Annu. Rev. Astron. Astrophys. 1994. 32:
319-70
Copyright © 1994 by Annual Reviews. All rights reserved |

**7.3. Effects of a Cosmological Constant**

Due to the emerging, but still controversial, discrepancy between the
lifetime of a matter-dominated
_{0} = 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
_{0} < 1
while retaining a "flat" cosmology and
avoid the "-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
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 can
be approximated as
30"
_{0}^{1/3}
(1 - _{0})^{-1/4}
(*h*^{-1}
Mpc)
(Blanchard 1984,
Stelmach et al 1990).
Hence, for the same normalization, the
0 model measures
anisotropies that correspond to smaller angular scales than in the
= 0 model. Secondly,
since the growth of fluctuations is different in a
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
_{0} < 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* ~ (_{0}^{-1} - 1)^{1/3}. For a fixed
power spectrum at *z* = 0 the potential in the early universe scales as
_{0} /
*D* (Peebles 1984),
where *D* is the growth factor and the extra factor
of _{0}
comes from the potential
_{0}. The
effects of geometry tend
to compensate the integrated Sachs-Wolfe contribution in an open
model. Consequently, in a flat
-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).]