98 An introduction to the physics of cosmology
Figure 2.20.For 10% baryons, the value ofnneeded to reconcile COBE and the cluster
normalization in CDM models.
The usual expression for the comoving angular-diameter distance isR 0 Sk(r)=c
H 0| 1 −|−^1 /^2 Sk[∫z0| 1 −|^1 /^2 dz′
√
( 1 −)( 1 +z′)^2 +v+m( 1 +z′)^3]
,
where=m+v. Definingωi≡ih^2 , this can be rewritten in a way that
has no explicithdependence:
R 0 Sk(r)=3000 Mpc
|ωk|^1 /^2Sk[∫z0|ωk|^1 /^2 dz′
√
ωk( 1 +z′)^2 +ωv+ωm( 1 +z′)^3]
,
whereωk ≡( 1 −m−v)h^2. This parameter describes the curvature of the
universe, treating it effectively as a physical density that scales asρ ∝ a−^2.
This is convenient for the present formalism, but it is important to appreciate that
curvature differs fundamentally from a hypothetical fluid with such an equation
of state: the value ofωkalso sets the curvature indexk.
The horizon distance at last scattering is 184ωm−^1 /^2 Mpc. Similarly, other
critical length scales such as the sound horizon are governed by the relevant
physical density,ωb. Thus, ifωmandωbare given, the shape of the spatial power
spectrum is determined. The translation of this into an angular spectrum depends
on the angular-diameter distance, which is a function of these parameters, plusωk
andωv. Models in whichω^1 m/^2 R 0 Sk(r)is a constant have the same angular horizon
size. There is therefore a degeneracy between curvature (ωk) and vacuum (ωv):
these two parameters can be varied simultaneously to keep the same apparent
distance, as illustrated in figure 2.21.