**2.2. Adiabatic vs. entropy perturbations**

Perturbations in a multi-component system can be of the *entropy*
or of the *adiabatic*
types. The first correspond to fluctuations in the form of the local
equation of state of
the system, e.g. fluctuations in the relative number densities of the
different particle
types present in the system, while the second correspond to fluctuations
in its energy density.
In the case of a perfect fluid composed of matter and radiation, pure
entropy perturbations are characterised by
_{r} =
- _{m},
while for pure adiabatic perturbations,
_{r} /
_{r} =
(4/3)(_{m} /
_{m}). The
entropy perturbations are also
called *isocurvature*, given that the total density of the system
remains homogeneous. In
contrast, the adiabatic perturbations are also known as
*curvature* perturbations, as they
induce inhomogeneities in the spatial curvature. The two types of perturbations
are orthogonal, in the sense that all other types of perturbations on
a system can be written as a combination of both adiabatic and entropy
modes.

On scales smaller than the Hubble radius any entropy perturbation rapidly becomes an adiabatic perturbation of the same amplitude, as local pressure differences, due to the local fluctuations in the equation of state, re-distribute the energy density. However, this change is slightly less efficient during the radiation dominated era than during the matter dominated era (and can only occur after the decoupling between photons and baryons, in the case of baryonic isocurvature perturbations). Causality precludes this re-distribution on scales bigger than the Hubble radius, and thus any entropy perturbation on these scales remains with constant amplitude. The end result is that initialy scale-invariant power spectra of adiabatic and isocurvature perturbations give rise, after matter-radiation equality, to power spectra of density perturbations with almost the same shape.

Entropy perturbations are not affected by either Silk or Landau damping, contrary to adiabatic perturbations, thus potentially providing a means of baryonic density perturbations existing below the characteristic Silk damping scale after recombination (note that this could also have been achieved if there were cold dark matter adiabatic perturbations at such scales, with the dark matter necessarily being the dominant matter component in the Universe).

However, presently the amplitude of any primordial entropy
perturbations is severely constrained by the level of anisotropy in the
temperature
of the CMBR as measured by *COBE*. In the case of an Universe with
critical-density and
scale-invariant perturbations, the total anisotropy on large angular
scales is six times bigger in
the case of pure entropy perturbations than in the case of pure
adiabatic perturbations, for the
same final matter density perturbation at those scales
(56).
We will later see that in a
critical-density universe with initial scale-invariant adiabatic
perturbations, the amplitude of the density
perturbation power spectrum needed to generate observed structures, like
rich galaxy clusters, and that
needed to generate the temperature anisotropies measured by *COBE*,
are roughly compatible. Therefore
if one requires small-scale density perturbations with high enough
amplitude to reproduce known structures in an universe with
critical-density and initial
scale-invariant isocurvature perturbations, one ends up with CMBR
anisotropies on COBE scales with much larger amplitude than those which
are measured.

Possible ways of escaping this handicap associated with entropy
perturbations are: breaking the
assumption of scale-invariance by assuming a steeper dependence with
*k*, i.e. decreasing the amount
of large-scale power relatively to small-scale power; and decreasing the
matter density in
the Universe, i.e. assuming
_{0} < 1. However,
the first possibility leads to values for
the spectral index which are in conflict with the constraints imposed by the
*COBE* data [see e.g.
(36)],
while the second solution demands unrealistically small values for
_{0}, well below 0.1
(12).
However, combinations of these two changes, together with the
introduction of more exotic forms of dark matter, like decaying
particles
(36), may provide
working models purely with isocurvature perturbations. Further, isocurvature
perturbations need
not have an inflationary origin, as cosmological defect models provide
an alternative means of generating structure from isocurvature initial
conditions.

In any case, clearly though the possibility of isocurvature perturbations is not yet ruled out from the point of view of structure formation, it is in much worse shape than the hypothesis of adiabatic perturbations, only surviving by appealing to a complex mixture of effects in the case of an inflationary origin, or by being associated with topological defects.