Annu. Rev. Astron. Astrophys. 1994. 32: 319-70
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7.3. Effects of a Cosmological Constant

Due to the emerging, but still controversial, discrepancy between the lifetime of a matter-dominated Omega0 = 1 universe inferred from the Hubble constant, and the lifetime inferred by stellar evolution (see e.g. Demarque et al 1991) there has been interest in the possibility that the cosmological constant may be nonzero (see also Caroll et al 1992). This would allow Omega0 < 1 while retaining a "flat" cosmology and avoid the "Omega-problem" (see e.g. Kolb & Turner 1990). If a large vacuum energy drove an inflationary phase in the early universe, the idea that it may be nonzero today is not wildly implausible.

Note that a nonzero cosmological constant has little effect on the large-scale structure and dynamics of the Universe (eg. Markevitch et al 1991, Lahav et al 1991), and will have negligible effect on the recombination process and the visibility function. However, microwave background fluctuations will be significantly different in Lambda neq 0 models. This is because of several effects. Firstly, there is a change in angular scales. The angle at recombination corresponding to a proper length lambda can be approximated as theta appeq 30" Omega01/3 (1 - Omega0)-1/4 lambda(h-1 Mpc) (Blanchard 1984, Stelmach et al 1990). Hence, for the same normalization, the Lambda neq 0 model measures anisotropies that correspond to smaller angular scales than in the Lambda = 0 model. Secondly, since the growth of fluctuations is different in a Lambda neq 0 model, the potential fluctuations are no longer constant in time. Hence the Sachs-Wolfe temperature fluctuations are augmented by the integrated potential term (see Equation 9). This modifies the (scalar) Sachs-Wolfe formula (Górski et al 1992, Vittorio & Silk 1992, Sugiyama et al 1990, Hu & Sugiyama 1994, Stompor & Górski 1994). (The formula for the tensor mode contribution is essentially unchanged.) Generally the integrated Sachs-Wolfe effect for a flat Omega0 < 1 universe is less important than in the case of an open universe. Finally, the fluctuations (assuming they are linear) stop growing when the the cosmological constant becomes important, which happens at 1 + z ~ (Omega0-1 - 1)1/3. For a fixed power spectrum at z = 0 the potential in the early universe scales as Omega0 / D (Peebles 1984), where D is the growth factor and the extra factor of Omega0 comes from the potential Phi propto Omega0. The effects of geometry tend to compensate the integrated Sachs-Wolfe contribution in an open model. Consequently, in a flat Lambda-dominated universe, the effective value of n is less than unity on large scales. [Similar conclusions can also be drawn (Sugiyama & Sato 1992) for models in which the cosmological constant decays with time (Freese et al 1987, Ratra & Peebles 1988, Overduin et al 1993).]

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