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April 3, 2005
The lens redshift distribution –
Constraints on galaxy mass evolution
Eran Ofek, Hans-Walter Rix, Dan Maoz (2003)
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April 3, 2005
Talk layout
Basics of gravitational lensing
The lens redshift distribution test
Sample selection
Constraints on galaxy mass evolution The maximum likelihood and bias
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April 3, 2005
Gravitational Lensing
Ds
DlsDl
22
42ˆ
c
GMd
c
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April 3, 2005
Gravitational Lensing
22
42ˆ
c
GMd
c
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April 3, 2005
Gravitational Lensing
22
42ˆ
c
GMd
c
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April 3, 2005
Lens Redshift Distribution
Based on observations: θ depends on lens mass(= velocity dispersion )
+
f/p
Numberdensity
velocity dispersion(=mass)
*
n*
Evolution?n* = n* (redshift)* = * (redshift)
Probabilityfor LensingP(zl | zs,)
~Number
density ofLenses
xGiven Lens:probability
of θxCosmology
How much“distance”in redshiftinterval
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April 3, 2005
7.03.0 m
7.03.0 m
9.01.0 m
Lens Redshift Distribution - Example
"12.1
49.0
72.2
l
s
z
z
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April 3, 2005
Selecting a sample
Doesn’t depend on angular selection completeness
Image separation is prior
Kochanek (1996): truncated the redshift probability distribution beyond the redshift at which the lens galaxy become too faint to have its redshift measured
Missing redshift information?
pr
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April 3, 2005
The Sample
Source redshift cut
1.2szSample I N=14
7.1szSample II N=11
More than 80 gravitational lenses known
Ignoring lenses without redshift information Bias
We need lenses with complete redshift information
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April 3, 2005
Probability vs. Lenses
,...,;,,1
msl
N
izzPL
JK
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April 3, 2005
Galaxy Mass Evolution
3.0
7.0
m
06.006.0
4.12.1
10.0
7.0
U
P
L
definitions
Uz
Pz
z
nzn
10
10
**
**
cos U/P/T
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April 3, 2005
Galaxy Mass Evolution
3.0
7.0
m
Not probable
rejected
Spirals for
S0 for
selliptical for
144
206
225
*
*
*
E
E
E
f
f
f
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April 3, 2005
29.059.049.0
CosmologyNo evolutionFlat Universe
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April 3, 2005
Jackknife - bias and outliners
Bias: 7.02.0
N
iKiKi TT
N
NStD
1
iKi TTNBias 1
Given a sample S1,....,Sn: calculate statistic T
Drop point number k (=Sk).
Calculate T(S1,...,Sk-1,Sk+1,...,Sn) = Ti≠k
Quenouille-Tukey jackknife bias:
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April 3, 2005
Jackknife - bias and outliners
Bias: 7.02.0
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April 3, 2005
Summary
Assuming “standard cosmology”
Sensitive to galaxy mass evolution
15.0,06.010.0,06.0* 10.0)]([log
dzzd 4.12.1* 7.0)]([log
dzznd
CL) (95% value current its of%651~@* z
Assuming no evolution CLskm %95@175 1*
19.0,08.012.0,07.090.0
Ef
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April 3, 2005
End
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April 3, 2005
Lens Redshift Distribution
sourcezdz
d, zn , A
dz
cdt
Number density of lenses =
galaxy mass function
Proper distanceinterval
Lensing cross section
Uz
Pz
z
nzn
10
10
**
**
+
f/p
Schechter function
Faber-Jackson relation n
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April 3, 2005
Sensitivity
definitions
Uz
Pz
z
nzn
10
10
**
**
FJ
"12
0.2
sz
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April 3, 2005
Future Work
Obtain additional lens redshiftMaoz, Rix, & Kochanek (@ VLT)
Use velocity dispersion functione.g., Sheth et al. (2003)
Larger sampleLarger sample Higher redshift (z~1.5)
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April 3, 2005
Spherical Isothermal Sphere 2
1
r
SIS assumption is consistent with data:- Galaxy dynamics (Rix et al. 1997)- X-ray emission from ellipticals (Fabbiano et al. 1989)- Lensing (Treu & Koopmans 2002)
SIS lenses Einstein radius = image separation2
1
For SIS the Einstein radius:s
lsE D
D
c2
2
4
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April 3, 2005
Previous Work
Kochanek 1992
Helbig & Kayser 1996
Kochanek 1996
Weak constraints on cosmology
N=4
N=6
N=8
Sample size
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April 3, 2005
Lens Redshift Distribution
2
**
2
2
1
*
112
1
*
2321
24
10exp110,
cn
dz
cdtD
D
Dzfz
dz
d
N
zUl
s
lsE
PUzN
]skm[
]Mpc[
1*
33*
hn
144
6.2
1046.1
16.12
206
0.4
1061.0
54.02
225
0.4
1039.0
54.02
Spiral S0 EllipticalParameters : 2dF - Madgwick et al. (2002)
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April 3, 2005
Problems
Lenses discovered based on lens-properties
Missing redshifts
Galaxy evolution
Clusters
Mass profile of galaxies
Q2237+0305
Photometric Redshift - FBQ0951+2635
Martel et al. (2002)Keeton et al. (2000)
Effect on separation %20Kochanek (1996) - small effect
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April 3, 2005
Galaxy EvolutionVan Dokkum et al. (1998) 0.83Keeton, Kochanek & Falco (1998) ~1Lin et al. (1999) 0.55
van Dokkum et al. (2001) 0.55Cohen (2002) ~1Rusin et al. (2002) ~1Im et al. (2002) ~1
20.026.0
3.08.0
04.056.0
6.0
15.059.0
2.08.0
2.07.0
5.0
45.0
P
Q
T
Q
T
Q
P
T
T
Tz
Uz
Pz
L
Mz
L
M
z
nzn
10
10
10
0
**
**
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April 3, 2005
Scatter of the Faber-Jackson
unperturbed
20% ln-normal scatter
40% ln-normal scatter
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April 3, 2005
Sensitivity - Cosmology
"1
2
sz
lzPmaxFlat U
niverse
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April 3, 2005
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April 3, 2005
Gravitational Lensing
A
Ds
DlsDl
22
42ˆ
c
GMd
c
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April 3, 2005
Gravitational Lensing
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April 3, 2005
Gravitational Lensing
Ds
DlsDl
22
42ˆ
c
GMd
c
A