© CAMBRIDGE UNIVERSITY PRESS 1999

1.3.3 Cosmological Constant

Inflation is the only known solution to the horizon and flatness problems and the avoidance of too many GUT monopoles. And inflation has the added bonus that at no extra charge (except the perhaps implausibly fine-tuned adjustment of the self-coupling of the inflaton field to be adequately small), simple inflationary models predict a near-Zel'dovich primordial spectrum (i.e., P_p(k) k^n_p with n_p 1) of adiabatic Gaussian primordial fluctuations - which seems to be consistent with observations. All simple inflationary models predict that the curvature is vanishingly small, although inflationary models that are extremely contrived (at least, to my mind) can be constructed with negative curvature and therefore ₀ 1 without a cosmological constant (see Section 1.6.6 below). Thus most authors who consider inflationary models impose the condition k = 0, or ₀ + = 1 where / (3H₀²). This is what is assumed in CDM models, and it is what was assumed in Figure 1.2. (Note that is used to refer only to the density of matter and energy, not including the cosmological constant, whose contribution in units is .)

The idea of a nonvanishing is commonly considered unattractive. There is no known physical reason why should be so small ( = 1 corresponds to ~ 10^-12 eV⁴, which is small from the viewpoint of particle physics), though there is also no known reason why it should vanish (cf. Weinberg 1989, 1996). A very unattractive feature of 0 cosmologies is the fact that must become important only at relatively low redshift - why not much earlier or much later? Also ₀ implies that the universe has recently entered an inflationary epoch (with a de Sitter horizon comparable to the present horizon). The main motivations for > 0 cosmologies are (1) reconciling inflation with observations that seem to imply ₀ < 1, and (2) avoiding a contradiction between the lower limit t₀ 13 Gyr from globular clusters and t₀ = (2/3)H₀^-1 = 6.52 h^-1 Gyr for the standard = 1, = 0 Einstein-de Sitter cosmology, if it is really true that h > 0.5.

The cosmological effects of a cosmological constant are not difficult to understand (Lahav et al. 1991; Carroll, Press, & Turner 1992). 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 ₀ = 0.3) the term finally becomes dominant. If it has been adjusted just right, can almost balance the attraction of the matter, and the expansion nearly stops: for a long time, 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 (see Figure 1.1). 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 at the redshift interval when the expansion slowed, and a correspondingly increased opportunity for lensing of quasars (which mostly lie at higher redshift z 2) by these galaxies.

The frequency of such 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. (Maoz & Rix 1993, Kochanek 1993), with more recent data giving even tighter constraints: < 0.66 at 95% confidence if ₀ + = 1 (Kochanek 1996b). 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 (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 1996b leads them to deduce a value = 0.64_-0.26^+0.15, which is barely consistent with Kochanek's upper limit. A recent paper - Malhotra, Rhodes, & Turner 1997 - presents evidence for extinction of quasars by foreground galaxies and claims that this weakens the lensing bound to < 0.9, but there is no quantitative discussion in the paper to justify this claim. 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 1997, which give a 2 limit < 0.73, will not be affected by extinction.)

Yet another constraint comes from number counts of bright E/S0 galaxies in HST images (Driver et al. 1996), since as was just mentioned these galaxies appear to have evolved rather little since z ~ 1. The number counts are just as expected in the = 1, = 0 Einstein-de Sitter cosmology. Even allowing for uncertainties due to evolution and merging of these galaxies, this data would allow as large as 0.8 in flat cosmologies only in the unlikely event that half the Sa galaxies in the deep HST images were misclassified as E/S0. This number-count approach may be very promising for the future, as the available deep HST image data and our understanding of galaxy evolution both increase.

A model-dependent constraint comes from a detailed simulation of CDM (Klypin, Primack, & Holtzman 1996, hereafter KPH96): a COBE-normalized model with ₀ = 0.3, = 0.7, and h = 0.7 has far too much power on small scales to be consistent with observations, unless there is unexpectedly strong scale-dependent antibiasing of galaxies with respect to dark matter. (This is discussed in more detail in Section 1.7.4 below.) For CDM models, the simplest solution appears to be raising ₀, lowering H₀, and tilting the spectrum (n_p < 1), though of course one could alternatively modify the primordial power spectrum in other ways.

Figure 1.2 shows that with 0.7, the cosmological constant does not lead to a very large increase in t₀ compared to the Einstein-de Sitter case, although it may still be enough to be significant. For example, the constraint that t₀ 13 Gyr requires h 0.5 for = 1 and = 0, but this becomes h 0.70 for flat cosmologies with 0.66.