238 The cosmic microwave background
in models producing perturbations causally on scales smaller than the horizon.
Defect models, for example, produce acoustic oscillations, but the oscillations
generically have incoherent phases and thus display no peak structure in their
power spectrum (Seljaket al1997). Simple models of inflation which produce
only adiabatic perturbations insure that all perturbations have the same phase at
η=0 because all of the perturbations are in the cosine mode of equation (7.28).
A glance at thekdependence of the adiabatic perturbation mode reveals how
the coherent peaks are produced. The microwave background images the radiation
density at a fixed time; as a function ofk, the density varies like cos(krs),where
rsis fixed. Physically, on scales much larger than the horizon at decoupling, a
perturbation mode has not had enough time to evolve. At a particular smaller
scale, the perturbation mode evolves to its maximum density in potential wells, at
which point decoupling occurs. This is the scale reflected in the first acoustic
peak in the power spectrum. Likewise, at a particular still smaller scale, the
perturbation mode evolves to its maximum density in potential wells and then
turns around, evolving to its minimum density in potential wells; at that point,
decoupling occurs. This scale corresponds to that of the second acoustic peak.
(Since the power spectrum is the square of the temperature fluctuation, both
compressions and rarefactions in potential wells correspond to peaks in the power
spectrum.) Each successive peak represents successive oscillations, with the
scales of odd-numbered peaks corresponding to those perturbation scales which
have ended up compressed in potential wells at the time of decoupling, while the
even-numbered peaks correspond to the perturbation scales which are rarefied in
potential wells at decoupling. If the perturbations are not phase coherent, then
the phase of a givenk-mode at decoupling is not well defined, and the power
spectrum just reflects some mean fluctuation power at that scale.
In practice, two additional effects must be considered: a given scale ink-
space is mapped to a range ofl-values; and radiation velocities as well as densities
contribute to the power spectrum. The first effect broadens out the peaks, while
the second fills in the valleys between the peaks since the velocity extrema will
be exactly out of phase with the density extrema. The amplitudes of the peaks
in the power spectrum are also suppressed by Silk damping, as mentioned in
section 7.2.5.
7.4.4 The effect of baryons
The mass of the baryons creates a distinctive signature in the acoustic oscillations
(Hu and Sugiyama 1996). The zero-point of the oscillations is obtained by setting
' 0 constant in equation (7.27): the result is
' 0
1
3 c^2 s=( 1 +a). (7.30)The photon temperature' 0 is not itself observable, but must be combined with
the gravitational redshift to form the ‘apparent temperature’' 0 −,which