Equations of State

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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.