Nothing compares to the measurement of the Hubble constant in bringing
out the worst in astronomers. As we discussed in the previous section on
lens modeling, many discussions of lens models seem obfuscatory rather
than illuminating, and the
tendency in this direction increases when the models are used to
estimate *H*_{0}. In this section we discuss the
relationship between time delay measurements, lens models and
*H*_{0}. All results in the literature are
consistent with this discussion, although it may take you several days and a
series of e-mails to confirm it for any particular paper. The basic idea is
simple. We see images at extrema of the virtual time delay surface
(e.g. Blandford & Narayan
[1986],
Part 1) so the propagation
time from the source to the observer differs for each image. The differences
in propagation times, known as time delays, are proportional to
*H*_{0}^{-1} because
the distances between the observer, the lens and the source depend on the
Hubble constant (Refsdal
[1964]).
When the source varies, the variations appear in the images separated by
the time delays and the delays are measured by cross-correlating the
light curves. There are recent reviews of time delays and the Hubble
constant by Courbin, Saha & Schechter
([2002])
and Kochanek & Schechter
([2004]).
Portions of this section are adapted from Kochanek & Schechter
([2004])
since we were completing that review at about the same time as we presented
these lectures.

To begin the discussion we start with our standard simple model, the
circular power law lens from Section B.3.
As a circular lens, we see two images at radii
_{A} and
_{B} from the
lens center and we will assume that
_{A} >
_{B}
(Fig. B.20).
Image A is a minimum, so source variability will
appear in image A first and then with a time delay
*t* in the
saddle point image B. We can easily fit this data with an SIS lens model
since (see Eqn. B.21 and B.22) to find that
_{A} =
+ *b*
and _{B} = *b*
- where
*b* = (_{A}
+ _{B}) / 2 is
the critical (Einstein) radius of the lens and
=
(_{A} -
_{B}) / 2 is the
source position. The light travel time for each image relative to a
fiducial unperturbed ray is (see Part 1)

(B.93) |

where the effective potential
= *b*
for the SIS lens. Remember
that the distances are comoving angular diameter distances rather than
the more familiar
angular diameter distances and this leads to the vanishing of the extra
1 + *z*_{l} factor that appears in the numerator if you
insist on using angular diameter distances. The propagation time scales as
*H*_{0}^{-1}
10*h*^{-1}
Gyr because of the *H*_{0}^{-1} scalings of the
distances. After substituting our lens model, and differencing the
delays for the two images, we find that

(B.94) |

The typical deflection angle *b* ~ 3 × 10^{-6} radians
(so *R*_{A}^{2} ~ 10^{-11})
converts the 10*h*^{-1} Gyr propagation time into a time
delay of months or years that can be measured by a graduate student.
Naively, this result suggests that the problem of interpreting time
delays to measure *H*_{0} is a trivial problem in astrometry.

We can check this assumption by
using our general power-law models from
Section B.3 instead of an
SIS. The power-law models correspond to density
distributions
*r*^{-n}, surface densities
*R*^{1-n}
and circular velocities
*v*_{c}
*r*^{(2-n) / 2} of which the SIS model is
the special case with *n* = 2. These models have effective potentials

(B.95) |

As we discussed in the Section B.4.1 we
can fit our simple, circular two-image lens
with any of these models to determine *b*(*n*) and
(*n*)
(Eqn. B.66), which we can then substitute into the expression for the
propagation time to find that the time delay between the images is

(B.96) |

where we have expanded the result as a series in the ratio between the mean
radius of the images
<> =
(_{A} +
_{B}) / 2 and the
thickness of the radial annulus separating them
=
_{A} -
_{B}. While
the expansion assumes that
/
<> ~
/ *b*
is small, we can usually ignore higher order terms even when
/
<> is of order
unity. We now see that the time delay depends critically on the density
profile, with more centrally concentrated mass distributions (larger
values of *n*) producing longer time delays or implying larger
Hubble constants for a fixed time delay.

The other idealization of the SIS model, the assumption of a circular
lens, turns out
to be less critical. A very nice analytic example is to consider a
singular isothermal model with arbitrary angular structure in which
=
*bF*() /
2 where
*F*() is
an arbitrary function
of the azimuthal angle. The singular isothermal ellipsoid (Eqn. B.37) is
an example of this class of potential. For this model family,
*t* =
*t*_{SIS}
independent of the actual angular structure
*F*()
(Witt, Mao & Keeton
[2000]).