236 The cosmic microwave background
respect to conformal time.)and+are the scalar metric perturbations in the
Newtonian gauge; if we neglect the anisotropic stress, which is generally small in
conventional cosmological scenarios, then+=−. But the details are not very
important. The equation represents damped, driven oscillations of the radiation
density, and the various physical effects are easily identified. The second term
on the left-hand side is the damping of oscillations due to the expansion of the
universe. The third term on the left-hand side is the restoring force due to the
pressure, sincecs^2 =dP/dρ. On the right-hand side, the first two terms depend
on the time variation of the gravitational potentials, so these two are the source
of the Integrated Sachs–Wolfe effect. The final term on the right-hand side is the
driving term due to the gravitational potential perturbations. As Hu and Sugiyama
emphasized, these damped, driven acoustic oscillations account for all of the
structure in the microwave background power spectrum.
A WKB approximation to the homogeneous equation with no driving source
terms gives the two oscillation modes (Hu and Sugiyama 1996)
' 0 (k,η)∝{
( 1 +R)−^1 /^4 coskrs(η)
( 1 +R)−^1 /^4 sinkrs(η)(7.28)
where the sound horizonrsis given by
rs(η)≡∫η0cs(η′)dη′. (7.29)Note that at times well before matter–radiation equality, the sound speed is
essentially constant,cs= 1 /
√
3, and the sound horizon is simply proportional to
the causal horizon. In general, any perturbation with wavenumberkwill set up an
oscillatory behaviour in the primordial plasma described by a linear combination
of the two modes in equation (7.28). The relative contribution of the modes will
be determined by the initial conditions describing the perturbation.
Equation (7.27) appears to be simpler than it actually is, becauseand+
are the total gravitational potentials due to all matter and radiation, including the
photons which the left-hand side is describing. In other words, the right-hand
side of the equation contains an implicit dependence on' 0. At the expense
of pedagogical transparency, this situation can be remedied by considering
separately the potential from the photon–baryon fluid and the potential from the
truly external sources, the DM and neutrinos. This split has been performed by
Hu and White (1996). The resulting equation, while still an oscillator equation,
is much more complicated, but must be used for a careful physical analysis of
acoustic oscillations.
7.4.2 Initial conditions
The initial conditions for radiation perturbations for a given wavenumberkcan
be broken into two categories, according to whether the gravitational potential