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