Cosmological Parameters with Gravitational Lens systems from the SDSS

Du-Hwan Han & Myeong-Gu Park

Kyungpook National UniversityDepartment of Astronomy and Atmospheric Sciences

1

INTRODUCTION

2

Gravitational lens

Deflection of light ray by gravitational fields of massive objects.

3

Gravitational lens

Q0957+561 (SDSS J1001+5553) 1979 Walsh et al.

Sloan Digital Sky Survey

4

Gravitational lens

Geometry of gravitational lens

Sd

dS

Sd

dS

dSSS

d

DD

DM

DD

D

c

GM

DDD

Dc

GM

22

2

4

ˆ

,4

ˆ

5

Statistical Analysis

Turner, Ostriker, and Gott (1984)Mean image separation of lens systems have dependence on source redshift.

Gott, Park, and Lee (1989) Calculation of the mean image separation

for various cosmology.

6

MEAN IMAGE SEPARATION

7

Mean Image Separation

Assumption : using lens model as Singular Isothermal Sphere & homogeneous & isotrophic universe described by

Robertson-Walker Metric

Comoving Distance

222220

2202

0

2222 sin)(

)( ddSadaa

tadtcds

21

0

21

23 1)1)(1()1(

wheredtttz

S

universeopen

universeclosedSwhere

universeflat

)sinh(

)sin()(

Gott, Park, & Lee, 1989, ApJ, 338, 1

8

Mean Image Separation

Flat Universe

Open Universe

Closed Universe

0

SSSSSSSS

SSSSSSSSSS

4sinh)1(coth321

coshsinh)coth(sinh21

)coth31(81

158

cosh54

cosh154

coth152

cosh31

cosh51

coth3coshsinh51

152

cosh54

sinh

22222

3335423

0

SSSSSSSS

SSSSSSSSSS

4sin)1(cot321

cossin)cot(sin21

)cot31(81

158

cos54

cos154

cot152

cos31

cos51

cot3cossin51

152

cos54

sin

22222

3335423

0

Gott, Park, & Lee, 1989, ApJ, 338, 1

9

• empty universe

• open universe

• flat universe

• closed universe

• closed universe

Park & Cott, 1997Park, 1998

43.1,2.0

0,0.2

0K

0,0

0,4.0

Cosmological Models

0,0

0,4.0

0,0.2

43.1,2.0

0K

10

LENS SYSTEMS FROM THE SDSS

11

Candidates for Lens systems from the SDSS

171

The SDSS is completed its 1st of operation. ⇒ ~ 90,000 quasars

Candidates selected by lens selection algorithm (Oguri et al.). ⇒ 171 candidates

12

Candidates for Lens systems from the SDSS

76

The SDSS is completed its 1st of operation. ⇒ ~ 90,000 quasars

Candidates selected by lens selection algorithm (Oguri, et. al.). ⇒ 171 candidates

Select 76 candidates between 1” and 5”

13

Candidates for Lens systems from the SDSS

The SDSS is completed its 1st of operation. ⇒ ~ 90,000 quasars

Candidates selected by lens selection algorithm (Oguri, et. al.). ⇒ 171 candidates

Select 76 candidates between 1” and 5”

Oguri, et. al., 2006, AJ, 132, 99914

Candidates for Lens systems from the SDSS

76 candidates

15

CURVATURE TEST

16

Curvature Test

Mean image separation Candidates for Lens systems

17

Spearman Rank Correlation Test

Test of the strength of a correlation.Using the rank of two quantities.

=+1 positive correlationcorrelation coefficient = 0 no correlation =-1 negative correlation

‘Normalized’⇒ Observed image separation divide by mean image separation of cosmological model.

Null hypothesisthere is no correlation between ‘normalized’ image separation versus source redshift.

Curvature Test

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Curvature Test

Cosmological Models Probability

K=-1

Empty universe0.149 0.22

Open universe0.100 0.38

K=0Flat universe

0.044 0.70

K=+1

Closed universe-0.008 0.94

Closed universe-0.528 0.00

0,0

0,4.0

0,0.2

43.1,2.0

0K

19

No Bi

g Ba

ng

0.01

0.05

0.32

K=0

0.70

Likelihood Map

0 ≤ Ω ≤ 3

0 ≤ ΩΛ ≤ 3

20

No

Big B

ang < 0.01

0.01

0.05

0.32

0.70

k=0

> 0.32

Ωm- ΩΛ TEST

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Lensing Probability

Probability distribution for image separation

For elliptical (i=E) and lenticular (i=SO) galaxies,

For spiral (i=S) galaxies,

Where, Ai=0.12, 0.19, 0.69 for E,S0,S galaxy type and

22

2

, ,

2

, ,0."1

( )

( )

iSi E SO S

L

iSi E SO S

L

df

dz ddP

ddf d

dz d

2 4 4 42

2 3 2*

12 exp

8i os

iL LS i

d D cA X

dz d D

1.3 3.3 2.6 2.62

1.3 2.3 1.3*

11.3 exp

8i os

iL LS i

d D cA X

dz d D

2

*8OS

LS i

D cX

D

Maximum Likelihood Methodis a useful tool to set constraints on parameters of the statistical model with a given data set

Likelihood function

For the JVAS/CLASS sample

For the SDSS sample

Ωm- ΩΛ TEST

23

1ni

i

dPL

d

* *1.0, 276, 254 ( / )SO E km s

*6.5, 88.8 ( / )km s

The JVAS/CLASS sample

24

1/ 2 0."15 0."25

29 / 31 0."25 0."65

99 /100 0."65 4."0

2 / 3 4."0 10."0

for

for

for

for

( )Sf

Lensed QSOs from JVAS/CLASS Survey

Case zL zs Δθobs(")

B0218 0.940 0.680 0.33 B1422 3.620 0.339 1.30

MG0414 2.630 0.468 3.00 B1030 1.535 0.599 1.60 B1600 1.589 0.414 1.40 B0712 1.339 0.406 1.27 B1608 1.390 0.640 2.20 B2045 1.280 0.870 2.00 B1152 1.019 0.439 1.90 B2144 0.588 0.316 2.40

No

Big

Ban

g

▬ 99% CL▬ 95% CL▬ 68% CL

The SDSS sample

25

No

Big B

ang

▬ 99% CL▬ 95% CL▬ 68% CL

Gravitational Lenses from the SDSS

Case zL zs Δθobs(")

SDSS J0145-0945 2.732 0.490 2.2

SDSS J0246-0825 1.685 0.724 1.0

SDSS J0746+4403 1.998 0.300 1.0

SDSS J0806+2006 1.537 0.570 1.4

SDSS J0903+5028 3.584 0.388 2.8

SDSS J0911+0550 2.783 0.770 3.2

SDSS J0913+5259 1.377 0.830 1.1

SDSS J0924+0219 1.523 0.400 1.3

SDSS J0951+2635 1.246 0.260 1.1

SDSS J1001+5027 1.839 0.300 2.9

SDSS J1004+4112 1.740 0.680 3.8

SDSS J1021+4913 1.720 0.500 1.0

SDSS J1118+0745 1.736 0.310 2.3

SDSS J1138+0314 2.442 0.445 1.1

SDSS J1155+6346 2.888 0.176 1.8

SDSS J1206+4332 1.790 0.748 3.0

SDSS J1226-0006 1.125 0.517 1.2

SDSS J1332+0347 1.445 0.191 1.0

SDSS J1335+0118 1.571 0.500 1.5

SDSS J1353+1138 1.623 0.300 1.3

SDSS J1406+6126 2.134 0.270 2.0

SDSS J1415+1129 2.561 0.900 1.7

SDSS J1650+4251 1.541 0.590 1.3

The SDSS sample

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No

Big B

ang

▬ 99% CL▬ 95% CL▬ 68% CL

Gravitational Lenses from the SDSS

Case zL zs Δθobs(")

SDSS J0145-0945 2.732 0.490 2.2

SDSS J0246-0825 1.685 0.724 1.0

SDSS J0746+4403 1.998 0.300 1.0

SDSS J0806+2006 1.537 0.570 1.4

SDSS J0903+5028 3.584 0.388 2.8

SDSS J0911+0550 2.783 0.770 3.2

SDSS J0913+5259 1.377 0.830 1.1

SDSS J0924+0219 1.523 0.400 1.3

SDSS J0951+2635 1.246 0.260 1.1

SDSS J1001+5027 1.839 0.300 2.9

SDSS J1004+4112 1.740 0.680 3.8

SDSS J1021+4913 1.720 0.500 1.0

SDSS J1118+0745 1.736 0.310 2.3

SDSS J1138+0314 2.442 0.445 1.1

SDSS J1155+6346 2.888 0.176 1.8

SDSS J1206+4332 1.790 0.748 3.0

SDSS J1226-0006 1.125 0.517 1.2

SDSS J1332+0347 1.445 0.191 1.0

SDSS J1335+0118 1.571 0.500 1.5

SDSS J1353+1138 1.623 0.300 1.3

SDSS J1406+6126 2.134 0.270 2.0

SDSS J1415+1129 2.561 0.900 1.7

SDSS J1650+4251 1.541 0.590 1.3

SUMMARY

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Summary

Calculation of the mean image separation as a function of source redshift for various cosmological models

Distribution of Candidates for lens systems from the SDSS show slightly positive correlation or do not show any correlation.

Curvature TestSpearman test shows the curvature of the universe is ‘0’, ‘negative’ or ‘not-excessively’ positive.

Ωm-ΩΛ TestCalculated the probability distribution and maximum likelihoodwith the JVAS/CLASS sample and the SDSS sample.

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