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Gravitational lensing:The number of images
Fernando Chamizo
Msc Theoretical Physics
May 31, 2016
F. Chamizo Gravitational lensing 2
Introduction Complex Elliptical I Elliptical II Disks
Gravitational lensing
A massive object deflects the light and it behaves as a lens.
The observer sees distorted images of the background objects.
F. Chamizo Gravitational lensing 3
Introduction Complex Elliptical I Elliptical II Disks
Einstein was rather skeptical in 1939 about the possibility ofmeasuring gravitational lensing:
[A. Einstein. Science 84 (2188): 506–507, 1936]
Fortunately, he was too skeptical and since its astronomicaldiscovery in 1979, we have amazing images like...
F. Chamizo Gravitational lensing 3
Introduction Complex Elliptical I Elliptical II Disks
Einstein was rather skeptical in 1939 about the possibility ofmeasuring gravitational lensing:
[A. Einstein. Science 84 (2188): 506–507, 1936]
Fortunately, he was too skeptical and since its astronomicaldiscovery in 1979, we have amazing images like...
F. Chamizo Gravitational lensing 4
Introduction Complex Elliptical I Elliptical II Disks
a cross
Credit: NASA, ESA, & STScI.
Source: http://hubblesite.org/newscenter/archive/releases/1990/20/image/a/
F. Chamizo Gravitational lensing 5
Introduction Complex Elliptical I Elliptical II Disks
a fried egg
Credit: ESA/Hubble & NASA.
Source: http://apod.nasa.gov/apod/image/1112/lensshoe_hubble_3235.jpg
F. Chamizo Gravitational lensing 6
Introduction Complex Elliptical I Elliptical II Disks
and even a smiley!
Credit: NASA & ESA.
Source: http://www.spacetelescope.org/images/potw1506a/
F. Chamizo Gravitational lensing 7
Introduction Complex Elliptical I Elliptical II Disks
A naive goal
Gravitational lensing is arguably the purest method to weighgalaxies and to “see” the dark matter, a fundamental ingredient ofthe cosmological models.
Can we say something about the structure of the lens fromthe image? This is not a well-posed problem. It is like tryingto reconstruct a human body from just one radiograph.
(from the Pioneer plaque)Magic machine?
Nevertheless, we can study if our educated guess matches theproperties of the observed lensing. The property considered here isthe number of images.
F. Chamizo Gravitational lensing 7
Introduction Complex Elliptical I Elliptical II Disks
A naive goal
Gravitational lensing is arguably the purest method to weighgalaxies and to “see” the dark matter, a fundamental ingredient ofthe cosmological models.
Can we say something about the structure of the lens fromthe image? This is not a well-posed problem. It is like tryingto reconstruct a human body from just one radiograph.
(from the Pioneer plaque)Magic machine?
Nevertheless, we can study if our educated guess matches theproperties of the observed lensing. The property considered here isthe number of images.
F. Chamizo Gravitational lensing 7
Introduction Complex Elliptical I Elliptical II Disks
A naive goal
Gravitational lensing is arguably the purest method to weighgalaxies and to “see” the dark matter, a fundamental ingredient ofthe cosmological models.
Can we say something about the structure of the lens fromthe image? This is not a well-posed problem. It is like tryingto reconstruct a human body from just one radiograph.
(from the Pioneer plaque)Magic machine?
Nevertheless, we can study if our educated guess matches theproperties of the observed lensing. The property considered here isthe number of images.
F. Chamizo Gravitational lensing 8
Introduction Complex Elliptical I Elliptical II Disks
Making the lens equation complex
[Straumann, 1997]
β = θ − DLS
DSα (lens equation)
DSβ = ~η, DLθ = ~ξ (ang. diam. dist.)
α = 4G
∫ ~ξ − ~ζ|~ξ − ~ζ|2
Σ(~ζ) d2ζ
(width of the lens � dist(O, L), dist(S , L))
Lens planeSource plane
}↔ copies of C
F. Chamizo Gravitational lensing 9
Introduction Complex Elliptical I Elliptical II Disks
Renaming, with some normalization, ~η, ~ξ, ~ζ (∈ R2) → w , z , ζ(∈ C) and using ξ−ζ
|ξ−ζ|2= 1
z∗−ζ∗
Complex form of the lens equation
w = z −∫L
dσ(ζ)
z∗ − ζ∗
where dσ/dArea is essentially the surface mass density.
Chang–Refsdal lens → Lensing of stars (or point-masses) in abackground galaxy (originally a quasar).
More complex form of the lens equation
w = z −∫L
dσ(ζ)
z∗ − ζ∗− γz∗
where γ is the background shear.
F. Chamizo Gravitational lensing 10
Introduction Complex Elliptical I Elliptical II Disks
The number of images
For point masses dσ = sum of Dirac deltas and we have
w = z −∑n
Mn
z∗ − z∗n− γz∗
where zn are the positions and Mn are related to the masses.
In any case
# Images = # solutions (in z) for w fixed
Perhaps, we could use the number of images not as a signature but as a hint
for some kind of structures.
F. Chamizo Gravitational lensing 11
Introduction Complex Elliptical I Elliptical II Disks
Idealized elliptical galaxies
F. Chamizo Gravitational lensing 12
Introduction Complex Elliptical I Elliptical II Disks
[Fassnacht, Keeton, Khavinson, 2009]
dσ = Kdx dy (evenly distrib.), lens = E ={x2a2
+y2
b2= 1}.
The integral can be computed explicitly!
The complex lens equation becomes (c = focal distance)w = z +
2ab
c2(z −√z2 − c2
)∗ − γz∗ if z 6∈ E
w =a2 + b2 − 2ab
c2z∗ − γz∗ if z ∈ E
The second equation is linear in x and y (z = x + iy). Then foreach w in the source there is at most one solution z ∈ E .
F. Chamizo Gravitational lensing 13
Introduction Complex Elliptical I Elliptical II Disks
The first equation is more complicated
w = z +2ab
c2(z −
√z2 − c2
)∗ − γz∗but it is clear that eliminating
√z2 − c2 we get two quadratic
equation in x and y . This the intersection of two conics.
{First equation → at most 4 solutions
Second equation → at most 1 solutions
In this idealized lensing with elliptical lenses
At most 4 clear images + 1 probably invisible image
F. Chamizo Gravitational lensing 13
Introduction Complex Elliptical I Elliptical II Disks
The first equation is more complicated
w = z +2ab
c2(z −
√z2 − c2
)∗ − γz∗but it is clear that eliminating
√z2 − c2 we get two quadratic
equation in x and y . This the intersection of two conics.{First equation → at most 4 solutions
Second equation → at most 1 solutions
In this idealized lensing with elliptical lenses
At most 4 clear images + 1 probably invisible image
F. Chamizo Gravitational lensing 14
Introduction Complex Elliptical I Elliptical II Disks
A real photo for the ideal model
Unfortunate low quality. Blame Hubble telescope!
Source: http://hubblesite.org/newscenter/archive/releases/1995/43/image/a/
F. Chamizo Gravitational lensing 15
Introduction Complex Elliptical I Elliptical II Disks
Less idealized elliptical galaxies
(Galaxy M60) Credit: NASA, & STScI.
F. Chamizo Gravitational lensing 16
Introduction Complex Elliptical I Elliptical II Disks
[Keeton, Mao, Witt, 2000]
A uniform distribution of the mass in a galaxy is far from beingrealistic. Very often for gravitational lensing it is assumedisothermal or isothermal elliptic density ∝ r−2
Some authors have studied the problem in a more general settingallowing the galaxies to be an ellipsoid.
In this general situation the caustics admit explicit formulas. Buttoo long to be displayed here.
F. Chamizo Gravitational lensing 17
Introduction Complex Elliptical I Elliptical II Disks
For (planar) elliptic galaxies with isothermal density. The complexlens equation leads to study the number of preimages of thefunction
F (z) = z − K
∫ c
0
((z∗)2 − u2
)−1/2du − γz∗.
Perhaps this is not so hard!
There is a result (the so-called odd number theorem) that implies that whenthere is no shear, the number of images is odd. It possible to give a simpleproof in the planar case, essentially an application of the argument principle
1
2πi
∫C
f ′(z)
f (z)= #zeros−#poles.
F. Chamizo Gravitational lensing 18
Introduction Complex Elliptical I Elliptical II Disks
Radial symmetry
[An, Evans, 2006]
Astronomically a star ( 6= Sun) is a point mass and the complexlens equation becomes
w = z − K
z∗− γz∗ (Chang–Refsdal lens)
(take K = 1 for simplicity).
For γ = 0, one gets readily the Einstein ring: w = 0 ⇒ |z |2 = 1.
In general, the Jacobian determinant is
∂w
∂z
∂w∗
∂z∗− ∂w
∂z∗∂w∗
∂z= 1−
∣∣γ − z−2∣∣.
F. Chamizo Gravitational lensing 18
Introduction Complex Elliptical I Elliptical II Disks
Radial symmetry
[An, Evans, 2006]
Astronomically a star ( 6= Sun) is a point mass and the complexlens equation becomes
w = z − K
z∗− γz∗ (Chang–Refsdal lens)
(take K = 1 for simplicity).
For γ = 0, one gets readily the Einstein ring: w = 0 ⇒ |z |2 = 1.
In general, the Jacobian determinant is
∂w
∂z
∂w∗
∂z∗− ∂w
∂z∗∂w∗
∂z= 1−
∣∣γ − z−2∣∣.
F. Chamizo Gravitational lensing 19
Introduction Complex Elliptical I Elliptical II Disks
Then the critical curve is z2 =(γ + e iθ
)−1and we get also the
caustics.
Caustics and critical curves for several values of the shear:
γ=0.1 γ=0.3 γ=0.5 γ=0.7
F. Chamizo Gravitational lensing 20
Introduction Complex Elliptical I Elliptical II Disks
One can count the number of images substituting the equationinto itself:
w = z − 1
z∗− γz∗ =⇒ z∗ = w +
1
z+ γz .
Then∑4n=0 an(w , γ)zn
z(1 + w∗z + γz2)= 0 =⇒ 4 images generically.
Surprisingly this analysis generalizes to any dσ with radialsymmetry. The key point is by Cauchy’s integral formula∫
|ζ|=r
1
z − ζdζ
ζ=
∫ 2π
0
ir
z − re iθ=
{−2πi/z if |z | < r
0 if |z | > r
F. Chamizo Gravitational lensing 21
Introduction Complex Elliptical I Elliptical II Disks
Then for dσ = ρ(r) dA,∫C
dσ(ζ)
z − ζ=
∫ ∞0
∫ 2π
0
ρ(r)r drdθ
z − re iθ= −2π
z
∫ |z|0
ρ(r)r dr .
If ρ is supported in r < R then the integral is a constant for|z | > R (outside of the lens) and we recover the Chang-Refsdallens equation (with a different constant).
The counting of the images reduces to study how many zeros ofthe polynomial
∑4n=0 an(w , γ)zn lie in the lens (the support of ρ).
F. Chamizo Gravitational lensing 22
Introduction Complex Elliptical I Elliptical II Disks
Bibliography I
J. H. An and N. W. Evans.The Chang-Refsdal lens revisited.Monthly Notices of the Royal Astronomical Society,369:317–334, June 2006.
C. D. Fassnacht, C. R. Keeton, and D. Khavinson.Gravitational lensing by elliptical galaxies, and the Schwarzfunction.In Analysis and mathematical physics, Trends Math., pages115–129. Birkhauser, Basel, 2009.
C.R. Keeton, S. Mao, and H.J. Witt.Gravitational lenses with more than four images I.Classification of caustics.Astrophysical Journal, 537:697–707, 2000.
F. Chamizo Gravitational lensing 23
Introduction Complex Elliptical I Elliptical II Disks
Bibliography II
P. Schneider, J. Ehlers, and E.E. Falco.Gravitational Lenses.Springer Study Edition. Springer-Verlag, Berlin, 1992.
N. Straumann.Complex formulation of lensing theory and applications.Helv. Phys. Acta, 70(6):894–908, 1997.
H. J. Witt.Investigation of high amplification events in light curves ofgravitationally lensed quasars.Astronomy and Astrophysics, 236:311–322, September 1990.
