Bernard Fort
Institut d’astrophysique de Paris
Gravitational lenses in the Universe
ESO-Vitacura November 14, 2006
Part 1: Strong Lensingmultiple images regime
Historical lensing observations Fermat principle and lens equations
Lensing by a point mass
Lensing by mass distributions
Galaxy and cluster lensing: astrophysical applications
Weak lensing principles
Lensing mass reconstruction
The flexion regime
The cosmic shear: an overview
Part 2: weak and highly
singly magnified image regime
November 16, 2006
Deflection of light
Metric for the weak field approximation
gravitational achromatic lens
Fermat principle+
Equivalent to an optical index n <1
1801 Soldner: are the apparent positions of stars
affected by their mutual light deflection?
hyperbolic passage of a photon bulet with v = c:
tan (/2) = GM/(c2r)
1911 Einstein: finds the correct General Relativity answer
= 4GM/(c2r)
=> and predicts 2 x the newton value
A short history of lensing
Light deflection by the sun
= 4 G Mo / (c2 r) = 1.75 ‘’
1919, Eddington
measures = 1.6“
at the edge of the sun, confirming GR
r
Mo
r
1937 Zwicky: galaxies can act as lenses 1964 Refsdal: time delay and Ho 1979 Walsh & Weyman: double QSO 0957+561 CCD cameras 1887 giant arcs in cluster and first Einstein ring1993 Macho and Eros microlensing 1995 the weak lensing regime 2000 cosmic shear measurements 2005 discovery of an extrasolar earth like planet 2010-15 the golden age of lensing
History of lensing
Discovery of the double quasar (Walsh et al. 1984)
Lensing by Galaxies: HST Images
An Einstein gravitational ring
The Giant arc in A370
The second giant arc
Cl 2244
The cosmic optical bench
(or multiple thin lenses)
SL
Calculating the deflection angle
geometrical term
n
deflection angle
=
equation 1
for weak gravitational field
light propagation time is reduced inpresence of a gravitational field
Fermat principle yields the deflection angle
are very small => Born's approximation can be used
to remember
The thin lens equation
A
Cosmic optical bench ~ Natural optical telescope
O S
L
Dol
Time delay and thin lens
tgeom. = Dol /(2 c) = (Dos Dol / Dls) ()^2 / (2c)
tgrav. = - (2/c^2) (Dol ) dz
Fermat’s principle: (tgeom. + tgrav.) = 0
gives: – 2 (Dls/Dos) (Dol ) (Dol )
identifying with: Dos + Dls = Dos
gives: () = (2 / c^2) (Dol )
From Blanford and Kochanek lectures «gravitational lenses », 1986
Point mass M
equation 2
Total deviation for a 2D mass distribution
O S
LGpcs Gpcs
kpcs
Surface mass density
equation 3
m
(1)
equation 3
Thin lens equation
A
SO
L
Uniform sheet of constant mass density o g/cm2
= (1-o/crit)
If o = crit = 0 for any
The plan focuses any beams onto the observer
~ - / 2~ o / 2
/ 2
equation 4
Reduced quantities
critical density (g/cm2)
convergence =reduced surface density
deflection angle
Reduced thin lens equation
A
Non-linear projection through the reduced deflection angle
s) i)
( 5) (6)
But non linear lens
From the Liege university lensing team
Caustic surfacesenvelopes of families of rays ~ focal surfaces
The 2D Poisson equation
3D Poisson equation
Using Green's function of the2D Laplacian operator gives the potential from the mass distribution
(3) equation 7
(10) (8)
Light Travel Time and Image Formation
1 image
3 images
detour
Time dilation
Total light travel time
=source
O L S
Multiple images formation
Convergence+ shear
~
Local image properties
A = Jacobian matrix of the projection through the lens equation
If the potential gradient does not vary on the image size
(9)
to remember
convergence
complex shear
projection matrix
(10)
(9)
Magnification matrix M
(10)
Etherington theorem
The elementary surface brightness (flux / dx.dy) on each position the source is conserved on the conjugated point of the projected image (but seeing effects).
consequence: one can detect the presence of a lense only from the magnification and distortion of a geometrical shape. A lens in front of a uniform brightness distribution (or random distribution of points) cannot be seen.
Magnification Abs
surface magnification
Two eingen values 2 caustic lines
(11)
(12)
from (10)
Convergence map only
Shear map: (amplitude and direction)
Map of a circular sources grid
Ar_1 r,rr 00 1 1rrr
Ax_, y_1 x,xx, yx,yx, yx,yx, y1 y,yx, y
r_1 r,rr1 1rrr̂1
Cylindrical projected potential
radial caustic tangential caustic
1 r,rr 0 1 1rrr 0
Solving the lens equation for a point mass M
Point mass lens equation
-2 -1 1 2
-2
-1
1
2
3
s = |i – 1 / i|
ri1,ri2
1/r projected potential Ln (r)
rs
with angular radial coordinates in e unit
two images but one is
very demagnified
Einstein radius e
(13)
ring configuration for point mass or spherical potential
Source, lens, observer perfectly aligned
~ 1-3” for a lens galaxy
~ 10-50” for a cluster of galaxies
Magnification for a point mass
In[66]:= PlotAbsr,r, 3, 3
-3 -2 -1 1 2 3
2
4
6
8
10
Out[66]= Graphics
f1/f2 =
Multi-site observations
Lensing by moving star mass
note that f1 / f2 = (2 / 1)4
12
DM = MACHOS ?
Nature of DM
Microlensing by MACHOS(dark stars, BH,.. )
t
Microlenlensing event by the binary star MACHO 98
Microlensing an observational challenge!
Data mining: Need to distinguish
microlensing from numerous variable
stars.
Candidate MACHOs: Late M stars, Brown Dwarfs, planets Primordial Black Holes Ancient Cool White Dwarfs
<10-20% of the galactic halo is made of compact objects of ~ 0.5 M
MACHO: 11.9 million stars toward the LMC observed for 7 yr >17 events
EROS-2: 17.5 million stars toward LMC for 5 yr >10 events (+2 events from EROS-1)
To be updated!
Dark Halo: Microlensing results
searching hearth like planet
Ar_1 r,rr 00 1 1rrr
Ax_, y_1 x,xx, yx,yx, yx,yx, y1 y,yx, y
r_1 r,rr1 1rrr̂1
Spherical potential
radial caustic tangential caustic
1 r,rr 0 1 1rrr 0
Spherical isothermal potentials
SIS particules in thermal equilibrium everywhere (DM, stars)
_ 2
2 G _
ReRe
4 2 Dls Dol
c2 Dos
2 k T
m
3Dr_ ^2
G M
deviation = constante
(13)
X-section ~ 4
to remember for SIS
central singularity
isothermal potential with core radius: SISrc
_oe^2 c^2
new Einstein radius e
e^2 c^2
deviation ~ if << c = constante if >> c
(14)
-3 -2 -1 1 2 3
-2
-1.5
-1
-0.5
0.5
1
1.5
2
Re
Isothermal potential with a core radius
Equivalent to a flat rotation curve
Parity changes
SIS
Universal Cold DM density profile
Numerical simulations gives:
~ 1
Navarro, Franck and White potential 1998
with
(15)
(16)
with galaxy & cluster potentials
also ellipsoidal dark matter halos
M(r >) converges
Central part:
DM+stars
effective deviation angle
Elliptical potential
q ~ ellipticity parameter
/Local surface magnification
Back to caustics and critical lines with projected elliptical potential
Locus of caustics lines in the source plan projected into critical lines in the image plan where become infinite
= 0 = 0
images for a non-singular elliptical lens.
Radial arc
Cusp arc
Einstein Cross
Fold arc
Singly magnified image
From Kneib et al 1993
rays
caustics
Caustics (focal surfaces)
rays: critical points of path length (Fermat-Hamilton)
field point
x . , z
initial wavefront, h(t)
t
path length
t;x, z
z h t 2 x t 2
t 0 and t2 0
t 0
rays and caustics
caustics are physical catastrophes described by the theory of Thom and Arnold
1 t;x t3 xt
variable
parameter
smooth function
1
t
x<01
t
x=0
1
t
x>0
critical points: ∂1/∂t=0
Multiple images of the sun on Villarica lake
images are places on the water where the distance sun-water-eye is stationary
Multiple caustics with merging
Caustics images drawn by a distant distant sun on the bottom of a swimming pool (a reverse light propagation with the sun as an observer )
Light curve of OGLE235
a binary system with a big jupiter like satellite
Binary events was first suggested by Mao & Paczynski, 1991, ApJL, 374, 40
Aplication of lensing in cosmology
Newtonian gravitational potentialCosmology Cosmology geometry Newtonian potential
Image magnification
Beyond z=6 with Strong Gravitational Lenses
From Kneib et al 2005
Measuring Ho from time delay
Cuevad Tello et al; 200670 +/- 10
Image location potential modelingdelay Ho
RCS1 giant arc sample from Gladders et al 2005
Some arcs have Einstein radius up to 50 "
A1689, RCS 0224
Specific X-ray Cluster surveys
Modeling A370
From Kneib et al 1993
LENSTOOL (strong and weak regime 1993 - 2006)people who wrote part of this project ( in chronological order ): Jean-Paul Kneib (1993), Henri Bonnet , Ghyslain Golse, David Sand, Eric Jullo, Phil Marshall
GRAVLENS 2005- Software for Gravitational Lensing by Chuck Keeton
Lensview 2006: Software for modelling resolved gravitational lens images B. Wayth & R. Webster
Many others: Rigaud, Kovner, Kochanek, Barthelmann, Gavazzi,Valls-Gabaud, Soyu..
Modelling softwares
Cf: seminar Marceau Limousin November 15
Probing the density profile of DM halos
inside ~ 10 kpcs ?10 < r < 2-300 kpcs r -2 r > 2-300 kpcs, maybe r -3
Cf. seminar Marceau Limousin Nov 15,2006
Results with MS2137-23
elliptic halo => collisionless DM, Miralda-escude 96; coupling a dual modeling SL-WL with a dynamical study of stars: profile compatible with NFW simulations for r > 10 kpcs; triaxial ellipsoïd projection effect (potential twist from radial to tangential images); MOND does not work ; Bartelmann 98, Gavazzi et al. 2002, 03, 04,..)
Testing DM halo shape with several arc systems
Several multiple image systems can probe a dark matter twist of ellipticity
Gavazzi et al 2004
Internal potential
external potential
Detection of dark Matter clumps
• Bonnet et al in Cl0024+1654 (WL)
• Weinberg & Kamionsky 2003 theoretical predictions for non virialized cluster mass still in the merging process.
Simulations: CDM halos are lumpy
(Bradac et al. 2002)
Substructure complicated catastrophes!
Dalal and Kochanek 2002
Fraction of the observed image brightnesses deviating from the
best smooth model fit?
(Dark) halo sub-structures can explain QSO anomalies !
Sub-halo analysis with simulations
The Einstein cross
No Dark Matter at the center of the galaxy!
Coupling lensing and stellar dynamics
Lens modelling give the mass at rEinstein andDM
Stars see the potential for r < reff Jeans equation
M* / Lv anisotropy= (M* / L, , v
anisotropyspectro
observation
(~ potential slope
from Koopmann & Treu 2005
SLACS
Lensing -> recovers the Ellipticals fundamental planeFor isolated E (external shear perturbation < 0.035)
<L/*> = 1.01 +/- 0.065 rms
(r) ~ r - 2.01 +/- 0.03 near Einstein Radius (~Flat Rot.Curve)
PA and ellipticity of light and DM trace each other ( M*~75%)
No evolution (<10%) of parameters with z (but more galaxies around <ZL>~0.2)
A SL2S cosmological tests with rings ?
Hypothesis: Treu's results
<L/*> =1. +/- 0.065
r(r) ~ r - 2.01+/-0.03 at Re ~ Flat Rot. Curve (DM light-conspiracy)
Re/L = Dol Dls /D os
Re/* = G (, or w0,w1)Log r
Re
Lens modeling
VLT spectroscopy
Simulations: CDM halos are lumpy
typical galaxy,~1012 Mo
should contain many sub-haloscorresponding to smallest satellite galaxies. Where
are they?
(Moore et al. 1999; Klypin et al. 1999)
QSO image anomalies
Fact
• In 4-image lenses, the image positions can be fit by smooth lens models:
positions determined by itrue i
smooth
• The flux ratios cannot; brightnesses determined by
ijtrue = ij
smooth + ijsub
• Interpretation• Flux ratios are perturbed by substructure in the lens
potential. (Mao & Schneider 1998; Metcalf & Madau 2001; Dalal & Kochanek 2002).
Is there Halo Sub-Structure?(e.g. Dalal and Kochanek 2001,2002)
1 image
3 images
B1555 radio
Images A and B should be equally bright!
Micro-lensing by stars? Maybe
Halo Sub-structure ?
Testing rotation curves
(Sanders & McGaugh 2002)
SIS mass distribution:
~ 1-3” for a lens galaxy ~ 10-50” for a cluster of galaxies
Where are the intermediate mass lenses ?
(3’’<< 7’’) ?
Cf ESO seminar Bernard Fort November 24
Does it exist cosmic strings lenses?
SNAPJoint Dark Energy Mission: NASA (75%) & DOE (25%) launch 2014-2015
6 years survey: super novae and weak lensing SNAP: 2m telescope, instrument FOV 1 deg2
Imaging / spectro. one deep field (15 deg2), one large field (~300 deg2 ?) ~ 1Billlion $
• DUNE (Dark Universe Explorer): similar survey but
1.2-1.5m telescope and imaging only instrument FOV 1 deg2
~ 300 M€
•Prediction snap n ~ 4000 and 14000 strong lenses
JWSTJWST: Le successeur de Hubble dans l’Infrarouge
• Un miroir de 6,6 m
• Lancement en 2011 mission de 5 à 10 ans
INSTRUMENT MIRISpectro-imageur, 5-28 μm
Participation française focalisée autour du banc optique de l’imageur (détecteur intégré au RAL, UK)
Responsabilité managériale de la partie française
Responsabilité « système » de l’ensemble