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A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
A primer on Gravitational Lensing
Eduardo S. Cypriano1
1Departamento de AstronomiaInstituto de Astronomia, Geofısica e Ciencias Atmosfericas
Universidade de Sao Paulo
IFT School on Dark Mater, 2016
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Outline
Gravitational Lensing basics
Micro lensing
Strong lensing
Weak Lensing
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Gravitational lensing geometry
Figure from Narayan & Bartelmann (1996; arXiv:astro-ph/9606001)
I Parameters:I Angular diameter distances: Ds , Dd and Dds
I Source position: β = η/Ds - Impact parameterI Image position: θ = ξ/Dd
I Deflexion angle: α = α Dds/Ds
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The Lensing Equation
I
η =Ds
Ddξ − Ddsα(ξ)
I Using the angular quantities instead of the physical oneswe get:
β = θ − Dds
Dsα(Ddθ)
I Then by using the physical deflexion angle we arrive tothe Lens Equation:
β = θ −α(θ)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The Lensing Equation
I
η =Ds
Ddξ − Ddsα(ξ)
I Using the angular quantities instead of the physical oneswe get:
β = θ − Dds
Dsα(Ddθ)
I Then by using the physical deflexion angle we arrive tothe Lens Equation:
β = θ −α(θ)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The Lensing Equation
I
η =Ds
Ddξ − Ddsα(ξ)
I Using the angular quantities instead of the physical oneswe get:
β = θ − Dds
Dsα(Ddθ)
I Then by using the physical deflexion angle we arrive tothe Lens Equation:
β = θ −α(θ)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The deflection angle
I From GR one can estimate the deflection angle due to amass point:
α =4 G M
c2ξ
I The projected angle is then:
α = |α| =Dds
Ds
4 G M
c2Dd |θ|
I Considering now a direction, as all the angles lies in theimage-lens direction, we have:
α =Dds
Ds
4 G M
c2Dd |θ|θ
|θ|
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The deflection angleI This prediction lead to the first observational proof of
the General Relativity
ANNOUNCEMENTOFEDDINGTON’SDISCOVERY.CREDIT:ILLUSTRATEDLONDON NEWS(1919)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The lensing effect
I Gravitational lensing causes several effects on theimages of the sources:
1. Radial displacement
2. Multiple imaging (given certain conditions)3. Magnification of the angular size
surface brightness conservation−−−−−−−−−−−−−−−−−−→ and flux
4. Distortion5. Time Delay
I The prevalence/interest of one or more of those effectsover the others have to do with the lensing regime.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The lensing effect
I Gravitational lensing causes several effects on theimages of the sources:
1. Radial displacement2. Multiple imaging (given certain conditions)
3. Magnification of the angular size
surface brightness conservation−−−−−−−−−−−−−−−−−−→ and flux
4. Distortion5. Time Delay
I The prevalence/interest of one or more of those effectsover the others have to do with the lensing regime.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The lensing effect
I Gravitational lensing causes several effects on theimages of the sources:
1. Radial displacement2. Multiple imaging (given certain conditions)3. Magnification of the angular size
surface brightness conservation−−−−−−−−−−−−−−−−−−→ and flux4. Distortion5. Time Delay
I The prevalence/interest of one or more of those effectsover the others have to do with the lensing regime.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The lensing effect
I Gravitational lensing causes several effects on theimages of the sources:
1. Radial displacement2. Multiple imaging (given certain conditions)3. Magnification of the angular size
surface brightness conservation−−−−−−−−−−−−−−−−−−→ and flux
4. Distortion5. Time Delay
I The prevalence/interest of one or more of those effectsover the others have to do with the lensing regime.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The lensing effect
I Gravitational lensing causes several effects on theimages of the sources:
1. Radial displacement2. Multiple imaging (given certain conditions)3. Magnification of the angular size
surface brightness conservation−−−−−−−−−−−−−−−−−−→ and flux4. Distortion
5. Time Delay
I The prevalence/interest of one or more of those effectsover the others have to do with the lensing regime.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The lensing effect
I Gravitational lensing causes several effects on theimages of the sources:
1. Radial displacement2. Multiple imaging (given certain conditions)3. Magnification of the angular size
surface brightness conservation−−−−−−−−−−−−−−−−−−→ and flux4. Distortion5. Time Delay
I The prevalence/interest of one or more of those effectsover the others have to do with the lensing regime.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Lensing regimes
I Different lensing regimes differentiate themselves by:
1. The lens mass distribution2. The distances involved3. The impact parameter
I The main lensing regimes are:
1. Micro lensing2. Strong lensing3. Weak lensing
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Lensing regimes
I Different lensing regimes differentiate themselves by:
1. The lens mass distribution2. The distances involved3. The impact parameter
I The main lensing regimes are:
1. Micro lensing2. Strong lensing3. Weak lensing
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Micro lensing
1. The lens mass distribution
: Compact, “point mass”
2. The distances involved
: O ∼ tenths of kpc
3. The impact parameter
: variable with very smallminimum
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Micro lensing
1. The lens mass distribution: Compact, “point mass”
2. The distances involved
: O ∼ tenths of kpc
3. The impact parameter
: variable with very smallminimum
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Micro lensing
1. The lens mass distribution: Compact, “point mass”
2. The distances involved: O ∼ tenths of kpc
3. The impact parameter
: variable with very smallminimum
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Micro lensing
1. The lens mass distribution: Compact, “point mass”
2. The distances involved: O ∼ tenths of kpc
3. The impact parameter: variable with very smallminimum
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The point mass lens
I The lens equation can be easily solved in this case.
I Lets first define a ‘natural scale’, the Einstein angle forthis lens:
θE =
√4GM
c2
Dds
DsDd
I The lens equation then becomes:
β = θ −θ2E
θ
I It has two solutions:
θ± =1
2
(β ±
√β2 + 4θ2
E
)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The point mass lens
I The lens equation can be easily solved in this case.
I Lets first define a ‘natural scale’, the Einstein angle forthis lens:
θE =
√4GM
c2
Dds
DsDd
I The lens equation then becomes:
β = θ −θ2E
θ
I It has two solutions:
θ± =1
2
(β ±
√β2 + 4θ2
E
)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The point mass lens
I The lens equation can be easily solved in this case.
I Lets first define a ‘natural scale’, the Einstein angle forthis lens:
θE =
√4GM
c2
Dds
DsDd
I The lens equation then becomes:
β = θ −θ2E
θ
I It has two solutions:
θ± =1
2
(β ±
√β2 + 4θ2
E
)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The point mass lens
I What is ‘natural’ about the Einstein angle or radius ?
I If β = 0→ θ = θE
I The distance between two images (solutions) is 2θEI The density inside the the Einstein radius is the critical
lensing density
Σcr =c2
4πG
Ds
Dd Dds
above which multiple imaging occur.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The point mass lens
I What is ‘natural’ about the Einstein angle or radius ?
I If β = 0→ θ = θE
I The distance between two images (solutions) is 2θEI The density inside the the Einstein radius is the critical
lensing density
Σcr =c2
4πG
Ds
Dd Dds
above which multiple imaging occur.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The point mass lens
I What is ‘natural’ about the Einstein angle or radius ?
I If β = 0→ θ = θE
I The distance between two images (solutions) is 2θE
I The density inside the the Einstein radius is the criticallensing density
Σcr =c2
4πG
Ds
Dd Dds
above which multiple imaging occur.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The point mass lens
I What is ‘natural’ about the Einstein angle or radius ?
I If β = 0→ θ = θE
I The distance between two images (solutions) is 2θEI The density inside the the Einstein radius is the critical
lensing density
Σcr =c2
4πG
Ds
Dd Dds
above which multiple imaging occur.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The point mass lensSolution
θ± =1
2
(β ±
√β2 + 4θ2
E
)x=θ/θE ; y=β/θE−−−−−−−−−−→ x± =
1
2
(y ±
√y + 4)
)Point source
Figura de Narayan & Bartelmann
1996, arXiv:astro-ph/9606001
Extended source
Figura de Narayan & Bartelmann
1996, arXiv:astro-ph/9606001
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The point mass lensSolution
θ± =1
2
(β ±
√β2 + 4θ2
E
)x=θ/θE ; y=β/θE−−−−−−−−−−→ x± =
1
2
(y ±
√y + 4)
)Point source
Figura de Narayan & Bartelmann
1996, arXiv:astro-ph/9606001
Extended source
Figura de Narayan & Bartelmann
1996, arXiv:astro-ph/9606001
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Magnification
I The flux magnification qis equal to the ration of theimage and source areas.
I For a circularly symmetric lens:
µ =θ
β
dθ
dβ
I For the point lens, then
µ± =
[1−
(θEθ±
)4]
,
I and
µ = |µ+|+ |µ−| =y2 + 2
y√y2 + 4
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Magnification
I The flux magnification qis equal to the ration of theimage and source areas.
I For a circularly symmetric lens:
µ =θ
β
dθ
dβ
I For the point lens, then
µ± =
[1−
(θEθ±
)4]
,
I and
µ = |µ+|+ |µ−| =y2 + 2
y√y2 + 4
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Magnification
I The flux magnification qis equal to the ration of theimage and source areas.
I For a circularly symmetric lens:
µ =θ
β
dθ
dβ
I For the point lens, then
µ± =
[1−
(θEθ±
)4]
,
I and
µ = |µ+|+ |µ−| =y2 + 2
y√y2 + 4
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Micro lenses
I In nature the micro lenses are usually stars, planets andother compact objects in or close to the Galaxy.
I Due to its proximity the proper motion of those objectsis relevant.
I Figura de Schneider 2006
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Micro lenses
I In nature the micro lenses are usually stars, planets andother compact objects in or close to the Galaxy.
I Due to its proximity the proper motion of those objectsis relevant.
I Figura de Schneider 2006
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Micro lenses
I Typical Einstein radius
θE = 0.902mas
(M
M�
)1/2( Dd
10kpc
)−1/2(1− Dd
Ds
)−1/2
I We cannot observationally resolve the multiple imaging,but we can measure the variation of the flux due to themagnification
I Note that microlensing should produce symmetric andachromatic light curves as above.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Micro lenses
I Typical Einstein radius
θE = 0.902mas
(M
M�
)1/2( Dd
10kpc
)−1/2(1− Dd
Ds
)−1/2
I We cannot observationally resolve the multiple imaging,but we can measure the variation of the flux due to themagnification
I Note that microlensing should produce symmetric andachromatic light curves as above.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Micro lenses
I Typical Einstein radius
θE = 0.902mas
(M
M�
)1/2( Dd
10kpc
)−1/2(1− Dd
Ds
)−1/2
I We cannot observationally resolve the multiple imaging,but we can measure the variation of the flux due to themagnification
I Note that microlensing should produce symmetric andachromatic light curves as above.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Astrophysical Applications of Microlensing
1. Extra solar planet detection
OGLE-2005-BLG-390; Credit:ESO
2. Stellar & Galactic astrophysics
3. Quantification of the MAssive Compact Halo Objects(MACHOs) in the Galaxy.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Astrophysical Applications of Microlensing
1. Extra solar planet detection
OGLE-2005-BLG-390; Credit:ESO
2. Stellar & Galactic astrophysics
3. Quantification of the MAssive Compact Halo Objects(MACHOs) in the Galaxy.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Astrophysical Applications of Microlensing
1. Extra solar planet detection
OGLE-2005-BLG-390; Credit:ESO
2. Stellar & Galactic astrophysics
3. Quantification of the MAssive Compact Halo Objects(MACHOs) in the Galaxy.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
MACHOs in the Galaxy
I Two main projects: MACHO & EROS: Observation ofthe LMC & SMC
Figure from Alcock et al. 2000
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
MACHOs in the Galaxy
I Macho project: 5.7 years; LMC.
→ 13-17 events ; 100%MACHO halo ruled out with 95% c.l.
I EROS project: 5 years; SMC
→ 5 events; 25% of less ofthe MW halo is composed by MACHOs (95% c.l.)
Figure from Afonso et al. 2003
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
MACHOs in the Galaxy
I Macho project: 5.7 years; LMC. → 13-17 events ; 100%MACHO halo ruled out with 95% c.l.
I EROS project: 5 years; SMC
→ 5 events; 25% of less ofthe MW halo is composed by MACHOs (95% c.l.)
Figure from Afonso et al. 2003
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
MACHOs in the Galaxy
I Macho project: 5.7 years; LMC. → 13-17 events ; 100%MACHO halo ruled out with 95% c.l.
I EROS project: 5 years; SMC
→ 5 events; 25% of less ofthe MW halo is composed by MACHOs (95% c.l.)
Figure from Afonso et al. 2003
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
MACHOs in the Galaxy
I Macho project: 5.7 years; LMC. → 13-17 events ; 100%MACHO halo ruled out with 95% c.l.
I EROS project: 5 years; SMC → 5 events; 25% of less ofthe MW halo is composed by MACHOs (95% c.l.)
Figure from Afonso et al. 2003
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
MACHOs in the Galaxy
Microlensing surveys, in practice, consolidate the notion thatdark matter must be constituted by some kind of (non-baryonic) particles, instead of microscopic objects
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Strong lensing
1. The lens mass distribution
: extended - galaxies andgalaxy clusters
2. The distances involved
: O ∼ Gpc
3. The impact parameter
: ≤ θE → multiple imaging
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Strong lensing
1. The lens mass distribution: extended - galaxies andgalaxy clusters
2. The distances involved
: O ∼ Gpc
3. The impact parameter
: ≤ θE → multiple imaging
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Strong lensing
1. The lens mass distribution: extended - galaxies andgalaxy clusters
2. The distances involved: O ∼ Gpc
3. The impact parameter
: ≤ θE → multiple imaging
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Strong lensing
1. The lens mass distribution: extended - galaxies andgalaxy clusters
2. The distances involved: O ∼ Gpc
3. The impact parameter: ≤ θE → multiple imaging
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Extendend lenses
I In practice the lens dimensions are much smaller thanthe other distances involved.
I In this limit the thin lens approximation is valid:
The relevant quantity is not the 3D mass distribution(ρ) but instead the projected (2D) mass density (Σ)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Extendend lenses
I In practice the lens dimensions are much smaller thanthe other distances involved.
I In this limit the thin lens approximation is valid:
The relevant quantity is not the 3D mass distribution(ρ) but instead the projected (2D) mass density (Σ)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Extendend lenses
I In practice the lens dimensions are much smaller thanthe other distances involved.
I In this limit the thin lens approximation is valid:The relevant quantity is not the 3D mass distribution(ρ) but instead the projected (2D) mass density (Σ)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Extendend lenses
I The deflection angle can then be calculated as the sumofthe contributions of every mass element:
α =∑i
4 G mi
c2
ξ − ξi|ξ − ξi |2
I For a continuous distribution mass distribution:
α(ξ) =4G
c2
∫Σ(ξ′)
ξ − ξ′
|ξ − ξ′|2d2ξ′
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Extendend lenses
I The deflection angle can then be calculated as the sumofthe contributions of every mass element:
α =∑i
4 G mi
c2
ξ − ξi|ξ − ξi |2
I For a continuous distribution mass distribution:
α(ξ) =4G
c2
∫Σ(ξ′)
ξ − ξ′
|ξ − ξ′|2d2ξ′
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Lens models
I Singular Isothermal Sphere (SIS)
ρ(r) =σ2v
2πGr2
- Elliptical galaxies
I Navarro, Frenk & White (NFW)
ρ(r) =ρs
(r/rs)(1 + r/rs)2
- Clusters of Galaxies
ρ(r < rs) ∝ r−1; ρ(r ∼ rs) ∝ r−2; ρ(r > rs) ∝ r−3
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Lens models
I Singular Isothermal Sphere (SIS)
ρ(r) =σ2v
2πGr2
- Elliptical galaxies
I Navarro, Frenk & White (NFW)
ρ(r) =ρs
(r/rs)(1 + r/rs)2
- Clusters of Galaxies
ρ(r < rs) ∝ r−1; ρ(r ∼ rs) ∝ r−2; ρ(r > rs) ∝ r−3
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Lens models
I Singular Isothermal Sphere (SIS)
ρ(r) =σ2v
2πGr2
- Elliptical galaxies
I Navarro, Frenk & White (NFW)
ρ(r) =ρs
(r/rs)(1 + r/rs)2
- Clusters of Galaxies
ρ(r < rs) ∝ r−1; ρ(r ∼ rs) ∝ r−2; ρ(r > rs) ∝ r−3
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Lens models
I Singular Isothermal Sphere (SIS)
ρ(r) =σ2v
2πGr2
- Elliptical galaxies
I Navarro, Frenk & White (NFW)
ρ(r) =ρs
(r/rs)(1 + r/rs)2
- Clusters of Galaxies
ρ(r < rs) ∝ r−1; ρ(r ∼ rs) ∝ r−2; ρ(r > rs) ∝ r−3
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Lens models
I Singular Isothermal Sphere (SIS)
ρ(r) =σ2v
2πGr2
- Elliptical galaxies
I Navarro, Frenk & White (NFW)
ρ(r) =ρs
(r/rs)(1 + r/rs)2
- Clusters of Galaxies
ρ(r < rs) ∝ r−1; ρ(r ∼ rs) ∝ r−2; ρ(r > rs) ∝ r−3
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Strong Lensing ModellingI Given a lens mass model and the position of the images
we can solve the lens equation
β = θ −α(θ)
I This is a mapping problem:
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Strong Lensing ModellingI Given a lens mass model and the position of the images
we can solve the lens equation
β = θ −α(θ)
I This is a mapping problem:
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Strong Lensing ModellingCircular Lens
Source Plane Image Plane
Figure from Hattori, Kneib & Makino, arXiv:astro-ph/9905009
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Strong Lensing ModellingCircular Lens
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Strong Lensing ModellingElliptical Lens
Source Plane Image Plane
Figure from Hattori, Kneib & Makino, arXiv:astro-ph/9905009
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Strong Lensing ModellingElliptical Lens
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Giants arcs in Galaxy ClustersResults
1. Reinforcement that clusters are dark matter dominated(M/L ∼ 100− 300M�/L�)
2. Arcs are large and relatively regular
→ The dark matteris smoothly distributed instead of attached to galaxies.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Giants arcs in Galaxy ClustersResults
1. Reinforcement that clusters are dark matter dominated(M/L ∼ 100− 300M�/L�)
2. Arcs are large and relatively regular
→ The dark matteris smoothly distributed instead of attached to galaxies.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Giants arcs in Galaxy ClustersResults
1. Reinforcement that clusters are dark matter dominated(M/L ∼ 100− 300M�/L�)
2. Arcs are large and relatively regular → The dark matteris smoothly distributed instead of attached to galaxies.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Giants arcs in Galaxy ClustersResults
1. Reinforcement that clusters are dark matter dominated(M/L ∼ 100− 300M�/L�).
2. The dark matter is smoothly distributed instead ofattached to galaxies.
3. Strong lensing clusters are not symmetrical and oftensubstructured.
4. The slope of the mass profile at the very centre isshallower than numerical predictions.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Giants arcs in Galaxy ClustersResults
1. Reinforcement that clusters are dark matter dominated(M/L ∼ 100− 300M�/L�).
2. The dark matter is smoothly distributed instead ofattached to galaxies.
3. Strong lensing clusters are not symmetrical and oftensubstructured.
4. The slope of the mass profile at the very centre isshallower than numerical predictions.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Giants arcs in Galaxy ClustersResults
1. Reinforcement that clusters are dark matter dominated(M/L ∼ 100− 300M�/L�).
2. The dark matter is smoothly distributed instead ofattached to galaxies.
3. Strong lensing clusters are not symmetrical and oftensubstructured.
4. The slope of the mass profile at the very centre isshallower than numerical predictions.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Galaxy strong lensing
Figure from Bolton et al. (2008)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Galaxy strong lensingResults for early-type galaxies
1. Dark matter is about 1/4 of the mass at the half-lightradius (70% at 5 re).
2. The mass profile is isothermal (ρ ∝ r−2) with very littlescatter.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Galaxy strong lensingResults for early-type galaxies
1. Dark matter is about 1/4 of the mass at the half-lightradius (70% at 5 re).
2. The mass profile is isothermal (ρ ∝ r−2) with very littlescatter.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Weak lensing
1. The lens mass distribution
: extended - galaxies, clustersand the large scale structure
2. The distances involved
: O ∼ Gpc
3. The impact parameter
: � θE
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Weak lensing
1. The lens mass distribution: extended - galaxies, clustersand the large scale structure
2. The distances involved
: O ∼ Gpc
3. The impact parameter
: � θE
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Weak lensing
1. The lens mass distribution: extended - galaxies, clustersand the large scale structure
2. The distances involved: O ∼ Gpc
3. The impact parameter
: � θE
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Weak lensing
1. The lens mass distribution: extended - galaxies, clustersand the large scale structure
2. The distances involved: O ∼ Gpc
3. The impact parameter: � θE
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Gravitational Image Distortion
I The gravitational lensing distorts the images
I The isotropic distortion is associated with theconvergence: κ = Σ(r)/Σcr
and anisotropic with the shear: γt × Σcr = Σ(r)− Σ(r)
I Both κ and γ are second derivatives of the lensingeffective potential.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Gravitational Image Distortion
I The gravitational lensing distorts the images
I The isotropic distortion is associated with theconvergence: κ = Σ(r)/Σcr
and anisotropic with the shear: γt × Σcr = Σ(r)− Σ(r)
I Both κ and γ are second derivatives of the lensingeffective potential.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Gravitational Image Distortion
I The gravitational lensing distorts the images
I The isotropic distortion is associated with theconvergence: κ = Σ(r)/Σcr
and anisotropic with the shear: γt × Σcr = Σ(r)− Σ(r)
I Both κ and γ are second derivatives of the lensingeffective potential.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
I Given the large impact parameter the distortion is verysmall and is dominated by the galaxies own shapes.
I The weak shear can be detected statistically, however,by using samples with large numbers of galaxies:〈e〉 = γ/(1− κ) ' γ
I Allow mass measurements far beyond the strong lensingarea.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
I Given the large impact parameter the distortion is verysmall and is dominated by the galaxies own shapes.
I The weak shear can be detected statistically, however,by using samples with large numbers of galaxies:〈e〉 = γ/(1− κ) ' γ
I Allow mass measurements far beyond the strong lensingarea.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
I Given the large impact parameter the distortion is verysmall and is dominated by the galaxies own shapes.
I The weak shear can be detected statistically, however,by using samples with large numbers of galaxies:〈e〉 = γ/(1− κ) ' γ
I Allow mass measurements far beyond the strong lensingarea.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
I Weak-lensing measurements are quite delicate as onehas to measure shapes of small source galaxies
Figure from Bridle et al. (2008; arXiv:0802.1214)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Weak lensing mass mapsI The grav. shear field can be used to reconstruct the
projected mass distribution
Figure from Richard Massey (CalTech)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
The Bullet Cluster
I On the merging cluster 1E 0657-56 gravity does notfollow the baryonic mass (mostly gas)
One of the most convincing evidences of the existence of darkmatter.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Cluster Mass Profiles
I General high quality results can be obtaining bystacking the weak lensing signal of samples of lenses.
Figure from Okabe et al. (2010)
I The cluster mass profile is curved in a log-log space andconsistent with numerical simulations such as NFW.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Cluster Mass Profiles
I General high quality results can be obtaining bystacking the weak lensing signal of samples of lenses.
Figure from Okabe et al. (2010)
I The cluster mass profile is curved in a log-log space andconsistent with numerical simulations such as NFW.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Mass measurementsI Weak grav. lensing cluster mass measurements are
regarded as the most trustworthy as they do not sufferfrom the hydrostatic mass bias.
I They are ideal to calibrate mass-observable relations.
Figure from Mantz et al. (2016; arXiv:1606.03407)
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Mass measurements
I Those scale relations are crucial for cluster-count basedcosmology. Constrains in the (dark) matter density andclumpiness
Figure from Mantz et al. (2014)
I With a larger redshift range constrains on dark energyas well.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Mass measurements
I Those scale relations are crucial for cluster-count basedcosmology. Constrains in the (dark) matter density andclumpiness
Figure from Mantz et al. (2014)
I With a larger redshift range constrains on dark energyas well.
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Cosmic shear
I Weak lensing by the large scale structure.
Figures from http://www.cfht.hawaii.edu/News/Lensing/
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Cosmic shear
I The signal is even smaller than the cluster weak lensing
I Measurements uses the 2-point correlation of shapes.
Figure from Kaiser et al. (2000; arXiv:astro-ph/0003338
A primer onGravitational
Lensing
Eduardo S.Cypriano
GravitationalLensing basics
Gravitational lensinggeometry
The lens equation
The deflection angle
Micro lensing
The point mass lens
Magnification
Micro lenses
AstrophysicalApplications ofMicrolensing
Strong lensing
Extendend lenses
Lens models
Strong LensingModelling
Main Results
Weak Lensing
Gravitational ImageDistortion
The Mass Distributionin Clusters
Cosmic shear
Cosmic shear
I Similar sensitive to cosmological parameters ascluster-counts.
Figure from Heymans et al. (2013)
I Cosmics shear and Cluster counts have the potential todiscriminate between dark energy and modifiedgravitation.
