Annu. Rev. Astron. Astrophys. 1992. 30:
311-358 Copyright © 1992 by . All
rights reserved |

*t*
from the source to the observer along two distinct rays is inversely
proportional to the Hubble constant *H*_{0}, if all the relevant
deflection angles are known. A simple geometrical construction
explains this result. Consider two spherical wavefronts, one
emanating from a point source, the other converging on the observer,
touching each other at the lens plane
(Figure 9). Images are located at
isolated points on the lens plane where the deflection angle
hat equals the angle
between the two wave normals. The
geometrical part of the time delay is expressible as (1 +
*z*_{d})'
hat / 2*c*,
measuring ' from
the tangent point of the
wavefronts. The gravitational time delay is -(1 +
*z*_{d}) (') /
*c*^{3}, where the potential is proportional to the mass of
the deflector, which in turn scales as ^{2}', where is
a characteristic velocity in the lens. The total time delay is thus
proportional to '
and hence inversely proportional to *H*_{0}.
Normalizing all angles to the image separation , we have
the scaling *H*_{0} = *K*^{2}/*t*, where the constant of
proportionality *K* depends upon the lens model
(Refsdal 1964b,
Dyer & Roeder 1980,
Kayser & Refsdal
1983,
Borgeest 1983,
Gaskell 1985,
Falco *et
al.* 1985,
Kayser 1986,
1990,
Narayan 1991).
Thus, given a reliable time delay measurement and a
well-constrained lens model, one can estimate *H*_{0}.

*et
al.* 1992a,
Press *et
al.* 1992a,
b,
cf also
Florentin-Nielsen 1984,
Falco *et
al.* 1990,
Beskin &
Oknyanski 1992),
giving *t* = 1.48 ±
0.03 yr (though Vanderriest *et al.* 1989 and Schild 1990 obtain
*t* = 1.1 yr). A
widely used model of this system is that developed by
Falco *et
al.* (1991a),
which employs five parameters to characterize the
lens; these consist of the one-dimensional velocity dispersion
of the lens (modified by
an additional convergence
as described below), two ``shape''parameters of the lens, namely an
angular core radius and a dimensionless compact core mass, and two
parameters describing the shear due to other mass in the lens
plane such as the surrounding cluster. We comment below on the
reliability of this model, but first we discuss a number of scenarios
which incorporate various levels of assumptions about the lensing mass
and the geometry of the universe. Although the following discussion
is focused on Q0957+561 and the
Falco *et
al.* (1991a)
model of this object, most
of the arguments will be valid also for other lensed quasars (such as
Q1115+080,
Narasimha *et
al.* 1992)
for which time delays may become available in future.

**Scenario 1**: Assume that, except for the scale *H*_{0},
the geometry of the universe and especially the deceleration
parameter *q*_{0} and the distance ratio
*D*_{s} / *D*_{ds} are exactly
known. Assume further that the model includes all relevant mass in
the lens plane; in particular, assume that there is no dark matter
that might contribute an extra convergence . [The shear
due to dark
matter could be included in the model (cf
Kayser 1990),
as Falco *et al.* have done for Q0957+561.] Two
deductions can then be made. First, the model will uniquely predict
the velocity parameter
of the lens. In Q0957+561, the Falco
*et al.* model gives = 390 km s^{-1}. Secondly, once
*t* is measured,
*H*_{0} will be uniquely determined. In
Q0957+561, assuming *t* = 1.48 yr,
this gives *H*_{0} = 61 ± 7 km s^{-1}
Mpc^{-1} for
_{0} = 1
(*q*_{0} = 1/2). The result is inversely proportional to
*t*.

**Scenario 2**: Allow now for an unknown amount of convergence
due to smooth dark
matter in the lens plane, which in
Q0957+561 would be due to the mass associated with
the cluster. This is equivalent to adding a quadratic lens with
(Equation 5) and will effectively
reduce the curvature of one of the two wavefronts discussed above,
say the source-lens wavefront (Figure 9). One
then finds that
*H*_{0} = K(1-)^{2} / *t*. This reveals the
following fundamental degeneracy in the model: as long as is
undetermined, there is no unique solution for *H*_{0}
(Falco *et
al.* 1985,
Gorenstein *et
al.* 1988).
There are two possibilities now.
First, since 0
(dark matter has positive density), one
can still obtain an *upper bound* on *H*_{0}
(Borgeest &
Refsdal 1984,
Falco *et
al.* 1985,
Kovner 1987c),
which for
Q0957+561 is the
result 61 ± 7 km s^{-1} Mpc^{-1} given under Scenario 1.
Secondly, it is easy to verify that ^{2} / (1-) is a
constant. Therefore, the degeneracy may be broken by obtaining
independently. The
velocity dispersion of the stars in
Q0957+561 has been measured by
Rhee (1991)
to be 303 ± 50 km s^{-1}. If the parameter in the Falco *et al.* model is
set equal to this, then one obtains *H*_{0} ~ 25 - 50
km s^{-1} Mpc^{-1} for *t* = 1.48 yr. However,
if the stars
are more centrally concentrated than the dark mass in the galaxy,
then could be larger
than the measured velocity dispersion
by a factor of up to (1.5)^{1/2}
(Turner *et
al.* 1984,
Kochanek 1991c,
Roberts *et
al.* 1991),
in which case *H*_{0}
~ 40-70 km s^{-1} Mpc^{-1}.

**Scenario 3**: Give up next any assumption on the
geometry of the universe, particularly knowledge of *q*_{0} or
*D*_{s}/*D*_{ds}. Alternatively, let the
source redshift *z*_{s} be
unavailable. Also, allow for large-scale mass inhomogeneities in the
line-of-sight between the lens and the source; in
Q0957+561, for
instance, there is evidence of a second cluster at redshift 0.5
(Garrett 1992).
However, assume that the additional inhomogeneities
are quadratic, i.e. that each is completely described by a convergence
and a shear (cf Equation 5). In this case, all the additional
uncertainties get absorbed into the parameters and
that have been introduced to describe the dark matter in the lens
plane. As a result it can be shown that one is still able to measure
the *angular diameter distance to the lens*
(Narayan 1991). For
Q0957+561, one obtains *D*_{d} =
1300-2400 Mpc without any correction
factor applied to the measured , and *D*_{d} = 900-1700 Mpc
including the factor of (1.5)^{1/2}. These distances can be
converted to estimates of *H*_{0} if a value of
*q*_{0} is assumed.

**Scenario 4**: Now include quadratic inhomogeneities
between the observer and the lens. The scalar angular diameter
distance *D*_{d} then needs to be generalized to the complex D
introduced in Section 3.3.
Ignoring rotation which factors out, this
means one needs three parameters to describe the mapping between
angles at the observer
and displacements at
the lens. Given a sufficiently well-constrained lens
(Q0957+561 is
probably inappropriate), one could in principle fit these parameters
while at the same time fitting the lens model
(Kovner 1987c,
Narayan 1991).
One can thus solve for D_{d}. Of course,
this characterizes only the local line-of-sight to the particular
lens
(Alcock &
Anderson 1985,
1986),
and one needs several good lenses
to obtain a global estimate of *D*_{d} or
*H*_{0}.

**Scenario 5**: Suppose there are significant levels of
small-scale inhomogeneity in the universe that are not consistent with
a quadratic model. Alternatively, suppose the dark matter in the lens
plane is not smooth but is lumpy on scales smaller than the image
separations. Even in this pessimistic scenario, the lens modeling is
still constrained by the requirement that the mass density be positive
and by the need not to create additional images in the field. How
extreme a Hubble constant can still be tolerated by the observations?
The answer depends to some extent on the subjective question of how
much latitude one allows oneself in designing an extreme model.
However, it would appear that even with considerable freedom one
cannot drastically modify the results. For instance, suppose one
considered adding a mass perturbation, say a (dwarf) galaxy of mass
*M* and angular size _{c}, at the position of image B in
Q0957+561. The time delay of this image will be
increased by an amount ~ 4*GM*(1 + *z*_{d})ln( / _{c}) */
c*^{3} ~ 0.008
(*M* / 10^{10}*M*_{})ln( / _{c}) yr,
the dominant contribution coming from
the gravitational component. However, the mass can be bounded above
by the requirement that the B VLBI image not be excessively magnified
or distorted. This is measured by the ratio of the perturbing galaxy
surface density to the critical density, ~
0.2(*M* / (10^{10} *M*_{})(_{c})^{-2}. For a perturbing
galaxy mass *M* that does not create extra images or significantly
modify the image positions or transformation matrix, only a small
change in the relative time delay will be allowed and hence only a
modest increase in the true Hubble constant will result.
A further encouraging
factor is that the relatively undistorted shapes of the long arcs in
Abell 370 and other clusters
(Section 2.2) argue against
significant levels
of small-scale mass fluctuations in these cases. If these clusters
and lines-of-sight are typical, then small-scale distortions must be
unimportant in the majority of gravitational lenses.

*is* sensitive
to the details of the mass distribution on the scale of the image
separation, i.e. to the particular parametrized form of the lens model
employed, and this is its chief weakness. In
Q0957+561, there are
effectively five constraints from the observations, two from the
relative image position, and three from the relative magnification
matrix (technically four constraints, but one of these is poorly
determined because the VLBI images are not well-resolved perpendicular
to the jet). Since two parameters are used to describe the shear of
the cluster, one is left with only three parameters for the main
galaxy, of which one is the parameter combination ^{2} / (1-). Although the Falco *et
al.* model appears to be quite robust,
nevertheless,
Kochanek (1991c)
has found other models that fit the
observations equally well. These give estimates of *H*_{0}
in the range
15-80 km s^{-1} Mpc^{-1} (for *t* = 1.48 yr). A more
comprehensive investigation of allowed models is needed.

^{1/2} mentioned above, the velocity distribution may also be
anisotropic (e.g.
Binney &
Tremaine 1987,
Foltz *et
al.* 1992),
which would further complicate the interpretation. Yet another problem is
that some of the relative magnifications observed at optical
wavelengths may be affected by microlensing
(Kayser 1990),
leading to errors in the derived lens model. However, this is probably not a
consideration for the radio transformation matrix
(Falco *et
al.* 1991b).
Even at radio wavelengths, the core and the jet regions of
Q0957+561 are known to have different brightness
ratios. This could
be interpreted as evidence for gradients in the lens magnification
across the source
(Garrett *et
al.* 1991),
which will provide additional
constraints, enabling the galaxy model to be improved. However, if
the discrepancy is due to some kind of microlensing, say due to an
intergalactic compact object, then it increases the uncertainties in
the model.