Abstract :
[en] In the early 1960s, gravitational lensing has received a special attention when S. Liebes and S. Refsdal have derived in detail some of the basic equations of the theory. While Liebes (1964) discussed the probability of detecting these effects and considered several astrophysical applications, Refsdal (1964) derived, inter alia, his famous relation which links the Hubble parameter (H0) to the expected time delays between pairs of lensed images. From that moment, the scientific community fully realized that gravitational lensing effects offer a new way of probing cosmology. However, an important fact has been to accept that the determination of H0 seems to be model dependent, not only on the universe model, but also on the mass distribution of the deflector.
The main topic of the present thesis constitutes a straight continuation of this inquiry. We have been sounding parts of the mathematical lensing framework on two fronts. First, considering to first order a very small misalignment between the source, the lens and the observer, we have derived the expressions of the lensed image positions along with their amplification ratios, for the case of power-law axially symmetric mass distributions, the so-called ε-γ family of models (Wertz, Pelgrims & Surdej, 2012). After combining these results, it has allowed us to derive an expression for H0 independently of the model parameters. We have extended this study to the ε-γ family of models with external shear, as well as to the singular isothermal ellipsoid (SIE) models. For both these types of models, we have obtained an expression of H0 which is once again independent, to first order, of the model parameters. Furthermore, even though these families of models remain rigorously distinct, except for the singular isothermal sphere (SIS) and the perfect alignment, the expression of H0 in terms of observable quantities and of the source position takes surprisingly the same simple form. In addition, we have demonstrated the feasibility of analytically constraining to first order the model parameters by only using the astrometric positions of the lensed images. Therefore, for the case of a small misalignment between the source, the deflector and the observer, it is straightforward to determine whether the ε-γ or SIE family of models constitutes a judicious representation of the mass distribution of the deflector. It is conceivable that similar results can be deduced for other families of models.
Secondly, we have developed a new analytical approach in order to determine the expression of the deflection angle, hereafter α. Since the latter depends on the deflector mass distribution, there exists no global explicit expression but only an implicit definition of α. Therefore, the analytical methods used to obtain the explicit expression differ for different types of mass distribution. However, using the Fourier transform theory, one may basically express α in terms of the Fourier transform of the surface mass density. Such a method allows us to approach any mass distribution in a unique way. As a first application, we have separately derived the expression of the two components of α for the case of homoeoidal symmetric lenses (Wertz & Surdej, 2013). This original result constitutes a first proof that the Fourier approach constitutes a promising alternative to the complex formalism introduced by Bourassa & Kantowski (1975, corrected by Bray 1984).
A particular case of homoeoidal symmetric lenses lies in the non-singular isothermal ellipsoid (NSIE) family of models for which the analytical treatment has been somewhat limited (Kormann & al. 1994). The use of the Fourier approach has made possible to derive a complete analytical treatment of the NSIE, i.e. the expressions of the deflection angle, the deflection potential, and the critical and caustic curves even off the axis (Wertz & Surdej, submitted to MNRAS in february 2014). This original result has allowed us to investigate and better understand this family of models. Furthermore, it is of great interest for mass distribution modeling and to rigorously determine the expected time delays between pairs of lensed images.
The previous analytical treatments mainly consisted of parametrical models for the deflector. An alternative way to grasp lenses consists in modeling their mass distribution using non-parametric models. With this aim in mind, we have proceeded as follows: we tessellate the lens plane with squared pixels, and associate to each of them a constant surface mass density. Making use of the Fourier approach, we have derived the expression of the deflection angle for the whole grid. This result contains the main advantage of the non-parametric models, i.e. to model any type of mass distribution without any preconception, and the usefulness of handling quantities which can be described with analytical functions.
