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Gravitational Lensing
Prof. Luke A. CorwinPHYS 792
South Dakota School of Mines & Technology
Jan. 23, 2014 (W3-1)
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 1 / 19
Outline
1 Gravitational Lensing Introduction
2 Strong Lensing
3 Weak Lensing
4 Microlensing
5 Reminders
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 2 / 19
Gravitational Lensing Introduction
Gravitational Lensing
Today, we will cover the concepts and formalism ofgravitational lensing. Thursday, we will study its implicationsfor dark matter.
“Following an early prediction of General Relativity,gravitational fields deflect the light path of photons andmodify the apparent flux and shape of astronomicalsources.”1
1G. Bertone, editor, Particle Dark Matter: Observations, Models andSearches, Cambridge University Press: Cambridge, UK (2010), p. 56
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 3 / 19
Gravitational Lensing Introduction
Figure : Diagram showing lensing configuration from the Source (S) tothe imaging lens (I) to the Observer (O). L denotes the plane of thelensing object (Bertone).
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 4 / 19
Gravitational Lensing Introduction
Three Categories of Lensing
Strong Lensing is created by objects of galactic mass orlarger producing arcs, rings and multiple images ofbackground objects.
Weak Lensing produces subtler effects such as distortions ofbackground galaxy shapes
Microlensing is produced by compact objects (e.g. stars orquasars) distorting images of background objects. Forexample, the lensing produced by the Sun was an earlyverification of GR.
All three kinds have made significant contributions to ourunderstanding of dark matter
We will focus mainly on Strong and Weak today.
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 5 / 19
Gravitational Lensing Introduction
Approximations
Rationale
The full mathematical formalism of gravitational lensing, likemuch of general relativity, is mathematically beyond the scope ofthis course. The systems important to dark matter can beapproximated quite accurately.
Assumptions
We use the thin lens approximation, as in the diagram, andassume that the lensing material is concentrated in a planeperpendicular to our line of sight.
The gravitational potential φ is proportional to the velocitydistribution σ2
v . We will use the weak field approximation,namely σ2
v � c2
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 6 / 19
Gravitational Lensing Introduction
Important Parameters
Gravitational Convergence
κ =Σ(DOLθ)
Σcrit
, Σcrit =c2
4πG
DOS
DOLDLS
(1)
Σ represents mass in the lensing plane divided by area in units ofkg/m2
γ Parameters
γ = (γ1; γ2)γ1 = 1
2(∂1∂1 − ∂2∂2)φ and γ2 = ∂1∂2φ,
where the ∂1 and ∂2 represent partial derivatives along the twocartesian axes of the lensing plane.
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 7 / 19
Gravitational Lensing Introduction
Gravitational lensing can distort, multiply, and magnify thesource. In our approximation, the magnification is
|µ| = 1
(1− κ)2 − |γ|2(2)
The magnification goes to infinity when the denominator iszero.
Where this occurs depends on the geometry of the source,lens, and their relationship to each other and the observer
The points on the lensing plane are called the critical lines
The points on the source plane are called the caustic lines
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 8 / 19
Strong Lensing
Strong Lensing
Formally defined as κ ≥ 0.5 and |γ| ≥ 0.5
Strong lensing occurs for sources near the caustic lines thatare lensed near the critical lines
This produces the most famous and spectacular instances ofgravitational lensing: bright arcs of distorted backgroundgalaxies, complete Einstein rings and multiple images.
Strongly lensed images are good tracers of critical lines andthus reveal much of the morphology of the lensing mass
A perfect ring will occur if the we and a lens with a circularmass distribution are exactly aligned with the source.
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 9 / 19
Strong Lensing
Einstein’sCross: fourimages of thesame quasar(Credt:NASA andSTScI)
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 10 / 19
Strong Lensing
Astrophys. J. 667 (2007) 176 [arXiv:astro-ph/0701589]L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 11 / 19
Strong Lensing
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 12 / 19
Strong Lensing
What might be some limitationsfor using strong lensing?
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 13 / 19
Strong Lensing
Limitations of Strong Lensing
Strong lensing is relatively rare
When it occurs, we can only learn about the massdistribution near the caustic lines
To learn about more extended mass distributions, we turn toweak lensing
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 14 / 19
Weak Lensing
Weak Lensing
Formally defined as κ� 1 and |γ| � 1
Approximately the regime experienced by all backgroundsources not near caustic lines, which is the majority of sources
The small magnifications and distortions caused by weaklensing are called gravitational shear
Too small to be detected for an individual galaxy; therefore, Ido not have any pretty pictures for this one.
Usually quantified as the statistical distortion in theellipticity of source galaxies
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 15 / 19
Weak Lensing
Important Parameters
εS is the true ellipticity of the source galaxy
εI is the ellipticity of the lensed image of the galaxy
In our approximations,εI ≈ εS + γ (3)
Since we can only detect this as a statistical effect, we areconcerned with 〈εI〉. If the ellipticities of the source galaxies arerandomly distributed, 〈εS〉 = 0 and
〈εI〉 ≈ γ (4)
Since γ is dependent on the gravitational potential φ, 〈εI〉 yieldsinformation on the mass distribution of the weak lens.
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 16 / 19
Weak Lensing
Weak Lensing Applications
Galaxy-Galaxy Lensing
Has been used to determine the mass distributions withingalaxy groups and clustersa
Most useful in the redshift range 0.1 < z < 0.8
Mass reconstructions are most limited by uncertainties inmeasurements in galaxy ellipticities, number density ofgalaxies, and red shift.
aarXiv:astro-ph/0605476
Universal Scale
Use statistical power of the entire sky
Compare hypothetical power spectra of mass distributionswith observed weak lensing
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 17 / 19
Microlensing
Microlensing
Gravitational lensing from compact sources with relativelysmall masses
Has been used to look for compact dark objects (CDOs, a.k.aMaCHOs) in the mass range 10−6 ≤ m/M� ≤ 106.2
Has even been used to find planets via their gravitationallensing of host stars3
2IAU Symposium 225 (2004) 3213arXiv:astro-ph/0404309
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 18 / 19
Reminders
Reminders
Choose your topic for mid-term presentation before Jan. 30
Doug Tiedt has chosen MaCHOsDan Rederth has chosen Sterile NeutrinosTyler Bogwardt has chosen Modified Gravity
Choose your topic for final presentation on or before Feb. 20
Since we have no USD students, for most of the rest of thesemester we will be in CB 110, except February 27 and April3, when we will be back in CB 108.
L. Corwin, PHYS 792 (SDSM&T) DM & Cosmic Structure Jan. 23, 2014 (W3-1) 19 / 19
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