In general relativity, the presence of matter (energy density) can curve spacetime, and the path of a light ray will be deflected as a result. This process is called gravitational lensing and in many cases can be described in analogy to the deflection of light by (e.g. glass) lenses in optics. Many useful results for cosmology have come out of using this property of matter and light.

For many of the cases of interest one does not need to fully solve
the general relativistic equations of motion for the
coupled spacetime and matter,
because the bending of spacetime by matter is *small*.
(Quantitatively the matter bending space is moving slowly relative to
c, the speed of light and
the "gravitational potential"
Phi induced by the matter obeys
|Phi|/c^{2} << 1 .)

A sketch of the paradigm of a lensed system is below
(
source):

In a system where lensing occurs there is a

- source: where the light comes from, can be a quasar, the cosmic microwave background, a galaxy, etc.
- lens(es): which deflect(s) the light by an amount related to its quantity of mass/energy, can be anything with mass/energy
- observer: who sees a different amount of light than otherwise because the lens has bent spacetime and thus the travel paths of the light
- image or images: what the observer sees

As a consequence of lensing, light rays that would have otherwise not reached the observer are bent from their paths and towards the observer. (Light can also be bent

Strong Lensing

The most extreme bending of light is when the lens is very massive
and the source is close enough to it: in this case light can
take different paths to the observer and more than
one image of the source will appear.
A multiple
image is shown at right
(source).
The first example of a double
image was found in 1979, of a quasar.
The number of lenses discovered
has been used to estimate the volume of space
back to the sources. This volume
depends strongly on cosmological parameters,
in particular the cosmological constant
(a classic reference is
here).

If the source varies with time, the multiple
images will vary with time as well.
However, the light doesn't travel the same distance to each image,
due to the bending of space. So there will be time delays for
the changes in the images. These time delays can be used to
calculate the hubble constant
H_{0}.
A few systems with these time delays have been
found and are under study. Much of the subtlety in this work
lies with constructing the model of the mass distribution
forming the lens
(see this
review for
technical detail).

In some special cases the alignment of the source and the lens will be
such that light will be deflected to the observer in
an "Einstein ring." Some examples and references can
be found here on
Wikipedia.
More often than a ring, the source may get stretched out and curved,
and form a tangential or radial arc. A lot of mass is needed to
cause an arc to appear, so that properties of arcs (numbers, size,
geometry) can often be used
to study massive objects like
clusters.
One can also, given a set of images, try to reconstruct the lens
mass distribution (for an example of reconstructing a cluster as a lens
see this
technical paper).

Weak Lensing

In many cases the lens is not strong enough to form multiple images or
arcs. However, the source can still be distorted: both stretched (shear)
and magnified (convergence). If all sources were well known in
size and shape, one could just use the shear and convergence to
deduce the properties of the lens. However, usually one does not know
the intrinsic properties of the sources, but has information about
the average properties. The statistics of the sources can then be used to
get information about the lens. For instance, galaxies in general
aren't perfectly spherical, but if one has a collection of galaxies
one doesn't expect them all to be lined up.
Thus, if this set of galaxies is lensed, on average, or statistically,
there will be some overall shear and/or convergence imposed on the
distribution, which will give information about the intervening lens(es).

There is a distribution of galaxies far enough away that can be
treated as sources, and thus
clusters nearby can be "weighed" (i.e. have their mass measured)
using their lensing. Superclusters have been considered as well.
In addition, theories of cosmology predict the
distribution of
large
scale structure, the distribution of matter in the universe.
The statistical properties of the large scale
structure (e.g. the probability of finding a galaxy at one place
when there is another a certain distance away) can also be measured by
weak lensing, because the matter will produce shear and convergence
in distant sources (which can be galaxies, or
the
cosmic
microwave background, for example).
Weak lensing is a useful complement to measures of the distribution of
luminous mass such as
galaxy surveys. Lensing measures all the mass, in particular
the
dark matter as well as the luminous matter.

There are ongoing searches to use lensing to find a type of dark matter called MACHOs (massive compact halo objects). Although MACHOs, as dark matter, cannot be seen themselves, if they pass in front of a source (e.g. a star nearby), they can cause the star to become brighter for a while, e.g. days or weeks. This effect has been observed, but determinations of the dark matter are not yet conclusive. Observations are underway by many groups. See these notes by Michael Richmond on Gravitational Microlensing: Searches and Results.

Some web references:

(See also the review at Wikipedia.)

- Simulation of a Gravitational Lens by Jim Lovell
- Basics of Gravitational Lensing, by Mellier, at IPAC, also see full article on Arcs
- One can lens an astrophysicist, set up by Pete Kernan. (this seems to be gone sadly)
- Lens a photo by Nick Kaiser
- Gravitational Lensing and Geometric Lensing, AMS feature column
- Lensing program by M. Boughen (needs to be downloaded)
- NASA has a note on gravitational lensing.
- A data base of lenses is maintained by CASTLes.
- More technical reviews on the web include Some lecture notes from Saas Fee by C.S. Kochanek on strong gravitational lensing, P. Schneider on weak gravitational lensing and J. Wambsganss on gravitational microlensing. See also the lectures by Bartelmann and Narayan. There is much recent interest in this technique and so this list is going to be quickly outdated!
- Constraints on galaxy mass distributions from strong lensing, by Kneib, (12/2001) review
- Bartelmann and Schneider review of weak lensing for physics reports
- Gravitational lens bibliography and data base.
- MPA Garching lensing information links
- discussion of a halo which lenses
- guided tour of gravitational lensing from MPA.
- B Jain's home page, many links to his pages on weak lensing
- A nice Scientific American Article (not on the web, unfortunately!) is by J. Wambsganss, "Gravity's Kaleidoscope", November 2001 issue.
- The gravitational lensing and the optics analogy is described in more detail here, page by Surdej et al
- Intro to weak lensing Santa Fe lectures, M. White.
- Chuck Keeton's GRAVLENS code for analyzing lenses.

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last updated Dec. 13, 2010.