9.1 Basics 357parameters that control the change of perturbation amplitudes after they enter the
Hubble scale.
The purpose of this chapter is to derive the spectrum of microwave background
fluctuations, assuming a nearly scale-invariant spectrum of primordial inhomo-
geneities, as occurs in inflationary models. Today, sophisticated computer programs
are used to obtain numerically precise predictions. Here, though, our purpose is to
understand from first principles the physics behind the characteristic features of the
spectrum and to determine how they depend on fundamental parameters. We are
willing to sacrifice a little accuracy to obtain a solid, analytic insight.
We first use an approximation of instantaneous recombination in which the
radiation behaves as a perfect fluid before recombination and as an ensemble of
the free photons immediately afterwards. This approximation is very good for
large angular fluctuations arising from inhomogeneities on scales larger than the
Hubble radius at recombination. In fact, recombination is a more gradual process
that extends over a finite range of redshifts and this substantially influences the
temperature fluctuations on small angular scales. The computations are then more
complicated, but the problem is still treatable analytically.
Throughout this chapter we consider a spatially flat universe predicted by in-
flation and favored by the current observations. The modifications of the most
important features of the CMB spectrum induced by the spatial curvature are rather
obvious and will be briefly discussed.
9.1 Basics
Before recombination, radiation is strongly coupled to ordinary matter and it is
well approximated as a perfect fluid. When sufficient neutral hydrogen has been
formed, the photons cease interacting with the matter and, therefore, they must be
described by a kinetic equation.
Phase volume and Liouville’s theoremThe state of a single photon with a given
polarization at (conformal) timeηcan be completely characterized by its position
in the spacexi(η)and its 3-momentumpi(η), wherei= 1 , 2 ,3 is the spatial index.
Since the 4-momentumpαsatisfies the equationgαβpαpβ=0, the “energy”p 0 can
be expressed in terms of the metricgαβandpi.
Theone-particlephase volume element is a product of the differentials of spatial
coordinates andcovariantcomponents of the momentum:
d^3 xd^3 p≡dx^1 dx^2 dx^3 dp 1 dp 2 dp 3. (9.1)