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).

Hubble expansion

In 1929 Edwin Hubble discovered after years of observations, that galaxies were moving away from the Earth at a velocity proportional to the distance from those galaxies to the Earth. This is

${\dot {x}}=H_{0}x$

where $H_{0}$ is the Hubble constant. The current accepted value of this constant is $H_{0}=70.1\pm 1.3$ (km/s)/Mpc.

Friedman equation

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

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

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.

Solution to the Friedman equation

Time evolution of temperature

Cosmic microwave background

Discovered (accidentally) in 1965, the CMB is a nearly perfect blackbody radiation spectrum visible in all directions at a temperature of ~2.75K. It marks the point in the history of the Universe at which recombination removed free electrons from the universe and thus decoupled the radiation field from the surrounding matter. This would've occured at about 3,000K, or 4,000 years after the Big Bang. After this point, photons continued without interaction, becoming redshifted with the cosmic expansion from their original temperature of 3,000K to the current CMB temperature.

Despite being almost isotropic, the CMB does