### 3. LENSED QUASARS AND H0

Quasar lensing helps to solves astrophysical puzzles in various ways, but one of its most beautiful applications is probably the determination of the Hubble parameter H0.

In 1964, the Norwegian astronomer Sjur Refsdal proposed an original method (Refsdal 1964) to use gravitational lensing as a tool to measure the size/age of the Universe. When photons propagate from a distant source toward the observer, they are under the effect of the gravity field of lenses along the line of sight. They do not follow a straight line anymore, but their trajectory is curved and longer than the original one. As a consequence, it takes more time for the photons to travel from a lensed source than from an unlensed one. The geometrical difference introduced by the lens between the two, lensed and unlensed paths, introduces a time-lag between the arrival times of the (lensed and unlensed) photons at the position of the observer. This time lag is called the geometrical "time-delay", tgeom. While passing in the immediate vicinity of the gravity field of the lens, the light is affected by a second delay: the gravitational time-delay, tgrav. A "lensed photon" will be seen by an observer with a total time-delay ttot = tgeom + tgrav, with respect to the observation of the same photon if it were not lensed.

The time-delay is a function of image position in projection on the plane of the sky. One can then define an arrival time surface that associates, to each position on the sky, a given a value of the time-delay. Most of this surface is missed by the observer who has only access to the few areas where the lensed images form. When two or more images of the source are observed it is possible to compare the arrival times at the positions of the lensed images and to determine a "relative time-delay". This is in fact the only truly measurable quantity, rather than the actual time-delay between the lensed and unlensed paths to the source, since the unlensed source is never visible.

In practice, time-delays are measured taking advantage of a lensed source with significant photometric variations. Due to the time-delay, the variations will be detected by the observer at different times in the light curves of each image. The shift in time between the light curves is simply the (total) time-delay between the images. Refsdal (1964) proposed to measure time-delays in lensed supernovae, but his method was published just when the first quasars were discovered (Schmidt 1963). Quasars, that later turned out to be very numerous in the sky, rather bright, and photometrically variable, were promising objects to measure time-delays if at least some of them were found to be lensed. They appeared in any case much more promising than rare and transient phenomena such as supernovae. Indeed, thousands of quasars are now known, and several tens of them are lensed. Measured quasar time-delays span over a broad range of values, between days and months. One is larger than a year: Q 0957+561 (e.g., Vanderriest et al. 1989).