There are a variety of possible systematic errors which afflict
gravitational lens
determinations of the Hubble constant. The first of these, which we have
already mentioned,
is the degeneracy between *H*_{0} and a uniform density
sheet in the lens plane. For example,
suppose that there is a uniform density circular disk of matter covering
all the images.
This will act like a simple (Gaussian) converging lens and bring the
same rays that would
have met at the observer to a common focus closer to the lens. In other
words, the
potential variation is quadratic and so the contribution to the
gravitational delay is also
quadratic, just like the geometrical delay, from which it is therefore
indistinguishable. It
has been argued that, as any sheet must have positive mass density, we
can only set an upper bound on the Hubble constant
(Falco et al. 1985).

A second uncertainty is associated with the choice of cosmographic world
model. Let
us suppose that the universe is of homogeneous Friedmann-Robertson-Walker (FRW)
type so that it is parameterized by the current density parameter
_{0}. The angular
diameter distances to high redshift sources and lenses depend quite
sensitively upon _{0}.
However, the combination *D*_{d}*D*_{s} /
*D*_{ds}, relevant for the inferred Hubble constant, is
relatively insensitive. For example, in the 0957+561 system, *K* =
39.9*h*^{-1} days arcsec^{-2} for
an Einstein-de Sitter universe with
_{0} = 1, and *K*
= 42.9*h*^{-1} days arcsec^{-2} for an open
universe with _{0} =
0.1. If we take an additional step and introduce a cosmological
constant while keeping the universe flat, the change in *K* is even
smaller. For 0 < _{} < 1,
*K* attains a maximum of 41.7*h*^{-1} days
arcsec^{-2} in the
_{0} = 0.25,
_{} = 0.75 model. In a
lens system with higher redshifts, however, the choice of cosmological
model is more
important. We can thus use high-redshift lenses to constrain the density
parameter _{0} once
*H*_{0} has been measured. For example, in the
*z*_{s} = 3.62 lensed BAL quasar B1422+231,
the difference in *K* between
_{0} = 1 and
_{0} = 0.1 models
amounts to 24% (assuming the
lens redshift of *z*_{d} = 0.65, as reported by
Hammer et al. 1995).

Now let us turn to inhomogeneous cosmological models. If the universe
has an overall
mean density sufficient to allow it to follow FRW dynamics on the
average but with this
mass confined to small concentrated lumps, none of which intersect the
line of sight,
then individual ray congruences will be subject to zero convergence as
they cannot pass
through any matter. In this case, the angular diameter distance must be
changed to the *affine parameter*
(Press & Gunn 1973).
If, for example, the mass in an Einstein-De Sitter
universe were concentrated in large, distant lumps, then *K* would
change by over 10%
for B1422+231. However, this assumption is quite
unrealistic, because the combined
tidal influence of these lumps must, on the average, reproduce the
cosmography of a
homogeneous universe, and, in fact, it does. In what follows we assume
that our line of
sight is not special in this sense and that individual dark masses are
small enough that we can treat them as smoothly distributed.

The next type of systematic error is starting to be taken more seriously
and has broader
implications. This is the error introduced in the measurement of
*H*_{0} due to large-scale
structure distributed along the line of sight from the source to the
observer. A simple and
standard description of large-scale structure in the universe is to form
the power spectrum
of relative density fluctuations, *P(k)*, that have supposedly
grown from perturbations
similar to those that we observe in the microwave background
fluctuations. The short
wavelength perturbations [*k > k*_{m} ~ (10
Mpc)^{-1}] have grown to non-linear strength; the
long wavelength perturbations should still be in the linear regime with
*P(k)* *k*. In the
linear regime the potential fluctuations therefore satisfy
*k*^{-2}
*k*^{-2}[*k*^{3} *P(k)*]^{1/2} ~
constant. In fact, the potential fluctuation is not just constant with
linear scale but also with time and has a value
~ 3 x 10^{-5}
*c*^{2} as normalized to the fluctuations measured by COBE.

The perturbations that are most important for perturbing gravitational
lens images
are those for which *k*^{2} *P(k)* is maximized,
i.e. with *k ~ k*_{m} ~ (10 Mpc)^{-1}. As typical
angular diameter distances are *D* ~ 1 Gpc, these subtend angles ~
30', much larger
than the strong lensing regions, and we expect to see *N ~
k*_{m} *D* ~ 100 perturbations, of
both signs, along the line of sight. If we consider a single ray passing
through a roughly
spherical perturbation, it will be deflected through an angle
~
/ *c*^{2} and as there are
*N* such deflections adding stochastically, the total deflection
will
~ *N*^{1/2}
~ 1'. (It is
amusing that although that there has been so much trouble taken to point
HST to a small
fraction of a pixel, the universe has a pointing error of about the
width of WFPC chip!)
In addition, each fluctuation will introduce a propagation time
fluctuation
/ *k*_{m}
*c*^{3} ~
(
/ *c*^{2})(*D /
Nc*). These fluctuations add stochastically giving a total time
difference
relative to a homogeneous universe of
*t* ~
(
/ *c*^{2})(*D /
N*^{1/2}*c*) ~ 10^{4} yr. However, none
of this matters because there is no way to detect the deflection or time
delay of a single ray.

The situation changes when we consider two rays separated by a small
angle
(Blandford &
Jaroszynski 1981).
On passing though a single perturbation, the angular
separation will change by an
angle ~ *k*_{m}
(*D*)
(
/ *c*^{2}). Again,
summing the effects of *N*
such fluctuations stochastically gives an estimate of the total
fractional change in , a
measure of the strength of both the magnification and the shear
fluctuations. We obtain

(5.31) |

These fluctuations should be coherent over angles ~
(1/*k*_{m} *D*) ~ 30' on the sky. There
have been attempts to measure this signal using the images of
faintgalaxies (e.g.
Mould et al. 1994).

There is a subtlety when we consider the observable time fluctuations. If the
separation of a pair of rays
is much less than the angular scale of the fluctuation, we
can Taylor expand the potential. We have already dismissed the constant
term and
might expect the linear term to be observable. However, this is not the
case as can
be seen by noting that its effect simply deflects both rays through the
same angle, just
like a prism, and does not increase the optical path at all. We have to
expand to
quadratic order across the rays to obtain a total time difference due to
the fluctuations
of *t* ~
(
/ *c*^{2})
(*D / N*^{1/2} *c*) (*k*_{m}
*D*)^{2} ~
(
/ *c*^{2})
*N*^{3/2}
(*D*^{2} /
*c*). Now suppose that these two
rays are associated with a common source and observer in a gravitational
lens. The time
delay in an otherwise homogeneous universe would be *t* ~
*D*^{2} / *c*
and so the relative change in the arrival time will satisfy

(5.32) |

as above. Therefore to order of magnitude, we find that a few percent error in the derived value of the Hubble constant will be introduced by the effects of large-scale structure and the effect deserves computing seriously.

This problem has been addressed by Seljak (1994) and Bar-Kana (1996), following earlier papers by Kovner (1987) and Narayan (1991). They find three distinct effects:

*(a)* The images of the source, their positions and their shapes,
are subject to a global
shear transformation. This effect will generally be absorbed in the
quadratic contribution
to shear from the large-scale distribution of mass in the deflector plane.

*(b)* The images of the deflecting galaxies, their positions
relative to the images of the
source and their shapes will be subject to an additional
shear. Naturally this shear is
only caused by fluctuations between us and the deflector. This effect
means that there
is a cosmic uncertainty in the image positions which must be taken into
account when we make the models.

*(c)* There will be quadratic potential fluctuations acting on the
unperturbed rays that
cause stochastic gravitational delays that create additional direct
changes in the measured
lags. These will translate directly into errors in
*H*_{0}. There is an interesting observational
possibility here. We can contemplate performing deep redshift surveys in
the vicinity of
the most promising lenses and rather than treat these effects as random
errors, try to remove them directly.