statistical analysis of caustic crossings in multiply imaged quasars

Statistical analysis Statistical analysis of caustic crossings of caustic crossings in multiply imaged in multiply imaged

quasarsquasarsTeresa Mediavilla GradolphTeresa Mediavilla Gradolph

Octavio Ariza SánchezOctavio Ariza Sánchez

Evencio Mediavilla GradolphEvencio Mediavilla Gradolph

Pilar Álvarez RuízPilar Álvarez Ruíz

IndexIndex

IntroductionIntroduction

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

S

iI

S

Si

S

iI

F

Fi

T. LIOUVILLE

X Y

Pixels-magnification mapPixels-magnification map

21 ,

2121 ),(),(

PSS I

X Y

Point-like lens magnification Point-like lens magnification mapmap

Binary lens magnification mapBinary lens magnification map

Magnification mapsMagnification maps

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

MicrolensesMicrolenses• AbundanceAbundance• MassMass

Lens systemLens system• Transversal velocityTransversal velocity

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

Caustics concentration analysisCaustics concentration analysis

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