**3.2. Constraints and Uncertainties**

Time-delays can be predicted from lens modeling, for any observed
image configuration and compared with the measured ones in order to
infer the value of H_{0}. The task requires detailed observations,
deep, and at high angular resolution, and a good mass model for the
lensing galaxy, as can be seen from the explicit expression for the
time-delay in equations (1-3). A full description of the calculation
can be found in
Schneider, Ehlers &
Falco (1992).
We only use the
result here to illustrate how observations help to achieve our goal.
As explained above, the total time delay is the sum of two
contributions, so that:

(1) |

Each contribution to the total time-delay writes as:

(2) |

(3) |

where *z*_{L} is the redshift of the lensing galaxy. As
illustrated in Fig. 3, the angle
(in 2D in
real cases) gives the position of the images on the plane of the sky and
is the
angular position of the source.

The Hubble parameter H_{0} is contained in the geometrical part
of the
time-delay, through the angular diameter distances to the lens
D_{L}, to the source D_{S} and through the distance
between the lens and the source, D_{LS}.

Equation (3), the gravitational part of the time-delay, depends only
on well known physical constants, and on the inverse of the 2D
Laplacian of the mass density profile in the lensing galaxy
(). In other
words, it strongly depends on the shape of
the 2D mass profile of the lens (ellipticity, position angle), and on
its slope. We will see later that the main source of uncertainty on
the gravitational part of the time-delay comes from the *radial
slope* of the mass distribution.

Several of the ingredients necessary to compute the time-delay can be precisely measured from observations. Although every lensed system has its own particularities, the positions of the lensed images defined by are usually the easiest quantities to constrain. With present day instrumentation, an accuracy of a few milli-arcsec is reached. The position of the lensing galaxy relative to the quasar images, when it is not double or multiple can be of the order of 10 milli-arcsec. As for the position of the source relative to the lens, it is usually free in lens models. No observation can constrain it.

In most cases, astrometry is not a major limitation to the use of
lensed quasars. However, image configurations that are very symmetric
about the center of the lens are more sensitive to astrometric errors
than assymetric configurations. Let us assume that
is very small compared with
(i.e.,
the source is almost
aligned with the lens and the observer). Lets then consider a doubly
imaged quasar with two images located at positions
_{1}
and
_{2}
away from the lens, and separated by
. The
geometrical time-delay between the two images is:

(4) |

If we now consider that the error on the image separation
is much
smaller than the error on the position of image
1 relative to the lens,
_{1},
we can approximate the error on the time-delay. Since the errors
*d*_{1} and
*d*
on
_{1}
and
are not much
correlated, they propagate on the time-delay as

(5) |

In symmetric configurations, where the lens is almost midway between
the images, 2|_{1}|
||, so
that the
denominator in equation (5) is zero or close to it, leading to large
relative errors on the time-delay, and hence on H_{0}, whatever mass
model is adopted for the lensing galaxy.

The advantage of symmetric configurations over assymetric ones is that they often have more than two lensed images (the source is more likely to be within the area enclosed by the radial caustic; see previous chapters on the basics of quasar lensing), offering the opportunity to measure several time-delays per system. The drawback is a larger sensitivity to astrometric errors.

Redshift information is also of capital importance in the calculation
of the time-delay. Time-delays are proportional to (1 +
*z*_{L}) as
seen in equation (4). The angular diameter distances also depend on
the redshift of the lens and source *z*_{L} and
*z*_{S}. Both
should therefore be measured carefully. Although the lens and source
redshifts are available for most know system, their measurement is not
as straightforward as one could expect. Given the small separation
between the lensed images and the high luminosity contrast between the
source and lens, obtaining a spectrum of the lens is often challenging
and may involve significant struggling with the data (e.g.,
Lidman et al. 2000).
In other cases, for example in systems discovered in the
radio, one faces the opposite situation: the optical counterpart of
the source is so faint that no spectrum can be obtained of it, while
the lens is well visible (e.g.,
Rusin et al. 2001).

Finally, the other cosmological parameters such as
_{}
and _{0}
also play a role in the calculation of the time-delay,
through the angular diameter distances. The dependence of the
distances on (_{},
_{0}) is
however very weak. In
addition, other methods (Supernovae, CMB) seem much better at pinning
down their values than quasar lensing does. One shall therefore use
the known values of
(_{},
_{0}) in
quasar lensing
and infer H_{0}, to which it is much more sensitive.