### 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.

### 4.1 Hubble Constant

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.

 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).

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.