Galaxy clusters as lenses 40914.5.1 Hubble constant from time delays
As first noted by Refsdal in 1964 [59], time delay measurements can yield, in
principle, the Hubble parameter. Unfortunately, the use of this method requires
a reliable lens model. This introduces systematic uncertainties. Moreover, the
cosmological Fermat potential involves the density parameter 0 and(set equal
to zero in equation (14.109)). The dependence on 0 andis, however, not
strong, at least in some redshift domains(zs≤ 2 ,zd≤ 0. 5 ).
Measuring the time delay is not an easy task as the history of the
famous double QSO0957+561 demonstrates. Fortunately, the time delay for
QSO0957+561 is now well known:t= 417 ±3 days [60]. Modellings lead to
a best estimate ofH 0 61 km s−^1 Mpc−^1. For this example there are constraints
for modelling the lens; nevertheless, it is difficult to assess an error for the value
ofH 0.
Another example is the Einstein ring system B0218+357. A single galaxy is
responsible for the small image splitting of 0. 3 ′′. The time delay was reported to
be 12±3 days and the valueH 0 ∼70 km s−^1 Mpc−^1 was deduced. The ongoing
surveys will hopefully find new lenses that possess the desirable characteristics
for a reliable determination ofH 0.
Besides having the above mentioned problems, the determination ofH 0
through gravitational lensing offers also some advantages compared to the other
methods. It can be directly used for large redshifts(∼ 0. 5 )and it is independent
of any other method. Moreover, it is based on fundamental physics, while other
methods rely on models for variable stars (Cepheids), or supernova explosions
(type II) or empirical calibrations of standard candles (Tully–Fisher distances,
type I supernovae).
Finally, we note that lensing can also lead to bounds on the cosmological
constant. The volume per unit redshift of the universe at high redshifts increases
for a large. This implies that the relative number of lensed sources for a given
comoving number density of galaxies increases rapidly with. This can be used
to constrainby making use of the observed probability of lensing. Various
authors have used this method and came up with a limit≤ 0 .6 for a universe
with 0 +=1. It remains to be seen whether such bounds, based on lensing
statistics, can be improved.
14.6 Galaxy clusters as lenses
Galaxy clusters similarly to galaxies can act as gravitational lenses for more
distant galaxies. One classifies the observed lensing effects due to clusters into
two types:
(1) rich centrally condensed clusters produce sometimes giant arc when a
background galaxy turns out to be almost aligned with one of the cluster
caustics (strong lensing) (see, for instance, figure 14.1); and