4. COSMOGRAPHY
The most important, and arguably the most difficult, application of
gravitational lenses is to cosmography - determining the geometry of
the universe on the largest scales. In this section, we consider
critically how the Hubble constant has recently been estimated using
Q0957+561, how the deceleration parameter of FRW
models may possibly
be measured, and how alternative cosmologies can currently be
constrained by observations.
As described in Section 3.4, the
difference in travel time t
from the source to the observer along two distinct rays is inversely
proportional to the Hubble constant H0, 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 +
zd)'
hat / 2c,
measuring ' from
the tangent point of the
wavefronts. The gravitational time delay is -(1 +
zd) (') /
c3, 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 H0.
Normalizing all angles to the image separation , we have
the scaling H0 = K2/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 H0.
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Figure 9. The upper panel uses wavefronts to show that the
geometrical time delay at the deflector is ' hat / 2c. The bottom
panel shows a transformation where (a) a constant density sheet
with convergence is
added, (b) the deflection due to the primary lens is reduced to
(1 - )hat, and (c) all
linear dimensions are expanded by a factor (1 - )-1. Since there
is no net time delay due to the sheet (the geometrical and
gravitational delays cancel), all observables, including the time
delay, remain unchanged in this transformation. Thus, in order to
determine the scale of the universe (i.e. Dd or
H0), either the convergence must be arbitrarily set to
zero (Scenario 1), or it must be independently estimated, e.g. by
measuring the velocity dispersion of the lens (Scenario 2).
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In the case of Q0957+561, the time delay appears to have been
convincingly measured at both radio and optical wavelengths
(Lehar 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 H0,
the geometry of the universe and especially the deceleration
parameter q0 and the distance ratio
Ds / Dds 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,
H0 will be uniquely determined. In
Q0957+561, assuming t = 1.48 yr,
this gives H0 = 61 ± 7 km s-1
Mpc-1 for
0 = 1
(q0 = 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
H0 = K(1-)2 / t. This reveals the
following fundamental degeneracy in the model: as long as is
undetermined, there is no unique solution for H0
(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 H0
(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 H0 ~ 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 H0
~ 40-70 km s-1 Mpc-1.
Scenario 3: Give up next any assumption on the
geometry of the universe, particularly knowledge of q0 or
Ds/Dds. Alternatively, let the
source redshift zs 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 Dd =
1300-2400 Mpc without any correction
factor applied to the measured , and Dd = 900-1700 Mpc
including the factor of (1.5)1/2. These distances can be
converted to estimates of H0 if a value of
q0 is assumed.
Scenario 4: Now include quadratic inhomogeneities
between the observer and the lens. The scalar angular diameter
distance Dd 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 Dd. 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 Dd or
H0.
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 ~ 4GM(1 + zd)ln( / c) /
c3 ~ 0.008
(M / 1010M)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 / (1010 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.
From the above discussion, we see that in general a gravitational lens
has the potential to provide important cosmographic information. The
technique has the virtue of being insensitive to mass fluctuations on
large scales because they can be adequately modeled by quadratic
lenses, as well as fluctuations on small scales whose effects are
likely to be limited. However, the method clearly 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 H0
in the range
15-80 km s-1 Mpc-1 (for t = 1.48 yr). A more
comprehensive investigation of allowed models is needed.
Looking to the future, it is clear that lenses that have no obvious
cluster surrounding the primary lens are preferable since one has the
option of applying Scenario 1. Also, lensed sources having more than
two images are likely to be superior to
Q0957+561. If the images have
resolved VLBI structure and one is able to measure complete
transformation matrices, then up to 18 constraints will be available
with a quadruply-imaged quasar, plus up to three relative time delays.
Potentially even better are the radio rings, where the number of
constraints is comparable to the number of resolution elements in the
multiply-imaged zones. However, to do useful cosmography, one
requires a time-variable core (to measure time delays), and a
well-identified lens with a measured redshift and velocity dispersion
(though the source redshift is not necessary, see Scenario 3). None
of the known quadruply-imaged quasars or radio rings satisfies all
these requirements yet.
Another serious uncertainty relates to how one interprets a measured
stellar velocity dispersion in the lens. Apart from the factor of
(1.5)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.