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Gravitational Lensing As a Probe for Dark Matier 171 GRAVITATIONALLENSING AS A PROBE FOR DARKMATIER Laurent Nottale CNRS. Departement d'Astrophysique Extragalactique et de Cosmologie. Observatoire de Meudon. F-92195 Meudon Cedex. France ABSTRACT We show in the present contribution how some statistical effects of lensing fllay yield unique information on the distribution of matter in the universe and may even probe the relative distributionof dark and visible matter. Such a goal may bereached provided detailed information is known on the observed foreground matter lying in the fields of any of the test sources in a sample of "standard candles". 172 1. INTRODUCTION. Gravitational lensing is a unique tool for probing the distribution of matter in the universe, and in particular to detect dark matter. This is due to the fact that either mass or surface density directly intervene in the various gravitational lensing effects, and that all levels of the hierarchy of structures, from compact objects and stars to the largest scales may act as lens and as source forlensing. But clearly, if gravitational lensing may be caused by all mass scales, the observable effects which are expected from them vary with the scale. In the present contribution, we shall consider some of these effects (which had been mainly studied up to now in the frame of a study of lensing itself), from the point of view of the information they may yield about the distribution of matter in the universe. A particular weight will be given here on statistical lensing effects. Neither the analysis of known multiple image configurations nor microlensing are considered here, since they have been the subject of many recent papers and reviews to which we send the interested reader.1 -4) 2. VARIOUS LENSING EFFECTS. The most common gravitational lens effect is luminosity magnification (or attenuation), since it is encountered whatever the physical configuration, provided it corresponds to a perturbation relative to the Friedmann-Robertson-Walker background average universe. To this magnification is always linked a diameter enlargement, since Etherington' s theorem5) ensures the conservation of +z)4, B(J where B is the surface brightness and z the redshift, this result holding in any Riemannian universe. It thus applies to various images of a same source (z=constant same expected surface brightness), and also allows to relate directly the 0=} magnitude-redshift relation to the diameter-redshift relation into any universe, whatever inhomogeneous. The diameter-redshift relation may be obtained in the most general frame by solving the Optical Scalar Equations (OSE) which govern 3,5 ) the propagation of the cross sectional area of a light beam -7 . In this equation, the effects of matter and energy (i.e. equivalently, thanks to Einstein's equations, of geometry) on the light beam propagation come from two terms: (i) A "matter" term, based on the Ricci tensor, which is often interpreted as describing the converging effe ct of a smooth density of matter included into the light beam. However it plays in fact a far more important role, since a lot of studies have shown that it also yields the effect of a clumpy distribution of matter lying in 9 the light beam 8, ), and the average effect of a clumpy distribution of matter lying outside the beam3). Though numerical simulations confirm this last result even for large average amplificationsS,10), unfortunately an analytical demonstration is still lacking in this case, due to the fact that it corresponds to multiple lensing. 173 (ii) A shear term, based on the complete Riemann tensor and mainly due to individual masses lying outside thebeam andto density anisotropy. It has beenshown in the most general way that the luminosity magnification may be written under the form:3l (1) where and y reduce to leading order respectively to the matter and shear terms K: obtained in Einstein linearized theory8,ll). We recall that the matter term writes typically: 4nG c d ( ) K'= [f op r] - L(zd) [1- 4tTl 2 -c 2 H 0 s where is the density difference between the lens and the cosmological op background universe and is integrated inside the lens along the line of sight, L(zd) is a characteristic function of the lens redshift and � is the so-called "optical distance"3,12), which is a monotonic function of redshift . The final term is another expression for the usual distance term written in terms of the angular diameter distances: (3) The luminosity magnification due to shear by a point mass lens M is defined, in terms of the lens-image impact parameter by a yterm which writes: r, 4G M c Y= - -L(zd) [l- 4t-] (4) -c2 r2 H o Lis For very large scales (superclusters, filaments, sheets of matter and most clusters of galaxies), "Ricci" luminosity (and diameter) magnification remain the only lensing effect. Not being related to a morphological, spectral or variability signature, it can be identified only by statistical methods. Two of them have been particularly considered: Perturbation of the Hubble diagram of distant sources by foreground matter,13,14) and change in the number surface density of galaxies or quasars,3,10,15) with the corresponding selection effects which are implied by and related to these phenomena. 3. PERTURBATION OF HUBBLEDIAGRAM BY FOREGROUND MATTER It has been suggested several years ago by Karoji and Nottalel3) that rich clusters of galaxies magnify in a statistically significant way the luminosity of background sources lying behind them. Their method consisted in dividing a sample of standard candles into two sub-samples, those which lie behind a 174 foreground cluster (B) and those whose light has not crossed any inhomogeneity on its way to the observer (A), then in comparing the relative distribution in the Hubble diagram of subsamples (A) and (B). The so-called "Hubble modulus": (5) is used for such a comparison, since it characterizes the deviation from the mean Hubble line. The effect was later confirmed on a sample of brightest cluster galaxies by a direct comparison with the prediction from gravitational lensing expectation.14) Each cluster in the sample was placed relatively to foreground Zwicky clusters in the same field, so that the distance term Dd.Dds!Ds (Eq. 3) could be computed for each configuration. From the observed impact parameters the density term f op.dr was also estimated (to a multiplicative constant depending on the absolute density of lensing clusters, which was assumed constant as a first approximation), so that the matter term could be estimated for each relative configuration (foreground K: cluster/background cluster). Note that most of the time the redshifts (or distance classes) of foreground and background objects are different enough to ensure that they actually do not lie at the same distance, but that anyway the form of the distance term (Eq.3) which vanishes when the lens comes close either of observer or of source, ensures the vanishing of the contribution of such configurations to statistics from the point of view of lensing. Another remark on which to stress here is about the validity of Zwicky clusters for leading such a study. While the Zwicky catalog may indeed be criticized if one is interested in the identification and analysis of individual clusters, since it is based on counts made up to a given limiting magnitude and on number surface density excess relative to the field, this method conversely implies that it is well adapted to statistical studying lensing effects, in particular by large scale structures. This is because it is surface density which intervenes in the intrinsic term of the lensing formula, so that the same global effect will be expected from several superposed small clusters ( lying at comparable distances) or from one rich cluster having the same integrated density. A third point to be made is that samples of brightest cluster galaxies, and more generally the Hubble diagram of "standard candles", have been given up as useful tool for cosmology, after the chess of the various attempts to get from it the value of q0• However it is important to notice that the Hubble diagram keeps all its value for statistical lensing studies. Indeed the rejection came from the fact that evolution effects were finally understood to be far larger than the searched effects of q0, thus preventing from its measurement. So the question here is the relation of 175 evolutionary effects and q0 effect to the statistical effect of magnification by lensing. First, while luminosity evolution depends only on the source redshift, it is one of the significant signature of lensing that it depends also of lens redshift and lens density. No other physical phenomenon may lead to correlations between what is thought to be intrinsic to a source (its "absolute" luminosity, when lensing is not accounted for) and the properties of foreground matter which may lie at cosmological distances from the source. Second the effect of q0 is also fixed for a given redshift, while a dispersion is introduced in the lensing effect for given source redshift by the various configurations which may be encountered (lens redshift, density and angular distance). The final result was a significant correlation between the deviation from mean Hubble line (Eq.5) and the expectation from lensing magnification (Eq.2).14) Unfortunately, though the existence of the effect itself was established independantly from the initially assumed value of q0 (or <J0 ), several uncertainties prevented from being able to state precisely at this stage on the amount of dark a matter required for it. In f ct information may be gained by this method on both q0 and CJ0, since q0 intervenes in the Mattig relation, while the unknown density term in the lensing formula (which is given by the slope of the regression line) is directly proportional to <J0 • The results are as follows: *The high value of q0 found for the whole sample (uncorrected for lensing), q0=d.6, falls down to q0<"'0.7 when only the "isolated" (i.e.
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