a119

How old is the universe?

There are several methods to evaluate the age of the universe.
The first is to refer to the Bible. Then you get something like 5600 years.
But radioactive decay forces people to enlarge this value.

The second method, based on the dynamics of globular clusters,
is that they contain primeval stars, the oldest in our galaxy. This
method will be described later.

Third one: based on some cosmological model. Then one
starts from a field equation. Einstein had his own (but, as mentioned in a preceding
section, Hilbert possibly invented it....).

(101)

S = c T

From this equation (1915) Einstein immediately tried to build
a model of the universe, where curvature could be identified with energy-matter
content. As he was unaware that the universe was non-steady, he tried to build a
steady-state model. But he ran into many problems. Then he visited the great French
mathematician Elie Cartan, who declared:

• My dear friend, I can suggest that you modify your field equation. What about:

(102)
S = c T -Lg

where g is your metric tensor and L
some constant. Notice that your equation still has tensor form, is coordinate-invariant
and divergenceless. Nice, isn't it?

• Yes, thanks a lot. But what could be the physical meaning
  of this new "cosmological" constant L?

• My good friend, that's your problem, not mine. I've done
  my job. You know, I'm a mathematician, not a physicist...

Einstein was puzzled and worried. He thought that the Newtonian
approximation could clear up the problem and shed some light on the ontological
significance of this mysterious constant.

Newtonian approximation:

1. Small space curvature, weak field.

2. Velocities small with respect to the speed of light c.

3. Quasi steady-state conditions (with respect to the
   general cosmic process: the universe, as a whole, is considered as if it
   were "frozen").

Then Newton's law gets a corrective term:

(103)

Notice that this corrective term is proportional to distance
r. It's a long-range force. Can it be attractive or repulsive? Depending on
the arbitrary sign of L
you choose. Considering this force to be repulsive, it was possible to build
a steady-state universe, which Einstein did immediately, where the mysterious
repulsive action of the void balanced the normal attractive Newtonian force.

But this model was fairly unstable. If its spatial extension
was increased, the Newtonian force became weaker and the repulsion of the void stronger,
and vice-versa. Einstein was more worried than ever.

Then two new facts occurred, almost at the same time.

• Edwin Hubble discovered the expansion of the universe.

• The Russian glider pilot Friedmann built a non-steady
  solution of the (101) field equation (no cosmological constant required).

Einstein got mad about it and declared:

• If I had been aware of the universe's non-steady features,
  I would have found it before Friedmann!

If, as the Lacedaemonians used to say.....

Then the cosmological constant fell into disuse, after
some time. Some astrophysicists built arguments in order to show that it should
necessarily be zero.

As this constant goes with a repulsive force which acts
only at very large distances, it modifies the evolution of the universe only
in a late period, in a second era of expansion.

Hubble's law simply says

(104)

The galaxies' recession velocity is proportional to their red shift z

The constant of proportionality is called the Hubble
constant Ho.

What about z?

A stationary atom, in the lab, can emit radiation, if hot enough
(is a Bunsen flame). This radiation corresponds to a nominal wavelength l.

If the atom is moving, with respect to the observer, the latter measures a different wavelength, due to the Doppler effect:

l' = l

• Dl

Or simply write:
(105)

If Dl >
0: the source moves away: red shift.

If Dl <
0, the source gets closer: "blue shift".

There are three Friedmann models, as shown on figure
(106), which differ in their description of the distant future. For the hyperbolic
and parabolic models, the expansion never stops. For the elliptic model, it finally
stops and the universe shrinks (Big Crunch).

(106)

Figure (107) corresponds to the time "from now to the beginning",
where the three curves are almost identical. Then the model sets a simple
relation between the age of the universe and the Hubble constant, shown
on the figure.
(107)

Imagine you take a picture of a grenade, just after explosion.
On your picture you can measure the velocities of the fragments, due to the
exposure time of your camera:

Notice that this velocity field does not correspond to
Hubble's law. The fragments are projected with similar velocities:

From the picture one can calculate the interval of time
between the beginning of the grenade's explosion and the time when the picture
was taken. Then we can calculate the "age of the explosion".

Same thing for the Universe, except the expansion law
(107) is different: the expansion velocity was higher in the past.

The universe is assimilated to a gas, whose molecules would
be the galaxies. An expanding gas, with an expansion velocity field, plus thermal
(random) velocities.