56 Propagation of light and horizons
Thus, the wavelength of the photon changes in proportion to the scale factor,λ(t)∝
a(t),and its frequency,ω∝ 1 /λ,decreases as 1/a.
The Planck distribution, characterizing blackbody radiation, has the important
property that it preserves its shape as the universe expands. However, because each
photon is redshifted,ω→ω/a, the temperatureTscales as 1/a. Therefore, the
energy density of radiation, which is proportional toT^4 , decreases as the fourth
power of the scale factor, in complete agreement with what we obtained earlier for
an ultra-relativistic gas with equation of statep=ε/ 3 .The number density of the
photons is proportional toT^3 ,and therefore decays as the third power of the scale
factor so that thetotalnumber of photons is conserved.
Redshift as Doppler shiftThe cosmological redshift can be interpreted as a Doppler
shift associated with the relative motion of galaxies due to Hubble expansion. If
we begin with two neighboring galaxies separated by distancelH−^1 ,then
there exists a local inertial frame in which spacetime can be considered flat. Ac-
cording to the Hubble law, the relative recessional speed of the two galaxies is
v=H(t)l 1. Because of this, the frequency of a photon,ω(t 1 ),measured
by an observer at galaxy “1” at the momentt 1 ,will be larger than the frequency
of the same photon,ω(t 2 ),measured att 2 >t 1 by an observer at galaxy “2”, by a
Doppler factor (Figure 2.10):
ω≡ω(t 1 )−ω(t 2 )≈ω(t 1 )v=ω(t 1 )H(t)l. (2.46)The time delay between measurements ist=t 2 −t 1 =land so we can rewrite
(2.46) as a differential equation:
ω ̇=−H(t)ω. (2.47)This has the solution
ω∝ 1 /a. (2.48)2∆l1v=H∆lω(t 2 ) ω(t^1 )Fig. 2.10.