3.3. The Mass Model and Degeneracies

Most unknowns in the calculation of the time-delay can be measured from deep and sharp images (and spectra), but we have not paid much attention, so far, on the gravitational part of the time-delay. It depends on the mass surface density distribution () of the lensing object(s), which can not be measured directly. It has therefore to be modeled, and getting a "realistic" estimate of () is not trivial.

A good lens model should in principle be able to reproduce the observables with as few free parameters as possible. Ideally, this model should be unique. It is "asked" to reproduce the astrometry of the quasar images with a very high accuracy, as well as their flux ratios. In fact we will not consider the latter as a strong constrain. Even if flux ratios can be measured with an accuracy of the order of the percent, they are affected by extinction by dust and by microlensing events due to the random motion of stars in the lensing galaxy (see Schechter 2003; Wambsganss 2003). Flux ratios may therefore vary with time and wavelength. In addition, they should be corrected for the effect of the time-delay, i.e., each quasar image should be measured when the quasar is seen in the same state of activity. This can be done for quasars with known time-delays and well measured light curves. Curiously, such a correction is not much taken into account in the literature, even for quasars with known time-delays.

When modeling lensed quasars, one is asked, on the basis of a few (usually 2 or 4) quasar images, to model the whole two-dimensional gravitational potential of the lensing galaxy or galaxies. There is of course no unique solution to the problem: too few observational constraints are available and several mass models giving each one a different time-delay can reproduce a given image configuration, its astrometry and flux ratios. In other words, lens models are degenerate.

Degeneracies have been described and blamed abundantly in the literature for being the main source of uncertainty in lens models (see for example Saha 2000; Wucknitz 2002). Whatever precision on the measured astrometry and time-delay, several mass models will predict several time-delays and hence several H₀. One must devise techniques to break the degeneracies or find quasars that are less affected by them.

The main degeneracy one has to face in quasar lensing is called the mass sheet degeneracy: when adding to a given mass model, a sheet of constant mass density (i.e., constant convergence , as defined in the preceding chapters), one does not change any of the observables, except for the time-delay. The additional mass can be internal to the lensing galaxy (e.g., ellipticity does not change the total mass within the Einstein radius, but does change at the position of the images) or due to intervening objects along the line of sight. The exact mass introduced by the mass sheet increases the total mass of the lens, but one can re-scale it and locally change its slope at the position of the images. The result is that the image configuration does not change, but the convergence , at the position of the images does change, and modifies the time-delay. Therefore, knowledge of the the slope of the mass profile of the lensing galaxy, whether it be under the form of a model or of a measurement, is one of the keys to the determination of a "good" model.

Changing the slope of the lens will change at the position of the images, but adding intervening objects along the line of sight to the lens has a similar effect. A group or cluster of galaxies, located angularly close to the lens, will add its own contribution to the total mass density at the position of the images. If the group/cluster has a constant density , rescaling the total mass of the lensing galaxy by 1 / (1 - ) will leave the observed images configuration unchanged, as illustrated in Fig. 4, but will change the time-delay.

Figure 4. Two ways of obtaining a given image configuration. The left panel displays a system with four images, with an elliptical lens that introduces convergence and shear at the position of the images. On the right panel, is shown the same image geometry and flux ratios, but the lens is now circular. One would in principle only obtain two images with such a lens. The shear required to obtain four images is introduced by the nearby cluster. The mass density of the cluster is represented through its convergence . The mass of the main lens is scaled accordingly by 1 / (1 - ) so that the image configuration remains the same as in the left panel: the mass in the main lens and in the cluster are degenerate. If no independent measurement is available for at least one of the components (main lens or cluster), it is often difficult to know, from the modeling alone, what exactly are their respective contributions.

Adding convergence also modifies the shear , hence the ellipticity of the main lens. There are therefore several ways of reproducing a given combination of shear and convergence at the position of the images, as illustrated in Fig. 4. In the left panel of the figure, the shear , is produced only by the main elliptical lensing galaxy. In the right panel, the total shear is a combination of the lens-induced shear and of that of the nearby galaxy cluster. In principle, it is even possible to model a given system equally with either one single elliptical lens or with a completely circular lens and an intervening cluster responsible for an "external" source of shear.

Both types of degeneracies can be broken or, at least, their effect can be strongly minimized, by constraining in an independent way (1) the mass profile of the main lens, and (2) the total mass (and possibly also the radial mass profile) of any intervening cluster along the line of sight. This work can be done with detailed imaging, spectroscopy of all objects along the line of sight, and by using numerical multi-components models for the total lensing potential. This is the topic of the next section, illustrated through the example of the well studied quadruple PG 1115+080.