The Cold Dark Matter Density Perturbation

The outstanding problem in cosmology today is undoubtedly the origin and evolution of large scale structure. In this context, no model has proved as successful as the standard Cold Dark Matter (CDM) model, based on a flat (scale-invariant) spectrum of density fluctuations growing under gravitational instability. Nevertheless, pressure has recently been exerted on the model both from analyses of large scale clustering in the galaxy distribution and from the measurement of the level of microwave background anisotropies by the Cosmic Background Explorer (COBE) satellite. In this report we aim to give a unified view of the Cold Dark Matter model, beginning with the creation of perturbations during an inflationary epoch and pursuing it right through to comparison with a host of observations. We discuss in detail the evolution of density perturbations in Friedmann universes, utilising the fluid flow approach pioneered by Hawking, and provide a simple derivation of the Sachs–Wolfe effect giving large angle microwave background anisotropies. We illustrate the means by which inflation provides an initial spectrum of inhomogeneities, the spectrum having a shape which can be readily calculated in a given inflationary model. We also include a discussion of the generation of long wavelength gravitational waves, which have recently been recognised as having a potentially important role with regard to microwave background anisotropies. Although the standard CDM model is based on a scale-invariant spectrum, the generic prediction of simple inflationary models is for a power-law spectrum, tilted away from scale-invariance to provide extra large-scale power. For many models such as chaotic inflation, this tilting is rather modest. However, in several models, such as power-law inflation, extended inflation and natural inflation, the tilting of the spectrum can be more dramatic, and potentially useful in the light of observations indicating a deficit of large-scale power in the galaxy distribution. In the former two of these, there is the interesting extra of a substantial production of gravitational waves. To appear, Physics Reports. Original version entitled ‘The Spectral Slope in the Cold Dark Matter Cosmogony’. We then discuss observational constraints on cold dark matter cosmogonies based on power-law spectra. We examine a range of phenomena, including large angle microwave background fluctuations, clustering in the galaxy distribution, peculiar velocity flows, the formation of high redshift quasars and the epoch of structure formation. One of our aims is to compute the current constraints on both the shape of the spectrum as defined by the spectral index n, and its normalisation as defined by the usual quantity σ8 ≡ 1/b8. We end by discussing briefly some variants on the CDM model, such as the incorporation of a hot dark matter component, or the introduction of a cosmological constant term. We do not however investigate them in the depth that we do the tilted CDM models, though the techniques illustrated throughout the paper provide the background required for such an undertaking. E-mail addresses: arl @ uk.ac.sussex.starlink ; lyth @ uk.ac.lancs.ph.v1