Gravitational lensing

This page will always be under development.

My work in the field of gravitational lenses covers several areas:
  1. Basic theory
  2. Classical lens modelling
  3. Lensing theory and model degeneracies
  4. Lens modelling with extended sources: LensClean
  5. Microlensing
  6. The golden lens B0218+357
  7. Lensed water and extreme VLBI structure: MG0414+0534
  8. A lensed LBG: The 8 o'clock arc
  9. RXS J1131-1231
  10. Lensing as a tool for other fields
  11. Future lens surveys
  12. More exotic questions in lensing
I also have an outdated list of my favourite lens systems on which I have been working on in the past. This is kept only for historical reasons.

Basic theory

[plot of function H] I started my work on gravitational lensing with my Master's thesis (Diplomarbeit), which comprises a self-contained derivation of the basic lens theory. The main approach in my thesis is to write all the observables (image separations, time delays, magnification matrices) as integrals of the surface-mass density multiplied with some weight functions. Other “highlights” include explicit definitions of the ‘thickness’ of lenses (which are not the thickness of the mass distribution) and a discussion of the idea to use individual lenses to constrain cosmological parameters even without time-delays. This idea is based on measuring the Dds/Ds ratio either using lensed sources at different redshifts or by combining lensing information with measurements of velocity dispersions. This concept was presented in some detail at the Golden Lenses conference at Jodrell Bank in 1997.

Classical lens modelling

[lensing geometry] With the term classical I refer to lens modelling with lensed point-like or very compact sources. Observables in this case are image positions and flux density ratios or relative magnification matrices. These are used as constraints for parametric mass models of the lenses. The advantage of classical modelling is the relative simplicity, which is entirely due to the simplicity of the source. In lens modelling, the true structure of the source is never known, so that it always has to be a (implicit or explicit) part of the model. With point-sources, the source model usually only consists of source position(s) and maybe true flux densities. For slightly resolved but compact sources, shape parameters can be added. The important point is that the source can be described by a very small number of parameters, which can be fitted simultaneously with the mass model parameters.

In order to fit a lens model to observed image positions, one has to find the observables that would correspond to a given lens+source model. In lensing this involves the highly non-trivial inversion of the lens equation. With a given lens model, it is usually easy to find a source position given an image position. The inverse process, finding all image positions for a given source position, on the other hand, is a real nightmare. Instead of comparing the predictions from the model with the measurements in the image plane, one can approximate the comparison by projecting it into the source plane. The idea is to back-project (easy!) the observed image positions into the source plane. For a correct model, the source positions of all images should coincide. In reality, one will have deviations, which can be minimised to find the best lens model. The deviations can be approximately projected back to the image plane by using the magnification matrices, but the accuracy of this approach is not always sufficient. Intermediate approaches can also be used, in which the comparison is made in the image plane, but the inversion of the lens equation is still avoided.

I developed my own software to perform different kinds of classical lens modelling using very general lens models. The goodness of fit can be measured in the source plane or in the image plane, with several different algorithms. Results of my work in this field have been published for several lens systems:

Lensing theory and model degeneracies

[plot for radial mass profile degeneracy] In order to understand which parameters of a mass model are well constrained by the observables, and which ones are affected by degeneracies, it is necessary to understand the properties of the lens mapping generally and for the class of models that is used in particular.

A well-known degeneracy is the so-called mass-sheet degeneracy. If the density of a mass model is scaled with 1-k, while at the same time a homogeneous mass-sheet with density k is added, the observables will not change if the size of the source is also scaled with 1-k. The additional constant mass density amplifies the effect of the scaled mass distribution, so that the total effect is the same. The only observables which are affected are the time-delays between images. That means that the mass-sheet degeneracy is a serious problem for the determination of the Hubble constant.

Another important degeneracy is the one of the radial mass profile. If the images are located at similar distances from the lens centre, it is often possible to fit the observables with models of very different mass profiles. This does again affect the time-delays, which become smaller for shallower profiles and larger for steeper ones. The supervisor of my PhD work, Sjur Refsdal, showed that this degeneracy is basically the same as the mass-sheet degeneracy. Scaling the mass profile with a factor 1-k with the addition of a constant density makes the profile shallower, while it keeps the image positions constant, if the source is also scaled. We presented this idea in a conference poster in 1999.

[critical shear ellipses] Later I extended this work to include the effects of ellipticity and external shear for quadruply lensed systems. In my publication about the subject I studied a very general family of lens models in which the potential follows a power-law rb in the radial direction and can have an arbitrary azimuthal shape. I found that when the external shear is kept fixed, the time-delays (or the Hubble constant, if the time-delays are measured) scale with (2-b)/b. If, on the other hand, the shear is fitted but the ellipticity kept constant, the scaling is weaker and goes like 2-b, a fact that has already been observed in the past with more special models.

In the same paper I introduced the concept of a “critical shear”. A shear of this value (and direction) has the effect that (when the ellipticity is fitted accordingly) all time-delays exactly vanish. This is still true if the shear is then varied orthogonally relative to this critical shear. The figure illustrates a nice geometric property. The critical shear is defined by the ellipticity of the ‘roundest’ ellipse going through all four images. This has a direct significant consequence for the determination of the Hubble constant from time-delays. Systems which are very ‘round’ have a small critical shear, which means that unknown contributions to the real shear have a large effect on the time-delays and the Hubble constant. More asymmetric systems are much more robust in this respect.

Lens modelling with extended sources: LensClean

[bias factor] The only way to break degeneracies in lens modelling is to include additional information. There are very good reasons to use exclusively information from lensing itself, because in this way we can avoid to depend on complicated additional astrophysics and untested or unjustified assumptions. In lensing, it is clear that each additional lensed source component contributes its own set of constraints for the lens models. As long as the components are all compact, classical lens modelling can be used to exploit the additional information.

Much more general is the use of general extended sources, in which the number of subcomponents and thus constraints can be very large. The disadvantage is that modelling such systems is a very complex task. The basic difficulty is that the true (unlensed) source structure is not known a priori but must be fitted simultaneously with the lens. In the case of radio observations, it is not a good idea to first create maps of the lensed source and then use these to model the lens, because the artifacts created by the deconvolution will affect the lens modelling results. A better approach is to combine the two pieces and try to solve the complete inverse problem. This has been tried before with the development of LensClean. I had data available of the lens system B0218+357, where the original LensClean algorithm proved to be insufficient to determine the position of the lens galaxy with good accuracy. The idea of LensClean is very simple. In the standard Clean algorithm, a radio source is decomposed into a collection of point-like components, which can be placed arbitrarily. In the lensed situation, it has to be taken into account that only certain combinations of multiply lensed components, with their corresponding magnification ratios, are allowed. LensClean builds a model by decomposing the source plane into point-like components, so that a consistent model of the source is found for a given lens model. In an outer loop, the lens model can then in turn be varied to produce a simultaneous fit of lens and source.

In order to extract all available information from the radio data, I developed a new version of LensClean and applied it to the case of B0218+357. One of the new concepts introduced is the correction for bias effects in LensClean. The standard algorithm preferably cleaned regions with higher multiplicity, because the residuals decrease faster there. This is corrected in my unbiased LensClean, which helped a lot to obtain good results.

[LenTil] LensClean relies on a good method to invert the lens equation, which means finding all image positions for a given source position and lens model. This has to be done so many times, that a reliable (failure less than one in 100 million) and fast method is essential. For this purpose I developed ‘LenTil”, a tiling algorithm that can find all images even for complicated models extremely reliably. The basic idea is simple, but the implementation became pretty complicated, as explained in my PhD thesis.


[B0218+357 lens
plane] As said above, this is one of the most interesting lens systems. In my LensClean modelling work, I was for the first time able to determine the position of the lensing galaxy with an accuracy sufficient for a serious application of Refsdal's method to determine the Hubble constant from a lensing time-delay. The effort neede for this was considerable, but still significantly less than that of projects like the HST key project to determine the Hubble constant. Results for the lens position and cosmology can be found in my publication. The lens models I found form the basis for most of the later work on this lens system.

One might be sceptic about the method to determine the lens position very indirectly using LensClean. It took me many tests to convince myself that the result is reliable. Finally it was possible to measure the position directly with a very deep HST exposure. The analysis of the maps confirms my results, even though the accuracy of the optical measurement is not comparable with my LensClean result. Higher resolution observations were later made with the VLA + Pie Town at 15 GHz. The analysis is still in progress.

source plane] The best lens model is the most important result of my LensClean work. On the way to this goal, LensClean also produces the optimal model of the (unlensed) source plane. I developed a new method to take into account the resolution of the instrument combined with the lens and ‘convolve’ the best model with what I have defined as the ‘Clean beam in the source plane’. This approach is superiour to further ideas, but still not optimal. I am working on methods in which the regularisation is incorporated directly into the mapping process and not applied afterwards.
[B0218+357 free-free absorption, model and observed] The question of frequency-dependent flux ratios of the bright images was (with contributions from me) investigated by Mittal et al. (2006) and Mittal, Porcas & Wucknitz (2007). We found out that the proposed structural changes of the source with frequency (together with magnification gradients) cannot be responsible for this effect. Instead we found a plausible explanation in free-free absorption in the ISM of the lensing galaxy.
[B0218+357 at 90cm and 2cm] The same lens was also target for a 90cm VLBI experiment that led to the very first VLBI map of an Einstein ring. We still do not understand the significant differences between the structures at 90cm from the ones known from higher frequencies (e.g. 2cm in comparison).
The data from this experiment also served as basis for the first wide-field VLBI project at low frequencies. Our results (published in Lenc at al., 2008) provide important input for future low-frequency work, in particular with LOFAR.


[MG0414+0534 composite] In this system we recently found a lensed water maser at a redshift of z=2.64, by far the most distant detection of water in the Universe. This discovery, which has motivated several surveys for lensed maser emission, motivates us to study this lens system in more detail, in particular concerning the mass distribution of the lens. For this purpose we carried out two major global continuum VLBI experiments (at 1.7 and 8.4 GHz) to produce better maps as input for LensCLEAN modelling. In addition, Andreas Brunthaler at the neighbouring MPIfR studied the source with global VLBI at the water line in order to determine the position (and maybe structure) of the water maser emission. Currently we also monitor this system with the Arecibo telescope.

The 1.7 GHz VLBI data are analysed by my student Filomena Volino, the 8.4 GHz I am doing myself. Preliminary maps are featured in a recent EVN newsletter.

The 8 o'clock arc

[The 8 o'clock arc, Allam et al. (2007)] This lensed LBG was discovered by Allam et al. (2007). Radio observations started by Mike Garrett and then continued by Filomena Volino and me show that the star-formation-rate is considerably lower than estimated in the discovery paper. The reason may be an overestimated dust extinction. This is work in progress.

RXS J1131-1231

[J1131-1231 with MERLIN (left) and the HST (right)] According to reports from a VLA snapshot observation done by another group, this interesting lens (a lensed star-forming galaxy with a Seyfert core) was not detected at radio wavelength. We reanalysed the same data and found a very clear detection. Subsequently, we re-observed the system with MERLIN (L-band), the VLA (C-band) and with the e-EVN (L-band). We clearly detect the lens and the lensed background source at the lower resolutions. With VLBI, however, only the core of the lens itself is detected. This means that the AGN core of the background object is too weak to be detected. Instead we see the lensed extended star-forming regions. [J1131-1231 with the VLA (C-band)]


[magnification map from ray-tracing simulation] A very elegant approach to calculate the variations induced by microlensing of large sources was published by Refsdal & Stabell already in 1991. Unfortunately that method can not be successfully generalised for situations with external shear. I developed an alternative approach that generalises well to the case with shear. The effect of shear (and also the shape of the source) can be very significant and change the expectation from the shear-less case by factors of a few. My analytical approach was confirmed by extensive numerical simulations.

Lensing as a tool for other fields

Gravitational lensing can also be used as tool for completely (and seemingly unrelated) fields, e.g. by providing almost identical copies of one and the same source or by magnifying and amplifying background source to make detailed studies possible.

Extinction and propagation effects

[HE0512-3329: Spectra (continuum and emission lines) and their ratios] Studies of extinction and other propagation effects always suffer from our ignorance of the true source structure and spectrum. In the case where lensing produces several images, we do at least know that their intrinsic spectra must be the same so that any differences in the observed spectra can be used to study differential propagation effects.

In a case study, I used HST spectra to determine the difference of the spectra of both images in the lens HE0512-3329. I found that both differential extinction and microlensing produce important differences. By studying the continuum separately from the emission lines, I was able to disentangle the effect for the first time. This sets the new standard for future projects in this field. Unfortunately many groups still study differential extinction neglecting microlensing or vice-versa. At least in the case of HE0512-3329 I showed that this is a very questionable approach, because both effects can be about equally strong.

Lenses as natural telescopes

The spectacular case of MG0414+0534, is discussed above. Here the total magnification of the lens (ca. 30) makes it possible to detect water maser emission at the redshift of the lensed background source (z=2.64) with reasonable observing time. Without the magnification, one would need at least one year of time at the Effelsberg telescope (one of the largest in the world) only to detect the emission.

[Cluster of galaxies MS0451.6-0305 with emission from lensed background
sources, Berciano-Alba et al., 2007] Another system where I am involved in the analysis is MS0451.6-0305, in which an ensemble of background sources is magnified by a cluster of galaxies in the foreground. The lens spreads the emission from several components of a merging system over such a wide area that resolved studies can be performed with current radio telescopes. A more detailed study was carried out later and a new paper was submitted.

Future lens surveys

[LOFAR low band antennas in Exloo] Together with Neal Jackson, Mike Garrett and others, I am working on a project to use LOFAR surveys to search for lenses. This is a very ambitious project, and the success will largely depend on how well the international baselines can be incorporated in the surveys. I am therefore deeply involved in the development of new analysis methods for LOFAR (and other projects) and in commissioning projects.

An e-MERLIN legacy project (PI: Neal Jackson) to study all known radio lenses was approved recently. My role in that project is the development of imaging methods for wide-band observations and the final lens modelling.

More exotic questions in lensing

Every now and then I start thinking about more academic topics in lensing. These are not always of direct relevance for astrophysical projects, but they are important to understand the nature of the lens effect better. Sometimes they also improve our understanding of relativity.

Relativistic lensing by moving lenses

[Deflection by a moving lens as function of its velocity v relative to a
lens at rest for test particles of different speeds w] The lens effect by moving object shows some surprising properties that seem to contradict common sense at first glance. One aspect is that radial motion in the same direction as the light propagation does decrease the light deflection, even though the "interaction time" seems to be less than for a lens at rest. For slow test particles, the effect is indeed the opposite. This also implies that there is a certain critical speed (c/sqrt 3) where the speed of the lens has no effect.
For transversal motion of the lens, an additional gravitational redshift is introduced. I explain this effect in the much simplified picture of an elastic collision of photon and lens. The very complicated calculations others used to describe this effect are in fact not necessary.
In a recent publication, my results on the effect of radial motion were disputed. The situation discussed in that paper, however, is not equivalent to the one we had discussed, because they are described in different reference frames. Once the proper translation is applied, the new publication does in fact confirm my result. A conference poster on the topic explains the differences and shows how the concept of angular diameter distances can be generalised to be measured between different reference frames that are moving with respect to each other. This leads to a very clean derivation of the effects of radial motion. A refereed publication is in preparation.

Lensing of gravity

[Local distortions of an external gravitational field as result of a
local mass concentration] It has been argued that gravitational lensing may have focusing effects on quasi-static gravitational fields in a similar way as it focuses light. My calculations do not confirm this view. Instead I show that static fields are only affected locally (without any long-range focusing). The situation is different for gravitational waves with wavelengths smaller than the typical scale of the lens. Gravitational lenses can thus focus gravitational waves.

Image coherence in gravitational lenses

[Young's double slit experiment] Gravitational lenses with multiple images act like a Young's slit experiment on (at least) galactic scales. The question arises if this situation can be used as an interferometer to resolve small structures of the source or (via correlation of electrical field signals from the images) to determine time-delays with extreme accuracy. My analysis of the situation shows that this would in principle be possible, provided that the sources are extremely small. For realistic extended sources, the time-delay varies over the area of the source so that the coherence is lost. This is a real pity, because otherwise we could use interferometric observations of lensed extended sources to determine the deflection field and potential without significant degeneracies.

Magnification theorem

[Correct magnification on the sphere compared to the (incorrect)
tangential plane approximation] There is a classical theorem in lensing according to which every lens produces at least one image with a magnification greater than unity. This seems to contradict total flux conservation if we consider closed surfaces around the source. The standard explanation (or excuse) is that the action of the lens modifies the geometry of the Universe, so that local flux density increase does not necessarily imply total global flux increase. In my publication about the subject, I show that this explanation misses an important point of the problem. The paradox can be resolved if we go beyond the usual approximation of small angles relative to the optical axis. In this spherical formalism, the deflection angle is modified, with the effect that the magnification can actually drop below unity so that the theorem is not valid anymore. Concerning the potential theory of lensing, this is related to a modification of Poisson's equation on the sphere. Additional important technical details can be found in the paper.

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This document last modified Wed Jul 15 13:08:07 CEST 2009