The need to model the gravitational potential of the lens is the aspect of interpreting time delays that creates the greatest suspicion. The most extreme view is that it renders the project "hopeless" because we will never be able to guarantee that the models encompass the degrees of freedom needed to capture all the systematic uncertainties. In order to address these fears we must show that we understand the specific properties of the gravitational potential determining time delays and then ensure that our parameterized models include these degrees of freedom.

The examples we considered in Section 2
illustrate the basic physics of time delays, but an extensive catalog of
(non)parametric models demonstrating
the same properties may not be convincing to the skeptic. We will instead
show, using standard mathematical expansions of the potential,
which properties of the lens galaxy are required to understand time delays
with accuracies of a few percent. While we can
understand the results of all models for the time delays of gravitational
lenses based on this simple theory, full numerical models should probably
be used for most detailed, quantitative analyses. Fortunately,
there are publically available programs for both the parametric and
nonparametric approaches.
^{(3)}
Our analysis uses the geometry of the schematic lens shown in
Figure 1. The two
images define an annulus bounded by their radii, *R*_{A}
and *R*_{B}, and with an interior region
for *R* < *R*_{B} and an exterior region for
*R* > *R*_{A}.

The key to understanding time delays comes from
Gorenstein, Falco, & Shapiro
(1988; see also
Saha 2000),
who showed that the time delay of
a circular lens depends only on the image positions and the
*surface density
( R) in the
annulus between the images*.
The mass of the interior region is implicit in the image positions
and accurately determined by the astrometry. From Gauss' law, we
know that the radial distribution of the mass in the interior
region and the amount or distribution of mass in the exterior
region is irrelevant. A useful approximation is
to assume that the surface density in the annulus can be

(7) |

Thus, the time delay is largely determined by the average density
<>, with
only modest corrections from the local shape of the surface density
distribution even when
*R* /
<*R*> 1. For
example, the second-order expansion is exact for an SIS lens
(<> = 1/2,
= 2) and
reproduces the time delay of a point mass lens
(<> = 0) to better
than 1% even when
*R* /
<*R*> = 1. This local model also explains the time delay scalings
of the global power-law models we discussed in
Section 2. A
*r*^{-} global power law has surface density
<> = (3 -
) / 2 near the
Einstein ring, so the leading term of the time delay is
*t* =
2
*t*_{SIS}(1 -
<>) =
( - 1)
*t*_{SIS},
just as in Equation (6).

The time delay is not determined by the global structure of the radial density profile but rather by the surface density near the Einstein ring.

Gorenstein et al. (1988)
considered only circular lenses, but a multipole expansion allows us to
understand the role of angular structure
(Kochanek 2002).
An estimate to the same order as in
Equation (7) requires only the quadrupole moments of the
regions interior and exterior to the annulus, provided the strengths
of the higher-order multipoles of the potential have the same order
of magnitude as for an ellipsoidal density distribution.
^{(4)}
This approximation can fail for the lenses in clusters (see
Section 4).
The complete expansion for
*t* when the
two quadrupole
moments have independent amplitudes and orientations is not very
informative. However, the leading term of the expansion when the two
quadrupole moments are aligned illustrates the role of angular structure.
Given an exterior quadrupole (i.e., an external shear) of amplitude
_{ext}
and an interior quadrupole of amplitude
_{int}
sharing a common axis
_{}, the
quadrupole potential is

(8) |

if we define the amplitudes at radius <*R*>. For images at
positions *R*_{A}(cos
_{A},
sin _{A}) and
*R*_{B}(cos
_{B},
sin _{B})
relative to the lens galaxy
(see Fig. 1),
the leading term of the time delay is

(9) |

where
_{AB} =
_{A} -
_{B} and
*f*_{int} =
_{int}
/ (_{ext} +
_{int})
is the fraction of the quadrupole due to the interior quadrupole moment
_{int}.
We need not worry about the possibility of a
singular denominator - successful global models of the lens
do not allow such configurations.

A two-image lens has too few astrometric constraints to fully
constrain a model with independent, misaligned internal and external
quadrupoles. Fortunately, when the lensed images lie on opposite
sides of the lens galaxy
(
_{AB}
+
,
|| << 1),
the time delay becomes insensitive to the quadrupole structure.
Provided the angular deflections are smaller than the radial
deflections
(||<*R*>
*R*), the
leading term of the time delay reduces to the result for a circular lens,
*t*
2
*t*_{SIS}(1 -
<>). There is,
however, one limiting case to remember. If the images and the lens
are colinear, as in a spherical lens, the component of the shear
aligned with the separation vector acts like a contribution to
the convergence. In most lenses this would be a modest additional
uncertainty - in the typical lens these shears must be small,
the sign of the effect should be nearly random, and it is only
a true degeneracy in the limit that everything is colinear.

A four-image lens has more astrometric constraints and can
constrain a model with independent, misaligned internal and external
quadrupoles. The quadrupole moments of the observed lenses are dominated
by external shear, with *f*_{int}
1/4 unless there
is more than
one lens galaxy inside the Einstein ring. The ability of the
astrometry to constrain *f*_{int} is important because the
delays depend strongly on *f*_{int} when the images do not
lie on opposite sides of the galaxy. If external shears dominate,
*f*_{int} 0
and the leading term of the delay becomes
*t*
2
*t*_{SIS}(1 -
<>)
sin^{2}
_{AB}/2.
If the model is isothermal, like the
=
*rF*() models we
considered in Section 2, then
*f*_{int} = 1/4 and we again
find that the delay is independent of the angle, with
*t*
2
*t*_{SIS}(1 -
<>).
The time delay ratios in a four-image lens are largely determined
by the angular structure and provide a check of the potential model.

In summary, if we want to understand time delays to an accuracy competitive
with studies of the local distance scale (5%-10%), the only important
variable is the surface density
<> of the lens in
the annulus between
the images. When models based on the same data for the time delay and
the image positions predict different values for *H*_{0},
the differences can always be understood as the consequence of different
choices for
<>. In parametric
models
<> is adjusted by
changing the central concentration of the lens (i.e., like
in the global
power-law models), and in the nonparametric models of
Williams & Saha
(2000)
it is adjusted directly. The expansion models of
Zhao & Qin (2003a,
b)
mix aspects of both approaches.

^{3} The *gravlens* and *lensmodel*
(Keeton 2003,
cfa-www.harvard.edu/~castles) packages include a very
broad range of parametric models for the mass distributions of lenses,
and the *PixelLens* package
(Williams & Saha
2000,
ankh-morpork.maths.qmw.ac.uk/~saha/astron/lens/pix/)
implements a nonparametric approach.
Back.

^{4} If the quadrupole potential,
_{2}
cos 2,
has dimensionless amplitude
_{2},
then it produces ray deflections of order
*O*(_{2}
*b*)
at the Einstein ring of the lens. In a four-image lens the quadrupole
deflections are comparable to the thickness of the annulus, so
_{2}
*R* /
<*R*>. In a two-image lens they are smaller than
the thickness of the annulus, so
_{2}
*R* /
<*R*>. For an ellipsoidal density distribution, the
cos(2*m*) multipole
amplitude scales as
_{2m} ~
_{2}^{m}
(*R* /
<*R*>)^{m}. This allows
us to ignore the quadrupole density distribution in the annulus and
all higher-order multipoles. It is important to remember that
potentials are much rounder than surface densities [with relative
amplitudes for a
cos(*m* ) multipole
of roughly *m*^{-2}:*m*^{-1}:1 for
potentials:deflections:densities], so the
multipoles relevant to time delays converge rapidly even for very
flat surface density distributions.
Back.