MODERN COSMOLOGY

Cosmic background fluctuations 95

clustering, in the limit of a selection function that goes to a delta function in
radius:

C"SW= 16 π

∫

(kDLS)−^4 ^2 k(zLS)j"^2 (kRH)

dk

k

,

where thej"arespherical Bessel functions(see chapter 10 of Abramowitz and
Stegun 1965). This formula, derived by Peebles (1982), strictly applies only to
spatially flat models, since the Fourier expansion of the density field is invalid in
an open model. Nevertheless, since the curvature radiusR 0 subtends an angle of

/[ 2 ( 1 −
)^1 /^2 ], even the lowest few multipoles are not seriously affected by this
point, provided
& 0 .1.
For simple mass spectra, the integral for the C" can be performed
analytically. The case of most practical interest is a scale-invariant spectrum
(^2 k∝k^4 ), for which the integral scales as

C"=

6

"("+ 1 )

C 2

(see equation (6.574.2) of Gradshteyn and Ryzhik 1980). The direct relation
between the mass fluctuation spectrum and the multipole coefficients of CMB
fluctuations mean that either can be used as a measure of the normalization of the
spectrum.

2.8.5 Predictions of CMB anisotropies

We are now in a position to understand the characteristic angular structure
of CMB fluctuations. The change-over from scale-invariant Sachs–Wolfe
fluctuations to fluctuations dominated by Doppler scattering has been shown
to occur atk  DLS. This is one critical angle (call itθ 1 ); its definition is
θ 1 =DLS/RH, and for a matter-only model it takes the value

θ 1 = 1. 8 
^1 /^2 degrees.

For flat low-density models with significant vacuum density,RHis smaller;θ 1
and all subsequent angles would then be larger by about a factor
−^0.^6 (i.e.θ 1 is
roughly independent of
in flat-dominated models).
The second dominant scale is the scale of last-scattering smearing set by
σr= 7 (
h^2 )−^1 /^2 Mpc. This subtends an angle

θ 2 = 4 
^1 /^2 arcmin.

Finally, a characteristic scale in many density power spectra is set by the horizon
atzeq.Thisis16(
h^2 )−^1 Mpc and subtends

θ 3 = 9 h−^1 arcmin,

independent of
. This is quite close toθ 2 , so that alterations in the transfer
function are an effect of secondary importance in most models.