Equations of State
Recall we had the following (Friedmann, Fluid, Acceleration) equations:
We introduced the Fluid and Acceleration equations in order to related and in the Friedmann equation. Unfortunately, in doing so, we introduced a new unknown: the pressure . To close the equations, we need to relate and with an equation of state. Equations of state generally have the form:
where pressure is proportional to energy density with some (dimensionless) constant of proportionality.
Using this generic equation of state (we haven’t decided what is yet) the Fluid equation becomes:
Note that we’ve assumed , which is okay most of the time. We don’t have any evidence so far that changes.
In general, can consist of multiple components: e.g.
To get density () as a function of time, want to solve for . Below we examine special cases of interest.
1 Matter (Non-relativistic particles)
Pressure-less “dust” , because volume goes as .
2 Radiation (Relativistic particles)
Relativistic particles (photons, bosons): , because , and energy is given by .
3 Dark Energy
()/Dark Energy: , constant in time.