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.