2. Lens population. 3. Lens model. 4. Cosmology. z=0.5. z=0

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1 Chuck Keeton Hubble Fellow, University of Chicago ffl Lens statistics probe the volume of the universe to z ο 1 3. ffl Current constraints on Λ. ffl Systematics! ffl Prospects for equation of state (w, _w).

2 z= Lens population 3. Lens model 4. Cosmology z=0 Optical depth : fi = Z dv Z dm dn dm A lens(m)

3 ffl Lens selection effects are important: Seeing ο typical lens size Extinction in lens galaxy Lens galaxy luminosity if L(gal) > L(QSO) Radio sources: ffl Flux distribution known. ffl Redshift distribution not well known. ffl Lens selection functions unimportant. Sample # Lenses # Sources Optical Radio 18 ß 12,000

4 2.1. Phenomenology Models ffl (All) observed lenses are produced by normal galaxies. (Not by group or cluster halos.) ffl Use observed Galaxy Luminosity Function (GLF). ffl Use local GLF and assume constant comoving number density. (Okay for ellipticals?) ffl Luminosity ψ! mass with Faber Jackson: L L Λ = 0 ff A ff Λ with ff Λ ' 220 km/s; fl ' 4 ffl Optical depth (flat cosmology): fi(z s ) = 8 ψ! 2 15 ß3 R 3 H ff 4 Λ ffiλ 41+ff + 4 c fl 3 5 D(zs ) 3 ffl Cosmology dependence: volume

5 ffl Sheth Tormen ffl Jenkins et al. These automatically include cosmology dependence and evolution. Problem: ffl Theoretical mass functions don't match observed luminosity functions. (c.f. semi-analytic models of galaxy formation) ffl Baryons! ffl It's the region around the baryons that matters for lensing. Cosmology dependence: volume, ΩM, growth factor

6 1 ρ / r(s + r) 2 Galaxies: Singular Isothermal Sphere (SIS) ρ / r 2 (inside 5 10 kpc) Cross section A / ff 4. The difference can be attributed to cooling: ffl M > M cool gas has not cooled, system remains NFW. ffl M < M cool gas has cooled and condensed into a galaxy, adiabatic compression has modified dark matter halo, result is SIS. ffl M cool ο M fi scale for which the gas can cool in a Hubble time. (See Kochanek & White 2001 ApJ 559:531.)

7 Li & Ostriker (astro-ph/ ) also Keeton (1998 thesis); Porciani & Madau (2000 ApJ 532:679)

8 ffl Galaxy luminosity function (for phenomenology models) ffl Phenomenology vs. theory mass function ffl Evolution Data: ffl Radio: source redshifts ffl Optical: Poisson statistics, selection effects

9 Autofib SSRS2 CfA EEP Durham/UKST 0.01 Stromlo/APM LCRS Absolute B-Band Magnitude Cross et al. (2001 MNRAS 324:825)

10 Phenomenology Models Study Sample GLF Ω Λ (if flat) Kochanek (1996) Opt+Rad K96 < 0.66 Falco et al. (1998) Opt/Rad K96 < 0.73 / 0.62 / 0.74 (samp Helbig et al. Rad K96 < 0.65 Waga & Miceli (1999) Opt K96 < 0.55 / 0.76 / 0.91 (extin Cooray et al. (1999) HDF HDF < Cooray (1999) Rad ESP < 0.79 EEP ' 0.5 Chiba & Yoshii (1999) Opt+Rad APM ' 0.7 CfA ' 0.8 uncertainties in the luminosity function of galaxies by type The as much to the uncertainties in the cosmological limits contribute from lens statistitics as the Poisson errors arising from the derived size of the samples." small Kochanek et al. (astro-ph/ )

11 Note: increasing Ω Λ increases N lens.

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13 Li & Ostriker (astro-ph/ ): CLASS survey SCDM 37 LCDM 16 OCDM 12 Observed 18 Higher Ω M =) more galaxies =) more lenses

14 Phenomenology: SCDM predicts fewer lenses than LCDM Theory: SCDM predicts more lenses than LCDM Phenomenology: + (All) normal lenses are produced by normal galaxies. Uncertainties in luminosity function and its evolution. Theory: + Automatically includes cosmology and evolution effects. But does it really describe galaxy populations?

15 ffl Claim little evolution to z ο 1 in early-type galaxies ) Assume constant comoving number density ) N lens / N gal / volume CNOC2 redshift survey (H. Lin et al.): ffl Observe sample 0:12 < z < 0:55, derive evolving Schechter function ffl See small, non-zero evolution in early-types ffl Evolution rate depends on cosmology Degeneracy between evolution and cosmology.

16 N lens (LCDM) N lens (SCDM) = 8 < : 3:1 traditional models 1:1 CNOC2 models If you use models normalized by deep number counts, do you lose most sensitivity to cosmology? Need to measure evolving dn=dff (e.g., DEEP).

17 Keeton & Zabludoff (in prep.) ffl Expect environments to affect the number of lenses by ο<10 20% ffl Working to identify lens environments, study their properties

18 0.3 p x =[ m 3-1]ρ x Ωm m Waga & Miceli (1999 PRD 59:103507); Cooray (1999 A&A 342:353) ffl Upper limits on Ω X ffl Lower limits on w ffl (Increasing w reduces N lens, allows higher Ω X.)

19 ffl project ο2000 lenses in LCDM ffl Redshifts known and binned: z max = 5, ff z = z = 0:3 ffl Flat universe, Ω M = 0:3 ± 0:1 (marginalized)

20 (dw/dz) z maz =5, dz=0.1 z max =5, dz=0.3 z max =3, dz=0.3 (dw/dz) w(z)= z w(z)= w(z)= w w 0

21 Sarbu, Rusin & Ma (2001 ApJL 561:L147) ffl Normal: normal galaxies ffl Wide-sep: groups/clusters more sensitive to growth factor

22 Sarbu, Rusin & Ma (2001 ApJL 561:L147)

23 ffl SDSS and 2dF seem to resolve many of the previous problems with the local GLF. ffl SDSS has dn=dff for ellipticals. ffl Compute robust optical depth (for normal lenses): fi = (const) Z dv 0 D D os 1 A 2 Z dff dn dff ff4 ffl Use with existing lens samples. ffl (Evolution?) 2. Wide-Separation Lenses ffl Define a statistical sample. (Expect a few per 10,000 QSOs.) ffl Know source population, selection functions. ffl Different dependence on cosmology =) complementary test.

24 Define the sample (normal or wide-sep lenses) ffl Galaxy luminosity function? Better data: SDSS, 2dF, etc. ffl Evolution? Use the data: CNOC2, DEEP, etc. Data: ffl Larger optical samples? SDSS(?), SNAP, etc. ffl Radio source redshifts? Hard work Proposal: normal vs. wide-separation lenses as a consistency check.