Matteo BarnabèMatteo BarnabèKapteyn Institute – Groningen UniversityKapteyn Institute – Groningen University

Joint Gravitational Lensing andJoint Gravitational Lensing andStellar Dynamics Analysis Stellar Dynamics Analysis

of Early-Type Galaxiesof Early-Type Galaxies

OZ Lens 2008 - Sydney, 29OZ Lens 2008 - Sydney, 29thth September September

Collaborators: Léon Koopmans (Kapteyn), Oliver Czoske (Kapteyn), Collaborators: Léon Koopmans (Kapteyn), Oliver Czoske (Kapteyn), Tommaso Treu (UCSB), Adam Bolton (IfA), and the SLACS teamTommaso Treu (UCSB), Adam Bolton (IfA), and the SLACS team

Detailed study of the inner mass density profile of (distant) early-type galaxies

Understand the internal structure of early-type galaxies: shape of dark matter halos and correlation with total mass, orbital state

Investigate the evolution of the density profile and structural properties with time = with redshifts

Goal: understand the formation and evolution of early-type galaxies

Ellipticals: great regularity in photometric, spectroscopic and kinematic

properties

Methods to study the mass profile of elliptical galaxies

Strong Lensing Inner z < 1

Stellar Dynamics Inner z < 0.1

Weak Lensing Outer z ~ 0.1 – 1

X-ray Haloes Outer z < 0.1

Discrete tracers: Inner/Outer z < 0.01

GC/PN dynamics

METHOD REGION RANGE

GRAVITATIONAL LENSINGGRAVITATIONAL LENSING Most direct probe to measure mass within the Einstein radius

Depends solely on gravity (no gastrophysics)

LIMITATIONS:LIMITATIONS:

Diagnostics of total mass: difficult to separate dark and luminous components

Mass-sheet degeneracy

STELLAR DYNAMICSSTELLAR DYNAMICS Can allow in principle very detailed analysis of the orbital structure of the

galaxy “dissect” galaxy in 3D

LIMITATIONS:LIMITATIONS: Scarcity of dynamical tracers at large radii

Mass-anisotropy degeneracy

at z > 0.1 the extraction of detailed kinematic information (higher order moments) is more difficult

Joint and self-consistent analysis:

+

GRAVITATIONALGRAVITATIONALLENSINGLENSING

STELLAR STELLAR DYNAMICSDYNAMICS

Determination of the mass inside the effective radius

(= inner regions)

Accurate and (nearly) model independent determination

of mass inside Einstein radius

REinst Reff

Breaking the degeneracies...

Sloan Lens ACS (SLACS) Survey ~80 early-type lens galaxies at z <= 0.35 HST images (F435W, F614W) Integral field spectroscopy for 17 systems

Analysis of 15 SLACS galaxies:Lensing + Dynamics as

INDEPENDENT PROBLEMS

Lensing: SIE model, MEinst imposed as a

constraint for the dynamical models

Dynamics: power-law density profile, r – , spherical Jeans equations

HIGHLIGHTS:

Total density profile very close to ISOTHERMAL: log. slope = 2.01 ± 0.03

Power-law: excellent description of density profile inside Reff

No evidence for evolution in range

z = 0.1 – 1 (SLACS + LSD)

Bolton et al. 2006 Treu et al. 2006

Koopmans et al. 2006

Image credit: Adam Bolton & the SLACS team

Motivation to develop a fully self-consistent approach The data contain a wealth of information: make full use of the

abundant information available from the data: lensed image structure, surface brightness profile and kinematic maps of the lens galaxy

Modeling: spherical axisymmetric

More detailed information about the lens galaxy potential

Information about the dynamical structure

CAULDRONCAULDRON: A SELF-CONSISTENT METHOD FOR : A SELF-CONSISTENT METHOD FOR JOINT JOINT LENSINGLENSING AND AND DYNAMICSDYNAMICS ANALYSIS ANALYSIS

(axisymmetric) density distribution: (R,z)

(axisymmetric) density distribution: (R,z)

Gravitational potential: (R,z,k)Gravitational potential: (R,z,k)

Maximize the bayesian evidence allows model comparison

automatically embodies Occam’s razor (MacKay 1992)

Maximize the bayesian evidence allows model comparison

automatically embodies Occam’s razor (MacKay 1992)

Best values for the non-linear parameters k

source reconstruction & DF reconstruction

Best values for the non-linear parameters k

source reconstruction & DF reconstruction

LENSED IMAGE REC.LENSED IMAGE REC.LENSED IMAGE REC.LENSED IMAGE REC. DYNAMICAL MODELDYNAMICAL MODELDYNAMICAL MODELDYNAMICAL MODEL

non-linearoptimization

vary kwhen converges

linear optimization linear optimization

Barnabè & Koopmans 2007

linear optimization

LENSED IMAGE REC.

CAULDRON: A SELF-CONSISTENT METHOD FOR JOINT LENSING AND DYNAMICS ANALYSIS

Axisymmetric density distribution: (R,z)Axisymmetric density distribution: (R,z)

Gravitational potential: (R,z,k)

Maximize the bayesian evidence allows model comparison

automatically embodies Occam’s razor (MacKay 1992)

Best values for the non-linear parameters k

source reconstruction & DF reconstruction

DYNAMICAL MODEL

non-linearoptimization

vary kwhen converges

linear optimization

Barnabè & Koopmans 2007

linear optimization

LENSED IMAGE REC.LENSED IMAGE REC.LENSED IMAGE REC.LENSED IMAGE REC.

Ls d

Lensed Image Reconstruction

s = source

d = observed lensed image (data)

L = lensing operator (describes how every source pixel is mapped onto the image plane)

• Pixelized source reconstruction method (Warren & Dye 2003, Koopmans 2005)

• Includes regularization, PSF blurring, oversampling• Expressed formally as a linear problem: L s = dL s = d

CAULDRON: A SELF-CONSISTENT METHOD FOR JOINT LENSING AND DYNAMICS ANALYSIS

Axisymmetric density distribution: (R,z)Axisymmetric density distribution: (R,z)

Gravitational potential: (R,z,k)

Maximize the bayesian evidence allows model comparison

automatically embodies Occam’s razor (MacKay 1992)

Best values for the non-linear parameters k

source reconstruction & DF reconstruction

non-linearoptimization

vary kwhen converges

linear optimization

Barnabè & Koopmans 2007

LENSED IMAGE REC. DYNAMICAL MODEL

linear optimization

DYNAMICAL MODELDYNAMICAL MODELDYNAMICAL MODELDYNAMICAL MODEL

linear optimization

TIC

2

+

+

=

surf. br. DF vlos los

TWO-INTEGRAL SCHWARZSCHILD METHOD (Verolme & de Zeeuw 2002) extended and sped up through a novel Monte Carlo approach: one full dynamical model in ~ 10 sec.

Dynamical Model

TIC

1TIC

3to

tal

Building blocks for the superposition: not orbits, but TICs: elementary systems (tori) derived from DF, completely specified by energy Ej and angular momentum Lz,j

The (unprojected) density and velocity moments of a TIC are analytical and easy to calculate.

HST-ACS image

zsrc = 0.5342

zlens = 0.0819

c = 245 km/s

REinst = 1.68’’

Reff,B = 5.50’’

SLACS lens galaxy J2321: a case study for joint lensing & dynamics analysis

velocity map velocity disp. map

(Czoske, Barnabè, Koopmans, Treu & Bolton 2008)

(m) = ,

0

m

m2 = Rc2 + R2 +

z2/q2

total mass density profile: axisymmetric POWER-LAWPOWER-LAW model

BEST MODELBEST MODEL inclination angle: 67o.8 [60.0 – 68.9]

lens strength0: 0.468 [0.467 – 0.475]

logarithmic slope : 2.061 [1.996 – 2.085]

axial ratio q: 0.739 [0.688 – 0.760]

core radius Rc ~ 0

Total density profile close to isothermal

J2321: combined analysis

LENSING

image grid = 100 × 100

source grid = 40 × 40

1 pixel = 0.05’’

blurring operator in the lensing matrix accounts for the PSF of the instrument (HST-ACS, F814W)

DYNAMICSsurf. bright. grid = 50 × 50(1 pixel = 0.10’’)

moments map grid = 9 × 9(1 pixel = 0.67’’)

Only data points with S/N > 8 are considered

NTIC = 10 × 5 × 2 = 100

surf. br.

vlos los

data

reco

nst

r.re

siduals

J2321: combined analysis

reconstr. weighted DF

J2321: dark and luminous mass

“Maximum bulge”: luminous mass rescaled to maximize the contribution of the stellar component Meff ~ 2 × 1011 M ; 5.2 (M/L)B

Dark matter fraction: ~15% at 5 kpc, ~30% at 10 kpc

SAURON: dark matter fraction of 30% within one Reff for local ellipticals (assumption: mass follows light, i.e. constant M/L ratio)

M(r

)

total massluminous massJaffe profileHernquist profile

Radial mass profile for the best model

SLACS sample: preliminary results

Barnabè et al. in preparation

J0037

zsrc = 0.632

zlens = 0.196

best model:

= 1.97

J0216

zsrc = 0.524

zlens = 0.332

best model:

= 2.13

J0912

zsrc = 0.324

zlens = 0.164

best model:

= 1.94

J0959

zsrc = 0.470

zlens = 0.241

best model:

= 1.79

A Crash Test for CAULDRON

CRASH TEST:the 2-Integral axisymmetric CAULDRON code is applied to a situation which severely violates its hypothesis (a non-symmetric N-body system)

Observables cannot be reproduced to the noise level: the single power-law model is an over-simplified description here

Barnabè, Nipoti, Koopmans, Vegetti & Ciotti 2008 (submitted)

LEN

SIN

GD

YN

AM

ICS

RESULTS: Total density slope recovered (<

10%) Total mass radial profile: within

~15% Total ang. momentum, V/ ratio,

anisotropy parameter : within 10-25% (if rotation in the kinematic maps)

Dark matter fraction within Reff reliably recovered (~10% of total mass),

limitations: flattening, lensed source(requires detailed potential corrections, e.g. adaptive lensing code of Vegetti & Koopmans)

CAULDRON IS RELIABLE EVEN IN A WORST-CASE SCENARIO

true total density profilerecovered profile: “face-on” data-setrecovered profile: “yz-plane” data-setrecovered profile: “zx-plane” data-set

Joint lensing & dynamics: powerful instrument for the study of the density profile of distant E/S0 galaxies The inclusion of stellar kinematics constraints allows to break

degeneracies that would arise if lensing alone was used

Several fundamental structural quantities are robustly recovered even in a worst case scenario

First in-depth analysis of a sample of elliptical galaxies at redshift beyond ~0.1 Power-law total density distribution: simple yet very satisfactory model

Total density profile close to isothermal (slope ~ 1.8 – 2.1)

dark matter fraction ~ 30-35% within Reff

Future work: Extend the analysis to the entire sample of SLACS lens galaxies

(17 with VLT-VIMOS IFU spectroscopy, 13 with Keck long-slit spectroscopy)

Extend CAULDRON flexibility: 3-integral models

Conclusions