The theory of linear fluctuations in an expanding universe is now a well established
subject. This thesis will concentrate on two aspects of this theory: the evolution of
small perturbations in a cosmological framework and the statistics of the peaks in such
a density field. Following the evolution of small fluctuations exactly is not possible
analytically and is also a difficult numerical task. It is desirable both to check the accuracy of the solutions obtained and to understand the physics behind these solutions.
This has particular relevance for modelling anisotropies in the cosmic background radiation. The constraints provided by the observational non-detection (and the possible
tentative detection) of any anisotropy at a significant level have consistently tightened
in the past few years. For this reason, much attention has focussed on universes dominated by exotic dark matter particles, which generally have small anisotropies. It is
not clear, however, that plain baryonic universes cannot satisfy the constraints. This
thesis re-examines Ω = 1 baryon dominated universes. The procedures followed in
constructing these models are an advance on most of those previously published since
new, approximate, post-recombination solutions are used, the normalisation applied is
conservative and well established and the model includes all the known species of particle in its framework. The solutions also vary from those previously published. With
the preferred normalisation scheme (J₃(10h⁻¹)), an isocurvature model with initial
power spectrum, |δk| α k⁻¹, agrees with the constraints, whilst if biasing is introduced
a scale-invariant adiabatic model is also marginally acceptable.
The other major topic tackled is the statistics of the fluctuations. It is a common
assumption that the initial fluctuations are a random Gaussian field. The properties of
such a field can be compared with the observations. Two possible areas are examined.
The first is the form of the correlation function for peaks in a Gaussian field, motivated
by the assumption that peaks may be related to the observed structure. This is examined in both one and three dimensions and certain general trends are identified. In
particular, the amplitude of the peak-peak correlation function is considerably amplified when compared to the autocorrelation function of the density field. A major new
feature though is that the location of the zeroes of the peak-peak correlation function do
not coincide in general with those of the autocorrelation function, in contrast to previous approximate calculations. A new approximate method of calculating the peak-peak
correlation function in three dimensions is derived from the one dimensional case and
this is found to agree well with accurate numerical simulations, and is much easier to
calculate. The techniques developed are used to discuss the observed cluster-cluster
correlation function, and, with the assumption that clusters can be suitably modelled
by filtering the power spectrum on some scale, the peak-peak correlation function is
compared with the observations. This test favours low density cold dark matter models,
and appears to exclude the favoured scale-invariant, Ω = 1, adiabatic cold dark matter
model and also a scale invariant baryonic model.
The second area relates to the observations of large scale peculiar velocities in the
universe. The assumption that we lie near a maximum in the density field leads to a
different answer from that obtained if we were to lie at an arbitrary point. The velocity
field near peaks is shown to differ by a small factor from points in the field. A statistic
is derived that reflects the probability of obtaining two peculiar velocities when the
velocity field is filtered on two separate scales. When combined with the constraint of
lying at a peak, this statistic is used to test various models. Using the latest published
all-sky surveys, this technique rules out mostly models with too much power. A low
density, cold dark matter model fits the data well, as it did for the clustering test.