## Gravitational lensing

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Dark halos as seen with gravitational lensing

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Dark matter is an important ingredient of galaxies, as was recognised early on by Ken Freeman himself! Evidence for dark matter halos is still indirect, based on analysing motions of tracers such as gas and stars. In a sense the visible galaxy is the mask through which we can study the dark matter. Light rays are also sensitive to gravitational fields, and dark haloes cause observable gravitational lensing effects. There are three regimes: microlensing (which probes the clumpiness of dark matter haloes), strong lensing (sensitive to the inner mass distribution) and weak lensing (which can probe haloes out to 100s of kpc from the center). This review will concentrate on weak lensing, and describe a new survey, the Kilo-Degree Survey (KiDS) that is designed to study galaxy halo masses, extents and shapes as a function of environment, galaxy type and redshift.

## 1Ken Freeman's Dark Side

Dark matter haloes are an important ingredient of galaxies, and indeed of the universe as a whole. They dominate the mass of material in galaxies, and drive the gravitational instability that leads to galaxies condensing and clustering in the first place.
The most convincing indications for dark matter haloes in galaxies are still from rotation curves of extended HI disks. In fact, the first mention of extra mass in the literature appears to be in [7], where exponential galaxy disks are described, their gravitational potentials are calculated analytically, and where, (in an appendix!), the calculations are compared with then available HI rotation curves of four bulgeless galaxies. For NGC 300, Ken prophetically wrote, without further comment, If the HI rotation curve is correct, then there must be undetected matter beyond the optical extent of NGC 300; its mass must be at least of the same order as the mass of the detected galaxy''.
Further influential work on dark matter haloes includes the work with van der Kruit (described elsewhere in this volume) on understanding the internal dynamics of stellar disks and hence their contribution to the total mass budget of the galaxy (are disks' contributions to the rotation curve maximal'' or not?), which included some of the first deep spectroscopy and integrated-light velocity dispersion measurements of galaxy disks ever attempted. See van der Kruit's chapter for more details.
A final contribution to be highlighted here is Ken's part in the MACHO project [1], which used the (now sadly burnt down) Mt. Stromlo 50-inch telescope to monitor millions of stars in the Large Magellanic Cloud for many years and discovered tens of gravitational microlensing events. This experiment probes the clumpiness of the matter distribution along the line of sight to the LMC, and the low number of events seen by MACHO (and by the similar EROS experiment [2]) proved that the Galactic dark halo does not consist predominantly of compact dark objects. Any substantial population of dark objects more massive than Earth is ruled out.

## 2Gravitational lensing and dark matter

Even though dynamical tracers were the first to provide evidence for dark matter, photons are a very useful alternative. Because of their speed they are only deflected slightly as they pass a massive galaxy lens', but the effect is large enough to be measurable and simple enough to be free from some of the uncertainties inherent in dynamical tracer studies. A brief comparison of dynamics and lensing as gravitational potential tracers is given below.

 Dynamics Lensing Sensitive to the 3-D potential of the lens Sensitive to the 2-D, projected potential on the sky Requires assumption of equilibrium Sensitive to the instantaneous mass distribution Orbit structure unknown Mass-sheet degeneracy Strong lensing: few data points Good S/N in baryon-dominated parts Weak lensing at large radii requires stacking Few good tracers at large radii

Both techniques have their fundamental limitations. For dynamics this is the need to understand the orbit structure of the tracer: the most famous example is the radial orbit anisotropy degeneracy: the same dispersion profile in a spherical galaxy can be explained with radial orbit anisotropy, or with a rising mass-to-light ratio [4]. For stellar tracers disentangling this degeneracy can be complicated requiring higher-order measurements of the shape of the velocity distribution. In the case of lensing, the main uncertainty is the mass-sheet degeneracy, which is a transformation of the lens that leaves the observed image positions intact but changes the (unobservable) source plane scale. The effect on the lens model is to trade some of the lens mass for a constant mass sheet, in such a way that the projected mass enclosed in the Einstein radius remains unchanged. The mass-sheet degeneracy makes it hard to measure the slope of the projected density of a lens accurately.
Lensing manifests itself in three different settings: microlensing, weak lensing and strong lensing. Microlensing is the twinkling that occurs when a source is viewed through a screen of effectively point-mass lenses. Occasionally a lens will pass very close to the line of sight to a source, and temporarily magnify it with a characteristic lightcurve. Observing the statistics and duration of these lightcurves allows the clumpiness of the foreground mass distribution to be studied. Strong lensing occurs when the lightrays are deflected sufficiently for several images of the same source to be observed. These special geometries allow some of the best mass determinations in cosmology to be made. Finally, weak lensing covers the other cases: a weak deflection that is not sufficient to cause multiple images, but can be detected statistically from the correlated distorting effect it has on the shapes of the sources.
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Figure 1: Top left: illustration of the asymmetry to be expected in the M31 halo microlensing rates between the near and far sides of the stellar disk. Right: field layout with detected events. Bottom left: Model predictions for number of events, and asymmetry in their distribution (curves) and measurements employing various cuts (data points). Different curves make different assumptions on extinction correction. From [5].

## 3Microlensing

As noted above, the MACHO and EROS experiments established that the Galactic dark halo is not predominantly composed of massive objects, at least along the sightline to the Magellanic clouds.
A later study of M31, modestly called the MEGA' project, found similar results [5]. In ground-based images of M31 the stars are terribly blended, so that variables can only be found with a different technique, difference image photometry. This involves subtracting exposures of the same field after carefully correcting for seeing and extinction differences, and it has the effect of turning a crowded field of stars into an uncrowded field of difference flux. The main difference in the analysis of difference image time series is that the baseline flux of the star in its unlensed state is lost, but it turns out that it is still possible to constrain the lens mass spectrum from such data. In fact, M31 offers several advantages over LMC studies because the entire M31 halo can be mapped with a single experiment. In particular, since M31 is inclined to the plane of the sky, sight lines to the near and far sides of the disk cross different path lengths of M31's halo. This implies that the two sides of the disk should see different microlensing rates, making it possible to distinguish disk self-lensing (which should be the same on both sides) from halo microlensing. The asymmetric lensing signal also affords (in principle) a constraint on the shape of the halo.
Results from a 4-year campaign using the Isaac Newton Telescope on La Palma, using about 200 nights of data, revealed 14 microlensing events over the face of M31's disk. Analysis is complicated because much of M31's disk is extincted to varying degree, and because the two fields monitored did not cover the full galaxy. A full Monte-Carlo simulation of the experiment concluded that the number of events, and the asymmetry in their distribution between both sides of the disk, are fully consistent with the expectation for self-lensing of/by disk and bulge stars (Fig. 1).

## 4Strong lensing

Many of the most spectacular astronomical images are caused by strong gravitational lensing, which splits the light from a distant source into multiple images. It turns out that these images can provide some of the best-measured masses in astronomy.
The angle through which a gravitational lens at distance(All distances are defined as angular diameter distances.)$D_l$ moves the image of a source at distance $D_s$ on the sky is determined by its surface mass density, scaled by the critical surface mass density' $\Sigma _{cr}$ given by
 (1)
where $D_{ls}$ is the distance of the source from the lens. In the ideal case in which the source lies directly behind the lens, and the lens has a circularly symmetric mass distribution on the sky, the source will be distorted into an Einstein Ring', inside which the average surface density is exactly equal to $\Sigma _{cr}$. Even for not-quite axisymmetric lenses, or not-quite perfect alignment, it is possible to estimate the Einstein radius of the lens and hence anchor the mass distribution of the lens. Typical values for the Einstein radius are an arcsecond, which corresponds roughly to an effective radius of the lens. In other words, strong lenses probe the mass distribution in the visible central parts of massive galaxies.
The most comprehensive study of the mass distribution of galaxies from strong lensing has been undertaken by the SLACS team [11]. They identified candidate strong lenses from the SDSS spectra, by looking for early-type galaxies with incompatible emission lines in the spectra that could be due to amplified background emission-line galaxies. Subsequent HST imaging revealed a very high success rate of the technique and resulted in a sample of dozens of new lenses. Unlike the existing lensed-quasar samples, many of these lenses have resolved sources so that the mass distribution of the lens can be probed at many positions. A complete modelling effort, including stellar dynamics measurements of many of the lenses, resulted in the surprising conclusion that a simple singular isothermal ellipsoid mass model is able to account for all the data.

## 5Weak lensing

Strong lensing probes the central, high surface mass density regions of galaxies. At larger radii gravitational lensing still occurs, but not at the strength required to split images. Instead the effect is to distort the background sources (since different parts of a source are deflected by different amounts). The leading-order effect of this distortion is expressed as a two-dimensional symmetric matrix that describes the mapping from image to (unlensed) source coordinates as a shear $\gamma$ and a convergence $\kappa$:
 (2)
Here $g\equiv \gamma /(1-\kappa )$ is called the reduced shear, and it is related to the axis ratio of $b/a$ of the ellipse into which a circular source is sheared, via $g=(a-b)/(a+b)$. It is the only quantity that can be measured from galaxy shapes, a fact that is related to the mass-sheet degeneracy.
The effect of a massive lens can be understood qualitatively as follows. Gravitational lensing acts to push images on the sky away from the lens, a consequence of the detour' that the light has to make around the lens. This distorts the background sky, effectively squeezing the sky around a lens radially. The result is a pattern of tangentially distorted background sources around every foreground mass concentration.
Weak lensing consists of extracting the shear field on the sky from measurements of galaxy shapes, essentially by trying to derive the shape of the average' galaxy, which is intrinsically round. This subject is technically complex, due to the many distorting effects in astronomical images that have nothing to do with gravitational lensing, but can be much stronger: smearing by the point spread function, distortion by the camera optics, pixelation of the image on the detector, charge transfer inefficiency, etc. The central issue is the need to average over large numbers of background galaxies in order to extract the shear against the intrinsic variety of galaxy shapes, the so-called shape noise.
Weak lensing is not only an effective way to study the mass distribution around galaxies and in clusters, it has also been recognized as a powerful cosmological probe. Lensing strength is sensitive to the angular-diameter distances of lens and source, and in combination with redshift measurements can be used to reconstruct the angular diameter distance-redshift relation, i.e., the expansion history of the universe. The statistical power of a large, three-D weak lensing map of the sky is sufficient to yield percent-level measurements of dark energy equation of state parameters, and is currently the subject of several design studies for space missions.
Figure 2: Left: cross-correlation functions (symbols) of the shear field in the COSMOS field, for different redshift slices. The amplitudes of the cross-correlation functions depend only on the angular-diameter distances to these redshift slices, and they scale as expected (solid lines). Right: Confidence intervals on the acceleration parameter, showing that these lensing data strongly favour an accelerating universe. From [14].

Recently we have demonstrated that it is possible to control the systematic errors to below the statistical errors, for what is currently the largest space-based weak lensing map, the COSMOS field [14]. Combining photometric redshifts with weak lensing measurements, we were able to show that the strength of the gravitational lensing signal increases with source redshift as expected in the standard cosmological model, and could even add independent, lensing-based confidence intervals on the acceleration parameter. Weak lensing is fast becoming established as a reliable probe for galactic structure and cosmology (Fig. 2).
Individual galaxy lenses have too few nearby background galaxies around them to yield useful mass measurements at large radii: the shape noise simply overwhelms any shear signal. A clear shear measurement can only by achieved by stacking large numbers of similar galaxies. Fortunately, modern wide-field imagers are capable of the necessary observations.
An impressive demonstration of the power of numbers is provided by the weak lensing analyses of the SDSS image survey. Already the early data release showed a convincing signal [6], but the more recent results have refined the result to the point where the average mass distribution can be studied as a function of galaxy stellar mass (derived from SDSS spectroscopy and SED fitting), luminosity, Hubble type and environmental density [12]. Tentative results on the shapes of galaxy haloes have also been reported . A separate study [8], using HST images, of the SLACS lenses discussed above has also been done, and shows that the singular isothermal lens model continues to fit this population well out to radii of several hundreds kpc.

### 5.1Future directions

Weak lensing focuses on the most visible effect of lensing, the shear, but these are not the only measures of lensing strength. Also the magnification, and the flexion, are showing promise as independent measures.
Figure 3: Illustration of the magnification effect on the luminosity function of a population of sources. Depending on the slope of the luminosity function the numbers of observed sources per unit area may increase or decrease.

Figure 4: Demonstration of magnification as a weak lensing measure, from [9]. Each panel shows the cross-correlation between foreground lenses and background galaxies identified with a dropout technique. Notice how in each case the brighter sources show a positive correlation, but the fainter ones anticorrelate, as expected from the shape of their luminosity function.

Magnification is in principle not possible to measure for an individual source whose intrinsic brightness is unknown. (In strong lensing the flux ratio of two images of the same source provides some of this information, but in the weak lensing regime there is only one image). However the brightness distribution of a population of sources can be measured, as can its change due to lensing. Magnifying a piece of sky containing a population of sources with a steep luminosity function will boost the individual source fluxes, but will also dilute the sky density by the same factor. For a sufficiently steep LF the boost will win, and the net result will be an increase in the number of sources above a given flux limit. For a flatter luminosity function, on the other hand, the dilution effect dominates and the number of sources will actually decrease as a result of the magnification (Fig. 3).
This effect is difficult to apply to individual cluster lenses because it can be affected by clustering of the background source populations. But it has recently been demonstrated to work very nicely on foreground galaxy lenses in the deep fields of the Canada-France-Hawaii Legacy Survey [9], using $u$, $g$, and $r$ dropout galaxies as the sources: brighter dropouts tend to show a positive correlation with the foreground sources, whereas the fainter ones (which have a flatter luminosity function) show an anticorrelation. Also qualitatively the strength of the signal is as expected (Fig. 4).
A second effect is flexion[3], the next-to-leading order distortion after shear, due to the fact that the shear will be different in different parts of a source. Flexion gives rise to the arc-like shapes of lensed galaxies, for example. While this is a weaker effect than shear, and it drops off faster with distance to the lens, it is a valuable addition of information because it probes the inner regions around lenses where there are not many background sources. Flexion also has the advantage that, unlike shear, it distorts galaxies into unnatural' shapes, so that the shape noise is considerably lower than it is for shear.

## 6The KiDS project

After many years of waiting, we are finally looking forward to the VLT Survey Telescope on Paranal being completed. It has been designed with image quality over a wide field in mind, and even though it is late to arrive it will still provide a unique capability in the Southern hemisphere. The hope (and design) is that the telescope will deliver natural seeing images over a full square degree field of view, well sampled with the small pixels of the OmegaCAM camera that was built by a consortium of institutes in the Netherlands (NOVA), Germany (Göttingen, Bonn, München and ESO) and Italy (Naples and Padua).
Figure 5: Field layout of the KiDS survey. The hashed areas indicate the survey area to be imaged, 1500 square degrees in total. Small and large dots delineate the areas where massive spectroscopic surveys of the brighter galaxies have taken place with SDSS and 2dF respectively.

The system allows an unprecedented weak lensing survey to be carried out, and this is the aim of the largest survey, KiDS (the Kilo-Degree Survey) that is planned for the VST as soon as it enters operations (Fig. 5). Together with a companion survey on the near-IR VISTA survey telescope, the project will make images of 1500 square degrees of sky, in nine photometric bands from $u$ to $K$ (Table 1). The survey area includes many galaxies with known redshift, ensuring that the foreground mass distribution is mapped in detail and can be weighed' with gravitational lensing. Because all observations will be done in queue-scheduling, it will be possible to optimize the use of the seeing distribution, so that the best seeing dark time will be used for deep $r$ band exposures suitable for weak lensing measurements, with brighter or worse seeing time being devoted to the other photometric bands. As a result the survey should be able to deliver high-fidelity weak lensing and photometric redshift measurements over a large part of the sky, with homogeneous data quality. In terms of the usual tradeoff between area and depth, the survey sits between the wide and shallower SDSS images (9000 square degrees) and the 170-square degree CFHTLS. But the unique combination of image quality and 9-band photometry should make it a very valuable resource.
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Main parameters of the KiDS+VIKING survey
 unknown command \hlineunknown command \noalign band exp.time seeing ($''$) 5-$\sigma$$2''$ limit telescope unknown command \noalignunknown command \svhlineunknown command \noalign$u$ 900 0.85-1.1 24.8 VST $g$ 900 0.7-0.85 25.4 VST $r$ 1800 $<0.7$ 25.2 VST $i$ 1080 0.7-1.1 24.2 VST $Z$ 500 $<1$ 23.1 VISTA $Y$ 400 $<1$ 22.4 VISTA $J$ 400 $<1$ 22.2 VISTA $H$ 300 $<1$ 21.6 VISTA $K$ 500 $<1$ 21.3 VISTA unknown command \noalignunknown command \svhlineunknown command \noalign
Table 1:
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Science goals for the survey are broad, but the design centered on using weak lensing to measure the mass distribution in and around galaxies as a function of environment, luminosity, and type, out to a radius of several 100 kpc. This is an interesting regime because it is far beyond the baryon-dominated regions of the galaxy and so can be compared to robust predictions from structure formation theory and simulations. A particularly interesting measurement will be the average flattening of galaxy haloes from edge-on disk galaxies: most alternatives to dark matter should make robust predictions for the variation of the quadrupolar field with radius, and these can be tested directly with such data.
The survey will also serve as a pathfinder for yet more ambitious lensing surveys for cosmology, including dark energy experiments from the ground and from space.

## 7Conclusions

Galaxies provide one of the best environments for studying the dark matter. In the parlance of this conference, they are the mask that covers the mystery, but that also shows us where to look.
Gravitational lensing is one of the most promising ways of studying the dark matter distribution, particularly at large radii where no classical dynamical tracers are available. New surveys such as KiDS will be able to map the dark matter in and between galaxy halos with great accuracy, which will enable new tests of structure formation as well as the cosmological model itself.
unknown environment acknowledgement
It is a pleasure to acknowledge the important role that Ken Freeman has played throughout my career as an astronomer, first as the author of a number of seminal papers, later as a source of occasional excellent advice, and now as a collaborator in the PN.S project, but foremost as a great person to be around. He and David Block organized a fantastic, memorable conference at Sossusvlei.
This paper benefited from the collaboration and hard work of my students and postdocs, in particular Jelte de Jong, Tim Schrabback and Hendrik Hildebrandt.
unknown environment acknowledgement

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