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

The Eulerian and Lagrangian second-order perturbation theories are solved explicitly in closed forms in 6= 1 and 6= 0 Friedmann-Lemâ tre models. I explicitly write the secondorder theories in terms of closed one-dimensional integrals. In cosmologically interested cases ( = 0 or + = 1), they reduce to elementary functions or hypergeometric functions. For arbitrary and , I present accurate tting formula which are su cient in practice for the observational cosmology. It is recon rmed for generic and of interest that second-order e ect only weakly depends on these parameters. Progress of Theoretical Physics (Letters), in press The gravitational instability presumably plays an essential role in the formation of the large-scale structure of the universe. The dynamics of such interesting phenomena involves the nonlinearity which is di cult to deal with. Any exact solution for nonlinear evolution is not known in general situation. We have been mainly resorted to N -body methods for fully nonlinear problems in this eld. Although such methods can shed light on strongly nonlinear regime, the resolution is fairly limited and they can barely survey only small fraction of parameter space of possible models. On large-scales where density uctuations are small, perturbation theories are quite powerful. They can analytically give results for large parameter space of possible models. Two formulations for higher-order perturbation theories are focused in the literature. One is Eulerian formulation [1{13] and the other is Lagrangian formulation [14{25]. The rst order in Eulerian formulation is well-known linear theory and that in Lagrangian formulation corresponds to well-known Zel'dovich approximation [26,27]. Since current observations seem to point to < 1 and/or 6= 0 [28], it is important to develop perturbation theories in general and . Perturbation theory in Eulerian space for Friedmann models (Throughout this letter, I mean models with arbitrary and = 0 by Friedmann models and models with arbitrary and by Friedmann-Lemâ tre models) are considered by several people [29, 16, 30{32]. Lagrangian-space counterpart is considered by some authors [16{18,21,23,24]. The second-order perturbation theories in Friedmann models can be expressed in closed forms. For Friedmann-Lemâ tre models, however, the analytical expression for the time-dependence in second-order perturbation theories has not been known except for numerical solutions [30,15,19,22,24]. In this Letter, I have obtained the explicit solution of the time-dependence in second-order perturbation theories both in Eulerian and Lagrangian space for the rst time in models with arbitrary and . Let us brie y review second-order perturbation theories rst. In Friedmann-Lemâ tre models, non-relativistic pressure-less self-gravitating uid are governed by the following continuity equation, Euler equation of motion and Poisson eld equation [33]: _ +r [(1 + )v] = 0; (1) _ v + 2Hv + (v r)v +r = 0; (2) r = 3 2 H ; (3) where x, v(x; t), (x; t) are position, peculiar velocity, peculiar potential in comoving coordi-