Abstract:
The peculiar velocities of galaxies distort the clustering pattern anisotropically in redshift space. This effect on the statistics of isodensity contours is examined by linear theory. The statistics considered in this paper are the three- and two-dimensional genus of isodensity contours, the area of isodensity contours, the length of isodensity contours in the 2-dimensional slice and the level crossing statistics on the line. We find that all these statistics in redshift space as functions of density threshold of contours have the same shape as in real space. The redshift space distortion affects only amplitudes of these statistics. The three-dimensional genus and the area hardly suffer from the redshift space distortion for $0 \leq \Omega b^{-5/3} \leq 1$, where $b$ is a linear bias parameter. The other statistics are defined in one- or two-dimensional slices of the sample volume and depend on the direction of these slices relative to the line of sight. This dependence on direction of these statistics provides a way to determine the density parameter of the universe.

Abstract:
We develop technique concerning statistical analyses of the large-scale structure of the universe in the presence of biasing in the structure formation. We formulate the diagrammatic method to calculate the correlation functions of the nonlocally biased field from a generally non-Gaussian density field. Our method is based on generalized Wiener-Hermite expansion of the density field. The present formalism has not only general applicability, but also practical significance, too. To show the effectiveness of this method, we revisit the problems on biasing that have been considered previously, i.e., various approximations for peak statistics and hierarchical structure of correlation functions of locally biased field. Further analyses which have not been possible so far on these problems can be performed by our formalism. The gravitational evolution of primordial fluctuation or other structure-forming processes can be described by nonlocal biasing, so can be treated by our new formalism in principle.

Abstract:
The gravitational evolution of the genus and other statistics of isodensity contours of the density field is derived analytically in a weakly nonlinear regime using second-order perturbation theory. The effect of final smoothing in perturbation theory on the statistics of isodensity contours is also evaluated. The resulting analytic expression for the genus is compared with $N$-body numerical simulations and exhibits a good agreement.

Abstract:
Analytic formulas of Minkowski functionals in two-dimensional random fields are derived, including effects of second-order non-Gaussianity in the presence of both the bispectrum and trispectrum. The set of formulas provides a promising method to constrain the primordial non-Gaussianity of the universe by temperature fluctuations in the cosmic microwave background radiation. In a case of local-type non-Gaussianity, the Minkowski functionals are analytically given by powers of quadratic and cubic parameters, $f_{\rm NL}$ and $g_{\rm NL}$. Our formulas are not restricted to this particular model, and applicable to a wide class of non-Gaussian models. The analytic formulas are compared to numerical evaluations from non-Gaussian realizations of temperature maps, showing very good agreements.

Abstract:
The Eulerian and Lagrangian second-order perturbation theories are solved explicitly in closed forms in $\Omega \neq 1$ and $\Lambda \neq 0$ {}Friedmann-Lema\^{\i}tre models. I explicitly write the second-order theories in terms of closed one-dimensional integrals. In cosmologically interested cases ($\Lambda = 0$ or $\Omega + \lambda = 1$), they reduce to elementary functions or hypergeometric functions. For arbitrary $\Omega$ and $\Lambda$, I present accurate fitting formula which are sufficient in practice for the observational cosmology. It is reconfirmed for generic $\Omega$ and $\Lambda$ of interest that second-order effect only weakly depends on these parameters.

Abstract:
We formulate a general method for perturbative evaluations of statistics of smoothed cosmic fields, and provide useful formulas in application of the perturbation theory to various statistics. This formalism is an extensive generalization of the method used by Matsubara (1994) who derived a weakly nonlinear formula of the genus statistic in a 3D density field. After describing the general method, we apply the formalism to a series of statistics, including genus statistics, level-crossing statistics, Minkowski functionals, and a density extrema statistic, regardless of the dimensions in which each statistic is defined. The relation between the Minkowski functionals and other geometrical statistics is clarified. These statistics can be applied to several cosmic fields, including 3D density field, 3D velocity field, 2D projected density field, and so forth. The results are detailed for second order theory of the formalism. The effect of the bias is discussed. The statistics of smoothed cosmic fields as functions of rescaled threshold by volume-fraction are discussed in the framework of second-order perturbation theory. In CDM-like models, their functional deviations from linear predictions plotted against the rescaled threshold are generally much smaller than that plotted against the direct threshold. There is still slight meat-ball shift against rescaled threshold, which is characterized by asymmetry in depths of troughs in the genus curve. A theory-motivated asymmetry factor in genus curve is proposed.

Abstract:
The standard nonlinear perturbation theory of the gravitational instability is extended to incorporate the nonlocal bias, redshift-space distortions, and primordial non-Gaussianity. We show that local Eulerian bias is not generally compatible to local Lagrangian bias in nonlinear regime. The Eulerian and Lagrangian biases are nonlocally related order by order in the general perturbation theory. The relation between Eulerian and Lagrangian kernels of density perturbations with biasing are derived. The effects of primordial non-Gaussianity and redshift-space distortions are also incorporated in our general formalism, and diagrammatic methods are introduced. Vertex resummations of higher-order perturbations in the presence of bias are considered. Resummations of Lagrangian bias are shown to be essential to handle biasing schemes in a general framework.

Abstract:
We present the general relationship between phase correlations and the hierarchy of polyspectra in the Fourier space, and the new theoretical understanding of the phase information is provided. Phase correlations are related to the polyspectra only through the non-uniform distributions of the phase sum $\theta_{\sbfm{k}_1} + ... + \theta_{\sbfm{k}_N}$ with closed wave vectors, $\bfm{k}_1 + ... + \bfm{k}_N = 0$. The exact relationship is given by the infinite series, which one can truncate in a consistent manner. The method to calculate the series to arbitrary order is explained, and the explicit expression of the first-order approximation is given. A numerical demonstration proves that the distribution of the phase sum is a robust estimator and provides an alternative statistic to search for the non-Gaussianity.

Abstract:
The gravitational lensing, as well as the velocity field and the cosmological light-cone warp, changes the observed correlation function of high-redshift objects. We present an analytical expression of 3D correlation function, simultaneously including those three effects. When two objects are separated over several hundreds Mpc along a line of sight, the observed correlation function is dominated by the effect of gravitational lensing rather than the intrinsic clustering. For a canonical lambda-CDM model, the lensing signals in the galaxy-galaxy and galaxy-QSO correlations are beyond noise levels in large-scale redshift surveys like the Sloan Digital Sky Survey.

Abstract:
Fourier methods are fundamental tools to analyze random fields. Statistical structures of homogeneous Gaussian random fields are completely characterized by the power spectrum. In non-Gaussian random fields, polyspectra, higher-order counterparts of the power spectrum, are usually considered to characterize statistical information contained in the fields. However, it is not trivial how the Fourier modes are distributed in general non-Gaussian fields. In this paper, distribution functions of Fourier modes are directly considered and their explicit relations to the polyspectra are given. Under the condition that any of the polyspectra does not diverge, the distribution function is expanded by dimensionless combinations of polyspectra and a total volume in which the Fourier transforms are performed. The expression up to second order is generally given, and higher-order results are also derived in some cases. A concept of N-point distribution function of Fourier modes are introduced and explicitly calculated. Among them, the one-point distribution function is completely given in a closed form up to arbitrary order. As an application, statistics of Fourier phases are explored in detail. A number of aspects regarding statistical properties of phases are found. It is clarified, for the first time, how phase correlations arise in general non-Gaussian fields. Some of our analytic results are tested against numerical realizations of non-Gaussian fields, showing good agreements.