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arXiv:1011.0372v1 [astro-ph.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, I-10125, Torino, Italy

2Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Torino, Italy

3Harvard-Smithsonian 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 1-σ 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 1-σ uncertainty

between 40 and 80 percent within the radial range (0.6 − 4)r200. When the correct

virial mass is known, the 1-σ 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 1-σ 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: large-scale structure

of Universe

⋆E-mail: 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 galaxy-environment

connection (e.g. Skibba et al. 2009; Huertas-Company et al.

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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 X-ray 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 steady-state 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

mass-X-ray 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 N-body/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 intra-cluster 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 X-ray

emission, and yields a correlation with mass which is tighter

than the mass-X-ray 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 Sunyaev-Zel’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/near-IR 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; X-ray estimates rarely

go beyond ∼ 0.5r200, where the X-ray surface brightness be-

comes smaller than the X-ray 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 N-body 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 X-ray 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).

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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 X-ray 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

X-ray 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 N-body

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 X-ray, Sunyaev-Zeldovich 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 non-radial 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 line-of-sight (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 non-increasing 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

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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 N-body 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 N-body/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 Plummer-equivalent 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 multi-phase 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

GADGET-2

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 two-step procedure: a friends-of-

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 three-dimensional (3D) real cumulative mass pro-

file with the NFW model

M(< r) = Ms

?

ln

?

1 +r

rs

?

−

r/rs

1 + r/rs

?

(7)

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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 non-isolated 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 large-scale 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

Page 7

Escape velocity and mass in the outskirts of galaxy clusters

7

upper-level threshold identifies the cluster candidate mem-

bers; the lower-level 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 Einstein-de

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

two-dimensional 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

Einstein-de 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 bi-unique 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 long-range 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

Page 8

8

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 left-hand 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.

Page 9

Escape velocity and mass in the outskirts of galaxy clusters

9

Figure 7. Line-of-sight 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 dot-dashed 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 5-7. 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.

Page 10

10

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.

4.3Redshift-space 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 two-dimensional

(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 5-7.

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 large-scale

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 dash-dotted 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 v-axis 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

Page 14

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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

left-hand panel shows the caustic amplitude when we ap-

ply a fine tuning to the parameters; the right-hand 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 top-right 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.