2.5. Scale-invariant perturbations ?

The nearly scale-invariant nature of the initial perturbations, as expected from inflation, can only be probed using large-scale structure data insofar as one assumes a certain matter content in the Universe. This unfortunate situation results from the fact that after horizon re-entry the subsequent evolution of the density perturbations is greatly affected by both the type of matter present in the Universe and the cosmic expansion rate, which is in turn a function of the total quantity of matter present in the Universe.

The changes in the initial power spectrum of density perturbations, which in our notation is given by _H²(k), are encapsulated in the transfer function, T(k, t). Thus, unless one assumes a certain transfer function, we have no hope of recovering the shape of _H²(k) from large-scale structure data. The evidence for _H²(k) k^{n - 1}, as expected in the simplest inflationary models (in particular n 1), is therefore dependent on assumptions regarding the parameters that affect the transfer function. However, the choice of certain values for some of these parameters may come at a price, and be in conflict with observations not directly related with the power spectrum. For example, the observational data regarding light element abundances implies that the present cosmic baryon density, _b, is about 0.02 h^-2, with an error at 95 per cent confidence of less than about 20 per cent, if homogeneous standard nucleosynthesis is assumed [see e.g. (77)]. Heavy tinkering with the value of _b is tantamount to throwing away the standard nucleosynthesis calculation, thus requiring a viable alternative to be put in its place. It is also not possible to have any type of dark matter one might want to consider. If all the dark matter particles had high intrinsic velocities, i.e. if they were hot, like neutrinos, then the effect of free-streaming would completely erase all the perturbations on small scales, e.g. in the case of standard neutrinos on all scales smaller than about 4 x 10¹⁴(₀ / ) ( h²)^-2 M. Not only it would then be impossible to reproduce the abundance of high-redshift objects, like proto-galaxies, quasars, or damped Lyman- systems, but the actual present-day abundance and distribution of galaxies would be radically different from that observed. It seems very improbable that neutrinos presently contribute with more than about 30 per cent of the total matter density in the Universe [see e.g. (78, 18, 66)].

What is needed is to reformulate slightly the original question and ask instead whether for the simplest, best observationally supported assumptions one can make, the present-day power spectrum of density perturbations is compatible with an initial power-law shape, and in particular with one which is nearly scale-invariant. These assumptions are: the total energy density in the Universe is equal to or less than the critical density, i.e. 1, and results from a matter component, plus a possible classical cosmological constant; the Hubble constant, in the form of h, has a value between 0.4 and 0.9; the baryon abundance, in the form of _bh², is within 0.015 to 0.025; the (non-baryonic) dark matter is essentially cold, with the possibility of any of the 3 known standard neutrinos having a cosmologically significant mass. Among all these assumptions, the one which could be more easily changed without conflicting with non-large scale structure data is the nature of the (non-baryonic) dark matter. There could be warm, decaying, or even self-interacting, dark matter, though usually the existence of some contribution by cold dark matter is found to be required to fit all the available large-scale structure data.

In order to constrain the above free parameters in this simplest model, one needs at least the same number of independent observational constraints. The most widely used are: the slope of the galaxy/cluster power spectrum; the present number density of rich galaxy clusters; the high-redshift abundance of proto-galaxies, quasars and damped Lyman- systems; the amplitude of velocity bulk flows; the amplitude of CMBR temperature anisotropies, both on large-angular scales, as measured by COBE, and on intermediate-angular scales, as presently measured by balloon experiments. While the first constraint directly limits the slope of the density power spectrum, the next four constraints do such only indirectly, by imposing possible intervals for the amplitude of the density power spectrum at specific scales.

Other constraint that has started to be used recently, is the slope and normalisation of the density power spectrum on Mpc scales as inferred from the abundance and distribution of Lyman- forest absorption features in the spectra of distant (z ~ 2.5) quasars (19). However, these calculations depend on assuming a relatively simple physical picture for the formation of such features, being presently still unclear if such a picture provides a good approximation to reality.

The comparison of the above defined simplest structure formation models with CMBR anisotropy data is also presently not as clean as one would like. The two most important culprits are: the possibility of a gravity wave contribution at the COBE scale, thus allowing one to arbitrarily decrease the amplitude imposed by the COBE result on the density power spectrum; the possibility of re-ionization, which allows models with too much intermediate-scale power in the CMBR anisotropy angular power spectrum to evade the observational limits on such scales.

Again taking refuge in the simplicity assumption, the simplest models would be those with a negligible contribution of gravity waves to the large-angle CMBR anisotropy signal, together with no significant re-ionization. In any case, it should be mentioned that relaxing these two further assumptions does not open up a large region of parameter space (78).

Unfortunately, the comparison of these simplest structure formation models with the observational constraints just described does not tell us much about the shape of the initial density power spectrum. The assumption of a power-law shape is perfectly compatible with the data, with the value of the spectral index being loosely constrained to be between 0.7 and 1.4 (78, 66). However, by imposing further restrictions on the type of structure formation model considered, stronger constraints can be obtained. For example, if the Universe was Einstein-de Sitter then the value of the Hubble constant would have to be smaller than about 0.55, in order for the Universe to be more than 12 Gyr old. This does not matter much, because high values for h increase the amount of small-scale power relative to large-scale power and at the same time suppress power at intermediate angular scales on the CMBR, and for h > 0.55 these two effects join together to exclude almost all viable Einstein-de Sitter structure formation models in the context of the simplest assumptions laid down before (78). Restricting ourselves to 0.5 < h < 0.55, then yields a preferred value for the spectral index n roughly between 0.9 and 1.1, with a total neutrino density of ~ 0.15 [see e.g. (78, 31, 66). Another example, is the case of the ₀ = 0.3 flat model, that preferred by the high-redshift type Ia supernova data of (68), for which values for the spectral index in excess of 1 tend to be preferred (78, 66). One should note however that both models are only marginally compatible with the observational data if n = 1.0.

Finally, it should be mentioned that, even within the simplified structure formation scenario we have been assuming, there is enough room for initial density power spectra with deviations from a power-law shape to be viable, as the survival of the broken scale-invariance model testifies (53). This only goes to show the still scarceness of good quality large-scale structure data at present.

In the near future the Sloan Digital Sky Survey (SDSS) will allow a much better constraint to be imposed on the value of n, by extending the measure of the slope of the galaxy power spectrum to larger scales, thus probing a region of the power spectrum which has not been in principle too much affected by the dark matter properties, retaining therefore more information about the shape of the initial density power spectrum (58).