MODERN COSMOLOGY

The Friedmann models 21

along a trajectory of fixedβandγwill eventually return to their starting point
(whenα= 2 π). In this respect, the positively curved 3D universe is identical to
the case of the surface of a sphere: it is finite, but unbounded. By contrast, the
k=−1 metric describes an open universe of infinite extent.
The Robertson–Walker metric (which we shall often write in the shorthand
RW metric) may be written in a number of different ways. The most compact
forms are those where the comoving coordinates aredimensionless.Definethe
very useful function

Sk(r)=

{sinr (k=1)

sinhr (k=−1)

r (k=0)

and its cosine-like analogue,Ck(r)≡

√

1 −kSk^2 (r). The metric can now be
written in the preferred form that we shall use throughout:

c^2 dτ^2 =c^2 dt^2 −R^2 (t)[dr^2 +Sk^2 (r)dψ^2 ].

The most common alternative is to use a different definition of comoving distance,
Sk(r)→r, so that the metric becomes

c^2 dτ^2 =c^2 dt^2 −R^2 (t)

(

dr^2

1 −kr^2

+r^2 dψ^2

)

There should of course be two different symbols for the different comoving radii,
but each is often calledrin the literature, so we have to learn to live with this
ambiguity; the presence of terms likeSk(r)or 1−kr^2 will usually indicate
which convention is being used. Alternatively, one can make the scale factor
dimensionless, defining

a(t)≡

R(t)

R 0

,

so thata=1 at the present.

2.4.2 The redshift

At small separations, where things are Euclidean, the proper separation of two
fundamental observers is justR(t)dr, so that we obtain Hubble’s law,v=Hd,
with

H=

R ̇

R

.

At large separations where spatial curvature becomes important, the concept
of radial velocity becomes a little more slippery—but in any case how could one
measure it directly in practice? At small separations, the recessional velocity
gives the Doppler shift
νemit
νobs

≡ 1 +z 1 +

v

c

.