099 | $\cdots$
Hennawi07 | all | WMAP1 | 12.3 | $1.3\times 10^{13}$ | 0.13 | 0.098
Gentile07 | all | WMAP3 | 13.6 | $10^{11}$ | 0.13 | $\cdots$
Bullock01 | all | WMAP1 | 9 | $1.3\times 10^{13}$ | 0.13 | 0.14aaWechsler et al. (2002, footnote 10) claim that the scatter of $\Delta\log_{10}c_{vir}=0.18$ reported by Bullock et al. (2001) was a bit too high and should actually be 0.14, thus bringing it in line with their own measured scatter.
Comerford07 | all | observed | $14.5\pm 6.4$ | $1.3\times 10^{13}$ | $0.15\pm 0.13$ | 0.15
Table 5 NFW $c(M,z)$ fit parameters for $0<z<2$: $c=c_{0}(M/M_{0})^{-\alpha}(1+z)^{-\beta}$ Sample |  | Cosmology | $\Delta$ | $c_{0}$ | $M_{0}[M_{\odot}/h]$ | $\alpha$ | $\beta$
---|---|---|---|---|---|---|---
Duffy08 | all | WMAP5 | 200 | $5.71\pm 0.12$ | $2\times 10^{12}$ | $0.084\pm 0.006$ | $0.47\pm 0.04$
Duffy08 | relaxed | WMAP5 | 200 | $6.71\pm 0.12$ | $2\times 10^{12}$ | $0.091\pm 0.009$ | $0.44\pm 0.05$
Duffy08 | all | WMAP5 | vir | $7.85^{+0.17}_{-0.18}$ | $2\times 10^{12}$ | $0.081\pm 0.006$ | $0.71\pm 0.04$
Duffy08 | relaxed | WMAP5 | vir | $9.23^{+0.17}_{-0.16}$ | $2\times 10^{12}$ | $0.090\pm 0.009$ | $0.69\pm 0.05$
Table 6 Observed $c(M)$ fit parameters ($z=0$):
$c=c_{0}(M/M_{0})^{-\alpha}(1+z)^{-\beta}$

|  |  |  | $M_{0}$ |  |  |  | $M_{\rm min}$ | $M_{\rm max}$ |
---|---|---|---|---|---|---|---|---|---|---
Sample | Analysis | $\Delta_{c}$ | $c_{0}$ | [$h^{-1}M_{\odot}$] | $\alpha$ | $\Delta\log_{10}c$ | N | [$h^{-1}M_{\odot}$] | [$h^{-1}M_{\odot}$] | $z$
Comerford07bbIncludes Buote07 and SchmidtAllen07 | compilation | vir | $14.5\pm 6.4$ | $1.3\times 10^{13}$ | $0.15\pm 0.13$ | 0.15 | 62 | $5\times 10^{13}$ | $4\times 10^{15}$ | $0.003-0.89$
Buote07 | X-ray | vir | $9.0\pm 0.4$ | $10^{14}$ | $0.172\pm 0.026$ |  $\cdots$ | 39 relaxed | $6\times 10^{12}$ | $2\times 10^{15}$ | $0.0033-0.2302$
SchmidtAllen07 | X-ray | vir |  $7.55\pm 0.90$ccQuoted uncertainties are 95% rather than 1-$\sigma$ | $8\times 10^{14}$ |  $0.45\pm 0.12$ccQuoted uncertainties are 95% rather than 1-$\sigma$ |  $\cdots$ | 34 relaxed | $2\times 10^{14}$ | $4\times 10^{15}$ |  $0.06-0.$7
Okabe09 | WL | vir | $8.45^{+3.91}_{-2.80}$ | $10^{14}$ | $0.41\pm 0.19$ | 0.19 | 30 | $2\times 10^{14}$ | $1.5\times 10^{15}$ | $0.15-0.30$
Johnston07 | stacked WL | 200 | $4.1\pm 1.2$ | $1.3\times 10^{13}$ | $0.12\pm 0.04$ |  $\cdots$ | 130,000 | $5\times 10^{12}$ | $5\times 10^{14}$ | $\sim 0.25$
Mandelbaum08ddMandelbaum08 used $\Delta_{vir}=200$ (not $\Delta_{c}=200$) and assumed $\Omega_{m}=0.27$, corresponding to $\Delta_{c}=\Omega_{m}\Delta_{vir}=54$. | stacked WL | 54 | $4.6\pm 0.7$ | $10^{14}$ | $0.13\pm 0.07$ |  $\cdots$ | 222,699 | $3\times 10^{13}$ | $6\times 10^{14}$ | $\sim 0.22$
Mandelbaum08eeConverted from previous line | stacked WL | 200 | $2.5\pm 0.4$ | $6\times 10^{13}$ | $0.13\pm 0.07$ |  $\cdots$ | 222,699 | $3\times 10^{13}$ | $6\times 10^{14}$ | $\sim 0.22$

Note. — SchmidtAllen07 find $\beta=0.71\pm 0.52$ccfootnotemark: , but all
others fix $\beta=1$ while fitting only $(c_{0},\alpha)$.

Table 7 Simulations  Simulation | Cosmology | $M_{\rm min}$ | $M_{\rm max}$ | N | N | particles
---|---|---|---|---|---|---
| (see Table 8) | [$h^{-1}M_{\odot}$] | [$h^{-1}M_{\odot}$] | all | relaxed | within $r_{vir}$
Duffy08 | WMAP5 | $10^{11}$ | $10^{15}$ | 1,269 | 561 | 10,000
Maccio08 | WMAP5 | $10^{10}$ | $10^{15}$ | 9,988 | 7,060 | 500
Neto07 (Millennium) | WMAP1 | $10^{12}$ | $10^{15}$ | 53,626 | 39,330 | 10,000
Hennawi07 | WMAP1 | $10^{14}$ | $10^{15}$ | 878 |  $\cdots$ |  $\cdots$
Bullock01 | WMAP1 | $10^{11}$ | $10^{14}$ | $\sim 5,000$ |  $\cdots$ | 150–120,000
Gentile07 (NFW96) | WMAP3 | $3\times 10^{11}$ | $3\times 10^{15}$ | 19 |  $\cdots$ | 5,000–10,000
Table 8 Cosmological parameters Author | WMAP | $\Omega_{m}$ | $\sigma_{8}$
---|---|---|---
Bullock01 | WMAP1 | 0.3 | 1.0
Hennawi07 | WMAP1 | 0.3 | 0.95
Neto07: Millennium | WMAP1 | 0.25 | 0.90
Maccio08 | WMAP1 | 0.268 | 0.90
Maccio08 | WMAP3 | 0.238 | 0.75
Maccio08, Duffy08 | WMAP5 | 0.258 | 0.796
$\cdots$ | WMAP7 | 0.26 | 0.803
Figure 8.— For a fixed NFW curve $(r_{s},\rho_{s})$, illustration of
conversion of $(c,M)$ between different values of $\Delta_{c}$. From Fig. 11,
we find $c_{100}\approx 1.37c_{200}$ and $M_{100}\approx 1.3M_{200}$. The two
(perfectly overlapping) black curves have identical $(r_{s},\rho_{s})$. Dots
mark $r_{s}$, while the dashed lines mark the overdensities and $r_{vir}$ for
$\Delta_{c}=200$ and 100. Note that for fixed $c$, $r_{s}$ and $r_{vir}$ vary
with $M_{vir}$ (black vs. yellow).  Figure 9.— Conversion from NFW $c_{200}$
to $c_{vir}$ for $\Delta_{c}=94,100,120,140,160,180,200$. The relations are
extremely linear for $c_{200}>2$.

Figure 10.— Parameters $a$ and $b$ for conversion of NFW $c_{vir}\approx
a\thinspace c_{200}+b$. Dashed lines are the relations given in Eqs. 19 & 20.

Figure 11.— Ratios of $r_{vir}$ (left) and $M_{vir}$ (right) to the
$\Delta_{c}=200$ values as a function of NFW ($c$, $\Delta_{c}$). Note that
$r_{vir}/r_{200}=c_{vir}/c_{200}$ since we hold $r_{s}$ (and $\rho_{s}$) fixed
and we define $r_{vir}=c_{vir}r_{s}$ and $r_{200}=c_{200}r_{s}$.

Figure 12.— The relation between $r_{vir}$ and $M_{vir}$ is independent of the
assumed profile, only dependent on the chosen overdensity, cosmology, and halo
redshift (Eq. 23). For a $z=0$ halo with $r_{vir}=1~{}{\rm Mpc}/h$,
$M_{vir}=1.16\times 10^{14}M_{\odot}/h~{}(\Delta_{c}/100)$.

Figure 13.— $\Omega_{m}(z)=1-\Omega_{\Lambda}(z)$ in a flat universe with a
cosmological constant

Figure 14.— $\Delta_{c}(z)$: The average overdensity above $\rho_{crit}$
within a collapsed halo

Figure 15.— $\Delta_{vir}(z)$: The average overdensity above
$\rho_{m}=\Omega_{m}\rho_{crit}$ within a collapsed halo  Figure 16.— Power
law slopes $\gamma^{\prime}$ for 3-D density $\rho\propto
r^{-\gamma^{\prime}}$. Here we compare the NFW, Einasto, and Sérsic profiles.
For Sérsic we use the Prugniel-Simien approximation. A realistic range of
$\alpha$ and $n$ are plotted for the profiles, as found in simulations
(Merritt et al., 2006; Gao et al., 2008; Navarro et al., 2010).  Figure 17.—
Power law slopes $\gamma^{\prime}$ for 3-D density $\rho\propto
r^{-\gamma^{\prime}}$. In this figure we add the Dehnen-McLaughlin and Stadel-
Moore profiles with $\gamma=7/9$ and $\lambda=0.10$, respectively.  Figure
18.— Power law slopes $\gamma^{\prime}$ for 3-D density $\rho\propto
r^{-\gamma^{\prime}}$. Here we compare various values of $\gamma$ for Dehnen-
McLaughlin as observed in simulations (Merritt et al., 2006). The value marked
with a * (7/9) is used in their 2-parameter fitting formula.  Figure 19.— NFW
power law slopes for projected mass within a cylinder $M(<R)$, projected mass
density $\kappa(R)$, and 3-D density $\rho(r)$.

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