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13 Cosmological Preliminaries
This beautiful result tells us that as the photon travels its wavelength stretches in
proportion to the scale factor of the universe. It is a very easy way to remember the
cosmological redshift relation.
Let us proceed to get the Hubble law (13.1) for cosmologically nearby galaxies.
Astronomers define a redshift parameter z as the fractional wavelength shift;
according to (13.21) z is related to the scale factor by
z =
λ
λ
=
λ o − λ e
λ e
=
λ o
λ e
− 1 =
a(t o )
a(t e )
− 1.
(13.22)
A cosmologically nearby galaxy then has z = 0.
A receding galaxy appears to be moving away from us at a velocity given by the
Doppler shift, so
ν
c
=
ν
ν
=
λ
λ
= z =
a(t o ) − a(t e )
a(t e )
.
(13.23)
For a cosmologically nearby galaxy the difference between the time of emission and
observation is small so we may expand a(t) to obtain
ν
c
=
a
(t o )
a(t o )
(t o − t e ), a
=
da
dt
.
(13.24)
The distance to such a nearby galaxy is approximately L = c(t o − t e ), so that the
velocity of recession is
v =
a
(t o )
a(t o )
L .
(13.25)
We have thus derived Hubble’s law (13.1) and identified the Hubble constant in
(13.25). Following current use we define a Hubble function H whose current value
is the Hubble constant,
H (t) ≡
a
(t)
a(t)
, H 0 = H (t 0 ) =
a
(t o )
a(t o )
.
(13.26)
Note that some authors refer to H (t) as the Hubble parameter, which somewhat
obscures its nature as a function of time.
To summarize, the FLRW metric with an increasing scale factor implies a cosmological redshift and the Hubble law, with the Hubble constant simply related to the
present scale factor. This gives a very useful observational constraint on the scale
factor.