Thermodynamics of The Universe - Energy Density in The Expanding Universe

Energy Density in The Expanding Universe

If the universe is not undergoing a phase transition, one can approximate its thermodynamics by neglecting interactions between particles, and assuming all the energy is in the form of heat. Then by the first law of thermodynamics:

Where is the total heat which is assumed to be constant, is the internal energy of the matter and radiation in the universe, is the pressure and the volume.

One then finds an equation for the energy density, and so

where in the last equality we used the fact that the total volume of the universe is proportional to, being the scale factor of the universe.

In fact this equation can be directly obtained from the equations of motion governing the Friedmann-Lemaître-Robertson-Walker metric: by dividing the equation above with and identifying (the energy density), we get one of the FLRW equations of motions.

In the comoving coordinates, is equal to the mass density . For radiation, whereas for matter and the pressure can be neglected. Thus we get:

For radiation thus is proportional to

For matter thus is proportional to

This can be understood as follows: For matter, the energy density is equal (in our approximation) to the rest mass density. This is inversely proportional to the volume, and is therefore proportional to . For radiation, the energy density depends on the temperature as well, and is therefore proportional to . As the universe expands it cools down, so depends on as well. In fact, since the energy of a relativistic particle is inversely proportional to its wavelength, which is proportional to, the energy density of the radiation must be proportional to .

From this discussion it is also obvious that the temperature of radiation is inversely proportional to the scale factor .