Physical Foundations of Cosmology

2.4 Redshift 59

where the definitions in (1.21) and (2.57) have been used. Atz= 0 ,this equation
reduces to
k
a^20

=(

0 − 1 )H 02 , (2.60)

allowing us to express the current value of the scale factora 0 in a spatially curved
universe (k=0) in terms ofH 0 and
0. Taking this into account, we obtain

H(z)=H 0

(

( 1 −
0 )( 1 +z)^2 + (^0)
ε(z)
ε 0

) 1 / 2

. (2.61)

Generically, the expressions fora(t) are rather complicated and one cannot di-
rectly invert (2.57) to express the cosmic timet≡temin terms of the redshift
parameterz.It is useful, therefore, to derive a general integral expression fort(z).
Differentiating (2.57), we obtain

dz=−

a 0

a^2 (t)

a ̇(t)dt=−( 1 +z)H(t)dt, (2.62)

from which it follows that

t=

∫∞

z

dz

H(z)( 1 +z)

. (2.63)

A constant of integration has been chosen here so thatz→∞corresponds to the
initial moment of time,t= 0 .Thus, to determinet(z), one should first findε(z)
and, after substituting (2.61) into (2.63), perform the integration.
Knowing the redshift of light from a distant galaxy we can unambiguously
determine its separation from us; that is, redshift can also be used as a measure
of distance. The comoving distance to a galaxy that emitted a photon at timetem
which arrives today is

χ=η 0 −ηem=

∫t^0

tem

dt

a(t)

. (2.64)

Substitutinga(t)=a 0 /( 1 +z)and the expression fordtin terms ofdzfrom (2.62),
we obtain

χ(z)=

1

a 0

∫z

0

dz

H(z)

. (2.65)

In a universe with nonzero spatial curvature (k=0), the current value of the scale
factora 0 can be expressed in terms ofH 0 and
0 via (2.60):

a− 01 =

√

|

0 − 1 |H 0.