Publications of the Astronomical Society of Australia (PASA), Vol. 32, e040, 16 pages (2015). C Astronomical Society of Australia 2015; published by Cambridge University Press. doi:10.1017/pasa.2015.39

Direct Shear Mapping: Prospects for Weak Lensing Studies of Individual Galaxy–Galaxy Lensing Systems

C. O. de Burgh-Day1,2,3,4, E. N. Taylor1,3,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), School of Physics, David Caro Building, The University of Melbourne, Parkville VIC 3010, Australia 4Email: [email protected]

(Received April 23, 2012; Accepted September 21, 2012)

Abstract Using both a theoretical and an empirical approach, we have investigated the frequency of low redshift galaxy-galaxy lensing systems in which the signature of 3D weak lensing might be directly detectable. We find good agreement between these two approaches. Using data from the Galaxy and Mass Assembly redshift survey we estimate the frequency of detectable weak lensing at low redshift. We find that below a redshift of z ∼ 0.6, the probability of a galaxy being weakly γ ≥ . ∼ . − lensed by 0 02 is 0 01. We have also investigated the feasibility of measuring the scatter in the M∗ Mh relation using shear statistics. We estimate that for a shear measurement error of γ = 0.02 (consistent with the sensitivity of the Direct Shear Mapping technique), with a sample of ∼50,000 spatially and spectrally resolved galaxies, the scatter in the − M∗ Mh relation could be measured. While there are currently no existing IFU surveys of this size, there are upcoming surveys that will provide this data (e.g The Hobby-Eberly Telescope Dark Energy Experiment (HETDEX), surveys with Hector, and the Square Kilometre Array (SKA)). Keywords: Gravitational lensing: weak

1 INTRODUCTION in the M∗–Mh relation using shear statistics, and we estimate the size of the statistical sample that would be required to Weak gravitational lensing is a powerful probe of dark mat- make this measurement. ter in the universe (e.g. Kaiser & Squires 1993). Following In this paper, the probability of weak lensing shear has initial investigations by Blain (2002) and Morales (2006), been estimated as a function of the redshifts of the source de Burgh-Day et al. (2015) have developed a new method and lensing galaxies, and a catalogue of candidate galaxy to measure the weak lensing signal in individual galaxies pairs is selected from the Galaxy and Mass Assembly Phase called Direct Shear Mapping (DSM). The primary scientific 1 Survey (GAMA I) Data Release 2 (DR2) catalogue (Driver application considered by de Burgh-Day et al. (2015)isthe et al. 2011, Liske et al., 2015). We also find that the dis- measurement of mass and mass distribution in dark matter tribution of shears in a galaxy sample is influenced by the halos around individual low-redshift galaxies. In particular, relationship between stellar mass and halo circular velocity, since the dark matter halo properties can be measured for and the scatter in this relation. With enough shear measure- individual galaxies, DSM will enable the measurement of ments, it may be possible to constrain this relationship, and the dispersion in the galaxy luminous matter to dark matter to measure the scatter. ratio, as a function of other galaxy observables. DSM uses spatially resolved velocity field information for The possibility of measuring individual galaxy dark matter an object to obtain a shear measurement from the velocity halo masses through DSM is an exciting prospect, however map. DSM assumes intrinsic rotational symmetry in the the the measurement itself is challenging, and potentially obser- velocity map, and searches for departures from this symme- vationally expensive. We have consequently developed the try. This requires either radio data cubes or spatially resolved approach presented here for identifying the most robust can- optical spectroscopy. To identify prospective targets, it is de- didates for such a measurement. We have also used this ap- sirable to first obtain an estimate of the shear signal present proach to investigate the possibility of measuring the scatter in a galaxy.

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While galaxy–galaxy lensing has been used to measure nates in the source and lens planes as halo masses in the past, those studies stack many galaxy– η = D β galaxy pairs statistically, to obtain average halo masses s ξ = θ. (Brainerd, Blandford, & Smail 1996; Hudson et al. 1998; Dd (3) Wilson et al. 2001; Mandelbaum et al. 2006). In addition, measurements of galaxy halo shapes have been made from In the case of a circularly symmetric lens, and perfect align- stacked galaxy–galaxy weak lensing measurements (Hoek- ment between the observer, lens and source, the lens equation stra, Yee, & Gladders 2004; van Uitert et al. 2012). can be solved to obtain the Einstein radius, a characteristic To test our target selection algorithm, the weak-lensing length scale statistics of a sample of galaxies in the Galaxy and Mass 4πGM(< θ D ) D θ 2 = E d ds , (4) E 2 Assembly Data Release 2 (GAMA-DR2) catalogue were in- c DsDd vestigated, using the stellar mass estimates from Taylor et al. M(< θ D ) (2011). The purpose of our lensing frequency algorithm is where E d is the mass enclosed within the Einstein D to estimate of the distribution of shear signals present in a radius, and d is the observer–deflector angular diameter dataset. This algorithm enables novel measurements to be distance. made, and will improve the success rate of any survey in- In the weak lensing regime, the light from the source passes tended to measure weak lensing via the DSM method. well outside the Einstein radius, and the source is singly- The rest of this paper is organised as follows. In Section 2, imaged. In this case, one can assume that so long as the length relevant weak lensing theory is described. In Section 3,a scale of the source is much less than that of the deflector, then theoretical estimate of the probability of weak lensing at low the lensing will be linear. Thus, it can be represented by a redshift is made, following Mortlock & Webster (2000). In first-order Taylor expansion, allowing Equation (2)tobere- Section 4, the inputs, structure, and outputs of the lensing expressed as a linear coordinate mapping between the lensed frequency algorithm are outlined. In Section 5, results of and un-lensed coordinate systems the application of the algorithm to a dataset obtained from β = A θ, (5) GAMA-DR2 are presented, along with an investigation of the possibility of using shear statistics to measure the scat- where A is the Jacobian of transformation; ⎛ ⎞ ter in the M∗–Mh relation. Conclusions and a summary are − κ − γ −γ ∂β 1 1 2 presented in Section 6. Throughout the paper, we assume A = = ⎝ −γ 1 − κ + γ ⎠ . (6)  = .  = . ∂θ 2 1 a flat Concordance cosmology, with m 0 3,  0 7, H = 70 km s−1 Mpc−1, and h = H /100 = 0.7. 0 70 0 κ γ γ Here, is the convergence, and 1 and 2 are the two com- γ = (γ ,γ ) ponents of the shear vector 1 2 . Equation (6) can 2 WEAK LENSING be inverted to obtain κ = − ( + )/ In this section, we will outline the relevant weak lensing 1 A11 A22 2(7) theory. Gravitational lensing is the deflection of light from γ =−( − )/ some source on its path to the observer by an intervening 1 A11 A22 2(8) mass. The deflection angle is given by γ =− =− . 2 A21 A12 (9) 4 G (ξ ) ξ − ξ  α(ξ) = d2ξ  , (1) c2 |ξ − ξ |2 The shear vector γ can be rewritten in polar co-ordinates as a function of a shear magnitude, γ , and an angle, φ where G is the gravitational constant,  is the projected mass ξ γ = γ φ; γ = γ φ, distribution of the deflector, c is the speed of light, and is 1 cos 2 sin (10) the distance from the deflector (i.e. the impact parameter). φ γ = γ 2 + γ 2 Since the angle of incidence of the light has been altered, where is the angle of the shear vector and 1 2 . the source will appear to be in a different location to its true In this work, we are interested in the value of γ for a given position. The true position of the source can be found by lens–source system, which is a function of the lens projected solving the lens equation mass density and the lens–source angular separation. D We assume a Singular Isothermal Sphere (SIS) lens profile β = θ − ds α, (2) D throughout this paper, whose projected surface density has s the form where θ is the apparent angular separation of the deflector σ 2 1 and source, β is the true angular separation of the deflector (ξ) = v , (11) 2G |ξ| and source, Dds is the angular diameter distance between the σ deflector and source, and Ds is the angular diameter distance where v is the halo velocity dispersion. Although the SIS between the observer and source. The angular coordinates β profile is a primitive lens model, it is sufficient for this work and θ can be related to the corresponding physical coordi- and allows for a simple shear estimation. The Einstein radius

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for an SIS lens is given by interested in the region where the source is singly imaged, 4πσ2 D yet sheared sufficiently such that it is still a measurable ef- θ 2 = v ds , (12) fect. This region takes the form of an annulus about the lens E c2 D s (assuming a spherically symmetric lens), the inner bound of and from e.g. Lasky & Fluke (2009), the shear components θ which is the Einstein radius, E, and the outer bound of which for an SIS are γ τ is a function of some limiting shear value, lim. Thus, is D θ D θ (γ ) γ = d E (ξ 2 − ξ 2 ); γ =− d E ξ ξ , (13) re-written as a function of a new area, a lim , defined as the 1 2|ξ|3 2 1 2 |ξ|3 1 2 area covered by this annulus: so that a(γ ) τ (γ ) = lim , D θ g lim (16) γ = d E . 4π |ξ| (14) 2 and Thus, we have an expression for the shear magnitude as a 2 1 function of lens–source projected separation and lens veloc- a(γ ) = π − 1 θ , (17) lim 2γ E ity dispersion, the latter of which can be related to the halo lim mass. and substituting in Equation (12) this becomes 1 2 σ 4 D 2 τ (γ ) = π 2 − 1 ds , (18) g lim γ 3 AN ESTIMATION OF THE PROBABILITY OF 2 lim c Ds LENSING where σ is the velocity dispersion of the deflector, c is the

In this section, we describe the process by which a theoretical speed of light, Dds is the deflector-source distance, and Ds is estimate may be made of the probability of a given source the observer-source distance. being weakly lensed, as a function of the source redshift. We If it is assumed that the population of lensing objects are begin by introducing a similar calculation for strong lens- non-evolving and have uniform volume density at all red- ing in analytically solvable cosmologies from Mortlock & shifts (which is reasonable at low redshift), then the differen- Webster (2000). We then adapt this work to the weak lensing tial number of objects at redshift z with velocity dispersion case, and for a more realistic Concordance cosmology. In the σ is following sections, we will assume two different lens popu- d2N dV dn d = 0 d . lations: a population of halos housing elliptical galaxies, as σ σ (19) dzdd dzd d in Mortlock & Webster (2000), and a Press–Schechter (Press / & Schechter 1974) population of halos. In both cases, we Here, dV0 dz is the co-moving volume element at redshift z, assume an SIS halo. We will discuss the general steps for ob- dV c (1 + z)2D (z)2 0 = A , taining the expression for the weak lensing optical depth, and ( ) d (20) dz H0 E z will then discuss the elliptical galaxy and Press–Schechter c D (z) halo cases individually. where is the speed of light, A is the angular diameter distance at redshift z, and Given an estimate for the spatial distribution and mass of lensing galaxies in a given volume, it is possible to make 3 E(z) = (1 + z)  + , (21) an estimate of the distribution of shears in the volume. This M can in turn be used to make an estimate of the probability assuming a flat universe. distribution of weak shears across the sky, assuming the dis- The optical depth to a redshift zs is then obtained by in- tribution of lensing galaxies is isotropic. Mortlock & Webster tegrating over the optical depths of the entire population of (2000) have used these arguments to make an estimate of the deflectors up to zs z ∞ 2 probability of lensing of quasars by elliptical galaxies for s d N τ(z ,γ ) = d τ (γ )dσ dz . (22) three simplified cosmologies. They define the lensing optical s lim dz dσ g lim d depth, τ, as the fraction of the source plane within which 0 0 d the lens equation has multiple solutions. It can be used as an We will now discuss using this relation to estimate the weak estimator for strong lensing probability. The contribution to lensing optical depth as a function of source redshift for a the total optical depth by one lensing galaxy is population of elliptical galaxies, ng, and a population of dark matter halos, n . πβ2 h τ = crit , (15) g 4π i.e. the fraction of the sky covered by its lensing cross-section. 3.1. A population of halos housing elliptical galaxies β Here, crit is the angular distance from the deflector where a Mortlock & Webster (2000) gives the local co-moving num- background source transitions between being singly or mul- ber density of elliptical galaxies as tiply imaged. δ σ δ(1+α)−1 σ δ In the case of weak lensing, rather than being interested dng n∗   = exp − , (23) in the region where the source is multiply imaged, we are dσ σ∗ σ∗ σ∗

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where σ is the observed line-of-sight velocity dispersion, α =− . ± . = ( . ± . ) 3 −3 σ = 1 07 0 05, n∗ 0 0019 0 003 h70 Mpc , ∗ −1 −1 225 ± 20 km s , and δ = 3.7 ± 1. The values of α and n∗ 10 are drawn from Efstathiou, Ellis, & Peterson (1988), and the − values of σ∗ and δ from de Vaucouleurs & Olson (1982). 10 2 02) Substituting this into Equation (22), the inner integral can be . solved analytically, to give =0 10−3 lim z ( + )2 2 s 1 z ,γ g DdDds s τ(z ,γ ) = C dz , (24) z − s lim g ( ) g ( 10 4 0 E zg Ds τ , M /M where Halo log10( min ) = 10.5 −5 , M /M 10 Halo log10( min )=10 σ 4 2 , M /M  1 4 Halo log10( min )=9.5 C = π 2n − 1  1 + α + . (25) g ∗ γ δ Ellipticals c 2 lim γ 0.2 0.6 1.0 1.4 1.8 For a given limiting shear value, lim, to obtain the probability z of measuring a shear of at leastthe limiting value, one must s solve Equation (24) numerically as a function of the source Figure 1. The lensing optical depth as a function of source redshift and z γ = . redshift, s. a limiting shear of lim 0 02. The coloured solid lines show the optical The dashed grey line in Figure 1 shows the lensing optical depths obtained when using a population of lenses drawn from a Press– depth as a function of source redshift for a limiting shear of Schechter halo mass function. The three lines show the effect of different γ = . minimum halo masses. The minimum halo mass is usually taken to be lim 0 02 based on a population of elliptical galaxies. The 10 10 M (e.g. in the Millennium simulation). The dashed grey line shows coloured lines in Figure 1 are discussed in the next section. the optical depth obtained when using a population of lenses drawn from an elliptical galaxy population. It is reassuring to see that the Press–Schechter curve which best matches the dashed curve is that which uses the commonly 3.2. A Press–Schechter halo population 10 used minimum halo mass of 10 M . The Press–Schechter mass function is given by z 2 s ( + )2 δ2 1 zh DdDds dn 2 ρ dlnσ(M) δ − crit × dz , (30) m crit σ 2( ) ( ) h = ×− exp 2 M (26) 0 E z D dM π M dM σ(M) h s where and gives the co-moving number density of dark matter halos 1 2 σ ∗ 4 as a function of halo mass (Press & Schechter 1974). C = π 2η2 − 1 , (31) h 2γ c Substituting this into Equation (22), lim σ ∗ −1 z ∞ 2 and 321 km s is a halo characteristic velocity disper- s d n τ( ,γ ) = d τ (γ ) , ∗ zs lim d lim dMdzd (27) sion, corresponding to a characteristic halo mass M defined 0 M dz dM ∗ min d σ( ) = δ ( ) δ ( ) as where M c z , where c z is the critical den- δ ( = ) . = ( ∗/ ) where Mmin is the mass of the smallest halo capable of con- sity and c z 0 1 686. At z 0, log10 M M 13 taining a galaxy, and (Barkana & Loeb 2001). γ 4 To obtain the probability of a shear of at least from 1 2 (Mh−1 )0.316 D 2 lim τ (γ ) = π 2η − 1 70 ds , halos, as a function of source redshift, the mass and redshift g lim γ c D 2 lim s integrals in Equation (30) must be solved numerically. In (28) order to illustrate the effect of varying the minimum halo where η = 6.14656 × 10−7 and we have used the relation mass, the mass integral in Equation (30) has been solved for . = . × −2 −1 0 316 , three values of the minimum halo mass: log (M /M ) = Vc 2 8 10 Mh70 (29) 10 min 9.5, 10, and 10.5. The results of this are shown by the three from Klypin, Trujillo-Gomez, & Primack (2011). Vc is the coloured curves in Figure 1. As expected, the probability

halo asymptotic circular velocity, and we have assumed Vc curve which best matches that of an elliptical population σ . The lower bound of the integral over M in Equation (27) ( / ) = is for the case log10 Mmin M 10. Not surprisingly, the has been truncated at Mmin because we are only interested in higher the minimum halo mass the lower the probability of halos large enough to contain at least one galaxy. The value lensing, since the majority of halos are of lower mass. ( / ) of Mmin at z 0 is usually taken to be log10 Mmin M 10 From Figure 1, roughly 1 in 1 000 sources at z ∼ 0.2 will (Barkana & Loeb 2001). be sheared by at least γ = 0.02. In the calculation of halo In this case, the integral over mass is not analytically solv- lensing probability, four major assumptions have been made: able, however the two integrals are still separable, giving the following expression: • The lensing cross-sections of each galaxy in the pop- ∞ dn . τ( ,γ ) = h −1 1 264 ulation do not overlap. This assumption breaks down zs lim Ch Mh70 dM M dM if: (1) The redshift is high, since the number of lenses min PASA, 32, e040 (2015) doi:10.1017/pasa.2015.39

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contributing to the optical depth increases with increas- the catalogue, assuming an SIS density distribution for each

ing source redshift. (2) Mmin is small, since this leads to galaxy, estimating the halo circular velocities of each galaxy, γ more lenses and an increasing cross-section. (3) lim is and ultimately returning an estimate of the shear each object small, since the cross-section of each lens is larger for imp