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The age of the universe

The Hubble constant measures how fast the universe expands right now. It tells nothing about how fast the universe was expanding in the past, or how old the universe is.

The age of the universe is inversely proportional to H₀, but a complex function of $\Omega_0$ . If

$\begin{displaymath}H_0 = 50 {{\rm\,km}/{\rm\,s}\over{\rm\,Mpc}} \end{displaymath}$

and

$\begin{displaymath}\Omega_0 = 1, \end{displaymath}$

then the universe is 13 billion years old.

$\framebox{\Huge\bf ?}$ How old is the universe with $\Omega_0=1$ and

$\begin{displaymath}H_0=70{{\rm\,km}/{\rm\,s}\over{\rm\,Mpc}}\mbox{\,?} \end{displaymath}$

A: 13 billion years old.
B: 18 billion years old.
C: 9 billion years old.

Models with the cosmological constant

Without a cosmological constant all cosmological models are decelerating, q₀>0.

Cosmological constant acts as a repulsive force, so the universe can become accelerating (thus q₀<0).

The cosmological constant is usually denoted as $\Lambda$ . The density parameter due to cosmological constant is denoted as

$\begin{displaymath}\Omega_\Lambda. \end{displaymath}$

In the presence of a cosmological constant, the condition for the universe to be flat changes:

$\begin{displaymath}\Omega_0 + \Omega_\Lambda = 1. \end{displaymath}$

This can be interpreted as $\Lambda$ being a special kind of matter with the repulsive gravity (but positive mass):

$\begin{displaymath}\Omega_0(\mbox{total}) = \Omega_0(\mbox{normal matter}) + \Omega_0(\mbox{lambda-matter}). \end{displaymath}$

Is there a ``lambda-matter''?

When Einstein introduced his cosmological constant, he did it on purely adhoc grounds. He did not think about it as a special kind of ``lambda-matter'' (there is no such a term, indeed).

However, the development of particle physics in 50s and 60s actually lead to theoretically predicting such a matter! An elementary particle which might play a role of such ``lambda-matter'' is the so called Higgs boson. It is not yet discovered experimentally, but we are close to finding it.

Even if the Higgs boson is found experimentally, it does not mean that the cosmological constant exists: there may be so few of them that $\Omega_\Lambda$ would be very small, equal to zero for any practical purpose.

So, after all, it may turn out that the cosmological constant was not Einstein's ``greatest blunder'', but one of his greatest successes!

In the presence of a cosmological constant, the expression for the deceleration parameter changes:

$\begin{displaymath}q_0 = {\Omega_0\over 2} - \Omega_\Lambda. \end{displaymath}$

Thus, the only way to obtain negative q₀, which means an accelerating expansion of the universe, is to have a cosmological constant.

There are also cosmological models which have negative $\Lambda$ . Nowadays they are disfavored, since this would imply lambda-matter with negative energy density and mass, which is unphysical.

The ultimate fate of the universe

The universe expands now. In the future it has only two possibilities:

It expands forever.
It turns around and collapses if $\Omega_0>1$ and $\Lambda=0$ .

If the universe expands forever:

In about another 10 billion years galaxies will use up their gas and new stars ceased to form.
After about 30 billion years last stars will die. The universe will be filled with stellar ``cinder'': white dwarfs, neutron stars, and black holes.
In another 1000 billion years most of the stellar remnants in galaxies will collapse to a central black hole, leaving a few stellar remnants to orbit it, and throwing a few more into the empty intergalactic space.
In about 10⁴⁰ years protons will decay, leaving the ``lepton desert''.
In about 10¹¹⁰ years all black holes will evaporate, and galaxies will dissolve.
After that the universe will be filled with very slowly moving electrons, neutrinos, very low energy photons, and a few unattached blobs of electrons and neutrinos, all that is left of stars.

If the universe recollapses:

The universe is still expanding, so it is not recollapsing for at least another 5 billion years.
By the time the universe stops expanding and begins recollapsing, most (perhaps all) of the stars will be dead.
About 10 billion years after it begins recollapsing, most of galaxies will collapse into huge black holes.
The universe will continue recollapsing until it becomes so small that those black holes begin to touch each other.
The black holes will be merging with each other as crazy. Perhaps, they will do it everywhere more or less uniformly, then the universe will collapse in a Big Crunch.
Perhaps, one of the black holes will become much larger than the others. Such a universe is not homogeneous and isotropic any more, and only God knows what will happen then...

No matter what the fate of the universe is, the perfect cosmological principle is wrong, the universe will not be like it is now.

Your book is saying: ``If we pass this way but once, we must make the most of it.''

Solution to the Olbers' paradox

Newton's universe was infinite in space and time and uniformly filled with stars. Kepler was the first to realize that in such a universe the sky should be ablaze: during infinite time stars emit infinite amount of light!

Several scientists discussed this paradox, which for some reason is now called Olbers' paradox, even if Heinrich Olbers was not the first to mention it.

In the expanding universe:

stars existed only for a finite time since they formed after the Big bang.
light from remote stars is redshifted and becomes invisible.

Any one of those two features is enough to explain the darkness of the night sky. Thus, in the Big Bang theory the Olbers' paradox is resolved with a large safe margin!