Book Editors: P. Benvenuti, F. D. Macchetto, and E. J. Schreier

Electronic Editor: H. Payne

**J.-F. Claeskens**

Institut d'Astrophysique de Liège, 5, Avenue de Cointe, B-4000
Liège, Belgium

**J. Surdej**

Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218
USA

Research Assistant, Belgian Fund for Scientific Research (FNRS) Member of the Astrophysics Division, Space Science Department of the European Space Agency; also Research Director, Belgian Fund for Scientific Research (FNRS)

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**Keywords: Gravitational Lensing - Dark Matter - Quasars**

Identifying the real nature of dark matter in the Universe constitutes one of
the most challenging goals in modern astrophysics (see Carr 1990 for a
review).
Gravitational lensing provides a very powerful tool to detect the various
signatures of baryonic dark matter. Indeed, the distribution of dark matter
associated with clusters of galaxies can be probed from the detailed analysis
of giant luminous arcs and of the shear deformations of background sources
(Fort & Mellier 1994); the effect of * local* compact dark matter,
belonging to the disk or to the halo of our galaxy, can be detected in
micro-lensing experiments such those being presently conducted by the DUO,
EROS, MACHO and OGLE teams (see the recent review by Paczynski 1995).
Finally, the *
cosmological* density of * isolated* dark compact objects can be indirectly
constrained by
the (non) observation of `micro-lensing induced' flux variations in background
QSOs or by the (non) detection of multiply imaged sources in a flux limited
sample of QSOs. The first technique is specially sensitive to the effects of
compact objects in the mass range -- (Canizares 1982,
Schneider 1993).

The second technique consists in trying
to resolve multiple macro-lensed QSO images using ground-based, radio
telescopes and/or the HST with adequate high angular resolution imaging
instruments. This class of techniques has a good sensitivity in the mass range
--, depending on the instrument being used. The
limits of this method are dictated by both the angular resolution * and*
the dynamics of the instrument. Using the point-mass lens model, we show in
this paper how statistical constraints can presently be derived in the above
mass range; the contribution due to HST is emphasized. To be complete, let us
mention here that by modelling the observed frequency of mutiply imaged Highly
Luminous Quasars with galaxy lenses (some examples are well known), it
has been possible to constrain the values of galactic parameters, the number
counts of QSOs and the cosmological constant (Claeskens et al. 1995a).

In the next section, we present and discuss the expression of the probability for observing doubly imaged QSOs lensed by compact objects; we also emphasize the influence of the instrumental characteristics (angular resolution and dynamics). We present the data in section 3. The results and a discussion are given in section 4.

The generic expression for the probability of observing a lensing event along the line-of-sight to a QSO at redshift , due to a population of compact objects with mass , uniformly distributed in space, is:

where represents the cosmological density of compact objects with
mass , in units of the present closure density of the Universe. Of
course, the probability **P** increases with . In the
above equation, is the so called magnification bias: because
macro-lensing amplifies the flux of a background source, observing *
bright* sources * enhances* the chance to discover a lens in a flux limited
sample of quasars. Because of these two latter effects, the probability to
discover a new lens is of course optimal when observing HLQs (typically ; see Turner 1984 & Surdej et al. 1988).
Finally, in Eq. (1) represents the lensing
cross-section for a compact object with mass at redshift **z** to produce
double images from a QSO at redshift , and be detected by the given
instrument.

At a given redshift, the general cross-section for multiple imaging is a ring whose thickness depends on three competing criteria:

- Because the angular separation between lensed images
increases with the degree of misalignment between the source, the lens and
the observer, and because must be larger than the angular
resolution of the instrument, the angular distance between the lens and the
source, measured at the observer, must be larger than a
*lower*limit . - The magnitude difference between double lensed images also
increases with the degree of misalignment between the source, the
lens and the observer; therefore, the finite dynamics of the instrument
implies an
*upper*limit on the angular distance between the lens and the source. So, the higher the dynamics of the instrument, the larger the cross-section for lensing. increases faster with when gets smaller, so, if is too small, then : the instrument is unable to resolve the lensed images. - In order to avoid contamination by cluster lensing and also by other
uncontrolled biases, we consider that must be smaller than
. So has to be smaller than an
*upper*limit . This condition sets an upper limit () on the mass range of the compact objects which can be probed.

The general expression for the lensing cross-section is the following:

where is the angular distance between the observer and the deflector. More formal details on this expression for the lensing probability can be found in Surdej et al. (1993a).

Figure 1a displays the relation existing between the dynamics and the angular resolution (the Angular Selection Function (ASF)) for typical good seeing observations obtained with a ground-based telescope (dotted line) and for the pre- and post-refurbished HST (dashed and continuous lines). In some way, these curves represent the joined performance of the instrument and of the method of image analysis for resolving multiple images. The latter method usually consists in carefully subtracting a scaled PSF from the QSO image (see Surdej et al. 1995, Remy 1996). The thick vertical line represents the maximal angular separation between the images (criterion 3). Figure 1b illustrates the corresponding lensing probability versus the angular separation between the lensed images, for two different compact lens masses (heavy lines: ; light lines: ). The optical depth is expressed in units. The better efficiency of HST to constrain the density of compact objects with is clearly seen from Fig. 1b.

**Figure:** a) The ASF b) The lensing probability (see text).

**Table 1:** Characteristics of HST and ground based imagery surveys of HLQs (1-
Maoz et al. 1993, 2- Surdej et al. 1993a, 3- Crampton et al. 1992, 4- Yee et
al. 1993, 5- Jaunsen et al. 1995)

The present sample contains 1680 observations of 1207 different HLQs. It has been compiled by merging the HST snapshot survey (before refurbishment) and four selected ground-based imagery surveys of HLQs. Table 1 summarizes the properties of each subsample. In the case of multiple observations of a same QSO, the most efficient one has been kept. But this choice depends on the assumed lens mass. The two last columns of Table 1 show how many observations per sample have been kept, for two different lens masses. The percentages represent the internal efficiency of each subsample.

The sample presented in Table 1 only contains three lens candidates which satisfy the hypothesis of being possibly lensed by a dark compact object (i.e., two lensed images and no already detected galaxy-type lens) and which present an angular separation between their images smaller than . These are Q1208+1011 (Magain et al. 1992, Maoz et al. 1992), Q1009-025 (Surdej et al. 1993b) and J03.13 (Claeskens et al. 1995b).

**Figure:** Upper limit on versus (with a 99.7 % confidence
level).

Comparing the observed and the predicted numbers of lenses in the sample, a firm upper limit on can be derived. With a 99.7% confidence level, and adopting an Einstein-de Sitter Universe (), we find that in the mass range --. The strongest constraint we have derived is for compact objects. The use of other cosmological models or the identification of a yet unseen galaxy-type lens among the three candidates would reinforce the present constraints. As shown in Fig. 2, our results are the most constraining ones for masses larger than , and a closure density of the Universe by any kind of compact objects in the mass range -- is ruled out.

The dotted line in Fig. 2 corresponds to a simulation of the constraints on if 500 additional observations of HLQs were obtained with the (presently refurbished) HST. This would drastically improve the constraint on in the mass range --. This would even surpass the best presently existing constraints derived from VLBI radio observations for compact objects more massive than (Kassiola et al. 1991). It is also important to note that the number counts of QSOs at radio wavelength is not known very accurately. So, the magnification bias for radio observations cannot be properly estimated and this makes optical observations presently more reliable in order to derive statistical values for the cosmological parameters. Therefore, HST can play a very important role in further constraining the cosmological density of compact dark matter. This goal could be achieved by carrying out a new snapshot survey.

This research has been supported by the PRODEX project for Gravitational Lens studies at the ``Institut d'Astrophysique de l'Université de Liège''.

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