Since the angular sizes of gravitational arc(let)s depends on the cosmological distances of the lens and sources, in principle arc(let)s can be used to probe the geometry of the universe itself. With well-known arc configurations it should be possible to determine the cosmological parameters H0, and = 0 + 0. In fact, we are unable to separate the relative contribution of 0 and 0. For the present time, we have some hope of measuring the mass density of the universe, 0, from independent observations, mainly from the large scale structure velocity field. Therefore we should be able to obtain a value for 0 = 1 - 0 if we adopt the theoretical hypothesis of an inflation period for the primeval universe. Additional observations of gravitational arcs may offer an opportunity to get around this hypothesis. In practice, a value of would be obtained by comparing the relative astrometric positions of arc(let)s predicted for various cosmological models with the observations. Unfortunately the positions of arcs vary very little with the cosmological parameter if its reduced value is not larger than 1., and we already know that it cannot be much larger (Carroll, Press & Turner 1992). Also, the positions of arcs are far more sensitive to the lens modeling. In conclusion, the determination of will be done only if an almost perfect modeling of the lens is achieved and if very good astrometric positions of the arcs are obtained. It will not be a surprise that this topic is still largely speculative.
In the following we must indeed assume that the general theoretical framework given in part 2 is valid. Note that Nottale (1988) emphasized that giant arcs with known redshift can test gravitation theories on cosmological scales, provided dynamical masses and gravitational masses are identical. The present-day observations do not contradict the predictions of General Relativity, and actually confirm the equality between the two masses (Nottale 1988, Dar 1992).
The time delay which appears between the successive observations of an intrinsic event in the source in each multiple image of an arc system can be used to infer H0. The original idea was introduced for multiple quasars by Refsdal (1964) and applied to the double QSO 0957+561. It has not succeeded so far in providing a better value of H0 than other methods (see Kochanek 1991): observational discrepancies were reported between the radio and optical measurements of the time delay, and the modeling of the lens appears still more complex than previously thought with the discovery of a fold arc in the field of the associated cluster (Bernstein et. al 1993). At first glance, the observational situation would seem better for arcs. A-priori we expect a large number of supernovae events in distant blue galaxies which have a large star formation rate. The magnified event that appears at different times on each image can be well recognized. Giraud (1992b) claimed to detected a local surface brightness variation in the giant arc observed in MS0302+17 which could be interpreted as such supernovae explosion. But his detection is marginal and is not confirmed by similar observations done by other observers during almost the same period. However, even if it is possible to search for and monitor supernovae, the time delay can reach hundreds of years for separate multiple images! Very often, only for the region near the critical line of two merging arcs is the time delay a few days or weeks (Kovner & Paczynski 1988). But even if by chance a supernova were observed with such an ideal geometrical configuration, there will still remain an uncertainty in the determination of H0 associated with the modelling of the cluster lens, even if it could be better determined than for small galaxy lenses. It is likely that a reliable value of the Hubble constant will need multiple observations of supernovae in a large number of clusters with giant arcs; an observational challenge which will be out of reach for a long time.
Paczynski & Gorki (1981) first suggested using the multiple images of a lensed quasar to constrain the cosmological constant provided the core radius and the velocity dispersion of the lens are known (and obviously the redshifts of the lens and the source). The technique assumes a model for the mass profile which relates the angular separation of split images to the velocity dispersion and the angular diameter distances. In that case, it is straightforward to find the best which is compatible with the observations. Recently, Breimer and Sanders (1992) used basically the same approach as Paczynski & Gorki but on clusters with giant arcs. They discussed simultaneously the gravitational lensing analysis and dynamical mass distributions inferred from both the galaxy behavior and the distribution of hot X-ray gas. They concluded that if light traces mass the observations of A370 are compatible with = 0. However, if light traces mass in a more complex way (M/L varies with distance), a wide range of cosmologies are also possible.
In a similar way, the cosmological parameters could be inferred by measuring the position and magnification of two arcs with different redshifts and observed in the same cluster. From equation (7) of section 2, the ratio of deviation angles generated by a singular isothermal sphere for two different sources with different redshifts is
and is independent of H0. Therefore, the ratio only
depends on the
curvature k (and the deceleration parameter q0)
and the cosmological constant :
In principle this technique should work. However, the
ratio of the angular distances
between the two arcs is strongly dependent on the lens modeling and the
assumptions made for the sources. It is likely that this approach will
need the discovery of at least two relatively bright arcs with different
redshifts in a cluster
lens with a simple geometry and
better spectroscopic capabilities coming from future large telescopes.
and is independent of H0. Therefore, the ratio only depends on the curvature k (and the deceleration parameter q0) and the cosmological constant :
In principle this technique should work. However, the ratio of the angular distances between the two arcs is strongly dependent on the lens modeling and the assumptions made for the sources. It is likely that this approach will need the discovery of at least two relatively bright arcs with different redshifts in a cluster lens with a simple geometry and better spectroscopic capabilities coming from future large telescopes.