So far we have been neglecting the baryons in the dynamics of the acoustic
oscillations. To see whether this is a reasonable approximation consider the
photon-baryon momentum density ratio . For typical values of the baryon density this
number is of order unity at recombination and so we expect baryonic effects
to begin appearing in the oscillations just as they are frozen in.
Baryons are conceptually easy to include in the evolution equations since
their momentum density provides extra inertia in
the joint Euler equation for pressure and potential gradients to overcome. Since
inertial and gravitational mass are equal, all terms in the Euler equation save
the pressure gradient are multiplied by leading to the revised oscillator equation [Hu & Sugiyama, 1995]
where we have used the fact that the sound speed is reduced by the baryons to
.
To get a feel for the implications of the baryons take the limit of constant
,
and
. Then
may be added to the left hand side to again put the oscillator equation
in the form of Equation (9) with
. The solution then becomes
Aside from the lowering of the sound speed which decreases the sound horizon,
baryons have two distinguishing effects: they enhance the amplitude of the oscillations
and shift the equilibrium point to (see Figure 1b).
These two effects are intimately related and are easy to understand since the
equations are exactly those of a mass
on a spring in a constant gravitational field. For the same initial
conditions, increasing the mass causes the oscillator to fall further in the gravitational
field leading to larger oscillations and a shifted zero
point.
The shifting of the zero point of the oscillator has significant phenomenological
consequences. Since it is still the effective temperature that is the observed temperature, the zero point shift breaks
the symmetry of the oscillations. The baryons enhance only the compressional
phase, i.e. every other peak. For the working cosmological
model these are the first, third, fifth... Physically, the extra gravity provided
by the baryons enhance compression into potential wells.
Figure: Idealized acoustic oscillations. (b) Baryon loading. Baryon loading boosts the amplitudes of every other oscillation. Plotted here is the idealization of Equation (16) (constant potentials and baryon loading
) for the third peak.
These qualitative results remain true in the presence of a time-variable . An additional effect arises due to the adiabatic damping of an oscillator
with a time-variable mass. Since the energy/frequency of an oscillator is an
adiabatic invariant, the amplitude must decay as
. This can also be understood by expanding the time derivatives
in Equation (16) and identifying
the
term as the remnant of the familiar expansion drag
on baryon velocities.