**3.1. Closed universe models ( = 1)**

Consider a particle moving with
negative total energy under the influence of the potential
*V*(*a*) shown in fig. (2),
then the following situations arise (the one dimensional particle
coordinate is equivalent to the value of `*a*' - the expansion
factor.)

(i) *Oscillating models*:
The particle moves from left to right (starting from *a* = 0)
but with insufficient energy to surmount the potential barrier.
Consequently the expansion factor *a*(*t*) first
increases then decreases describing a universe which, after expanding,
contracts into a singularity. Such models are called *oscillating
models of the first kind*
[142].

(ii) *Bouncing models*:
The particle moves from right to left (starting from
*a* = ) again with
insufficient energy to surmount the potential,
in this case *a*(*t*) first decreases then increases and the
universe rebounds after collapsing without ever reaching a singular state.
Such models are called *bouncing models* or
*oscillating models of the second kind*,
an example of such a model is provided by the complete de Sitter
space-time

(10) |

where - < *t*
< , 0
,
0
,
0
2.

(iii) *Static Einstein Universe (SE)*:
The particle is placed at the top of the potential with exactly zero
kinetic energy:
=
= 0. This situation,
describes the static Einstein universe. Setting
=
= 0,
= 1 in (3) and
assuming for simplicity that
matter is pressureless (*w* = 0) we obtain

(11) |

which relates the value of the cosmological constant to the density of
matter *and* the curvature of space. The volume and mass of a SE
universe are respectively
*V* = 2^{2}
*a*_{0}^{3}, *M* = *V*
_{m}
= 2^{2}
*a*_{0}^{3}
_{m}. As a
result
*M* = (*c*^{2}
/ 2*G*)*a*_{0}, and one finds
lim_{a00}
*M* 0,
*i.e.* the mass of the static Einstein universe decreases as its
radius shrinks
to zero, consequently a static empty universe simply cannot exist !
This feature of SE found favour with the proponents of Mach's
principle as discussed in section 2.

(iv) *Loitering Universe*:
The static Einstein universe is clearly unstable: small fluctuations can
make it either contract or expand (these correspond to tiny perturbations
of a particle located at the hump of *V*(*a*) in
fig. (2)
which cause it to roll either towards the left
(*a* 0)
or towards the right (*a*
).
Based on this observation, an interesting new model of the universe was
proposed by Eddington and Lemaitre in which the value of
was kept slightly
larger than
_{crit}.
In this case the universe begins from the Big Bang,
approaches the static Einstein universe and remains
close to it for a substantial period of time before re-expanding
[53,
122].
(If <
_{crit}
the universe will contract instead of expanding.)
The quasi-static or *loitering*
phase, during which the universe remains close to
*a*
*a*_{0}, has
several appealing features not present in models which expand monotonically
[169]:
(i) density perturbations grow at the exponential (Jeans) rate
exp[4
*G*
]^{1/2}
*t* and not at the weaker rate
*t*^{2/3} characteristic of an Einstein-de Sitter universe;
(ii) a prolonged quasi-static phase results in an older universe,
ameliorating the `age' problem which can arise in matter dominated flat
cosmologies
if the value of the Hubble parameter turns out to be large (see
section 4.1).

Interest in loitering models rose dramatically in the late 1960's
when observations suggested the existence of an excess of quasars near
redshift
*z*_{l}
2. To explain these observations the Lemaitre model
with a quasi-static (loitering) phase at
*z*_{l}
2 was invoked
[160,
175,
166].
(Loitering at *z*_{l} arises if the cosmological constant
exactly balances
_{m}
leading to the relation: (1 + *z*_{l})^{3} =
_{} /
_{m},
where _{} =
/
3*H*^{2}.
A decaying cosmological constant will lead to loitering at higher values of
*z*_{l} which has certain advantages from the standpoint of
current observations
[169].)

(v) *Monotonic Universe*:
The particle approaches the potential from the left (*a* = 0) with
sufficient energy to surmount it and travel on towards
*a*
.
In such a situation the scale factor will have an inflection point
at
0,
> 0. By adjusting
initial conditions so that the particle remains close to the hump of the
potential for a sufficiently long duration, one recovers the `loitering'
models discussed in (iv).

(vi) *Nonsingular Oscillating model*:
Another cosmological model deserving mention consists of a form of
matter which behaves as a
-term when the
universe is small, as the universe expands
the -term decays
into either radiation or matter.
The energy density in such a model can be phenomenologically described by
8 *G*
=
/
(1 +
*a*^{p} / ),
so that
lim_{a0}
8 *G*
,
lim_{a}
8 *G*
/
*a*^{p}, *p* = 3, 4
for matter and radiation respectively. The potential
*V*(*a*) = - 4
*G* / 3
*a*^{2} associated with this model
has a broad minimum
which leads to a non-singular oscillatory motion of the expansion factor
*a*(*t*). This toy model is interesting since it exhibits an
infinite number of
expansion and contraction cycles without ever becoming singular.

(vii) Other possibilities not shown in Figure (1) include `asymptotic models' in which the universe asymptotically approaches or moves away from the static Einstein universe. The reader is referred to [62, 142] for a more quantitative discussion of these issues.

Although the above discussion referred to cosmological models
filled with matter having non-negative pressure and a
cosmological constant, it is
easy to show that the qualitative behaviour of the
universe described in (i) - (vii) remains valid, if we generalize
the definition of the
-term to include
any form of matter which violates
the *strong energy condition* so that
_{} +
3*P*_{} < 0
[169].