Gravitational Lensing. A Brief History, Theory, and Applications

Gravitational Lensing A Brief History, Theory, and Applications

A Brief History Einstein (1915): light deflection by point mass M due to bending of space-time = 2x Newtonian light tangentially grazing Sun s surface deflected by 1.7 1919: apparent angular shift of stars at Sun s limb measured during eclipse Eddington (1920): may be multiple light paths connecting source, observer Einstein (1936): little chance of observing lensing phenomena caused by stellar-mass lenses

A Brief History (cont.) Zwicky (1937): galaxies can split images of background sources by large enough angle to be observed galaxies commonly believed to have masses of 10^9 solar virial analysis of Coma cluster --> 10^11 solar not only additional test of GR, but also magnify distant galaxies and allow accurate mass determination even calculated the probability of lensing by galaxies

A Brief History (cont.) Refsdal (1964): Hubble constant through gravitational lensing of a variable source light travel times for images unequal --> intrinsic variations of source observed at different times in images time delay between images proportional to difference in absolute lengths of light paths, proportional to 1/H_0 H_0 if time delay measured + accurate model of lensed source developed

A Brief History (cont.) Walsh, Carswell, & Weymann (1979): discovered first example of gravitational lensing, quasar QSO 0957+561A,B two images, A and B, separated by 6 evidence for twin lensed images of single QSO: (i) similarity of spectra of two images, (ii) flux ratio between images similar in optical and radio, (iii) foreground galaxy between images, (iv) VLBI observations show correspondence between knots of emission in two radio images

A Brief History (cont.) Paczynski (1986): at any time one in million stars in Large Magellanic Cloud (LMC) measurably magnified by star in Milky Way s halo microlensing events have time scales 2h-2yr for lens masses 10^-6 to 10^2 solar proposed experiment of monitoring light curves of million stars feasible? light curves must be sampled frequently and distinguished from intrinsically variable stars

A Brief History (cont.) Hewitt et al. (1987): Einstein rings discovered first in radio, detailed modeling of galaxy mass distribution Paczynski (1987): arcs = background galaxies strongly distorted and elongated by foreground cluster, confirmed when first arc redshifts were significantly > than that of cluster

A Brief History (cont.) why care about lensing? magnification: lenses act as cosmic telescopes, can infer source properties far below resolution limit or sensitivity limit of other methods depends solely on projected, 2-D mass distribution of lens, independent of lens luminosity or composition --> ideal way to detect dark matter, growth and structure of mass condensations properties of lensed objects depend on age, scale, and overall geometry of universe (eg, H_0, dark energy)

Lensing by Point Masses b dls dl ds

Lensing by Point Masses (cont.)

Lensing by Point Masses: Summary ^ b solve for theta in terms of beta using GR determination of small deflection alpha dls dl looks Euclidean -- but not ds

Lensing by Point Masses: Summary negative image is inverted (if extended), longer route, lies within Einstein radius, can be dimmer than source positive image further from lens, lies outside Einstein radius, always brighter than source

Lensing by Point Masses: Summary as beta --> 0, images brighter, more symmetric, extended tangentially (shear) as beta increases, source moves from lens, negative image --> lens and dims, positive image --> true source position and mu goes to 1 see simulation... at beta=0, mu infinite!?! but no true pt sources: only one pt at ctr, so 0 area, 0 flux, finite flux in Einstein radius

Lensing by Point Masses: Microlensing Application vs. variable stars, which change their colors in contrast, variable stars have asymmetric light curves

Lensing by Point Masses: Microlensing Application

Lensing by Point Masses: Microlensing Application consider lensing of star in Galaxy: if closest approach between point mass lens and source is within Einstein radius, peak magnification in light curve is at least 1.34 corresponds to brightening by 0.32 magnitudes, observable

Lensing by Point Masses: Microlensing Application time scale for microlensing-induced variations in terms of the typical angular scale (Einstein radius), relative velocity v between source and lens, and distance to the lens: L L distance ratio ~ 1 for MW-LMC LS if light curves are sampled over intervals between 1h and 1yr, MACHOs in the mass range 10^-6 to 10^2 solar detectable measurement of t_0 for event does not directly give M, but only a combination of M, distances, and v --> degeneracy!

Lensing by Point Masses: Microlensing Application chance of microlensing event is an optical depth, probability that at any time given star is within Einstein radius of lens optical depth is integral over the number density n(dl) of lenses x area enclosed by Einstein ring of each lens: where L L L is volume of infinitesimal spherical shell with radius DL which covers solid angle dw integral gives the solid angle covered by Einstein circles of lenses, and probability is obtained from dividing this quantity by solid angle dw observed --> bottom line: tau depends on mass density of MACHOs, not enough

Lensing by Extended Sources axially symmetric (but extended) lens, deflections still radial, only mass interior to b matters: ^ ^

Lensing by Extended Sources (cont.) axially symmetric (but extended) lens, deflections still radial, only mass interior to b matters: ^ ^ but deflection no longer simple function... find roots where critical density

Lensing by Extended Sources: Example axially symmetric, singular isothermal sphere lens: projected mass surface density: thus, mass interior to r:

Lensing by Extended Sources: Example axially symmetric, singular isothermal sphere lens: projected mass surface density: thus, mass interior to r: so, deflection angle: ^ independent of radius!

Lensing by Extended Sources: Example axially symmetric, singular isothermal sphere lens: projected mass surface density: thus, mass interior to r: so, deflection angle: ^ independent of radius! Einstein radius: again ring radius for beta=0, +/- depends on which side of lens light travels

Lensing by Extended Sources: Example for object at cosmological distance, impact parameters are 1-2 kpc in galaxy and 50 kpc in cluster probing the central regions of the halo

Lensing by Extended Sources: Example what images do we get from? caustic crossing: number of images changes discontinuously source location

Lensing by Extended Sources: Example what images do we get from? caustic crossing: number of images changes discontinuously source location magnification: so, outer image is brighter, but as beta--> 0, images become symmetric, each image still magnified by up to 2x highly magnified image is tangentially elongated into giant arc, since, arcs determine Einstein radius with source and lens z, get mass of lensing cluster or galaxy

Lensing by Extended Sources: Example imaging of extended source by non-singular circularly symmetric lens source close to point caustic at lens center produces two tangential arc-like images close to outer critical curve and faint image at lens center source on outer caustic produces radially elongated image on inner critical curve, and tangential image outside outer critical curve

Lensing by Extended Sources: Example real mass distributions are not axially symmetric allows 4 or even more images source plane has regions where 1, 3, 5 or more images created boundaries are caustics, number of images odd for nonsingular density distribution one of images typically demagnified, leaving 2 or 4

Lensing by Extended Sources: Cluster Mass Application

Lensing by Extended Sources: Cluster Mass Application arc location in cluster gives projected cluster mass within circle traced by arc for circularly symmetric lens, average surface mass density within arc radius equals critical surface mass density radius of circle traced by arc gives estimate of Einstein radius of cluster L

Lensing by Extended Sources: Cosmological Application for beta > 0, light travel time between two images is different light from the outer, brighter image arrives first quasars vary, so by measuring light curve of two (or more) images separately, get time delay time delay is part geometric and part gravitational time dilation Einstein radius, D s all have 1/H_0, so delta t goes like 1/H_0

Lensing by Extended Sources: Weak Lensing Application Einstein radius about 50 kpc for cluster at 5 Mpc, mu only 1%, background galaxy 1% streched for any given galaxy, undetectable (galaxies vary in their luminosities and ellipticities) but, average over many background galaxies, position angles of their intrinsic ellipticities cancel, so need to measure 1% ellipticities, requires 10^4 galaxies! 5 Mpc radius corresponds to 1000, 1/3 of degree, so enough galaxies on sky, can reconstruct cluster mass on scales well beyond virial radius! but distortions in camera, atmosphere...