MODERN COSMOLOGY

244 The cosmic microwave background

the agreement between the predicted and observed large-scale structure power
spectrum. Massive neutrinos have several small effects on the microwave
background, which have been studied systematically by Dodelsonet al(1996).
They can slightly increase the sound horizon at decoupling due to their transition
from relativistic to non-relativistic behaviour as the universe expands. More
importantly, free-streaming of massive neutrinos around the time of last scattering
leads to a faster decay of the gravitational potentials, which in turn means more
forcing of the acoustic oscillations and a resulting increase in the monopole
perturbations. Finally, since matter–radiation equality is slightly delayed for
neutrinos with cosmologically interesting masses of a few eV, the gravitational
potentials are less constant and a larger Integrated Sachs–Wolfe effect is induced.
The change in sound horizon and shift in matter–radiation equality due to massive
neutrinos cannot be distinguished from changes in
bh^2 and
0 h^2 ,butthe
alteration of the gravitational potential’s time dependence due to neutrino free-
streaming cannot be mimicked by some other change in parameters. In principle
the effect of neutrino masses can be extracted from the microwave background,
although the effects are very small.

7.5.4 Idealized experiments

Remarkably, the microwave background power spectrum contains enough
information to constrain numerous parameters simultaneously (Jungmanet al
1996). We would like to estimate quantitatively just how well the space of
parameters described earlier can be constrained by ideal measurements of the
microwave background. The question has been studied in some detail; this
section outlines the basic methods and results, and discusses how good various
approximations are. For simplicity, only temperature fluctuations are considered
in this section; the corresponding formalism for the polarization power spectra is
developed in Kamionkowskiet al(1997a, b).
Given a pixelized map of the microwave sky, we need to determine the
contribution of pixelization noise, detector noise, and beam width to the multipole
moments and power spectrum. Consider a temperature map of the skyTmap(nˆ)
which is divided intoNpixequal-area pixels. The observed temperature in pixel
jis due to a cosmological signal plus noise,T
map
j =Tj+T

noise
j. The multipole
coefficients of the map can be constructed as

dTlm=

1

T 0

∫

dnˆTmap(nˆ)Ylm(nˆ)



1

T 0

N∑pix

j= 1

4 π

Npix

TjmapYlm(nˆj), (7.31)

wherenˆj is the direction vector to pixel j. The map moments are written
asdlmto distinguish them from the moments of the cosmological signalalm;
the former include the effects of noise. The extent to which the second line