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 = ω0a(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-ω/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.
A remote galaxy resides at a coordinate χ (assume θ=φ=0)
from the Earth (χ=0). Assume that a(η-χ)≈a(η).
What is the "proper" distance to that galaxy?
Calculate the velocity
with which the galaxy appears to move relative to the Earth.
Interpret the cosmological red shift
ω0-ω/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.
Name the exercises which you
did not learn much from and which you'd rather drop.
did learn something from.
Hints
[l = aχ; v = Hl (v = dl/dt, H = da/adt);
v = Hl]