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How old is the universe?
There are several methods to evaluate the age of the universe. The first is to refer to the Bible, which gives about 5600 years. However, radioactive decay forces people to increase this value.
The second method, based on the dynamics of globular clusters, is that they contain primordial stars, the oldest of 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 previous section, Hilbert possibly invented it first...).
(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. However, he encountered many problems. Then he visited the great French mathematician Elie Cartan, who said:
- My dear friend, I could suggest you modify your field equation. What about:
(102) S = c T - Lg
where g is your metric tensor and L a constant. Notice that your equation still has a tensor form, is coordinate-invariant and divergenceless. Isn't it elegant?
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Yes, thank you very much. But what could be the physical meaning of this new "cosmological" constant L?
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My good friend, it'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 clarify the problem and bring some insight into the ontological meaning of this mysterious constant.
Newtonian approximation:
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Weak spatial curvature, weak field.
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Velocities much smaller than the speed of light c.
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Quasi-steady-state conditions (with respect to the general cosmic process: the universe, as a whole, is considered as if it was "frozen").
In this case, Newton's law gets a corrective term:
(103)
Notice this corrective term is proportional to distance r. It is a long-range force. It can be attractive or repulsive, depending on the arbitrary sign of L you choose. Considering this force was repulsive, it was possible to build a steady-state universe, which Einstein did immediately, where the mysterious repulsive action of the vacuum balanced the normal attractive Newtonian force.
But this model was fairly unstable. If its spatial extension increased, the Newtonian force became weaker and the repulsion of the vacuum stronger, and vice versa. Einstein was more worried than ever.
Then two new discoveries occurred almost simultaneously:
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Edwin Hubble discovered the expansion of the universe.
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The Russian glider pilot Alexander Friedmann built a non-steady solution of the (101) field equation (no cosmological constant required).
Einstein was shocked and declared:
- If I had known that the universe was not steady, 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 developed arguments to show that it had to be zero.
As this constant corresponds to a repulsive force that acts only at very large distances, it only affects the evolution of the universe in a late phase, in a second era of expansion.
Hubble's law simply states (104)
The velocity of galaxies moving away is proportional to their redshift z.
The constant of proportionality is called the Hubble constant, denoted H₀.
What does z represent?
A stable atom, in the laboratory, can emit radiation if it is hot enough (for example, in a Bunsen flame). This radiation corresponds to a nominal wavelength λ.
If the atom is moving relative to the observer, the latter measures a different wavelength, due to the Doppler effect:
λ′ = λ + Δλ
or simply:
(105)
If Δλ > 0: the source is moving away → redshift.
If Δλ < 0: the source is approaching → "blueshift".
There are three Friedmann models, illustrated in 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 eventually stops and the universe collapses ("Big Crunch").
(106)
Figure (107) corresponds to the time "from now to the beginning", where the three curves are almost identical. Then the model establishes a simple relationship between the age of the universe and the Hubble constant, shown in the figure.
(107)
Imagine you take a photo of a grenade, just after its explosion. On your photo, you can measure the velocities of the fragments, thanks to the exposure time of your camera:
Notice that this velocity field does not correspond to Hubble's law: the fragments are projected with approximately the same velocities:
From the photo, one can calculate the time interval between the beginning of the grenade's explosion and the moment the photo was taken, then deduce the "age of the explosion".
It is the same for the universe, except that the expansion law (107) is different: the expansion velocity was higher in the past.
The universe is likened to a gas, whose molecules would be the galaxies. An expanding gas, with an expansion velocity field, superimposed on thermal (random) velocities.