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Gravitational Lensing: Mass Reconstruction Methods and Results

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##### Gravitational Lensing: Mass Reconstruction Methods and Results

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**Gravitational Lensing:**Mass Reconstruction Methods and Results Liliya L.R. Williams (U Minnesota) Prasenjit Saha (QMW, London & Univ. of Zurich)**Outline**Galaxy cluster Abell 1689 • Brief, non-technical introduction to strong (multiple image) lensing • Bayesian approach to the reconstruction of lens mass distribution • Overview of mass reconstruction methods and results • Non-parametric (free-form) lens reconstruction method: PixeLens • Open questions and future work**A Brief Introduction to Lensing**Goal: find positions of images on the plane of the sky How? use Fermat’s Principle - images are formed at the local minima, maxima and saddle points of the total light travel time (arrival time) from source to observer position on the sky total travel time**A Brief Introduction to Lensing**Plane of the sky Circularly symmetric lens On-axis source Circularly symmetric lens Off-axis source Elliptical lens Off-axis source**All the Information about Imagesis contained in the Arrival**Time Surface Positions: Images form at the extrema, or stationary points (minima, maxima, saddles) of the arrival time surface. Time Delays: A light pulse from the source will arrive at the observer at 5 different times: the time delays between images are equal to the difference in the “height” of the arrival time surface. Magnifications: The magnification and distortion, or shearing of images is given by the curvature of the arrival time surface. [Schneider 1985] [Blandford & Narayan 1986]**Substructure and Image Properties**Maxima, minima, saddles of the arrival time surface correspond to images smooth elliptical lens … with mass lump (~1%) added**Examples of Lens Systems**Galaxy Clusters Galaxies ~ 1 arcminute ~1 arcsecond • Properties of lensed images provide precise information about the total (dark and light) mass distribution can get dark matter mass map. • Clumping properties of dark matter the nature of dark matter particles. • We would like to reconstruct mass distribution without any regard to how light is distributed.**Bayesian approach to lens mass reconstruction**prior likelihood posterior • P(H|I) choices: • maximum entropy • min. w.r.t. observed light • smoothing (local, global) • … evidence parametric methods 5-10 parameters #data > #model parameters P(D|H,I) dominates P(H|I) not important #data < #model parameters P(H|I) is important ! D is data with errors P(D|H,I) is the usual c2-typefcn P(H|I) provides regularization D is exact (perfect data) P(D|H,I) is replaced by linear constraints P(H|I): can use additional constraints P(H|I) can also provide regularization D is exact (perfect data) P(D|H,I) is replaced by linear constraints P(H|I) is replaced by linear constraints no regularization -> ensemble average PixeLens**Mass Modeling Methods**Parametric–unknowns:masses, ellipticities, etc. of individual galaxies sufficient for some purposes, but not general enough Kneib et al. (1996), Natarajan et al. (2002), Broadhurst et al. (2004) Free-form – unknowns:usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)? lensing potential – automatically accounts for external shear mass – ensures mass non-negativity what data and errors to use? strong lensing (multiply imaged sources), weak lensing (singly imaged) data with errors: P(D|H,I) is usually a c2-type function data without errors: P(D|H,I) replaced by linear constraints how many model parameters (# pixels) to use? comparable to # observables greater than # observables what prior P(H|I) to use? regularization prior (MaxEnt; minimize w.r.t light; smoothing) linear constraints motivated by knowledge of galaxies, clusters how to estimate errors? if regularization – several possibilities if ensemble average – dispersion between individual models AbdelSalam et al. (1997,98), Bradac et al. (2005a,b), Diego et al. (2005a,b) PixeLens: Saha & Williams (2004), Williams & Saha (2005)**Parametric mass reconstruction:Kneib et al. (1996),**Natarajan et al. (2002) Question: what is the size of cluster galaxies? Each galaxy’s mass, radius are fcn (Lum) galaxy + cluster mass are superimposed Maximize P(D|H,I) likelihood fcn Abell 2218, z=0.175 collisionless DM predictions Best fit to 25 galaxies collisional fluid-like DM predictions Within 1 Mpc of cluster center galaxies comprise 10-20% of mass; consistent with collisionless DM 520 kpc**Mass Modeling Methods**Parametric–unknowns:masses, ellipticities, etc. of individual galaxies sufficient for some purposes, but not general enough Kneib et al. (1996), Natarajan et al. (2002), Broadhurst et al. (2004) Free-form – unknowns: usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)? lensing potential – automatically accounts for external shear mass – ensures mass non-negativity what data and errors to use? strong lensing (multiply imaged sources), weak lensing (singly imaged) data with errors: P(D|H,I) is usually a c2-type function data without errors: P(D|H,I) replaced by linear constraints how many model parameters (# pixels) to use? comparable to # observables greater than # observables what prior P(H|I) to use? regularization prior: minimize w.r.t light; smoothing linear constraints motivated by knowledge of galaxies, clusters how to estimate errors? if regularization: dispersion bet. scrambled light reconstructions if ensemble average – dispersion between individual models AbdelSalam et al. (1997,98), Bradac et al. (2005a,b), Diego et al. (2005a,b) PixeLens: Saha & Williams (2004), Williams & Saha (2005)**Free-form mass reconstruction withregularization:**AbdelSalam et al. (1998) Lens eqn is linear in the unknowns: mass pixels, source positions Image elongations also provide linear constraints. Data: coords, elongations of 9 images (4 sources) & 18 arclets Pixelate mass distribution ~ 3000 pixels (unknowns) Regularize w.r.t. light distribution Errors: rms of mass maps with randomized light distribution P(D|H,I) replaced by linear constraints P(H|I) Cluster Abell 2218 (z=0.175) 260 kpc**Free-form mass reconstruction withregularization:**AbdelSalam et al. (1998) Cluster Abell 2218 (z=0.175) center of mass center of light are displaced by ~ 30 kpc (~ 3 x Sun’s dist. from Milky Way’s center) Overall, mass distribution follows light, but: Mass/Light ratios of 3 galaxies differ by x 10 Chandra X-ray emission elongated “horizontally”; X-ray peak close to the predicted mass peak. centroid peak Machacek et al. (2002)**Free-form mass reconstruction withregularization:**AbdelSalam et al. (1997) Cluster Abell 370 (z=0.375) Color map: optical image of the cluster Contours: recovered surface density map Regularized w.r.t. observed light image Regularized w.r.t. a flat “light” image**Free-form mass reconstruction withregularization:**AbdelSalam et al. (1997) Cluster Abell 370 (z=0.375) Contours of constant fractional error in the recovered surface density**Mass Modeling Methods**Parametric–unknowns:masses, ellipticities, etc. of individual galaxies sufficient for some purposes, but not general enough Kneib et al. (1996), Natarajan et al. (2002), Broadhurst et al. (2004) Free-form – unknowns:usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)? lensing potential – automatically accounts for external shear mass – ensures mass non-negativity what data and errors to use? strong lensing (multiply imaged sources), weak lensing (singly imaged) data with errors: P(D|H,I) is usually a c2-type function data without errors (perfect data): P(D|H,I) replaced by linear constraints how many model parameters (# pixels) to use? comparable to # observables greater than # observables what prior P(H|I) to use? regularization prior: smoothing linear constraints motivated by knowledge of galaxies, clusters how to estimate errors? if regularization: bootstrap resampling of data if ensemble average – dispersion between individual models AbdelSalam et al. (1997,98), Bradac et al. (2005a,b), Diego et al. (2005a,b) PixeLens: Saha & Williams (2004), Williams & Saha (2005)**Free-form potential reconstruction withregularization:**Bradac et al. (2005a) Known mass distribution:N-body cluster Solve for the potential on a grid: 20x20 50x50 Minimize: Error estimation: bootstrap resampling of weakly lensed galaxies likelihood moving prior regularization Reconstructions: starting from three input maps; using 210 arclets, 1 four-image system**Free-form potential reconstruction withregularization:**Bradac et al. (2005b) Cluster RX J1347.5-1145 (z=0.451) Reconstructions: starting from three input maps; using 210 arclets, 1 three-image system Essentially, weak lensing reconstruction with one multiple image system to break mass sheet degeneracy Cluster mass, r<0.5 Mpc = 1.3 Mpc**Mass Modeling Methods**Parametric–unknowns:masses, ellipticities, etc. of individual galaxies sufficient for some purposes, but not general enough Kneib et al. (1996), Natarajan et al. (2002), Broadhurst et al. (2004) Free-form – unknowns:usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)? lensing potential – automatically accounts for external shear mass – ensures mass non-negativity what data and errors to use? strong lensing (multiply imaged sources), weak lensing (singly imaged) data with errors: P(D|H,I) is usually a c2-type function data without errors (perfect data): P(D|H,I) replaced by linear constraints how many model parameters (# pixels) to use? comparable to # observables; adaptive pixel size greater than # observables what prior P(H|I) to use? regularization prior: source size linear constraints motivated by knowledge of galaxies, clusters how to estimate errors? if regularization: the intrinsic size of lensed sources is specified if ensemble average – dispersion between individual models AbdelSalam et al. (1997,98), Bradac et al. (2005a,b), Diego et al. (2005a,b) PixeLens: Saha & Williams (2004), Williams & Saha (2005)**Free-form mass reconstruction withregularization: Diego et**al. (2005b) Known mass distribution: 1 large + 3 small NFW profiles Lens equations: N = [N x M matrix] M N – image positions M – unknowns: mass pixels, source pos. Pixelate mass: start with ~12 x 12 grid, end up with ~500 pixels in a multi-resolution grid. Sources: extended, few pixels each Minimize R2: R = N – [N x M] M; residuals vector Inputs: Prior R2 Initial guess for M unknowns P(D|H,I) replaced by linear constraints Contours: input mass contours Gray scale: recovered mass P(H|I)**Abell 1689, z=0.183**106 images from 30 sources [Broadhurst et al. 2005]**Free-form mass reconstruction withregularization: Diego et**al. (2005b) Cluster Abell 1689 (z=0.183) Errors: rms of many reconstructions using different initial conditions (pixel masses, source positions, source redshifts – within error) Data: 106 images (30 sources) but 601 data pixels Mass pixels: 600, variable size map of S/N ratios 1 arcmin 185 kpc contour lines: reconstructed mass distribution**Mass Modeling Methods**Parametric–unknowns:masses, ellipticities, etc. of individual galaxies sufficient for some purposes, but not general enough Kneib et al. (1996), Natarajan et al. (2002), Broadhurst et al. (2004) Free-form – unknowns:usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)? lensing potential – automatically accounts for external shear mass – ensures mass non-negativity what data and errors to use? strong lensing (multiply imaged sources), weak lensing (singly imaged) data with errors: P(D|H,I) is usually a c2-type function data without errors (perfect data): P(D|H,I) replaced by linear constraints how many model parameters (# pixels) to use? comparable to # observables greater than # observables what prior P(H|I) to use? regularization prior (MaxEnt; minimize w.r.t light; smoothing) linear constraints motivated by knowledge of galaxies, clusters how to estimate errors? if regularization – several possibilities if ensemble average: dispersion between individual models AbdelSalam et al. (1997,98), Bradac et al. (2005a,b), Diego et al. (2005a,b) PixeLens: Saha & Williams (2004), Williams & Saha (2005)**Free-form mass reconstruction withensemble averaging:**PixeLens Known mass distribution • Solve for mass: • ~30x30 grid of mass pixels • Data: • P(D|H,I) replaced by linear • constraints from image pos. • Priors P(H|I): • mass pixels non-negative • lens center known • density gradient must point within of radial • -0.1 < 2D density slope < -3 • (no smoothness constraint) • Ensemble average: • 200 models, each reproduces • image positions exactly. 5 images (1 source) Blue – true mass contours Black – reconstructed Red – images of point sources 13 images (3 sources)**Free-form mass reconstruction withensemble averaging:**PixeLens • Fixed constraints: positions of 4 QSO images • Priors: • external shear PA = 10 45 deg. (Oguri et al. 2004) • -0.25 < 2D density slope < -3.0 • density gradient direction constraint: must point within 45 or 8 deg. from radial SDSS J1004, zQSO =1.734 15’’ 115 kpc blue crosses: galaxies (not used in modeling) red dots: QSO images [Oguri et al. 2004] [Inada et al. 2003, 2005] [Williams & Saha 2005]**Free-form mass reconstruction withensemble averaging:**PixeLens SDSS J1004, zQSO =1.734 19 galaxies within 120 kpc of cluster center: comprise <10% of mass, have 3<Mass/Light<15 galaxies were stripped of their DM Mass maps of residuals for 2 PixeLens reconstructions 15’’ 115 kpc blue crosses: galaxies (not used in modeling) red dots: QSO images density slope -1.25 density slope -0.39 [Oguri et al. 2004] [Inada et al. 2003, 2005] [Williams & Saha 2005] contours: …-6.25, -3.15, 0, 3.15, 6.25… x 109MSun/arcsec2 dashed solid**Conclusions**Galaxy clusters: In general, mass follows light Galaxies within ~20% of the virial radius are stripped of their DM Unrelaxed clusters: mass peak may not coincide with the cD galaxy Results consistent with the predictions of cold dark matter cosmologies Mass reconstruction methods: Parametric models sufficient for some purposes, but to allow for substructure, galaxies’ variable Mass/Light ratios, misaligned mass/light peaks, and other surprises need more flexible, free-form modeling Open questions in free-form reconstructions: Influence of priors – investigate using reconstructions of synthetic lenses Reducing number of parameters: adaptive pixel size/resolution Principal Components Analysis How to avoid spatially uneven noise distribution in the recovered maps PixeLens – easy to use, open source lens modeling code, with a GUI interface (Saha & Williams 2004); use to find it.