Measuring the escape velocity and mass profiles of galaxy clusters beyond their virial radius
ABSTRACT The caustic technique uses galaxy redshifts alone to measure the escape velocity and mass profiles of galaxy clusters to clustrocentric distances well beyond the virial radius, where dynamical equilibrium does not necessarily hold. We provide a detailed description of this technique and analyse its possible systematic errors. We apply the caustic technique to clusters with mass M_200>=10^{14}h^{1} M_sun extracted from a cosmological hydrodynamic simulation of a LambdaCDM universe. With a few tens of redshifts per squared comoving megaparsec within the cluster, the caustic technique, on average, recovers the profile of the escape velocity from the cluster with better than 10 percent accuracy up to r~4 r_200. The caustic technique also recovers the mass profile with better than 10 percent accuracy in the range (0.64) r_200, but it overestimates the mass up to 70 percent at smaller radii. This overestimate is a consequence of neglecting the radial dependence of the filling function F_beta(r). The 1sigma uncertainty on individual escape velocity profiles increases from ~20 to ~50 percent when the radius increases from r~0.1 r_200 to ~4 r_200. Individual mass profiles have 1sigma uncertainty between 40 and 80 percent within the radial range (0.64) r_200. We show that the amplitude of these uncertainties is completely due to the assumption of spherical symmetry, which is difficult to drop. Alternatively, we can apply the technique to synthetic clusters obtained by stacking individual clusters: in this case, the 1sigma uncertainty on the escape velocity profile is smaller than 20 percent out to 4 r_200. The caustic technique thus provides reliable average profiles which extend to regions difficult or impossible to probe with other techniques. Comment: MNRAS accepted, 20 pages
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Article: Principles of Physical Cosmology
American Journal of Physics 03/1994; 62:381381. · 0.78 Impact Factor  Technometrics 03/2012; 29(4). · 1.42 Impact Factor
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ABSTRACT: We analyse a catalogue of simulated clusters within the theoretical framework of the Spherical Collapse Model (SCM), and demonstrate that the relation between the infall velocity of member galaxies and the cluster matter overdensity can be used to estimate the mass profile of clusters, even though we do not know the full dynamics of all the member galaxies. In fact, we are able to identify a limited subset of member galaxies, the 'fair galaxies', which are suitable for this purpose. The fair galaxies are identified within a particular region of the galaxy distribution in the redshift (lineofsight velocity versus skyplane distance from the cluster centre). This 'fair region' is unambiguously defined through statistical and geometrical assumptions based on the SCM. These results are used to develop a new technique for estimating the mass profiles of observed clusters and subsequently their masses. We tested our technique on a sample of simulated clusters; the mass profiles estimates are proved to be efficient from 1 up to 7 virialization radii, within a typical uncertainty factor of 1.5, for more than 90 per cent of the clusters considered. Moreover, as an example, we used our technique to estimate the mass profiles and the masses of some observed clusters of the Cluster Infall Regions in the Sloan Digital Sky Survey catalogue. The technique is shown to be reliable also when it is applied to sparse populated clusters. These characteristics make our technique suitable to be used in clusters of large observational catalogues. Comment: 11 pages, 11 figures, 5 tables  Slightly revised to match the version published on MNRAS; abstract updatedMonthly Notices of the Royal Astronomical Society 10/2009; · 5.52 Impact Factor
Page 1
arXiv:1011.0372v1 [astroph.CO] 1 Nov 2010
Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed 2 November 2010(MN LATEX style file v2.2)
Measuring the escape velocity and mass profiles of galaxy
clusters beyond their virial radius
Ana Laura Serra1,2⋆, Antonaldo Diaferio1,2,3, Giuseppe Murante4
& Stefano Borgani5,6,7
1Dipartimento di Fisica Generale “Amedeo Avogadro”, Universit` a degli Studi di Torino, Via P. Giuria 1, I10125, Torino, Italy
2Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Torino, Italy
3HarvardSmithsonian Center for Astrophysics, MS20, 60 Garden St., Cambridge, MA 02138, USA
4INAF, Osservatorio Astronomico di Torino, Torino, Italy
5Dipartimento di Astronomia, Universit` a di Trieste, Trieste, Italy
6INAF, Osservatorio Astronomico di Trieste, Trieste, Italy
7Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste, Trieste, Italy
2 November 2010
ABSTRACT
The caustic technique uses galaxy redshifts alone to measure the escape velocity and
mass profiles of galaxy clusters to clustrocentric distances well beyond the virial ra
dius, where dynamical equilibrium does not necessarily hold. We provide a detailed
description of this technique and analyse its possible systematic errors. We apply the
caustic technique to clusters with mass M200? 1014h−1M⊙extracted from a cosmo
logical hydrodynamic simulation of a ΛCDM universe. With a few tens of redshifts
per squared comoving megaparsec within the cluster, the caustic technique, on aver
age, recovers the profile of the escape velocity from the cluster with better than 10
percent accuracy up to r ∼ 4r200. The caustic technique also recovers the mass profile
with better than 10 percent accuracy in the range (0.6 − 4)r200, but it overestimates
the mass up to 70 percent at smaller radii. This overestimate is a consequence of ne
glecting the radial dependence of the filling function Fβ(r). The 1σ uncertainty on
individual escape velocity profiles increases from ∼ 20 to ∼ 50 percent when the radius
increases from r ∼ 0.1r200to ∼ 4r200. Individual mass profiles have 1σ uncertainty
between 40 and 80 percent within the radial range (0.6 − 4)r200. When the correct
virial mass is known, the 1σ uncertainty reduces to a constant 50 percent on the
same radial range. We show that the amplitude of these uncertainties is completely
due to the assumption of spherical symmetry, which is difficult to drop. Other poten
tial refinements of the technique are not crucial. We conclude that, when applied to
individual clusters, the caustic technique generally provides accurate escape velocity
and mass profiles, although, in some cases, the deviation from the real profile can be
substantial. Alternatively, we can apply the technique to synthetic clusters obtained
by stacking individual clusters: in this case, the 1σ uncertainty on the escape velocity
profile is smaller than 20 percent out to 4r200. The caustic technique thus provides
reliable average profiles which extend to regions difficult or impossible to probe with
other techniques.
Key words: gravitation – galaxies: clusters: general – techniques: miscellaneous –
cosmology: miscellaneous – cosmology: dark matter – cosmology: largescale structure
of Universe
⋆Email: serra@ph.unito.it
1INTRODUCTION
Clusters of galaxies are valuable tools to measure the cosmo
logical parameters and test structure formation models (e.g.
Voit 2005; Diaferio et al. 2008) and the galaxyenvironment
connection (e.g. Skibba et al. 2009; HuertasCompany et al.
Page 2
2
Serra et al.
2009; Mart´ ınez et al. 2008; Dom´ ınguez et al. 2001). The
evolution of the cluster abundance is a sensitive probe of
the cosmological parameters because clusters populate the
exponential tail of the mass function of virialised galaxy sys
tems. Accurate mass measurements are however required to
avoid the propagation of systematic errors into the estima
tion of the cosmological parameters. There are two families
of mass estimators: those which estimate the mass profiles
and those that measure the mass enclosed within a specific
projected radius.
Traditionally, the estimation of the cluster mass is based
on the assumptions of spherical symmetry and dynamical
equilibrium: either the cluster galaxies move accordingly to
the virial theorem (Zwicky 1937), or the hot intracluster
plasma which emits in the Xray band is in hydrostatic equi
librium within the gravitational potential well of the clus
ter (Sarazin 1988). More sophisticated approaches apply the
Jeans equation for a steadystate system, with the velocity
anisotropy parameter β as a further unknown (The & White
1986; Merritt 1987; see Biviano 2006 for a comprehensive re
view of various improvements of this method).
Dynamical equilibrium is also assumed when using the
massXray temperature relation to estimate the cluster
mass (e.g. Pierpaoli et al. 2003; see also Borgani 2006).
Both the slope and the normalization of the observed mass–
temperature relation is grossly reproduced by synthetic
clusters obtained with Nbody/hydrodynamical simulations
(Borgani et al. 2004). However, the presence of non–thermal
pressure support (e.g., related to turbulent gas motions) and
the complex thermal structure of the intracluster medium
can significantly bias cluster mass estimates based on the
application of hydrostatic equilibrium to the X–ray esti
mated temperature (e.g. Rasia et al. 2006; Nagai et al. 2007;
Piffaretti & Valdarnini 2008). In principle, one can improve
the mass estimate by exploiting the integrated Sunyaev
Zel’dovich effect, which depends on the first power of the gas
density, rather than the square of the density as in the Xray
emission, and yields a correlation with mass which is tighter
than the massXray temperature relation (e.g. Motl et al.
2005; Nagai 2006). Only recently, such results have been
confirmed by observations of real clusters (Rines et al. 2010;
Andersson et al. 2010). However, their confirmation over a
large statistical ensemble has to await the results from ongo
ing large SunyaevZel’dovich surveys. Alternatively, one can
bypass the problematic gas physics and use the correlation
between mass and optical richness, which is relatively easy
to obtain from observations in the optical/nearIR bands,
but returns mass with a poorer accuracy (Andreon & Hurn
2010).
The dynamical equilibrium assumption can be dropped
in the gravitational lensing techniques, because the lensing
effect depends only on the amount of mass along the line of
sight and not on its dynamical state (e.g. Schneider 2006).
It is relevant to emphasize that all these methods do not
measure the cluster mass on the same scale: optical obser
vations measure the mass within ∼ r200, where r200 is the
radius within which the average mass density is 200 times
the critical density of the universe; Xray estimates rarely
go beyond ∼ 0.5r200, where the Xray surface brightness be
comes smaller than the Xray telescope sensitivity; lensing
measures the central mass within ∼ 0.1r200 or the outer re
gions at radii larger than ∼ r200 depending on whether the
strong or weak regime applies. Scaling relations do not pro
vide any information on the mass profile and give the total
mass within a radius depending on the scaling relation used,
but still within ∼ r200.
Diaferio & Geller (1997) (DG97, hereafter) suggested a
novel method, the caustic technique, to estimate the mass
from the central region out to well beyond r200 with galaxy
celestial coordinates and redshifts alone and without as
suming dynamical equilibrium (Diaferio 2009). Prompted by
the Nbody simulations of van Haarlem & van de Weygaert
(1993), DG97 noticed that in hierarchical models of struc
ture formation, the velocity field in the cluster outskirts is
not perfectly radial, as expected in the spherical infall model
(Reg¨ os & Geller 1989; Hiotelis 2001) but has a substantial
random component. They thus suggested to exploit this fact
to extract the galaxy escape velocities as a function of ra
dius from the distribution of galaxies in redshift space. In the
caustic method, the velocity anisotropy parameter β and the
mass density profile in hierarchical clustering models com
bine in such a way that their knowledge is largely unneces
sary in estimating the mass profile. This property explains
the power of the method.
This method is particularly relevant because it is an
alternative to lensing to measure mass in the cluster external
regions and, unlike lensing, it can be applied to clusters at
any redshift, provided there are enough galaxies to sample
the redshift diagram properly. We will see below that a few
tens of redshifts per squared comoving megaparsec within
the cluster are sufficient to apply the technique reliably. This
request might have appeared demanding a decade ago, but
it is perfectly feasible for the large redshift surveys currently
available.
Geller, Diaferio, & Kurtz (1999) were the first to ap
ply the caustic method: they measured the mass profile
of Coma out to 10h−1Mpc from the cluster centre and
were able to demonstrate that the Navarro, Frenk, & White
(1997) (NFW) profile fits well the cluster density profile
out to these very large radii, thus ruling out the isother
mal sphere as a viable cluster model; a few years later, the
failure of the isothermal model was confirmed by the first
analysis based on gravitational lensing applied to Cl 0024
(Kneib et al. 2003). The goodness of the NFW fit out to
5−10h−1Mpc was confirmed by applying the caustic tech
nique to a sample of nine clusters densely sampled in their
outer regions, the Cluster And Infall Region Nearby Survey
(CAIRNS, Rines et al. 2003), and later to a complete sample
of 72 Xray selected clusters with galaxy redshifts extracted
from the Fourth Data Release of the Sloan Digital Sky Sur
vey (Cluster Infall Regions in the Sloan Digital Sky Survey:
CIRS, Rines & Diaferio 2006) and from the Fifth Data Re
lease (Rines & Diaferio 2010). CIRS is currently the largest
sample of clusters whose mass profiles have been measured
out to ∼ 3r200; Rines & Diaferio (2006) were thus able to
obtain a statistically significant estimate of the ratio be
tween the masses within the infall and the virial regions:
they found a value of 2.2 ± 0.2, in agreement with current
models of cluster formation. These analyses have been ex
tended to a sample of groups of galaxies (Rines & Diaferio
2010). Rines, Diaferio, & Natarajan (2007) also used the
CIRS sample to estimate the virial mass function of nearby
clusters and determined cosmological parameters consistent
with WMAP values (Dunkley et al. 2009).
Page 3
Escape velocity and mass in the outskirts of galaxy clusters
3
A good fit with the NFW profile out to ∼ 2r200 was
also found by Biviano & Girardi (2003) who applied the
caustic technique to an ensemble cluster obtained by stack
ing 43 clusters from the Two Degree Galaxy Redshift Sur
vey (2dGFRS, Colless et al. 2001). Here, unlike the previous
analyses, the caustic method was not applied to individual
clusters because the number of galaxies per cluster was rela
tively small. Recently, Lemze et al. (2009) have applied both
the caustic technique and the Jeans analysis to ∼ 500 clus
ter members of A1689. The estimated virial mass from both
methods agrees with previous lensing and Xray analyses.
Thecausticmethoddoes
namical state of the cluster and of its external re
gions: there are therefore estimates of the mass of
unrelaxedsystems, namely
(Reisenegger et al. 2000; Proust et al. 2006), the Fornax
poor cluster, which shows two distinct dynamical compo
nents (Drinkwater, Gregg, & Colless 2001), the A2199 com
plex (Rines et al. 2002), and two clusters, A168 and A1367,
which are undergoing major mergers (Rines et al. 2003).
More recently, Diaferio et al. (2005b) analysed Cl 0024, a
cluster that is likely to have suffered a recent merging event
(Czoske et al. 2002), and showed that the caustic mass and
the lensing mass agree with each other, but disagree with the
Xray mass, which is the only estimate relying on dynamical
equilibrium.
The method was shown to return reliable mass profiles
when applied to synthetic clusters extracted from Nbody
simulations. However, these analyses of the performance of
the method either were done when the operative procedure
of the technique was not yet completely developed (DG97)
or the results of a fully detailed analysis of the systematic er
rors was not provided (Diaferio 1999) (D99, hereafter). The
dense redshift surveys currently available, especially around
clusters, provide ideal data sets where the caustic technique
can now be applied. With this perspective, it is therefore
timely to reconsider the possible systematic errors and bi
ases of this technique, in view of a robust application to the
new data sets.
The purpose of this paper is to provide both a transpar
ent description of the technique, with some improvements to
what is described in D99, and a thorough statistical analy
sis. The caustic technique represents a powerful method to
infer cluster mass profiles and complements other methods
based on Xray, SunyaevZeldovich and lensing observations
which probe different scales of the clusters.
In Section 2 we describe the basic idea of the caustic
technique. Section 3 describes the simulated cluster sam
ple used, and Section 4 describes the implementation of the
technique. In Sections 5.2 and 5.3 we estimate the accuracy
of the escape velocity and mass profiles of galaxy clusters
estimated with the caustic technique. In Section 6 we inves
tigate the systematics due to the choice of the parameters.
We summarize our main results and draw our conclusions
in Section 7.
notrelyonthedy
theShapleysupercluster
2BASICS
In this section, we briefly review the simple physical idea
behind the caustic technique. More details are given in DG97
and D99.
In hierarchical clustering models of structure formation,
clusters form by the aggregation of smaller systems falling
onto the cluster from the surrounding region. The accretion
does not take place purely radially (e.g. White et al. 2010),
therefore galaxies or dark matter particles within the falling
clumps have velocities with a substantial nonradial compo
nent. Specifically, the r.m.s. ?v2? of these velocities is due
to the gravitational potential of the cluster and the groups
where the galaxy resides, and to the tidal fields of the sur
rounding region. When viewed in the redshift diagram, viz.
the plane of the lineofsight (l.o.s.) velocity w.r.t. the clus
ter centre and the clustrocentric radius r, galaxies populate
a region with a characteristic trumpet shape with decreasing
amplitude A with increasing r. This amplitude is related to
?v2?. The breakthrough of DG97 was to identify this ampli
tude with the escape velocity from the cluster region cor
rected for a function depending on the velocity anisotropy
parameter β.
Assuming a spherically symmetric system, the escape
velocity v2
potential originated by the cluster, is a nonincreasing func
tion of r, because gravity is always attractive and dφ/dr > 0.
At any given radius r, we expect that observing a galaxy
with a velocity larger than the escape velocity is unlikely.
Thus, we identify the escape velocity with the maximum ve
locity that can be observed. It follows that the amplitude
A at the projected radius r⊥ measures the average com
ponent along the l.o.s. of the escape velocity at the three
dimensional radius r = r⊥. To determine this average com
ponent of the velocity, we consider the velocity anisotropy
parameter β(r) = 1−(?v2
vr are the longitudinal, azimuthal and radial components
of the velocity v of a galaxy, respectively, and the brack
ets indicate an average over the velocities of the galaxies in
the volume d3r centred on position r. If the cluster rota
tion is negligible, ?v2
component of the velocity, and ?v2
stituting in the definition of β, we obtain ?v2? = ?v2
where
esc(r) = −2φ(r), where φ(r) is the gravitational
θ?+?v2
φ?)/2?v2
r?, where vθ, vφ, and
θ? = ?v2
φ? = ?v2
los?, where vlosis the l.o.s.
r? = ?v2?−2?v2
los?. By sub
los?g(β)
g(β) =3 − 2β(r)
1 − β(r)
By applying this relation to the escape velocity at radius r,
?v2
obtain the fundamental relation between the gravitational
potential φ(r) and the observable caustic amplitude A(r)
− 2φ(r) = A2(r)g(β) ≡ φβ(r)g(β) .
Note that the gravitational potential profile is related to the
caustic amplitude by the function g(β). Therefore, after the
caustic amplitude estimation, β becomes the only unknown
function for the gravitational potential estimation.
The further novel suggestion of DG97 is to use this rela
tion to infer the cluster mass to very large radii. To do so, one
first notices that the mass of an infinitesimal shell can be cast
in the form Gdm = −2φ(r)F(r)dr = A2(r)g(β)F(r)dr
where
.(1)
esc(r)? = −2φ(r), and assuming that A2(r) = ?v2
esc,los?, we
(2)
F(r) = −2πGρ(r)r2
φ(r)
.(3)
Therefore the mass profile is
Page 4
4
Serra et al.
GM(< r) =
?r
0
A2(r)Fβ(r)dr(4)
where Fβ(r) = F(r)g(β).
Equation (4) however only relates the mass profile to
the density profile of a spherical system and a profile can
not be inferred without knowing the other. DG97 solve this
impasse by noticing that in hierarchical clustering scenarios,
F(r) is not a strong function of r. This is easily seen in the
case of the NFW model which is an excellent description
of dark matter density profiles in the hierarchical clustering
scenario:
FNFW(r) =
r2
2(r + rs)2
1
ln(1 + r/rs), (5)
where rs is a scale factor. One expects that Fβ(r) is also
a slowly changing function of r if g(β) is. DG97 and D99
show that this is the case, and here we confirm their result.
The final, somewhat strong, assumption is then to consider
Fβ(r) = Fβ = const altogether and adopt the recipe
?r
0
GM(< r) = Fβ
A2(r)dr .(6)
It is appropriate to emphasize that equations (2) and
(4) are rigorously correct, whereas equation (6) is a heuris
tic recipe to estimate the mass profile. Based on equation
(2) the caustic technique can also estimate a combination
of the gravitational potential profile and velocity anisotropy
parameter β. Below, we show the accuracy of these esti
mates.
3THE SIMULATED CLUSTER SAMPLE
Before analysing the systematic errors of the caustic tech
nique, we need to check how well the physical assumptions
of the method are satisfied. We do so by using Nbody sim
ulations, assuming that these simulations are a faithful rep
resentation of the real universe.
Here we use the cluster sample extracted from the cos
mological Nbody/hydrodynamical simulation described in
Borgani et al. (2004). They simulate a cubic volume, 192
h−1Mpc on a side, of a flat ΛCDM universe, with mat
ter density Ω0 = 0.3, Hubble parameter H0 = 100h km
s−1Mpc−1with h = 0.7, power spectrum normalization
σ8 = 0.8, and baryon density Ωb = 0.02h−2. The den
sity field is sampled with 4803dark matter particles and
an initially equal number of gas particles, with masses
mDM = 4.6 × 109h−1M⊙ and mgas = 6.9 × 108h−1M⊙, re
spectively. The Plummerequivalent gravitational softening
is set to 7.5 h−1physical kpc at z < 2, while being fixed in
comoving units at higher redshift.
Thesimulationwas
(Springel, Yoshida, & White 2001), a massively parallel
Tree+SPH code with fully adaptive time–stepping. Besides
standard hydrodynamics, it also contains a treatment of
radiative gas cooling, star formation in a multiphase inter
stellar medium and feedback from supernovae in the form
of galactic ejecta (e.g., Borgani et al. 2004; Diaferio et al.
2005a). Here we are only interested in the gravitational
dynamics of the matter distribution.
The simulation volume yields a cluster sample large
enough for statistical purposes. We identify clusters in the
runwith
GADGET2
Figure 1. Concentration parameters c vs. the cluster mass
MNFW
200
of our simulated cluster sample.
Figure 2. Profiles of the ratio between the mass profile of each
cluster predicted by the NFW fit and its true mass profile: 90
(50) percent of the profiles are within the upper and lower dotted
(solid) curves. The central solid curve is the median profile. For
each cluster, the NFW fit is only performed to the mass distribu
tion within 1h−1Mpc.
simulation box with a twostep procedure: a friendsof
friends algorithm applied to the dark matter particles alone
provides a list of halos whose centres are used as input to
the spherical overdensity algorithm which outputs the final
list of clusters (Borgani et al. 2004). Centred on the most
bound particle of each cluster, the sphere with virial over
density 200, with respect to the critical density, defines the
virial radius r200. At redshift z = 0 the simulation box con
tains 100 clusters with mass M(< r200) ? 1014h−1M⊙. This
cluster set is our reference sample.
3.1Cluster properties in 3D
We fit the threedimensional (3D) real cumulative mass pro
file with the NFW model
M(< r) = Ms
?
ln
?
1 +r
rs
?
−
r/rs
1 + r/rs
?
(7)
Page 5
Escape velocity and mass in the outskirts of galaxy clusters
5
Figure 3. Profiles of the ratio between the gravitational potential
profile of each cluster predicted by the NFW fit and the numerical
gravitational potential profile derived from the true mass distri
bution within rmax = 10h−1Mpc from the cluster centre: 90
(50) percent of the profiles are within the upper and lower dot
ted (solid) curves. The central solid curve is the median profile.
The lower and upper dashed curves are the median profiles for
rmax= 8 and 6h−1Mpc, respectively.
where Ms = M(< rs)/(ln2 − 1/2) = 4πδcρcrr3
(200/3)c3/[ln(1 + c) − c/(1 + c)], and c = rNFW
the mass profile rather than the density profile to resemble
the procedure adopted with real clusters (e.g. Diaferio et al.
2005b). Similarly, for the fit, we only consider the mass dis
tribution within r = 1h−1Mpc. Moreover, the parameters
Ms and rs are less correlated than δc and rs and adopting
them in the fit provides more robust results (Mahdavi et al.
1999).
From the NFWfit
rNFW
200
and c. The relation between c and MNFW
(4π/3)(rNFW
tersampleprovidesthe percentile
[0.77,0.86,1.17]h−1Mpc and c = [2.06,4.62,6.50].1The ra
dius rNFW
200
is basically identical to the true r200 derived from
the actual mass distribution. Their ratio is rNFW
[0.99,1.00,1.02]. The NFW model is an excellent fit to the
true mass profile (see Figure 2) within r200. Therefore the
ratio between the mass MNFW
200
spondingly close to 1: MNFW
200
/M200 = [0.96,1.00,1.05]. The
NFW fit is on average very good also at radii larger than
r200, although the scatter substantially increases. In the very
centre, the NFW model underestimates the mass by 50 per
cent. This is due to the fact that the total mass includes dark
matter, gas, and stars and in the simulation there is always
a central galaxy which is unrealistically massive: the stellar
particles, on average, contribute 50 (15) percent of the total
mass within 0.02 (0.1) r200. When the NFW fitting routine
tries to accomodate the mass profile within 1h−1Mpc, un
derestimating the mass profile in this very central region
yields the minimum χ2.
s, δc =
200 /rs. We fit
we derivethe parameters
200
=
200 )3200ρcr is shown in Figure 1. Our clus
rangesrNFW
200
=
200 /r200 =
and the actual M200 is corre
1Throughout this paper the notation [q1,q2,q3] shows the me
dian q2 and the range [q1,q3] which contains 90 percent of the
sample.
The caustic amplitude is related to the gravitational
potential φ(r) and we are thus interested in determining
φ(r) in our simulation. For an isolated spherical system with
density profile ρ(r), the potential φ(r) obeying the Poisson
equation can be cast in the form
φ(r) = −4πG
?1
r
?r
0
ρ(x)x2dx +
?∞
r
ρ(x)xdx
?
.(8)
In the real universe, the relevant quantity is the gravita
tional potential originating from the mass density fluctua
tions around the mean density ?ρ?. We can thus use equa
tion (8) for nonisolated clusters by replacing ρ(r) with
ρ(r)−?ρ? ≡ ?ρ?δ(r). The second integral is finite, because at
sufficiently large r, δ(r) ∼ 0. In the simulation, we replace
the upper limit of the second integral with rmax = 10h−1
Mpc. Figure 3 compares φnum(r) computed from the actual
mass distribution around each cluster in the simulation, with
the gravitational potential
φNFW(r) = −GMs
r
ln
?
1 +r
rs
?
(9)
expected from an isolated cluster described by the NFW
model. The NFW potential well is deeper than the numer
ical φnum(r), because it neglects the mass surrounding the
cluster. This mass exerts a pull to the mass within the clus
ter that makes the actual φnum(r) to be 10–30 percent shal
lower. The upper limit rmax of the second integral of equa
tion (8), adopted in the numerical estimate, plays a neg
ligible effect when it is large enough: the median φnum(r)
for rmax = 8h−1Mpc is indistinguishable from the profile
computed with rmax = 10h−1Mpc.
Figure 4 shows the profiles of the velocity anisotropy
parameter β(r) and the other functions defined in Section 2
for our sample of 100 simulated galaxy clusters. This figure
confirms the results of the ΛCDM model presented in Figure
3 of D99 (see also Figure 25 in Diaferio et al. 2001, which
shows the velocity field of simulated galaxies rather than
dark matter particles). Specifically, Figure 4 shows that, on
average, β(r) ? 0.7 at r ? 3r200 and consequently, on these
scales, the median profile of g(β) varies by less than 30 per
cent. Individual clusters may of course have wider variations.
We remind that β(r) is derived from the velocities of
all the particles in the simulations (dark matter, gas and
stars). These velocities are set in the rest frame of each
cluster and are augmented of the Hubble flow contribution
H0r. This term provides an increasing contribution to the
particle velocity which is not negligible at r ? (1 − 2)r200.
Specifically, by assuming dynamical equilibrium and setting
v2
velocity at r, we see that vp decreases with r; therefore,
the total velocity v(r) = vp(r) + H0r reaches a minimum
value when vp(r) ∼ H0r, viz. when r ∼ [GM(< r)/H2
4.01h−1Mpc, where we have set M = 1.49 × 1014h−1M⊙,
the median M200 of our cluster sample. Since the tangential
component of the total velocity is unaffected by the Hubble
flow, the minimum of vp(r), and consequently of ?v2
minimum of β. For our sample the median r200 = 0.86h−1
Mpc, and we have a minimum β at r/r200 = 4.66 which is
in rough agreement with Figure 4.
Finally, Figure 4 shows that F(r) and Fβ(r) are slowly
varying functions of r, as expected. We discuss these profiles
in Section 5.3.
p(r) ∼ GM(< r)/r, where vp(r) is the average peculiar
0]1/3∼
r?, is a
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Serra et al.
Figure 4. Profiles of the functions β(r), g(β), F(r) and Fβ(r) described in the text: 50, 68 and 90 percent of the profiles are within the
upper and lower dotted, solid and dashed curves. The central solid curves are the median profiles.
3.2The mock cluster catalogue
We compile mock redshift catalogues from our 100 simu
lated clusters as follows. We project each cluster along ten
random lines of sight; for each line of sight, we also project
the clusters along two additional lines of sight in order to
have a set of three orthogonal projections for each random
direction. We thus have a total of 100×10×3 = 3000 mock
redshift surveys. We locate the cluster centre at the celes
tial coordinate (α,δ) = (6h,0◦) and redshift cz = 32000
km s−1. We consider a field of view of 12h−1Mpc on a side
at the cluster distance. The simulation box is 192h−1Mpc
on a side, and the mock survey contains a large fraction of
foreground and background largescale structure. We only
consider a random sample of 1000 dark matter particles in
each mock catalogue. Only a fraction of these 1000 parti
cles are within 3r200 of the cluster centre, specifically this
number has percentile range [96,185,408] in our 3000 mock
catalogues. In Sect. 6.4 we investigate the effect of varying
the number of particles in the mock survey.
4THE CAUSTIC TECHNIQUE
The celestial coordinates and redshifts of the galaxies within
the cluster field of view are the input data of the caustic
technique. The technique proceeds along four major steps:
(1) arrange the galaxies in a binary tree according to a
hierarchical method;
(2) select two thresholds to cut the tree twice, at an up
per and a lower level: the largest group obtained from the
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Escape velocity and mass in the outskirts of galaxy clusters
7
upperlevel threshold identifies the cluster candidate mem
bers; the lowerlevel threshold identifies the substructures
of the cluster; the cluster candidate members determine the
cluster centre, its velocity dispersion and size;
(3) with the cluster centre of the candidate members,
build the redshift diagram of all the galaxies in the field; with
the velocity dispersion and size of the candidate members
determine the threshold κ which enters the caustic equation,
locate the caustics, and identify the final cluster members;
(4) the caustic amplitude determines the escape velocity
and mass profiles.
We detail these steps in turn.
4.1Binary tree
Each galaxy is located by the vector ri = (αi,δi,ri), where
αi and δi are the celestial coordinates, and ri is the comov
ing distance to the observer of a source at redshift zi, i.e.
the spatial part of the geodesic travelled by a light signal
from the source to the observer. The comoving radial coor
dinate ri is determined by the relation?ri
?zi
curvature and H2(z) = H2
ΩΛ] is the Hubble constant at redshift z. In the Einsteinde
Sitter universe (κ = 0, Ω0 = 1)
0dx/√1 − κx2=
0cdz/H(z), where κ = (H0/c)2(Ω0+ΩΛ−1) is the space
0[Ω0(1+z)3+(1−Ω0−ΩΛ)(1+z)2+
ri =2c
H0
?
1 −
1
(1 + zi)1/2
?
.(10)
The comoving separation r12 between two sources with co
moving radial coordinates r1 and r2 and angular separation
θ as observed from O is
r2
12
=r2
1+ r2
?
2− κr2
1 − κr2
1r2
?
2(1 + cos2θ) − 2r1r2cosθ×
1 − κr2
×
12.(11)
This equation is a generalization of the cosine law on the
twodimensional surface of a sphere (e. g. Peebles 1993, equa
tion 12.42).
For the construction of the binary tree, we are interested
in defining an appropriate similarity based on the pairwise
separation r12. In general, the rank of the separation r12 of a
number of sources with different redshift zi is not preserved
when we vary Ω0 and ΩΛ. However, a sufficient condition
for having the ranks preserved is that the universe is never
collapsing and κ ? 0. This condition is satisfied when Ω0 ? 1
and Ω0+ ΩΛ ? 1.
Based on this argument, we can compute r12 in the
Einsteinde Sitter universe, and use the ordinary Euclidean
geometry to derive the binary tree similarity. For each galaxy
pair, we define the l.o.s. vector l = (r1 + r2)/2 and the
separation vector r12 = r1 − r2. We then determine the
component of r12 along the line of sight,
π =r12· l
l
=
r2
r1+ r2,
1− r2
2
(12)
and the component perpendicular to the line of sight
rp = (r2
12− π2)1/2.(13)
We now consider the proper l.o.s. velocity difference Π =
π/(1+zl) and the proper projected spatial separation Rp =
rp/(1 + zl) of the galaxy pair at the intermediate redshift
zl that satisfies the relation rl= (r1+ r2)/2. We adopt the
similarity
Eij = −Gmimj
Rp
+1
2
mimj
mi+ mjΠ2.(14)
This similarity reduces to the one adopted by D99 at small
z. In equation (14), the galaxy mass is a free parameter.
D99 assumes mi = mj = 1012h−1M⊙, as we do here.
Varying the galaxy mass changes the details of the binary
tree, because one differently weights Π and Rp. However, in
the range (1011,1013)h−1M⊙, the centre location and the
galaxy membership remain substantially unchanged. One
can of course include different galaxy masses proportional
to the galaxy luminosities. This prescription turns out to be
unnecessary, because it would not increase the accuracy of
the determination of the mass and escape velocity profiles.
In fact, we will see below (Section 6.3) that projection ef
fects completely dominate the uncertainties of the caustic
method.
For the sake of completeness, we note that, in a non
flat universe, with Ω0 < 1 and Ω0 + ΩΛ < 1, one can still
perform the component separation with equations (12) and
(13) on a flat space tangent to the observer’s location point
O, provided one defines an appropriate biunique function
between a geodesic distance ri and a vector ri on the flat
space.
The binary tree is built as follows: (i) each galaxy can
be thought as a group gα; (ii) to each group pair gα,gβ it is
associated the binding energy Eαβ = min{Eij}, where Eij
is the binding energy between the galaxy i ∈ gα and the
galaxy j ∈ gβ; (iii) the two groups with the smallest binding
energy Eαβare replaced with a single group gγ and the total
number of groups is decreased by one; (iv) the procedure is
repeated from step (ii) until we are left with only one group.
Figure 5 shows the binary tree of a simulated cluster as an
example.
4.2Cutting the binary tree: The σ plateau
Gravity is a longrange force and defining a system of galax
ies is somewhat arbitrary. The most widely used approach is
to select a system according to a density threshold, either in
redshift space, for real data, or in real space, for simulated
data.
When applied to real data, the binary tree described in
the previous section has the advantage of building a hierar
chy of galaxy pairs with increasing, albeit projected, binding
energy. The tree automatically arranges the galaxies in po
tentially distinct groups based on a single parameter, the
galaxy mass. To get effectively distinct groups, however, we
need to choose a threshold, i.e. the node from which the
group members hang, where we cut the tree. This choice has
been traditionally arbitrary (Materne 1978; Serna & Gerbal
1996).
D99 suggests a more objective criterion based on the fol
lowing argument. We first identify the main branch of the bi
nary tree, that is the branch that links the nodes from which,
at each level of the tree, the largest number of leaves hang.
The velocity dispersion σx of the galaxies hanging from a
given node x shows a characteristic behaviour when walking
towards the leaves along the main branch (Figure 6): ini
tially it decreases rapidly, it then reaches a long plateau and
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Serra et al.
Figure 5. Dendrogram representation of the binary tree of a subsample of the particles in the field of a simulated cluster. The particles
are the leaves of the tree at the bottom of the plot. The particles within 3r200 in real space are highlighted in black. The thick path
highlights the main branch. The horizontal lines show the upper and lower thresholds used to cut the tree. Only as a guide, some nodes
are labelled on the lefthand side, with their number of descendants in brackets.
again drops rapidly towards the end of the walk. The plateau
is a clear indication of the presence of the nearly isothermal
cluster: at the beginning of the walk, σx is large because of
the presence of background and foreground galaxies; at the
end of the walk the tree splits into different substructures
and σx drops again.
The two nodes x1 and x2 that limit the σ plateau are
good candidates for the cluster/substructure identification.
To locate x1 and x2, we proceed as follows. We consider
the density distribution of the σx. The mode of this density
distribution is σpl. We then identify the Nδ nodes whose
velocity dispersion σx fulfills the inequality
σpl− σx
σpl
? δ . (15)
Clearly, the number of nodes Nδdepends on the param
eter δ. We limit the parameter δ in the range [0.03,0.1], but
we also compute the number of nodes N0.3 when δ = 0.3.
We determine the number of nodes Nδ corresponding to in
creasing values of δ ∈ [0.03,0.1] by step of 0.01 until Nδ is
larger than 0.8N0.3. This sets the final number of nodes Nδ.
If all the Nδ with δ ∈ [0.03,0.1] are smaller than 0.8N0.3,
we choose 0.8N0.3 as the final number of nodes. The up
per limit of the range of δ, δ = 0.1, is chosen because it
always provides a sufficiently large number of nodes (15 in
our sample); the arbitrary threshold 0.8N0.3 is chosen be
cause it enables to deal efficiently with particularly peaked
density distributions of σx. Among the Nδ nodes, we locate
the five nodes closest to the root and the five nodes closest
to the leaves; among the former set, the final node x1 has σx
with the smallest discrepancy from σpl, and similarly for the
final node x2among the five nodes closest to the leaves. This
procedure, illustrated in Figure 7, guarantees a sufficiently
large number of nodes along the σ plateau and prevents us
from locating the extrema of the σ plateau on nodes whose
σx are too discrepant from σpl.
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Escape velocity and mass in the outskirts of galaxy clusters
9
Figure 7. Lineofsight velocity dispersion of the leaves of each node along the main branch of the binary tree shown in Figure 5. The
filled red circles are the Nδnodes. The dashed line indicates σpl, whereas the dotdashed lines show the range defined by equation (15):
δ = 0.05 in this case. The ten circles with a yellow centre indicate the five nodes closest to the root and the five nodes closest to the
leaves. These nodes also appear in the two insets. The two blue squares denote the two final nodes x1 and x2.
Figure 8. Left panel: the function fq(r,v) on the redshift diagram (colour map and contours) of the cluster shown in Figures 57. The
thick line shows where fq(r,v) = κ. Right panel: the redshift diagram of the same cluster with the upper and the lower caustics. The
black and cyan lines are the estimated and true caustics respectively; the error bars are estimated with the procedure described in section
5.5. The symbols are the particles in the catalogue: the red dots are the particles within a sphere of 3r200 centred on the cluster centre,
in real space; the green diamonds are the particles whose l.o.s. velocity exceeds 3.5 times the velocity dispersion ?v2?1/2of the candidate
members.
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Serra et al.
Figure 6. Velocity dispersion of the leaves of each node along
the main branch of the binary tree shown in Figure 5. The filled
square and triangle indicate the final thresholds (nodes x1 and
x2) chosen by the algorithm. The dashed line shows the l.o.s.
velocity dispersion of the particles within a 3D sphere of radius
3r200.
4.3Redshiftspace diagram, caustics and final
members
The candidate cluster members are the galaxies hanging
from node x1 and constitute the main group of the binary
tree. These galaxies determine the centre and the size R
of the cluster. The redshift zc of the cluster is the me
dian of the candidate redshifts. The celestial coordinates
of the centre are the coordinates of the twodimensional
(2D) density peak of the candidates. To find the peak, we
compute the 2D density distribution fq(α,δ) of the can
didates on the sky with the kernel method described be
low (equation 18). In general, the smoothing parameter
hc is automatically chosen by the adaptive kernel method.
However, to save substantial computing time, here we set
hc = 0.15(DA/320h−1Mpc) rad, where DA(zc) is the an
gular diameter distance of the cluster. This choice yields
accurate results. The cluster size R is the mean projected
separation of the candidate cluster members from the cluster
centre.
We then build the redshift diagram, which is the plane
(r,v) of the projected distance r and the l.o.s. velocity v of
the galaxies from the cluster centre. If ψ is the angular sep
aration between the cluster centre and a galaxy at redshift
z, then
r =cDA(zc)
H0
sinψ ,(16)
and
v = cz − zc
1 + zc
,(17)
where we have assumed that the galaxy velocity within the
cluster is much smaller than the speed of light c and we have
neglected the peculiar velocity of the cluster. Note that to
avoid artificial depletion of the caustic amplitude at small r
due to the small number of galaxies in the central region, the
galaxy distribution is mirrored to negative r. As an exam
ple, in Figure 8 we show the redshift diagram of the system
shown in Figures 57.
It is now necessary to locate the caustics. The distribu
tion of N galaxies in the redshift diagram is described by
the 2D distribution:
fq(x) =
1
N
N
?
i=1
1
h2
i
K
?x − xi
hi
?
(18)
where x = (r,v), the adaptive kernel is
K(t) =
?4π−1(1 − t2)3
0
t < 1
otherwise;
(19)
hi
[γ/f1(xi)]1/2, f1 is equation (18) where hc = λi = 1 for
any i, and logγ =?
adaptive kernel method automatically determines hc.
The optimal smoothing length
=hchoptλi
is a local smoothing length, λi
=
ilog[f1(xi)]/N. D99 describes how the
hopt =6.24
N1/6
?σ2
r+ σ2
2
v
?1/2
(20)
depends on the marginal standard deviations σr and σv of
the galaxy coordinates in the (r,v) plane. These two σ’s
must have the same units and we require the smoothing
to mirror the uncertainty in the determination of the posi
tion and velocity of the galaxies. We therefore rescale the
coordinates such that q = σv/σr assumes a fixed value. We
choose q = 25. With this value, an uncertainty of 100 km s−1
in the v direction, for example, weights like an uncertainty
of 0.02h−1Mpc in the r direction. The optimal smoothing
length hopt adopted by D99 is half the value we use here.
However, that hopt is derived in the context of the proba
bility density estimate under the assumption that x = (r,v)
is a normally distributed random variate with unit variance
(Silverman 1986, sects. 4.3 and 5.3). However, this assump
tion clearly does not apply to our context here, and we find
that we obtain more accurate results with equation (20).
The caustics are now the loci of the pairs (r,v) that
satisfy the caustic equation
fq(r,v) = κ ,(21)
where κ is a parameter we determine below. In general, the
density distribution fq(r,v) is not symmetric around the
straight line v = 0 in the (r,v) plane, and equation (21) de
termines two curves vup(r) and vdown(r), above and below
this line. Because we assume spherical symmetry, we define
the two caustics to be V±
This choice limits the number of interlopers. The caustic
amplitude is Aκ(r) = [V+
els of galaxy clusters has dlnA/dlnr ? 1/4. A value of
this derivative much larger than 1/4 indicates that the algo
rithm has found an incorrect location of the caustics, which
includes an excessive number of foreground and/or back
ground galaxies. Therefore, whenever dlnA/dlnr > ζ at a
given r, the algorithm replaces A(r) with a new value that
yields dlnA/dlnr = 1/4. We choose ζ = 2, rather than
ζ = 1/4, to keep this constrain loose.
We choose the parameter κ, that determines the correct
caustic location, as the root of the equation
κ(r) = ±min{vup(r),vdown(r)}.
κ(r)−V−
κ(r)]/2. Any realistic mod
S(κ) ≡ ?v2
where ?v2
esc?κ,R− 4?v2? = 0 ,
esc?κ,R =?R
(22)
0A2
κ(r)ϕ(r)dr/?R
0ϕ(r)dr is the mean
Page 11
Escape velocity and mass in the outskirts of galaxy clusters
11
parametersymboldefault value
Binary tree construction and candidate members
galaxy mass
cluster threshold node
substructure threshold node
m
x1
x2
1012h−1M⊙
−
−
Caustic location
rescaling
smoothing length
threshold
derivative limit of A
q25
−
−
2
hc
κ
ζ
Mass profile
filling factorFβ
0.7
Table 1. List of the tunable parameters.
caustic amplitude within the main group size R, ϕ(r) =
?fq(r,v)dv, and ?v2?1/2is the velocity dispersion of the
candidate cluster members with respect to the median red
shift. When S(κ) = 0, the escape velocity inferred from the
caustic amplitude equals the escape velocity of a system in
dynamical equilibrium with a Maxwellian velocity distribu
tion within R. However, it is important to emphasize that
the prescription for determining κ works equally well for
any cluster independently of its dynamical status within R.
Therefore this prescription appears to be just a convenient
recipe to locate the caustics properly and does not limit the
caustic technique to systems in equilibrium within R.
The outcome of the process described in the preceding
paragraphs is outlined in Figure 8, where V±
together with the expected caustic amplitude vesc,los(r) =
±?2  φ(r)  /g(β) ≡ ±?φβ(r). Figure 8 shows a good
agreement between the expected and estimated caustics; in
Sect. 5.2, we will see that this is a general result.
κ(r) are shown
4.4Tunable parameters
The caustic technique requires two sets of parameters: a set
for building the binary tree and identifying the cluster can
didate members and a set for the location of the caustics
in the redshift diagram. Most parameters are automatically
determined by the method with the prescriptions described
above. A few remaining parameters, listed in Table 1, are
chosen in input. In most cases, keeping these parameters to
the default value returns accurate results. In fact, in Section
6.2 we show how, in general, the improvement provided by
a fine tuning of these parameters is modest.
5THE CAUSTIC TECHNIQUE AT WORK
We now apply the caustic method to the mock redshift cata
logues described in Sect. 3.2. We describe how the technique
recovers the centre and velocity dispersion of a cluster, and
its escape velocity and mass profiles. Finally, we provide a
simple recipe to estimate the uncertainties on these profiles.
5.1Global properties of the cluster
In order to remove foreground and background largescale
structure from the redshift diagram, we remove all the par
Figure 10. Profiles of the ratio between the caustic ampli
tude A(r) and the l.o.s. component of the true escape velocity
?v2
esc,los(r)? = −2φ(r)/g(β) ≡ φβ(r). The numerical gravitational
potential profile φ(r) is derived from the true mass distribution
within rmax = 10h−1Mpc from the cluster centre: 50, 68, and
90 percent of the profiles are within the upper and lower solid,
dashed, and dotted curves, respectively. The solid squares show
the median profile. The darkness of the shaded areas is propor
tional to the profile number density on the vertical axis.
ticles with l.o.s. velocity larger than 3.5 times the velocity
dispersion of the candidate members. Nevertheless, when
the cluster is embedded in a particularly crowded area of
the sky, it can also happen that the main branch of the tree
identifies a cluster different from the target cluster. These
cases can happen, for example, when the target cluster is less
rich than nearby systems; they are easily solved by limiting
the input particle catalogue to a small enough region cen
tred on the target cluster. In our sample, the main branch of
the binary tree identifies a cluster different from our target
326 times out of 3000, in other words only 11 percent of the
times. For the sake of simplicity, we limit our sample to the
2674 clusters which are identified without further limitation
of the cluster field of view.
The input centre [α,δ,cz] = [90◦,0◦,32000kms−1],
is recovered very well: the percentile ranges are α =
[89.96,90.00,90.04]◦, δ = [−0.040,0.000,0.036]◦, and cz =
[31740,31998,32222] km s−1.
Figure 9 shows the agreement between the estimated ve
locity dispersion σcaus(the velocity dispersion of the galaxies
hanging from the node x1) and the true velocity dispersion
σtrue, defined as the l.o.s. velocity dispersion of the parti
cles within a sphere of radius 3r200 in real space: in 50 (95)
percent of the systems the estimated velocity dispersion is
within 5 (30) percent of the real one.
5.2The escape velocity profile
The caustic amplitude is related to φβ(r) (equation 2), the
mean component along the line of sight of the escape ve
locity, which is a combination of the gravitational poten
tial and the velocity anisotropy parameter. Figure 10 shows
how well the caustic amplitude recovers φβ(r). On average,
Page 12
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Serra et al.
Figure 9. Comparison between the true velocity dispersion σtrue and the velocity dispersion of the cluster candidate members σcaus.
The red solid line in the histogram in the right panel represents the median and the dashdotted orange (dashed violet) lines represent
the 50 (90) percent range.
the potential is slightly underestimated in the central region
(r ? 0.2r200), but in the outer regions the agreement with
the true escape velocity is remarkable. The slight systematic
underestimate at the centre is due to the small number of
particles at very small radii in the redshift diagram, that
leads to an underestimate of the caustic amplitude. In the
cluster outskirts, the location of the caustics is less accurate
because the number of particles sampling the density distri
bution fq(r,v) is smaller. This fact increases the scatter of
the profiles at large radii. We will see below that increasing
the number of particles in the mock catalogues indeed re
duces the scatter (Figure 20). We remind, however, that the
statistical significance of the scatter decreases at increasing
radii: in fact, the number of clusters contributing to the me
dian profile decreases at large r, because we exclude those
clusters that have a null caustic amplitude A(r) = 0 beyond
any given radius r.
5.3The mass profile
DG97 show that the caustic amplitude can be related to the
cumulative mass profile of the cluster by the relation (4):
GM(< r) =
?r
0
Fβ(r)A2(r)dr .
The bottom right panel of Figure 4 shows the function Fβ(r)
in our simulations. At radii in the range ∼ (0.5−4)r200, the
average Fβ(r) has a mild variation, between 0.5 and 0.8.
This result led DG97 and D99 to assume Fβ(r) = const
tout court and assume that the mass profiles of real clusters
can be estimated with the expression (6):
GM(< r) = Fβ
?r
0
A2(r)dr .
We can choose the correct value of the factor Fβ by con
sidering the contribution of the filling function Fβ(r) in
the integral of equation (4). Figure 11 shows ?Fβ(r)? =
?r
cluster. At radii larger than ∼ 0.5r200, ?Fβ(r)? is basically
constant and supports the validity of equation (6).
We see that the most appropriate value is Fβ = 0.7.
This choice disagrees with the value Fβ = 0.5 adopted by
0Fβ(x)dx/r, where Fβ(x) is the profile of each individual
Figure 11. Profiles of the integral?r
text; 90, 68, and 50 percent of the profiles are within the upper
and lower dashed, solid and dotted curves. The central solid curve
is the median profile.
0Fβ(x)dx/r described in the
DG97 and D99. In this early work, the algorithm for the
determination of the σ plateau was less accurate than our
algorithm here and systematically provided slightly larger
caustic amplitudes. This overestimate was compensated by
a smaller Fβ, that, on turn, returned the correct mass pro
file, on average. Here, our improved algorithm appears to be
more appropriate because it returns the correct φβ(r) pro
file (Figure 10) and, in order to estimate the correct mass
profile, requires a value of Fβ in agreement with what can
be expected by inspecting Figure 11.
Figure 12 shows that, on average, the mass profile is
estimated at better than 10 percent at radii larger than ∼
0.6r200. Clearly, at smaller radii, the assumption Fβ(r) =
const breaks down and the mass is severely overestimated.
As already suggested by DG97, if we assume that the
cluster is in virial equilibrium in the central region, we can
use the virial theorem to estimate the mass there and limit
the use of the caustic method to the cluster outskirts alone,
where the equilibrium assumption does not hold. Here we
Page 13
Escape velocity and mass in the outskirts of galaxy clusters
13
Figure 12. Profiles of the ratio between the caustic and the true
mass profile, adopting equation (6) and Fβ= 0.7. The lines and
shaded areas are as in Figure 10.
use the virial theorem and the median and average mass
estimators from Heisler et al. (1985) to estimate the mass
within αR, where R is the mean clustrocentric separation
of the candidate cluster members from the binary tree (see
Section 4.3) and α is a free parameter. In our sample, the
percentile range is R = [0.50,1.23,1.68]h−1Mpc. We com
pute the ratio between the estimated mass and the true mass
for different values of α. We find that the best estimates are
obtained when α = 0.7. In this case, the ratio between the
estimated and true masses is, on average, 1.03 for the virial
theorem, 1.30 and 1.49 for the median and average mass
estimators, respectively. Different values of α yield worse
mass estimates. This result indicates that the radius 0.7R
generally contains the cluster region in approximate virial
equilibrium.
When we estimate the mass with the virial theorem
within 0.7R and use equation (6) with 0.7R as the lower
limit of the integral, we still obtain a very good estimate of
the real mass (Figure 13).
5.4The gravitational potential profile
Within r = 1h−1Mpc, we fit the caustic mass profile of each
individual cluster with an NFW profile. The scale factor
r200 derived from this fit agrees with the true r200 derived
in 3D; the percentile range of their ratio is [0.84,1.03,1.33].
Consequently, for M200, which is proportional to r3
have [0.59,1.10,2.36] (Figure 14).
However, the concentration parameter c derived from
the NFW fit to the caustic mass profile tends to overesti
mate by 36 percent, on average, the concentration param
eter of the NFW fit to the 3D mass profile: in fact, the
caustic c has percentile range [3.59,5.92,10.23], whereas we
find [2.06,4.62,6.50] in 3D; the ratio of these two parame
ters has percentile range [0.73,1.36,3.26]. This result is an
obvious consequence of the mass overestimate at small radii
(Figure 12).
The NFW fit parameters can be used to derive the grav
200, we
Figure 13. Profiles of the ratio between the caustic and the true
mass profile, when combined with the virial masses. The lines and
shaded areas are as in Figure 10.
itational potential profile of the cluster. The left panel of
Figure 15 shows that the estimated gravitational potential
profile returns the true profile within 10 percent, on aver
age, and, in 50 percent of the cases, the estimated profile
is within 25 percent from the real one out to 4r200. This
result is impressive and derives from the mild radial varia
tion of the functions F(r) and g(β) in hierarchical clustering
models.
The right panel of Figure 15 shows the ratio between
the caustic mass profile derived from the NFW fit and the
true mass profile. The agreement is again good, within 20
percent, although not as good as in Figure 12, because the
caustic estimate is biased low due to the overestimate of c.
5.5 Uncertainty estimate
Figures 10 and 12 show the spread derived from the distribu
tion of the profiles of the individual clusters. This spread is
mostly due to projection effects, as we will see below (Sect.
6.3). We remind here the simple recipe suggested in D99 to
estimate this spread when observing a real cluster of galax
ies.
The uncertainty in the measured value of A(r) depends
on the number of galaxies contributing to the determination
of A(r); we thus define the relative error
δA(r)/A(r) = κ/max{fq(r,v)},
where the maximum value of fq(r,v) is along the vaxis at
fixed r. The resulting error on the cumulative mass profile
is
(23)
δMi =
?
j=1,i
2mjδA(rj)/A(rj),(24)
where mj is the mass of the shell [rj−1,rj] and i is the index
of the most external shell.
The shaded bands in Figure 16 are the median spreads
derived from the above equations. The median uncertainty
on the escape velocity profile increases up to r ∼ 1.5r200
and remains within ∼ 20 − 30 percent at larger radii. The
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14
Serra et al.
Figure 14. Distribution of the ratio between the parameters of the best NFW fit to the caustic mass profile and the true M200and r200
(left and central panels) and distribution of the concentration parameter of the best NFW fit to the caustic mass profile (right panel).
The dashed lines indicate the median while the solid lines show the 90 percent range.
Figure 15. Left panel: Profiles of the ratio between the NFW gravitational potential profile, with parameters derived from the NFW
best fit to the caustic mass profile, and the numerical gravitational potential profile derived from the true mass distribution within
rmax = 10h−1Mpc from the cluster centre. Right panel: Profiles of the ratio between the NFW best fit to the caustic mass profile,
and the true mass profile. In both panels, 90 and 50 percent of the profiles are within the upper and lower dotted and solid curves,
respectively. The central solid curves are the median profiles. For each cluster, the NFW fit to the mass profile is only performed to the
mass distribution within 1h−1Mpc.
Figure 16. The shaded areas show the median uncertainties estimated from equations (23) and (24). For comparison, we also show the
90 (dotted lines), 68 (dashed lines), and 50 (solid lines) percent ranges of the profiles from Figure 10. The solid squares show the median
profiles.
Page 15
Escape velocity and mass in the outskirts of galaxy clusters
15
Figure 17. Profiles of the ratio between the caustic and the true
mass profile when the centre is forced to be the true one. The
lines and shaded areas are as in Figure 16.
mass profile has a slightly larger uncertainties but remains
below ∼ 30 − 40 percent out to r ∼ 3r200. At radii smaller
than ∼ 0.6r200 the assumption of a constant filling function
Fβ(r) breaks down, as we mentioned above, but this mass
overestimate is systematic and can be easily taken into ac
count.
Figure 16 shows that our recipes for the estimate of
the uncertainties yield values in good agreement with the
50 percent range of the profiles. Therefore, when applied to
real clusters, this algorithm provides uncertainties with 50
percent confidence levels.
6SYSTEMATICS
In this section we investigate the systematic errors affecting
the estimate of the escape velocity and mass profiles: the
systematic errors can originate from the centre location, the
parameters of the algorithm, and the projection effects. We
also show how the accuracy of the estimate depends on the
number of objects in the redshift catalogue.
6.1Centre identification
In Sect. 5.1, we show how the cluster centre generally is well
recovered. The deviations are smaller than 0.07◦on the sky
and 250 km s−1along the line of sight in 90 percent of the
clusters. These deviations produce negligible effects on the
mass profile estimate. Figure 17 shows the ratio between
the caustic mass profile and the true mass profile, when the
centre of the cluster is imposed to be the true one. A com
parison with Figure 12 shows that the average profile is still
correct at radii larger than 0.6r200: forcing the method to
use the correct centre only slightly improves the estimate
and reduces the scatter at radii larger than ∼ 2.5r200. It is
worth pointing out that forcing the code to use the correct
centre alters the quantities involved in the construction of
the redshift diagram, specifically we recompute the veloc
ity dispersion and the cluster size R of the candidate cluster
members with respect to the new centre. We then derive the
new redshift diagram and locate the caustics. Despite these
variations, the mass profile estimate does not substantially
change. This result confirms the robustness of the method.
6.2 Tuning the parameters
The algorithm allows the user to choose some parame
ters, namely the thresholds of the binary tree, the optimal
smoothing length hc for the galaxy density distribution in
the redshift diagram, and the threshold κ. This freedom is
necessary when the target cluster is in particularly crowded
regions, or the galaxy sampling is too sparse. In these cases,
the algorithm is either unable to locate the caustics or it
returns unrealistic caustic amplitudes. If one is interested
in the average profiles of a large cluster sample, tuning the
caustic parameters for each individual cluster can be very
time consuming. To quantify the impact of this possible fine
tuning, for each of our 100 simulated clusters, we randomly
draw one of the 30 projections and tune the input parame
ters by hand until the caustic location appears to be close
enough to where we might expect them to be by eye. It turns
out that most clusters need a rather minor fine tuning, as
demonstrated by the final result shown in Figure 18. The
lefthand panel shows the caustic amplitude when we ap
ply a fine tuning to the parameters; the righthand panel is
for the same sample without fine tuning. Clearly, the me
dian profile remains unchanged, but the scatter is slightly
reduced.
6.3Projection effects
The analysis provided above clearly shows that the uncer
tainties on the cluster centre determination and the freedom
on the algorithm parameters are not responsible for most of
the spread of the escape velocity and mass profiles.
This spread originates from the assumption of spher
ical symmetry. In hierarchical clustering, this assumption
does not hold in general, but observationally our informa
tion is limited to the galaxy distribution on the sky alone,
although various techniques can in principle provide infor
mation on the 3D shape of the cluster (e.g. Zaroubi et al.
2001; Ameglio et al. 2009).
To show the impact of the projection effects on the caus
tic method, we take our 100 clusters and plot the escape
velocity profiles derived from each of the 30 lines of sight.
Figure 19 shows four randomly chosen clusters as examples.
The caustic technique returns a median profile systemati
cally large for the cluster in the topright panel. In the other
three cases, however, the median profile is within 30 percent
of the correct one out to ∼ 3r200.
The relevant result of this test is the fact that the spread
due to the different lines of sight is comparable to the spread
of the entire sample (shaded area) taken from Figure 10.
This result clearly indicates that the projection effects are
the major responsible for the systematic uncertainties of the
caustic technique and further refinements of the technique,
which still assume spherical symmetry, appear to be unable
to improve the mass estimate.
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