# University of Virginia Library

#### VIII. MODEL UNIVERSES WITH THE COSMICAL CONSTANT EQUAL TO ZERO

We have seen that the Einstein field equations lead
to both expanding and oscillating models of the uni-
verse, but these field equations do not permit us to
determine which one of the eleven models listed in
Table I corresponds to the actual universe. The reason
for this is that three unknowns, viz., the cosmical con-
stant λ, the sign of the curvature k, and the scale of
the universe (the units in which R and the time are
to be expressed) appear in the final solutions, whereas
direct observations of the galaxies can give us only the
rate of expansion of the universe (Hubble's law) and
its average density. Another possible observation is the
deceleration of the expansion of the universe, and some
work has been done on that possibility which we shall
discuss later. If the deceleration could be measured
accurately, we could decide among the various models,
but until we have reliable observational evidence on
this point, we must proceed by making some assump-
tion about either λ or k.

For the time being, we proceed as Einstein did after
Friedmann's work and place λ = 0. Einstein was very
unhappy about the introduction of λ in the first place
since he considered it to be an ad hoc modification
of the theory which spoiled “its logical simplicity”;
he therefore felt that the models with λ = 0 were the
ones to be favored. From Table I we see that λ = 0
leads to two expanding models of type I for k = -1
and k = 0, and to a single oscillating model for k > 0.
To decide between the expanding and oscillating
models, we must have the equation that tells us just
how k depends on the density of the universe and its
rate of expansion when λ = 0. This relationship, which
is obtained from the solution of the field equations,
is the following:
k = R2/c2 (8/3πGρ - H2 , (3)

where G is the gravitational constant, c is the speed
of light, ρ is the average density of the universe, and
H is Hubble's constant—that is, the rate of expansion
of the universe.

The important quantity in equation (3) is that con-
tained in the parenthesis on the right hand side; for
it determines whether k is negative, zero, or positive,
and hence whether the universe is expanding or oscil-
lating. If we express distance in centimeters, mass in
grams, and time in seconds, the quantity (8/3)πG equals
5.58 × 10-7 and the parenthesis in (3) becomes (5.58 ×
10-7 ρ - H2). If we knew ρ and H accurately, we could
see at once from this expression whether our universe
(with λ = 0) is expanding or oscillating, but neither
ρ nor H is well known. Hubble was the first to measure
H by analyzing the recession of the galaxies and placed
it equal to 550 km per sec per million parsecs; but
we now know that this is too large. According to A.
Sandage (1961), observations on the recession of the
galaxies indicate that H is about 100 km per sec per
million parsecs. If we use this value, H2 becomes (in
cm-gm-sec units) 9 × 10-36 and the quantity in the
critical parenthesis becomes (5.58 × 10-7 ρ - 9 ×
10-36) or 5.58 × 10-7 (ρ - 1.61 × 10-29).

This is a most remarkable result, for it tells us that
the model of the universe (for a given value of the
recession) is determined by the density of matter in
the universe. In our particular case (the cosmical con-
stant zero) the density ρ must be larger than 1.61 ×
10-29 gms per cc (one proton per 100,000 cubic cm.
of space) for the universe to be an oscillating one. If
the density just equals this value, the universe is ex-
panding and Euclidean (no curvature), and if the den-
sity is less than this value, the universe is expanding
but it has negative curvature. It is precisely here that
we run into difficulty in drawing a definite conclusion
because the density ρ is not accurately known.

In terms of our present data, the density appears
to be about 7 × 10-31, which would make k = -1,

565

and the universe an expanding hyperbolic one (nega-
tive curvature). But there may be great quantities of
undetected matter that can increase ρ considerably.
One must therefore try to get other observational
evidence which can permit us to decide between ex-
panding and oscillating models. This can be done if
one determines (from observational evidence) whether
the Hubble constant H is changing with time, and, if
so, how rapidly. If the value of H, as determined from
the recession of nearby galaxies, is sufficiently smaller
than the value as determined from the recession data
of the distant galaxies, we must conclude that H was
considerably larger when the universe was younger (the
distant galaxies show us a younger universe) than it
is now. This would mean that the rate of expansion
had decreased and that ultimately the universe must
stop expanding and begin to collapse. This means that
the universe is oscillating. This sort of analysis has been
carried out jointly by Humason, Mayall, and Sandage
(1956) and the evidence favors an oscillating universe.
This means either that the value of the density ρ has
been greatly understimated or that the correct model
of the universe is one in which λ is different from zero.
Of course, it may be that H is even smaller than 100
km per sec per M pc, but it cannot be much smaller
than this value, and reducing H by a small amount
does not help.

Before leaving these Friedmann models with λ = 0,
we briefly consider the principal properties of the
models associated with the three different values of
k. For k = 0 there is no curvature and space is infinite.
The age of the universe (as measured from some initial
moment t = 0 when the expansion began) is then equal
to 2/3(1/H), and we obtain about 8 × 109 years, which
appears to be too small to account for the evolution
of the stars and galaxies. For this kind of universe, the
expansion parameter R increases as the 2/3 power of
the time.

For k = -1, space is negatively curved and infinite;
the expansion is continuous and endless, so that the
universe finally becomes completely empty and Eu-
clidean. At some initial moment, t = 0, the universe
was in an infinitely condensed state and then began
to expand. According to this model, the age of the
universe is 1/H or 1.2 × 1010 years, which gives ample
time for stellar evolution.

For k = +1, we obtain the oscillating universe
which began from an infinitely condensed state at
t = 0. This is a positively curved, closed universe,
whose radius R will reach a maximum value and then
decrease down to zero again. A similar expansion will
then begin again and this will be repeated ad infinitum.
The age of this model of the universe is smaller than
that of the other two.