PRINCIPLES AND APPLICATIONS OF

GRAVITATIONAL LENSING

Georges Meylan

Laboratoire d’Astrophysique

Ecole Polytechnique Fédérale de Lausanne

http://lastro.epfl.ch

Cours de Cosmologie observationnelle Master 2010-2011 EPFL 18 May 2011

1 April 2011 : gravitational lensing effect by a black hole of 1 mJ

The precursors of XVIII et XIX centuries

•! 1783: John Michell in England

•! 1796: Pierre Simon Laplace in France

•! 1801: Johann von Soldner in Germany

The escape velocity ve at the surface

of a spherical mass M of radius r :

!

ve

=2 GM

r

Schwarzschild radius

deflection angle (Newton)

deflection angle (Einstein RG)

In 1919, during a total solar eclipse, first test of a GR prediction :

GR prediction at solar limb ! = 1.75" confirmed by Eddington (1920)

The Sun : first gravitational lens

rv

GM

rv

GM

22

2

2tan =!= "

"

kmM

M

c

GMR

sol

S95.2

2

2!"

r

R

rc

GMS

24

2==!

The phenomenon of gravitational lensing

1919 : the Sun : first gravitational lens

After the solar eclipse of 1919 and papers in 1930’s

subject ~ totally abandoned for about 30 years

1960’s – 1970’s

• Theoretical discussions •

• Inventions of CCDs •

• Discovery of quasars •

!

In 1979, first case of a gravitational lens

(extragalactic, i.e., at cosmological distance)

QSO 0957+561 Walsh, Carswell, Weymann, 1979, Nature, 279, 381

The disruptive action of the lensing galaxy splits

the single image of the quasar into two or more componants

QSO A image

QSO B image

QSO

ionmagnificat=!

!=

s

i

d

dµ

QSO 0957+561: first gravitational

lens at cosmological distance

•! HST/WFPC2

•! "(A,B) = 6.1"

•! VA = VB # 17.1

•! z(source)=1.41

•! VG=19.1

•! z(lens)=0.36

•! "(A,B) = 6.1"

•! VA = VB # 17.1

•! z(source)=1.41

Image A

Image B

Walsh, Carswell, Weymann

1979, Nature, 279, 381 CIV 1549 Å CIII] 1909 Å MgII 2798 Å

QSO 0957+561: first gravitational

lens at cosmological distance

All pairs of quasars observed in the

Universe do not always proceed

from the effect of a gravitational lens

PKS 1145-071: first tight pair of quasars

Djorgovski & Meylan, 1987, ApJ, 321, L17

A

B

z = 1.345

PKS 1145-071: first tight pair of quasars

Djorgovski & Meylan, 1987, ApJ, 321, L17

Radio image from VLA

IA/IB = 2.7 in V

IA/IB > 104 in radio

"(A,B) = 4.2"

PHL 1222: second tight pair of quasars

Meylan & Djorgovski (1994)

QSO A

QSO B

KECK U KECK V

"(A,B) = 4.2" z ! 1.91

PHL 1222: second tight pair of quasars

Meylan & Djorgovski (1991) "(A,B) = 4.2" z ! 1.91

LBQS 1429-008

•! Discovered by Hewett et al. (1989), two QSO components (A and B), proposed as a gravitational lens; z = 2.076

•! Suggested as a binary QSO by Kochanek et al. (1999), Mortlock et al. (1999), and Faure et al. (2003) –! No obvious lensing galaxy

–! Difficult to model as a lens

–! Faure et al. find no weak lensing distortion in the field

•! deep Keck and VLT images reveal additional components, one of which (C) is a QSO at the same redshift

CASTLES HST image

A

B

C

Hewett et al.

QQQ 1429-008 : first tight triplet of quasars

Djorgovski et al. (2007) "(A,B) = 4.2" z = 2.076

1

A

B

C 4

3

8

2

9

5 arcsec

I band (Keck)

1

A

B

C 4

3

8

2 9

5 arcsec

D?

K band (Keck + VLT)

Keck Spectra of the QSO Components

Absorbers: za1 = 1.512 (A,B), za2 = 1.662 (A), za3 = 1.837 (B,C)

$obs (Å)

%

%

%

%

% a1

a1

a1

a1

a1 a2

a1

a3

a3 a3 a3 a3

Gravitational Lens Models

•! Assume a singular isothermal sphere + external shear

–! A standard model which reproduces most known lenses

–! Use C. Keeton’s gravlens software

–! Explore the parameter space, seek the best fit solutions

•! Model always produces four QSO images; assume two

viable scenarios:

–! Model L1: the faint image D is the 4th component

–! Model L2: image A is an unresolved blend, "& < 0.05”

•! Both scenarios fail: –! L1: best reduced '2 = 1941 (!), image D is the brightest, images B

and C about equal, positions off by ~ 0.5”

–! L2: best reduced '2 = 74, image A is ~ 1.2” displaced

•! Conclude that the lensing hypothesis is unlikely

What about the lensing galaxy? Our “best” lensing model L2 predicts a massive and

luminous lens galaxy, which is not seen, even if placed

in an optimal position:

Putative

lens

Putative

lens

A B

C

Observed L2: z lens = 0.5 L2: z lens = 1.4

K lens > 24 mag K lens = 18.5 mag K lens = 17.1 mag

Flux Ratios of the QSO Spectra

Spectrum Differences •! Component C has a bluer UV continuum, but redder optical to

IR colors:

•! Spectrum differences between components A and B are about as

expected for a random pair of QSOs at this redshift (Mortlock et

al. 1999)

•! Different shape of the C IV line; possibly C III] as well

•! Marginal redshift differences from cross-correlation:

"VAB = 280 ± 160 km/s, "VBC = 100 ± 400 km/s

•! While the optical and IR flux ratio is A/B = 25 ± 3, but in X-rays

it is A/B = 5.3 ± 1.8 (from ChaMP; Kim et al. 2006)

A B C

(R-K) 2.49 ± 0.03 2.27 ± 0.03 3.23 ± 0.21

(J-K) 1.13 ± 0.03 0.85 ± 0.03 1.87 ± 0.13

(Due to a

contamination by the host

galaxy?)

Triple QSO vs. Gravitational Lens •! We are unable to reproduce the observed geometry and

intensities of images using plausible range of lensing

models

•! No evidence for a massive lensing galaxy in the images

•! No weak lensing distortions in the field (Faure et al.), even

if there was a dark, massive lens present

•! Observed spectroscopic and color differences are naturally

much easier to explain if these were physically distinct

AGN

•! Therefore, we conclude that this is most likely a case of a

physical close triple QSO

–! Projected separations are typical for interacting galaxy systems:

"&AB = 43 kpc, "&AC = 36 kpc, "&BC = 30 kpc (proper units, for

h = 0.7, (m = 0.3, () = 0.7 cosmology)

1

A

B

C

4

3

8

2

9

D?

R band (VLT)

MCS deconvolution

3 arcsec Disturbed host galaxy?

QQQ 1429-008 : first tight triplet of quasars

Djorgovski et al. (2007) "(A,B) = 4.2" z = 2.076

QQQ 1429-008 : first tight triplet of quasars

Djorgovski et al. (2007) "(A,B) = 4.2" z = 2.076

Conclusion about LBQS 1429-008

•! We see this system at a peak epoch of QSO activity and galaxy merging

–! Binary QSOs at comparable redshifts are known to occur with frequencies up to ~ 100 times higher than what may be expected from galaxy clustering alone

–! This can be understood if galaxy interactions are responsible for an onset of QSO activity

–! In this case, we may be witnessing a 3-galaxy interaction, with AGN occurring in all of them

•! Further studies of this system, and discoveries of more such QSO triples may provide useful new insights into a joint hierarchical formation of galaxies and SMBHs

•! For more details, please see astro-ph/0701155

3 directions: - to the lens

- to the source

- to the image

3 angles: - ! " #

3 plans: - of the source $

- of the image %

- of the observer

3 distances: - Ds Dd Dds

Theoretical bases Friedman-Lemaître-Robertson-Walker metrics

The inhomogeneities creating the effect

of gravitational lens are only local perturbations *

the luminous path is made of three independent parts :

The inhomogeneities creating the effect of gravitational lens are

only local perturbations * the luminous path is made of three

independent parts : i, ii, iii.

In a way similar to a prism, the light rays are deflected (by a very

small angle) while they travel through the gravitational field of a

point-like mass :

The deflection happens essentially for " z ~ ± b where "z « D

* the mass distribution can be projected along the line of sight

and replaced by the surface mass density + ( , ).

SRbbc

MGdz

c

242

22==!"=

#$!"

%

!

Theoretical bases

i

ii

iii

The lens equation

The positions of the source and of the images

are related by a non linear equation

providing the possibility of multiple images:

multiple image positions #

corresponding to a unique source position "

!

!

)(!"!#!!!!

$=

Distances at cosmological scale

The three distances Dd , Ds and Dds are

defined in such a way that the relation is true

in the space-time of GR : such distances are

called “angular-diameter distances”.

(in general)

with

dssdDDD !"

22

0

0

0 )1)(1(

))(1(2),(

ji

jiji

i

j

jizz

GGGG

H

czzD

++!

""!"==

#

$

2/1

0 )1(iizG !+=

Einstein radius in the case of a lens with axial symmetry

Thanks to the axial symmetry, a source on

the optical axis (" = 0) creates an image

with the shape of a ring with a radius value :

Einstein radius

!

!

" =!

# $! % (!

# ) & "(#) = # $Dds

DdDs

4GM(#)

c2#

!

"E

=4GM("

E)

c2

Dds

DdDs

#

$ %

&

' (

1/ 2

In many models, #E represents the frontier between the positions of

the sources generating either multiple images or a single image.

A star as a lens (Einstein’s pessimism)

of masse M # 1 M! at D # 10 Kpc:

A galaxy as a lens (Zwicky’s optimism)

of masse M # 10 11 M! at D # 1 Gpc:

2/12/1

109000.0

!

""#

$%%&

'""#

$%%&

'((=

Kpc

D

M

M

sol

E)

2/12/1

11110

9.0

!

""#

$%%&

'""#

$%%&

'((=

Gpc

D

M

M

sol

E)

B1938+666

HST-NICMOS

zsource = ??

zlentille = 0.881

King et al. 1998, MNRAS, 295, L41

Full Einstein ring

in the IR

(diameter # 1")

MG 0414+0534

HST-WFPC2 B

A2

A1 C

G diameter # 2.12"

zsource = 2.64

zlentille = 0.96

Courbin et al. 1999

HST-ACS image of RXJ 1131-1231!

HST imaging:!

- light profile of the lens !

- ellipticity and PA!

- astrometry of quasar images!

- lensed quasar host!

- other lensed objects!

VLT spectroscopy:!

- lens redshift!

-! chromatic microlensing!

Sluse et al. (2006)!

Abell 2218 HST WFPC2 zamas = 0.175 zsources ~ 0.7-5.6

Kneib et al., 1996, ApJ, 471, 643 -amas = 1370 ± 140 km s-1

235 large and small arcs

The gravitational lenses classified following three regimes:

•! STRONG: the source is imaged into several

components, their shapes and luminosities are

strongly perturbed

•! WEAK: one single image of the source, with

its shape and luminosity strongly perturbed

•! MICRO: one single image of the source, with

only its luminosity strongly perturbed

QSO 2237+0305: La Croix d’Einstein

HST-WFPC2

"(A,B)=1.7"

zsource = 1.69

zlentille = 0.04 A

B C

D

galaxie . macro lentille

étoiles . micro lentilles

QSO 2237+0305: La Croix d’Einstein

Adam et al., 1989, A&A, 208, L15

z = 1.69

slight complication from microlensing

QSO 2237+0305: La Croix d’Einstein

Les étoiles de la galaxie à z = 0.04 agissent comme autant

de microlentilles, induisant des variations des luminosités

dans les quatre composantes du quasar à z = 1.69.

Moyen très direct de mesurer la taille d’un QSO dans le visible

C B

D A

microlensing events

Einstein Cross ESO-VLT June 2006

Usefulness of gravitational lenses via microlensing

•! Mass distribution in our Galaxy

•! Constraints on the size of quasar sources

•! Upper limits on (PM from point masses (neutron stars, black holes)

•! Search for, and study of, exosolar planets

•! Frequency of binary stars

•! Enormous samples of variable stars

A galaxy acting as a lens: Isothermal sphere with central singularity (SIS)

Stars considered as the particles of a perfect gas, con-

fined by their own mean gravitational potential, with

spherical symmetry :

equation of state

thermal equilibrium

hydrostatic equilibrium 2

2

)(

r

drrMGg

dp

kTm

m

Tkp

v

=

=

=

!

"

!

A galaxy acting as a lens: Isothermal sphere with central singularity (SIS)

A simple solution : SIS

surface density

deflection angle

Multiple images only if the source verifies : " < #E

Solutions of the lens equation : #± = " ± #E

2

2 1

2)(

rGr

v

!

"# =

!

"!

1

2)(

2

G

v=#

2

12

2

)220(4.14

!""==

kmsc

vv##

$%!

A galaxy acting as a lens: Isothermal sphere with central core (CIS)

A simple solution : CIS

surface density

22

2 1

2)(

c

v

rrGr

+=

!

"#

22

2 1

2)(

c

v

rG +=!

"

#"

sc

dsdv

Dr

DD

cD

2

2

4!

"#

0

2

0

00

11

!

!"!

#++= D

defines the number of images

lens equation

A galaxy acting as a lens: Isothermal sphere with central core (CIS)

0

2/12

000 /]1)1[( !!!" #+#= D

multiple

images

if D > 2

critical lines and

positions of the images

(lens plane)

caustics and

position of the source

(source plane)

The local properties of the mapping source plane – lens plane are described

by its jacobian matrix : A . &" /&#

The locus of the points # in the lens plane where strongly disturbed images

are created is the set of points where the matrix A cannot be locally inver-

ted, i.e., where its jacobian is null * critical lines et caustics.

Surface brightness preserved :

photons neither created nor distroyed

The magnification µ is the ratio of the solid angles

of the images and of the sources, with A . &" /&#

!

µ =d"

I

d"S

= det(#!

$

#!

% )

&1

Adet

1=µ

d'I+

d'I- d'S

QSO HE 1104-1805 ESO-MPI 2.2-m IRAC J Courbin et al., 1998, ApJ, 330, 57 "(A,B)=3.19" zsource = 2.32 zlens = 0.73

Observations 0.7" After deconvolution 0.3"

The deconvolution provides an essential step

•

•

• • • •

•

• • •

•

• • •

•

•

•

•

critical lines caustics

HE1104-1805

H1413+117

PG1115+080

B1422+231

The local

properties of

the application

source plane – lens plane are described

by its jacobian

matrix :

A . &" /&#

The locus

of a point

where A cannot be

locally

inverted,

zero jacobian:

critical lines

and caustics.

movie

The Hubble constant and

the age of the Universe

http://cfa-www.harvard.edu/~huchra

Evolution of the Hubble constant Ho with time !

http://cfa-www.harvard.edu/~huchra

Evolution of the Hubble constant H0 with time !

Efstathiou, 2003: H0 between 37 and 72

WMAP data do not independantly constrain H0

(Spergel et al. 2003, 2006)

Data from WMAP Spergel et al. 2003

The age of the Universe from

the gravitational lenses

method based on cosmological distances

independant of any local calibrations contrary to the HST Cepheid Key Program

Time delay between two different

light paths with different lengths

* *

*

*

Intrinsic QSO light variations * time delay "/

obs

Time delay the travel time of a photon (Refsdal 1964, 1966)

•! The geometric term tgeom represents the time delay induced by the longer light path followed by the deflected photons.

•! The gravitational term tgrav represents the time delay due to the relativistic time dilation induced by the gravitational field of the deflector.

•! The term in front of the brackets ensures that the measured quantities correspond to the time delay as measured by the observer.

intrinsic variations * time delay "/ * H0

!

t(!

" ) =(1+ zd )

c

DdDs

Dds

1

2

!

" #!

$ ( )2

#%(!

" )&

' ( )

* + = tgeom + tgrav

Measure of the time delay in radio

QSO 0957+561 Haarsma et al., 1997, ApJ, 479, 102

Visible: "/ = 417 ± 3 days

H0 via QSO 0957+561

model: redshifts, positions, magnitudes, mass profile

Observations:

-v (lens) = 279 ± 12 km s-1

"/BA = 417 ± 3 days

!

H0 = 67 ± 8 km s-1 Mpc-1

Falco et al. 1997, ApJ, 484, 70

!

H0

= 98"11+12 # v

330km s"1

$

% &

'

( )

2

1.1yr

*+BA

$

% &

'

( ) km s

"1Mpc

"1

H0 via photometric monitoring for QSO RX 0911+05

•! 17

Burud et al., 2003

"/AB = 146 days

H0 = 74 ± 9 km s-1 Mpc-1

H0 via photometric monitoring for QSO RX 0911+05

•! 17

"/AB = 146 days

H0 = 74 ± 9 km s-1 Mpc-1

Burud et al., 2003

intrinsic luminosity fluctuations

! time delay measurement

! Hubble constant determination

!

age of the Universe

The Hubble constant from quasar time delays!

10 gravitational lenses * H0 = 61 ± 7 km s-1 Mpc-1

H0 = 72 ± 8 km s-1 Mpc-1

H0 = 61 ± 7 km s-1 Mpc-1

Goal: production of 20 time delays over the next few years

COSMOGRAIL COSmological MOnitoring of GRAvItational Lenses

For the monitoring:

•! Euler Swiss telescope, La Silla, Chile

•! Mercator Belgian-Swiss telescope, La Palma, Canary Islands

•! Maïdanak telescope, Uzbekistan

•! Manchester Robotic telescope, La Palma, Canary Islands

•! Himalayan Chandra telescope, Bangalore, India

For high-resolution photometry and spectroscopy:

•! ESO-VLT, KECK, GEMINI 8-10 meter-class telescopes

•! Hubble Space Telescope NASA/ESA

Hawaii

Paranal

COSMOGRAIL

COSmological Monitoring GRAvitatIonal Lenses

currently, 22 quasars observed as frequently as possible

Observations

La Palma Spain

Paranal La

Silla Cerro

Tololo Chili

Maidanak

Ouzbekistan Keck

Gemini

Hawaii

Himalayan

Chandra

Telescope India

In order to constrain models

we need a good knowledge of:

- the position of each image

- the luminosity of each image

- distance of the source

- distance of the lens

- masse of the lens

Determination of the redshifts of lensing galaxies with VLT !

Eigenbrod et al. 2006b, A&A 451, 759!

1.5 arcsec!

COSMOGRAIL : gravitational lens and time delay!

SDSS J1650+4251!

Image A!

Image B!

Vuissoz et al. 2006a, A&A, submitted, see astro-ph/0606317

Image A!

Image B!

Vuissoz et al. 2007, A&A, 464, 845!H0 = 52 ± 4 km s-1 Mpc-1

COSMOGRAIL : gravitational lens and time delay!

SDSS J1650+4251!QSO RXJ 1131-123

Claeskens et al. 2006 A&A 451 865!

Sluse et al. 2006 A&A 449 539

Detailed study of gravitational lenses!

HST imaging:!

- light profile of the lens !

- ellipticity and PA!

- astrometry of quasar images!

- lensed quasar host!

- other lensed objects!

VLT spectroscopy:!

- lens redshift!

- chromatic microlensing!

HST-ACS image of RXJ 1131-1231 (Sluse et al. 2006)!

HE 0435-1223

Lens with 4 images, zs = 1.69, zl = 0.45, separation = 2.6”

one clear Einstein ring connecting all four images

about 10 galaxies within 40”

HST IR images NIC 2

HE 0435-1223 5 seasons with Euler+Mercator+Maidanak (01/04-02/08) 1pt / 5j

+ measurements 2 seasons from SMARTS (Kochanek et al. 2006) HE 0435-1223

5 seasons with Euler+Mercator+Maidanak (01/04-02/08) 1pt / 5j

+ measurements 2 seasons from SMARTS (Kochanek et al. 2006)

Status in 2010 : The Hubble constant from quasar time delays!

18 time delays * H0 = 63.4 ± 8.4 km s-1 Mpc-1

HST KP : H0 = 74.2 ± 3.6 km s-1 Mpc-1

Lensing : H0 = 63.4 ± 8.4 km s-1 Mpc-1

Conclusions

•! Time delays are not so cheap :

- it takes time to observed for 5 seasons …

•!Time delays now with uncertainties smaller than 4 %

- including systematic

- at least twice better than before

- now main uncertainties coming from the slope of

the mass profile

•! H0 0 63.4 ± 8.4 km s-1 Mpc-1

•! If lenses are on average isothermal, then H0 0 72 ± 6 km s-1 Mpc-1

~ 15 more time delays in hand,

however, slow careful interpretation

“You may delay, but time will not.”

Benjamin Franklin

1706 - 1790

Convergence and shear

The local properties of the application source plane ! lens plane are described by its jacobian matrix: A . &" /&#

With the convergence 1 and the shear 2 :

The convergence 1 has a magnification action on the light rays:

the image conserve the shape of the source, but with a different size.

The shear induces an anisotropy with intensity 2 and orientation 3.

!!"

#$$%

&

''!!"

#$$%

&'=

((

(()*

2cos2sin

2sin2cos

10

01)1(A

*

Field of deformation

of background galaxies

without deformation with deformation

Abell 1689 HST ACS (2003)

zamas = 0.182

(a= 1848 ± 166 km s-1

Deep HST image

tint = 13.2 hours

Abell 1689 HST ACS

Abell 1689 HST ACS

Abell 1689 HST ACS (2003)

zamas = 0.182

(a= 1848 ± 166 km s-1

Deep HST image

tint = 13.2 heures

Thousands of mirages in this image !!!

z = 3.04

z = 3.04

Shear 2

as a function of

the convergence 1

weak lensing see lectures by P. Schneider

Seitz & Schneider,

1997, A&A, 318, 687

Reconstruction of the mass distribution

via the gravitational distortions

image colombi IAP

Weak Gravitational Lensing

•! Map the 3D distribution of DM in the Universe

•! Measures the mass without assumptions in relation between mass and light

•! Very sensitive to DE through both geometry and growth

Massey et al. Nature 2007

Based on Cosmos data

Sarah Bridle Great08

The shape of a galaxy at ~ 1% accuracy

The gravitational lensing phenomenon is ubiquitous

everywhere in the Universe on galactic scales as well as on cosmological scales

A new astrophysical tool :

since 1979, the phenomenon of gravitational lensing

is the subject of intense research activities, both theoretical and observational,

which have created a new tool for the study of the whole Universe,

from nearby planets to the most distant galaxies

Usefulness of gravitational lenses via strong and weak lensing

• Direct determination of the total mass of the lensing galaxy

• Direct determination of cosmological parameters:

- Hubble constant H0

- density parameters (m and ()

• Study of mass distribution of dark matter:

- in galaxies

- in clusters of galaxies

- in large scale structures

• Natural telescopes for the observations of very distant objects at very high redshifts