PREDRAG JOVANOVIĆAND

LUKA Č. POPOVIĆ

ASTRONOMICAL OBSERVATORY

BELGRADE, SERBIA

Gravitational Lensing Statistics and Cosmology

Outline

Observational cosmology: basics and parameters Cosmological experiments:3. Cosmic Microwave Background Radiation (CMBR)4. Type Ia supernovae5. Gravitational lensing

Strong: detection of distant galaxies Weak: detection of dark matter Time delay: determination of H0

Statistics: constraining Ω0 and ΩΛ

Problems with gravitational lensing statistics

Conclusions

Cosmology basics

The current models of cosmology are based on the field equations of general relativity:

Friedmann-Lemaître-Robertson-Walker (FLRW) metric: a solution of the Einstein field equations in the case of a simply connected, homogeneous, isotropic expanding or contracting universe:

r, ϕ, ϑ - comoving polar coordinates k - the scalar curvature of the 3-space: k = 0, > 0, or < 0 corresponds to

flat, closed, or open universe a(t) - the dimensionless scale factor of the universe ΛCDM model uses the FLRW metric, the Friedmann equations and the

cosmological equation of state to describe the universe

4

1 8,

2

GR g R g T

cµν µν µν µνπ− + Λ =

( )2

2 2 2 2 2 2 22

( ) sin ,1

drds dt a t r d d

krϑ θ ϕ

= − + + + −

Cosmological parameters

H - the Hubble constant ρ - the mass density of the universe Λ - the cosmological constant k - the curvature of space a - the expansion factor of universe dimensionless density

parameters:

where the subscript “0” indicate the quantities which in general evolve with time and which are referring to the present epoch

several observational techniques are used for their estimation

020

8,

3M

G

H

π ρΩ =2

20

,3

c

HΛΛΩ =

2 20 0

,k

k

a HΩ = − 1M kΛΩ + Ω + Ω =

Wilkinson Microwave Anisotropy Probe (WMAP)

The "angular spectrum" of the fluctuations in the WMAP full-sky map, showing the relative brightness of the "spots" in the map vs. the size of the spots. The shape of this curve contain a wealth of information about the history of the universe

Supernova Cosmology Project

Type Ia supernovae: the standard candles

Intrinsic luminosity is known Apparent luminosity can be measured The ratio of above two luminosities

can provide the luminosity-distance (dL) of a SN

The red shift z can be measured independently from spectroscopy

Using dL (z) or equivalently the magnitude(z) one can draw a Hubble diagram

Constraining the cosmological parameters

• Riess et al. 2004, ApJ, 607, 665• Tonry et al. 2003, ApJ, 594, 1

Content of the Universe

Gravitational lensing

Einstein Ring Radius of a gravitational lens

QSO 2237+030 (z=1.695), also known as “Einstein cross” and lensing galaxy ZW2237+030 (z=0.0394)

RXJ1131-1231

PG 1115+080

Examples:

Strong lensing: detection of distant galaxies

•The orange arc: an elliptical galaxy at z=0.7, •the blue arcs: star forming galaxies at z= 1 - 2.5 •the red arc and the red dot: the farthest known galaxy at z~7 (13 billion ly away, i.e.

only 750 million years after the big bang

Weak lensing: detection of dark matter

unlensed lensed

Distribution of dark matter

The Hubble constant from gravitational lens time delays

Kochanek & Schechter, 2003, astro-ph/0306040

Courbin, 2003,astro-ph/0304497

HST Key Project: determination of the H0 by the systematic observations of Cepheid variable stars in several galaxies using HST

Gravitational lensing statistics

More details about history and basics in the book: P. Schneider, C. Kochanek and J. Wambsganss, 2006, “Gravitational Lensing: Strong, Weak and Micro”, Saas-Fee Advanced Courses, Springer Berlin Heidelberg (http://www.springerlink.com/content/n37347/)

Optical depth for gravitational lensing, i.e. the probability to observe such effects (Turner et al. 1984, ApJ, 284, 1; Turner, 1990, ApJ, 365, L43):

where zS and zL are the source and lens redshifts, σ is lens velocity dispersion, φ(σ; zL) is the velocity function, A is the cross section for multiple imaging, B is the magnification bias, dV is the differential comoving volume element

The Current State: lens statistics constraints on ΩΛ and Ω0 are in good agreement with results from Type Ia supernovae

for a spatially flat universe: ΩΛ = 0.72 - 0.78 (Mitchell et al. 2005, ApJ, 622, 81)

Likelihood contours at the 68%, 90%, 95%, and 99% confidence levels. The dotted line marks spatially flat cosmologies

The separation distribution of the 12 CLASS lenses

Mitchell et al. 2005, ApJ, 622, 81

Differential (thick) and cumulative (thin) probability along the line of spatially flat cosmologies

Gravitational macrolensing optical depth

The effective optical depth is related to the number NGL(z) of multiply imaged quasars within a sample of NQSO(z) quasars with redshifts z by:

( ) ( )( )

GLGL

QSO

N zz

N zτ =

Zakharov, Popović and Jovanović, 2004, A&A, 881

Distribution of all QSOs and lensed QSOs in Veron & Veron Catalogue

Veron-Cetty & Veron, 2006, A&A, 455, 773: a sample of 85221 (NQSO) quasars among which 69 (NGL) are gravitationally lensed

The ratio of lensed to total number of quasars and optical depth for three different flat cosmological models as a function of quasar redshift

Optical depth of cosmologically distributed gravitational microlenses

(Zakharov, Popović and Jovanović, 2004, A&A, 881)

Optical depth of cosmologically distributed gravitational microlenses for three different values of ΩL

Problems with gravitational lensing statistics

Small number of observed gravitational lenses (~100) is insufficient for reliable statistics. Solution not later than 2015: LSST, SNAP, SKA and JWST projects will drastically increase the number of detected gravitational lenses

Large Synoptic Survey Telescope (LSST): 2013

SuperNova/Acceleration Probe (SNAP): 2013

Square Kilometre Array (SKA): 2015

James Webb Space Telescope (JWST): 2013

Extinction by dust in the lens galaxies leads to artificially low number of observed lenses Galaxy evolution: decrease of lensing population for higher redshifts would lower the

number of observed lenses Ellipticity and clustering: mass distributions of lenses is not circularly symmetric Cosmology

Conclusions

We demonstrated constraining the cosmological parameters by gravitational lens statistics on a sample of lensed quasars from Veron & Veron catalogue of quasars and active nuclei

Obtained results are in satisfactory agreement with those obtained from CLASS and SDSS surveys (Mitchell et al. 2005, ApJ, 622, 81)

Optical depth of cosmologically distributed gravitational microlenses also depends on assumed cosmological model (Zakharov, Popović and Jovanović, 2004, A&A, 881)