Acoustic oscillations 237perturbation from the baryon–photon fluid,bγ, is non-zero or zero asη→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
thatnb/nγ, 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 =0as
η→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
perturbationbγ, 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: nb/nγvaries, and thus
these perturbations are in entropy, not curvature. At an early enough time, the
temperature perturbation in a givenkmode 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 section 7.2.2. Setting the correct initial
conditions involves specifying the value of each variable in the limit asη→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 behaviour for all of the variables is only valid in
the high-temperature limit. Since the equations are stiff, small departures from
this limiting behaviour 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.
7.4.3 Coherent oscillations
The characteristic ‘acoustic peaks’ which appear in figure 7.1 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
forall k-modes, including those with wavelengths longer than the horizon. Such
a condition arises naturally for inflationary models, but is very hard to reproduce