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 .
Read more about this topic: Thermodynamics Of The Universe
Famous quotes containing the words expanding universe, energy, expanding and/or universe:
“In an expanding universe, time is on the side of the outcast. Those who once inhabited the suburbs of human contempt find that without changing their address they eventually live in the metropolis.”
—Quentin Crisp (b. 1908)
“The chief function of the city is to convert power into form, energy into culture, dead matter into the living symbols of art, biological reproduction into social creativity.”
—Lewis Mumford (1895–1990)
“Children’s view of the world and their capacity to understand keep expanding as they mature, and they need to ask the same questions over and over, fitting the information into their new level of understanding.”
—Joanna Cole (20th century)
“Man is but a reed, the feeblest one in nature; but he is a thinking reed. The entire universe need not arm itself to crush him—a vapor, a drop of water suffices to kill him. But if the universe were to crush him, man would still be nobler than that which killed him, because he knows that he dies and the advantage which the universe has over him; the universe knows nothing of this.”
—Blaise Pascal (1623–1662)