58 Propagation of light and horizons
One can always go to a coordinate system in which only the radial peculiar
velocity of the particle,uχ,is different from zero. Taking into account that in an
isotropic, homogeneous universe the metric componentsgηηandgχχdo not depend
onχ,we infer from (2.54) thatuχ=const. Hence, the peculiar velocity,
w=auχ=agχχuχ∝a−^1 , (2.55)decays in inverse proportion to the scale factor.
2.4.1 Redshift as a measure of time and distance
Theredshift parameteris defined as the fractional shift in wavelength of a photon
emitted by a distant galaxy at timetemand observed on Earth today:
z=λobs−λem
λem. (2.56)
According to (2.45), the ratioλobs/λemis equal to the ratio of the scale factors at
the corresponding moments of time, and hence
1 +z=a 0
a(tem), (2.57)
wherea 0 is the present value of the scale factor.
The light detected today was emitted at some earlier timetemand, according
to (2.57), there is a one-to-one correspondence betweenzandtem.Therefore, the
redshiftzcan be used instead of timetto parameterize the history of the universe.
Agivenzcorresponds to a time when our universe was 1+ztimes smaller than
now. We can express all time-dependent quantities as functions ofz.For exam-
ple, the formula for the energy densityε(z)follows immediately from the energy
conservation equationdε=− 3 (ε+p)dlna:
∫ε(z)ε 0dε
ε+p(ε)=3ln( 1 +z). (2.58)To obtain the expression for the Hubble parameterHin terms ofzand the present
values ofH 0 and
0 , it is convenient to rewrite the Friedmann equation (1.67) in
the form
H^2 (z)+
k
a^20( 1 +z)^2 =
0 H 02
ε(z)
ε 0