IX. MODELS OF THE UNIVERSE
WITH THE COSMICAL
CONSTANT
DIFFERENT FROM ZERO
We saw in the last section that placing λ = 0 se-
verely restricts the number of models, and that these
models represent ages that are somewhat too small for
stellar evolutionary
comfort. For this reason, a group
of investigators, particularly
Lemaître, Eddington,
Robertson, Tolman, and McVittie, in the early
days (all
independently of each other and without knowledge
of
Friedmann's work) and Gamow (1946) later, con-
structed various models with λ different from zero.
There are many more such models than one can obtain
with λ = 0,
and among them are both the expanding
and oscillating types, as we have
already noted. The
most popular of these models during the earlier
period
of this work is the one first proposed by Lemaître in
1927 and strongly supported by Eddington. This is the
expanding II model
listed in Table I, for which both
λ and k
are positive. In this model the universe is
always closed and finite and
began its expansion from
some finite nonzero value of R. But the moment of
the beginning of the expansion was not the
moment
of zero time (that is, the moment of the origin of the
universe) because in this model the universe could have
remained in a
nonexpanding, static state for as long
as one might desire—in
fact, for an infinite time in
the past.
Since this model starts expanding from a static
model, both
Lemaître and Eddington assumed this
initial static model to be the
original Einstein static
model. In this model the value of λ and
the radius R
are chosen (in relationship to the mass
M of the uni-
verse)
in such a way as to give a closed spherical
universe in which the cosmical
repulsion is just bal-
anced by the
gravitational attraction. However, this
Einstein universe is unstable, as
we have already noted,
so that any initial expansion reduces the density
and
causes this model to expand still more, with further
reduction in
density, and so on. The expansion thus
proceeds faster and faster until the
universe is infinitely
expanded and the density is everywhere zero. On
the
other hand, a slight compression could have caused the
Einstein
model to have contracted indefinitely, finally
ending up as an infinitely
condensed point of matter.
If, then, we accept this Lemaître-Eddington picture,
the universe
was in a static Einstein state for an infinite
time in the past and then at
some finite time in the
past, for some unknown reason, began to expand,
at-
taining its present rate of expansion
after a few billion
years. Although Eddington never abandoned this con-
cept and fought for it vigorously to the end
of his life,
Lemaître revised his thinking in 1931 and replaced
this
type II expanding model by a type I expanding model.
As we can see from Table I, three possible models of
the
universe can be constructed with λ positive and
k = 1: an oscillating type, an expanding I type, and
an expanding II type. If we reject the last of these
(which corresponds to
the original Lemaître-Eddington
model, which we have just
discussed) we still have the
oscillating and the expanding I models.
The reason Lemaître replaced the expanding II
model by the
expanding I model is that he had no
reasonable explanation for the start of
the initial ex-
pansion of the actual universe
from an Einstein static
state. Although his own theoretical investigations
and
those of McCrea and McVittie (1931) strongly sug-
gested that any local condensation of the matter in
the Einstein static universe (for example, the formation
of a single galaxy
or star) would cause it to start ex-
panding,
these investigations left unanswered the
question as to why other galaxies
were formed. If
expansion began after the formation of a single
galaxy,
the density of the universe would immediately begin
to
decrease and other condensations into galaxies would
be precluded. This
would mean, of course, that the
cosmological principle defined in Section V
would be
untenable, since the distribution of matter in the
neighborhood of this initial condensation would be
different from that
elsewhere in the universe. More-
over, it is
difficult to see how the heavy elements such
as iron, lead, and uranium
could have originated in
an Einstein static-state universe, since we know
from
nuclear theory that the formation of such elements
from hydrogen
in great abundance requires extremely
high temperatures and pressures. This
means that the
entire universe, or at least parts of it, must have
passed
through a high temperature-high pressure phase. Thus
the very
existence of the stars and heavy elements
argues against the Einstein
static state as the initial
phase of our present universe.
Owing to these difficulties, inherent in the assump-
tion that our present universe evolved from an Einstein
static
universe of finite radius, Lemaître introduced the
assumption that
we live in an expanding universe of
type I, which began its expansion from
a highly con-
densed state. He referred to
this initial condensation
as the primordial atom or nucleus and assumed
that
a vast, radioactive explosion occurred in this atom and
that what
we now see in the recession of the galaxies
all about is the result of this
explosion. In this picture,
the expanding universe is always finite in
size, but
closed like a sphere. The initial condensed state (that
is,
the Lemaître primordial atom) may be pictured as
having been
present for an infinite time in the past
or we may suppose that the
universe began its life in
the Einstein static state and then collapsed
violently
into a primordial atom from which it began to expand.
According to Lemaître, this expansion carried the uni-
verse back to its initial Einstein state,
but it did not
stop there. Its velocity of expansion carried it beyond
this static phase, and after that its expansion proceeded
with ever
increasing speed.
Whether we are discussing an Einstein-Friedmann
expanding model, with
λ = 0; or an oscillating model,
with λ = 0; or a
Lemaître model, with λ > 0 and
k = +1 (expanding II or oscillating), we are dealing
with a group of models that are referred to as the “big
bang” models of the universe, since all of them picture
the
universe as having originated explosively from a
point. The term
“big bang” was first introduced by
Gamow (1948) who,
together with Alpher and Herman
(1950), sought to account for the origin of
the heavy
elements by supposing that they were formed from the
original protons and neutrons in the very early and
very hot stage of the
explosion. According to this
picture of the origin of the universe,
neutrons were
the principal components of the original material
ejected from the primordial atom or point source. Some
of these neutrons
quickly decayed into protons and
electrons, and these protons then captured
other neu-
trons to build up the heavy
elements. This whole
buildup of heavy elements must have occurred
during
the first thirty minutes after the initial explosion, for
the
temperature of this primordial material dropped
very rapidly after that and
everything then remained
frozen.
Gamow's theory was very appealing at first since
no other theory of the
elements was available then;
the theory of stellar structure and evolution
had not
yet reached a point of development where it could
be shown
that heavy elements can be and are built
up inside stars, as they evolve
from structures like the
sun into red giants like Antares and Betelgeuse,
with
their internal temperatures rising to billions of degrees.
Gamow's theory of the buildup of the heavy elements
during the first thirty
minutes of the life of the universe
had to be discarded, however, since
there are no stable
nuclei of atomic masses 5 and 8, so that neutron cap-
ture alone could not have bridged the nuclear
gap
between the light and heavy nuclei. Even if some heavy
nuclei were
formed by neutron capture in this early
fireball stage of the universe (and
all nuclei capture
neutrons very readily) a half hour would hardly
have
been long enough for the heavy elements to have been
formed in
their present abundances. Since we now
know that the heavy elements can all
be baked in the
stellar furnaces at various stages of evolution, this
phase
of the Gamow “big bang” theory is not essential
and
one can discard it without invalidating the overall
concept.
If we then accept this Lemaître-Gamow hot “big
bang” hypothesis, the universe must have passed
through a very high temperature phase (about 1010 to
1011 degrees K) soon
after the initial explosion, and
some observable evidence of this may still
be around.
That this should be so was first pointed out by Gamow
himself, who argued that there must have been a con-
siderable amount of very hot black body radiation
present in this initial phase of the universe and most
of it must still be
around, but in a very much red-shifted
form. He estimated that its
temperatures would now
be 6°K. Without knowing about Gamow's
suggestion,
Dicke proposed the same idea in 1964 (he called it
the
“primordial fireball radiation”) and later, in collab-
oration with Peebles, Roll, and
Wilkinson (1965), dem-
onstrated that the
initial hot black body radiation (at
a temperature of 1010 degrees K) must
now be black
body radiation (at a temperature of 3.5°K). The
general
idea behind this deduction is the following: if the
universe
was initially filled with very hot black body
radiation (that is, of very
short wavelength), this radia-
tion would
remain black body radiation during the
expansion of the universe, but it
would become redder
and redder owing to the Doppler shift imparted to
it
by the expansion. This is similar to radiation that is
reflected
back and forth from the walls of an expanding
container. This
3.5°K black body radiation was de-
tected by Penzias and Wilson in 1965 and has since
been verified by
other observers. It is present in the
form of isotropic, unpolarized
microwave background
radiation in the wavelength range from 1/10 to 10 cm.
One other residual feature of the “big bang” should
still be visible, or at least amenable to verification—the
present helium abundance. During the initial fireball
period when the
temperature was considerably larger
than 1010 degrees K, the thermal
electrons and neu-
trinos that were present
would have resulted in very
nearly equal abundances of neutrons and
protons.
When the temperature of the fireball dropped to 1010
degrees
K these neutrons and protons would have
combined to form deuterium, which,
in turn, would
have been transformed into He4, and no heavier ele-
ments would have been formed. Two questions
then
arise. (1) Is the helium that we now observe all about
us, in our
own galaxy and in others, still this primordial
helium? (2) If so, what can
this tell us about the models
of our universe?
The evidence relating to the first question is some-
what ambiguous because we know that helium burning
occurs during
the giant stage of a star's evolution, so
that some of the original helium
must certainly have
been transformed into heavy elements in stellar inte-
riors, and thus disappeared. But we may
assume that
the helium that is found in stellar atmospheres is pri-
mordial and the evidence here is that
although there
is an overall helium abundance of about 25%, some
stars have
been observed with very weak helium lines.
In spite of these, however, the
overall evidence favors
the 25% abundance, which is in agreement with
the
“big bang” hypothesis.
Taking all of the observed data into account (the
3°K black body
radiation and the helium abundance)
the preponderance of the evidence
favors the “big
bang” theory and points to an age of
at least 1010,
i.e., ten billion years for our universe. The observed
helium abundance (if we accept 25% as the primeval
abundance) also
indicates that the density of matter
in the universe must be at least 4
× 10-31 grams per
cc. But if the density of matter in the universe
is no
larger than this, we run into difficulty with the obser-
vations on the rate at which the
expansion of the
universe is decelerating. We have already noted that
Humason, Mayall, and Sandage have given a value for
this deceleration which
indicates that the universe must
ultimately stop expanding and begin to
collapse. This
means that the correct model of the universe is an
oscillating one, rather than expanding, but, as we have
seen, this requires
the density of matter to be about
10-29 gms/cc, as compared to the observed
density of
7 × 10-31
In spite of this, the evidence for an oscillating uni-
verse has been greatly strengthened recently by the
analysis of
the distribution of quasars and of quasi-
stellar radio sources in general. Since these objects
(according to
their red shifts) are at enormous distances
from us, they give us the rate
of expansion of the
universe in its earliest stages. By comparing this
with
the present rate of expansion, we obtain a very reliable
value
for the deceleration, which shows the universe
to be oscillating. To
account for the discrepancy be-
tween the
observed and required density of matter for
such a model of the universe,
we must suppose that
there are large quantities of dark matter in inter-
galactic space—in the form
of hydrogen clouds, black
dwarf stars, and streams of neutrinos. But until
we have
direct evidence of this, we cannot be sure about the
validity
of the oscillating model.
