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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=8π3GρR2−kc2\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: ρm∝R−3\rho_\text{m}\propto R^{-3}. Solving for R(t)R\left(t\right), we get

R(t)=At2/3R\left(t\right)=At^{2/3}

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 dλ∝Rd\lambda\propto R and λ5∝R5\lambda^5\propto R^5. Therefore, ρr∝R−4\rho_\text{r}\propto R^{-4}. Integrating, we find

R(t)=A′t1/2R\left(t\right)=A't^{1/2}

where A′A' 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:

H=1RdRdtH=\frac{1}{R}\frac{dR}{dt}