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.
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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
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LENS SYSTEMS FROM THE SDSS
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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
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CURVATURE TEST
16
Curvature Test
Mean image separation Candidates for Lens systems
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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
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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
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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
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1ni
i
dPL
d
* *1.0, 276, 254 ( / )SO E km s
*6.5, 88.8 ( / )km s
The JVAS/CLASS sample
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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
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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
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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