**The Age Of The Universe**

As we noted in the cosmology lecture the observed
increase in redshift Z with distance is interpreted, by most
scientists, as evidence of the expansion of the universe. This
led to the idea that the universe may have gone through a dense
hot phase, with the whimsical name of the **big bang**, a
finite time ago. The hot dense phase is generally regarded as the
beginning of the universe, and the time since the beginning is,
by definition, the age of the universe. A basic, and natural,
question to ask is: How old is the universe?

It is important to realize that this question
is undefined until one specifies 1) a definition of what one
means by the *age of the universe* and 2) how it should be
measured; that is, one needs a theory for the age of the
universe. This was recognized as long ago as 1658, when Bishop
James Ussher determined the age of the
universe thus: he defined the age of the universe as the time
since creation (by God) and measured the age by adding the
periods specified in the Bible, suitably corrected using
available astronomical and historical records. He determined that
the universe was created in 4004 BC, at a time that would have
been sunset in Jerusalem, October 22.

Today, however, there is overwhelming evidence that the universe is considerably more ancient that previously thought. But, like Bishop Ussher we need a theory of the age of the universe. At present the best theory we have is Einstein's general relativity. When applied to the universe, assuming the Cosmological Principle (the universe is uniform and isotropic on very large scales), the theory provides a relationship between the mass density and the age of the universe.

Acccording to general relativity, the universe
will either expand forever or recollapse after a finite time,
depending on the value of its mass density, r. If the density is too
great, that is, greater than some critical density, r_{c}, the
universe will recollapse; that is, it will be *closed*. If
the density is below, or equal to, the critical density it will
expand forever; the universe will be *open*. The critical
density is given by

(1) r

_{c }= 3H_{0}^{2}/8pG= 8 * 10

^{-}^{27}(H_{0}/65) kg/m^{3}= just over 4 hydrogen atoms per cubic meter,

where H_{0 }is the Hubble constant and
G is Newton's gravitational constant. We have assumed a Hubble
constant of H_{0 }= 65 km/sec/Mpc. It is customary to
discuss the density of the universe in terms of the parameter W, defined by

(2) W = r/r

_{c}

In the Einstein-de Sitter model the universe is assumed to have a flat spatial geometry. For this case the age of the universe is predicted to be

(3) t

_{0}= (2/3)/H_{0}

Current models of the universe assume that W is close to one.
For example, in the **inflationary model **a truly stupendous
expansion is assumed to have occurred early in the history of the
universe. About 10^{-34 }seconds after the creation, when
the temperature was perhaps one thousand trillion trillion
degrees, a bubble no more than 10^{-24} cm might have
undergone an expansion by a factor of maybe 10^{50}! At
about 10^{-32} seconds this hyper-expansion stopped,
leaving the bubble 10^{26}cm across (that is, 100 million
light years). However, the bit of the universe we can observe
would have been, at that early time, only a few centimeters
across! After 10^{-32} seconds the normal big bang
evolution takes over.

This may sound utterly crazy, but it explains a
lot. The fantastic stretching of space renders the observable
universe geometrically flat (with W=1). This is because the *observable*
universe would be but a tiny patch of a universe that may be a
trillion trillion times larger. Even if the geometry of the
universe were highly curved the observable part of it is such a
small patch that it would nonetheless appear to us as spatially
flat. Moreover, any weird particles or spacetime distortions that
may have been created then would have been diluted practically
out of existence; so we would not expect to find any such
oddities in our neck of the woods. Lastly, the microwave
background radiation, which was already uniform and isotropic
within the micro-bubble, would have remained so after the
hyper-expansion. The latter explains why the radiation at
opposite sides of the sky have the same temperature; these
patches of sky were once much less than 10^{-24} cm
apart.

Alas, W is observed to be about 0.25! Therefore, if we nonetheless insist that W = 1, we are forced to conclude that the luminous matter we observe forms only a part of the total energy in the universe. The deficit must therefore be made up by some other unknown source of energy. This could be dark matter. Much more interesting and much more mysterious: the possible existence of vacuum energy; or perhaps an admixture of the two.

When we use Eq. (3) to estimate the age of the universe we get an answer of about 10 billion years. Unfortunately, this is less than the age of the oldest stars found in globular clusters, like M13. It is thought that these stars are about 15 billion years old. Presumably, however, the universe cannot be younger than its contents. This problem has been the source of much controversy and debate recently. Some scientists have suggested that the big bang idea is wrong and should be abandoned. However, it could be that the ages of the oldest stars are too high, or that the assumptions of the inflationary model are wrong, or the measurement of Hubble's constant is in error. Or perhaps there is a missing ingredient. Or some combination of all of these.

Within the context of a general relativistic
description of the the universe there are two critical parameters
that determine the global properties of the universe: The Hubble
constant H_{0} and the deceleration parameter q_{0}.
If the universe is expanding today, it must have been expanding
faster in the past. This is because gravity has been slowing down
the expansion. (Of course, if an anti-gravitational force exists
it is conceivable that the expansion is actually accelerating.)
The deceleration parameter, as the name suggests,
is a measure of how much the expansion has slowed since the big
bang.

We have already encountered Hubble's law

(4) Z = d H

_{0}/c,

where Z is the redshift of a galaxy and d its (proper) distance from us. This law is valid only for small redshifts, say less than about 0.2; that is, for galaxies that are not too distant. For very distant galaxies we need a more accurate formula. We need this because when we look at very distant galaxies we are looking into the distant past when the expansion rate was, presumably, greater. A more accurate formula that takes account of the universal deceleration is

(5) Z = d H

_{0}/c + (1+q_{0})(d H_{0}/c)^{2}/2.

In practice, cosmologists do not use the proper
distance d, directly, but rather use a distance scale, called the
luminosity distance, d_{L}, which is determined by
requiring that the inverse square law

(6) f = L/4pd

_{L}^{2}

hold true even for an expanding universe. (In a static, flat universe both distance scales coincide.) Here f is the measured flux (energy per square meter per second) and L is the absolute luminosity (energy per second). The luminosity distance is convenient because it can be determined from measuring f and determining L, only.

If we measure d (or d_{L}) and Z for
many galaxies we should, in principle, be able to determine the
two constants H_{0} and q_{0}.
The chief difficulty is measuring the distances. The basic
technique is to use objects of known intrinsic luminosity L
(standard candles) and use the inverse square law along with the
apparent luminosity to infer the distance. But to measure
distances of billions of light years one needs extraordinarily
bright objects. Amongst the brightest objects in the universe are
exploding stars, called supernovas.

In recent years there has been growing evidence
that Type Ia supernovas are standard candles. Type Ia supernovas,
thought to be exploding carbon-oxygen white dwarf stars, can be as
bright as an entire galaxy, therefore, they can be seen over
great distances. The white dwarf star forms a binary with a
larger star from which it draws matter. When the total mass of
the white dwarf and its accreted material exceeds the **Chandresekhar
limit **of 1.4 solar masses the white dwarf star detonates,
reaching a temperature in excess of a billion degrees.

The hypothesis that all Type Ia supernovae have
the same intrinsic brightness is suggested by the idea that the
stars explode when they reach the *same* critical mass. In
practice, there is some variation in luminosity, as well as in
the shape of the light curves. But by calibrating the light
curves, that is, by checking the luminosity-distance relationship
using nearer supernovas it is possible to take into account some
of this variation. The distances to the nearer Type Ia supernovas
is measured using Cepheid variables that reside in the same
galaxy as the supernovas.

There is now ample evidence that W, due to the
observable energy, is rather less than one. One way out of this
difficulty, assuming one wants a unit value for W, is to assume that
the energy density of the universe consists of ordinary
observable energy plus the energy of, say, the vacuum; that
is, we assume that W = W_{M} + W_{L}, where W_{L }is the contribution from the vacuum, and W_{M }is
that from ordinary matter and energy. (The parameter L was introduced by
Einstein in his (not fully successful) attempt to predict a
static universe. Later he removed L from his equations, calling
its prior introduction the greatest mistake of his life.)

But how can mere vacuum have energy? Well,
according to modern physics the vacuum is a cauldron of virtual
particles that come in and out of existence on very short
timescsales. In fact, when an honest calculation is done of the
energy of the vacuum comes out to a gigantic value; alas, the
answer is wrong by about a factor of 10^{100}! This is
surely the worst prediction in the history of physics. It is a
complete mystery why the energy of the vacuum is so much smaller
than the prediction.

Nonetheless, if vacuum energy exists it can
have a marked effect on the evolution of the universe. Many
groups have tried to measure the W parameters using Type Ia
supernovas. Here is a plot from a paper by S. Perlmutter et al., which
shows how the distance (measured in terms of the magnitude---if
you like, the faintness---m_{B}) is related to the
redshift.

The three black (solid) curves are predictions
for a *curved* universe assuming (W_{M}, W_{L}) =
(0, 0), top curve, (1, 0), middle curve and (2, 0), bottom curve.
The dashed curves are calculated for a *flat* universe. From
the top, the dashed curves correspond to (W_{M}, W_{L}) = (0,
1), (0.5, 0.5), (1, 0) and (1.5, -0.5). So, if the universe
has a globally flat geometry it would appear that it contains
some amount of vacuum energy. Careful fits by the Supernova
Cosmology Project, of all the available
data, assuming a flat universe, gives a value of W_{M} = 0.25
+/- 0.06
(statistical) +/- 0.4 (systematic). The alternative model is a curved
(open) universe. If this model is true then the data seems to
suggest that the universe is empty, which it clearly is not! Stay
tuned.