1. Statistical analysis of caustic crossings in multiply imaged quasars
The title of thisspeech is Statistical analysis of caustic crossings in multiply imaged quasars. This work have been done by
Teresa Mediavilla Gradolph
Octavio Ariza Sánchez
Evencio Mediavilla Gradolph
Pilar Álvarez Ruíz 1

2. Index
Introduction
Statistical analysis of the caustics concentration based on caustic crossings counts. Application to QSO
Conclusions
I´m going to begin with an introduction to the basics of gravitational lensing and I will present the reasons which have led this work. Then I will describe the analysis of caustics concentration based on the account of caustic crossings in an observation window and I will comment on the results of applying statistical analysis to a QSO I will finish with the main conclusions of this work 2

3. Introduction
First I´ll introduce the basics of gravitational lensing and will expose the motives of this work 3

4. In this slide we can see two images of one of NASA's space shuttles
In this slide we can see two images of one of NASA's space shuttles. the bottom image is inverted and distorted. This phenomenon is a mirage. 4

5. Terrestrial mirage
Terrestrial mirages are phenomena produced by the bending of light in the atmosphere due to changes in refractive index resulting from changes in temperature..
The upper light ray goes from object to observer producing a direct image of the shuttle. the second ray would go to ground and in principle, would not reach the observer, but due to high temperature of air close to ground, the ray of light bends looking for cooler areas of atmosphere, reaching observer and leading to formation of the bottom image 5

6. Light deflection by the Sun –1919 eclipse
Not only variation in refractive index in atmosphere causes the bending of light rays, but, as revealed in 1919 eclipse experiment, gravity bends light rays too.
In solar system we can observe a weak effect of the bending of light by gravity of sun, the variation in apparent position of a star.
But in other cosmological scenarios gravity can lead to a strong effect of curvature of light forming multiple images from a single source. 6

7. Gravitational mirage Without gravity With gravity
the object of our study is deflection of light rays from a quasar, a bright source located in the depths of universe, by the gravitational field of a galaxy almost aligned with quasar and observer.
If there were no gravity, ray wouldn´t be bent and observer would see galaxy and a single source image.
When gravity is present, the paths of the light ray are bent and light can follow more than one way to reach observer who can see more than one image of the source. Then observer would see galaxy and several quasar images around it. 7

8. First discovered gravitational lens
This is the first gravitational lens system discovered in 1979
We can see two images of the quasar and the lens galaxy
(QSO ) 8

9. QSO
We are interested in studying the quasar 2237, Einstein´s Cross. It is a multiple imaged system. We can see the four images through the center of the galaxy, where density of matter is greater. This causes the phenomenon of microlensing what is the aim of our study 9

10. Microlensing
In the scenario we are considering a galaxy acts as a lens to produce two images of a distant object (quasar), so far we have assumed that the mass distribution of the galaxy is continuos, but in fact, the mass of the galaxy is discretely distributed in stars. The beam of light from one of the images may meet with a star that will act as a lens. We can´t see the images generated by this microlens, because the mass of the stars are very small , and images can´t be separated with current measurement methods. However, microlenses produce an observable effect:magnification of image flux. 10

11. One Source several images MagnificationX Y
T. LIOUVILLE
This is an sketch of lensing.
This is source plan and this is image plane
To a given source, lensing associates in general more than one image.
Magnification is defined as the ratio between the sum of fluxes of all images with respect to source flux.
Lioville´s Theorem says fluxes are proportional to areas
With this formulae we can calculate magnification of a given source in a given position in the source plane. However, in most cases we would like to obtain light curves, that is to study the change of the magnification when we change the position of the source or to study the change of the magnification when we change the parameters of source, for instance the size of source.
To separate the computation of magnification from properties and position of source we can divide source plane in pixels 11

12. Pixels-magnification mapY
And compute magnification of each pixel. Then magnification of a source will be obtained as convolution of its brightness with magnification of the pixels covered by source.
Thus, our objective is to calculate magnification maps, that is, the magnification at each pixel in the source plane. 12

13. Point-like lens magnification map
Point like lens magnification map is the simplest case. Magnification incrisses from outside to inside and reaches its maximun at the center. This is a singular point because it cancels the Jacobian and magnification is theoretically infinite. The geometrical place in source plane where Jacobian cancels is called caustic curve. In this case it is degenerate at a point 13

14. Binary lens magnification map
This is a Binary lens magnification map. We can see caustic curve with typical diamond shape.. 14

15. Caustics are related to the appearance and disappearance of images
Caustics are related to the appearance and disappearance of images. In this slide we see a galaxy and a caustic in source plane. Depending on alignment between source and galaxy the number of images changes. When source is outside caustic we only see two images, when source cross the caustic two more images appear that will dissappear when the source cross the caustic again. 15

16. Magnification maps
If we add more lens, magnification map will be more complicate. It will have more and more caustics. Like we see in these examples 16

17. Simulation and statistical analysis
Comparison between observed and simulated microlensed effect allows us to study:
Source
Size at different wavelengths.
Quasar luminosity profile
Lens galaxy
Mass distribution
Microlenses
Abundance
Mass
Lens system
Transversal velocity
Determination of these parameters can be only statistically done.
Fluctuations induced by microlensing in observed light curves of quasars contain information about lensing objects (masses, density), about the unresolved source structure (size of the continuum or broad line regions of the quasar, brightness profile of the quasar) and about lens system (transversal velocity). So from a comparison between observed and simulated quasar microlensing, one can obtain information about the physical parameters of interest. These conclusions can only be done in a statistical sense . 17

18. Statistical study problems
Experimental errors and intrinsical variability can affect data and results
Thus, simulating magnification maps for some values of physical parameters of interest we can try to calculate probability that microlensing magnification takes a value determined from observations. This statistical analysis faces a problem: experimental errors and sources of variability other than microlensing can significantly affect data and results. 18

19. Objectives
Simplify the problem reducing microlensing to a series of discrete events, caustic crossings. If the source size is small enough :
They appear well separated
They are of high magnification
They are difficult to mistake with other variability features
To solve this problem we propose to simplify it reducing microlensing effect to a series of discrete events, the caustics crossings. If the source size is small enough (X-ray emitting source) each caustics crossing will appear as a single event. In addition, caustics crossings are events of high magnification (little affected by measurement errors), and they are difficult to mistake with other kind of variability. 19

20. Statistical analysis of caustics concentration based on caustic crossings counts. Application to QSO
Now i´m going to resume the most important results of analysis of caustics concentration 20

21. Caustics concentration analysis
In this slide we see a magnification map. The straight green line corresponds to the path of a pixel size source. On the right we have represented source’s light curve. Whenever source crosses a caustic a large increase in magnification that only can be associated with the phenomenon of caustic crossing is produced. That is, for a source of this size the light curve would provide direct information of an observable very interesting from an statistical point of view: the number of caustic crossings. One source of this size corresponds, in practice, to X-ray emission. However, if the source were larger, red line, it wouldn´t be possible to identify caustic crossings as isolated events. 21

22. Analysis steps
Simulate magnification maps for different densities of matter, different mass distribution and shear.
Identify caustic curves
Count the number of caustics detected in a one-dimensional window of certain size in pixels for each axis
Estimate probability of detecting a caustic in a pixel for each axis
Compare experimental distributions obtained in simulations with theoretical binomial distribution.
We have used the method of Inverse Polygon Mapping to carry out two first steps.
Distribution of caustics in magnification maps depends on the characteristics of the distribution of stars and dark matter in the lensing galaxy. In this work, we study how the existence of a range of masses in the distribution of stars affect caustics concentration.
Statistical analysis of the caustics spatial distribution is based on the following steps: (i) simulate magnification maps for different densities of matter and different mass distribution, (ii) identify caustic curves, (iii) count the number of caustics detected in a one-dimensional window of certain size in pixels for each axis, (iv) estimate probability of detecting a caustic in a pixel for each axis and (v) compare experimental distributions obtained in simulations with theoretical binomial distribution.
We have used the method of Inverse Polygon Mapping to carry out two first steps.

23. Application to QSO
We have simulated magnification and caustic maps for the four images of QSO We have considered two extreme cases in the stars mass distribution: stars distributed in a range of masses (from 0.01 to 1 solar masses with an m-1 density law; hypothesis A) and simple distribution of identical stars of one solar mass (hypothesis BI).
In next slide we can see magnification maps 23

24. Magnification Maps 1 solar mass microlenses
A Y B C D
Microlenses distributed in a range of masses
We use the same simulation for images A and B because their convergence and shear are similar
A Y B C D 24

25. Caustics 1 solar mass microlenses
A Y B C D
Microlenses distributed in a range of masses
Looking at slide we can see that for the four images the number of caustics is larger when the microlenses are distributed in a range of masses..
A Y B C D 25

26. Comparison with the binomial distribution (D image)
We have counted the number of caustics detected in one-dimensional windows of 4, 40, 200 and 400 pixels for each axis (1 pixel corresponds to Einstein radii for a solar mass star) and we have calculated experimental and theoretical binomial probability distributions. Comparing the different distributions we have derived the following results for image D
Differences between probability distributions are very significant. For example considering a 400 pixel size window for the X axis, the peak and centroid of the number count distribution correspond to 6 or 7 detections when stars are distributed in a range of masses but only to one detection in the case of 1 solar mass stars. In a 200 pixel size window we also can see diferences
From an experimental point of view, a single measure of the number of caustics detected in a window of 400 pixels may be enough to distinguish between hypotheses A and B.
Unimodal distribution
Peak
Centroid
400 pixels X axis 1
200 pixels X axis
400 pixels Y axis 2
200 pixels Y axis
Masses in a range
Peak
Centroid
400 pixels X axis 6 7
200 pixels X axis 3
400 pixels Y axis 9 10
200 pixels Y axis 4 26

27. We can distinguish between A and B hypothesis
Results (I)
D IMAGE
X AXIS
n=7, error= 3
P(7 3/A)=0.63
P(7 3/B)=0.22
n=1, error= 1
P(1 1/A)=0.049
P(1 1/B)=0.66
P(A/7)=0.75
P(B/7)=0.25
P(A/1)=0.07
P(B/1)=0.93
Y AXIS
n=10, error= 3
P(10 3/A)=0.37
P(10 3/B)=0.12
n=2, error= 1
P(2 1/A)=0.12
P(2 1/B)=0.38
P(A/10)=0.76
P(B/10)=0.24
P(A/2)=0.24
P(B/2)=0.76
From an experimental point of view, a single measure of the number of caustics detected in a window of 400 pixels may be enough to distinguish between hypotheses A and B.
In the X axis case let´s assume that measurement is 7 ( expected value for hypothesis A) whit a mistake of Applying Bayes Theorem and assuming that P(A) = P(B) we obtain p(A/7)=0.75 and p(B/7)=0.25. let´s now assume that measurement is 1 ( expected value for hypothesis B) whit a mistake of Applying Bayes Theorem we obtain p(A/1)=0.07 an p(B/1)=0.93
In the Y axis case let´s assume that measurement is 10 ( expected value for hypothesis A) whit a mistake of Applying Bayes Theorem we obtain p(A/10)=0.76 and p(B/10)=0.24. let´s now assume that measurement is 2 ( expected value for hypothesis B) whit a mistake of Applying Bayes Theorem we obtain p(A/2)=0.24 an p(B/2)=0.93
We can distinguish between A and B hypothesis 27

28. Can we solve the size / transversal velocity degeneracy?
Results (II)
Can we solve the size / transversal velocity degeneracy?
We need to know transversal velocity of quasar in order to define observing window in suitable Einstein radii units. It´s dificult to meassure transversal velocity and in many experimental studies the estimates of parameters depend on it.
Obviusly the number of caustic crossings depends on source´s velocity moving over magnification map 28

29. Results (II) In the case of image D hypothesis A
in X axis a number of caustics greater than 6 implies a window greater than 1.2 einstein radii, a number of caustics less than 3 implies a window less than 1.2 einstein radii
in Y axis a number of caustics greater than 9 implies a window greater than 1.2 einstein radii, a number of caustics less than 3 implies a window less than 1.2 einstein radii 29

30. Results (II) D image microlenses distributed in a range of masses
Number of caustics (X axis) > Window > 1.2 Einstein radii
Number of caustics (X axis) < Window < 1.2 Einstein radii
Number of caustics (Y axis) > Window > 1.2 Einstein radii
Number of caustics (Y axis) < Window < 1.2 Einstein radii
In the case of image D hypothesis A
in X axis a number of caustics greater than 6 implies a window greater than 1.2 einstein radii, a number of caustics less than 3 implies a window less than 1.2 einstein radii
in Y axis a number of caustics greater than 9 implies a window greater than 1.2 einstein radii, a number of caustics less than 3 implies a window less than 1.2 einstein radii 30

31. Bayesian Analysis D image 400 pixels X axis 400 píxels Y axis
These are the functions of probability conditionated to n caustics crossing for hypotesis A (red) an B (blue)
In X axis probability of distinguishing between the two hypotheses with more than 80% of likelihood is 76% in a window of 400 pixeles
In Y axis probability of distinguishing between the two hypotheses with more than 70% of likelihood is 77% . In this case we coudn´t apply the rule of 80%. This diference is due to shear influence.
In a 76% of cases we can distinguish between both hypothesis with more than 80% of likelihood
In a 77% of cases we can distinguish between both hypothesis with more than 70% of likelihood 31

32. Conclusions
The main conclusions of this work are: 32

33. Conclusions
Caustic crossing statistics is affected by the microlenses mass function and by shear.
For QSO D detection of a small number of events will allow us to distinguish between unimodal and distributed in a range mass distributions.
We could determinate the size of the observing window 33