Structure observed in redshift surveys of galaxies contains information about fundamental properties of the universe, but these maps are distorted by the peculiar motions of galaxies. This distortion can, however, be accounted for and used to help constrain the density parameter Ωo, usually via (3 = Q°-6/b where b is the bias parameter.

In chapter 1 a brief introduction is given to cosmology in general and to the origin, evolution and statistical analysis of large-scale structure in particular. Methods of measuring Ωo from large-scale velocities are briefly reviewed.

Chapter 2 introduces redshift distortions, the alteration of 3-dimensional redshift survey maps by peculiar velocities. Linear and nonlinear effects are reviewed both qualitatively and quantitatively. A simple nonlinear correction to a linear theory β estimator is tested and found to be moderately successful. It is clear, however, that for linearity to be reliable, very large scale modes must be analysed which subtend large angles in present surveys, preventing the use of conventional Fourier analysis. Instead spherical harmonic analysis must be used, which is pursued in chapters 4 and 5.

At high redshift the assumption of the incorrect cosmological model can lead to an additional geometric effect which can be confused with redshift distortions. If these can be disentangled, limits can be placed on the cosmological constant A in a way which is independent of source evolution. Chapter 3 introduces a detailed power-spectrum model including A with linear and nonlinear redshift distortions. The effects of evolution of the bias parameter are considered and a full statistical analysis is performed, showing that the next generation of redshift surveys may be just about capable of putting limitsxix on A.

A spherical harmonic and spherical Bessel function transform is introduced in chapter 4. Following and refining the analysis of Heavens & Taylor (1995), the effects of linear and nonlinear redshift distortions are modelled along with the effects of incomplete sky coverage and a radial selection function. The equivalence of this method to a conventional Fourier analysis and the advantages this entails are discussed. In chapter •5 the methods are applied to the IRAS 1.2Jy survey and the new PSCz survey. A non- parametric measurement of the shape and amplitude of the real-space {i.e. undistorted) power spectrum was introduced and applied to both surveys. In both spectra there is clear evidence for a turnover and Cold Dark Matter models fit fairly well, although there is marginal evidence for a tighter break. In addition, the values (β = 1.04 ± 0.3 and β = 0.61 ± 0.17 (marginal errors) were measured respectively for the two surveys, simultaneously with the power spectrum. The latter result - from the superior survey - implies that IRAS galaxies must be biased if a flat Do = 1 universe is required.

The likelihood methods discussed and/or used in chapters 3-5 can be computationally expensive to carry out, requiring the repeated inversion of large matrices. In the future, giant datasets could make the task of parameter estimation almost impossible. To deal with this, a method for compressing datasets while retaining information about ■multiple, correlated parameters is introduced and tested. The method appears to be very promising, and should prove very useful if applied to future surveys.