# Pressure of (cold matter)

The first pages of Weinberg's Cosmology state that matter, cold matter like dust has zero pressure. I find it very disturbing. Does 'cold matter' also include fluids since hot matter is radiation, which have pressure in other fields of physics? What am I missing?

If you consider a gas then it has a pressure because the molecules of the gas are whizzing around rebounding off the walls of whatever container it's in. For an ideal gas the pressure is proportional to the temperature, and the temperature is proportional to the internal energy, so the pressure is proportional to the internal energy i.e. the kinetic energy of the gas molecules.

If we allow the gas to expand reversibly then the gas does work, and that work comes from the kinetic energy of the gas molecules. So as the gas expands the kinetic energy of the molecules decreases and ultimately falls to zero. However the rest mass of the molecules is of course unchanged.

The significance of this is that in cosmology both the rest mass and the kinetic energy of the gas contributes to the stress energy tensor. As the universe expands the rest mass density scales as $$1/V$$ i.e. as $$a^{-3}$$ because it's a constant mass spread over an increasing volume. However the kinetic energy density scales as $$a^{-4}$$ because not only is the matter getting diluted but it also does work as it expands.

We call matter cold if the kinetic energy is small compared to the rest mass. In this case the energy density is dominated by the rest mass so it scales as $$a^{-3}$$. Since the kinetic energy (and hence the pressure) is small compared to the rest mass we can ignore the pressure and approximate the cold matter as pressureless.

By contrast we call matter hot if the kinetic energy dominates over the rest mass, and in this case we'll get the $$a^{-4}$$ scaling. As you say, light is the obvious example of this since it has no rest mass.