← Cosmology

Cosmology and General Relativity

Friday, April 22, 2022

Measuring the Universe

The equations of general relativity can be used to measure certain properties of the universe. However, the mass-energy density must be measured for the whole universe, at a scale large compared to the spacing between galaxies. However, the universe's density decreases as it expands.

Solving the equation for the large-scale structure of the universe gives the Friedmann equation:

(dRdt)2=8π3GρR2kc2\left(\frac{dR}{dt}\right)^2=\frac{8\pi}{3}G\rho R^2-kc^2

where R(t)R\left(t\right) is the size or distance scale factor of the universe at time tt and ρ\rho is the total mass-energy density at that time. The kk is a geometrical constant of the universe. If k=0k=0, the universe is flat. If k=+1k=+1, the universe is curved and closed. If k=1k=-1, the universe is curved and open.

At large scales, the universe appears to be flat, so k=0k=0 is used. The ρ\rho term must include both matter and radiation. Since the universe expands, the matter density ρm\rho_\text{m} decreases with radius: ρmR3\rho_\text{m}\propto R^{-3}. Solving for R(t)R\left(t\right), we get


where AA is a constant. We can use this to eliminate RR, getting

t=16πGρmt=\frac{1}{\sqrt{6\pi G\rho_\text{m}}}

In the early universe, radiation dominated. We know the energy density depends on dλ/λ5d\lambda/\lambda^5, and since all wavelengths scale with RR, we know dλRd\lambda\propto R and λ5R5\lambda^5\propto R^5. Therefore, ρrR4\rho_\text{r}\propto R^{-4}. Integrating, we find


where AA' is a constant, so

t=332πGρrt=\frac{3}{\sqrt{32\pi G\rho_\text{r}}}

Finally, we can define the Hubble parameter in terms of the time variation of the scale factor: