Model-independent cosmological constraints 255or isocurvature. Quite generally, equation (7.28) shows that adiabatic and
isocurvature power spectra must have peaks which are out of phase. While
current measurements of the microwave background and large-scale structure rule
out models based entirely on isocurvature perturbations, some relatively small
admixture of isocurvature modes with dominant adiabatic modes is possible. Such
mixtures arise naturally in inflationary models with more than one dynamical field
during inflation (see, e.g., Mukhanov and Steinhardt 1998).
7.6.4 Gaussian primordial perturbations
If the temperature perturbations are well approximated as a Gaussian random
field, as microwave background maps so far suggest, then the power spectrumCl
contains all statistical information about the temperature distribution. Departures
from Gaussianity take myriad different forms; the business of providing general
but useful statistical descriptions is a complicated one (see, e.g., Ferreiraet
al1997). Tiny amounts of non-Gaussianity will arise inevitably from the
nonlinear evolution of fluctuations, and larger non-Gaussian contributions can be
a feature of the primordial perturbations or can be induced by ‘stiff’ stress–energy
perturbations such as topological defects. As explained later, defect theories of
structure formation seem to be ruled out by current microwave background and
large-scale structure measurements, so interest in non-gaussianity has waned.
But the extent to which the temperature fluctuations are actually Gaussian is
experimentally answerable and, as observations improve, this will become an
important test of inflationary cosmological models.
7.6.5 Tensor or vector perturbations
As described in section 7.3.3, the tensor field describing microwave background
polarization can be decomposed into two components corresponding to the
gradient-curl decomposition of a vector field. This decomposition has the same
physical meaning as that for a vector field. In particular, any gradient-type tensor
field, composed of the G-harmonics, has no curl, and thus may not have any
handedness associated with it (meaning the field is even under parity reversal),
while the curl-type tensor field, composed of the C-harmonics, does have a
handedness (odd under parity reversal).
This geometric interpretation leads to an important physical conclusion.
Consider a universe containing only scalar perturbations, and imagine a single
Fourier mode of the perturbations. The mode has only one direction associated
with it, defined by the Fourier vectork; since the perturbation is scalar, it must
be rotationally symmetric around this axis. (If it were not, the gradient of the
perturbation would define an independent physical direction, which would violate
the assumption of a scalar perturbation.) Such a mode can have no physical
handedness associated with it and, as a result, the polarization pattern it induces
in the microwave background couples only to the G harmonics. Another way of