The universe is Homogenous (looks the same from every location) and Isotropic (looks the same in every direction)

Olbers paradox: if the universe was infinite the sky would be infinitely bright, therefore the universe is finite

The Hubble-Lemaitre law

$$ v=H_0r $$

• $v$ is the rate at which the universe expands at a given $r$
• $H_0$ is the Hubble constant = $70 \pm 5$ km s$^{-1}$ Mpc$^{-1}$

The redshift of a galaxy

$$ z=\frac vc=\frac{\Delta\lambda}\lambda=\frac{\lambda_\text{obs}-\lambda_{rest}}{\lambda_\text{rest}} $$

• More distant galaxies have a higher value of $z$

Hubble values

$$ t_\text{H}=\frac rv=\frac1{H_0}\approx10^{10} \,\text{yr} $$

• $t_\text{H}$ is the age of the universe assuming constant expansion

$$ r_\text{H}=ct_\text{H}\approx 4\,\text{Gpc} $$

• $r_\text{H}$ is the Hubble radius, the size of the universe using a constant expansion

$$ \rho_\text{cr}=\frac{3H^2_0}{8\pi G} $$

• $\rho_\text{cr}$ is the Critical density, it is the density that will balance out the expansion and the gravitational force at a time infinetly far in the future

• Currently $\Omega_0\approx0.3$

The above equation proposes 3 possible future,

$$ \Omega_0=\frac\rho{\rho_\text{cr}} $$

• if $\Omega_0$<1 it is an open universe
• if $\Omega_0$=0 it is a critical universe (flat universe)
• if $\Omega_0$>1 it is a closed universe

Curvature constnant $k$

• $k$<0 for an open universe
• $k$=0 for a critical universe
• $k$>0 for a closed universe

Scale factor

$$ a=\frac{r(t)}{r(t_0)}=\frac1{1+z} $$