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The strongest evidence for a positive Lambda comes from high-redshift SNe Ia, and independently from a combination of observations indicating that Omegam ~ 0.3 together with CMB data indicating that the universe is nearly flat. We will discuss these observations in the next section. Here we will start by looking at other constraints on Lambda.

The cosmological effects of a cosmological constant are not difficult to understand (Felton & Isaacman 1986; Lahav et al. 1991; Carroll, Press, & Turner 1992). In the early universe, the density of energy and matter is far more important than the Lambda 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 Omegam = 0.3, OmegaLambda = 0.7) the Lambda term finally becomes dominant. Around this redshift, the Lambda term almost balances the attraction of the matter, and the scale factor a ident (1 + z)-1 increases very slowly, although it ultimately starts increasing exponentially as the universe starts inflating under the influence of the increasingly dominant Lambda term. The existence of a period during which expansion slows while the clock runs explains why t0 can be greater than for Lambda = 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 gtapprox 2).

The observed frequency of such lensed quasars is about what would be expected in a standard Omega = 1, Lambda = 0 cosmology, so this data sets fairly stringent upper limits: OmegaLambda ltapprox 0.70 at 90% C.L. (Maoz & Rix 1993, Kochanek 1993), with more recent data giving even tighter constraints: OmegaLambda < 0.66 at 95% confidence if Omegam + OmegaLambda = 1 (Kochanek 1996). 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 ltapprox 1 (Steidel, Dickinson, & Persson 1994; Lilly et al. 1995; Schade et al. 1996). An alternative analysis by Im, Griffiths, & Ratnatunga (1997) of some of the same optical lensing data considered by Kochanek (1996) leads them to deduce a value OmegaLambda = 0.640.26+0.15, which is barely consistent with Kochanek's upper limit. Malhotra, Rhodes, & Turner (1997) presents evidence for extinction of quasars by foreground galaxies and claims that this weakens the lensing bound to OmegaLambda < 0.9, but this is not justified quantitatively. Maller, Flores, & Primack (1997) shows that edge-on disk galaxies can lens quasars very effectively, and discusses a case in which optical extinction is significant. But the radio observations discussed by Falco, Kochanek, & Munoz (1998), which give a 2sigma limit OmegaLambda < 0.73, are not affected by extinction. Recently Chiba and Yoshii (1999) have suggested that a reanalysis of lensing using new models of the evolution of elliptical galaxies gives OmegaLambda = 0.7+0.1-0.2, but Kochanek et al. (1999) (see especially Fig. 4) show that the available evidence disfavors the models of Chiba and Yoshii.

A model-dependent constraint appeared to come from simulations of LambdaCDM (Klypin, Primack, & Holtzman 1996) and OpenCDM (Jenkins et al. 1998) COBE-normalized models with h = 0.7, Omegam = 0.3, and either OmegaLambda = 0.7 or, for the open case, OmegaLambda = 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 (Klypin et al. 1999) 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 LambdaCDM model. Similar results have been found using simulations plus semi-analytic methods (Benson et al. 1999, but cf. Kauffmann et al. 1999).

Another constraint on Lambda from simulations is a claim that the number of long arcs in clusters is in accord with observations for an open CDM model with Omegam = 0.3 but an order of magnitude too low in a LambdaCDM model with the same Omegam (Bartelmann et al. 1998). 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 (Meneghetti et al. 1999; Flores, Maller, & Primack 1999).

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