Parameter information from non-linear cosmological fields

We develop a general formalism for analysing parameter information from non-Gaussian cosmic fields. The method can be adapted to include the non-linear effects in galaxy redshift surveys, weak lensing surveys and cosmic velocity field surveys as part of parameter estimation. It can also be used as a test of non-Gaussianity of the cosmic microwave background. Generalizing maximum-likelihood analysis to second order, we calculate the non-linear Fisher information matrix and likelihood surfaces in parameter space. To this order we find that the information content is always increased by including non-linearity. Our methods are applied to a realistic model of a galaxy redshift survey, including non-linear evolution, galaxy bias, shot-noise and redshift-space distortions to second order. We find that including non-linearities allows all of the degeneracies between parameters to be lifted. Marginalized parameter uncertainties of a few per cent will then be obtainable using forthcoming galaxy redshift surveys.     

Phase information and the evolution of cosmological density perturbations

The Fourier transform of cosmological density perturbations can be represented in terms of amplitudes and phases for each Fourier mode. We investigate the phase evolution of these modes using a mixture of analytical and numerical techniques. Using a toy model of one-dimensional perturbations evolving under the Zel'dovich approximation as an initial motivation, we develop a statistic that quantifies the information content of the distribution of phases. Using numerical simulations beginning with more realistic Gaussian random-phase initial conditions, we show that the information content of the phases grows from zero in the initial conditions, first slowly and then rapidly when structures become non-linear. This growth of phase information can be expressed in terms of an effective entropy. Gaussian initial conditions are a maximum entropy realization of the initial power spectrum; gravitational evolution decreases the phase entropy. We show that our definition of phase entropy results in a statistic that explicitly quantifies the information stored in the phases of density perturbations (rather than their amplitudes), and that this statistic displays interesting scaling behaviour for self-similar initial conditions.     

The lepton asymmetry: the last chance for a critical-density cosmology?

We use a wide range of observations to constrain cosmological models possessing a significant asymmetry in the lepton sector, which offer perhaps the best chance of reconciling a critical-density Universe with current observations. The simplest case, with massless neutrinos, fails to fit many experimental data and does not lead to an acceptable model. If the neutrinos have mass of order 1 eV (which is favoured by some neutrino observations), then models can be implemented which prove a good fit to the microwave anisotropies and large-scale structure data. However, taking into account the latest microwave anisotropy results, especially those from BOOMERANG, we show that the model can no longer accommodate the observed baryon fraction in clusters. Together with the observed acceleration of the present Universe, this puts considerable pressure on such critical-density models.     

The size distribution of void filaments in a ΛCDM cosmology

The size distribution of minifilaments in voids has been derived from the Millennium Run halo catalogues at redshifts   z = 0, 0.5, 1  and 2. It is assumed that the primordial tidal field originated the presence of filamentary substructures in voids and that the void filaments have evolved only little, keeping the initial memory of the primordial tidal field. Applying the filament-finding algorithm based on the minimal spanning tree (MST) technique to the Millennium voids, we identify the minifilaments running through voids and measure their sizes at each redshift. Then, we calculate the comoving number density of void filaments as a function of their sizes in the logarithmic interval and determine an analytic fitting function for it. It is found that the size distribution of void minifilaments in the logarithmic interval,  d N /d log  S   , has an almost universal shape, insensitive to the redshift. In the short-size section, it is well approximated as a power law,  d N /d log  S ≈ S   , while in the long-size section it decreases exponentially as  d N /dlog  S ≈ exp(− S ^α)  . We expect that the universal size distribution of void filaments may provide a useful cosmological probe without resorting to the rms density fluctuations.     

General relativistic self-similar solutions in cosmology

We present general relativistic solutions for self-similar spherical perturbations in an expanding cosmological background of cold pressure-less gas. We focus on solutions having shock discontinuities propagating in the surrounding cold gas. The pressure, p , and energy density, μ, in the shock-heated matter are assumed to obey   p = w μ  , where w is a positive constant. Consistent solutions are found for shocks propagating from the symmetry centre of a region of a positive density excess over the background. In these solutions, shocks exist outside the radius marking the event horizon of the black hole which would be present in a shock-less collapse. For large jumps in the energy density at the shock, a black hole is avoided altogether and the solutions are regular at the centre. The shock-heated gas does not contain any sonic points, provided the motion of the cold gas ahead of the shock deviates significantly from the Hubble flow. For shocks propagating in the uniform background, sonic points always appear for small jumps in the energy density. We also discuss self-similar solutions without shocks in fluids with   w < −1/3  .     

Large-scale cosmic homogeneity from a multifractal analysis of the PSCz catalogue

We investigate the behaviour of galaxy clustering on large scales using the PSC z catalogue. In particular, we ask whether there is any evidence of large-scale fractal behaviour in this catalogue. We find the correlation dimension in this survey varies with scale, consistent with other analyses. For example, our results on small and intermediate scales are consistent those obtained from the QDOT sample, but the larger PSC z sample allows us to extend the analysis out to much larger scales. We find firm evidence that the sample becomes homogeneous at large scales; the correlation dimension of the sample is D ₂=2.992±0.003 for r >30  h ⁻¹ Mpc. This provides strong evidence in favour of a universe that obeys the cosmological principle.     

Topology of the Las Campanas Redshift Survey

The topology of the Las Campanas Redshift Survey is analysed using Minkowski functional statistics, taking into account the selection effects of the survey. The results are compared with the predictions of some toy models of structure formation, including the standard cold dark matter model and topological defect-based models. All of the toy models have a scale invariant primordial spectrum of perturbations, but quite different topologies. The Minkowski functionals can discriminate between the predictions of the models with high significance. Amongst the four functionals, the integrated mean curvature statistic appears to be the most powerful discriminant, followed by the genus statistic. These two functionals can be used to differentiate between the signatures of primordial Gaussian and non-Gaussian phases. None of the models considered gives an acceptable fit to the data.     

The importance of Fourier phases for the morphology of gravitational clustering

The phases of the Fourier modes appearing in a plane-wave expansion of cosmological density fields play a vital role in determining the morphology of gravitationally developed clustering. We demonstrate this qualitatively and quantitatively using simulations. In particular, we use cross-correlation and rank-correlation techniques to quantify the agreement between a simulated distribution and phase-only reconstructions. The phase-only reconstructions exhibit a high degree of correlation with the original distributions, showing how meaningful spatial reconstruction of cosmological density fields depends more on phase accuracy than on amplitudes.     

Evolution of the cosmological density distribution function

We present a new calculation for the evolution of the one-point probability distribution function (PDF) of the cosmological density field based on an exact statistical treatment. Using the Chapman–Kolmogorov equation and second-order Eulerian perturbation theory we propagate the initial density distribution into the non-linear regime. Our calculations yield the moment generating function, allowing a straightforward derivation of the skewness of the PDF to second order. We find a new dependence on the initial perturbation spectrum. We compare our results with other approximations to the one-point PDF, and with N -body simulations. We find that our distribution accurately models the evolution of the one-point PDF of dark matter.     

Unbiased reconstruction of the large-scale structure

We present a new unbiased minimal variance (UMV) estimator for the purpose of reconstructing the large-scale structure of the Universe from noisy, sparse and incomplete data. Similar to the Wiener filter (WF), the UMV estimator is derived by requiring the linear minimal variance solution given the data and an assumed a priori model specifying the underlying field covariance matrix. However, unlike the WF, the minimization is carried out with the added constraint of an unbiased reconstructed mean field. The new estimator does not necessitate a noise model to estimate the underlying field; however, such a model is required for evaluating the errors at each point in space. The general application of the UMV estimator is to predict the values of the reconstructed field in unsampled regions of space (e.g. interpolation in the unobserved Zone of Avoidance), and to dynamically transform from one measured field to another (e.g. inversion of radial peculiar velocities to over-densities). Here, we provide two very simple applications of the method. The first is to recover a 1D signal from noisy, convolved data with gaps, for example CMB time-ordered data. The second application is a reconstruction of the density and 3D peculiar velocity fields from mock SEcat galaxy peculiar velocity catalogues.     

Boundary corrections in fractal analysis of galaxy surveys

The analysis of redshift surveys with fractal tools requires one to apply some form of statistical correction for galaxies lying near the geometric boundary of the sample. In this paper we compare three different methods of applying such a correction to estimates of the correlation integral in order to assess the extent to which estimates may be biased by boundary terms. We apply the corrections to illustrative examples, including a simple fractal set (Lévy flight), a random β -model, and a subset of the CfA2 Southern Cap survey. This study shows that the new 'angular' correction method we present is more generally applicable than the other methods used to date: the conventional 'capacity' correction imposes a bias towards homogeneity, and the 'deflation' method discards large-scale information, and consequently reduces the statistical usefulness of data sets. The 'angular' correction method is effective at recovering true fractal dimensions, although the extent to which boundary corrections are important depends on the form of fractal distribution assumed as well as the details of the survey geometry. We also show that the CfA2 Southern sample does not show any real evidence of a transition to homogeneity. We then revisit the IRAS PSC z survey and 'mock' PSC z catalogues made using N -body simulations of two different cosmologies. The results we obtain from the PSC z survey are not significantly affected by the form of boundary correction used, confirming that the transition from fractal to homogeneous behaviour reported by Pan & Coles is real.