The cosmological effects of a cosmological constant are not difficult
to understand
[42,
74,
21].
In the early universe, the density of energy
and matter is far more important than the
term on the
r.h.s. of the Friedmann equation. But the average matter density
decreases as the universe expands, and at a rather low redshift
(z ~ 0.2 for
m = 0.3,
= 0.7) the
term
finally becomes dominant. Around this redshift, the
term
almost balances the attraction of the matter, and the scale factor
a
(1 +
z)-1 increases very slowly, although it ultimately
starts increasing exponentially as the universe starts inflating under
the influence of the increasingly dominant
term. The
existence of a period during which expansion slows while the clock
runs explains why t0 can be greater than for
= 0, but this
also shows that there is an increased likelihood of finding galaxies
in the redshift interval when the expansion slowed, and a
correspondingly increased opportunity for lensing by these galaxies of
quasars (which mostly lie at higher redshift z
2).
The observed frequency of such optical lensed quasars is about what
would be expected in a standard
= 1,
= 0 cosmology, so
this data sets fairly stringent upper limits:
0.70 at 90% C.L.
[81,
69],
with more recent data giving
even tighter constraints:
< 0.66 at 95%
confidence
if
m +
= 1
[70].
This limit could
perhaps be weakened if there were (a) significant extinction by dust
in the E/S0 galaxies responsible for the lensing or (b) rapid
evolution of these galaxies, but there is much evidence that these
galaxies have little dust and have evolved only passively for
z
1
[120,
78,
112].
An alternative analysis
[58]
of some of the same optical lensing data gives a value
=
0.64-0.26+0.15. My group
[80]
(cf. [7])
showed that edge-on disk galaxies can lens quasars very effectively,
and discussed a case in which optical extinction is significant. But
the radio observations discussed by
[39],
which give a 2
limit
< 0.73, are not
affected by
extinction, so those are the ones quoted in the Table above. Recently
a reanalysis
[25]
of lensing using new models of the evolution
of elliptical galaxies gave
=
0.7+0.1-0.2, but Kochanek et al.
[71]
(see especially Fig. 4) show that the
available evidence disfavors such models.
A model-dependent constraint appeared to come from simulations of
CDM
[67]
and OpenCDM
[60]
COBE-normalized models with h = 0.7,
m = 0.3, and
either
= 0.7 or, for the open
case,
= 0.
These models have too much power on small scales to be consistent with
observations, unless there is strong scale-dependent antibiasing of
galaxies with respect to dark matter. However, recent high-resolution
simulations
[68]
find that merging and destruction of
galaxies in dense environments lead to exactly the sort of
scale-dependent antibiasing needed for agreement with observations for
the
CDM model. Similar
results have been found using simulations plus semi-analytic methods
[8]
(but cf.
[62]).
Another constraint on
from simulations is a claim
[6]
that the number of long arcs in clusters is in accord with
observations for an open CDM model with
m
= 0.3 but an order of
magnitude too low in a
CDM
model with the same
m. This
apparently occurs because clusters with dense cores form too late in
such models. This is potentially a powerful constraint, and needs to
be checked and understood. It is now known that including cluster
galaxies does not alter these results
[83,
44].