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arXiv:0904.4381v2 [astro-ph.CO] 12 Nov 2009

Density profile, velocity anisotropy and line-of-sight mass

contamination of SLACS gravitational lenses

Antonio C. C. Guimar˜ aes and Laerte Sodr´ e Jr.

Departamento de Astronomia, Universidade de S˜ ao Paulo,

Rua do Mat˜ ao 1226,CEP 05508-090 S˜ ao Paulo - SP, Brazil

aguimaraes@astro.iag.usp.br

ABSTRACT

Data from 58 strong lensing events surveyed by the Sloan Lens ACS Survey is used to esti-

mate the projected galaxy mass inside their Einstein radii by two independent methods: stellar

dynamics and strong gravitational lensing. We perform a joint analysis of these two estimates in-

side a model with three degrees of freedom with respect to the lens density profile, stellar velocity

anisotropy and a possible line-of-sigh (l.o.s.) mass contamination. We also consider two possibili-

ties for the lens light distribution (Jaffe and Hernquist profiles). A Bayesian analysis is employed

to estimate the model parameters, evaluate their significance and compare models. We find that

the data favor Jaffe’s profile over Hernquist’s, but that any particular choice between these two

does not change the qualitative conclusions with respect to the features of the system that we

investigate. The data do not strongly constrain the l.o.s. mass contamination, yielding no contri-

bution from secondary matter at maximum likelihood and an average contamination of the order

of 10%, when considering the full range of probabilities. The density profile is compatible with an

isothermal, being sightly steeper and having an uncertainty in the logarithmic slope of the order

of 5%. We identify a degeneracy between the density profile slope and the anisotropy parameter,

but we encounter no evidence in favor of an anisotropic velocity distribution on average for the

whole sample.

Subject headings: dark matter — galaxies: elliptical and lenticular, cD — galaxies: kinematics and

dynamics — galaxies: structure — galaxies: fundamental parameters — gravitational lensing

1. Introduction

The observation of strong gravitational lens-

ing events has allowed many studies about the

mass, density profile and structure of the galaxies

that act as lenses (Bolton et al. 2006; Treu et al.

2006; Koopmans et al. 2006; Gavazzi et al. 2007;

Bolton et al. 2007, 2008a,b; Czoske et al. 2008;

Treu et al. 2008), which could also have impor-

tant implications for dark matter and cosmol-

ogy studies – see also Kochanek’s contribution in

Meylan et al. (2006) for a more general review in

the field. Therefore it is fundamental to control

for possible systematic effects, such as line-of-sight

(l.o.s.) mass contamination.

these works consider a free path from source to

However most of

lens and from lens to observer, even though there

are observational and theoretical suggestions that

the l.o.s. mass contamination may be significant.

Bar-Kana (1996) investigated theoretically the

effect of the Large-Scale Structure on strong lens-

ing events, finding that it can be significant.

Keeton et al. (1997) observed that external shear

due to galaxies and clusters associated with the

primary lens or along the l.o.s.

portant perturbation in individual lens models.

Premadi & Martel (2004) used ray-tracing to in-

vestigate the effect of density inhomogeneities

along the l.o.s.of strong lenses and concluded

that it can be of order 10% on the magnifica-

tion of the sources. Dalal et al. (2005) found

that significant errors can arise from l.o.s. pro-

can be an im-

1

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jections when using giant arcs generated by clus-

ter strong lensing to constrain the cosmologi-

cal parameters. Wambsganss et al. (2005) noted

that secondary matter along the l.o.s. of strong

lenses can lead in some circumstances to an over-

estimate of 10-15% of the primary lens mass

if ignored. Using spectroscopy, various groups

(Tonry & Kochanek 2000; Momcheva et al. 2006;

Fassnacht et al. 2006; Williams et al. 2006; Auger et al.

2007) discovered a significant l.o.s.

strong lens galaxies.Moustakas et al. (2007)

observed that even in under-dense local envi-

ronments, the l.o.s. contamination may give a

considerable contribution to galaxy-scale strong

lenses. Using ray-tracing thought the Millennium

Simulation, Hilbert et al. (2007) determined that

strong lensing lines-of-sight are biased towards

higher than average mean densities, contributing

a few percent to the total surface density, and

Puchwein & Hilbert (2009) found that secondary

matter along the l.o.s. has a large effect on the

strong-lensing optical depth and the cross-section

for cluster strong lensing. Treu et al. (2008) mea-

sured the over-density of galaxies around SLACS

lenses and observed that typical contributions

from external mass distributions are of order of

few percent, but reaching 10-20% in some cases.

Faure et al. (2008) considered strong and weak

lensing observations in the COSMOS survey and

compared with simulations, finding that strong

lensed images with large angular separation were

in the densest regions. On the other hand, Auger

(2008) did not find an over-density of photomet-

ric sources along the l.o.s.

of SLACS strong lenses in comparison with other

SDSS massive early-type galaxies and interpreted

that as evidence against a possible l.o.s. contami-

nation.

In this paper we use two independent galaxy

mass estimate methods, strong gravitational lens-

ing and stellar dynamics, to examine the presence

of l.o.s. mass contamination in the lens set and

its effect on the determination of the density pro-

file. We use all the suitable events in the SLACS

sample, considering realistic brightness functions

for the lens-galaxies, and incorporating our prior

ignorance on their velocity anisotropy.

In Section 2 we describe how we calculate the

mass of SLACS lenses using strong gravitational

lensing and stellar dynamics, and how we deter-

effect on

of a limited sample

mine the l.o.s. mass contamination. In Section 3

we show our results, which are discussed in Section

4, together with our conclusions.

2. Data and Methods

The analysis in this paper is based on the com-

parison of galaxy masses calculated through two

different methods: gravitational lensing and dy-

namical analysis. In sec. 2.1 we present the data

used in the analysis, collected from the SLACS

survey.

In sections 2.2 and 2.3, we discuss the lens-

ing and dynamical mass determinations, respec-

tively. We will assume simple models for the

galaxy mass distribution (e.g., spherical symme-

try, power-law density distribution) because they

have few free parameters and allow to illustrate

well the two methods. For a similar approach see

Koopmans et al. (2006).

We want to examine whether the two mass esti-

mates are indeed equivalent and/or if there is ev-

idence of systematic differences between them. In

particular, we consider the plausibility of a con-

tamination affecting the lensing mass. Sec. 2.4

presents a Bayesian analysis of this problem.

2.1.Data

The selected set of galaxies is part of the Sloan

Lens ACS Survey, SLACS (Bolton et al. 2006),

which is a Hubble Space Telescope (HST) Snap-

shot imaging survey for strong gravitational galac-

tic lenses.The candidates for the HST imag-

ing were selected spectroscopically from the SDSS

database and are a sub-sample of the SDSS Lumi-

nous Red Galaxy (LRG) sample.

We use data compiled from Koopmans et al.

(2006), Gavazzi et al. (2007) and Bolton et al.

(2008a), constructing a sample of 58 strong gravi-

tational lensing events where the lenses are iso-

lated early-type galaxies (E+S0).

SLACS is especially suitable for joint strong lens-

ing and dynamical analysis because they allow

precise determination (5% error) of the Einstein

radius for each lens-galaxy in a relatively homo-

geneous sample of early-type galaxies. And, at

the same time, SDSS has precise stellar veloc-

ity dispersion measurements (6% average error)

for the lenses, as well as redshifts for lenses and

background sources.

Data from

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For each lens system we are interested in the

redshift of the background lensed source zs, the

redshift of the lens zl, the average stellar velocity

dispersion inside an aperture σap, the effective an-

gular radius θef and the Einstein angular radius

θE. The sample average values for these quanti-

ties are ?zl? = 0.2, ?zs? = 0.6, ?σap? = 250kms−1,

?θef? = 2.2′′, and ?θE? = 1.2′′.

The source and lens redshifts were determined

from the SDSS spectra, and the stellar velocity

dispersion corresponds to the light-weighted aver-

age inside the 3′′diameter SDSS fiber.

2.2. Lensing Mass

The estimated projected mass inside the Ein-

stein radius RE= θEDL, is given by

ML=

c2

4G

DLDS

DLS

θ2

E, (1)

where D[L,S,LS]is the angular-diameter distance

of the lens, source, and between lens and source,

respectively.These distances are calculated as-

suming a redshift-distance relation derived inside a

chosen cosmological model that in the present pa-

per is a concordance ΛCDM model with Ωm= 0.3,

ΩΛ= 0.7 and h = 0.7.

The Einstein radii were determined from HST

images using strong lensing modeling of the

lenses and reconstruction of the unlensed sources

(Koopmans et al. 2006; Gavazzi et al. 2007). The

uncertainties on θE were reported to be around

5%, so we use this value for all Einstein radii

when calculating the error on ML.

the lensing modeling uses a Singular Isothermal

Ellipsoid (SIE) mass model, but the resulting pro-

jected mass distribution is parameterized by an

Eintein radius so that the enclosed mass in the

projected ellipse is the same that would be en-

closed in a projected circle from an equivalent

Singular Isothermal Sphere.

we adopt here. Indeed, the Einstein radius deter-

mined this way is a robust attribute of the lens,

being little sensitive to the lens model used [see

Kochanek’s contribution in Meylan et al. (2006)].

Note that

This is the radius

2.3.Dynamical Mass

We call the dynamical mass, mD, the mass esti-

mated from the observed velocity dispersion. Here

we are interested in examining the case of a power

law for the density profile, ρ = Arγ, where A is

a constant that has to be determined from the

Jeans equation and the observed velocity disper-

sion. The mass within the cylinder CEof Einstein

radius REis then

mD(γ,β) =

?

CE

ρ(r)dV =2π3/2

3 + γ

Γ?−1+γ

2

?

Γ?−γ

2

? AR3+γ

E

.

(2)

The spherical Jeans equation (Binney & Tremaine

1987) can be written as

1

ν

d(νσ2

dr

r)

+ 2βσ2

r

r

= −dΦ

dr= −πG

3 + γAr1+γ, (3)

where σr is the radial velocity dispersion, ν(r)

is the luminosity density profile (Jaffe 1983;

Hernquist 1990), β ≡ 1 − σ2

parameter of the velocity distribution (σt is the

tangential velocity dispersion), and Φ is the grav-

itational potential produced by the assumed den-

sity profile.

Since what is available is the luminosity-

weighted average velocity dispersion within a

given aperture, σ2

ap, the following constraint is

necessary for the determination of the constant A:

t/σ2

ris the anisotropy

σ2

ap=

?

Capνσ2

?

rdV

CapνdV

, (4)

where the integration volume is an infinite cylinder

of radius Rapwith axis along the l.o.s.

To simplify, and since there is very little prior

knowledge on the velocity anisotropy parameter,

we assume that β is a constant. In Appendix A.1

we give more details on the solution of the Jean’s

equation (3), and in Appendix A.2 we examine the

correction due to seeing effects.

Figure 1 displays the general behavior of the

dynamical mass as a function of the density pro-

file slope and velocity anisotropy parameter. We

examine mD(γ,β = 0) and mD(γ = −2,β). Other

combinations around these fixed values give quali-

tatively similar results. The doted line depicts the

strong lensing mass for this hypothetical system,

so it is possible to glimpse from the intersection

of the curves the expected value of the dynamical

parameters, γ and β. The difference of the curves

calculated using the Jaffe or the Hernquist light

profiles are just quantitative. The use of a con-

stant mass to light ratio, ν(r) ∝ ρ(r) ∝ rγ, dis-

played as the dot-dashed line on the left panel of

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figure 1, gives a very distinct and interesting be-

havior for mD(γ), reproducing a result obtained

by Guimar˜ aes & Sodr´ e (2007). However, a single

power law is not a realistic approximation for the

light distribution of the lens-galaxies in the sam-

ple.

2.4.Statistical Model

We want to compare the estimates of lensing

and dynamical masses taking into account the pos-

sibility that masses obtained through gravitational

lensing are affected by some type of contamina-

tion. Specifically, we consider a model where the

real lensing mass mL (assumed equal to the dy-

namical mass) is related to the measured lensing

mass by

mL(fc) = (1 − fc)ML,

where fcgives the fractional mass contamination.

Our model, then, has three free parameters:

fc, γ and β. These parameters are determined

through the Bayes theorem. Since, however, we

shall assume uniform priors for the parameters

(see below), the posterior probability of the pa-

rameters is equivalent to the likelihood.

To construct a likelihood for the system, which

has the mission of describing the probabilities of

the model parameters given the data, we define

the quantity

(5)

F =mL

mD

−mD

mL. (6)

Note that both mLand mDrefer to the projected

mass withing the Einstein radius. The likelihood

for each lens system is then written as

Li=

1

√2πσ2

F,i

exp

?

−(Ft− Fobs,i)2

2σ2

F,i

?

, (7)

where Fobs = Fobs(fc,γ,β;data) is the measured

F given the model and the observational data,

and Ft is the expected value for it, which, in

the desired case where both mLand mDare esti-

mates of the same true galaxy mass, corresponds

to Ft = 0. Note that other quantities may be

defined to construct the likelihood, for example

F = mL/mD+ mD/mL (Ft = 2), F = mL/mD

(Ft = 1), F = mL− mD (Ft = 0). The next to

last definition gives a likelihood that is not sym-

metrical between mLand mD, what is not desir-

able, and the last example has the inconvenience of

maximizing the likelihood not only in the desired

region of the parameter space in which mL∼ mD,

but also in the region where both mass estimates

are small, what introduces artificial solutions that

give maximum l.o.s. contamination. However if

fc= 0 is fixed, then all definitions for F, includ-

ing the last, give very similar results.

The variance in F is estimated as being (an

index i is implied in all quantities)

σ2

F=

?m2

L+ m2

m4

D

?2

Lm4

D

?m2

Lσ2

D+ m2

Dσ2

L

?, (8)

where σ{D,L}is the uncertainty in m{D,L}.

The joint likelihood for the whole set of N

galaxies is then given by

L(fc,γ,β) =

N

?

i=1

Li, (9)

which allows us to find the maximum likelihood

estimator for the free parameters that we denote

by putting a hat over the parameter (ˆfc, ˆ γ,ˆβ),

and the Bayes estimator (also called the posterior

mean) defined by (all priors used are flat)

?p? ≡

?L(p)pdp

?L(p)dp.

(10)

We also calculate the root mean square deviation,

rms(p) =

??p2? − ?p?2, for each free parameter

?

where P(p) is the prior probability distribution

in the parameter space p. We adopt a flat prior

probability distribution for all the parameters con-

sidered: fc∈ [0,0.5], γ ∈ [−3,−1] and β ∈ [−1,1].

We sample the parameter space using a grid and

use it to make the various calculations.

The main appeal of this approach for model

comparison is that the Bayesian evidence auto-

matically implements Occam’s razor by penalizing

more strongly more complex models, those with

more free parameters.

and the Bayesian evidence (Trotta 2008)

E ≡L(p)P(p)dp , (11)

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-3

-2.5

-2

γ

-1.5

-1

1010

1011

1012

mD(γ,β)

Jaffe

Hernquist

constant M/L

-1

-0.5

0

β

0.5

1

1010

1011

1012

β=0 γ=−2

mL

Fig. 1.— Dynamical mass behavior in relation to density profile logarithmic slope, γ, and the velocity

anisotropy parameter, β, in units of h−1M⊙. The curves were calculated for an “average” system with

zl= 0.2, zs= 0.6, σap= 250kms−1, θE= 1.2′′, θef = 2.2′′. Solid lines use Jaffe and dashed use Hernquist

light distribution profiles. The dotted line depicts the strong lensing mass, equation (1), for the “average”

system: ML= 1.34·1011h−1M⊙. The dot-dashed line on the left panel was obtained using ν(r) ∝ ρ(r) ∝ rγ.

3. Results

We have analyzed the data described in Section

2.1 with a model with three degrees of freedom,

one (fc) corresponding to a possible l.o.s. mass

contamination that would affect the strong lensing

estimate of the lens mass and two (γ and β) cor-

responding to intrinsic properties of the lens that

determine its dynamical mass estimate. Two light

distribution profiles (Jaffe and Hernquist) were ex-

amined.

Table 1 summarizes the results for the maxi-

mum likelihood and Bayes estimators for each free

parameter, as well as for their root mean square

deviations, which give a measure of their disper-

sion. For the fractional mass contamination the

maximum likelihood estimator is zero (no l.o.s.

contamination), but its mean value is of the order

of 10%. The density profile logarithmic slope is es-

timated to be close to an isothermal profile, using

the Jaffe’s light distribution, or somewhat steeper,

using Hernquist’s. In both cases it is found a ∼ 5%

dispersion, what is considerably higher than what

was found by Koopmans et al. (2009) (∼ 1%), ne-

glecting l.o.s. contamination and anisotropy. The

overall effective velocity anisotropy parameter is

compatible with the null value (isotropy).

note that Koopmans et al. (2009) found a positive

anisotropy (significantly distinct from isotropy)

for the same sample, but using for that an inde-

pendent determination based on scaling relations

of the density profile logarithmic slope. That is a

different method from ours, which relies solely on

the joint strong lensing and dynamical analysis.

Figure 2 shows the likelihood marginalizations

over one free parameter. The top row plots use

Jaffe’s profile and the bottom ones, Hernquist’s.

In figure 3 the marginalizations are over two free

parameters.Qualitatively, the results for both

profiles are the same. The likelihood distribution

for the fractional mass contamination indicates a

peak in fc= 0 with a tail falling to fc∼ 0.5 in the

3σ confidence region.

For γ and β the likelihood follows a near Gaus-

sian behavior, their maximum likelihood and mean

values are about the same. Jaffe’s light distribu-

tion implies a slightly flatter density profile and

higher anisotropy compared to Hernquist’s. The

fc marginalized likelihoods (last row of figure 2)

show a degeneracy between γ and β that was

We

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Table 1: Maximum likelihood parameters, posterior means and root mean squares.

light profile

ˆfc

?fc?

Jaffe 0. 0.13 0.11

Hernquist 0.0.100.09

rms(fc)ˆ γ?γ?

-2.08

-2.13

rms(γ)

0.12

0.11

ˆβ?β?

0.20

0.05

rms(β)

0.21

0.17

-2.04

-2.18

0.11

-0.01

already hinted in figure 1, since the curves for

mD(γ,β = 0) and mD(γ = −2,β) have mono-

tonically crescent and decrescent behavior, respec-

tively.

If the l.o.s. contamination is neglected, fix-

ing fc = 0 in our model, there is no significant

change in the determination of the density profile

logarithmic slope. However if the velocity distri-

bution is assumed to be isotropic (β = 0), then

there is a small steepening of the density pro-

file, ˆ γ = −2.12, and the probability distribution

dispersion is smaller, rms(γ) = 0.06 when using

Jaffe’s light profile, compared with the model with

three degrees of freedom (Table 1). Similarly, for

Hernquist’s profile, there is no significant change

in ˆ γ and rms(γ) = 0.07.

If both fractional contamination and anisotropy

are fixed, fc = β = 0, then ˆ γ = 0.12 for both

light profiles and rms(γ) = 0.019 using Jaffe and

rms(γ) = 0.027 using Hernquist. Therefore, the

neglect of our ignorance on β and, to a lower ex-

tent, fc, by fixing them equal to zero, implies a

considerable underestimate of the dispersion in γ

and possibly the introduction of a systematical er-

ror. In the central panel of figure 3 it is shown the

likelihood (narrower curves) for a model with fixed

fc= β = 0, free γ and Hernquist profile, reproduc-

ing closely what was obtained by Koopmans et al.

(2009). In the same figure we examine the im-

pact of the seeing correction (see Appendix A.2).

The two narrow and very close curves differ just in

that in their calculation one takes into account the

seeing correction (dot-dashed line) and the other

(dotted line) does not. The seeing correction is

negligible.

To compare models we calculate the ratio of the

Bayesian evidences, also known as Bayes factor,

B(M1,M2) =E(M1)

E(M2), (12)

which values can be interpreted qualitatively using

Jeffrey’s scale (Trotta 2008).

Table 2 gives a summary of several model com-

parisons. The Bayes factor between the models

using Jaffe’s or Hernquist’s light profile indicates

that the first is favored over the second. We also

compare the gain in Bayesian evidence yielded by

each free parameter separately in relation to a

model with fixed values (fc= 0, γ = −2, β = 0).

In general, there is no or weak evidence favoring

the simpler models (with fixed parameter values)

over the more complex ones. However, if all pa-

rameter values are fixed the resulting model with

no free parameter is strongly (Jaffe) or moderately

(Hernquist) disfavored over the model with three

free parameters

4.Discussion and Conclusions

Two independent methods, strong gravitational

lensing and stellar dynamics, were used to deter-

mine the projected galaxy mass within its Einstein

radius for a set of 58 galaxies from SLACS. From

the comparison of the two masses, the lens den-

sity profile, velocity anisotropy and l.o.s.

contamination were probed.

Even thought the maximum likelihood esti-

mation of the l.o.s. mass contamination gives

a null contribution from secondary matter along

the l.o.s., the distribution of possible contami-

nation fraction values is broad, yielding an av-

erage contamination of the order of 10%. This

result is compatible with theoretical expecta-

tions coming from large-scale structure simula-

tions (Premadi & Martel 2004; Hilbert et al. 2007;

Faure et al. 2008; Puchwein & Hilbert 2009) and

also with some observationalworks (Momcheva et al.

2006; Williams et al. 2006; Treu et al. 2008) that

do find l.o.s. contamination. Auger (2008) did not

find an over-density of photometric sources along

the l.o.s. of a sample of SLACS strong lenses in

comparison with other SDSS massive early-type

galaxies. However, it is worth pointing out that

this is not in contradiction with the existence of

mass over-densities along the l.o.s. since SLACS

mass

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?3.0

?2.5

?2.0

Γ

?1.5

?1.0

0.0

0.1

0.2

0.3

0.4

0.5

fc

?1.0

?0.5 0.00.51.0

0.0

0.1

0.2

0.3

0.4

0.5

Β

fc

?1.0

?0.5 0.00.5 1.0

?3.0

?2.5

?2.0

?1.5

?1.0

Β

Γ

?3.0

?2.5

?2.0

Γ

?1.5

?1.0

0.0

0.1

0.2

0.3

0.4

0.5

fc

?1.0

?0.5 0.00.51.0

0.0

0.1

0.2

0.3

0.4

0.5

Β

fc

?1.0

?0.50.0 0.5 1.0

?3.0

?2.5

?2.0

?1.5

?1.0

Β

Γ

Fig. 2.— Marginalized likelihood contours, corresponding to the 1σ, 2σ and 3σ confidence levels. The

parameter values of maximum marginalized likelihood are shown as points in the interior of the contours.

Upper panels use Jaffe’s light distribution profile and bottom panels use Hernquist’s. The marginalization

is over one of the three free parameters: β in the left row, γ in the center and fcin the right.

0 0.10.2 0.30.4

0.5

fc

0

0.2

0.4

0.6

0.8

1

L / Lmax

Jaffe

Hernquist

-3

-2.5

-2

γ

-1.5

-1

0

0.2

0.4

0.6

0.8

1

L / Lmax

-1

-0.5

0

β

0.5

1

0

0.2

0.4

0.6

0.8

1

L / Lmax

Fig. 3.— Marginalized likelihood over two parameters. The narrower curves at the center panel is for a

model with fixed fc= β = 0 and free γ, using a Hernquist luminosity density profile: the dotted line does

not take into account the seeing correction and the dot-dashed line does.

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Table 2: Comparison of models.

model 1: M1

J, free (fc, γ, β)

J, free (fc, γ, β)

H, free (fc, γ, β)

J, free (fc, γ, β)

H, free (fc, γ, β)

J, free (fc, γ, β)

H, free (fc, γ, β)

J, free (fc, γ, β)

H, free (fc, γ, β)

model 2: M2

H, free (fc, γ, β)

J, fc= 0, free (γ, β)

H, fc= 0, free (γ, β)

J, γ = −2, free (fc, β)

H, γ = −2, free (fc, β)

J, β = 0, free (fc, γ)

H, β = 0, free (fc, γ)

J, fc= 0, γ = −2, β = 0

H, fc= 0, γ = −2, β = 0

Lmax(M1)

Lmax(M2)

14

1.0

1.0

1.1

2.4

1.3

1.0

4.6·107

1.6·104

lnE(M1)

E(M2)

2.9

-1.1

-1.4

-1.7

-0.69

-0.70

-1.6

11

3.1

Jeffreys’ scale

moderate evidence for M1

weak evidence for M2

weak evidence for M2

weak evidence for M2

inconclusive

inconclusive

weak evidence for M2

strong evidence for M1

moderate evidence for M1

Note.—J stands for Jaffe’s light distribution, H for Hernquist’s. Jeffreys’ scale according to Trotta (2008).

lenses were selected to be isolated and excess pho-

tometric sources would only trace rare high den-

sity peaks, but not more common diffuse mass

concentrations.

The shape of the fc× γ and fc× β degenera-

cies indicates that the determination of the density

profile and anisotropy parameter are not very sen-

sitive to the value of the l.o.s. contamination. This

can be understood from the functional dependence

of the dynamical mass with γ and β (figure 1). A

small variation in γ or β implies a large change in

mD.

The joint strong lensing and stellar dynam-

ics analysis seems to do not strongly constrain

the anisotropy parameter, its likelihood distribu-

tion being broad and statistically compatible with

isotropy on average. Nevertheless, the degeneracy

between γ and β gives an indetermination that

correlates a larger anisotropy with a flatter den-

sity profile, which can also be understood from

the functional behavior of mD(γ,β) (figure 1). An

increment in γ can be annulled, in terms of a vari-

ation in the dynamical mass, by a decrement in β,

and vice-versa.

The inclusion of two degrees of freedom in the

model with respect to the l.o.s.

and velocity anisotropy allows us to take into ac-

count and examine the effect of our prior igno-

rance on these features of the lens system. The

most visible effect of fc and β on the determi-

nation of the density profile logarithmic slope

is the considerable broadening of its likelihood

contamination

distribution (posterior probability), what means

a more uncertain determination of γ.

this uncertainty, the density profile is statisti-

cally compatible with an isothermal profile. This

very particular density profile is also found by

other authors, apparently as a result of the com-

plementarity of baryonic and dark matter pro-

files (Hamana et al. 2005; Ferreras et al. 2005;

Lintott et al. 2006; Baltz et al. 2007; Czoske et al.

2008).

We used standard assumptions and approxi-

mations in our modeling of galaxies and anal-

ysis.Nevertheless, we can identify several ar-

eas where further work can be done to refine

the understanding of these strong lensing systems,

which can also be seem as caveats to the present

works in the area. Among them we highlight

(i) the triaxiality and substructure of lens ha-

los, whose importance was already suggested by

Meneghetti et al. (2005) and Yencho et al. (2006)

in the context of simulations on cluster scales;

(ii) the correction of the dynamical mass esti-

mate due to rotational support. It is well known

that some early-type galaxies can have a signifi-

cant rotational component (e.g., Emsellem et al.

2007) and two-dimensional kinematics for some

few SLACS lenses are already becoming available

(Czoske et al. 2008; Barnabe et al. 2009); (iii) the

brightness distribution, which could be treated

more realistically with the observed full surface

luminosities for the individual lenses, instead of

individualized fits of an universal profile; and (iv)

the velocity anisotropy parameter, which was as-

Within

8

Page 9

sumed to be a constant, but that more realistically

must be a function of radius. However very little

is known about the velocity anisotropy of early-

type galaxies, be it observationally, theoretically

and even from simulations.

The relaxation of some of our assumptions, with

the almost inevitable addition of extra free param-

eters, could prove a fruitful source of investiga-

tion, however would likely require a larger galaxy

sample to reduce the likely increased degeneracies

among the degrees of freedom of the model. As

we have illustrated, models with a large number

of parameters may have a higher likelihood but

lower Bayesian evidence, since the added complex-

ity must pay its price in a Bayesian sense.

The authors thank CNPq and FAPESP for fi-

nancial support and the SLACS and SDSS teams

for the databases used in this work.

9

Page 10

Appendix

A.1. Solving Jean’s Equation

The spherical Jeans equation (3) can be rewritten, defining x ≡ r/Ref (dimensionless) and

y ≡3 + γ

4πGAR−(2+γ)

ef

(νσ2

r), (A1)

as

dy

dx= −2βy

x− νx1+γ. (A2)

The luminosity distribution is well approximated by the profiles (Koopmans et al. 2006)

ν(x) ∝

1

xγ∗(x + x∗)4−γ∗, (A3)

where γ∗= 1 and x∗= 1/1.8153 (Hernquist 1990), or γ∗= 2 and x∗= 1/0.7447 (Jaffe 1983). We examine

both profiles.

The first-order linear differential equation (A2) has as solution

y(x) = x−2β

?

C −

?

ν(x)x1+γ+2βdx

?

, (A4)

where C is an arbitrary constant and the most evident boundary condition is y(x → ∞) = 0. The analytical

solution for the integral in (A4) with (A3) for ν(r) has the hypergeometric function 2F1, which has a

computationally demanding solution. Therefore we solve (A2) using a fourth order Runge Kutha algorithm,

starting at y(x = 1000) = 0 and evolving y(x) down to x = 0.01.

The dynamical mass (2) within the Einstein radius REis then

mD(γ,β) =π1/2

2G

Γ?−1+γ

2

?

Γ?−γ

2

? σ2

apRE

?RE

Ref

?2+γ?

CEνdV

?

CEydV,

(A5)

and its error is estimated from the observational errors on σapand REthrough error propagation.

A.2. Seeing

We model the effect of the seeing through a Gaussian smoothing of the galaxy projected luminosity.

Therefore, the observed surface brightness profile is related to an intrinsic (no seeing) profile by

Iobs(θ) =e−θ2/2σ2

s

σ2

s

?∞

0

I(θ′)I0

?θθ′

σ2

s

?

sis the Gaussian seeing variance. We use

e−θ′2/2σ2

sθ′dθ′, (A6)

where I0 is the modified Bessel function of first kind, and σ2

σs= 0.64′′, which corresponds to a FWHM of 1.5′′.

The seeing correction of the average velocity dispersion within the observational aperture is then given by

σ2

ap

?

?σ2

ap

obs

=

?Rap

0

σ2

p(R)I(R)RdR

?Rap

0

?σ2

p(R)I(R)?

obsRdR

?Rap

0

?Rap

obsis defined in an analogous way to Iobs(R)

Iobs(R)RdR

0

I(R)RdR

, (A7)

where σ2

in Eq.(A6) and the projection is calculated through (Binney & Tremaine 1987)

pis the projected velocity dispersion profile,?σ2

p(R)I(R)?

1 − βR2

I(R)σ2

p(R) = 2

?∞

R

?

r2

?

νσ2

rr

√r2− R2dr.(A8)

10

Page 11

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