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

Abstract

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