Physical Foundations of Cosmology

2.5 Kinematic tests 67

a moment of time characterized by redshiftzis spatially uniform and equal ton(z).
Then, the number of galaxies with redshifts betweenzandz+z,and within a
solid angle
,is

N=n(z)a^3 (z) 2 (χ)χ
=n(z)( 1 +z)−^3 a 02 H−^1 (z) 2 (χ)z
,
(2.83)
where we have used the relation betweenχandz(see (2.65)). Substituting
2 (χ) from (2.67), we find that in a dust-dominated universe

N

z

=

4 n(z)

H 03

40

[ 

0 z+( 

0 −2)((1+ 

0 z)^1 /^2 −1)]^2

( 1 +z)^6 (1+ 

0 z)^1 /^2

. (2.84)

If we known(z), then measurement ofN/z
can be used as a test of cos-
mological models. The difficulty in applying this method is that the number of
galaxies varies with redshift not only because of the expansion but also as a result
of dynamical evolution. For example, small galaxies merge to form large ones.
Conceivably, this problem can be avoided if the number density of some subset of
galaxies has predictable evolution.

2.5.4 Redshift evolution

The redshift of a given object drifts slowly with time due to the acceleration (or
deceleration) of the universe. The effect is so small that it is not possible to measure
it using today’s technology. However, we introduce the concept as an example of a
measurement that could be possible in the coming decades.
Light from a source located at comoving distanceχthat we observe today at
conformal timeη 0 was emitted at conformal timeηe=η 0 −χ.The appropriate
redshift depends onη 0 and is equal to

z(η 0 )=

a 0

ae

=

a(η 0 )

a(η 0 −χ)

. (2.85)

This redshift depends on the time of observationη 0 and sinceχis constant, its time
derivative is

̇z≡

dz

dt

=

1

a(η 0 )

∂z

∂η 0

=

a ̇ 0

ae

−

a ̇e

ae

=( 1 +z)H 0 −H(z). (2.86)

Taking into account thatε(z)=εcr 0 ( (^) + (^) m(1+z)^3 ) in a universe with a mixture
of matter and vacuum density, and using this expression in (2.61) forH(z), we obtain
̇z=( 1 +z)H 0

{

1 −[1−
0 + (^) m(1+z)+ (^) (1+z)−^2 ]^1 /^2

}

. (2.87)