In a previous post I derived the first of Friedmann's equations but the equation cannot be solved unless we know how the density of the universe changes over time. In cosmology the Roberston-Walker metric is formulated with fixed coordinates for fundamental particles which do not vary with time. We can therefore assume that the matter and radiation which fills the universe is describable as an ideal fluid. The evolution of the density is going to depend on the pressure present in the universe, which depends on the material which fills the cosmos.
The assumption I make is that there is a unique pressure for every density. This is the simplest assumption because under this condition pressure does not contribute any additional force.
This is the equation of state and gives us enough to solve the equation once we derive it. Consider the first law of thermodynamics
This applies to an expanding volume, V the energy of this volume is given by
The change in energy of the system then requires us to use the product from differential calculus
If we assume this process is adiabatic such that TdS equals zero then plug dE/dt back into the equation, taking dV as
Rearrange the equation slightly and you should get
I've substatued a for radius, as this is the form found in text books, the terms in the brackets form the most essential part of the equation. The first of these terms corresponds to the density, (mass per volume) which decreases as the universe expands. The second term has to do with energy, as the universe increases in size, pressure of the material has done work, which transfers into potential energy. This equation and the earlier derived Friedman equation are all we need to describe the evolution of the universe but it will be helpful for is to derive an equation for acceleration, the second Friedmann equation. Which will follow in an up coming post.
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