April 3, 2005

The lens redshift distribution –

Constraints on galaxy mass evolution

Eran Ofek, Hans-Walter Rix, Dan Maoz (2003)

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

April 3, 2005

Gravitational Lensing

Ds

DlsDl

22

42ˆ

c

GMd

c

April 3, 2005

Gravitational Lensing

22

42ˆ

c

GMd

c

April 3, 2005

Gravitational Lensing

22

42ˆ

c

GMd

c

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

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

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

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

April 3, 2005

Probability vs. Lenses

,...,;,,1

msl

N

izzPL

JK

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

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

April 3, 2005

29.059.049.0

CosmologyNo evolutionFlat Universe

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:

April 3, 2005

Jackknife - bias and outliners

Bias: 7.02.0

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

April 3, 2005

End

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

April 3, 2005

Sensitivity

definitions

Uz

Pz

z

nzn

10

10

**

**

FJ

"12

0.2

sz

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)

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

April 3, 2005

Previous Work

Kochanek 1992

Helbig & Kayser 1996

Kochanek 1996

Weak constraints on cosmology

N=4

N=6

N=8

Sample size

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)

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

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

**

**

April 3, 2005

Scatter of the Faber-Jackson

unperturbed

20% ln-normal scatter

40% ln-normal scatter

April 3, 2005

Sensitivity - Cosmology

"1

2

sz

lzPmaxFlat U

niverse

April 3, 2005

April 3, 2005

Gravitational Lensing

A

Ds

DlsDl

22

42ˆ

c

GMd

c

April 3, 2005

Gravitational Lensing

April 3, 2005

Gravitational Lensing

Ds

DlsDl

22

42ˆ

c

GMd

c

A