Annu. Rev. Astron. Astrophys. 1992. 30: 499-542 Copyright © 1992 by Annual Reviews. All rights reserved

3.1 Expansion Dynamics

If a 1/(1 + z) R/R₀ is the expansion factor relative to the present (z being the redshift), and if H₀t is a dimensionless time variable (time in units of the measured Hubble time 1/H₀), then Equation 1 can be rewritten in terms of measurable quantities as

9.

Note that _M and here serve as constants that parametrize the past (or future) evolution in terms of quantities at the present epoch. Equivalently, it was formerly common to parametrize the evolution by _M (or _{0 M}/2) and the deceleration parameter q₀ = - (R/ ²)₀. Equation 9 then readily yields the relation

10.

We will often use the parametrization _M and _{tot M} + = 1 - _k, since it is _tot < 1 (> 1) that makes the universe spatially open (closed) - a fundamental issue in cosmology. For different assumed values of _M and _tot (or any other parametrization) one gets qualitatively different expansion histories. Figure 1 displays the various regimes. Felten & Isaacman (1986) show graphs of a() for various values of _M and .

Figure 1. Qualitative behavior of cosmological models in the (M, tot) plane. Flat models, with tot = 1 and non-zero , are on the vertical line ACE. Models with = 0 lie on the diagonal line ABD. Use this figure as a ``finding chart'' for Figures 4, 7, and 9.

Qualitatively, the effect of a non-zero can be described as follows: Looking from now towards the future, a positive value of (or ) tries to drive the universe towards unbounded exponential expansion - asymptotically becoming a DeSitter spacetime. It can fail at this only if the matter density _M is so large as to cause the universe to recollapse before it reaches a sufficiently large size for the -driven term (which scales asymptotically as a² in Equation 9) to become significant - the narrow wedge in the upper-right corner of Figure 1. A universe fated to recollapse has some value a greater than 1 (the present value), such that the right hand side of Equation 9 vanishes. Some manipulation of the resulting cubic equations (Glanfield 1966, Felten & Isaacman 1986) yields an analytic formula for the boundary between recollapsing and perpetually expanding universes in the (_M, ) plane (see Figure 1): Unbounded expansion occurs when

11.

Otherwise the universe recollapses. In particular, negative implies inevitable recollapse, even for spatially open universes, because the effect of is in the same direction as gravity (attraction) rather than opposing it (repulsion).

For large, positive values of , the universe has a turning point in its past, that is, it collapsed from infinite size to a finite radius and is now reexpanding. This occurs when

12.

where ``coss'' is defined as being cosh when <sub>M</sub> < 1/2 and cos when <sub>M</sub> > 1/2. (The join at <sub>M</sub> = 1/2 is perfectly analytic. The need for two formulas to represent a single function is an artifact of solving cubic equations. Here and below it is sometimes useful to use the identities sinh<sup>-1</sup>x = ln[x + (x<sup>2</sup> + 1)<sup>1/2</sup>] and cosh<sup>-1</sup> x = ln[x + (x<sup>2</sup> - 1)<sup>1/2</sup>].) The redshift z<sub>c</sub> of the ``bounce'' [which is the maximum redshift of any object in the universe, since the universe never gets smaller than a = (1 + z_c)^-1] satisfies

13.

(see, e.g. Börner & Ehlers 1988). Inequality 13 can be solved for z_c, giving

14.

where ``coss'' is as defined above. In general, such ``bounce'' cosmologies are ruled out by the mere existence of high redshift quasars and (even more strongly) by the cosmic microwave background (see Section 4.1).

First noted by Lemaitre (1931), so-called ``hesitating'' or ``loitering'' universes occur when is close to, but barely outside, the bounce region of Equation 12. These are big-bang universes that are now expanding, exponentially in fact, but formerly had an epoch of indecision about whether to recollapse (from their matter content) or to continue expanding (due to their large positive cosmological constant). They thus spent a period of proper time loitering at a nearly constant value of a. (The closer Equation 12 is to an equality, the longer they coast.) The redshift of the loiter satisfies Equations 13 and 14 as equalities. This redshift is plotted in Figure 1 as a parameter along the loitering boundary. One sees that, analogously with bouncing universes, a high redshift loiter requires unreasonably small _M today. The present value in a universe that had a loitering phase is related to z_c (or _M) by

15.

(cf Equations 12 and 14).

In view of the above arguments, and the observations described in Section 4.1, it should be no surprise that universes with large, positive values for are presently out of fashion. We think they will remain so. Universes with _M > 1 are of course out of fashion, since all evidence is that there is a ``missing mass problem,'' and not an ``excess mass problem.'' Our attention henceforth will therefore focus on the ``fashionable'' region in Figure 1, bounded approximately by 0 < _M < 1 and 0 < _tot < 1. In this region the big questions are: (a) Is _tot exactly equal to 1, as is required by inflationary scenarios? And, (b) if it is equal to 1, is _tot made of _M (cold matter), (vacuum energy), or some more exotic form of matter (Peebles 1984, M.S. Turner 1991)?

Figure 2 shows the past and future expansion history of the models of Table 1, found by integrating Equation 9. Model A shows the familiar t^2/3 expansion law. Nearly empty (non-) models B and D show nearly identical histories, except close to a = 0 where B's larger matter content has an effect. Models C and E - flat models with a cosmological constant - have nearly identical future histories, since both have already entered their exponential expansion phase. Model E, being emptier of matter, shows a longer exponential phase to the past, while C's matter content asserts itself more readily and drives the expansion to a more recent big bang (a = 0).

Figure 2. Expansion history of the five models A-E shown in Table 1 and Figure 1. -dominated models like C and E have already entered an exponential expansion phase; also, they are older than the open = 0 models B and D.