Hubble expansion law:
v = hubble constant x distance
X(t) = X(0) * (1 + Ht)
X(t): position of galaxy at time t
X(0): position of galaxy at time 0
H: hubble constant
Hubble flow
Redshift z = (H / c) * D for z <<1, D << cH^-1

Olbers' Paradox: Why the sky is dark?
- In a universe of finite age, sky is dark even if size is infinite.
- Light from farthest visible stars is redshifted.
- We have not seen everything yet.
- We never will.

Uniform Expansion
- Objects at distance D0 now are at D(t) = a(t) * D0, a(0) = 1
- If a(t) ~ 1 + H0 * t for small H0 * t, D(t) ~ D0 + H0 * D0 * t
- So v0 = H0 * D0. Write this as H0 = a(0)
- Cosmological redshift 1 + z = observed_wavelength / emitted_wavelength = a(t-obs) / a(t-emited) = 1 / a(t-emited)
1 / 1 + H0 * t-emitted ~ 1 - H0 * t-emitted (Newton's approximation)

Kinematics of expansion
density(t) = a(t) ^-3 * density(0) (deduced from mass conservation)

angular distance (deduced from small angle approx.)
Da = D0 / (1 + z)
object appears smaller
brightness
b = (L / (4 * pi * D0^2)) * (1 / (1+z)) * (1 / (1+z))
first (1 / (1+z)): E = hf = hc / frequency, each photon brings (1 / (1+z)) less energy than it was emitted
second (1 / (1+z)): since we are moving away, relativistic redshift