Cosmological redshift. Hubble constant.
In an isotropic universe 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−ω)/(ω0)]=[(a′)/(a2)]l ≡ H l,
where H is the so called Hubble constant,
H=[(a′)/(a2)]=[1/(a)][(da)/(dt)].
The current value of the Hubble constant is H ≈ [1/((13 bil.years))].
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 "dark energy". The visible matter constitutes only about 3%
of the density.
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On 11 Oct 2007, 15:25.