Friedmann Equation

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1 The Friedmann Equation

The Friedmann Equation is an equation of motion for the scale factor in a homogeneous universe. A rigorous derivation requires General Relativity, but we can fake it with a quasi-Newtonian derivation. We will model the universe as an adiabatically () expanding, isotropic, homogeneous medium. Isotropy allows us to use as a scalar. Consider a thin, expanding spherical shell of radius . Birkhoff’s Theorum states that even in GR, the motion of such a shell depends only on the enclosed mass . Thus, the energy per unit mass per unit length is:

We define , and we will show later that is a measure of the curvature of the universe:

where has units of if is dimensionless. Substituting into the above energy equation, and solving for , we get:

In GR, we understand that it is not just mass that curves space, but also energy. Thus, we can generalize this equation to use an energy density , with , giving us:

This is the Friedmann Equation.

1.1 Critical Density

As discussed above, corresponds to a flat, critically bound universe. If we assume , we can solve for the critical density of the universe: the value of that leads to a critically bound universe. To do this, we need a measure of what is. A lot of effort has gone into measuring , the value of the Hubble parameter at the present day, with km s Mpc. Using that

we find that kg m Mpc. This is called the critical density of our (present day) universe. Note that generally, the critical density is time dependent because the Hubble parameter changes with .

1.2 In terms of the Dimensionless Density Parameter

The critical density is an important yardstick for determining how a universe behaves. It is common to measure energy densities relative to this yardstick. We define

to be the dimesionless density parameter. Using this definition and remembering that , we can rewrite the Friedmann equation as: