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Cosmological redshift and Hubble constant

In an isotropic universe (ds2 = a2(dη2 - dχ2 - Sin2χ dΩ2) the radial (dθ=dφ=0) propagation of light (ds2=0) is described by χ = +/- η + const, from where one can deduce that along the light ray there remains a constant product ωa = const. A light ray with frequency ω0 emitted at a distance χ and observed at the origin (χ=0) at time η should then have the frequency
ω = ω0 a(η-χ)/a(η) ≈ ω0 (1 - χ a'/a),
that is redshifted, if the universe expands (a'>0). The proper distance l to the source of light is l = χa. Thus the frequency shift z can be written as
z ≡ ω0/ω0 = a'/a2l ≡ Hl,
where H is the so called Hubble constant, H = a'/a2 = 1/ada/dt. The current value of the Hubble constant is H ≈ 1/(13 bil. yeas). Inserting a'/a2 = H into Friedmann's equations leads to
1/a2 = H2 - κμ/3
for a closed universe, and to
1/a2 = κμ/3 - H2
for an open universe. For the critical density μc, such that κμc/3 = H2, the universe is flat. The current measurements show that the relative density Ω = μ/μc is close to one with an error about few per cent (flatness problem). About 30% of it is "dark matter" and about 70% is cosmological constant. The visible matter constitutes only about 3% of the density.

Quantum gravity

  1. Covariant formulation = String theory, blah-blah-blah...
  2. Canonical formulation = Quantum geometry = Loop quantum gravity, blah-blah-blah...

Exercises

  1. A remote galaxy resides at a coordinate χ (assume θ=φ=0) from the Earth (χ=0). Assume that a(η-χ)≈a(η).
    1. What is the "proper" distance to that galaxy?
    2. Calculate the velocity with which the galaxy appears to move relative to the Earth.
  2. Interpret the cosmological red shift ω0/ω0 = Hl (l is the distance to the "red shifted" galaxy) as a Doppler effect and calculate the velocity with which a galaxy appears to be moving relative to the observer.
  3. Name the exercises which you
    1. did not learn much from and which you'd rather drop.
    2. did learn something from.
Hints
[l = aχ; v = Hl (v = dl/dt, H = da/adt); v = Hl]
Copyleft © 2003 D.V.Fedorov (fedorov@ifa.au.dk)