Annu. Rev. Astron. Astrophys. 1999. 37: 127-189
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Matter intervening along the light paths of photons causes a displacement and a distortion of ray bundles. The properties and the interpretation of this effect depend on the projected mass density integrated along the line of sight and on the cosmological angular distances to the observer, the lens and the source.

The sensitivity to mass density implies that gravitational lensing effects can probe the mass of deflectors, without regard to their dynamical stage and the nature of the deflecting matter. This is therefore a unique tool to probe the dark matter distribution in gravitational systems as well as to study the dynamical evolution of structures with redshift. The dependence on the various angular distances involved in the lens configuration means that the deviation angle depends on the cosmological parameters, Ho, Omega and lambda, so that the analysis of gravitational lensing can potentially provide a diagnosis on cosmography. Of course, the sensitivity to cosmological parameters is not unique to gravitational lensing, and many other astrophysical phenomena depend on them. However, owing to magnification, image multiplicity and deflection angle produced by lensing, it is possible to use the lensing effect as a bonus when compared with other experiments: image magnification permits observation of the high-redshift universe, study of the evolution of galaxies with look-back time and comparison with theoretical cosmological scenarios. Image multiplicity probes different light paths taken by photons emitted by one source. By computing time delays of the same transient event observed in each individual image, one can measure Ho. Finally, for high-redshift sources the deflecting angle depends on the geometry of the universe and provides a unique tool for measuring the cosmological parameters.

The interest in gravitational lensing for cosmology started very early, after Zwicky's discovery (Zwicky 1933) of the apparent contradiction between the visible mass of the Coma cluster and its virial mass, which could not be explained without recognizing that it is dominated by unseen mass. This surprising statement could not be confirmed without an independent mass estimator, which could probe the total mass directly, without using the light distribution or critical assumptions on the dynamical stage of the cluster components. Four years later, Zwicky (1937) envisioned that extragalactic nebulae could be efficient gravitational lenses and provide an invaluable tool for weighting the gravitational systems of the Universe.

The other works that raised interest in lensing for cosmology are more contemporary. Refsdal (1964) first emphasized that time delays in multiple images could be used to measure Ho, and the very first considerations of light propagation and deformation of ray bundles in inhomogeneous universes were discussed initially by Sachs (1961), Zel'dovich (1964) and later by Gunn (1967). From an observational point of view, the discoveries of the first multiply imaged quasar (Walsh et al 1979) and the first distorted galaxies (Soucail et al 1987, Lynds & Petrosian 1986) were major steps that boosted theoretical and observational investigations of gravitational lenses.

Most of the cosmological interest in gravitational lenses has already been reviewed by Blandford & Narayan (1992), Schneider et al (1992), Refsdal & Surdej (1994). Fort & Mellier (1994) presented the first review which focused particularly on the use of arc(let)s in cosmology, and the interest in the use of lensed galaxies to probe the deep universe has been recently reviewed by Ellis (1997). With the amazing observational and theoretical developments in the field, in particular in weak lensing, it seems timely to review all these results and to address the new and future issues in the area.

During the last five years, thanks to the seminal work on mass reconstruction from weak lensing analyses (Tyson et al 1990, Kaiser & Squires 1993), mass reconstruction algorithms have provided new and robust tools for studying the mass distribution of gravitational systems and have permitted the establishment of a link between theoretical investigations of weak lensing and the observations of weakly distorted galaxies. In particular, there have been impressive developments in cosmological diagnoses from the analysis of weak lensing induced by large-scale structures. Theoretical and numerical studies demonstrate that the statistical analysis of gravitational lensing will provide valuable insights on the mass distribution as well as on the cosmological parameters. With the coming of new wide field surveys with subarcsecond seeing [such as Megacam at the Canada-France-Hawaii Telescope (CFHT) or the VLT-Survey-Telescope (VST) at Paranal] or very wide field shallow surveys [such as the VLA-Faint Images of the Radio Sky at Twenty-Centimeters (FIRST) survey or the Sloan Digital Sky Survey (SDSS)], weak-lensing analysis should probe the power spectrum of the projected mass density, from arcminutes up to degree scales. Visible weak lensing surveys should also be capable of providing a projected mass map of the universe, just as the Automated Plate Machine (APM) survey provides the visible light distribution (Maddox et al 1990). From the observational point of view, the outstanding images coming from Hubble Space Telescope (HST) had a considerable positive impact on our intuitions about the potential usefulness of gravitational distortion. The wonderful shear pattern around lensing cluster A2218 is visual proof that weak lensing works and that it directly reveals the mass distribution. One of the most spectacular uses of HST images for lensing was done by Kneib et al (1996), also in A2218. The superb HST images allowed them to demonstrate, from the morphology of only one arclet, and without the need of a spectroscopic redshift, that it must be a lensed image associated with the same source as the giant arc. The similarity of the morphologies of the giant arc and the counter-image is so impressive that it cannot be questioned that they are images of the same source. In parallel, the Keck telescope, which is currently detecting the most distant galaxies, reveals the obvious importance of giant gravitational telescopes. Finally, the impressive results obtained by the Submillimetre Common-User Bolometer Array (SCUBA) in the submillimeter wavebands have shown that the joint use of a submillimeter instrument with magnification of high-redshift galaxies is an ideal tool for studying the evolution and content of distant galaxies.

In the following I review most of these recent works and discuss their impact for cosmology. Although this review focuses on weak lensing, the distinction between arclets and the weak lensing regime is somewhat arbitrary, and both are relevant for our purpose. Furthermore, because some of the results cannot be discussed without referring to strong lensing, I often include new results from arcs and multiple image studies. Section 2 recalls the basic equations useful in gravitational lensing which help in the understanding of this review. The definitions for strong lensing cases are not presented again, and I will refer to the review by Fort & Mellier (1994) for all these aspects. In Section 3 I focus on the mass distribution in clusters of galaxies from arc(let)s or mass reconstruction from weak lensing inversion. I also address the issues concerning the measurement of weak shear because it appears to be a major challenge for observers. Section 4 presents weak lensing induced by large-scale structures, and Section 5 presents weak lensing induced by foreground galaxies on the background sources (the so-called galaxy-galaxy lensing analysis). I then move toward the high-redshift universe in Section 6. Sections 7 and 8 are devoted to cosmological parameters and weak lensing on the cosmological microwave background (CMB), respectively. Conclusions and future prospects are discussed in the last section.

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