Abstract:
These lectures cover the basics of inflationary models for the early universe, concentrating particularly on the generation of density fluctuations from scalar-field dynamics. The subsequent gravitational dynamics of these fluctuations in dark matter in a Friedmann model are described, leading to a review of the current situation in confronting inflationary models with the latest data on the clustering of galaxies and other measures of large-scale structure.

Abstract:
Some comments are made on the usefulness or otherwise of the concept of `expanding space' in cosmology. These notes are an expanded version of material first published in 2001 but not previously available online except at www.roe.ac.uk/japwww. Since that personal webpage has been referred to in published work, it seems sensible to give these notes a more permanent home.

Abstract:
This paper investigates whether nonlinear gravitational instability can account for the clustering of galaxies on large and small scales, and for the evolution of clustering with epoch. No CDM-like spectrum is consistent with the shape of the observed nonlinear spectrum. Unbiased low-density models greatly overpredict the small-scale correlations; high-density models would require a bias which does not vary monotonically with scale. The true linear power spectrum contains a primordial feature at $k\simeq 0.1 \hompc$, and must break quite abruptly to an effective slope of $n\ls -2.3$ on smaller scales This empirical fluctuation spectrum also fits the CFRS data on the evolution of clustering, provided the universe is open with $\Omega\simeq 0.3$. Only this case explains naturally how the small-scale spectrum can evolve at the observed rate while retaining the same power-law index. An unbiased open model also matches correctly the large-scale COBE data, and offers an attractively simple picture for the phenomenology of galaxy clustering.

Abstract:
This paper reviews selected aspects of the growth of cosmological structure, covering the following general areas: (1) expected characteristics of linear density perturbations according to various candidate theories for the origin of structure; (2) low-order theory for statistical measures of fluctuations; (3) formation of nonlinear structures and nonlinear evolution of the mass distribution; (4) the relation between the density field and the galaxy distribution; (5) constraints on cosmological models from galaxy clustering and its evolution.

Abstract:
The 2dF Galaxy Redshift Survey is the first to observe more than 100,000 redshifts, making possible precise measurements of many aspects of galaxy clustering. The spatial distribution of galaxies can be studied as a function of galaxy spectral type, and also of broad-band colour. Redshift-space distortions are detected with a high degree of significance, confirming the detailed Kaiser distortion from large-scale infall velocities, and measuring the distortion parameter beta = Omega_m^{0.6}/b = 0.49 +- 0.09. The power spectrum is measured to <10% accuracy for k > 0.02 h Mpc^{-1}, and is well fitted by a CDM model with Omega_m h =0.18 +- 0.02 and a baryon fraction of 0.17 +- 0.06. A joint analysis with CMB data requires Omega_m = 0.31 +- 0.05 and h = 0.67 +- 0.04, assuming scalar fluctuations. The fluctuation amplitude from the CMB is sigma_8 = 0.76 +- 0.04, assuming reionization at z < approx 10, so that the general level of galaxy clustering is approximately unbiased, in agreement with an internal bispectrum analysis. Luminosity dependence of clustering is however detected at high significance, and is well described by a relative bias of b/b^* = 0.85 + 0.15(L/L^*). This is consistent with the observation that L^* in rich clusters is brighter than the global value by 0.28 +- 0.08 mag.

Abstract:
This review discusses three ways in which radio galaxies and other high-redshift objects can give us information on the nature and statistics of cosmological inhomogeneities, and how they have evolved between high redshift and the present: (1) The present-day spatial distribution and clustering of radio galaxies; (2) The evolution of radio-galaxy clustering and biased clustering at high redshift; (3) Measuring density perturbation spectra from the abundances of high-redshift galaxies.

Abstract:
These lectures cover various aspects of the statistical description of cosmological density fields. Observationally, this consists of the point process defined by galaxies, and the challenge is to relate this to the continuous density field generated by gravitational instability in dark matter. The main topics discussed are (1) nonlinear structure in CDM models; (2) statistical measures of clustering; (3) redshift-space distortions; (4) small-scale clustering and bias. The overall message is optimistic, in that simple assumptions for where galaxies should form in the mass density field allow one to understand the systematic differences between galaxy data and the predictions of CDM models.

Abstract:
This paper reviews the measurements of galaxy correlations at high redshifts, and discusses how these may be understood in models of hierarchical gravitational collapse. The clustering of galaxies at redshift one is much weaker than at present, and this is consistent with the rate of growth of structure expected in an open universe. If $\Omega=1$, this observation would imply that bias increases at high redshift, in conflict with observed $M/L$ values for known high-$z$ clusters. At redshift 3, the population of Lyman-limit galaxies displays clustering which is of similar amplitude to that seen today. This is most naturally understood if the Lyman-limit population is a set of rare recently-formed objects. Knowing both the clustering and the abundance of these objects, it is possible to deduce empirically the fluctuation spectrum required on scales which cannot be measured today owing to gravitational nonlinearities. Of existing physical models for the fluctuation spectrum, the results are most closely matched by a low-density spatially flat universe. This conclusion is reinforced by an empirical analysis of CMB anisotropies, in which the present-day fluctuation spectrum is forced to have the observed form. Open models are strongly disfavoured, leaving $\Lambda$CDM as the most successful simple model for structure formation.

Abstract:
We describe an attempt to reconstruct the initial conditions for the formation of cosmological large-scale structure. The power spectrum of the primordial fluctuations is affected by bias, nonlinear evolution and redshift-space distortions, but we show how these effects can be corrected for analytically. Using eight independent datasets, we obtain excellent agreement in the estimated linear power spectra given the following conditions. First, the relative bias factors for Abell clusters, radio galaxies, optical galaxies and IRAS galaxies must be in the ratios 4.5:1.9:1.3:1. Second, the data require redshift-space distortion: $\Omega^{0.6}/b_{\ss I} = 1.0 \pm 0.2$. Third, low values of $\Omega$ and bias are disfavoured. The shape of the spectrum is extremely well described by a CDM transfer function with an apparent value of the fitting parameter $\Omega h =0.25$. Tilted models predict too little power at 100 Mpc wavelengths.

Abstract:
Hamilton et al. have suggested an invaluable scaling formula which describes how the power spectra of density fluctuations evolve into the nonlinear regime of hierarchical clustering. This paper presents an extension of their method to low-density universes and universes with nonzero cosmological constant. We pay particular attention to models with large negative spectral indices, and give a spectrum-dependent fitting formula which is of significantly improved accuracy by comparison with an earlier version of this work. The tendency of nonlinear effects to increase power on small scales is stronger for spectra with more negative spectral indices, and for lower densities. However, for low-density models with a cosmological constant, the nonlinear effects are less strong than for an open universe of the same $\Omega$.