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Direct Shear Mapping – a New Weak Lensing Tool MNRAS 451, 2161–2173 (2015) doi:10.1093/mnras/stv1083 Direct shear mapping – a new weak lensing tool C. O. de Burgh-Day,1,2,3‹ E. N. Taylor,1,3‹ R. L. Webster1,3‹ and A. M. Hopkins2,3 1School of Physics, David Caro Building, The University of Melbourne, Parkville VIC 3010, Australia 2The Australian Astronomical Observatory, PO Box 915, North Ryde NSW 1670, Australia 3ARC Centre of Excellence for All-sky Astrophysics (CAASTRO), Australian National University, Canberra, ACT 2611, Australia Accepted 2015 May 12. Received 2015 February 16; in original form 2013 October 24 ABSTRACT We have developed a new technique called direct shear mapping (DSM) to measure gravita- tional lensing shear directly from observations of a single background source. The technique assumes the velocity map of an unlensed, stably rotating galaxy will be rotationally symmet- ric. Lensing distorts the velocity map making it asymmetric. The degree of lensing can be inferred by determining the transformation required to restore axisymmetry. This technique is in contrast to traditional weak lensing methods, which require averaging an ensemble of background galaxy ellipticity measurements, to obtain a single shear measurement. We have tested the efficacy of our fitting algorithm with a suite of systematic tests on simulated data. We demonstrate that we are in principle able to measure shears as small as 0.01. In practice, we have fitted for the shear in very low redshift (and hence unlensed) velocity maps, and have obtained null result with an error of ±0.01. This high-sensitivity results from analysing spa- tially resolved spectroscopic images (i.e. 3D data cubes), including not just shape information (as in traditional weak lensing measurements) but velocity information as well. Spirals and rotating ellipticals are ideal targets for this new technique. Data from any large Integral Field Unit (IFU) or radio telescope is suitable, or indeed any instrument with spatially resolved spectroscopy such as the Sydney-Australian-Astronomical Observatory Multi-Object Integral Field Spectrograph (SAMI), the Atacama Large Millimeter/submillimeter Array (ALMA), the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) and the Square Kilometer Array (SKA). Key words: gravitational lensing: weak – galaxies: kinematics and dynamics – dark matter. lensing survey depends on the total survey area and the number 1 INTRODUCTION density of perfectly measured galaxies. The dominant source of Weak gravitational lensing maps matter distributions in the Uni- uncertainty in all of these methods is shape noise: the error in the verse, both baryonic and dark (e.g. Kaiser & Squires 1993). This measurement due to the intrinsic and random orientations of the paper explores the enhanced potential of weak lensing in three di- images in the sample. Using these techniques, one would typically mensions (Blain 2002; Morales 2006). We describe a methodology need ∼10 objects to measure a shear of 10 per cent on arcminute to obtain a shear measurement from a 3D data cube of a single scales. weakly lensed galaxy. This technique will allow us to measure the A powerful enhancement of the weak lensing method was pro- size and shape of dark matter distributions around individual galax- posed by Blain (2002), followed by Morales (2006). Theoretically, ies at low redshifts. the rotation curves of regularly rotating elliptical and spiral galaxies Conventional 2D weak lensing techniques rely on measuring the will have maximum and minimum values in the projected velocity (two-dimensional) shapes of many ( 100) images in a field. With maps. These coincide with the major and minor axes of the pro- the assumption that the images in the field should have no pre- jected 2D image. In gravitational lensing, the equivalence principle ferred orientation, one can infer the presence of a shear field if any requires that photons of different energies are affected similarly. correlation in alignments is detected. There are a number of statis- Thus weak lensing will shear the velocity maps and distort the 2D tical approaches (Kaiser, Squires & Broadhurst 1995; Refregier & image, causing the angle between maximum and minimum rotation Bacon 2003; Heymans et al. 2006;Bridleetal.2010)usedtoper- axes to deviate from 90◦. While there are other kinematic effects form this analysis. The statistical uncertainty of a particular weak which also introduce perturbations into the velocity map of a galaxy, weak gravitational lensing has a unique signature. An unsheared ve- locity map is symmetrical about the major and minor axis, while a E-mail: [email protected] (COdeB-D); [email protected] sheared velocity map loses these symmetries. Since we can predict (ENT); [email protected] (RLW) the properties expected in an unsheared velocity map (namely that C 2015 The Authors Downloaded fromPublished https://academic.oup.com/mnras/article-abstract/451/2/2161/1749032 by Oxford University Press on behalf of the Royal Astronomical Society by Swinburne University of Technology user on 21 March 2018 2162 C. O. de Burgh-Day et al. the angle between the major and minor axis is orthogonal, or put generation of synthetic galaxy data, and present the results of a suite another way, that the velocity is symmetrical about these axes), a of systematic tests of the DSM method. In Section 5, we present metric which measures the presence and strength of the shear field fits to low-redshift data from the literature, demonstrating the abil- between the observer and the background galaxy can be designed. ity of DSM to recover a null result. Conclusions are presented in Blain (2002) suggested using the distortion in the rotation curve Section 6. to measure shear. He fitted for shear in an inclined ring model using a Monte Carlo routine, concluding that while currently a typical shear could not be measured by this method, it would be possible 2 WEAK LENSING to measure it with future higher resolution surveys. Morales (2006) Lensing theory is well developed, and there are many derivations of suggested a similar method, but fitted for the entire velocity map. the relevant equations (e.g. Mortlock 1999; Bartelmann & Schneider This has the advantage of utilizing additional information from the 2001; Bartelmann 2010). velocity map, and potentially avoiding problems the concentric- Light travels along geodesics, and in the presence of massive ring method would encounter when fitting for shear in warped or bodies, geodesics are curved. As light passes massive bodies, its path disturbed discs (since in that case each concentric ring will recover is deflected, and a background image will be distorted, magnified, a different shear). Morales’ method involved measuring the angle shifted and duplicated. Weak gravitational lensing is the term used between the major and minor rotation axes, and obtaining a measure to describe minimal distortion of the background source with no of the shear strength from the deviation of the fitted angle from observable multiplication of the source image. orthogonal. For this paper, the focus is on linearizable weak lensing, that is, We have extended Morales’ work, and developed a technique to lensing where the distortions are small, and do not vary across the measure the shear vector directly from the 3D data cube of a single source. This class of lensing can be expressed as a single transfor- weakly lensed galaxy. Our technique is called direct shear map- mation matrix. It is this feature which forms the basis of the method ping (DSM) and utilizes a Monte Carlo Markov Chain (MCMC) by which the DSM shear values are determined. We will also restrict algorithm to search for asymmetries in the 3D data. The MCMC ourselves to individual lenses (e.g. galaxies or clusters), however 1 function used is called EMCEE (Foreman-Mackey et al. 2013). We this method is equally valid for measuring the cosmic lensing signal. have performed a suite of tests on simulated data to characterize the If one assumes the length-scales of the lensing mass distribution efficiency and accuracy of the fitting algorithm, and to understand are much smaller than the observer-lens distance, then the thin lens the limits of its fitting range. Additionally, we have tested the tech- approximation can be used. In the thin lens approximation, we nique on unsheared data to ensure we can recover a null result at project the lensing mass distribution on to a plane perpendicular to low redshift, and determine realistic systematic errors. the observer’s line of sight to obtain the surface mass density: Using a symmetry search method rather than focusing on the velocity axes directly allows us to use all the available data, giving (ξ) = dzρ(ξ, z), (1) a statistically better fit. This method is unique among weak lensing measurement methods in that it uses the velocity map of an object where ξ = x2 + y2 is a set of coordinates in the lens plane, and z to measure the shear field, and does not rely on fitting for the shape is the third coordinate parallel to the observer’s line of sight. Using of the galaxy. Rather than obtaining a single measure of the shear the weak field approximation, the deflection angle of light passing over a wide area of sky, we can determine a shear value for a single the lensing mass is then given by background galaxy. Hence, we are able to measure the mass of an G ξ − ξ individual foreground dark matter halo. 4 2 α(ξ) = d ξ (ξ ) , (2) We have demonstrated that we can measure shears as small as c2 |ξ − ξ |2 γ ∼ 0.01 in realistic simulations and expect to extend this to ob- where (ξ) is the (projected) surface mass density and servational data sets (Taylor et al, in preparation). Two-dimensional weak lensing methods are able to statistically measure the mass and M(< ξ) = d2ξ (ξ )(3) structure of an individual dark matter halo and establish its relation- ship to other observables, for example baryonic mass (eg Velander is the mass enclosed within ξ.
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