gravitational lensing - sdsm&todessa.phy.sdsmt.edu/~lcorwin/phys792dm_spring2014/... ·...

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


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



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).



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.



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



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.



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


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.


Strong Lensing

Einstein’sCross: fourimages of thesame quasar(Credt:NASA andSTScI)


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

arXiv:astro-ph/0701589

Strong Lensing


Strong Lensing

What might be some limitationsfor using strong lensing?


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


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


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.


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



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



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.


gravitational lensing - sdsm&todessa.phy.sdsmt.edu/~lcorwin/phys792dm_spring2014/... ·...

Documents