Cosmology and Strongly Lensed QSOs

Transcription

1 Cosmology and Strongly Lensed QSOs Andy Friedman Astronomy 200, Harvard University, Spring

2 Ho Outline ΩΛ Determining Ho from Time Delays History/Theoretical Framework Observations of Time Delay Systems Current Results/Controversy Determining ΩΛ from Quasar Lensing Statistics History/Theoretical Framework Observations and Current Constraints Conclusions Comparison to other Cosmographic Methods

3 Ho from Time Delays - History

4 Ho from Time Delays - History First Gravitationally Lensed Quasar discovered Walsh et. al Q First Time Delay Measurements For Same Lens Q , reported in Florentin-Nielsen Initial Controversy: Δt t ~ 1.1 years (Schild( & Cholfin 1986, Vanderriest et al. 1989) Δt t ~ 1.5 years (Press, Rybicki,, Hewitt 1992) Settled in favor of Δ t ~ 1.1 years after 5 more years of observing (Kundic( et al. 1997, Schild & Thomson 1997, Haarsma et al. 1999).

5 Ho from Time Delays - Theory Angular Diameter Distances An extra factor of Comes from the ratios of Matter Density Parameter Lens Redshift Source Redshift Proper length at redshift Angle subtended by proper length observed from redshift <

6 Schematic 2-Image 2 Time Delay Lens If the source lies inside the Einstein ring of the lens, you ll get multiple images A,B. As we ll see, in the simplest lens model, the time delay and Ho depend most strongly on the average surface density <κ> inside the annulus formed at the radius of the images. Kochanek & Schechter, 2003

7 Singular Isothermal Sphere (SIS) SIS Potential (parametric model) By solving the 3D or 2D Poisson Eq. SIS Time Delay Where and For the SIS lens produces 2 Colinear images. Basically, the diagram on the left with.

8 Power Law Monopole SIS potential is special case of power law monopole for. approaches potential of a point mass. Larger η more centrally concentrated mass distributions, falling rotation curves. Approximate Time Delay Expansion Where and

9 Lessons From (SIS) Model Image astrometry of simple 2 and 4 image lenses generally cannot constrain the radial mass distribution of the lens. More centrally concentrated mass distributions (larger η ) predict longer time delays, resulting in a larger Hubble constant for a given time delay measurement. Assuming that the surface density near the Einstein ring goes as a power law in η,, where is the average surface density in that region, we find Once you know the image positions & the total mass inside the Einstein ring, the time delay is largely determined not by the global structure of the radial density profile, but rather by the surface density near the Einstein ring!

10 What we ve ignored so far deviations from power law monopole radial density profile. deviations from circular symmetry i.e. ellipticity of lensing galaxy. Mass-Sheet degeneracy (lenses in galaxy clusters) angular structure of lens. Shear from local tidal gravity field, internal and external to lens. lenses with satellite galaxies.

11 Observational Procedures for Measuring Time Delays Monitoring campaign must produce light curves for the individual images that are well sampled compared to the time delays. Source quasar must be variable on timescales shorter than monitoring period. Resulting light curves are cross correlated to measure time delays and their uncertainties. Care must be taken to identify uncorrelated variability microlensing,, systematic photometric errors, (radio scintillation).

12 Light Curves for B Left Panel Light Curves for B, A, C, D images (top down) Right Panel Composite Light Curve after accounting for 3 Independent Time Delays: Δ t AB: 31.5+/-1.5 days, Δ t CB: 36.0+/1.5 days, Δ t DB: 77.0+/- 1.5 days (VLA Radio Map)

13 Observational Uncertainties Want Δt accurate to 3-5% 3 (~1% later). If Δt is known to ~5%, the uncertainty can be reduced since one can predict flux variations and plan the monitoring campaign accordingly. Errors in source and lens redshifts are negligible. Dependence of angular diameter distances on cosmology is unimportant until total errors ~5%. Astrometry errors limited by resolution of (VLA, Merlin, VLBA), (HST), dust in lens galaxy, perturbations from, microlensing by stars (~1 µas), CDM halos, satellite galaxies (~1 mas).

14 Uncertainties In Lens Models Try to fit models of gravitational potential constrained by available data: images and lens positions, relative image fluxes, relative time delays (parametric & non-parametric). Lens model should include: microlensing - motion of stars effects time delays through uncorrelated variability. Also millilensing for a satellite galaxy in lens, or for perturbations of image positions due to CDM halos. Both microlensing & millilensing effect magnification, flux ratios, image positions. harder to constrain models. In the end, however, uncertainty in value of <κ> dominates in regard to determining Ho.

15 Time Delay Measurements Kochanek & Schechter 2003

16 Ho Results Models for 4 simple time delay lenses yield: Ho = 43 +/- 3 km/smpc Isothermal Density Profiles, (smaller η ~2 -less centrally concentrated, flat rotation curves ) Ho = 71 +/- 3 km/smpc Constant M/L models (larger η > 2 - more centrally concentrated, falling Keplerian rotation curves.) Ho = 72 +/- 8 km/smpc (HST Key Project) Lensing results agree with local estimates only if the lenses have little dark matter in their halos!

17 Ho Results Values of Ho for all lenses with known time delays, taken directly from the literature. Courbin 2003 Mean value of Ho is only very marginally compatible with local estimate from HST key project. (Also WMAP, SNe Ia, etc.)

18 How to Improve the Ho Results We must break the degeneracy between Ho and <κ>, the central concentration problem. Kochanek & Schechter 2003 note several approaches to solving this: We can apply constraints on density profiles derived from observations of other early-type galaxies. We can make new observations of the existing time delay systems to improve errors, further constrain density profiles. Measure the time delays in systems where the lens galaxies already have well constrained densities. Use the statistical properties of time delay lenses to break the degeneracies seen in individual lenses. Find more time delay lenses w/ large scale surveys.

19 So Who Do You Believe? If Ho from lensing is right, assuming galaxies have isothermal profiles w/ DM halos, then local estimates are too high. If Ho from local estimates is right, then galaxies have less DM than expected. Learn something about mass profiles of galaxies rather than Ho. How then, do we reconcile this with measured velocity dispersions in E/S0 lens galaxies, which appear to be isothermal, closer to SIS models? Which cosmographic method do you trust?

20 Part II: Constraining ΩΛ From Quasar Lensing Statistics Cosmological models with large values of the cosmological constant produce more lenses. A flat cosmology with ΩΛ = 1 has ~10 times as many gravitational lenses as a flat cosmology with ΩΛ = 0. Number of lenses is very sensitive to ΩΛ. This is a purely geometric effect which deals with how the comoving volume element dv/dz depends on ΩΛ. High ΩΛ dv/dz increases with redshift z.

21 Dependence of dv/dz on Redshift (ΩM, ΩΛ) = (0.2,0.8) (ΩM, ΩΛ) = (0.05,0) above (ΩM, ΩΛ) = (1,0) below Valid for flat cosmologies Flat cosmologies w/ large ΩΛ yield larger comoving volume elements at higher redshift! More volume at higher z implies a larger path length between observer and quasar, higher likelihood of hitting an intervening lens galaxy, (assuming constant galaxy density).

22 Λ From Lensing Statistics Natural Approach: Find out what percentage of quasars you would expect to see lensed in a given quasar survey, as a function of ΩΛ. Do the survey, count the observed lenses, divide by the total number of quasars in the sample to find the lensing fraction, which then constrains ΩΛ. Turner, Ostriker,, & Gott 1984 (TOG) worked out an expression for the probability of quasar lensing, also termed the lensing optical depth,, which takes into account cosmology, the various survey selection effects, and the lens model.

23 History of From Lens Statistics ΩΛ Turner 1990, using (TOG) explored the limits on ΩΛ,, needed for flatness concluded that ΩΛ could not make a dominant contribution in a flat cosmology. Kochanek 1996, w/ better lensing survey data, galaxy mass modelling.. Obtained upper limit of ΩΛ < 0.66 (95% confidence). Mitchell et al. 2004, using JVAS/CLASS survey data, find ΩΛ consistent with SNe Ia results, WMAP, concordance cosmology. Now, how do we actually constrain ΩΛ?

24 Back to the SIS Lens Model Density Profile Einstein Radius *Sources located at angle θs < θe on sky from an SIS lens yield 2 images on opposite side of lens at angular positions: Image magnifications Total magnifications of both images with Bright/Faint Image Flux Ratio Only sources w/ f < fmax are detected.

25 Optical Depth For Lensing Comoving Volume Element Velocity Function of Lenses Cross Section of Lens Magnification Bias To get probability that a quasar at redshift z s will be lensed in a given cosmology, integrate over volume out to source, multiply by the integral over all lenses out to some redshift z l, each with a certain cross section, all with the survey magnification bias

26 Lens Cross Section & Magnification Bias Cross Section: : The angular size of the lens. Without a flux ratio cut, it is effectively the radius of the Einstein Ring... Only weakly dependent on cosmology. Magnification Bias: : Intrinsically faint sources can appear in a flux-limited limited survey by virtue of lensing magnification. Product of Cross Section and Magnification Bias Number of sources brighter than flux Flux limit of survey Total magnification Source position corresponding to fmax. Cross section depends on σ, but in SIS model, there is a 1-1 relation between σ and θ max (which is less than the Einstein radius)

27 Velocity Distributions of Lensing Galaxies Previous studies relied on combination of E/S0 galaxy luminosity function with Faber-Jackson (FJ) relation (L~σ 4 ) to arrive at velocity dispersion distribution. Ignoring dispersion in FJ relation led to biased estimate of a and thus, overconfident cosmographic constraints. However, we don t t need FJ anymore. Can measure velocity dispersions of E/S0 galaxies (i.e. lens population) directly w/ Sloan Digital Sky Survey (SDSS), construct velocity dispersion distribution. Or you can use Press-Schechter Formalism.

28 Number of Lensed Quasars in Sample Quasar Redshift Distribution Observer s Optical Depth for Lensing (includes Maximum redshift in quasar sample magnification bias) JVAS (Jodrell( Bank/Very Large Array Astrometric Survey)/CLASS (Cosmic Lens All- Sky Survey) has found 22 lenses so far out of a sample of ~16,000 radio selected quasars. ~80+ lensed quasars have been found overall, by all surveys (see CASTLes website). CASTLe - CfA-Arizona Arizona Space Telescope Lens Survey CLASS Cosmic Lens All-Sky Survey -

29 Quasar Lens Surveys JVAS/CLASS Want survey of radio quasars to be fairly complete. Best w/ Radio selected sample (~16,000 objects). Not affected by dust. Higher angular resolution. (0.3 ) Can automate, for example, using VLA, MERLIN. Drawbacks of a Radio Survey Need to get separate optical redshifts of lens, source. Drawbacks of any survey Flux limited survey (not volume limited). Limited dynamic range flux ratio cutoff (fmax( = 10).

30 What We Wont Discuss Redshift evolution of the lensing galaxies. How CLASS does likelihood analysis of several statistical distribution to constrain ΩΛ. Distribution of image separations. Redshift distribution of lens galaxies. Joint Image Separation/Redshift distrib. Detailed calculations of predicted number of lenses for the CLASS survey and comparison to observations to constrain ΩΛ.

31 CLASS Lensing Constraints on ΩΛ ΩΛ = (ΩΛ( < 0.86 at 95% confidence) CLASS sample using measured velocity dispersion function Φ(σ) ) of large sample of E/S0 galaxies. Reflects uncertainty in number of lenses in sample. ΩΛ = (ΩΛ( < 0.89 at 95% confidence) Assumes velocity function Φ(σ) ) evolves in accord with extended Press-Schechter theory. Now in good accord with SNe Ia data. Provides independent confirmation of nonzero ΩΛ even w/o assuming flatness! With flat universe prior, this method also constrains ΩM.

32 Conclusions: Comparison to Other Cosmographic Methods CMB, SNeIa,, LSS, Cepheids vs. Strong QSO Lensing QSO Lensing now gives ΩΛ consistent with other estimates in concordance model. Lensing still gives inconsistent Ho. Unlikely it will reach precision of WMAP, HST key project, others anytime soon. If lensing Ho value is right, HST, WMAP Ho value is too large, cosmology has fundamental inconsistencies. However, if that Ho is correct, maybe one should use lensing to try to learn about galaxies, dark + light matter distributions. s. Maybe there is less DM than we think. Maybe we don t understand DM at all. Either way, QSO lensing is still a useful cosmographic tool.

33 References General * Blandford & Narayan,, 1992, ARA&A. Vol. 30 (A ), 90), p Blandford & Narayan,, 1986, Astrophysical Journal, Part 1, vol. 310, Nov. 15, 1986, p Courbin, Saha,, & Schechter,, 2002, Gravitational Lensing: : An Astrophysical Tool, Edited by F. Courbin D. Minniti,, Lecture Notes in Physics, vol. 608, p.1 Hogg, 2000, astro-ph/ , Distance Measures in Cosmology The Hubble Constant Ho Kochanek & Schechter,, 2003, eprint arxiv:astro-ph/ , to appear in Measuring and Modeling the Universe (Carnegie Observatories Astrophysics Series, Vol. 2), ed. e W.L. Freedman Courbin,, F. eprint arxiv:astro-ph/ * Blandford & Narayan,, 1992, ARA&A. Vol. 30 (A ), 90), p Refsdal,, S. 1964, MNRAS, Vol. 128, p.307. The Cosmological Constant -ΩΛ Mitchell, Keeton, Frieman,, & Sheth,, 2004, eprint arxiv:astro-ph/ , submitted to ApJ. Kochanek,, C. 1996, Astrophysical Journal v.466, p.638 Kochanek,, C. 1993, MNRAS, vol. 261, no. 2, p Carroll, Press, & Turner, 1992, ARA&A. Vol. 30 (A ), 90), p Turner, E. 1990, Astrophysical Journal, Part 2 Letters, vol. 365, Dec. 20, 1990, p. L43-L46. L46. Gott,, Park, & Lee, Astrophysical Journal, Part 1, vol. 338, March 1, 1989, p Turner, Ostriker,, & Gott,, Astrophysical Journal, Part 1, vol. 284, Sept. 1, 1984, p

34 Websites Astronomy 200, Spring Randall Cooper History, Overview and Basic Theory of Gravitational Lensing Xavier Koenig General Theory of Gravitational Lensing Kaloyan Penev Strong Lensing Web link CASTLe - CfA-Arizona Arizona Space Telescope Lens Survey - CLASS Cosmic Lens All-Sky Survey -

35 Ho from Time Delays - Theory Lensing Geometry Blandford & Narayan 1992

36 Model Dependence of Ho Results Different Ho values in these limits is entirely explained by the differences in <κ> and η between the SIS (small η=2) and constant M/L models (large 3>η>2). Can reduce Ho estimates to a simple approx. formula. A and B ~A/10 are derived from image positions using SIS. For 5 lenses PG , SBS , B , PKS *, HE , and a fixed slope η, the following relation holds. (Kochanek & Schechter 2003) In all such parametric models, <κ> is adjusted by changing η. In non-parametric models, <κ> is adjusted directly.

37 Time Delay Equation Ho from Time Delays - Theory Geometric Time Delay Shapiro Time Delay (Effective Grav. Potential) Image Position Source Position (No Lens) Lens Redshift Distance To Lens Distance To Source Distance From Source To Lens Lens Equation Fermat s Principle - Images Formed When