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Statistical analysis Statistical analysis of caustic crossings of caustic crossings in multiply imaged in multiply imaged
Statistical analysis of the caustics Statistical analysis of the caustics concentration based on caustic crossings concentration based on caustic crossings counts. Application to QSO 2237+0305counts. Application to QSO 2237+0305
ConclusionsConclusions
IntroductionIntroduction
Terrestrial mirageTerrestrial mirage
Light deflection by the Sun –1919 Light deflection by the Sun –1919 eclipseeclipse
With gravity
Without gravity
Gravitational mirageGravitational mirage
First discovered gravitational lensFirst discovered gravitational lens
(QSO 0957+561)
QSO 2237+0305QSO 2237+0305
MicrolensingMicrolensing
One Source several imagesOne Source several imagesMagnificationMagnification
Simulation and statistical Simulation and statistical analysisanalysis
Comparison between observed and simulated microlensed Comparison between observed and simulated microlensed effect allows us to study:effect allows us to study: SourceSource
• Size at different wavelengths.Size at different wavelengths.• Quasar luminosity profileQuasar luminosity profile
Lens galaxyLens galaxy• Mass distributionMass distribution
Determination of these parameters can be only statistically Determination of these parameters can be only statistically done.done.
Statistical study problemsStatistical study problems
Experimental errors and intrinsical Experimental errors and intrinsical variability can affect data and resultsvariability can affect data and results
ObjectivesObjectives
Simplify the problem reducing Simplify the problem reducing microlensing to a series of discrete microlensing to a series of discrete events, caustic crossings. If the events, caustic crossings. If the source size is small enough :source size is small enough : They appear well separatedThey appear well separated They are of high magnificationThey are of high magnification They are difficult to mistake with other They are difficult to mistake with other
variability features variability features
Statistical analysis of Statistical analysis of caustics concentration caustics concentration
based on caustic based on caustic crossings counts. crossings counts.
Application to QSO Application to QSO 2237+03052237+0305
Analysis stepsAnalysis steps Simulate magnification maps for different densities Simulate magnification maps for different densities
of matter, different mass distribution and shear.of matter, different mass distribution and shear. Identify caustic curvesIdentify caustic curves Count the number of caustics detected in a one-Count the number of caustics detected in a one-
dimensional window of certain size in pixels for dimensional window of certain size in pixels for each axiseach axis
Estimate probability of detecting a caustic in a Estimate probability of detecting a caustic in a pixel for each axis pixel for each axis
Compare experimental distributions obtained in Compare experimental distributions obtained in simulations with theoretical binomial distribution.simulations with theoretical binomial distribution.
We have used the method of Inverse Polygon We have used the method of Inverse Polygon Mapping to carry out two first steps.Mapping to carry out two first steps.
Application to QSO 2237+0305Application to QSO 2237+0305
Magnification MapsMagnification Maps1 solar mass microlenses
Microlenses distributed in a range of massesA Y B
A Y B
C
C
D
D
CausticsCaustics1 solar mass microlenses
Microlenses distributed in a range of massesA Y B
A Y B
C
C
D
D
Comparison with the binomial Comparison with the binomial distribution (D image)distribution (D image)
Unimodal distributionUnimodal distribution PeakPeak CentroidCentroid
400 pixels X axis400 pixels X axis 11 11
200 pixels X axis200 pixels X axis 00 00
400 pixels Y axis400 pixels Y axis 00 22
200 pixels Y axis200 pixels Y axis 00 00
Masses in a rangeMasses in a range PeakPeak CentroidCentroid
400 pixels X axis400 pixels X axis 66 77
200 pixels X axis200 pixels X axis 33 33
400 pixels Y axis400 pixels Y axis 99 1010
200 pixels Y axis200 pixels Y axis 33 44
Results (I)Results (I)
X AXISX AXIS
n=7, error= 3n=7, error= 3
P(7 3/A)=0.63P(7 3/A)=0.63
P(7 3/B)=0.22P(7 3/B)=0.22
n=1, error= 1n=1, error= 1
P(1 1/A)=0.049P(1 1/A)=0.049
P(1 1/B)=0.66P(1 1/B)=0.66
P(A/7)=0.75P(A/7)=0.75
P(B/7)=0.25P(B/7)=0.25
P(A/1)=0.07P(A/1)=0.07
P(B/1)=0.93P(B/1)=0.93
Y AXISY AXIS
n=10, error= 3n=10, error= 3
P(10 3/A)=0.37P(10 3/A)=0.37
P(10 3/B)=0.12P(10 3/B)=0.12
n=2, error= 1n=2, error= 1
P(2 1/A)=0.12P(2 1/A)=0.12
P(2 1/B)=0.38P(2 1/B)=0.38
P(A/10)=0.76P(A/10)=0.76
P(B/10)=0.24P(B/10)=0.24
P(A/2)=0.24P(A/2)=0.24
P(B/2)=0.76P(B/2)=0.76
We can distinguish between A and B hypothesis
D IMAGE
Results (II)Results (II)
Can we solve the size / transversal velocity degeneracy?Can we solve the size / transversal velocity degeneracy?
Results (II)Results (II)
Results (II)Results (II)
D image microlenses distributed in a range of massesD image microlenses distributed in a range of masses
Number of caustics (X axis) > 6 Window > 1.2 Einstein radiiNumber of caustics (X axis) > 6 Window > 1.2 Einstein radii
Number of caustics (X axis) < 3 Window < 1.2 Einstein radiiNumber of caustics (X axis) < 3 Window < 1.2 Einstein radii
Number of caustics (Y axis) > 9 Window > 1.2 Einstein radiiNumber of caustics (Y axis) > 9 Window > 1.2 Einstein radii
Number of caustics (Y axis) < 3 Window < 1.2 Einstein radiiNumber of caustics (Y axis) < 3 Window < 1.2 Einstein radii
Bayesian AnalysisBayesian Analysis400 pixels X axis 400 píxels Y axis
In a 76% of cases we can In a 76% of cases we can distinguish between both hypothesis distinguish between both hypothesis
with more than 80% of likelihoodwith more than 80% of likelihood
In a 77% of cases we can In a 77% of cases we can distinguish between both hypothesis distinguish between both hypothesis
with more than 70% of likelihoodwith more than 70% of likelihood
D imageD image
ConclusionsConclusions
Conclusions Conclusions
Caustic crossing statistics is affected by Caustic crossing statistics is affected by the microlenses mass function and by the microlenses mass function and by shear.shear.
For QSO 2237+0305D detection of a small For QSO 2237+0305D detection of a small number of events will allow us to number of events will allow us to distinguish between unimodal and distinguish between unimodal and distributed in a range mass distributions.distributed in a range mass distributions.
We could determinate the size of the We could determinate the size of the observing windowobserving window