Determination of H0 with gravitational lenses Sjur Refsdal
Hamburger Sternwarte
Gojenbergsweg 112
21029 Hamburg
Abstract:
The determination of the Hubble parameter with the time delay method
based on gravitational lensing has now reached an accuracy comparable
to classical methods. Theoretical modelling based on optical and radio
observations of QSO 0957+561 now gives H0= 63 ±
12 km s-1 Mpc-1 where the error bars are 2{sigma}, in good
agreement with the results obtained with classical methods.
The possibility of determining H0 by making use of the time delay
in multiply imaged systems was first pointed out by Refsdal (1964,
1966). The wave front method used to calculate the time delay was
discussed in more detail in later papers (Chang & Refsdal 1976,
Kayser & Refsdal 1983 and Refsdal & Surdej 1994). Alternative and
useful methods to calculate the time delay were developed by Cooke &
Kantowski (1975), who separately considered the potential and geometrical
time delay, and by Schneider (1985) who made elegant use of Fermat's
principle.
The first gravitational lens system to be discovered was the double
QSO 0957+561 A,B (Walsh et al. 1979). After this first detection the
interest in gravitational lensing increased enormously and new systems
are now discovered every year. For the determination of H0
QSO 0957+561 is still the most interesting system.
From statistical analysis of extensive optical and radio monitoring
data of that system the time delay is now rather well determined to be
417 ±3 days (Vanderriest et al. 1989, Schild 1990, Pelt et al. 1996,
Kundic et al. 1997).
The major remaining source of error is then the uncertainty in the
mass distribution of the lens which is in this case
complicated by the fact that not only a galaxy acts as a lens but also
its surrounding cluster. The mass distribution can
therefore only be described with a relatively large number of
parameters. The number of constraints due to optical observations
(image positions, position and velocity dispersion of the lensing
galaxy and observations of the cluster) is not sufficient to model the
system reasonably. Several additional constraints can however be
obtained from radio observations of a jet which is seen in both
images. The most recent modelling based on optical and radio
observations gives H0= 63 ±12 km s-1 Mpc-1 (Grogin &
Narayan 1996a, 1996b, and Kundic et al. 1997). Very recently, however,
Bernstein et al. (1997) obtained more accurate relative positions of
the images and the lensing galaxy with new HST data and found that the
model of Grogin & Narayan must be reevaluated.
Among the many contributions leading up to the current situation
should be mentioned Borgeest & Refsdal (1984), Gorenstein et al.
(1988) and Falco et al. (1991). There are several other systems which
can be used for the determination of H0, but the accuracy is still
not comparable to that obtained from QSO 0957+561. P. Schechter will
discuss this in his contribution to this workshop.
References
Bernstein, G., Fischer, P., Tyson, J. A., Rhee, G. 1997,
ApJ 483, L79
Borgeest, U., Refsdal, S. 1984, A&A 141, 318
Chang, K., Refsdal, S. 1976, Colloques internationaux du
C.N.R.S. No.263, 369
Cooke, J. H., Kantowski, R. 1975, ApJ 195, L11
Falco, E. E., Gorenstein, M. V., Shapiro, I. I. 1991, ApJ
372, 364
Gorenstein, M. V., Falco, E. E., Shapiro, I. I. 1988, ApJ
327, 693
Grogin, A., Narayan, R. 1996a, ApJ 464, 92
Grogin, A., Narayan, R. 1996b, ApJ 473, 570 (Erratum zu 1996a)
Kayser, R., Refsdal, S. 1983, A&A 128, 156
Kundic, T., Turner, E. L., Colley, W. N., Gott III, J. R., Rhoads,
J. E. et al. 1997, ApJ 482, 75
Pelt, J., Kayser, R., Refsdal, S., Schramm, T. 1996, A&A 305, 97
Refsdal, S. 1964, MNRAS 128, 307
Refsdal, S. 1966, MNRAS 132, 101
Refsdal, S., Surdej, J. 1994, Rep. Prog. Phys. 57, 117
Schild, R. E. 1990, AJ 100, 1771
Schneider, P. 1985, A&A 143, 413
Vanderriest, C., Schneider, J., Herpe, G., Chevreton, M., Moles, M.,
Wlérick, G. 1989, A&A 215, 1
Walsh, D., Carswell, R. F., Weymann, R. J. 1979, Nature 279, 381