6th Week: The Early Universe and Big Bang Nucleosynthesis B

Thermal History of the Big Bang

The cosmological principle

This principle states the following:

When averaged over a large enough volume, the universe appears the same in all locations.

Here large enough means larger than 500 Mpc (recall that 1 pc or parsec is equal to $30.857{\mbox{x}}10^{12}$ km).

Friedman equation

If one consider the Universe as an isotropic and uniform gas, the expansion in time of this gas is given by

$\left({\frac {{\dot {R}}(t)}{R(t)}}\right)=H^{2}={\frac {8}{3}}\pi G\rho -{\frac {kc^{2}}{R^{2}(t)}}$

which is called the Friedman equation. Here $\rho$ is the density of the universe, G is the gravitational constant, c is the speed of light, and k is the curvature parameter. The so-called scale factor R, comes from the assumption that the metric of the universe is given by

$ds^{2}=R(t)^{2}dx^{2}-dt^{2}$

where $dx^{2}$ is the three dimensional metric. The curvature parameter yield to three physically different solutions of the Friedman equation

• Whit k<0 the Universe is expanding with a positive acceleration and will expand forever. There is not enough density to attract back all the matter in the Universe. The end result of this case is called Big Chill.

• With k=0 we get a flat Universe. The density of the Universe is at its critical value so there is an equilibrium between the gravitational attraction and the repulsion due to the Big Bang. The Universe will continue expanding forever.

• The case k>0 represents a closed Universe. The density is high enough so that the gravitational force will attract all matter back to a single point. This is referred as Big Crunch.