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Dictionary of the History of Ideas | ||

####
*VIII. MODEL UNIVERSES WITH THE*

COSMICAL CONSTANT
EQUAL TO ZERO

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)π*G* equals

5.58 × 10-7
and the parenthesis in (3) becomes (5.58 ×

10-7 ρ - *H*2). 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, *H*2 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,

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.

Dictionary of the History of Ideas | ||