## Thursday, 24 December 2020

### The Friedmann equations

 Expanding Universe
In section 3.8 we start with the Robertson-Walker metric, which expands at a rate $a^2\left(t\right)$ ($a$ is the dimensionless scale factor), and the energy-momentum tensor of a perfect fluid which we use to model the universe, we then use the equation of state $p=w\rho$ and the conservation of the energy-momentum tensor $\nabla_\mu T^{\mu\nu}=0$ to find that the energy density is proportional to some power of the scale factor: $\rho\propto a^{-3\left(1+w\right)}$. Energy conditions give us an idea of possible values of $w$.

But then we go on to find values of $w$ for a
• matter dominated universe (now) which gives $\rho_M\propto a^{-3}$
• radiation dominated universe (early) which gives $\rho_R\propto a^{-4}$
• vacuum dominated (late / de Sitter and anti-de Sitter) $\rho_\Lambda\propto a^0$
The first has an energy-momentum tensor for dust, the second for electromagnetism and the third from another sort of perfect fluid where $p=-\rho$ is the equation of state.

We then apply Einstein's equation and using the metric, the energy-momentum tensor and Ricci tensor for this metric that we found in the previous section we get the Friedmann equations which define Friedmann-Robertson-Walker (FRW) universes and can determine the evolution of the scale factor $a$.

Finally I tried evolving the $a$ in the matter dominated universe and rewrote the Friedmann equations using the conventional metric

Find out what the equations are and how to get them Commentary 8.3 Friedmann equation.pdf (7 pages)