Any lens system allows the projected mass distribution of the lens to be probed. Since the lens is typically an elliptical galaxy at significant (~ 0.5) redshift, this is an intrinsically interesting observation, as detailed study of stellar dynamics at this distance is a painful operation involving large telescopes and long integration times. Moreover, lensing gives the projected mass of the lens within the Einstein radius with unique accuracy. There is less good news, however; the probes of the lens potential are obtained only at points where an image is formed, and typically these are limited because the emission from quasars is usually very compact. The resulting problem of mass reconstruction is thus typically underconstrained, giving a serious problem with degeneracies when trying to deduce other mass properties of the lens, particularly the radial mass profile (e.g. Saha 2000, Wucknitz 2002, Kochanek 2002, Kochanek 2004). Probably the most depressing degeneracy is the so-called mass sheet degeneracy (Falco et al. 1985, Gorenstein et al. 1988), which results from the observation that it is possible for an intervening mass distribution to leave the lensed image positions and fluxes undisturbed, while rescaling the overall potentials and time delays together with the (unobservable) source position. This is a potentially serious problem for cosmological investigations.
One approach which alleviates some of the degeneracy problems is to abandon the use of point-source lenses in favour of lensed galaxies, which have extended source structure and thus give constraints at multiple radii in the image plane. This has had considerable success in the investigation both of overall mass profiles of galaxies and of sub-galactic level substructure. The most significant work in this direction stems from the SLACS survey (Bolton et al. 2006, Koopmans et al. 2006, Bolton et al. 2008). A second approach is to concentrate on a small sample of high-value objects and do the followup observations which are necessary to break the degeneracies; this is mainly to derive additional constraints on the mass profile from observations in regimes where extended sources are visible in addition to the point-like quasar. This approach also requires extensive investigation of the foreground (Fassnacht et al. 2006, Momcheva et al. 2006), in order to derive the information about nearby objects required to ease the mass-sheet degeneracy. (Alternatively, very rare objects, such as lenses with two different sources at different redshifts, can be used to determine the source position and thus break the MSD completely). The third approach is to use quantities derived from the observations which are sensitive to a limited subset of the mass properties of the lens. Both the second and third approaches have been used for quasar lenses.
3.2. Lensed substructure
The main attraction of quasar lenses is that they provide a probe of sub-galactic-scale matter structure, which is in turn relevant to a strong prediction of CDM models of structure formation. In such scenarios, structure in the Universe forms in a hierarchical way, with small dark matter clumps coalescing into larger haloes as time passes to form steadily larger agglomerations (White & Rees 1978). Baryons lose potential energy by non-gravitational means and therefore settle in to the resulting potential wells, forming galaxies, groups and clusters. Because many physical processes are involved, how this happens is quite complicated, and in practice semi-empirical recipes are used for describing the process (Blumenthal et al. 1986, Ryden 1988, Gnedin et al. 2004). There are also ongoing processes which may rearrange the baryonic material in galaxy-sized haloes, including the influence of supernovae in lower-mass haloes (Heckman et al. 1990) and of periodic ejections of matter from active nuclei in the centres (e.g. Begelman et al. 1991, Croton et al. 2006); these processes are collectively known as feedback. Complications associated with baryon physics cannot be avoided in lensing systems, because typical gravitationally lensed images form at radii of about an arcsecond, corresponding to 5-10 kpc in projection against the lens galaxy. At this radius, dark matter is expected to contribute at the level of a few tens of percent to the projected matter distribution, and baryon processes are therefore dominant.
The substructure debate matters, because CDM works so well on cluster and supercluster scales that its predictions on smaller scales are one of its few possible failure modes. In our own Galaxy, the observation that fewer luminous satellites were found than CDM subhalo models would predict (Moore et al. 1999, Klypin et al. 1999), generated a vast literature reflecting the importance of the problem. Possible ways out of the problem included finding at least some of the missing satellites (Belokurov et al. 2006, 2007; Zucker et al. 2006a, b), or finding some way in which they might be present but not accrete gas and form stars (Bullock et al. 2000). The current situation is that it is difficult to explain both the incidence of substructure and its dynamical properties and stellar content (e.g. Boylan-Kolchin, Bullock & Kaplinghat 2011, Boylan-Kolchin, Bullock & Kaplinghat 2012) although possibilities exist including detailed adjustments corresponding to detailed treatments of the physics, or gross changes such as an alteration in the overall mass of the Milky Way and thus of its expected subhalo content. Curiously, despite the apparent lack of substructure on sub-Magellanic Cloud mass scales, the presence of substructure on scales as massive as the Magellanic Clouds themselves is mildly anomalous, unless the Milky Way has a mass towards the upper end of the allowed range (Boylan-Kolchin et al. 2010).
3.3. Lensed substructure in other galaxies
Relief from detailed arguments about CDM substructure in our own Galaxy can be had by considering substructure probes in other galaxies, in which details can be swept under the observational carpet - or less cynically, a larger number of objects can be probed in less detail, thus accounting for the possibility that our Galaxy may be untypical. The main mass probe in other galaxies at cosmological distance is gravitational lensing.
The observation of a gravitational lens system yields a set of observed positions and flux densities of lensed images. As already discussed, this set of observables does not give unique information about the mass distribution, and in general a large number of macromodels (a term generally used for overall mass distributions, or at any rate for components of spatial frequency in mass distribution of a few kpc or larger) are compatible with images of a single object. Some plausibility arguments can be used to restrict the available set of macromodels, mainly the observation that well-constrained lens systems, with stellar dynamics and extended images, suggest approximately isothermal 1 distributions of mass (e.g. Cohn et al. 2001, Rusin et al. 2003, Koopmans et al. 2006). This density profile corresponds to a flat rotation curve. On top of the macromodel, any smaller-scale perturbations will affect the image positions and fluxes. Since image positions depend on the first derivative of the projected potential distribution and fluxes on the second, smaller perturbations are expected to be detectable in image fluxes.
A number of relations between fluxes of individual lensed images exist which are independent of, or at least relatively insensitive to, the details of the macromodel. For example, "cusp" configuration lens systems, in which the source lies close to the cusp of the astroid caustic produced by the lens galaxy (Fig. 2), produce three bright images close together on the opposite side of the Einstein ring from a single faint image. The brightness of the central image of these three should be equal to the combined brightness of the outer two (Schneider & Weiss 1992; see also Keeton, Gaudi & Petters 2003, 2005; Congdon, Keeton & Nordgren 2008 for more detailed treatment of other cases), and any departure from this indicates a non-smooth mass model. The relation holds because the images form at a very similar, and relatively flat, part of the Fermat surface 2 in which unphysically large changes in the macromodel would be needed to produce disagreements – known in the literature as "cusp violations" – with the expected relation. On the other hand, small-scale structure can produce cusp violations relatively easily. Constraining small-scale structure is in principle a matter of counting the number and magnitude of cusp violations in a sample of quasar lenses.
Figure 2. Geometry of cusp (left) and fold (right) lenses. In each case the galaxy is centred on the green dot, and the projected position of the source is the red dot. The astroid caustic (the red diamond-shaped feature) separates 2-image from 4-image systems. The greyscale represents the observed images of the source, which form at positions which are stationary in the time-delay surface (indicated by green contours).
There are a number of reasons why the problem is not that easy. The first is that anomalous fluxes can be produced not only by CDM substructure on 106 - 109 M scales, but also by the movements of individual stars in the lensing galaxy which create a caustic pattern of differential magnification which tracks across the field in timescales of years, with individual events happening on shorter timescales. This phenomenon, microlensing, described in detail later, is itself extraordinarily interesting, but a contamination for the current purpose. It can be got around by using sources which have large sizes relative to the scale of the microlensing caustic pattern, which in practice is about 1 µas. The cores and VLBI-scale jets of radio sources, with a typical intrinsic angular size 3 of about 1 mas, fulfil this condition, but optical quasars do not. The second problem, even for radio quasars, concerns propagation effects. Radio waves can be scattered while propagating through ionized media; the details are complicated (Rickett 1977) but the effect is to produce flux variations, which can be seen on timescales of weeks to years if the scattering screen is in our own Galaxy. By definition, there is a possible source of foreground screen in gravitational lens systems, in the form of the lensing galaxy, as well as a nearer screen in our own galaxy. Koopmans et al. (2003) found evidence for this in at least one object during a monitoring campaign of some radio lenses, but it is likely to be a small effect compared with most of the observed flux anomalies. The third possible problem is the effects of intrinsic variation of the quasar, coupled with a differential time delay between the images. Even though time delays and flux density variation are useful for measuring the Hubble constant, for the present purpose they are a nuisance. Again, however, the level of variation of most radio sources does not seem to be significant enough to be a major problem, and can be averaged out if observations are made for periods much longer than the time delay. In the optical, extinction is present, and can be used to probe the properties of the dust in the lensing galaxy by using the fact that the same object's light path passes through two different regions of the galaxy (Elíasdóttir et al. 2006).
The first obvious flux anomaly was pointed out by Mao & Schneider (1998) in the CLASS lens system B1422+231; this is a cusp system which produces a violation which requires a significant amount of substructure (about that predicted by CDM) in order to give a significant chance of reproducing the observed anomaly. Other examples of flux anomalies which defied smooth macromodels soon followed (Fassnacht et al. 1999, Metcalf & Zhao 2002, Chiba 2002, Saha, Williams & Ferreras 2007), leading to the first attempt to address the overall statistics (Dalal & Kochanek 2002, see also Kochanek & Dalal 2004). Using seven four-image lenses, Dalal & Kochanek derived an overall substructure contribution of 0.6-7% (2 confidence) in substructures between 106 M and 109 M, in rough agreement with the overall predictions of CDM. However, this substructure appears to be in the wrong place (Mao et al. 2004); dark matter, and hence dark matter substructure, should in CDM models be less centrally concentrated than the baryons, and such levels of substructure at projected radii of 5-10 kpc are thus surprisingly high – a curious contrast to the "missing satellite" problem in our own Galaxy. Incidentally, the presence of a tension between lensing observations and CDM probably gives a severe problem for models involving significant amounts of warm dark matter (WDM), which would predict even less substructure (Miranda & Macciò 2007).
A more sophisticated approach to CDM testing can be taken, if instead of calculating an average contribution of substructure "expected" at the projected Einstein radius, we instead take an actual CDM halo simulation and investigate its lensing properties. Early attempts, with lower resolution simulations, produced mixed results (Bradac et al. 2004, Macciò et al. 2006, Amara et al. 2006), but generally confirmed the picture of an excess of flux anomalies compared to the expected incidence in CDM. As better simulations became available, they have been used for these comparisons (Xu et al. 2009, see also Chen, Koushiappas & Zentner 2011 for more detailed treatment of halo-to-halo variations) using, for example, the Aquarius dark-matter simulations. There are a number of limitations with such investigations. The two main problems are that the simulations are being pushed to the limit of their resolution, since they are being asked questions about mass condensations on scales down to 106 M, comparable to the lowest masses considered in the simulations, and that the effect of baryons in modifying the structure of the subhaloes is not taken into account. Until higher-resolution simulations with extra physics are available, however, this is the best that can be done. Xu et al.'s conclusion was that the cusp violations in existing lenses clearly exceeded the level of violation which would be expected in the dark matter simulations. Two important caveats to this emerged in subsequent work, however. Firstly, detection of substructure along the line of sight means just that, and the substructures which produce the flux anomalies do not have to be within the lensing galaxy (Metcalf 2005a, b, Inoue & Takahashi 2012). Xu et al. (2012) suggested that 20-30% of the substructure could be outside the lensing galaxy, somewhere along the line of sight. If correct, this would potentially alleviate the tension between lensing observations and CDM, although it is probably fair to say that more work, both theoretical and observational, is needed before this can be regarded as well established. Secondly, finite source-size effects may modify the statistics of substructure detection (Dobler & Keeton 2006; Metcalf & Amara 2012).
The sample of seven radio-loud lenses used for substructure studies has remained largely unchanged in the last decade, owing to the current difficulty of finding significant numbers of new radio lenses with existing telescopes. There are a number of alternative approaches. The first involves the use of observational brute force; to target radio-quiet lenses, but observe flux densities in parts of the electromagnetic spectrum where the source has significantly greater size than the microlensing characteristic size of 1 µas. The obvious choice is the mid infra-red, where the source is expected to consist of a more extended thermal component than the accretion disk which radiates in the optical and ultraviolet. Despite the difficulties of observing in this waveband, a number of successful programmes have been carried out (Chiba et al. 2005, Fadely & Keeton 2011; Fadely & Keeton 2012) resulting in the detection of a number of other flux anomalies, and measurement of their likely masses. These range from the 107.3 M and 107.7 M clumps found in MG0414+0534 and HE 0435-1223 respectively (MacLeod et al. 2012, Fadely & Keeton 2012) to larger perturbations (109 M in SDSS J1029+2623 and 2 × 108 M in 1938+666; Kratzer et al. 2011, Vegetti et al. 2012). By contrast, the substructure identified in galaxy-galaxy lenses is often larger (e.g. Vegetti et al. 2010). Radio-quiet quasars are also not radio-silent, and flux densities have been measured for a number of such lens systems (Kratzer et al. 2011, Jackson 2011). Indeed, one would expect that all quasars emit measurable flux density at radio frequencies (White et al. 2007) with current instruments such as the EVLA and e-MERLIN.
An alternative approach is to use the presence of additional observational constraints, such as those provided by radio jets in quasars, to give additional observational constraints, in the case where the jets can be detected in more than one lensed image. This was first attempted by Metcalf (2002) in the case of the lens system CLASS B1152+199 (Myers et al. 1999, Rusin et al. 2002) and a detection was claimed in this case. With further investigations using VLBI, other lenses have been shown to require substructure (MacLeod et al. 2012) and this may be a promising path to more detailed substructure measurements in the future, given sensitive VLBI observations and high resolution (Zackrisson et al. 2012). Currently, one of the most puzzling cases is the four-image lens system CLASS B0128+437 (Phillips et al. 2000), in which the source consists of three radio components separated by a few milliarcseconds and resolvable with VLBI. Attempts to fit the positions of the twelve resulting images fail badly (Biggs et al. 2004). In this object, also, the SIE macromodel which properly fits the four images on arcsecond scales contains an implausibly large amount of external shear, which is inconsistent with the observed number of surrounding galaxies, and also does not fit the extended structure around the images seen by adaptive optics observations (Lagattuta et al. 2010). A fruitful area of future investigation may well be to try and combine the flux and astrometric anomalies in a sample of lenses; although in the case of astrometric anomalies, unlike flux anomalies, the anomaly is always likely to be underestimated since it can be absorbed into the macromodel (Chen et al. 2007). An alternative issue for the future is the proposal that time delay measurements may also be useful for measuring the effects of substructure, which can in extreme cases change the sign of the differential time delay between two images (Keeton & Moustakas 2009).
Having detected mass substructures, we can ask whether they consist purely of dark matter or whether they contain stars. In many cases, flux anomalies can be explained by a mass contribution from a substructure which corresponds to an observed luminous satellite galaxy (Schechter & Moore 1993, Ros et al. 2000, McKean et al. 2007, Macleod, Kochanek & Agol 2009), although in some cases (McKean et al. 2007) the mass model of the satellite is contrived, indicating that further mass structures may be needed. The number of bright subsidiary deflectors may be larger than expected from simulations (Shin & Evans 2008), a problem which may be resolvable if some of them are actually line-of-sight structures, or explicable as a selection effect if brighter condensations are rendered more effective at causing flux anomalies because they have higher central densities.
3.4. Black holes and central potentials of lens galaxies
A further important astrophysical application of quasar lens systems - and in particular, of radio quasar lens systems - is the detection and study of "odd" images. All gravitational lens systems have an odd number of images, usually 3 or 5, resulting from the properties of the lens Fermat surface. One of these images is always a Fermat maximum which forms very close to the centre of the lens galaxy. For most realistic mass distributions, this maximum in the surface is very sharp, which implies that the corresponding image is very faint (Wallington & Narayan 1993, Rusin & Ma 2001, Keeton 2003). How faint it is depends on the geometry of the lens system and the degree to which the central potential is singular; if the potential is dominated by a massive black hole, the corresponding image can be hugely demagnified and for all practical purposes invisible. The geometry of the lens system has an effect because this determines the separation of the central image from the centre of the lensing galaxy. Three-image lens systems, particularly those with high primary-secondary flux ratios, create central images further from the lens centre and which are thus less demagnified. Five-image systems, with four bright images, are expected to nearly always contain an undetectably faint fifth image because the symmetry of the lens configuration places it close to the galaxy centre.
Propagation effects are likely to be particularly acute when trying to detect central images, because the light path passes straight through the lens galaxy centre where the concentration of dust and ionized gas is high. The use of radio lenses, where the galaxy is unlikely to be visible, is required, but relies on the expectation or hope that the central image will not be scattered out of existence. Nevertheless, detection of a faint radio image is not the end of the story, since it may result from low-level radio emission from the core of the lensing galaxy. In principle this can be distinguished by observation at different frequencies and examination of the radio spectrum to see if it differs from the other images.
Observations to detect odd images are very difficult, because they require a combination of high resolution and extreme radio sensitivity. A comprehensive theoretical study (Keeton 2003) of likely lens mass profiles, based upon HST observations of Virgo ellipticals (Faber et al. 1997) showed that central image detections were likely only once flux density levels of 10-100 µJy were reached. This level is only now becoming routine thanks to high-bandwidth upgrades to the VLA (now the JVLA) and MERLIN (now e-MERLIN). With older instruments, there is only one secure detection of a central image in a galaxy-mass lensing system, namely PMN J1632-0033 (Winn et al. 2002, 2003, 2004), although other systems in which a softer cluster potential is the primary deflector have shown central images (SDSS 1004+4112, Inada et al. 2005). In other cases, considerable effort with older instruments has yielded only upper limits (Boyce et al. 2006, Zhang et al. 2007).
The central image in PMN J1632-0033 implies two limits; on the mass of the central black hole (which must be less than 2 × 108 M) and a lower limit on the central surface mass density. These can alternately be rewritten as joint limits on the index of the central mass power law and black hole mass. In principle, the degeneracy between these two parameters can be broken in the case where the third image can itself be split by the combined lensing effect of the black hole and central stellar cusp into two images (Mao, Witt & Koopmans 2001). Such a detection, although very much more difficult and requiring another factor of 10 in sensitivity, would be very exciting because it would enable the immediate measurement of the black hole mass and central stellar cusp density separately. Even the detection of third images, or significant limits thereon, in a number of radio lenses would give a powerful indication of the evolution of the central regions of elliptical galaxies between z = 0.5 and the present day.
1 This corresponds to a surface density profile r-1, or a 3-dimensional density profile r-2. Back.
2 The Fermat surface is a very useful way of thinking about gravitational lens optics. Imagine a source, viewed by the observer in projection on to the lens plane, with contours drawn according to the light travel time of rays originating in the source, bending in the lens plane and reaching the observer. These contours are simply concentric circles centred on the source, with a central stationary point (a minimum) at which Fermat's principle dictates the formation of an image. If we then introduce a galaxy, which distorts these contours, we eventually reach a point at which further stationary points (a maximum and saddle point) simultaneously form. Back.
3 The size of a compact radio source is controlled by where the optical depth to synchrotron self-absorption becomes 1. This is typically about 1 mas for a source of around 1 Jy, although it becomes smaller with increasing frequency, and it decreases as the square root of the flux. Sources typically found by the Square Kilometre Array, which will be sensitive to sources of 1 µJy, may therefore show radio microlensing. Back.