Before decoupling, the matter in the universe has significant pressure because it is tightly coupled to radiation. This pressure counteracts any tendency for matter to collapse gravitationally. Formally, the Jeans mass is greater than the mass within a horizon volume for times earlier than decoupling. During this epoch, density perturbations will set up standing acoustic waves in the plasma. Under certain conditions, these waves leave a distinctive imprint on the power spectrum of the microwave background, which in turn provides the basis for precision constraints on cosmological parameters. This section reviews the basics of the acoustic oscillations.

In their classic 1996 paper, Hu and Sugiyama transformed the basic
equations describing the evolution of perturbations into an oscillator
equation. Combining the zeroth moment of the photon Boltzmann equation
with the baryon Euler equation for a given *k*-mode
in the tight-coupling approximation
(mean baryon velocity equals mean radiation velocity) gives

(27) |

where _{0} is the
zeroth moment of the temperature distribution
function (proportional to the photon density perturbation),
*R* = 3_{b}
/ 4_{}
is proportional to the scale factor *a*,
*H* = / *a* is the
conformal Hubble parameter, and the sound speed is
given by *c*_{s}^{2} = 1 / (3 + 3*R*). (All overdots
are derivatives with 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 side is the damping
of oscillations due to the expansion of the universe. The third
term on the left side is the restoring force due to the pressure,
since *c*_{s}^{2} = *dP* /
*d*. On the
right 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 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)

(28) |

where the sound horizon *r*_{s} is given by

(29) |

Note that at times well before matter-radiation equality, the sound speed is
essentially constant,
*c*_{s} = 1 / 3, and
the sound horizon is
simply proportional to the causal horizon.
In general, any perturbation with wavenumber *k* will set up an
oscillatory behavior in the primordial plasma described by a linear
combination of the two modes in Eq. (28). The relative
contribution of the modes will be determined by the initial conditions
describing the perturbation.

Equation (27) appears to be simpler than it actually
is, because and
are the total gravitational potentials
due to all matter and radiation, including the photons which the
left side is describing. In other words, the right 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 dark matter 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.

The initial conditions for radiation perturbations for a given
wavenumber *k* can be broken into two categories, according to whether
the gravitational potential perturbation from the baryon-photon fluid,
_{b}, is nonzero or zero as
-> 0. The
former case is known as ``adiabatic'' (which is somewhat of a misnomer
since adiabatic technically refers to a property of a time-dependent process)
and implies that *n*_{b} /
*n*_{}, the ratio of baryon to photon number
densities, is a constant in space. This case must couple to the cosine
oscillation mode since it requires
_{0}
0 as
-> 0. The simplest
(i.e. single-field) models of
inflation produce perturbations with adiabatic initial conditions.

The other case is termed ``isocurvature'' since the fluid
gravitational potential perturbation
_{b}, and hence the
perturbations to the spatial curvature, are zero.
In order to arrange such a perturbation,
the baryon and photon densities must vary in such a way that they
compensate each other: *n*_{b} /
*n*_{} varies, and thus these
perturbations are in entropy, not curvature. At an early enough time,
the temperature perturbation in a given *k* mode must arise entirely
from the Sachs-Wolfe effect, and thus isocurvature perturbations
couple to the sine oscillation mode. These perturbations arise from
causal processes like phase transitions: a phase transition cannot
change the energy density of the universe from point to point, but it
can alter the relative entropy between various types of matter
depending on the values of the fields involved.
The potentially most interesting cause of isocurvature perturbations
is multiple dynamical fields in inflation. The fields will exchange
energy during inflation, and the field values will vary stochastically
between different points in space at the end of the phase transition,
generically giving isocurvature along with adiabatic perturbations
(Polarski and
Starobinsky 1994).

The numerical problem of setting initial conditions is somewhat tricky. The general problem of evolving perturbations involves linear evolution equations for around a dozen variables, outlined in Sec. 3.2. Setting the correct initial conditions involves specifying the value of each variable in the limit as -> 0. This is difficult for two reasons: the equations are singular in this limit, and the equations become increasingly numerically stiff in this limit. Simply using the leading-order asymptotic behavior for all of the variables is only valid in the high-temperature limit. Since the equations are stiff, small departures from this limiting behavior in any of the variables can lead to numerical instability until the equations evolve to a stiff solution, and this numerical solution does not necessarily correspond to the desired initial conditions. Numerical techniques for setting the initial conditions to high accuracy at temperaturesare currently being developed.

The characteristic ``acoustic peaks'' which appear in Figure 1 arise
from acoustic oscillations which are phase coherent:
at some point in time, the phases of all of the acoustic oscillations
were the same. This requires the same initial
condition for *all* *k*-modes, including those with wavelengths
longer than the horizon. Such a condition arises naturally for
inflationary models, but is very hard to reproduce 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
(Seljak *et
al.* 1997).
Simple models of inflation which produce only adiabatic perturbations
insure that all perturbations have the same phase at
= 0 because
all of the perturbations are in the cosine mode of Eq. (28).

A glance at the *k* dependence 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 of *k*,
the density varies like
cos(*kr*_{s}), where *r*_{s} is
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 given *k*-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 in *k*-space is mapped to a range of
*l*-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 Sec. 3.5.

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 Eq. (27): the result is

(30) |

The photon temperature
_{0} is not itself
observable, but must
be combined with the gravitational redshift to form the ``apparent
temperature''
_{0} -
, which oscillates around
*a*.
If the oscillation amplitude is much larger than
*a* =
3_{b}
/
4_{}, then the
oscillations are effectively about
the mean temperature. The positive and negative oscillations are of
the same amplitude, so when the apparent temperature is squared to form the
power spectrum, all of the peaks have the same height. On the other
hand, if the baryons contribute a significant mass so that
*a* is
a significant fraction of the oscillation amplitude, then the zero
point of the oscillations are displaced, and when the apparent
temperature is squared to form the power spectrum, the peaks arising
from the positive oscillations are higher than the peaks from the
negative oscillations. If
*a* is larger than the
amplitude of the
oscillations, then the power spectrum peaks corresponding to the
negative oscillations disappear entirely. The physical interpretation
of this effect is that the baryon mass deepens the potential well in
which the baryons are oscillating, increasing the compression of the
plasma compared to the case with less baryon mass. In short, as the baryon
density increases, the power spectrum peaks corresponding to
compressions in potential wells get higher, while the alternating
peaks corresponding to rarefactions get lower. This alternating peak
height signature is a distinctive signature of baryon mass, and allows
the precise determination of the cosmological baryon density with the
measurement of the first several acoustic peak heights.