4
The standard model of cosmology
Because the variation of R(I) on the time scale of an electromagnetic wave period
is very small, this equation may be approximated by
Ato
All
- - = - -
(1.15)
R(to)
R(II)
But Alo and Atl are the times between adjacent crests; in other words, they are
the periods of the waves. Thus, the waves have frequencies
and
(1.16)
vo = Alo
VI = All
respectively and, in units where c = I, wavelengths
Aa = AIO
and
AI = All
(1.17)
respectively. The redshift is usually defined by
Aa-AI
Z=
( 1.18)
Al
and, from (1.15), we conclude that
1
_ R(IO)
(1.19)
+Z - R(II)'
Equations (1.19) and (1.17), reinterpreted in terms of photons, mean that a photon
emitted at time 1I undergoes a redshifting of its wavelength as the universe
expands, such that its wavelength at time 10 is increased by a factor R(IO)/ R(I).
Since the momentum (or energy) of the photon is inversely proportional to its
wavelength, the momentum (or energy) of the photon is reduced by a factor
R(I)/ R(lo) as a result of the expansion of the universe. This is often expressed
as energy of photons being redshifted away.
When III - tol is not too large, we can make the expansion
R(I» = R(to) + (11 - 'o)R(lo) + !(I\ - '0)2R(lo) + .. .
= R(lo)(1 + Ho(tl -10) - !qOHJ(11 - 10)2 + ... )
(1.20)
where
=
R(tO)
(1.21)
Ho
R(lo)
is the present value of the Hubble parameter and qO is the present deceleration
parameter
R(IO) _
R(IO)R(IO)
(1.22)
qO = -R(lo)HJ ­ R(tO)2
The redshift may also be expanded in powers of I) - 10:
1+ Z = (1 + Ho{tl - (0) - ~qOHJ(11 - 10)2 + ... )-1
(1.23)