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 $\Omega \neq 1$ 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.