On Second-order Perturbation Theories of Gravitational Instability in Friedmann-Lemaitre Models

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.Comment: 9 pages, LaTeX, to appear in Progress of Theoretical Physics (Letters

Effects of Smoothing Functions in Cosmological Counts-in-Cells Analysis

A method of counts-in-cells analysis of galaxy distribution is investigated with arbitrary smoothing functions in obtaining the galaxy counts. We explore the possiblity of optimizing the smoothing function, considering a series of $m$-weight Epanechnikov kernels. The popular top-hat and Gaussian smoothing functions are two special cases in this series. In this paper, we mainly consider the second moments of counts-in-cells as a first step. We analytically derive the covariance matrix among different smoothing scales of cells, taking into account possible overlaps between cells. We find that the Epanechnikov kernel of $m=1$ is better than top-hat and Gaussian smoothing functions in estimating cosmological parameters. As an example, we estimate expected parameter bounds which comes only from the analysis of second moments of galaxy distributions in a survey which is similar to the Sloan Digital Sky Survey.Comment: 33 pages, 10 figures, accepted for publication in PASJ (Vol.59, No.1 in press