Non-linear vorticity-density coupling in Lagrangian dynamics

We present a general Lagrangian formalism that allows the treatment of vorticity. We give solutions for the rotational perturbations up to the third-order in a flat background universe. We show how the primordial vorticity affects the evolution of the density fluctuation in high-density regions. The study of the galaxy formation based on gravitational instability has not revealed the origin of the vorticity of galaxies. The possibility that galactic spins are the relics of primordial vorticity has not been taken into account. Because the standard linear Eulerian theory (see, for example, Peebles' book 1) for the detail of the Eulerian theory) shows that the vorticity decays at late time and does not couple to the density enhancement. For this reason, the methods that generate the vorticity posteriorly are investigated. Doroshkevich 2) explained the acquisition of the vorticity by the formation of shock fronts in the pancake. The explanation by tidal torques was made by Hoyle. 3) However, can we actually ignore the effects of the primordial vorticity in the galaxy formation process? To answer this question, we must reconsider the validity of the Eulerian theory. In the Eulerian treatment, density fluctuation is perturbative quantity, so it must be small. Therefore, the Eulerian approach is limited to the description of weak density fluctuation. For this reason, to discuss the behavior of the vorticity in the high-density regions, we must prepare the theory beyond the Eulerian theory. The Lagrangian theory improves such shortcomings of the Eulerian theory. The most advantageous point of the Lagrangian theory is that the density fluctuation is non-perturbative quantity. Therefore, it is unnecessary that the density fluctuation is expanded in a perturbation series and that we impose the condition of the smallness of the density fluctuation. By using the property of the Lagrangian theory, we can investigate the behavior of the primordial vorticity. Buchert 4) solved the first-order Lagrangian perturbative equations and showed that the primordial vorticity was amplified in proportion to the enhancement of the density fluctuation by deriving the relation between the vorticity and the density fluctuation. Barrow and Saich 5) solved the vorticity equation on the inhomogeneous background and pointed out that the growing directions of the vorticity were lying in the plane of the pancake. These analyses based on the Lagrangian theory shows that the primordial vorticity is not negligible in the high-density regions such as the pancake. In this paper, for the purpose of …