# Gravitational lenses

When the light of distant galaxies or quasars travels close to massive objects, such as other galaxies or galaxy clusters, it suffers a bending called gravitational lensing.

This phenomenon can be fully understood using Einstein's General Relativity, but can be described phenomenologically in terms of a variable refraction index of space, modulated as

*n* = 1 – 2*U/c*^{2}

where *U* is the gravitational potential of the matter responsible for the deflection and *c* is the speed of light.

In many ways, gravitational lenses can be interpreted in terms of standard optics: for example, lenses do not change the surface brightness of background sources, but they magnify and distort them; additionally, under certain conditions the lens can produce multiple images from the same source (much like a mirage).

One important difference with respect to standard optics is that gravitational lenses are completely *acromatic*: the refraction index *n* above does not depend on the wavelength, but only on the lens matter distribution.

The "Einstein cross", a quadruply imaged quasar observed through the bulge of a spiral galaxy (only the bulge of the galaxy is visibile in this image as the diffuse object at the center).

The galaxy cluster Abell 1689 is probably the best known gravitational lens. This image, obtained by the Hubble Space Telescope, shows more than one hundred multiple images, including many spectacular giant arcs.

- Gravitational lenses have many different astrophysical applications:Determination of the mass and mass distribution of the deflector: gravitational lensing is affected by the
*total*mass of the deflector (cf. above the expression for the refraction index*n*, which includes the total gravitational potential*U*). Therefore, gravitational lensing is an ideal technique to study dark matter. - Use of gravitational lensing as "cosmic telescopes", taking advantage of the magnification they produce to study distant and faint sources located behind them.
- Cosmological studies: gravitational lenses can be used to measure the speed of expansion of the Universe, i.e. Hubble constant
*H*_{0}, though the*time delay*observed in the multiple images of the same source, or to investigate other cosmological parameters (such as the matter density Ω_{m}of the Universe). - Cosmic shear, i.e. weak lensing by large-scale structures, allows us to obtain information on cosmological parameters (including the energy density equation of state w) and on the structure growth.

In general terms, two main forms of lensing can occur: **strong lensing**, where multiple images are produced, and **weak lensing**, where background images are distorted but no multiple images are produced. These two cases correspond to different approaches in the study of the lensing system:

**Strong lenses** are studied by modeling the mass distribution of the lens (typically using parametric models) and the source position and shape (again with parametric models, but more recently also with non-parametric ones), and by modifying the model until the configuration predicted is in agreement with the observed images (and eventually with other data available).

**Weak lenses** are in contrast studied using a statistical approach, by measuring the anisotropy in the orientation of background galaxies and by inferring properties of the lens from it. Although the problem could be studied using parametric models, there is a well-known technque to infer in a non-parametric way the mass distribution of the lens.

A third form of gravitational lensing, called **microlensing**, occurs when multiple images are created, or at least the image is strongly distorted and magnified, but these effects are not visible because they occur on extremely small angular scales. In this case, the presence of the lens is revealed by a particular variation of the luminosity of a background star (due to a change in the angular positions of the background star and the object responsible for the magnification).

The probability that such an even occurs at a given moment is extremely small, and therefore millions of stars need to be monitored. Recently, microlensing has found also applications for extra-solar planet searches.

A standard microlensing event is characterized by a symmetric, bell-shaped curve for the luminosity as a function of time of the background star.

Many review articles are available on Gravitational lensing, including

- A general review on Gravitational Lensing by Matthias Bartelmann (2010)
- A more theoretical paper on Gravitational Lensing by Volker Perlick (2004)
- An overview of Gravitational Lensing from a mathematical perspective, by Arlie Petters (2010)
- An older review on Weak Gravitational Lensing by Matthias Bartelmann and Peter Schneider (2001)
- A review focusing on the Cosmological Applications of Weak Gravitational Lensing by Henk Hoekstra and Bhuvnesh Jain (2008)
- A review on Cosmic Shear by Alexandre Refrieger (2004)
- A specialized review on Gravitational Microlensing by Shude Mao (1998)