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)

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